Descriptor Systems of Integer and Fractional Orders (Studies in Systems, Decision and Control, 367) 3030724794, 9783030724795

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Studies in Systems, Decision and Control 367

Tadeusz Kaczorek Kamil Borawski

Descriptor Systems of Integer and Fractional Orders

Studies in Systems, Decision and Control Volume 367

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

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Tadeusz Kaczorek Kamil Borawski •

Descriptor Systems of Integer and Fractional Orders

123

Tadeusz Kaczorek Faculty of Electrical Engineering Bialystok University of Technology Białystok, Poland

Kamil Borawski Faculty of Electrical Engineering Bialystok University of Technology Białystok, Poland

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-72479-5 ISBN 978-3-030-72480-1 (eBook) https://doi.org/10.1007/978-3-030-72480-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This monograph covers some selected problems of the descriptor integer- and fractional-order positive continuous-time and discrete-time systems. The monograph consists of three chapters, four appendices and the list of references. Chapter 1 is devoted to descriptor integer-order continuous-time and discrete-time linear systems. The state equations and their solutions of the linear systems are presented. Necessary and sufficient conditions for the positivity, stability and superstability of the systems are established. The considerations are based on the Laurent series expansion method, Drazin inverse matrix method and Weierstrass–Kronecker decomposition method. In addition, for each of the methods and for the assumed approximation of the derivative (Euler method) formulae linking together, the matrices of the continuous-time model and its discrete-time counterpart are given and the influence of the discretization period on the properties of descriptor systems are examined. In Chapter 2, descriptor fractional-order continuous-time and discrete-time linear systems are considered. The Euler gamma function, Mittag–Leffler functions, Caputo derivative integral and Grünwald–Letnikov fractional-order difference and their basic properties are recalled. The state equations and their solutions of continuous-time and discrete-time linear systems are presented. The positivity, stability and superstability of the factional continuous-time and discrete-time linear systems are investigated. Similarly, to Chapter 1, the considerations are based on the Laurent series expansion method, Drazin inverse matrix method and Weierstrass–Kronecker decomposition method. Chapter 3 is devoted to the stability of descriptor continuous-time and discrete-time systems of integer and fractional orders. Some stability tests for positive descriptor linear systems are considered. Necessary and sufficient stability conditions for positive descriptor interval linear systems are given. New sufficient conditions for the global stability of nonlinear feedback systems with positive descriptor linear parts are established. These considerations are limited to the Drazin inverse matrix method and Weierstrass–Kronecker decomposition method.

v

vi

Preface

In Appendix A, extensions of the Caley–Hamilton theorem for descriptor linear systems are given. Some methods for computation of the Drazin inverse are presented in Appendix B. In Appendix C, some basic definitions and theorems on Laplace transforms and Ƶ-transforms are given. Some properties of the nilpotent matrices are given in Appendix D. The monograph contains some original results of the authors. These considerations are an extension of their research presented in articles and the Ph.D. thesis [14]. It is dedicated to scientists and Ph.D. students from the field of control systems theory. January 2021

Taedusz Kaczorek Kamil Borawski

Acknowledgment

This research was supported by National Science Centre in Poland under work No. 2017/27/B/ST7/02443.

vii

Contents

1 Descriptor Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 State Equations of Descriptor Continuous-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Solution to the State Equation of Descriptor Continuous-Time Linear Systems . . . . . . . . . . . . . . . . 1.1.3 Positive Descriptor Continuous-Time Linear Systems . 1.1.4 Stability of Descriptor Continuous-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Superstability of Descriptor Continuous-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 State Equations of Descriptor Discrete-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Solution to the State Equation of Descriptor Continuous-Time Linear Systems . . . . . . . . . . . . . . . . 1.2.3 Positive Descriptor Discrete-Time Linear Systems . . . . 1.2.4 Stability of Descriptor Discrete-Time Linear Systems . 1.2.5 Superstability of Descriptor Discrete-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fractional Descriptor Linear Systems . . . . . . . . . . . . . . . . . . . . . 2.1 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Euler Gamma Function and Its Properties . . . . . . . . . . 2.1.2 Mittag-Leffler Function . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Caputo Definition of Fractional Derivative-Integral . . . 2.1.4 State Equations of Fractional Descriptor Continuous-Time Linear Systems . . . . . . . . . . . . . . . . 2.1.5 Solution to the State Equation of Fractional Descriptor Continuous-Time Linear Systems . . . . . . . . . . . . . . . .

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25

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31 40

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79 79 79 80 81

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ix

x

Contents

2.1.6 Positive Fractional Descriptor Continuous-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.7 Stability of Fractional Descriptor Continuous-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Superstability of Fractional Descriptor Continuous-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Grünwald-Letnikov Fractional-Order Backward Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 State Equations of Descriptor Discrete-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Solution to the State Equation of Fractional Descriptor Discrete-Time Linear Systems . . . . . . . . . . . . . . . . . . . 2.2.4 Positive Fractional Descriptor Discrete-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Stability of Fractional Descriptor Discrete-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Superstability of Fractional Descriptor Discrete-Time Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Stability of Positive Descriptor Systems . . . . . . . . . . . . . . . 3.1 Stability Tests for Positive Linear Systems . . . . . . . . . . . 3.1.1 Continuous-Time Systems . . . . . . . . . . . . . . . . . 3.1.2 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . 3.1.3 Fractional Continuous-Time Systems . . . . . . . . . 3.1.4 Fractional Discrete-Time Systems . . . . . . . . . . . . 3.2 Stability of Positive Interval Systems . . . . . . . . . . . . . . . 3.2.1 Continuous-Time Systems . . . . . . . . . . . . . . . . . 3.2.2 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . 3.2.3 Fractional Continuous-Time Systems . . . . . . . . . 3.2.4 Fractional Discrete-Time Systems . . . . . . . . . . . . 3.3 Stability of Nonlinear Systems with Positive Linear Parts 3.3.1 Continuous-Time Systems . . . . . . . . . . . . . . . . . 3.3.2 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . 3.3.3 Fractional Continuous-Time Systems . . . . . . . . . 3.3.4 Fractional Discrete-Time Systems . . . . . . . . . . . . 3.3.5 Analysis of Global Stability of Descriptor Continuous-Time Nonlinear Feedback Systems by the Use of Nyquist Plots . . . . . . . . . . . . . . . . 3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . .

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92

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97

. . 103 . . 111 . . 111 . . 113 . . 114 . . 126 . . 129 . . 135 . . 147 . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

149 149 149 154 158 162 168 168 173 178 180 184 184 192 198 203

. . . . . . . 212 . . . . . . . 218

Contents

A

xi

Extensions of the Cayley-Hamilton Theorem for Descriptor Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 A.1 Cayley-Hamilton Theorem for Descriptor Linear Systems with Commuting Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 A.2 Cayley-Hamilton Theorem for Descriptor Linear Systems with Noncommuting Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 221 A.3 Cayley-Hamilton Theorem for Drazin Inverse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

B

Computation of the Drazin Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 B.1 Method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 B.2 Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 B.3 Method 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

C

Laplace Transforms of Continuous-Time Functions and Z-Transforms of Discrete-Time Functions . . . . . . . . . . . . . . . . . 237 C.1 Convolutions of Continuous-Time and Discrete-Time Functions and Their Transforms . . . . . . . . . . . . . . . . . . . . . . . . 237 C.2 Laplace Transforms of Derivative-Integrals . . . . . . . . . . . . . . . 238 C.3 Z-Transforms of Discrete-Time Functions . . . . . . . . . . . . . . . . 240

D

Nilpotent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

List of Symbols

R Rþ Z Zþ C N Rnm Rnm þ Mn In dðiÞ ðtÞ detA rankA A1 AT AD ImA KerA kxk kAk deg pðsÞ diag½a1 ; . . .; an  n! CðxÞ Ea ðzÞ Ea;b ðzÞ a 0 It f ðtÞ a 0 Dt f ðtÞ n 0 D k xi

the set of real numbers the set of real nonnegative numbers the set of integers the set of nonnegative integers the set of complex numbers the set of natural numbers the set of n  m real matrices the set of n  m real matrices with nonnegative entries the set of n  n Metzler matrices (real matrices with nonnegative off-diagonal entries) the n  n identity matrix i-th distributional derivative of the Dirac impulse determinant of the matrix A rank of the matrix A inverse of the matrix A transpose of the matrix A Drazin inverse of the matrix A Image of the matrix A kernel of the matrix A norm of the vector x norm of the matrix A degree of the polynomial pðsÞ diagonal matrix with a1 ; . . .; an on the diagonal factorial of the natural number n Euler gamma function one parameter Mittag-Leffler function of z two parameters Mittag-Leffler function of z fractional (a-order) integral of the function f(t) fractional (a-order) derivative of the function f(t) n-order backward difference of xi on the interval [0, k]

xiii

xiv

List of Symbols

L½f ðtÞ ¼ FðsÞ Z½fi  ¼ FðzÞ f ðtÞ  f ðtÞ f_ ðtÞ ¼ dfdtðtÞ f ðnÞ ðtÞ ¼ d dtf ðtÞ n n

Laplace transform of the function f(t) Z-transform of the function fi convolution of functions f(t) and g(t) first derivative of the function f(t) n-th derivative of the function f(t)

Chapter 1

Descriptor Linear Systems

In this chapter descriptor continuous-time and discrete-time linear systems will be analyzed. In the subsequent sections solutions of the state equation and analytical conditions of the positivity, stability and superstability will be established considering the Laurent series expansion method, Drazin inverse matrix method and WeierstrassKronecker decomposition method. In addition, for each of the methods and for the assumed approximation of the derivative (Euler method) formulae linking together the matrices of the continuous-time model and its discrete-time counterpart will be given and the influence of the discretization period on the properties of descriptor systems will be examined.

1.1 Continuous-Time Systems 1.1.1 State Equations of Descriptor Continuous-Time Linear Systems Consider the continuous-time linear system described by state equations E x(t) ˙ = Ax(t) + Bu(t),

(1.1a)

y(t) = C x(t) + Du(t),

(1.1b)

where x(t) ∈ Rn , u(t) ∈ Rm , y(t) ∈ R p are state, input and output vectors and E, A ∈ Rn×n , B ∈ Rn×m , C ∈ R p×n , D ∈ R p×m . We assume that detE = 0 and det[Es − A] = 0 for some s ∈ C.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Kaczorek and K. Borawski, Descriptor Systems of Integer and Fractional Orders, Studies in Systems, Decision and Control 367, https://doi.org/10.1007/978-3-030-72480-1_1

(1.2)

1

2

1 Descriptor Linear Systems

If the matrix E is nonsingular (detE = 0), then we can transform (1.1a) to the state equation of a standard continuous-time linear system through premultiplication of (1.1a) by E −1 . The system (1.1) with the singular matrix E (detE = 0) will be called a descriptor continuous-time linear system.

1.1.2 Solution to the State Equation of Descriptor Continuous-Time Linear Systems Let U be the set of admissible input vectors u(t) ∈ U ⊂ Rm and X 0 ⊂ Rn the set of consistent initial conditions x(0) = x0 ∈ X 0 for which the Eq. (1.1a) has the solution x(t) with u(t) ∈ U .

1.1.2.1

Laurent Series Expansion Method

In this section the solution to the Eq. (1.1a) will be given using the Laurent series expansion method [26, 47, 49, 111, 147]. Two forms of the solution will be considered: the one that contain Dirac impulses and their derivatives and the one without these components. Theorem 1.1. Solution to the Eq. (1.1a) for x0 ∈ X 0 and u(t) ∈ U is given by the formula ⎞ ⎛ t ∞ k k  T Et T (t − τ ) k ⎝ k x0 + Bu(τ )dτ ⎠ x(t) = k! k! k=0 0 (1.3a) n n0 0 −1  + T−k E x0 δ (k−1) (t) + T−k Bu (k−1) (t), k=1

k=1

where the matrices Tk ∈ Rn×n are determined by ⎧ ⎨ E Tk − ATk−1 = In for k = 0, E Tk − ATk−1 = 0 for k = 0, ⎩ E Tk = 0 for k ≤ −n 0 ,

(1.3b)

δ (k) (t) is the k-th distributional derivative of the Dirac impulse, u (k) (t) is the k-th time derivative of the input vector and n 0 ∈ Z+ is called the index of the system. Proof. Using the Laplace transform on (1.1a) we obtain [Es − A]X (s) = E x0 + BU (s). If the condition (1.2) holds, then we have

(1.4)

1.1 Continuous-Time Systems

3

X (s) = [Es − A]−1 [E x0 + BU (s)].

(1.5)

Expanding the matrix [Es − A]−1 in the Laurent series we obtain [Es − A]−1 =

∞ 

Tk s −(k+1) .

(1.6)

k=−n 0

If detE = 0, then n 0 = 0. From definition of the inverse matrix [Es − A]−1 [Es − A] = [Es − A][Es − A]−1 = In

(1.7)

and (1.6) we have ∞ 

Tk [Es − A]s −(k+1) =

k=−n 0

∞ 

[Es − A]Tk s −(k+1) = In .

(1.8)

k=−n 0

Comparison of the coefficients at the same powers of s in (1.8) yields (1.3b). Substituting (1.6) into (1.5) we obtain ∞  

X (s) =

Tk E x0 s −(k+1) + Tk Bs −(k+1) U (s) .

(1.9)

k=−n 0

The transform (1.9) can be written as a sum X (s) = X 1 (s) + X 2 (s), where X 1 (s) =

∞  

(1.10a)

Tk E x0 s −(k+1) + Tk Bs −(k+1) U (s) ,

(1.10b)

k=0

X 2 (s) = =

−1   k=−n 0 n0 



Tk E x0 s −(k+1) + Tk Bs −(k+1) U (s)

(1.10c)

T−k E x0 s k−1 + T−k Bs k−1 U (s) .

k=1

Applying to (1.10b) the inverse Laplace transform we obtain the first two components k! , k ∈ Z + . Applying to (1.10c) the inverse of the solution (1.3) since L{t k } = s k+1 Laplace transform and taking into account that T−n 0 E = 0 we obtain the third and   fourth component of the solution (1.3) since L{δ (k) (t)} = s k , k = 0, 1, . . .. Remark 1.1. If E = In , then n 0 = 0 and Tk = Ak , k = 0, 1, . . .. In this case from (1.3) we have

4

1 Descriptor Linear Systems ∞ 

⎛

k k ⎝ A t x0 + x(t) = k! k=0

t = e At x0 +

⎞ Ak (t − τ )k Bu(τ )dτ ⎠ k!

t 0

(1.11)

e A(t−τ ) Bu(τ )dτ,

0

Ak t k since e At = ∞ k=0 k! . The formula (1.11) is the solution to the state equation of a standard continuous-time linear system [47]. Note that the solution (1.3a) contains the component with Dirac impulse and its derivatives, which will not appear in physically realizable systems. It should be taken into account that this component affects the state vector only for t = 0 and it does not have impact on the trajectory of state vector for t > 0. Thus we can neglect this part of the solution and assume that for t = 0 the initial condition x0 ∈ X 0 . We will show that the formula (1.3a) without the third component, which will be called the non-impulse solution to the state equation, also satisfies the equality (1.1a). Theorem 1.2. Non-impulse solution to the Eq. (1.1a) for x ∈ X 0 and u(t) ∈ U has the form ⎞ ⎛ t n0 ∞ k k   T Et T (t − τ ) k k ⎠ ⎝ x0 + Bu(τ )dτ + T−k Bu (k−1) (t), (1.12) x(t) = k! k! k=0 k=1 0

and the consistent initial conditions of the system (1.1) are given by the formula x0 = T0 Ev +

n0 

T−k Bu (k−1) (0),

(1.13)

k=1

where the matrices Tk ∈ Rn×n can be calculated from (1.3b) or (1.6), v ∈ Rn is an arbitrary vector, u (k) (t) is the k-th time derivative of the input vector and n 0 ∈ Z+ is called the index of the system. Proof. From Eqs. (1.1a) and (1.12) we have ∞ 

⎛

k−1 ⎝ E Tk t E x(t) ˙ = E x0 + (k − 1)! k=1

+ E T0 Bu(t) +

n0  k=1

t 0

⎞ E Tk (t − τ )k−1 Bu(τ )dτ ⎠ (k − 1)!

E T−k Bu (k) (t)

(1.14)

1.1 Continuous-Time Systems

5

and ∞ 

⎛

k ⎝ ATk t E x0 + Ax(t) = k! k=0

t 0

⎞ n0  ATk (t − τ )k ⎠ Bu(τ )dτ + AT−k Bu (k−1) (t) k! k=1 (1.15)

Taking into account that E Tk − ATk−1 = 0 for k = 0 we obtain E x(t) ˙ − Ax(t) = E T0 Bu(t) +

n0 

(k)

E T−k Bu (t) −

k=1

n0 

AT−k Bu (k−1) (t) = Bu(t),

k=1

(1.16) since E Tn 0 = 0 i E T0 − AT−1 = In . The solution (1.12) satisfies the Eq. (1.1a). We obtain the formula (1.13) assuming in (1.12) t = 0.   Example 1.1. Consider the descriptor continuous-time linear system (1.1a) with ⎡

−0.4 ⎢ −0.2 E =⎢ ⎣ 0.4 0.2

⎤ ⎡ 0 −0.5 0 −0.2 ⎢ 0.4 0 0 0⎥ ⎥, A = ⎢ ⎣ 0.2 1 0.5 0 ⎦ 0 0 0 −0.4 ⎤ ⎡ −1 −3.6 ⎢ 0 −0.8 ⎥ ⎥ B=⎢ ⎣ −1 2.6 ⎦ . 0 −0.2

⎤ 1.8 0.5 0 0.4 0 0 ⎥ ⎥ −1.8 −0.5 0.5 ⎦ 0.6 0 0

(1.17)

The matrix pencil (E, A) of (1.17) is regular since det[Es − A] = −0.05(s + 1)(s + 2) = 0.

(1.18)

The matrix [Es − A]−1 has the form ⎡

[Es − A]−1

⎤ −0.4s + 0.2 −1.8 −0.5s − 0.5 0 ⎢ −0.2s − 0.4 −0.4 0 0 ⎥ ⎥ =⎢ ⎣ 0.4s − 0.2 s + 1.8 0.5s + 0.5 −0.5 ⎦ 0.2s + 0.4 −0.6 0 0 ⎤ ⎡ 3 2 0 − s+2 0 s+2 ⎢ 0 −1 0 −1 ⎥ ⎥ ⎢ =⎣ 2 2(s+4) ⎦ . 6 0 (s+1)(s+2) − s+1 s+2 −2 −2s −2 −2s

(1.19)

Using (1.6) for n 0 = 2, the matrix (1.19) can be written in the form −1

[Es − A]

=

∞  k=−2

Tk s (k+1) ,

(1.20a)

6

1 Descriptor Linear Systems

where ⎧⎡ ⎤ 0 0 0 0 ⎪ ⎪ ⎪ ⎢0 0 0 0 ⎥ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎣0 0 0 0 ⎦ ⎪ ⎪ ⎪ ⎪ 0 −2 0 −2 ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ 0 0 0 0 ⎪ ⎪ ⎨ ⎢ 0 −1 0 −1 ⎥ ⎥ Tk = ⎢ ⎣ 0 0 0 0 ⎦ ⎪ ⎪ ⎪ ⎪ −2 0 −2 0 ⎪ ⎪ ⎡ ⎪ ⎪ ⎪ 0 −3(−2)k ⎪ ⎪ ⎪ ⎢ ⎪ 0 0 ⎪ ⎢ ⎪ ⎪ ⎣ −2(−1)k 6(−2)k ⎪ ⎪ ⎪ ⎪ ⎩ 0 0

for k = −2,

for k = −1, ⎤ 0 2(−2)k ⎥ 0 0 ⎥ k k ⎦ for k = 0, 1, 2, . . . . 0 6(−1) − 4(−2) 0 0 (1.20b)

Using (1.17) and (1.20b) we have ⎤ ⎧⎡ 0 0 00 ⎪ ⎪ ⎪ ⎢0 0 0 0⎥ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎣0 0 0 0⎦ ⎪ ⎪ ⎪ ⎪ ⎨ 0 −2 0 0 Tk E = ⎡ (−2)k ⎪ ⎪ ⎪ ⎪⎢ 0 ⎪ ⎪ ⎢ ⎪ ⎪ ⎣ 2(−1)k − 2(−2)k ⎪ ⎪ ⎪ ⎩ 0

for k = −1, ⎤ 0 0 0 0 0 0⎥ ⎥ for k = 0, 1, 2, . . . , 0 (−1)k 0 ⎦ 0 0 0

⎧⎡ ⎤ ⎪ ⎪ 00 ⎪ ⎢0 0⎥ ⎪ ⎪ ⎢ ⎥ ⎪ for k = −2, ⎪ ⎪ ⎣0 0⎦ ⎪ ⎪ ⎪ ⎪ 02 ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ 00 ⎪ ⎪ ⎨⎢0 1⎥ ⎥ for k = −1, Tk B = ⎢ ⎣0 0⎦ ⎪ ⎪ ⎪ ⎪ 42 ⎪ ⎪ ⎤ ⎪⎡ ⎪ ⎪ 0 2(−2)k ⎪ ⎪ ⎪ ⎢ 0 ⎥ ⎪ 0 ⎪ ⎢ ⎥ ⎪ ⎪ k k k ⎦ for k = 0, 1, 2, . . . . ⎣ ⎪ 6(−1) − 4(−2) 2(−1) ⎪ ⎪ ⎪ ⎩ 0 0

(1.21a)

(1.21b)

1.1 Continuous-Time Systems

7

Next, we compute ⎡

∞  k=0

0 −3e−2t k ⎢ Tk t 0 0 =⎢ ⎣ −2e−t 6e−2t k! 0 0

⎤ 0 2e−2t ⎥ 0 0 ⎥. −2t t 0 2e (3e − 2) ⎦ 0 0

(1.22)

Using (1.3a), (1.17), (1.21) and (1.22) we obtain ⎞ ⎛ t ∞ k k  Tk (t − τ ) ⎝ Tk Et x0 + Bu(τ )dτ ⎠ x(t) = k! k! k=0 0

+ T−1 E x0 δ(t) + T−1 Bu(t) + T−2 Bu (1) (t) ⎤ ⎡ 0 0 0 e−2t ⎢ 0 0 0 0 ⎥ ⎥ =⎢ ⎣ 2e−2t (et − 1) 0 e−t 0 ⎦ x0 0 0 0 0 ⎡ ⎤ 0 2e−2(t−τ ) t ⎢ ⎥ 0 0 ⎥ + ⎢ ⎣ 2e−(t−τ ) 2e−2(t−τ ) (3et−τ − 2) ⎦ u(τ )dτ 0 0 0 ⎡ ⎡ ⎤ ⎤ ⎡ 0 0 0 0 0 0 ⎢ 0 0 0 0 ⎥ ⎢ 0 1 ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ +⎢ ⎣ 0 0 0 0 ⎦ x0 δ(t) + ⎣ 0 0 ⎦ u(t) + ⎣ 0 −2 0 0 4 2

(1.23)

0 0 0 0

⎤ 0 0 ⎥ ⎥ u (1) (t). 0 ⎦ 2

The equation above shows the solution (1.3a) of the state equation of the descriptor linear system (1.1a) with (1.17). In further considerations we will use the solution (1.12), which would have been obtained neglecting the third component of (1.23). 1.1.2.2

Drazin Inverse Matrix Method

In this section the solution to the Eq. (1.1a) will be given using the Drazin inverse matrix method [26, 27, 34, 47]. Two forms of the solution will be considered: the one that contain Dirac impulses and their derivatives and the one without these components. Consider the Eq. (1.1a) and assume that the condition (1.2) is met. There exists a number c ∈ C such that det[Ec − A] = 0. Premultiplying the Eq. (1.1a) by the matrix [Ec − A]−1 we obtain ¯ ¯ E¯ x(t) ˙ = Ax(t) + Bu(t),

(1.24a)

where E¯ = [Ec − A]−1 E, A¯ = [Ec − A]−1 A, B¯ = [Ec − A]−1 B. Note that Eqs. (1.1a) and (1.24a) have the same solution x(t).

(1.24b)

8

1 Descriptor Linear Systems

Definition 1.1. [47] The smallest nonnegative integer q satisfying rank E¯ q = rank E¯ q+1

(1.25)

is called the index of the matrix E¯ ∈ Cn×n . Definition 1.2. [47] A matrix E¯ D is called the Drazin inverse of E¯ if it satisfies the conditions: ¯ (1.26a) E¯ E¯ D = E¯ D E, E¯ D E¯ E¯ D = E¯ D ,

(1.26b)

E¯ D E¯ q+1 = E¯ q .

(1.26c)

The Drazin inverse of a square matrix always exists and is unique. If det E¯ = 0 then E¯ D = E¯ −1 . Some methods for computation of the Drazin inverse are given in [47, 68, 69]. Lemma 1.1. [47] The matrices E¯ and A¯ defined by (1.24b) have the following properties: ¯ A¯ D E¯ = E¯ A¯ D , E¯ D A¯ = A¯ E¯ D , A¯ D E¯ D = E¯ D A¯ D , A¯ E¯ = E¯ A, ker A¯ ∩ ker E¯ = {0}, (In − E¯ E¯ D ) A¯ A¯ D = In − E¯ E¯ D , (In − E¯ E¯ D )( E¯ A¯ D )q = 0. ¯ E¯ D and A¯ can be written in the form Also, the matrices E,      −1  J 0 A1 0 J 0 −1 −1 D ¯ ¯ ¯ T −1 , T , E =T E=T T , A=T 0 A2 0 0 0 N

(1.27a) (1.27b) (1.27c)

(1.27d)

where T ∈ Rn×n , detT = 0, J ∈ Rn 1 ×n 1 is a nonsingular matrix, N ∈ Rn 2 ×n 2 is a nilpotent matrix with the nilpotency index q, i.e. N q−1 = 0 i N q = 0, A1 ∈ Rn 1 ×n 1 , A2 ∈ Rn 2 ×n 2 and n 1 + n 2 = n. Lemma 1.2. If x1 (t) = E¯ E¯ D x(t), x2 (t) = (In − E¯ E¯ D )x(t),

(1.28a)

x1 (t) + x2 (t) = x(t),

(1.28b)

then the Eq. (1.24a) is equivalent to the following equations: ¯ 1 (t) + E¯ D Bu(t), ¯ x˙1 (t) = E¯ D Ax

(1.29a)

1.1 Continuous-Time Systems

9

¯ n − E¯ E¯ D )x˙2 (t) = x2 (t) + (In − E¯ E¯ D ) A¯ D Bu(t). ¯ A¯ D E(I

(1.29b)

Proof. Premultilpying (1.24a) by E¯ E¯ D we obtain ¯ ˙ = A¯ E¯ E¯ D x(t) + E¯ E¯ D Bu(t), E¯ 2 E¯ D x(t)

(1.30)

which is equivalent to ¯ 1 (t) + E¯ E¯ D Bu(t). ¯ E¯ x˙1 (t) = Ax

(1.31)

Next, multiplying (1.31) by E¯ D we get ¯ 1 (t) + E¯ D E¯ E¯ D Bu(t) ¯ E¯ E¯ D x˙1 (t) = E¯ D Ax

(1.32)

¯ 1 (t) + E¯ D Bu(t), ¯ x˙1 (t) = E¯ D Ax

(1.33)

and

since E¯ E¯ D x1 (t) = E¯ E¯ D E¯ E¯ D x(t) = E¯ E¯ D x(t) = x1 (t). Then, subtracting (1.30) from (1.24a) we have ¯ n − E¯ E¯ D )x(t) + (In − E¯ E¯ D ) Bu(t) ¯ ¯ n − E¯ E¯ D )x(t) ˙ = A(I E(I and

¯ 2 (t) + (In − E¯ E¯ D ) Bu(t). ¯ E¯ x˙2 (t) = Ax

(1.34)

(1.35)

Premultiplying (1.35) by A¯ D (In − E¯ E¯ D ) we obtain ¯ n − E¯ E¯ D )x˙2 (t) = A¯ A¯ D (In − E¯ E¯ D )x2 (t) + (In − E¯ E¯ D )2 A¯ D Bu(t). ¯ A¯ D E(I (1.36) Using (1.27c) and the fact that (In − E¯ E¯ D )k = In − E¯ E¯ D for k = 1, 2, . . . we have ¯ n − E¯ E¯ D )x˙2 (t) = x2 (t) + (In − E¯ E¯ D ) A¯ D Bu(t), ¯ A¯ D E(I

(1.37)

since (In − E¯ E¯ D )x2 (t) = (In − E¯ E¯ D )2 x(t) = (In − E¯ E¯ D )x(t) = x2 (t).

 

Theorem 1.3. Solution to the Eq. (1.24a) for x0 ∈ X 0 and u(t) ∈ U has the form x(t) = e

¯ E¯ D At

E¯ E¯ D x0 +

t 0

+ ( E¯ E¯ D − In )

¯ D A(t−τ ¯ )

eE

 q−1  k=1

¯ E¯ D Bu(τ )dτ

 q−1  ¯ k x0 + ¯ k A¯ D Bu ¯ (k) (t) , δ (k−1) ( A¯ D E) ( A¯ D E) k=0

(1.38)

10

1 Descriptor Linear Systems

where δ (k) is the k-th distributional derivative of the Dirac impulse, u (k) (t) is the k-th ¯ time derivative of the input vector and q is the index of E. Proof. The equality (1.29a) is the state equation of a standard continuous-time system, hence its solution has the form [47] x1 (t) = e

¯ E¯ D At

t x10 +

¯ D A(t−τ ¯ )

eE

¯ E¯ D Bu(τ )dτ

0 ¯ D At ¯

= eE

E¯ E¯ D x0 +

(1.39)

t

¯ D A(t−τ ¯ )

eE

¯ E¯ D Bu(τ )dτ,

0

where x10 = E¯ E¯ D x0 . ¯ n − E¯ E¯ D ) is a nilpotent matrix with Next, taking into account that the matrix E(I q D ¯ ¯ ¯ the nilpotency index q, i.e. E (In − E E ) = 0, since according to (1.27d) we have ¯ n − E¯ E¯ D ) = T E(I

 

=T

J 0 0 N 0 0 0N



 In −



J 0 0 N



J −1 0 0 0



T −1 (1.40)

T −1 .

Using the Laplace transform on (1.29b) we obtain ¯ n − E¯ E¯ D )s]−1 X 2 (s) = −[In − A¯ D E(I ¯ n − E¯ E¯ D )x20 + (In − E¯ E¯ D ) A¯ D BU ¯ (s)], × [ A¯ D E(I

(1.41)

where x20 = (In − E¯ E¯ D )x0 . Using the equality [47] ¯ n − E¯ E¯ D )s]−1 = [In − A¯ D E(I

q−1 

¯ n − E¯ E¯ D )]k s k [ A¯ D E(I

(1.42)

k=0

and (1.41) we get X 2 (s) = −

q−1 

¯ n − E¯ E¯ D )]k s k−1 x20 [ A¯ D E(I

k=1

−

q−1 

¯ n − E¯ E¯ D )]k (In − E¯ E¯ D ) A¯ D Bs ¯ k U (s) [ A¯ D E(I

k=0

= ( E¯ E¯ D − In )

 q−1  q−1   D ¯ k k−1 D ¯ k ¯D ¯ k ¯ ¯ ( A E) s x20 + ( A E) A Bs U (s) . k=1

k=0

(1.43)

1.1 Continuous-Time Systems

11

Applying to (1.43) the inverse Laplace transform and using (1.28) we obtain x2 (t) = ( E¯ E¯ D − In )

 q−1 

 q−1  ¯ k x0 + ¯ k A¯ D Bu ¯ (k) (t) . δ (k−1) ( A¯ D E) ( A¯ D E)

k=1

(1.44)

k=0

Taking into account that x1 (t) + x2 (t) = x(t), from (1.39) and (1.44) we get (1.38).   Note that the solution (1.38) contains the component with Dirac impulse and its derivatives. Similarly to the Laurent series expansion method, we assume that for t = 0 the initial condition x0 ∈ X 0 and neglect the third component of (1.38). This form of the solution to the Eq. (1.24a) have been considered in [26, 47] and it will be called the non-impulse solution to the state Eq. (1.24a). Theorem 1.4. [26, 47] Non-impulse solution to the Eq. (1.24a) for x0 ∈ X 0 and u(t) ∈ U has the form x(t) = e

¯ E¯ D At

E¯ E¯ D x0 + E¯ D

t

¯ D A(t−τ ¯ )

eE

¯ Bu(τ )dτ

0

 ¯ (k) (t), ( E¯ A¯ D )k A¯ D Bu + ( E¯ E¯ D − In )

(1.45)

q−1

k=0

¯ where u (k) (t) is the k-th time derivative of the input vector and q is the index of E. From (1.45) for t = 0 we have x0 = E¯ E¯ D v + ( E¯ E¯ D − In )

q−1  ¯ (k) (0), ( E¯ A¯ D )k A¯ D Bu

(1.46)

k=0

where v ∈ Rn is an arbitrary vector. In the case when u(t) = 0 from (1.46) we have x0 = E¯ E¯ D v. A homogenous equation E x(t) ˙ = Ax(t) has the solution if and only if x0 ∈ Im E¯ E¯ D . ¯ E¯ D A, ¯ Lemma 1.3. [47] The matrices E¯ D [Ec − A]−1 , A¯ D [Ec − A]−1 , E¯ D E, D D D ¯ A¯ B¯ and the index q of the matrix E¯ do not depend of the choice E¯ A¯ , E¯ B, of c. Theorem 1.5. [47] The solution (1.45) of the Eq. (1.1a) and the set of consistent initial conditions X 0 do not depend of the choice of c. Example 1.2. (Continuation of Example 1.1) Consider the descriptor continuoustime linear system (1.1a) with (1.17). According to (1.18) the matrix pencil (E, A) is regular. We choose c = 0 and using (1.24b) we obtain

12

1 Descriptor Linear Systems

⎡

⎤ 0.5 0 0 0 ⎢ 0 0 0 0⎥ ⎥ E¯ = [−A]−1 E = ⎢ ⎣ 1 0 1 0⎦, 0 −2 0 0 ⎡ ⎤ −1 0 0 0 ⎢ 0 −1 0 0 ⎥ ⎥ A¯ = [−A]−1 A = ⎢ ⎣ 0 0 −1 0 ⎦ , 0 0 0 −1 ⎡ ⎤ 01 ⎢0 1⎥ ⎥ B¯ = [−A]−1 B = ⎢ ⎣2 4⎦. 42

(1.47)

It is easy to see that rank E¯ 2 = rank E¯ 3 , so q = 2. The Drazin inverse of E¯ can be computed using (1.27d):   J 0 E¯ = T T −1 0 N ⎡ ⎤⎡ 0 1 00 1 ⎢0 0 1 0⎥⎢0 ⎥⎢ =⎢ ⎣ 1 −2 0 0 ⎦ ⎣ 0 0 0 01 0 ⎡ ⎤ 0.5 0 0 0 ⎢ 0 0 0 0⎥ ⎥ =⎢ ⎣ 1 0 1 0⎦, 0 −2 0 0

0 0.5 0 0

 −1  J 0 T −1 E¯ D = T 0 0 ⎡ ⎤⎡ 0 1 00 1 ⎢0 0 1 0⎥⎢0 ⎥⎢ =⎢ ⎣ 1 −2 0 0 ⎦ ⎣ 0 0 0 01 0 ⎡ ⎤ 2 000 ⎢ 0 0 0 0⎥ ⎥ =⎢ ⎣ −2 0 1 0 ⎦ . 0 000

0 0 0 −2

0 2 0 0

0 0 0 0

⎤⎡ 0 2 ⎢1 0⎥ ⎥⎢ 0⎦⎣0 0 0

⎤⎡ 0 2 ⎢1 0⎥ ⎥⎢ 0⎦⎣0 0 0

0 0 1 0

0 0 1 0

¯ Next, we compute the matrices Note that A¯ D = A¯ −1 = A.

1 0 0 0

1 0 0 0

⎤ 0 0⎥ ⎥ 0⎦ 1

⎤ 0 0⎥ ⎥ 0⎦ 1

(1.48a)

(1.48b)

1.1 Continuous-Time Systems

13

⎡

⎡ ⎤ ⎤ 1000 −2 0 0 0 ⎢0 0 0 0⎥ ⎢ ⎥ ⎥ ¯D ¯ ⎢ 0 0 0 0⎥ E¯ E¯ D = ⎢ ⎣ 0 0 1 0 ⎦ , E A = ⎣ 2 0 −1 0 ⎦ , 0000 0 0 0 0 ⎡ ⎡ ⎤ ⎤ 0 0 0 0 02 ⎢ 0 −1 0 0 ⎥ ⎢0 0⎥ ⎢ ⎥ ⎥ ¯ ¯D E¯ D B¯ = ⎢ ⎣ 2 2 ⎦ , ( E E − I4 ) = ⎣ 0 0 0 0 ⎦ , 0 0 0 −1 00 ⎡ ⎡ ⎤ ⎤ −0.5 0 0 0 0 −1 ⎢ 0 0 0 0⎥ ⎢ ⎥ ⎥ ¯ D ¯ ⎢ 0 −1 ⎥ A¯ D E¯ = ⎢ ⎣ −1 0 −1 0 ⎦ , A B = ⎣ −2 −4 ⎦ 0 2 0 0 −4 −2 ⎡

and ¯ D At ¯

eE

0 e−2t ⎢ 0 1 =⎢ ⎣ 2e−2t (et − 1) 0 0 0

0 0 e−t 0

⎤ 0 0⎥ ⎥. 0⎦ 1

(1.49)

(1.50)

Taking into account that q = 2 and using (1.38), (1.49) and (1.50) we obtain x(t) = e

¯ E¯ D At

E¯ E¯ D x0 +

t

¯ D A(t−τ ¯ )

eE

¯ E¯ D Bu(τ )dτ + ( E¯ E¯ D − I4 ) A¯ D E¯ x0 δ(t)

0

¯ ¯ (1) (t) + ( E¯ E¯ D − I4 ) A¯ D E¯ A¯ D Bu + ( E¯ E¯ D − I4 ) A¯ D Bu(t) ⎡ ⎤ 0 0 0 e−2t ⎢ 0 0 0 0⎥ ⎥ =⎢ ⎣ 2e−2t (et − 1) 0 e−t 0 ⎦ x0 0 0 0 0 ⎡ ⎤ 0 2e−2(t−τ ) t ⎢ ⎥ 0 0 ⎥ + ⎢ ⎣ 2e−(t−τ ) 2e−2(t−τ ) (3et−τ − 2) ⎦ u(τ )dτ 0 0 0 ⎡ ⎡ ⎤ ⎤ ⎡ ⎤ 0 0 00 00 00 ⎢0 0 0 0⎥ ⎢0 1⎥ ⎢ 0 0 ⎥ (1) ⎢ ⎥ ⎥ ⎢ ⎥ +⎢ ⎣ 0 0 0 0 ⎦ x0 δ(t) + ⎣ 0 0 ⎦ u(t) + ⎣ 0 0 ⎦ u (t), 0 −2 0 0 42 02 (1.51) which is equivalent to (1.23). The equation above shows the solution (1.38) of the state equation of the descriptor linear system (1.1a) with (1.17). In further considerations we will use the solution (1.45), which would have been obtained neglecting the third component of (1.51).

14

1 Descriptor Linear Systems

1.1.2.3

Weierstrass-Kronecker Decomposition Method

In this section the solution to the Eq. (1.1a) will be given using the WeierstrassKronecker decomposition method [45, 47, 96, 176]. Two forms of the solution will be considered: the one that contain Dirac impulses and their derivatives and the one without these components. It is well-known [47, 48] that if the condition (1.2) holds, then there exists a pair of nonsingular matrices P, Q ∈ Rn×n such that  0 In 1 s − A1 , A1 ∈ Rn 1 ×n 1 , N ∈ Rn 2 ×n 2 , P[Es − A]Q = 0 N s − In 2 

(1.52)

where n 1 = deg{det[Es − A]}, n 2 = n − n 1 and N is a nilpotent matrix with the nilpotency index μ, i.e. N μ−1 = 0 and N μ = 0. The matrices P and Q can be computed using one of procedures given in [47, 48, 168]. Premultiplying (1.1a) by the matrix P and introducing the new state vector  x(t) ¯ = we obtain

 x¯1 (t) = Q −1 x(t), x¯1 (t) ∈ Rn 1 , x¯2 (t) ∈ Rn 2 , x¯2 (t)

(1.53)

˙ = P AQ Q −1 x(t) + P Bu(t), P E Q Q −1 x(t)

(1.54a)

y(t) = C Q Q −1 x(t) + Du(t)

(1.54b)

x˙¯1 (t) = A1 x¯1 (t) + B1 u(t),

(1.55a)

N x˙¯2 (t) = x¯2 (t) + B2 u(t),

(1.55b)

y(t) = C1 x¯1 (t) + C2 x¯2 (t) + Du(t),

(1.55c)

and using (1.52) we get



where PB =

 B1 , B1 ∈ Rn 1 ×m , B2 ∈ Rn 2 ×m , B2

 C Q = C1 C2 , C1 ∈ R p×n 1 , C2 ∈ R p×n 2 .

(1.56a) (1.56b)

Theorem 1.6. [47] Solution to the Eq. (1.55a) for given initial condition x¯10 = x¯1 (0) ∈ Rn 1 and input u(t) ∈ U has the form t x¯1 (t) = e

A1 t

x¯10 + 0

e A1 (t−τ ) B1 u(τ )dτ.

(1.57)

1.1 Continuous-Time Systems

15

Theorem 1.7. [47] Solution to the Eq. (1.55b) for consistent initial condition x¯20 = x¯2 (0) ∈ Rn 2 and input u(t) ∈ U is given by the formula x¯2 (t) = −

μ−1 

δ

(k−1)

(t)N x¯20 − k

k=1

μ−1 

N k B2 u (k) (t),

(1.58)

k=0

where δ (k) (t) is the k-th distributional derivative of the Dirac impulse and u (k) (t) is the k-th time derivative of the input vector. Similarly to the previously considered methods, the solution (1.58) contains the component with Dirac impulse and its derivatives. We will show that the formula (1.58) without this component, which will be called the non-impulse solution to the state equation, also satisfies the equality (1.55b). Theorem 1.8. Non-impulse solution to the Eq. (1.55b) for u(t) ∈ U has the form x¯2 (t) = −

μ−1 

N k B2 u (k) (t)

(1.59)

k=0

and the consistent initial conditions of the subsystem (1.55b) are given by the formula x¯20 = −

μ−1 

N k B2 u (k) (0),

(1.60)

k=0

where u (k) (t) is the k-th time derivative of the input vector. Proof. From (1.55b) and (1.59) we have N x˙¯2 (t) = −

μ−1 

N

k+1

B2 u

(k+1)

(t) = −

k=0

μ−1 

N k B2 u (k) (t) + B2 u(t)

k=0

(1.61)

= x¯2 (t) + B2 u(t) since N μ = 0. Substituting t = 0 into (1.59) we obtain (1.60).

 

Assume that the matrix N in (1.55b) has the form ⎡

0 ⎢0 ⎢ ⎢ N = ⎢ ... ⎢ ⎣0 0

⎤ 0 ... 0 1 ... 0⎥ ⎥ .. . . .. ⎥ . . .⎥ ⎥ 0 0 ... 1⎦ 0 0 ... 0 1 0 .. .

(1.62)

16

1 Descriptor Linear Systems

⎡

and

⎢ ⎢ x¯2 (t) = ⎢ ⎣

x¯21 (t) x¯22 (t) .. .

⎤

⎡ ⎥ ⎥ ⎢ ⎥ , B2 = ⎣ ⎦

x¯2n 2 (t)

⎤ B21 .. ⎥ . . ⎦ B2n 2

(1.63)

From (1.55b), (1.62) and (1.63) we have ⎡

0 ⎢0 ⎢ ⎢ .. ⎢. ⎢ ⎣0 0 and

⎤ 0 ... 0 1 ... 0⎥ ⎥ .. . . .. ⎥ d . . .⎥ ⎥ dt 0 0 ... 1⎦ 0 0 ... 0 1 0 .. .

⎡ ⎢ ⎢ ⎢ ⎣

x¯21 (t) x¯22 (t) .. .

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎦ ⎣

x¯2n 2 (t)

x¯21 (t) x¯22 (t) .. .

⎤

⎡ ⎥ ⎥ ⎢ ⎥+⎣ ⎦

x¯2n 2 (t)

⎤ B21 .. ⎥ u(t) . ⎦ B2n 2

(1.64)

0 = x¯2n 2 (t) + B2n 2 u(t), x˙¯2n 2 (t) = x¯2n 2 −1 (t) + B2n 2 −1 u(t), .. . x¯˙22 (t) = x¯21 (t) + B21 u(t).

(1.65)

Solving the Eq. (1.65) with respect to components of the vector x¯2 (t) we obtain x¯2n 2 (t) = −B2n 2 u(t), x¯2n 2 −1 (t) = −B2n 2 u(t) ˙ − B2n 2 −1 u(t), .. .

(1.66)

x¯21 (t) = −B2n 2 u (n 2 −1) (t) − . . . − B21 u(t), where u (k) (t) is the k-th time derivative of the input vector. Consistent initial conditions of the system (1.55) can be computed from (1.66) for t = 0. Solution to the Eq. (1.55b) for u(t) ∈ U and consistent initial condition x¯20 = x¯2 (0) ∈ Rn 2 is given by (1.66). Considerations can be easily extended to the case when the matrix N has the form  N = blockdiag N1 . . . N j , j > 1,

(1.67)

where Nk , k = 1, 2, . . . , j are given by (1.62). Example 1.3. (Continuation of Examples 1.1 and 1.2) Consider the descriptor continuous-time linear system (1.1a) with (1.17). According to (1.18) the matrix pencil (E, A) is regular. In this case

1.1 Continuous-Time Systems

17

⎡

−1 ⎢ 0 P=⎢ ⎣ 1 0 and

3 −3 0 1

0 0 1 0

⎤ ⎡ 1 01 ⎢0 0 2⎥ ⎥, Q = ⎢ ⎣2 0 0⎦ 1 00

0 0 0 2

⎤ 0 1⎥ ⎥ 0⎦ 0

⎡

⎤ 1000 ⎢0 1 0 0⎥ In 1 0 ⎥ = PEQ = ⎢ ⎣0 0 0 1⎦, 0 N 0000 ⎡ ⎤ −1 1 0 0   ⎢ 0 −2 0 0 ⎥ A1 0 ⎥ = P AQ = ⎢ ⎣ 0 0 1 0⎦, 0 In 2 0 0 01 ⎡ ⎤ 1 1   ⎢ 0 2 ⎥ B1 ⎥ = PB = ⎢ ⎣ −2 −1 ⎦ . B2 0 −1 



(1.68)

(1.69)

Next we compute the transition matrix  e−t e−2t (et − 1) . = 0 e−2t 

e

A1 t

(1.70)

Using (1.57), (1.69) and (1.70) we obtain  e−t e−2t (et − 1) x¯10 x¯1 (t) = 0 e−2t   t  −(t−τ ) −(t−τ ) 3e − 2e−2(t−τ ) e u(τ )dτ. + 0 2e−2(t−τ ) 

(1.71)

0

Taking into account that μ = 2 and using (1.58), (1.69) we obtain x¯2 (t) = −N δ(t)x¯20 − B2 u(t) − N B2 u (1) (t)       21 0 −1 0 1 (1) = u(t) + δ(t)x¯20 + u (t). 01 0 0 00 From (1.53), (1.71) and (1.72) we have

(1.72)

18

1 Descriptor Linear Systems

⎡

⎤ e−t e−2t (et − 1) 0 0 ⎢ 0 0 0⎥ e−2t ⎥ x¯ x(t) ¯ =⎢ ⎣ 0 0 0 0⎦ 0 0 0 00 ⎡ −(t−τ ) −(t−τ ) ⎤ 3e − 2e−2(t−τ ) t e ⎢ 0 ⎥ 2e−2(t−τ ) ⎥ u(τ )dτ + ⎢ ⎣ 0 ⎦ 0 0 0 0 ⎡ ⎡ ⎤ ⎤ ⎡ ⎤ 000 0 00 00 ⎢0 0 0 0 ⎥ ⎢0 0⎥ ⎢ 0 0 ⎥ (1) ⎢ ⎥ ⎥ ⎢ ⎥ +⎢ ⎣ 0 0 0 −1 ⎦ x¯0 δ(t) + ⎣ 2 1 ⎦ u(t) + ⎣ 0 1 ⎦ u (t). 000 0 01 00

(1.73)

Taking into account that x(t) ¯ = Q −1 x(t) we have to premultiply the Eq. (1.73) by −1 Q and substitute x¯0 = Q x0 . Thus we obtain the formula (1.23) (or (1.51)), which means that all considered methods of the descriptor systems analysis give the same result. In further considerations the component with Dirac impulse in (1.73) will be neglected. Remark 1.2. Comparing all methods of the descriptor systems analysis we can notice that the numbers: n 0 in (1.3a) and (1.12), q in (1.38) and (1.45), μ in (1.58) and (1.59) are equivalent.

1.1.3 Positive Descriptor Continuous-Time Linear Systems In positive systems inputs, outputs and state variables take only nonnegative values [40, 49]. The following definition of the positivity is suitable for all methods of the descriptor systems analysis. Definition 1.3. The descriptor continuous-time linear system (1.1) is called (interp nally) positive if x(t) ∈ Rn+ and y(t) ∈ R+ , t ∈ [0, +∞) for any consistent initial n condition x0 ∈ X 0 ⊂ R+ and all admissible inputs u(t) ∈ U ⊂ Rm + , t ∈ [0, +∞) , k = 1, . . . , l − 1, t ∈ [0, +∞), where l = n such that u (k) (t) ∈ Rm 0 = q = μ. + 1.1.3.1

Laurent Series Expansion Method

In this section sufficient positivity conditions of the descriptor continuous-time linear system (1.1) will be established using the Laurent series expansion method. It is assumed that the solution to the state Eq. (1.1a) has the form (1.12). Theorem 1.9. The descriptor continuous-time linear system (1.1) is (internally) positive if

1.1 Continuous-Time Systems

Tk B ∈

19

Tk E n×m R+ ,

∈ Rn×n + , k = 0, 1, . . . , k = −n 0 , −n 0 + 1, . . . , 0, 1, . . . ,

C∈

p×n R+ ,

D∈

(1.74)

p×m R+ ,

where the matrices Tk ∈ Rn×n are given by (1.3b). k

Proof. From (1.12) we have tk! ≥ 0 for k ∈ Z+ and t ∈ [0, +∞). According to Definition 1.3 we have x0 ∈ Rn+ and u (k) (t) ∈ Rm + , k = 0, 1, . . . , n 0 − 1, t ∈ [0, +∞). n×m Therefore, x(t) ∈ Rn+ if Tk E ∈ Rn×n + , k = 0, 1, . . . and Tk B ∈ R+ , k = −n 0 , −n 0 + 1, . . . , 0, 1, . . .. Substituting (1.12) into (1.1b) we obtain y(t) = C +C

∞ 

⎛ k

⎝ Tk Et x0 + k! k=0

n0 

t

⎞ Tk (t − τ ) Bu(τ )dτ ⎠ k! k

0

(1.75)

T−k Bu (k−1) (t) + Du(t).

k=1 p

p×n

From (1.75) it follows that y(t) ∈ R+ , t ∈ [0, +∞) if C ∈ R+

p×m

and D ∈ R+

. 

Example 1.4. Consider the descriptor continuous-time linear system (1.1) with ⎡

⎤ ⎡ ⎤ ⎡ ⎤ 100 02 0 0 E = ⎣ 0 1 0 ⎦ , A = ⎣ 2 0 −1 ⎦ , B = ⎣ 2 ⎦ , 000 00 1 −1  C = 1 2 0 , D = 0.

(1.76)

The matrix pencil (E, A) of (1.76) is regular since    s −2 0    det[Es − A] =  −2 s 1  = −s 2 + 4 = 0.  0 0 −1 

(1.77)

The matrix [Es − A]−1 has the form ⎡

[Es − A]−1

⎤−1 ⎡ s s −2 0 s 2 −4 = ⎣ −2 s 1 ⎦ = ⎣ s 22−4 0 0 −1 0

2 2 s 2 −4 s 2 −4 s s s 2 −4 s 2 −4

0

−1

⎤ ⎦.

(1.78)

Using (1.6) the matrix (1.78) can be written in the form [Es − A]−1 =

∞  k=−1

Tk s −(k+1) ,

(1.79a)

20

where

1 Descriptor Linear Systems

⎧⎡ ⎤ ⎪ ⎪ 00 0 ⎪ ⎪ ⎣ 0 0 0 ⎦ for k = −1, ⎪ ⎪ ⎪ ⎪ 0 0 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ 2k 0 0 ⎤ ⎪ ⎨ k k Tk = ⎣ 0 2 2 ⎦ for k = 0, 2, 4, 6, . . . , ⎪ ⎪ 0 0 0 ⎪ ⎪ ⎤ ⎡ ⎪ ⎪ ⎪ 0 2k 2k ⎪ ⎪ ⎪ ⎣ 2k 0 0 ⎦ for k = 1, 3, 5, 7, . . . . ⎪ ⎪ ⎪ ⎪ ⎩ 0 0 0

(1.79b)

Using (1.76) and (1.79b) we have ⎧⎡ k 2 ⎪ ⎪ ⎪ ⎣ ⎪ 0 ⎪ ⎪ ⎪ ⎨ 0 Tk E = ⎡ 0 ⎪ ⎪ ⎪ ⎪ ⎣ 2k ⎪ ⎪ ⎪ ⎩ 0

⎤ 0 0 2k 0 ⎦ for k = 0, 2, 4, 6, . . . , 0 0 ⎤ 2k 0 0 0 ⎦ for k = 1, 3, 5, 7, . . . , 0 0

⎧⎡ ⎤ 0 ⎪ ⎪ ⎪ ⎪ ⎣ 0 ⎦ for k = −1, ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ ⎨ 0 k Tk B = ⎣ 2 ⎦ for k = 0, 2, 4, 6, . . . , ⎪ ⎪ 0 ⎪ ⎪ ⎡ k⎤ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎣ 0 ⎦ for k = 1, 3, 5, 7, . . . . ⎪ ⎪ ⎪ ⎪ ⎩ 0

(1.80a)

(1.80b)

3×1 It is easy to see that Tk E ∈ R3×3 + , k = 0, 1, . . . and Tk B ∈ R+ , k = −1, 0, 1, . . ., 1×3 1×1 C ∈ R+ , D ∈ R+ . Therefore, by Theorem 1.9 the descriptor continuous-time linear system (1.1) with (1.76) is positive.

1.1.3.2

Drazin Inverse Matrix Method

In this section necessary and sufficient positivity conditions of the descriptor continuous-time linear system (1.24) (or equivalently (1.1)) will be established using the Drazin inverse matrix method. This problem has been also considered in [79, 141, 142, 172]. It is assumed that the solution to the state Eq. (1.24a) has the form (1.45).

1.1 Continuous-Time Systems

21

¯D ¯ Lemma 1.4. [13] The matrices E¯ D and e E At satisfy the equality ¯D ¯ ¯D ¯ E¯ D e E At = e E At E¯ D . ¯ D At ¯

Proof. The matrix e E

can be expanded into a series

¯ D At ¯

eE

(1.81)

¯ + = In + E¯ D At

¯ 2t 2 ( E¯ D A) + .... 2!

(1.82)

Substituting (1.82) into (1.81) we have   ¯ 2 E¯ D A¯ E¯ D At ¯ D E¯ D At D D ¯ ¯ ¯ ¯ E e + ... In + E At + =E 2! ¯D ¯D ¯ ¯D ¯ 2 ¯ + E E A E At + . . . = E¯ D + E¯ D E¯ D At 2!   D ¯ ¯D ¯ 2 ¯ A E At E ¯D ¯ D ¯ ¯ + . . . E¯ D = e E At E¯ D , = In + E At + 2! since E¯ D A¯ = A¯ E¯ D .

(1.83)

 

Lemma 1.5. [49] Let E¯ D A¯ ∈ Mn . Then ¯ D At ¯

eE

∈ Rn×n + , t ∈ [0, +∞).

(1.84)

Lemma 1.6. The matrices E¯ D A¯ and E¯ D A¯ + G(In − E¯ E¯ D ), where G ∈ Rn×n is arbitrary, satisfy the equality ¯ D At ¯

eE

¯D ¯ ¯ ¯D E¯ E¯ D = e[ E A+G(In − E E )]t E¯ E¯ D .

¯ D A+G(I ¯ ¯D ¯ n − E E )]t

Proof. The matrix e[ E ¯ D A+G(I ¯ ¯D ¯ n − E E )]t

e[ E

(1.85)

E¯ E¯ D can be written in the form

E¯ E¯ D

  [ E¯ D A¯ + G(In − E¯ E¯ D )]2 t 2 + . . . E¯ E¯ D = In + [ E¯ D A¯ + G(In − E¯ E¯ D )]t + 2!   D ¯ 2 2 ¯ ¯ + ( E A) t + . . . E¯ E¯ D = e E¯ D At¯ E¯ E¯ D , = In + E¯ D At 2! (1.86)   since (In − E¯ E¯ D ) E¯ E¯ D = 0. Theorem 1.10. Let ¯ k = 0, 1, . . . , q − 1. F¯k = ( E¯ A¯ D )k A¯ D B,

(1.87)

22

1 Descriptor Linear Systems

The descriptor continuous-time linear system (1.24) (or equivalently (1.1)) is positive if and only if there exists a matrix G ∈ Rn×n [79, 141] such that E¯ D A¯ + G(In − E¯ E¯ D ) ∈ Mn and

p×n p×m , Im E¯ E¯ D ⊂ Rn+ , E¯ D B¯ ∈ Rn×m + , C ∈ R+ , D ∈ R+

( E¯ E¯ D − In ) F¯k ∈ Rn×m + , k = 0, 1, . . . , q − 1.

(1.88a)

(1.88b) (1.88c)

Proof. Using the superposition principle we will consider components of the solution (1.45) independently: n ¯ ¯D 1. If Im E¯ E¯ D ⊂ Rn×n + , then E E x 0 ⊂ R+ . Therefore, for the matrix G such that ¯D ¯ ¯ ¯D D D E¯ A¯ + G(In − E¯ E¯ ) ∈ Mn according to Lemma 1.5 we have e[ E A+G(In − E E )]t D D ¯ ¯ ¯ ¯ ¯ n×n [ E¯ D A+G(I − E E )]t D E n ∈ R+ and using Lemma 1.6 we obtain e E¯ E¯ x0 = e At E¯ D n E¯ x0 ∈ R+ , t ∈ [0, +∞).  t ¯D ¯ 2. Using Lemma 1.4 we can write the second component in the form 0 e E A(t−τ ) E¯ D ¯ and u(t) ∈ U ⊂ Rm Bu(τ )dτ . If E¯ D A¯ + G(In − E¯ E¯ D ) ∈ Mn , E¯ D B¯ ∈ Rn×m + +  D ¯ t ¯ ¯ we have E¯ D 0 e E A(t−τ ) Bu(τ )dτ ∈ Rn+ . ¯ 3. If ( E¯ E¯ D − In ) F¯k ∈ Rn×m + , k = 0, 1, . . . , q − 1, where the matrices Fk are given (k) m by (1.87) and u (t) ∈ R+ , k = 0, 1, . . . , q − 1, t ∈ [0, +∞), then the last component of the solution (1.45) is also nonnegative.

¯D ¯ Therefore, we have x(t) ∈ Rn+ , t ∈ [0, +∞) if Im E¯ E¯ D ⊂ Rn×n + , E A + G(In − and ( E¯ E¯ D − In ) F¯k ∈ Rn×m E¯ E¯ D ) ∈ Mn , E¯ D B¯ ∈ Rn×m + + , k = 0, 1, . . . , q − 1. Subp p×n stituting (1.45) into (1.1b) it is easy to see that y(t) ∈ R+ , t ∈ [0, +∞) if C ∈ R+ p×m and D ∈ R+ .   Remark 1.3. [79] Note that the descriptor continuous-time linear system (1.24) can be positive even though the matrix E¯ D A¯ is not a Metzler matrix. In the special case for G = 0 from (1.88a) we obtain the positivity condition E¯ D A¯ ∈ Mn . Example 1.5. (Continuation of Example 1.4) Consider the descriptor continuoustime linear system with (1.76). According to (1.77) the matrix pencil (E, A) is regular. We choose c = 0 and using (1.24b) we obtain ⎡

⎤ ⎡ ⎤ 0 −0.5 0 −1 0 0 E¯ = [−A]−1 E = ⎣ −0.5 0 0 ⎦ , A¯ = [−A]−1 A = ⎣ 0 −1 0 ⎦ , 0 0 0 0 0 −1 ⎡ ⎤ (1.89) −0.5 B¯ = [−A]−1 B = ⎣ 0 ⎦ . 1

1.1 Continuous-Time Systems

23

¯ To compute the Drazin inverse can use Method  of E we    1 from Appendix B. Note 0 −0.5 0 that E¯ = E¯ 1 , where E¯ 11 = , E¯ 12 = and the index of E¯ is q = 1 −0.5 0 0 since rank E¯ = rank E¯ 2 = 2. Using (B.2) we compute E¯ D =



⎤ 0 −2 0 = ⎣ −2 0 0 ⎦ , 0 0 0 ⎡ ⎤ −1 0 0 = A¯ = ⎣ 0 −1 0 ⎦ 0 0 −1

−1 ¯ −2 ¯ E 11 E 12 E¯ 11 0 0

A¯ D = A¯ −1



⎡

⎡

and Im E¯ E¯ D

⎤ ⎡ ⎤ 100 v1 3 ⎣ v = Im ⎣ 0 1 0 ⎦ ⊂ R3×3 for v = 2 ⎦ ⊂ R+ , + 000 0

(1.90)

⎤ ⎡ ⎤ 020 0 E¯ D A¯ = ⎣ 2 0 0 ⎦ ∈ M3 , E¯ D B¯ = ⎣ 1 ⎦ ∈ R3×1 + . 000 0

(1.91a)

⎡

(1.91b)

According to Lemma 1.3 the matrices (1.91) and the index of E¯ do not depend of the choice of c. From (1.87) we have ⎡

⎤ 0.5 F¯0 = ( E¯ A¯ D )0 A¯ D B¯ = ⎣ 0 ⎦ . −1

(1.92)

From (1.91a) and (1.92) we have ⎡ ⎤ 0 ( E¯ E¯ D − I3 ) F¯0 = ⎣ 0 ⎦ ∈ R3×1 + . 1

(1.93)

Therefore, the descriptor continuous-time linear system with (1.76) is positive.

1.1.3.3

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient positivity conditions of the descriptor continuous-time linear system (1.55) (or equivalently (1.1)) will be established using the Weierstrass-Kronecker decomposition method with the assumption that the matrix Q determined by (1.52) is a monomial matrix [59, 84]. This problem has

24

1 Descriptor Linear Systems

been also considered in [19]. It is assumed that the solution to the Eq. (1.55b) has the form (1.59). Definition 1.4. [49] A square matrix A ∈ Rn×n is called monomial if each its row + and each its column contains only one positive entry and the remaining entries are zero. Remark 1.4. [49] An inverse of a monomial matrix is also monomial. Let the decomposition (1.52) of the positive descriptor continuous-time linear system (1.1) be possible for a monomial matrix Q ∈ Rn×n ¯ ∈ Rn+ + . In this case x(t) = Q x(t) n if and only if x(t) ¯ ∈ R+ . It is also well-known that premultiplying the Eq. (1.1a) by a nonsingular matrix P does not change the solution x(t). Theorem 1.11. [19, 59, 84] Let the decomposition (1.52) of the system (1.1) be possible for a monomial matrix Q ∈ Rn×n + . The descriptor continuous-time linear system (1.55) (or equivalently (1.1)) is positive if and only if A1 ∈ Mn 1 , B1 ∈ Rn+1 ×m , C ∈ R+ , D ∈ R+ p×n

−N B2 ∈ k

Rn+2 ×m ,

k = 0, 1, . . . , μ − 1.

p×m

,

(1.94)

n Proof. For a monomial matrix Q ∈ Rn×n ¯ ∈ Rn+ . Accord+ we have x(t) ∈ R+ if x(t) n×n A1 t ing to Lemma 1.5 for A1 ∈ Mn 1 we have e ∈ R+ . If A1 ∈ Mn 1 and B1 ∈ Rn+1 ×m , then from (1.57) we obtain x¯1 ∈ Rn+1 since from Definition 1.3 x¯10 = Q −1 x10 ∈ Rn+1 n 2 ×m k and u(t) ∈ Rm , k = 0, 1, . . . , μ − 1, then from (1.59) we obtain + . If −N B2 ∈ R+ n2 x¯2 ∈ R+ since from Definition 1.3 u (k) (t) ∈ Rm + , k = 0, 1, . . . , μ − 1. From (1.54b) p×n p×m p for C ∈ R+ and D ∈ R+ we have y(t) ∈ R+ since x(t) ∈ Rn+ and u(t) ∈ Rm  +. 

Example 1.6. (Continuation of Example 1.4) Consider the descriptor continuoustime linear system with (1.76). According to (1.77) the matrix pencil (E, A) is regular. In this case ⎤ ⎡ ⎤ ⎡ 100 100 P = ⎣0 1 1⎦, Q = ⎣0 1 0⎦ (1.95) 001 001 and

⎡

⎤ 100 In 1 0 = PEQ = ⎣0 1 0⎦, 0 N 000 ⎡ ⎤   020 A1 0 = P AQ = ⎣ 2 0 0 ⎦ , 0 In 2 001 ⎡ ⎤   0 B1 = PB = ⎣ 1 ⎦. B2 −1 



(1.96)

1.1 Continuous-Time Systems

25

Hence, n 1 = 2 and n 2 = 1. The matrix Q ∈ R3×3 + in (1.95) is monomial. From (1.76) and (1.96) we have  A1 =

   02 0 1×1 ∈ M2 , B1 = ∈ R2×1 + , −B2 = 1 ∈ R+ , 20 1  1×1 C = 1 2 0 ∈ R1×3 + , D = 0 ∈ R+ .

(1.97)

By Theorem 1.11 the descriptor continuous-time linear system with (1.76) is positive.

1.1.4 Stability of Descriptor Continuous-Time Linear Systems In the following considerations it is assumed that det A = 0, i.e. the descriptor continuous-time linear system (1.1) has exactly one equilibrium point [47].

1.1.4.1

Laurent Series Expansion Method

In this section necessary and sufficient stability conditions of the descriptor continuous-time linear system (1.1) will be established using the Laurent series expansion method. It is assumed that the solution to the state Eq. (1.1a) has the form (1.12). Definition 1.5. The descriptor continuous-time linear system (1.1) is called asymptotically stable if (1.98) lim x(t) = 0 t→∞

for all consistent initial conditions x0 ∈ X 0 and u(t) = 0. Let (t) be the matrix given by (t) =

∞  Tk Et k . k! k=0

(1.99)

Definition 1.6. The characteristic equation of the matrix pair (E, A) has the form p(E,A) (s) = det[Es − A] = ar s r + ar −1 s r −1 + . . . + a1 s + a0 = 0,

(1.100)

where r = rank(t) and p(E,A) (s) is the characteristic polynomial of the matrix pair (E, A). Theorem 1.12. [47] The matrix pair (E, A) (or equivalently the system (1.1)) is asymptotically stable if and only if its eigenvalues sk , k = 1, . . . , r (roots of the characteristic equation) satisfy the condition

26

1 Descriptor Linear Systems

Resk < 0, k = 1, . . . , r.

(1.101)

Theorem 1.13. The descriptor continuous-time linear system (1.1) is asymptotically stable if and only if (1.102) lim (t) = 0. t→∞

Proof. Substituting u(t) = 0 into (1.12) we obtain x(t) =

∞  Tk Et k x0 . k! k=0

(1.103)

Next, by Definition 1.5 we have ∞  Tk Et k x0 = lim (t)x0 . t→∞ t→∞ k! k=0

lim x(t) = lim

t→∞

(1.104)

Therefore, the descriptor system (1.1) is asymptotically stable for x0 ∈ X 0 if and only if the condition (1.102) holds.   From the above considerations we have the following theorem. Theorem 1.14. The descriptor continuous-time linear system (1.1) is asymptotically stable if and only if one of the following equivalent conditions is satisfied: 1. Roots of the characteristic Eq. (1.100) satisfy the condition (1.101). 2. The matrix (t) defined by (1.99) satisfies the condition (1.102). Example 1.7. (Continuation of Example 1.4) Consider the descriptor continuoustime linear system (1.1) with (1.76). Using (1.80a) and (1.99) we have ⎤ ⎡ 2k 2k 2 t 22k+1 t 2k+1 0 ∞ ∞ k (2k)! (2k+1)!   Tk Et ⎥ ⎢ 22k+1 t 2k+1 22k t 2k (t) = = ⎣ (2k+1)! 0⎦ (2k)! k! k=0 k=0 0 0 0 ⎡ ⎤ cosh(2t) sinh(2t) 0 = ⎣ sinh(2t) cosh(2t) 0 ⎦ . 0 0 0

(1.105)

According to the second condition of Theorem 1.14 the descriptor system (1.1) with (1.76) is unstable since limt→∞ cosh(2t) = ∞ and limt→∞ sinh(2t) = ∞. The characteristic equation of the system (1.76) has the form    s −2 0    p(E,A) (s) = det[Es − A] =  −2 s 1  = −s 2 + 4 = 0.  0 0 −1 

(1.106)

1.1 Continuous-Time Systems

27

Eigenvalues of the pair (E, A) are s1 = −2, s2 = 2. Therefore, by the first condition of Theorem 1.14 the descriptor system (1.1) with (1.76) is unstable since Res2 = 2 > 0. Example 1.8. Consider the descriptor continuous-time linear system (1.1) with 

   10 −2 1 E= , A= . 00 1 −2

(1.107)

The matrix pencil of (1.107) is regular since    s + 2 −1   = 2s + 3 = 0. det[Es − A] =  −1 2 

(1.108)

The matrix [Es − A]−1 has the form −1

[Es − A]



s + 2 −1 = −1 2

−1

 =

1 2 2s+3 2s+3 1 s+2 2s+3 2s+3

 .

(1.109)

Using (1.6) for n 0 = 1 the matrix (1.109) can be written in the form ∞ 

[Es − A]−1 =

Tk s −(k+1) ,

(1.110a)

k=−1

where  ⎧ 0 0 ⎪ ⎪ for k = −1, ⎪ ⎨ 0 0.5  Tk =  (−1.5)k 0.5(−1.5)k ⎪ ⎪ for k = 0, 1, 2, . . . . ⎪ ⎩ 0.5(−1.5)k 0.25(−1.5)k

(1.110b)

Using (1.107) and (1.110b) we have  Tk E =

(−1.5)k 0 0.5(−1.5)k 0

 for k = 0, 1, 2, . . .

(1.111)

and     ∞ ∞ (−1.5)k t k   Tk Et k 0 e−1.5t 0 k! . = = (t) = k k t 0.5e−1.5t 0 k! 0.5 (−1.5) 0 k! k=0 k=0

(1.112)

By the second condition of Theorem 1.14 the descriptor system (1.1) with (1.107) is asymptotically stable since limt→∞ e−1.5t = 0. The characteristic equation of the system (1.107) is given by

28

1 Descriptor Linear Systems

   s + 2 −1   = 2s + 3 = 0. p(E,A) (s) = det[Es − A] =  −1 2 

(1.113)

From (1.113) we can see that the pair (E, A) has one eigenvalue s = −1.5 and by Theorem 1.14 the descriptor system (1.1) with (1.107) is asymptotically stable since Res = −1.5 < 0.

1.1.4.2

Drazin Inverse Matrix Method

In this section necessary and sufficient stability conditions of the descriptor continuous-time linear system (1.24) (or equivalently (1.1)) will be established using the Drazin inverse matrix method. It is assumed that the solution to the state Eq. (1.24a) has the form (1.45). Definition 1.5 of the asymptotic stability of the descriptor system is suitable for this method since the solutions of (1.1) and (1.24) are equivalent. ¯ A) ¯ is given by Definition 1.7. The characteristic equation of the matrix pair ( E, ¯ − A] ¯ = a¯ r s r + a¯ r −1 s r −1 + . . . + a¯ 1 s + a¯ 0 = 0, p( E, ¯ A) ¯ (s) = det[ Es

(1.114)

where r = rank E¯ D A¯ and p( E, ¯ A) ¯ (s) is the characteristic polynomial of the matrix pair ¯ A). ¯ ( E, ¯ A) ¯ Lemma 1.7. The characteristic polynomials of the matrix pairs (E, A) and ( E, are related by p(E,A) (s) . (1.115) p( E, ¯ A) ¯ (s) = det[Ec − A] ¯ A) ¯ has the form Proof. The characteristic polynomial of the pair ( E, ¯ − A] ¯ = det{[Ec − A]−1 Es − [Ec − A]−1 A} p( E, ¯ A) ¯ (s) = det[ Es = det{[Ec − A]−1 [Es − A]} = det[Ec − A]−1 det[Es − A] = {det[Ec − A]}

−1

(1.116)

p(E,A) (s),

which is equivalent to (1.115).

 

¯ A) ¯ Remark 1.5. The characteristic equations of the matrix pairs (E, A) and ( E, have the same form. Therefore, both pairs have the same set of eigenvalues and Theorem 1.12 can be used for the roots of the Eq. (1.114). ¯ A) ¯ (or (E, A)) and Lemma 1.8. The matrix E¯ D A¯ has r eigenvalues of the pair ( E, additionally n − r zero eigenvalues, i.e. its characteristic equation has the form ¯ = s n−r p( E, p E¯ D A¯ (s) = det[In s − E¯ D A] ¯ A) ¯ (s) = 0,

(1.117)

1.1 Continuous-Time Systems

29

¯ where p E¯ D A¯ (s) is the characteristic polynomial of the matrix E¯ D A. Proof. Using (1.27d) we have     −1 ¯ = det T In 1 s − J A1 0 T −1 p E¯ D A¯ (s) = det[In s − E¯ D A] 0 In 2 s = detT det[In 1 s − J −1 A1 ]det[In 2 s]detT −1

(1.118)

= s n 2 det[In 1 s − J −1 A1 ], since detT detT −1 = In . It is easy to see that n 1 = r and n 2 = n − r . Using again (1.27d) we can write     0 J s − A1 −1 ¯ ¯ T p( E, ¯ A) ¯ (s) = det[ Es − A] = det T 0 N s − A2 = detT det J s − A1 ]det[N s − A2 ]detT −1

(1.119)

= det[J s − A1 ]det[N s − A2 ], since detT detT −1 = In . ¯ Then from (1.27d) we have N A2 = A2 N if and only if Note that E¯ A¯ = A¯ E. A2 = αIn 2 , where α ∈ R, i.e. A2 is a scalar matrix (a nilpotent matrix N can not be scalar). Hence, the Eq. (1.119) can be written in the form p( E, ¯ A) ¯ (s) = det[J s − A1 ]det[N s − αIn 2 ] = (−α)n 2 det[J s − A1 ] = (−α)n 2 det[In 1 s − J −1 A1 ],

(1.120)

since det[N s − αIn 2 ] = (−α)n 2 . Equating (1.118) and (1.120) to zero we have p E¯ D A¯ (s) = s n 2 det[In 1 s − J −1 A1 ] = 0,

(1.121a)

−1 A1 ] = 0, p( E, ¯ A) ¯ (s) = det[In 1 s − J

(1.121b)

which is equivalent to (1.117).

 

Theorem 1.15. The descriptor continuous-time linear system (1.1) is asymptotically stable if and only if the matrix E¯ D A¯ has r stable eigenvalues (satisfying the condition (1.101)) and n − r zero eigenvalues. Proof. The proof follows directly from Lemma 1.8.

 

Based on the above considerations we have the following Theorem. Theorem 1.16. The descriptor continuous-time linear system (1.1) is asymptotically stable if and only if one of the following equivalent conditions is satisfied: 1. Roots of the characteristic Eq. (1.114) satisfy the condition (1.101).

30

1 Descriptor Linear Systems

2. The matrix E¯ D A¯ has r stable eigenvalues and n − r zero eigenvalues. Example 1.9. (Continuation of Examples 1.5 and 1.7) Consider the descriptor continuous-time linear system (1.1) with (1.76). Using (1.89) and (1.114) we have    1 −0.5s 0    ¯ − A] ¯ =  −0.5s 1 0  = −0.25s 2 + 1 = 0. p( E, ¯ A) ¯ (s) = det[ Es    0 0 1

(1.122)

¯ A) ¯ are s1 = −2, s2 = 2 and are equal to the eigenvalues of Eigenvalues of the pair ( E, the pair (E, A). By Theorem 1.16 the descriptor system (1.1) with (1.76) is unstable since Res2 = 2 > 0. Using (1.91) and (1.117) we have    s −2 0    ¯ =  −2 s 0  = s(s 2 − 4) = 0. p E¯ D A¯ (s) = det[I3 s − E¯ D A]    0 0 s

(1.123)

It is easy to see that r = rank E¯ D A¯ = 2. According to (1.123) the matrix E¯ D A¯ has ¯ A) ¯ equal to s1 = −2, s2 = 2 and n − r = 1 zero r = 2 eigenvalues of the pair ( E, eigenvalue s3 = 0. Therefore, by Theorem 1.16 the considered system is unstable.

1.1.4.3

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient stability conditions of the descriptor continuous-time linear system (1.55) (or equivalently (1.1)) using the WeierstrassKronecker decomposition method. It is assumed that the solution to the Eq. (1.55b) has the form (1.59). Note that the state vector of the subsystem (1.55b) x¯2 (t) = 0 for u(t) = 0. Therefore, the stability of the system (1.55) (or equivalently (1.1)) depends entirely on the subsystem (1.55a). Definition 1.8. The descriptor continuous-time linear system (1.55) (or equivalently (1.1)) is called asymptotically stable if lim x¯1 (t) = 0

t→∞

(1.124)

for all initial conditions x¯10 ∈ Rn 1 and u(t) = 0. Definition 1.9. [47] The characteristic equation of the matrix A1 is given by p A1 (s) = det[In 1 s − A1 ] = s n 1 + a˜ n 1 −1 s n 1 −1 + . . . + a˜ 1 s + a˜ 0 = 0,

(1.125)

where n 1 = deg{det[Es − A]} and p A1 (s) is the characteristic polynomial of the matrix A1 .

1.1 Continuous-Time Systems

31

Lemma 1.9. [11] The characteristic polynomials of the matrix pair (E, A) and of the matrix A1 are related by p(E,A) (s) =

(−1)n 2 p A (s). det(P Q) 1

(1.126)

Proof. From (1.52) we have det{P[Es − A]Q} = det[In 1 s − A1 ]det[N s − In 2 ]

(1.127)

det Pdet[Es − A]det Q = (−1)n 2 det[In 1 s − A1 ],

(1.128)

and since det[N s − In 2 ] = (−1)n 2 . The matrices P and Q are nonsingular, and therefore det P = 0 and det Q = 0. Dividing both sides of the Eq. (1.128) by det Pdet Q = det(P Q) we obtain (1.126).   Remark 1.6. The characteristic equations of the matrix pair (E, A) and of the matrix A1 have the same form. Therefore, the matrix pair (E, A) and the matrix A1 have the same set of eigenvalues and Theorem 1.12 can be used for the roots of the Eq. (1.125). Theorem 1.17. [47] The matrix A1 (or equivalently the system (1.55)) is asymptotically stable if and only if its eigenvalues satisfy the condition (1.101). Example 1.10. (Continuation of Examples 1.6 and 1.7) Consider the descriptor continuous-time linear system (1.1) with (1.76). Using (1.97) and (1.125) we have    s −2   = s 2 − 4. p A1 (s) = det[I2 s − A1 ] =  −2 s 

(1.129)

Eigenvalues of the matrix A1 are s1 = −2, s2 = 2 and are equal to the eigenvalues of the pair (E, A). By Theorem 1.17 the considered descriptor system is unstable.

1.1.5 Superstability of Descriptor Continuous-Time Linear Systems The asymptotic stability of a dynamical system ensures that its free response decreases to zero for t → ∞, however its value may increase significantly in the initial part of the state vector trajectory (it is so-called “peak effect”). In superstable systems the norm of the state vector decreases monotonically to zero for t → ∞, which prevents such undesirable effects [70, 137, 138].

32

1.1.5.1

1 Descriptor Linear Systems

Properties of Superstable Continuous-Time Linear Systems

The following norms will be used: 1) ∞-norm of a vector x = [xi ] ∈ Rn x = max |xi |,

(1.130)

1≤i≤n

2) 1-norm of a matrix A = [ai j ] ∈ Rn×n ⎞ ⎛ n  A = max ⎝ |ai j |⎠ . 1≤i≤n

(1.131)

j=1

Consider the standard continuous-time linear system x(t) ˙ = Ax(t),

(1.132)

where x(t) ∈ Rn is the state vector and A ∈ Rn×n . Definition 1.10. [137] The matrix A ∈ Rn×n of the continuous-time linear system (1.132) is called superstable if ⎛

⎞

n  ⎜ ⎟ −a − |ai j |⎟ σ (A) = σ = min ⎜ ii ⎠ > 0. 1≤i≤n ⎝

(1.133)

j=1 j=i

Quantity σ (A) is called the superstability degree of the matrix A. If the matrix is superstable, then it is also stable, however the reverse implication does not hold. Lemma 1.10. [137] For a superstable matrix A we have e At ≤ e−σ t .

(1.134)

Theorem 1.18. [137] If the system (1.132) is superstable, then x(t) ≤ x0 e−σ t , t ∈ [0, +∞).

(1.135)

Note that the superstability ensure monotonic decrease of the norm of the state vector, however particular state variables can oscillate. The main difference is that for asymptotic stable systems the Eq. (1.135) is replaced by x(t) ≤ c(A, ν) x0 e−νt , 0 < ν < min {−Resi }, 1≤i≤n

(1.136)

1.1 Continuous-Time Systems

33

where the constant c(A, ν) can take significant values in the initial part of the state vector trajectory. In superstable systems there are no such undesirable “peaks” [137]. Example 1.11. Consider the continuous-time linear system (1.132) with  A=

 −3 2 . 0 −2

(1.137)

The system is asymptotically stable since the eigenvalues of (1.137) are s1 = −3, s2 = −2. By the condition (1.133) we have σ = 1 > 0 and the system is also superstable. Assuming the initial condition x0 = [ 1 2 ]T we present time plots of the state variables in Fig. 1.1, from which we can see that the norm of the state vector decreases monotonically for t → ∞. Now let us consider the continuous-time linear system (1.132) with   −3 6 A= . (1.138) 0 −2 Note that the eigenvalues of (1.137) and (1.138) are the same (i.e. s1 = −3, s2 = −2), but in the second case the system is not superstable since by the condition (1.133) we have σ = −3 < 0. Thus, this property of dynamical systems does not depend of their eigenvalues. From Fig. 1.2 we can see that the value of the state variable x1 (t) increases highly above the initial condition and the norm of the state vector does not decrease monotonically for t → ∞. 2

2 x1 (t)

||x(t)||

x2 (t)

1.5

1.5

1

1

0.5

0.5

0

0

1

2

t [s]

3

4

0

0

1

2

3

4

t [s]

Fig. 1.1 State variables (on the left) and norm of the state vector (on the right) of the system (1.132) with (1.137) for x0 = [ 1 2 ]T

34

1 Descriptor Linear Systems 2.5

2.5

||x(t)||

x1 (t) x2 (t)

2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

4

0

0

1

2

3

4

t [s]

t [s]

Fig. 1.2 State variables (on the left) and norm of the state vector (on the right) of the system (1.132) with (1.138) for x0 = [ 1 2 ]T

1.1.5.2

Laurent Series Expansion Method

In this section necessary and sufficient superstability conditions of the descriptor continuous-time linear system (1.1) will be established using the Laurent series expansion method. It is assumed that the solution to the state Eq. (1.1a) has the form (1.12). Consider the descriptor continuous-time linear system (1.1). Solution to the Eq. (1.1a) for u(t) = 0 is given by (1.103). Let (t) be the matrix defined by (1.99). Theorem 1.19. The descriptor continuous-time linear system (1.1) is superstable if and only if (t1 ) > (t2 ) for any t1 < t2 , t1 , t2 ∈ [0, +∞).

(1.139)

Proof. From (1.103) we have x(t) = (t)x0 ≤ (t) x0 .

(1.140)

From the equation x(t) = (t)x0 we can see that the set of consistent initial is given by x0 ∈ Im(0). Using (1.12) and (1.13) for u(t) = 0 it is easy to see that if x0 ∈ Im(0), then x0 = (0)x0 . If the condition (1.139) holds, then we have max (t)x0 = (0)x0 = x0 ≤ (0) x0

(1.141)

x(t1 ) > x(t2 ) for any t1 < t2 , t1 , t2 ∈ [0, +∞).

(1.142)

t

and From the condition (1.139) it also follows that limt→∞ (t) = 0. Therefore, the norm of the state vector decreases monotonically to zero for t → ∞.  

1.1 Continuous-Time Systems

35

Example 1.12. Consider the descriptor continuous-time linear system (1.1) with ⎡

⎤ ⎡ ⎤ 0 −1 0 −1 3 0 E = ⎣ 0.5 0.5 0 ⎦ , A = ⎣ 0 −1.5 0 ⎦ . 0 0 1 0 0 0

(1.143)

The matrix pencil (E, A) of (1.143) is regular since    1 −(s + 3) 0    det[Es − A] =  0.5s 0.5(s + 3) 0  = −0.5s 2 − 2s − 1.5 = 0.  0 0 −1 

(1.144)

The matrix [Es − A]−1 has the form ⎡

[Es − A]−1

⎤−1 ⎡ 1 1 −(s + 3) 0 s+1 s = ⎣ 0.5s 0.5(s + 3) 0 ⎦ = ⎣ − (s+1)(s+3) 0 0 −1 0

2 s+1 2 (s+1)(s+3)

0

⎤ 0 0 ⎦. −1 (1.145)

Using (1.6) the matrix (1.145) can be written in the form [Es − A]−1 =

∞ 

Tk s −(k+1) ,

(1.146a)

k=−1

where ⎤ ⎧⎡ 00 0 ⎪ ⎪ ⎪ ⎪⎣0 0 0 ⎦ for k = −1, ⎪ ⎪ ⎪ ⎨ 0 0 −1 ⎤ Tk = ⎡ (−1)k 2(−1)k 0 ⎪ ⎪ ⎪ ⎪ ⎣ 0.5(−1)k+1 (3k+1 − 1) (−1)k+1 (3k − 1) 0 ⎦ for k = 0, 1, 2, . . .. ⎪ ⎪ ⎪ ⎩ 0 0 0 (1.146b) Using (1.143) and (1.146b) we have ⎡

⎤ (−1)k 0 0 Tk E = ⎣ 0.5(−1)k+1 (3k − 1) (−3)k 0 ⎦ for k = 0, 1, 2, . . . 0 0 0

(1.147)

36

1 Descriptor Linear Systems 4

4 x1 (t)

3.5

x3 (t)

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

4

||x(t)||

3.5

x2 (t)

5

0

0

1

2

3

4

5

t [s]

t [s]

Fig. 1.3 State variables (on the left) and norm of the state vector (on the right) of the system (1.1a) with (1.143) for x0 = [ 4 1 0 ]T

and

⎤ ⎡ (−1)k t k ∞ ∞ 0 0   k! Tk Et k ⎣ 0.5 (−1)k+1 (3k −1)t k (−3)k t k 0 ⎦ = (t) = k! k! k! k=0 k=0 0 0 0 ⎡ ⎤ e−t 0 0 = ⎣ 0.5(e−t − e−3t ) e−3t 0 ⎦ . 0 0 0

(1.148)

Analyzing (1.148) we can see that !⎡ ⎤! ! 100 ! ! ! ⎣ 0 1 0 ⎦! = 1 max (t) = (0) = ! ! ! t ! 000 !

(1.149)

and (t1 ) > (t2 ) for any t1 < t2 , t1 , t2 ∈ [0, +∞). By Theorem 1.19 the descriptor continuous-time linear system (1.1) with (1.143) is superstable. Using (1.13) we can determine consistent initial conditions of the system ⎡

⎤⎡ ⎤ ⎡ ⎤ 100 v1 v1 x0 = T0 Ev = ⎣ 0 1 0 ⎦ ⎣ v2 ⎦ = ⎣ v2 ⎦ , 000 v3 0

(1.150)

where v1 , v2 ∈ R are any numbers. From the time plots presented in Fig. 1.3 for x0 = [ 4 1 0 ]T we can see that the norm of the state vector decreases monotonically to zero for t → ∞.

1.1 Continuous-Time Systems

1.1.5.3

37

Drazin Inverse Matrix Method

In this section necessary and sufficient superstability conditions of the descriptor continuous-time linear system (1.24) (or equivalently (1.1)) will be established using the Drazin inverse matrix method. It is assumed that the solution to the state Eq. (1.24a) has the form (1.45). Consider the descriptor continuous-time linear system (1.24). Solution to the Eq. (1.24a) for u(t) = 0 has the form ¯ D At ¯

E¯ E¯ D x0 .

(1.151)

¯D ¯ ¯ ¯D E¯ E¯ D x0 = e[ E A+G(In − E E )]t E¯ E¯ D x0 ,

(1.152)

x(t) = e E Using Lemma 1.6 we can write ¯ D At ¯

x(t) = e E

where G ∈ Rn×n is an arbitrary matrix. Taking into account (1.45) and (1.46) for u(t) = 0 it is easy to see that if x0 ∈ Im E¯ E¯ D , then x0 = E¯ E¯ D x0 . Therefore, from (1.152) we have ¯ D A+G(I ¯ ¯D ¯ n − E E )]t

x(t) ≤ e[ E

x0 .

(1.153)

If the matrix E¯ D A¯ + G(In − E¯ E¯ D ) is superstable (i.e. it satisfies the condition ¯D ¯ ¯ ¯D (1.133)), then by Lemma 1.10 we have e[ E A+G(In − E E )]t ≤ e−σ t . From the inequality (1.153) we obtain x(t) ≤ x0 e−σ t ,

(1.154)

and the norm of the state vector decreases monotonically to zero. Therefore, the following theorem has been proved. Theorem 1.20. The descriptor continuous-time linear system (1.24) (or equivalently (1.1)) is superstable if and only if there exists a matrix G ∈ Rn×n such that the matrix E¯ D A¯ + G(In − E¯ E¯ D ) satisfies the condition (1.133). Example 1.13. (Continuation of Example 1.12) Consider the descriptor continuoustime linear system (1.1) with (1.143). Using (1.24) for c = 0 we obtain ⎡

E¯ = [−A]

−1

⎡ ⎤ ⎤ 100 −1 0 0 −1 E = ⎣ 13 31 0 ⎦ , A¯ = [−A] A = ⎣ 0 −1 0 ⎦ . 0 0 −1 000

(1.155)

Next, from (B.1) and (B.2) we have ⎡

E¯ D

⎤ 1 00 = ⎣ −1 3 0 ⎦ 0 00

(1.156)

38

1 Descriptor Linear Systems

⎡

⎤ −1 0 0 E¯ D A¯ = ⎣ 1 −3 0 ⎦ . 0 0 0

and

(1.157)

For the matrix G = [gi j ] ∈ R3×3 we obtain ⎤ −1 0 g13 E¯ D A¯ + G(I3 − E¯ E¯ D ) = ⎣ 1 −3 g23 ⎦ . 0 0 g33 ⎡

(1.158)

Assuming g13 = 0, g23 = 0 and g33 < 0 it is easy to see that the matrix (1.158) satisfies the condition (1.135) since σ = 1 > 0 for g33 ≤ −1 and σ = g33 > 0 for g33 ∈ (−1; 0). Therefore, the considered descriptor system is superstable. Consistent initial conditions of the system are determined by (1.150). The time plots of the state variables and the norm of the state vector for x0 = [ 4 1 0 ] are presented in Fig. 1.3.

1.1.5.4

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient superstability conditions of the descriptor continuous-time linear system (1.55) (or equivalently (1.1)) using the WeierstrassKronecker decomposition method. It is assumed that the solution to the Eq. (1.55b) has the form (1.59). Note that every superstable system is also asymptotically stable. State vector of the subsystem (1.55b) x¯2 = 0 for u(t) = 0. Using (1.53) and (1.57) we obtain 

  At  x¯1 (t) e 1 x¯10 x(t) = Q x(t) ¯ =Q =Q 0 x¯2 (t)   At    At e 1 0 x¯10 e 1 0 =Q Q −1 x0 =Q 0 e Mt 0 e Mt 0     A1 0 −1 Q x0 = e A M t x0 , = exp Q 0 M

(1.159)

where the matrix M ∈ Rn 2 ×n 2 is arbitrary and  AM = Q

A1 0 0 M



Q −1 .

(1.160)

From the Eq. (1.159) we have x(t) = e A M t x0 ≤ e A M t x0 .

(1.161)

1.1 Continuous-Time Systems

39

If the matrix A M is superstable (i.e. it satisfies the condition (1.133)), then by Lemma 1.10 we have e A M t ≤ e−σ t . From the inequality (1.161) we obtain x(t) ≤ x0 e−σ t ,

(1.162)

and the norm of the state vector decreases monotonically to zero. Therefore, the following theorem has been proved. Theorem 1.21. The descriptor continuous-time linear system (1.55) (or equivalently (1.1)) is superstable if and only if there exists a matrix M ∈ Rn 2 ×n 2 such that the matrix A M defined by (1.160) satisfies the condition (1.135). Example 1.14. (Continuation of Example 1.12) Consider the descriptor continuoustime linear system (1.1) with (1.155). In this case ⎡

⎤ ⎡ ⎤ −1 0 0 010 P = ⎣ 1 2 0⎦, Q = ⎣1 0 0⎦ 0 01 001 ⎤ 100 In 1 0 = PEQ = ⎣0 1 0⎦, 0 N 000 ⎡ ⎤   −3 1 0 A1 0 = P AQ = ⎣ 0 −1 0 ⎦ , 0 In 2 0 0 1

and





(1.163)

⎡

(1.164)

where n 1 = 2 and n 2 = 1. Using (1.160) for the matrices (1.163), (1.164) and M = [m] ∈ R1×1 we obtain ⎡

AM

⎤⎡ ⎤⎡ ⎤ ⎡ ⎤ 010 −3 1 0 010 −1 0 0 = ⎣ 1 0 0 ⎦ ⎣ 0 −1 0 ⎦ ⎣ 1 0 0 ⎦ = ⎣ 1 −3 0 ⎦ . 001 0 0 m 001 0 0 m

(1.165)

Assuming m < 0 it is easy to see that the matrix (1.165) satisfies the condition (1.135) since σ = 1 > 0 for m ≤ 1 and σ = m > 0 for m ∈ (−1; 0). Therefore, the considered descriptor system is superstable.

40

1 Descriptor Linear Systems

1.2 Discrete-Time Systems 1.2.1 State Equations of Descriptor Discrete-Time Linear Systems A discrete approximation of the derivative of the state vector in (1.1a) can be obtained using the Euler method: xi+1 − xi , (1.166) x(t) ˙ ≈ h where h > 0 is the discretization period and xi , i ∈ Z+ are the values of x(t) for t = 0, h, 2h, . . .. Other methods of approximation are given in [100]. Discretizing the Eq. (1.1) and using the approximation (1.166) we obtain E xi+1 = Ad xi + Bd u i ,

(1.167a)

yi = C xi + Du i ,

(1.167b)

Ad = Ah + E ∈ Rn×n , Bd = Bh ∈ Rn×m ,

(1.167c)

where xi ∈ Rn , u i ∈ Rm , yi ∈ R p are state, input and output vectors and E ∈ Rn×n , C ∈ R p×n , D ∈ R p×m . It is assumed that detE = 0 and det[E z − Ad ] = 0 for some z ∈ C.

(1.168)

Consider the matrix Ad defined by (1.167c). The condition (1.168) takes the form det[E z − Ad ] = det[E(z − 1) − Ah] = h n det[E(z − 1)h −1 − A] = 0. (1.169) If the condition (1.2) holds, then the condition (1.169) is also met for h > 0. If the matrix E is nonsingular (det E = 0), then we can transform (1.167a) to the state equation of a standard discrete-time linear system through premultiplication of (1.167a) by E −1 . The system (1.167) with the singular matrix E (detE = 0) will be called a descriptor discrete-time linear system.

1.2.2 Solution to the State Equation of Descriptor Continuous-Time Linear Systems Let U be the set of admissible input vectors u i ∈ U ⊂ Rm and X 0 ⊂ Rn the set of consistent initial conditions x(0) = x0 ∈ X 0 for which the Eq. (1.167a) has the solution xi with u i ∈ U .

1.2 Discrete-Time Systems

1.2.2.1

41

Laurent Series Expansion Method

In this section the solution to the Eq. (1.167a) will be given using the Laurent series expansion method [26, 47, 49, 111, 147]. In addition, formulae linking together the matrices of the continuous-time system (1.1) and its discrete-time equivalent (1.167) will be given. Theorem 1.22. [47] Solution to the Eq. (1.167a) for x0 ∈ X 0 and u i ∈ U , i ∈ Z+ is given by the formula xi = Td,i E x0 +

n 0 +i−1

Td,i−k−1 Bd u k ,

(1.170a)

k=0

where the matrices Td,k ∈ Rn×n are determined by ⎧ ⎨ E Td,k − Ad Td,k−1 = In for k = 0, E Td,k − Ad Td,k−1 = 0 for k = 0, ⎩ for k ≤ −n 0 E Td,k = 0

(1.170b)

and n 0 is called the index of the system. From (1.170a) for i = 0 we have x0 = Td,0 Ev +

n 0 −1

Td,−k−1 Bd u k ,

(1.171)

k=0

where v ∈ Rn is an arbitrary vector. Remark 1.7. If the matrix E = In , then n 0 = 0 and Td,i = Aid . Therefore, from the Eq. (1.170) we have i−1  Ai−k−1 Bd u k . (1.172) xi = Aid x0 + d k=0

The formula (1.172) is the solution to the state equation of a standard discrete-time linear system. Similarly as for continuous-time systems, the matrices Td,k can be computed from the equality ∞  −1 [E z − Ad ] = Td,k z −(k+1) . (1.173) k=−n 0

Using (1.167c) we can write the above equation in the form

42

1 Descriptor Linear Systems

[E z − Ad ]

−1

= [E z − (Ah + E)] ∞ 

= h −1

−1

=h

−1



z−1 E −A h

Tk (z − 1)−(k+1) h k+1

−1

(1.174)

k=−n 0

=

∞ 

h k Tk (z − 1)−(k+1) ,

k=−n 0

where Tk are determined by (1.3b) (they are matrices of the continuous-time system). Expanding the expressions (z − 1)−(k+1) in the series with respect to z −k , k = −n 0 , . . . , 0, 1, . . . we can write (1.174) in the form (1.173), where Td,k =

k    k j=0

Td,−k =

n0 

 (−1)

j−k

j=k

and

j

h j Tj ,

 j − 1 −j h T− j k−1

  k! k = . j (k − j)! j!

(1.175a)

(1.175b)

(1.175c)

The formulae (1.175) link together the matrices Tk of the continuous-time system and the matrices Td,k of the discrete-time system for the assumed approximation of the derivative (1.166). Example 1.15. (Continuation of Example 1.1) Consider the descriptor discretetime linear system (1.1) with (1.17). The discrete-time equivalent of the continuoustime system is described by (1.167a), where the matrix E is determined by (1.17) and ⎡ ⎤ −0.2h − 0.4 1.8h 0.5h − 0.5 0 ⎢ 0.4h − 0.2 0.4h 0 0 ⎥ ⎥ Ad = Ah + E = ⎢ ⎣ 0.2h + 0.4 −1.8h + 1 −0.5h + 0.5 0.5h ⎦ , −0.4h + 0.2 0.6h 0 0 ⎡ ⎤ (1.176) −h −3.6h ⎢ 0 −0.8h ⎥ ⎥ Bd = Bh = ⎢ ⎣ −h 2.6h ⎦ . 0 −0.2h The matrix pencil (E, Ad ) is regular since

1.2 Discrete-Time Systems

43

det[E z − Ad ] = − 0.05h 2 z 2 + (−0.15h 3 + 0.1h 2 )z − 0.1h 4 + 0.15h 3 − 0.05h 2 = 0.

(1.177)

The matrix [E z − Ad ]−1 has the form ⎡ ⎢ [E z − Ad ]−1 = ⎢ ⎣−

3 − z+2h−1 0 1 −h 0 6 0 z+2h−1 2(z−1) − h 2 − h2

0 0 2 z+h−1 − h2

2 z+2h−1 − h1 2(z+4h−1) (z+h−1)(z+2h−1) − 2(z−1) h2

⎤ ⎥ ⎥. ⎦

(1.178)

Using (1.173) the matrix (1.178) can be written in the form [E z − Ad ]

−1

=

∞ 

Td,k z k+1 ,

(1.179a)

k=−2

where

Td,k

⎤ ⎧⎡ 0 0 0 0 ⎪ ⎪ ⎪ ⎢0 0 0 0 ⎥ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎣0 0 0 0 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ 0 − h22 0 − h22 ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ ⎨ 0 0 0 0 0 − h1 0 − h1 ⎥ = ⎢ ⎢ ⎥ ⎪ ⎣ 0 0 0 0 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ − h2 h22 − h2 h22 ⎪ ⎪ ⎪⎡ ⎪ ⎪ 0 −3(1 − 2h)k 0 ⎪ ⎪ ⎪ 0 0 0 ⎪ ⎣ ⎪ ⎪ −2(1 − h)k 6(1 − 2h)k 0 ⎪ ⎩ 0 0 0

for k = −2,

for k = −1, 2(1 − 2h)k 0 6(1 − h)k − 4(1 − 2h)k 0

(1.179b)

⎤ ⎦ for k = 0, 1, . . ..

Using (1.17) and (1.179b) we have ⎡

(1 − 2h)k ⎢ 0 Td,k E = ⎢ ⎣ 2(1 − h)k − 2(1 − 2h)k 0

0 0 0 0 0 (1 − h)k 0 0

⎤ 0 0⎥ ⎥ for k = 0, 1, 2, . . . , 0⎦ 0 (1.180a)

44

1 Descriptor Linear Systems

⎤ ⎧⎡ 00 ⎪ ⎪ ⎪ ⎢0 0 ⎥ ⎪ ⎪ ⎢ ⎥ ⎪ for k = −2, ⎪ ⎣0 0 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ 0 2 ⎪ ⎪ ⎪⎡ h ⎤ ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎨⎢0 1 ⎥ ⎥ for k = −1, Td,k Bd = ⎢ ⎣0 0 ⎦ ⎪ ⎪ ⎪ ⎪ 4 2 − h2 ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ 0 2h(1 − 2h)k ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ 0 0 ⎪ ⎢ ⎥ ⎪ ⎪ k k k ⎣ ⎪ 2h(1 − h) 6h(1 − h) − 4h(1 − 2h) ⎦ for k = 0, 1, 2, . . .. ⎪ ⎪ ⎪ ⎩ 0 0 (1.180b) Using (1.170a), (1.17), (1.176), (1.180) and taking into account that n 0 = 2 we obtain xi = Td,i E x0 +

i+1  k=0

+ Td,−2 Bd u i+1 ⎡

Td,i−k−1 Bd u k = Td,i E x0 +

Td,i−k−1 Bd u k + Td,−1 Bd u i

k=0

⎡

(1 − 2h)i ⎢ 0 =⎢ ⎣ 2(1 − h)i − 2(1 − 2h)i 0

0 ⎢ 0 ⎢ + ⎣ 2h(1 − h)i−k−1 k=0 0 ⎤ ⎡ ⎡ 0 0 0 ⎢0 1 ⎥ ⎢0 ⎥ ⎢ +⎢ ⎣ 0 0 ⎦ ui + ⎣ 0 4 2 − h2 0 i−1 

i−1 

0 0 0 0 0 (1 − h)i 0 0

⎤ 0 0⎥ ⎥x 0⎦ 0 0 ⎤

2h(1 − 2h)i−k−1 ⎥ 0 ⎥u 6h(1 − h)i−k−1 − 4h(1 − 2h)i−k−1 ⎦ k 0 ⎤ 0 0⎥ ⎥u . 0 ⎦ i+1 2 h

(1.181) The equation above shows the solution (1.170a) of the state equation of the discretetime equivalent (1.167a) of the system (1.1a) with (1.17).

1.2.2.2

Drazin Inverse Matrix Method

In this section the solution to the Eq. (1.167a) will be given using the Drazin inverse matrix method [26, 27, 34, 47]. In addition, formulae linking together the matrices of the continuous-time system (1.24) and its discrete-time equivalent (1.182) will be given. Consider the Eq. (1.167a) and assume that the condition (1.168) is met. There exists a number c ∈ C such that det[Ec − Ad ] = 0. Premultiplying the Eq. (1.167a) by the matrix [Ec − Ad ]−1 we obtain

1.2 Discrete-Time Systems

45

E¯ d xi+1 = A¯ d xi + B¯ d u i ,

(1.182a)

where E¯ d = [Ec − Ad ]−1 E, A¯ d = [Ec − Ad ]−1 Ad , B¯ d = [Ec − Ad ]−1 Bd . (1.182b) Note that the Eqs. (1.167a) and (1.182a) have the same solution xi . Basic definitions and properties concerning Drazin inverses are given in Sect. 1.1.2.2. Theorem 1.23. [47] Solution to the Eq. (1.167a) for x0 ∈ X 0 and u i ∈ U , i ∈ Z+ has the form xi = ( E¯ dD A¯ d )i E¯ dD E¯ d x0 +

i−1 

E¯ dD ( E¯ dD A¯ d )i−k−1 B¯ d u k

k=0

(1.183)

q−1  D ¯ ¯ + ( E d E d − In ) ( E¯ d A¯ dD )k A¯ dD B¯ d u i+k , k=0

where q is the index of E¯ d . From (1.183) for i = 0 we have x0 = E¯ d E¯ dD v + ( E¯ d E¯ dD − In )

q−1  ( E¯ d A¯ dD )k A¯ dD B¯ d u k ,

(1.184)

k=0

where v ∈ Rn is an arbitrary vector. For discrete-time systems we can formulate a similar theorem to Theorem 1.5, i.e. the solution (1.183) of the Eq. (1.167a) and the set of consistent initial conditions X 0 do not depend of the choice of c [47]. Using (1.167c) the matrices (1.182b) can be written in the form  −1 c−1 −A E¯ d = [Ec − (Ah + E)]−1 E = h −1 E E = h −1 [E c¯ − A]−1 E, h (1.185a)  −1 c − 1 −A (Ah + E) A¯ d = [Ec − (Ah + E)]−1 (Ah + E) = h −1 E h = [E c¯ − A]−1 A + h −1 [E c¯ − A]−1 E, B¯ d = [Ec − (Ah + E)]−1 Bh = h −1



c−1 E −A h

−1

(1.185b) Bh = [E c¯ − A]−1 B, (1.185c)

where c¯ =

c−1 . h

46

1 Descriptor Linear Systems

Taking into account that we can always find c, c¯ ∈ C such that [Ec − A] = [E c¯ − A] and det[Ec − A] = det[E c¯ − A] = 0 from (1.185) we obtain ¯ ¯ A¯ d = A¯ + h −1 E, ¯ B¯ d = B, E¯ d = h −1 E,

(1.186)

¯ A, ¯ B¯ are the matrices of the continuous-time system defined by (1.24b). where E, ¯ A, ¯ B¯ of the continuous-time The formulae (1.186) link together the matrices E, system and the matrices E¯ d , A¯ d , B¯ d of the discrete-time system for the assumed approximation of the derivative (1.166). Example 1.16. (Continuation of Examples 1.2 and 1.15) Consider the descriptor continuous-time linear system (1.24a) with (1.47). The discrete-time equivalent of the continuous-time system is described by (1.182a). Using (1.186) we obtain ⎤ 0 00 ⎢ 0 0 0 0⎥ ⎥ E¯ d = h −1 E¯ = ⎢ ⎣ 1 0 1 0⎦, h h 0 − h2 0 0 ⎤ ⎡ 1 −1 0 0 0 2h ⎢ 0 −1 0 0 ⎥ ⎥, A¯ d = A¯ + h −1 E¯ = ⎢ 1 ⎣ 1 0 h −1 0 ⎦ h −2 0 0 −1 h ⎡ ⎤ 01 ⎢0 1⎥ ⎥ B¯ d = B¯ = ⎢ ⎣2 4⎦. 42 ⎡

1 2h

(1.187)

The matrix E¯ D of the continuous-time system has the form (1.48b). Taking into ¯ D = h E¯ D we have account that E¯ dD = (h −1 E) ⎡

E¯ dD = h E¯ D

2h ⎢ 0 =⎢ ⎣ −2h 0

0 0 0 0

0 0 h 0

⎤ 0 0⎥ ⎥. 0⎦ 0

(1.188)

The matrix A¯ d determined by (1.187) is nonsingular and then ⎡

2h − 2h−1 0

⎢ ⎢ A¯ dD = A¯ −1 2h d = ⎣− (2h−1)(h−1) 0 Next, we compute the matrices

0 0 −1 0 h 0 − h−1 2 0 h

⎤ 0 0 ⎥ ⎥. 0 ⎦ −1

(1.189)

1.2 Discrete-Time Systems

⎡

47

⎡ ⎤ ⎤ 0 1 − 2h 0 0 0 ⎢ ⎥ 0⎥ ⎥ , E¯ D A¯ d = ⎢ 0 0 0 0 ⎥ , E¯ d E¯ dD d ⎣ ⎦ 0 2h 0 1 − h 0 ⎦ 0 0 0 0 0 ⎡ ⎤ ⎤ 0 2h 0 0 0 0 ⎢ 0 0 ⎥ ⎢ 0 −1 0 0 ⎥ ⎢ ⎥ ⎥ ¯ ¯D E¯ dD B¯ d = ⎢ ⎣ 2h 2h ⎦ , ( E d E d − I4 ) = ⎣ 0 0 0 0 ⎦ , 0 0 0 0 0 −1 ⎤ ⎡ 1 0 0 0 − 2h−1 ⎢ 0 0 0 0⎥ ⎥, A¯ dD E¯ d = ⎢ 2h ⎣− 0 − 1 0⎦ 1 ⎢0 =⎢ ⎣0 0 ⎡

0 0 0 0

0 0 1 0

(h−1)(2h−1)

⎡

0

2 h

2h − 2h−1

(1.190)

h−1

0

0 ⎤

0 ⎥ ⎢ 0 −1 ⎥ A¯ dD B¯ d = ⎢ ⎣ − 2h − 2h(4h−1) ⎦ . h−1 (2h−1)(h−1) 2 −4 h−2 Taking into account that q = 2 and using (1.183), (1.190) we obtain xi = ( E¯ dD A¯ d )i E¯ dD E¯ d x0 +

i−1 

E¯ dD ( E¯ dD A¯ d )i−k−1 B¯ d u k + ( E¯ d E¯ dD − I4 ) A¯ dD u i

k=0

+ ( E¯ d E¯ dD − I4 ) E¯ d A¯ dD A¯ dD B¯ d u i+1 ⎤ ⎡ 0 0 0 (1 − 2h)i ⎢ 0 0 0 0⎥ ⎥ =⎢ ⎣ 2(1 − h)i − 2(1 − 2h)i 0 (1 − h)i 0 ⎦ x0 0 0 0 0 ⎡ ⎤ 0 2h(1 − 2h)i−k−1 i−1 ⎢ ⎥ 0 0 ⎢ ⎥ + ⎣ 2h(1 − h)i−k−1 6h(1 − h)i−k−1 − 4h(1 − 2h)i−k−1 ⎦ u k k=0 0 0 ⎤ ⎤ ⎡ ⎡ 0 0 00 ⎢0 1 ⎥ ⎢0 0 ⎥ ⎥ ⎥ ⎢ +⎢ ⎣ 0 0 ⎦ u i + ⎣ 0 0 ⎦ u i+1 . 4 2 − h2 0 h2

(1.191) The equation above shows the solution (1.183) of the state equation of the discretetime equivalent (1.182a) of the system (1.24a) with (1.47). Note that the Eqs. (1.181) and (1.191) are equivalent.

48

1 Descriptor Linear Systems

1.2.2.3

Weierstrass-Kronecker Decomposition Method

In this section the solution to the Eq. (1.167a) will be given using the WeierstrassKronecker decomposition method [45, 47, 96, 176]. In addition, formulae linking together the matrices of the continuous-time system (1.55) and its discrete-time equivalent (1.208) will be given. If the condition (1.168) holds, then there exists a pair of nonsingular matrices Pd , Q d ∈ Rn×n such that [47, 48]  0 In 1 z − A1d , A1d ∈ Rn 1 ×n 1 , N ∈ Rn 2 ×n 2 , 0 N z − In 2 (1.192) where n 1 = deg{det[E z − Ad ]}, n 2 = n − n 1 and N is a nilpotent matrix with the nilpotency index μ, i.e. N μ−1 = 0 and N μ = 0. The matrices Pd and Q d can be computed using one of procedures given in [47, 48, 168]. Premultiplying (1.167a) by the matrix Pd and introducing the new state vector 

Pd [E z − Ad ]Q d =



we obtain

x¯1,i x¯2,i



n1 n2 = Q −1 d x i , x¯ 1,i ∈ R , x¯ 2,i ∈ R ,

(1.193)

−1 Pd E Q d Q −1 d x i+1 = Pd Ad Q d Q d x i + Pd Bd u i ,

(1.194a)

yi = C Q d Q −1 d x i + Du i

(1.194b)

x¯1,i+1 = A1d x¯1,i + B1d u i ,

(1.195a)

N x¯2,i+1 = x¯2,i + B2d u i ,

(1.195b)

yi = C1d x¯1,i + C2d x¯2,i + Du i ,

(1.195c)

x¯i =

and using (1.192) we get



where Pd Bd =

 B1d , B1d ∈ Rn 1 ×m , B2d ∈ Rn 2 ×m , B2d

 C Q d = C1d C2d , C1d ∈ R p×n 1 , C2d ∈ R p×n 2 .

(1.196a) (1.196b)

Theorem 1.24. [47] Solution to the Eq. (1.195a) for given initial condition x¯10 ∈ Rn 1 and input u i ∈ U , i ∈ Z+ has the form

1.2 Discrete-Time Systems

49

x¯1,i = Ai1d x¯10 +

i−1 

Ai−k−1 B1d u k . 1d

(1.197)

k=0

Theorem 1.25. Solution to the Eq. (1.195b) for given input u i ∈ U , i ∈ Z+ has the form μ−1  N k B2d u i+k , (1.198) x¯2,i = − k=0

and the consistent initial conditions of the subsystem (1.195b) are given by the formula μ−1  x¯20 = − N k B2d u k . (1.199) k=0

Proof. From (1.195b) and (1.198) we have N x¯2,i+1 = −

μ−1 

N k+1 B2d u i+k+1 = −

k=0

μ−1 

N k B2d u i+k + B2d u i

k=0

(1.200)

= x¯2,i + B2d u i , since N μ = 0. Substituting i = 0 into (1.198) we obtain (1.199).

 

Assume that the matrix N in (1.195b) has the form (1.62) and ⎡ x¯2,i

⎢ ⎢ =⎢ ⎣

x¯21,i x¯22,i .. .

x¯2n 2 ,i

⎤

⎤ ⎡ B21d ⎥ ⎥ ⎢ . ⎥ ⎥ , B2d = ⎣ .. ⎦ . ⎦ B2n 2 d

(1.201)

From (1.195b), (1.201) and (1.62) we have ⎡

0 ⎢0 ⎢ ⎢ .. ⎢. ⎢ ⎣0 0 and

⎤ ⎤ ⎡ ⎤ 0 ⎡ ⎡ ⎤ x¯21,i+1 x¯21,i ⎥ 0⎥⎢ B21d ⎥ ⎢ x¯22,i ⎥ x ¯ 22,i+1 ⎥ ⎢ ⎥ ⎢ . ⎥ .. ⎥ ⎢ ⎢ . ⎥ = ⎢ . ⎥ + ⎣ .. ⎦ u i .⎥ ⎥ ⎣ .. ⎦ ⎣ .. ⎦ 0 0 ... 1⎦ B2n 2 d x¯2n 2 ,i+1 x¯2n 2 ,i 0 0 ... 0 1 0 .. .

0 1 .. .

... ... .. .

0 = x¯2n 2 ,i + B2n 2 d u i , x¯2n 2 ,i+1 = x¯2n 2 −1,i + B2n 2 −1d u i , .. . x¯22,i+1 = x¯21,i + B21d u i .

(1.202)

(1.203)

50

1 Descriptor Linear Systems

Solving the Eq. (1.203) with respect to components of the vector x¯2,i we obtain x¯2n 2 ,i = −B2n 2 d u i , x¯2n 2 −1,i = −B2n 2 d u i+1 − B2n 2 −1d u i , .. .

(1.204)

x¯21,i = −B2n 2 d u i+n 2 −1 − . . . − B21d u i . Consistent initial conditions of the system (1.195) can be computed from (1.204) for i = 0. Solution to the Eq. (1.195b) for u i ∈ U and consistent initial condition x¯20 ∈ Rn 2 is given by (1.204). Considerations can be easily extended to the case when the matrix N has the form (1.67). Assuming Pd = P and Q d = Q determined by (1.52) and using (1.167c) the equality (1.192) takes the form  P[E z − (Ah + E)]Q =

 0 In 1 z − (A1 h + In 1 ) , 0 N z − (In 2 h + N )

(1.205)

where n 1 and n 2 are defined by (1.192). Premultiplying (1.167a) by the matrix P and introducing the new state vector  x˜i =

x˜1,i x˜2,i



= Q −1 xi , x˜1,i ∈ Rn 1 , x˜2,i ∈ Rn 2 ,

(1.206)

we obtain P E Q Q −1 xi+1 = (P AQh + P E Q)Q −1 xi + P Bhu i ,

(1.207a)

yi = C Q Q −1 xi + Du i

(1.207b)

x˜1,i+1 = (A1 h + In 1 )x˜1,i + B1 hu i ,

(1.208a)

N x˜2,i+1 = (In 2 h + N )x˜2,i + B2 hu i ,

(1.208b)

yi = C1 x˜1,i + C2 x˜2,i + Du i ,

(1.208c)

and using (1.52) we get

where A1 ∈ Rn 1 ×n 1 , B1 ∈ Rn 1 ×m , B2 ∈ Rn 2 ×m , C1 ∈ R p×n 1 , C2 ∈ Rn 2 × p are the matrices of the continuous-time system determined by (1.52) and (1.56). Theorem 1.26. Solution to the Eq. (1.208a) for given initial condition x˜10 ∈ Rn 1 and input u i ∈ U , i ∈ Z+ has the form

1.2 Discrete-Time Systems

51

x˜1,i = (A1 h + In 1 )i x˜10 +

i−1  (A1 h + In 1 )i−k−1 B1 hu k .

(1.209)

k=0

Proof. The proof can be accomplished in a similar way to the proof of Theorem 1.24 [47].   Theorem 1.27. Solution to the Eq. (1.208b) for given input u i ∈ U , i ∈ Z+ has the form μ−1  (1.210a) N¯ k B2 u i+k , x˜2,i = − k=0

where N¯ k =

μ−1    j j=k

" j#

The coefficients

k

k

h − j N j (−1) j−k .

(1.210b)

are given by (1.175c).

Proof. The proof can be accomplished in a similar way to the proof of Theorem 1.25. Without loss of generality we will show that the solution (1.210) satisfies the equality (1.208b) for μ = 3. From (1.208b) and (1.210) we have N x˜2,i+1

⎛ ⎞ 2 2     j ⎝ h − j N j+1 (−1) j−k B2 u i+k+1 ⎠ =− k k=0 j=k =−

2    j

0

j=0

h − j N j+1 (−1) j B2 u i+1 −

2    j j=1

−1

−1

1

h − j N j+1 (−1) j−1 B2 u i+2

= (−N B2 + h N B2 )u i+1 − h N B2 u i+2 , ⎛ ⎞ 2 2     j ⎝ h − j+1 N j (−1) j−k B2 u i+k ⎠ In 2 h x˜2,i = − k k=0 j=k =−

2    j j=0

−

2   j=2

0

2

2

h − j+1 N j (−1) j B2 u i −



2    j j=1

1

(1.211a)

h − j+1 N j (−1) j−1 B2 u i+1

j − j+1 j h N (−1) j−2 B2 u i+2 2

= (−h B2 + N B2 − h −1 N 2 B2 )u i + (−N B2 + 2h −1 N 2 B2 )u i+1 − h −1 N 2 B2 u i+2 , (1.211b)

52

1 Descriptor Linear Systems

N x˜2,i

⎛ ⎞ 2 2     j ⎝ =− h − j N j+1 (−1) j−k B2 u i+k ⎠ k k=0 j=k =−

2    j j=0

0

h − j N j+1 (−1) j B2 u i −

= (−N B2 + h

2    j j=1

−1

N B2 )u i − h 2

−1

1

h − j N j+1 (−1) j−1 B2 u i+1

2

N B2 u i+1 (1.211c)

and (In 2 h + N )x˜2,i + B2 hu i = (−N B2 + h −1 N 2 B2 )u i+1 − h −1 N 2 B2 u i+2 = N x˜2,i+1 . (1.212) In a similar way we can show that the solution (1.210) satisfies the equality (1.208b) for any μ.   Example 1.17. (Continuation of Examples 1.3, 1.15 and 1.16) Consider the descriptor continuous-time linear system (1.55) with (1.69). The discrete-time equivalent of the continuous-time system is described by (1.208), where  A1 h + I2 =

   1−h h h h , B1 h = . 0 1 − 2h 0 2h

(1.213)

Using (1.209) and (1.213) we obtain  (1 − h)i (1 − h)i − (1 − 2h)i x˜10 = 0 (1 − 2h)i  i−1   h(1 − h)i−k−1 3h(1 − h)i−k−1 − 2h(1 − h)i−k−1 uk . + 0 2h(1 − 2h)i−k−1 

x˜1,i

(1.214)

k=0

Taking into account that μ = 2 and using (1.210), (1.69) we get x˜2,i = − N¯ 0 B2 u i − N¯ 1 B2 u i+1 = (−B2 + h −1 N B2 )u i − h −1 N B2 u i+1    1 2 1 − h1 0 h ui + u . = 0 1 0 0 i+1 From (1.206), (1.214) and (1.215) we have

(1.215)

1.2 Discrete-Time Systems

53

⎡

⎤ (1 − h)i (1 − h)i − (1 − 2h)i 0 0 ⎢ 0 0⎥ 0 (1 − 2h)i ⎥ x˜ x˜i = ⎢ ⎣ 0 0 0 0⎦ 0 0 0 00 ⎡ ⎤ i−k−1 i−k−1 3h(1 − h) − 2h(1 − h)i−k−1 h(1 − h) i−1 ⎢ ⎥ 0 2h(1 − 2h)i−k−1 ⎢ ⎥ uk + ⎣ ⎦ 0 0 k=0 0 0 ⎤ ⎤ ⎡ ⎡ 0 0 00 ⎢0 0 ⎥ ⎢0 0 ⎥ ⎥ ⎥ ⎢ +⎢ ⎣ 2 1 − 1 ⎦ u i + ⎣ 0 1 ⎦ u i+1 . h h 0 1 00

(1.216)

The equation above shows the solutions (1.209) and (1.210) of the state equations of the discrete-time equivalent (1.208) of the system (1.55) with (1.69). Taking into account that x˜i = Q −1 xi we have to premultiply the Eq. (1.216) by Q and substitute x˜0 = Q −1 x0 . Thus we obtain the formula, which is equivalent to (1.181) and (1.191).

1.2.3 Positive Descriptor Discrete-Time Linear Systems In positive systems inputs, outputs and state variables take only nonnegative values [40, 49]. The following definition of the positivity is suitable for all methods of the descriptor systems analysis. Definition 1.11. The descriptor discrete-time linear system (1.167a) is called (interp nally) positive if xi ∈ Rn+ and yi ∈ R+ , i ∈ Z+ for any consistent initial condition n x0 ∈ X 0 ⊂ R+ and all admissible inputs u i ∈ U ⊂ Rm + , i ∈ Z+ . 1.2.3.1

Laurent Series Expansion Method

In this section necessary and sufficient positivity conditions of the descriptor discretetime linear system (1.167) will be established using the Laurent series expansion method. Considerations on the influence of the discretization period h on the positivity of the discrete-time equivalent (1.167) of the continuous-time model (1.1) will be presented. Theorem 1.28. The descriptor discrete-time linear system (1.167a) is (internally) positive if and only if Td,k E ∈ Rn×n + , k = 0, 1, . . . , n×m Td,k B ∈ R+ , k = −n 0 , −n 0 + 1, . . . , 0, 1, . . . C∈

p×n R+ ,

D∈

p×m R+ ,

(1.217)

54

1 Descriptor Linear Systems

where the matrices Td,k ∈ Rn×n are given by (1.170b) (or (1.175)). Proof. By Definition 1.11 x0 ∈ X 0 ⊂ Rn+ and u i ∈ U ⊂ Rm + , i ∈ Z+ . From (1.170) we have xi ∈ Rn+ , i ∈ Z+ for u i = 0, i ∈ Z+ if and only if Td,i E ∈ Rn×n + , i ∈ Z+ and . Similarly, from the Eq. (1.167b) it follows for x0 = 0 if and only if Td,i B ∈ Rn×m + p p×n p×m   that yi ∈ R+ if and only if C ∈ R+ i D ∈ R+ . Lemma 1.11. If the matrices Tk E and Tk B of the continuous-time system (1.1) satisfy the conditions (1.74) for k ≥ 0, then the matrices Td,k E and Td,k B of its discrete-time equivalent (1.167) satisfy the conditions (1.217) for k ≥ 0. Proof. The proof follows immediately from (1.175a). The matrices Td,k are linear   combinations (with nonnegative coefficients) of the matrices Tk for k ≥ 0. From the above considerations it follows that if the positive continuous-time system (1.1) satisfies the conditions (1.74), then analysis of the positivity of its discrete-time equivalent (1.167) should be done only for the matrices Td,−k , k = 1, . . . , n 0 since by (1.175b) they may contain negative elements. Example 1.18. (Continuation of Example 1.4) Consider the descriptor continuoustime linear system (1.1) with (1.76). The discrete-time equivalent of the continuoustime system is described by (1.167), where the matrices E, C, D are determined by (1.76) and ⎡

⎤ ⎡ ⎤ 1 2h 0 0 Ad = Ah + E = ⎣ 2h 1 −h ⎦ , Bd = Bh = ⎣ 2h ⎦ . 0 0 h −h

(1.218)

The matrix pencil (E, Ad ) is regular since det[E z − Ad ] = −hz 2 + 2hz + 4h 3 − h = 0.

(1.219)

The matrices Td,k of the discrete-time system can be computed using (1.173) or (1.175) basing on the matrices Tk of the continuous-time system given by and Tk B ∈ Rn×m for k ≥ 0 and (1.79b). From (1.80) it follows that Tk E ∈ Rn×n + + n×n by Lemma 1.11 the matrices Td,k E ∈ R+ and Td,k B ∈ Rn×m for k ≥ 0. Therefore, + to determine the positivity of the discrete-time system we should calculate the matrix Td,−1 using (1.79b) and (1.175b): ⎡

Td,−1 = h −1 T−1 Hence, we have

⎤ 00 0 = ⎣0 0 0 ⎦. 0 0 −h −1

(1.220)

⎡

⎤ 0 Td,−1 B = ⎣ 0 ⎦ ∈ R3×1 + . h −1

(1.221)

1.2 Discrete-Time Systems

55

Therefore, the discrete-time equivalent (1.167) of the continuous-time system (1.1) with (1.76) and (1.218) is positive for any discretization period h > 0.

1.2.3.2

Drazin Inverse Matrix Method

In this section necessary and sufficient positivity conditions of the descriptor discretetime linear system (1.182) (or equivalently (1.167)) will be established using the Drazin inverse matrix method. This problem has been also investigated in [19, 142, 172]. Considerations on the influence of the discretization period h on the positivity of the discrete-time equivalent (1.182) of the continuous-time model (1.24) will be presented. Theorem 1.29. Let F¯d,k = ( E¯ d A¯ dD )k A¯ dD B¯ d , k = 0, 1, . . . , q − 1.

(1.222)

The descriptor discrete-time linear system (1.182) (or equivalently (1.167)) is positive if and only if there exists a matrix G d ∈ Rn×n [142] such that E¯ dD A¯ d + G d (In − E¯ d E¯ dD ) ∈ Rn×n + , and

(1.223a)

p×n p×m Im E¯ d E¯ dD ⊂ Rn+ , E¯ dD B¯ d ∈ Rn×m + , C ∈ R+ , D ∈ R+

(1.223b)

( E¯ d E¯ dD − In ) F¯d,k ∈ Rn×m + , k = 0, 1, . . . , q − 1.

(1.223c)

Proof. The proof can be accomplished in a similar way to the proof of Theorem 1.10. Using the superposition principle we will consider components of the solution (1.183) independently: n ¯ ¯D 1. If Im E¯ d E¯ dD ⊂ Rn×n + , then E d E d x 0 ⊂ R+ . Therefore, for the matrix G d such that n×n D D we have [ E¯ dD A¯ d + G d (In − E¯ d E¯ dD )]i ∈ E¯ d A¯ d + G d (In − E¯ d E¯ d ) ∈ R+ n×n D R+ . Taking into account that [ E¯ d A¯ d + G d (In − E¯ d E¯ dD )]i E¯ d E¯ dD = ( E¯ dD A¯ d )i E¯ d E¯ dD we obtain [ E¯ dD A¯ d + G d (In − E¯ d E¯ dD )]i E¯ d E¯ dD x0 = ( E¯ dD A¯ d )i E¯ d E¯ dD x0 ∈ Rn+ , i ∈ Z+ .

¯ D ¯ i−k−1 2. Using (1.27a) we can write the second component in the form i−1 k=0 ( E d Ad ) n×n n×m D ¯ D ¯ D D ¯ ¯ ¯ ¯ ¯ ¯ E d Bd u k . If E d Ad + G d (In − E d E d ) ∈ R+ and E d Bd ∈ R+ , then for u i ∈

i−1 D D ¯ ¯ ¯ i−k−1 B¯ d u k ∈ Rn+ , i ∈ Z+ . U ⊂ Rm + , i ∈ Z+ we have k=0 E d ( E d Ad ) n×m D 3. If ( E¯ d E¯ d − In ) F¯d,k ∈ R+ , k = 0, 1, . . . , q − 1, where the matrices F¯d,k are given by (1.222) and u i ∈ U ⊂ Rm + , i ∈ Z+ , then the last component of the solution (1.183) is also nonnegative.

¯D ¯ ¯ ¯D Therefore, we have xi ∈ Rn+ , i ∈ Z+ if Im E¯ d E¯ dD ⊂ Rn×n + , E d Ad + G d (In − E d E d ) n×n n×m n×m D ¯ D ¯ ¯ ¯ ¯ and ( E d E d − In ) Fd,k ∈ R+ , k = 0, 1, . . . , q − 1. ∈ R+ , E d Bd ∈ R+

56

1 Descriptor Linear Systems p

p×n

Substituting (1.183) into (1.167b) it is easy to see that yi ∈ R+ , i ∈ Z+ if C ∈ R+ p×m and D ∈ R+ .   Remark 1.8. Note that the descriptor discrete-time linear system (1.182) can be positive even though the matrix E¯ dD A¯ d contains negative entries. In the special case for G d = 0 from (1.223a) we obtain the positivity condition E¯ dD A¯ d ∈ Rn×n + . Now let us consider the continuous-time system (1.24). It is assumed that the system is positive, i.e. it satisfies the conditions (1.88). Using (1.186) we can write ¯ E¯ D = E¯ E¯ D , E¯ d E¯ dD = h −1 Eh

(1.224a)

¯ D = h E¯ D , which follows from (1.27d). Moreover, we have since E¯ dD = (h −1 E) ¯ = h E¯ D A¯ + E¯ E¯ D E¯ dD A¯ d = h E¯ D ( A¯ + h −1 E)

(1.224b)

¯ E¯ dD B¯ d = h E¯ D B.

(1.224c)

and

Using (1.224) we can easily obtain the formulae linking together the positivity conditions of the continuous-time system (1.88) and its discrete-time equivalent (1.223). Analysis of the conditions (1.88c) and (1.223c) is much more difficult. Using (1.186) and (1.222) we can write ( E¯ d E¯ dD − In )Fd,k = ( E¯ d E¯ dD − In )( E¯ d A¯ dD )k A¯ dD B¯ d ¯ + E) ¯ D B¯ ¯ Ah ¯ + E) ¯ D ]k h( Ah = ( E¯ E¯ D − In )[h −1 Eh(

(1.225)

¯ Ah ¯ + E) ¯ D ]k ( Ah ¯ + E) ¯ D B, ¯ = h( E¯ E¯ D − In )[ E( ¯ + E)] ¯ D = h( Ah ¯ + E) ¯ D. since [h −1 ( Ah Theorem 1.30. If the descriptor continuous-time linear system (1.24) (or equivalently (1.1)) is positive, i.e. it satisfies the conditions (1.88), then its discrete-time counterpart (1.182) (or equivalently (1.167)) is also positive if and only if 0 < h ≤ min

1≤i≤n

−e¯iid for a¯ iid < 0 a¯ iid

(1.226a)

or h > 0 for a¯ iid > 0,

(1.226b)

¯ where e¯idj i a¯ idj , i, j = 1, . . . , n are the elements of the matrices E¯ E¯ D and E¯ D A, respectively and ¯ Ah ¯ + E) ¯ D ]k ( Ah ¯ + E) ¯ D B¯ ∈ Rn×m ( E¯ E¯ D − In )[ E( + , k = 0, 1, . . . , q − 1 (1.227) for h satisfying (1.226).

1.2 Discrete-Time Systems

57

¯D ¯ Proof. If the continuous-time system (1.24) is positive, then Im E¯ E¯ D ⊂ Rn×n + ,E A+ n×m D D G(In − E¯ E¯ ) ∈ Mn and E¯ B¯ ∈ R+ . Therefore, from (1.224a) and (1.224c) n×m ¯D ¯ for any h > 0. In the case of the matrix we have Im E¯ d E¯ dD ⊂ Rn×n + and E d Bd ∈ R+ D D (1.224b) we have E¯ d A¯ d + G d (In − E¯ d E¯ d ) ∈ Rn×n + for G d = G if and only if h a¯ iid + e¯iid ≥ 0, i = 1, . . . , n.

(1.228)

Solving (1.228) with respect to h we obtain (1.226). If there exists such a value of h in the range (1.226), for which the condition (1.227) is met, then the discrete-time equivalent (1.182) of the positive continuous-time system (1.24) is also positive.   Example 1.19. (Continuation of Example 1.5) Consider the descriptor continuoustime linear system (1.24) with (1.89). According to (1.91) and (1.93) the system is positive. Its discrete-time equivalent is described by (1.182). Using (1.186) we have ⎡

Im E¯ d E¯ dD = Im E¯ E¯ D

⎤ ⎡ ⎤ 100 v1 3 ⎣ v = Im ⎣ 0 1 0 ⎦ ⊂ R3×3 for v = 2 ⎦ ⊂ R+ , + 000 0 ⎡

E¯ dD A¯ d = h E¯ D A¯ + E¯ E¯ D

⎤ 1 2h 0 = ⎣ 2h 1 0 ⎦ ∈ R3×3 + , 0 0 0

(1.229a)

⎡ ⎤ 0 E¯ dD B¯ d = h E¯ D B¯ = ⎣ h ⎦ ∈ R3×1 + 0

(1.229b)

(1.229c)

for any h > 0. By (1.186) the indices of the matrices E¯ d and E¯ are equivalent and so q = 1. According to (1.227) we shall check if all entries of the matrix ( E¯ E¯ D − ¯ + E) ¯ D B¯ are nonnegative. It is easy to show that I3 )( Ah ⎤ 0 ¯ + E) ¯ D B¯ = ⎣ 0 ⎦ ∈ R3×1 ( E¯ E¯ D − I3 )( Ah + . h −1 ⎡

(1.230)

Therefore, the discrete-time equivalent (1.229) of the continuous-time system (1.89) is positive for any discretization period h > 0.

1.2.3.3

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient positivity conditions of the descriptor discretetime linear system (1.195) (or equivalently (1.167)) will be established using the Weierstrass-Kronecker decomposition method with the assumption that the matrix Q d determined by (1.192) is a monomial matrix [59, 66]. This problem has been also

58

1 Descriptor Linear Systems

investigated in [19]. Considerations on the influence of the discretization period h on the positivity of the discrete-time equivalent (1.208) of the continuous-time model (1.55) will be presented. Let the decomposition (1.192) of the positive descriptor discrete-time linear system (1.167) be possible for a monomial matrix Q d ∈ Rn×n + . In this case x i = Q d x¯i ∈ Rn+ if and only if x¯i ∈ Rn+ . It is also well-known that premultiplying the Eq. (1.167a) by a nonsingular matrix P does not change the solution xi . Theorem 1.31. [19, 59, 66] Let the decomposition (1.192) of the system (1.167) be possible for a monomial matrix Q d ∈ Rn×n + . The descriptor discrete-time linear system (1.195) (or equivalently (1.167)) is positive if and only if A1d ∈ Rn+1 ×n 1 , B1d ∈ Rn+1 ×m , C ∈ R+ , D ∈ R+ p×n

−N B2d ∈ k

Rn+2 ×m ,

p×m

k = 0, 1, . . . , μ − 1.

,

(1.231)

n n Proof. For a monomial matrix Q d ∈ Rn×n + we have x i ∈ R+ , i ∈ Z+ if x¯i ∈ R+ , i ∈ n 1 ×n 1 n 1 ×m n1 and B1d ∈ R+ , then from (1.197) we obtain x¯1,i ∈ R+ , i ∈ Z+ . If A1d ∈ R+ k Z+ since by Definition 1.11 x¯10 = Q −1 x10 ∈ Rn+1 and u i ∈ Rm + , i ∈ Z+ . If −N B2d ∈ n 2 ×m n2 R+ , k = 0, 1, . . . , μ − 1, then from (1.198) we obtain x¯2,i ∈ R+ since u i ∈ Rm +, p×n p×m p i ∈ Z+ . From (1.194b) for C ∈ R+ and D ∈ R+ we have yi ∈ R+ , i ∈ Z+ since   xi ∈ Rn+ and u i ∈ Rm + , i ∈ Z+ .

Now let us consider the continuous-time system (1.1), for which the decomposition (1.52) is possible with a monomial matrix Q ∈ Rn×n + . The system (1.55) is positive if it satisfies the conditions (1.94). The discrete-time equivalent of the system is described by (1.208). In this case xi = Q x˜i ∈ Rn+ if and only if x˜i ∈ Rn+ . Also, premultiplying the Eq. (1.167a) by a nonsingular matrix P does not change the solution xi . Theorem 1.32. If the descriptor continuous-time linear system (1.55) (or equivalently (1.1)) is positive, i.e. it satisfies the conditions (1.94), then its discrete-time counterpart (1.208) (or equivalently (1.167)) is also positive if and only if 0 < h ≤ min

1≤i≤n

−1 for aii1 < 0 aii1

(1.232a)

or h > 0 for aii1 > 0,

(1.232b)

where ai1j , i, j = 1, . . . , n are the entries of the matrix A1 and − N¯ k B2 ∈ Rn×m + , k = 0, 1, . . . , μ − 1

(1.233)

for h satisfying (1.232). Proof. If the continuous-time system (1.55) is positive, then A1 ∈ Mn and B1 ∈ Rn+1 ×m . From (1.205) we have B1 h ∈ Rn×m for any h > 0 and A1 h + In 1 ∈ Rn+1 ×n 1 +

1.2 Discrete-Time Systems

59

if and only if 1 + haii1 ≥ 0, i = 1, . . . , n.

(1.234)

Solving (1.234) with respect to h we obtain (1.232). If there exists such a value of h in the range (1.232), for which the condition (1.233) is met, then the discrete-time equivalent (1.208) of the positive continuous-time system (1.55) is also positive.   Example 1.20. (Continuation of Example 1.6) Consider the positive descriptor continuous-time linear system (1.55) with (1.97). The discrete-time equivalent of the system is described by (1.208), where 

1 2h A1 h + I2 = 2h 1

 ∈

R2×2 + ,

  0 B1 h = ∈ R2×1 + h

(1.235)

for any h > 0. From (1.210b) and (1.233) we have − N¯ 0 B2 = −h 0 N 0 B2 = −B2 = 1 ∈ R1×1 + .

(1.236)

Therefore, the discrete-time equivalent (1.235) of the continuous-time system (1.97) is positive for any discretization period h > 0.

1.2.4 Stability of Descriptor Discrete-Time Linear Systems In the following considerations it is assumed that det[E − Ad ] = 0, i.e. the descriptor discrete-time linear system (1.167) has exactly one equilibrium point [47]. For the assumed approximation of the derivative (1.166) this condition takes the form det Ah = 0. If the continuous-time system (1.1) has one equilibrium point, i.e. det A = 0, then its discrete-time equivalent (1.167) also has one equilibrium point since det Ah = h n det A = 0 for h > 0.

1.2.4.1

Laurent Series Expansion Method

In this section necessary and sufficient stability conditions of the descriptor discretetime linear system (1.167) will be established using the Laurent series expansion method. Considerations on the influence of the discretization period h on the stability of the discrete-time equivalent (1.167) of the continuous-time model (1.1) will be presented. Definition 1.12. The descriptor discrete-time linear system (1.167) is called asymptotically stable if (1.237) lim xi = 0 i→∞

for all consistent initial conditions x0 ∈ X 0 and u i = 0, i ∈ Z+ .

60

1 Descriptor Linear Systems

Definition 1.13. The characteristic equation of the matrix pair (E, Ad ) has the form p(E,Ad ) (z) = det[E z − Ad ] = ad,r z r + ad,r −1 z r −1 + . . . + ad,1 z + ad,0 = 0, (1.238) where r = rankTd,i E and p(E,Ad ) (z) is the characteristic polynomial of the matrix pair (E, Ad ). Theorem 1.33. [47] The matrix pair (E, Ad ) (or equivalently the system (1.167)) is asymptotically stable if and only if its eigenvalues z k , k = 1, . . . , r (roots of the characteristic equation) satisfy the condition |z k | < 1, k = 1, . . . , r.

(1.239)

Note that for Ad = Ah + E we have det[E z − Ad ] = h n det[E(z − 1)h −1 − A]. Comparing this polynomial with (1.100) we obtain s = (z − 1)h −1

(1.240)

z = sh + 1.

(1.241)

and therefore

From the above considerations we have the following theorem. Theorem 1.34. [82] If the descriptor continuous-time linear system (1.1) is asymptotically stable, then its discrete-time equivalent (1.167) is also asymptotically stable if the discretization period h satisfies the condition 0 < h < min

2αk , + βk2

1≤k≤r α 2 k

(1.242)

where sk = −αk + jβk , k = 1, 2, . . . , r are the eigenvalues of the matrix pair (E, A). Proof. The eigenvalues of the matrix pairs (E, A) and (E, Ad ) are related by (1.241). The discrete-time equivalent (1.167) of the continuous-time system (1.1) is asymptotically stable if and only if the condition (1.239) is satisfied, i.e. |z k | = |sk h + 1| = |1 − hαk + j hβk | < 1, k = 1, 2, . . . , r.

(1.243)

From the inequality (1.243) we have (1 − hαk )2 + (hβk )2 < 1. Solving (1.244) with respect to h we obtain (1.242).

(1.244)  

The following theorem can also be used to test the stability of descriptor discrete-time systems.

1.2 Discrete-Time Systems

61

Theorem 1.35. The descriptor discrete-time linear system (1.167) is asymptotically stable if and only if (1.245) lim Td,i E = 0. i→∞

Proof. Substituting u i = 0, i ∈ Z+ into (1.170a) we obtain xi = Td,i E x0 .

(1.246)

lim xi = lim [Td,i E x0 ].

(1.247)

By Definition 1.12 we have i→∞

i→∞

Therefore, the descriptor system (1.167) is asymptotically stable for x0 ∈ X 0 if and only if the condition (1.245) is met.   Example 1.21. (Continuation of Example 1.7) Consider the descriptor continuoustime linear system (1.1) with (1.76). According to (1.106) the eigenvalues of the pair (E, A) are s1 = −2 and s2 = 2 and therefore the system is unstable. The characteristic equation of the pair (E, Ad ) of the discrete-time equivalent (1.167) with (1.218) has the form    z − 1 −2h 0    det[E z − Ad ] =  −2h z − 1 h  = −hz 2 + 2hz + 4h 3 − h = 0, (1.248)  0 0 −h  and by (1.241) its roots are z 1 = 2h + 1, z 2 = −2h + 1,

(1.249)

From the above considerations it follows that the discrete-time equivalent of the system (1.76) is unstable for any h > 0. Example 1.22. (Continuation of Example 1.8) Consider the descriptor continuoustime linear system (1.1) with (1.107). The system is stable since it has one pole s = −1.5. Its discrete-time equivalent (1.167) has one pole z = −1.5h + 1. Using (1.242) we have 4 (1.250) 0 0.

1.2.4.3

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient stability conditions of the descriptor discretetime linear system (1.195) (or equivalently (1.167)) will be established using the Weierstrass-Kronecker decomposition method. Considerations on the influence of the discretization period h on the stability of the discrete-time equivalent (1.208) of the continuous-time model (1.55) will be presented. Note that the state vector of the subsystem (1.195b) x¯2,i = 0 for u i = 0, i ∈ Z+ . Therefore, the stability of the system (1.195) (or equivalently (1.167)) depends entirely on the subsystem (1.195a). Definition 1.15. The descriptor discrete-time linear system (1.195) (or equivalently (1.167)) is called asymptotically stable if lim x¯1,i = 0

i→∞

(1.255)

for all initial conditions x¯10 ∈ Rn 1 and u i = 0, i ∈ Z+ . Definition 1.16. [47] The characteristic equation of the matrix A1d is given by p A1d (z) = det[In 1 z − A1d ] = z n 1 + a˜ d,n 1 −1 z n 1 −1 + . . . + a˜ d,1 z + a˜ d,0 = 0, (1.256) where n 1 = deg{det[E z − Ad ]} and p Ad1 (z) is the characteristic polynomial of the matrix A1d . Lemma 1.14. The characteristic polynomials of the matrix pair (E, Ad ) and of the matrix A1d are related by p(E,Ad ) (z) =

(−1)n 2 p A (z). det(Pd Q d ) 1d

(1.257)

64

1 Descriptor Linear Systems

 

Proof. The proof is similar to the proof of Lemma 1.9.

Remark 1.10. The characteristic equations of the matrix pair (E, Ad ) and of the matrix A1d have the same form. Therefore, the matrix pair (E, Ad ) and the matrix A1d have the same set of eigenvalues and Theorem 1.33 can be used for the roots of the Eq. (1.256). Theorem 1.37. [47] The matrix A1d (or equivalently the system (1.195)) is asymptotically stable if and only if its eigenvalues satisfy the condition (1.239). In a similar way we can analyze the system (1.208). For u i = 0, i ∈ Z+ we have x˜2,i = 0, i ∈ Z+ and the stability of the system (1.208) (or equivalently (1.167)) depends entirely on the vector x˜1,i , i ∈ Z+ . From (1.205) we have 

0 I z − (A1 h + In 1 ) det P[E z − (Ah + E)]Q = det n 1 0 N z − (In 2 h + N )

 (1.258)

and 

   z−1 z−1 n2 n det(P Q)h det E − A = (−1) h det In 1 − A1 , h h n

(1.259)

since det[N z − (In 2 h + N )] = (−h)n 2 = (−1)n 2 h n 2 ,   z−1 − A1 , det[In 1 z − (A1 h + In 1 )] = h n 1 det In 1 h h n 1 h n 2 = h n 1 +n 2 = h n .

(1.260a) (1.260b) (1.260c)

From (1.259) we obtain     (−1)n 2 z−1 z−1 −A = det In 1 − A1 . det E h det(P Q) h

(1.261)

Comparing (1.126) and (1.261) we obtain the formula (1.241). To test the stability of the discrete-time equivalent (1.208) of the stable continuous-time system (1.55) we can use Theorem 1.34 for the eigenvalues of the matrix A1 . Example 1.24. (Continuation of Examples 1.6 and 1.10) Consider the descriptor continuous-time linear system (1.55) with (1.97). According to (1.129) the eigenvalues of the matrix A1 are s1 = −2 i s2 = 2. The matrix A1 h + I2 of the discrete-time equivalent (1.208) has the form (1.235) and its characteristic equation is given by    z − 1 −2h   = z 2 − 2z + 1 − 4h 2 . p(A1 h+I2 ) (z) = det[I2 z − (A1 h + I2 )] =  −2h z − 1  (1.262)

1.2 Discrete-Time Systems

65

From (1.262) we have z 1 = −2h + 1 and z 2 = 2h + 1. Therefore, the discrete-time equivalent of the continuous-time system is unstable for any h > 0.

1.2.5 Superstability of Descriptor Discrete-Time Linear Systems The asymptotic stability of a dynamical system ensures that its free response decreases to zero for i → ∞, however its value may increase significantly in the initial part of the state vector trajectory (it is so-called “peak effect”). In superstable systems the norm of the state vector decreases monotonically to zero for i → ∞, which prevents such undesirable effects [70, 137, 138].

1.2.5.1

Properties of Superstable Discrete-Time Linear Systems

In further considerations we will use the norms (1.130) and (1.131) defined in Sect. 1.1.5.1. Consider the standard discrete-time linear system xi+1 = Ad xi , i ∈ Z+ ,

(1.263)

where xi ∈ Rn is the state vector and Ad = [ad, jk ] ∈ Rn×n . Definition 1.17. [137] The matrix Ad ∈ Rn×n of the discrete-time linear system (1.263) is called superstable if σd (Ad ) = σd = 1 − Ad > 0.

(1.264a)

Ad < 1.

(1.264b)

or equivalently Quantity σd is called the superstability degree of the matrix Ad . If the matrix is superstable, then it is also stable, however the reverse implication does not hold. Theorem 1.38. [137] If the system (1.263) is superstable, then xi ≤ σdi x0 , i ∈ Z+ .

(1.265)

Note that the superstability ensure monotonic decrease of the norm of the state vector, however particular state variables can oscillate. The main difference is that for asymptotic stable systems the Eq. (1.265) is replaced by xi ≤ c(ε)(ρ + ε)i x0 , ε > 0, ρ + ε < 1,

(1.266)

66

1 Descriptor Linear Systems

where the constant c(ε) can take significant values in the initial part of the state vector trajectory. In superstable systems such a situation does not occur [137]. Now let us consider the discrete-time equivalent of the continuous-time system (1.132). Discretizing the Eq. (1.132) and using the approximation (1.166) we obtain (1.263), where (1.267) Ad = Ah + In .

Lemma 1.15. For the matrix norm A defined by (1.131) and the coefficient σ determined by (1.133) the following relationships are true: σh = σ h − 1,

(1.268a)

|σ | ≤ A and |σh | ≤ Ah + In ,

(1.268b)

A

A

A < 2|aii | for aii

< 0, i = 1, . . . , n and σ > 0,

(1.268c)

where σh = σ (Ah + In ) is the quantity (1.133) computed for the matrix (1.267) and A aii is the element on the main diagonal of the matrix A in its i-th row corresponding to its norm. Proof. Using (1.133) for the matrix (1.267) we obtain ⎛

⎞

⎜ −(aii h + 1) − σh = min ⎜ 1≤i≤n ⎝ ⎛

n  j=1 j=i

⎟ |ai j h|⎟ ⎠

⎞

⎜ −aii − = h min ⎜ 1≤i≤n ⎝

n  j=1 j=i

(1.269)

⎟ |ai j |⎟ ⎠ − 1 = σ h − 1.

From (1.133) and (1.131) we have  ⎛ ⎛ ⎞  ⎞         n n    ⎜ ⎜ ⎟  ⎟    ⎜ ⎜ ⎟ ⎟ |σ | =  min ⎝−aii − |ai j |⎠ = − max ⎝aii + |ai j |⎠ 1≤i≤n  1≤i≤n   j=1 j=1     j=i j=i and

⎛ A = max ⎝ 1≤i≤n

n  j=1

⎞

⎛

(1.270)

⎞

n  ⎜ ⎟ |a |ai j |⎠ = max ⎜ | + |ai j |⎟ ii ⎝ ⎠. 1≤i≤n j=1 j=i

(1.271)

1.2 Discrete-Time Systems

67

Basing on (1.270) and (1.271) we have A = −σ for aii ≥ 0, i = 1, . . . , n and A > σ for aii < 0, i = 1, . . . , n. Therefore, |σ | ≤ A . In a similar way it can be shown that |σh | ≤ Ah + In . Using again (1.271) we have ⎛ ⎜ |aii | + A = max ⎜ 1≤i≤n ⎝

⎞ n  j=1 j=i

n  ⎟ A A ⎟ |ai j |⎠ = |aii | + |ai j |,

(1.272)

j=1 j=i

A

where ai j , i, j = 1, . . . , n are components of the norm of the matrix A. If the system (1.132) is superstable, then n  A A |ai j |. (1.273) |aii | > j=1 j=i

Taking into account (1.272) and (1.273) we obtain the inequality (1.268c).

 

Theorem 1.39. If the continuous-time linear system (1.132) is not superstable, then its discrete-time equivalent (1.263) with (1.267) is also not superstable. Proof. If the continuous-time system (1.132) is not superstable, then σ < 0 and |σh | = |σ h − 1| > 1 and therefore from (1.268b) we have Ah + In ≥ |σh | > 1. According to (1.264b) the discrete-time equivalent (1.263) of the system (1.132) is also not superstable.   Theorem 1.40. If the continuous-time linear system (1.132) is superstable, then its discrete-time equivalent (1.263) with (1.267) is also superstable if and only if 0 2, 3) is less than 1 for p ∈ (0, 2). Therefore, the discrete-time equivalent of the superstable continuous-time system (1.132) is also superstable for h satisfying (1.274).   Example 1.25. (Continuation of Example 1.11) Consider the superstable continuous-time linear system (1.132) with (1.137), which norm (1.131) is A = 5. According to (1.274) the discrete-time equivalent (1.263) of the system with  Ad = Ah + I2 =

−3h + 1 2h 0 −2h + 1

 (1.278)

will also be superstable if and only if 0 < h < 0.4.

(1.279)

It is easy to see that for h = 0.4 we have 

 −0.2 0.8 Ad = 0.4 A + I2 = , Ad = 1. 0 0.2

(1.280)

We will consider two cases. First we assume x0 = [ 1 2 ]T and h = 0.3. By (1.279) the system is superstable. From Fig. 1.4 we can see that the norm of the state vector decreases monotonically. In second case we assume x0 = [ 1 2 ]T and h = 0.6. By (1.279) the system is not superstable. From Fig. 1.5 we can see that the norm of the state vector does not decrease monotonically and the state variable x1,i highly oscillate.

1.2 Discrete-Time Systems

69

2

2 x1,i

||xi ||

x2,i

1.5

1.5

1

1

0.5

0.5

0

0

0.5

1

1.5

2

2.5

0

3

0

0.5

1

t [s]

1.5

2

2.5

3

t [s]

Fig. 1.4 State variables (on the left) and norm of the state vector (on the right) of the system (1.263) with (1.278) for x0 = [ 1 2 ]T and h = 0.3 2

2

||xi ||

x1,i

1.5

x2,i

1

1.5

0.5 0

1

-0.5 -1

0.5

-1.5 -2

0

5

10

15

0

0

t [s]

5

10

15

t [s]

Fig. 1.5 State variables (on the left) and norm of the state vector (on the right) of the system (1.263) with (1.278) for x0 = [ 1 2 ]T and h = 0.6

1.2.5.2

Laurent Series Expansion Method

In this section necessary and sufficient superstability conditions of the descriptor discrete-time linear system (1.167) will be established using the Laurent series expansion method. Considerations on the influence of the discretization period h on the superstability of the discrete-time equivalent (1.167) of the continuous-time model (1.1) will be presented. Consider the descriptor discrete-time linear system (1.167). Solution to the Eq. (1.167a) for u i = 0, i ∈ Z+ is given by (1.246). Theorem 1.41. The descriptor discrete-time linear system (1.167) is superstable if and only if Td,i E > Td,i+1 E for i ∈ Z+ . (1.281)

70

1 Descriptor Linear Systems

Proof. From (1.246) we have xi = Td,i E x0 ≤ Td,i E x0 .

(1.282)

Note that the set of consistent initial conditions is given by x0 ∈ ImTd,0 E. If x0 ∈ ImTd,0 E, then x0 = Td,0 E x0 . If the condition (1.281) is satisfied, then max Td,i E = Td,0 E x0 = x0

(1.283)

xi > xi+1 for i ∈ Z+ .

(1.284)

i

and

Therefore, the norm of the state vector decreases monotonically to zero.

 

If the descriptor continuous-time linear system (1.1) is superstable, then testing the superstability of its discrete-time equivalent (1.167) should be done for the matrices Td,i , i ∈ Z+ determined by (1.175a). Using (1.281) and (1.175a) we obtain the set of inequalities T0 E > T0 E + hT1 E , T0 E + hT1 E > T0 E + 2hT1 E + h 2 T2 E , .. . ! ! ! ! ! ! ! !    i+1  ! ! ! ! i i i + 1 j j ! ! ! h Tj E! > ! h Tj E! !, ! j j ! ! j=0 ! ! j=0

(1.285)

where Tk are the matrices of the continuous-time system. The discrete-time equivalent (1.167) of the continuous-time system (1.1) is superstable for such a value of discretization period h, for which the inequalities (1.285) are satisfied. Example 1.26. (Continuation of Example 1.12) Consider the superstable descriptor continuous-time linear system (1.143). The matrices Tk E are given by (1.147). Analyzing the form of these matrices and using (1.285) we can check that for h = 0.5 we have ! ! ! ! ! ! i+1  ! ! i    i + 1 ! ! ! ! i j j ! ! ! ! = (1.286) h h T E T E j ! j ! , i = 0, 2, 4, . . . ! ! j ! ! j=0 ! ! j=0 j and for 0 < h < 0.5

(1.287)

1.2 Discrete-Time Systems

71

we obtain ! ! ! ! ! ! i+1  ! ! i    i + 1 ! ! ! ! i j j ! ! ! h Tj E! > ! h Tj E! ! , i ∈ Z+ . ! j ! ! j=0 ! ! j=0 j

(1.288)

Therefore, the discrete-time equivalent of the superstable continuous-time linear system (1.143) is also superstable for h satisfying (1.287). Taking into account (1.171) and the fact that Td,0 = T0 consistent initial conditions of the system can be computed from (1.150). In Figs. 1.6 and 1.7 we present time plots of the state variables and the norm of the state vector for x0 = [ 2 3 0 ]T and two values of the discretization period: h = 0.3 satisfying (1.287) and h = 0.6 not satisfying (1.287). In the first case the norm of 3

3 x1,i

2.5

x3,i

2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

||xi ||

2.5

x2,i

4

0

0

1

2

t [s]

3

4

t [s]

Fig. 1.6 State variables (on the left) and norm of the state vector (on the right) of the discrete-time equivalent (1.167a) of the continuous-time system (1.1a) with (1.143) for x0 = [ 2 3 0 ]T and h = 0.3 3

3 x1,i

||xi ||

2.5

x2,i

2

x3,i

2

1

1.5 0

1

-1 -2

0.5 0

2

4

6

t [s]

8

10

12

0

0

2

4

6

8

10

12

t [s]

Fig. 1.7 State variables (on the left) and norm of the state vector (on the right) of the discrete-time equivalent (1.167a) of the continuous-time system (1.1a) with (1.143) for x0 = [ 2 3 0 ]T and h = 0.6

72

1 Descriptor Linear Systems

the state vector decreases monotonically to zero and the discrete-time equivalent preserves superstability (Fig. 1.6), while in the second case too high value of the discretization period causes the system to lose this property (Fig. 1.7).

1.2.5.3

Drazin Inverse Matrix Method

In this section necessary and sufficient superstability conditions of the descriptor discrete-time linear system (1.182) (or equivalently (1.167)) will be established using the Drazin inverse matrix method. Considerations on the influence of the discretization period h on the superstability of the discrete-time equivalent (1.182) of the continuous-time model (1.24) will be presented. Consider the descriptor discrete-time linear system (1.182). Solution to the Eq. (1.182a) for u i = 0, i ∈ Z+ has the form xi = ( E¯ dD A¯ d )i E¯ dD E¯ d x0 .

(1.289)

The Eq. (1.289) can also be written in the form xi = [ E¯ dD A¯ d + G d (In − E¯ d E¯ dD )]i E¯ dD E¯ d x0 ,

(1.290)

where G d ∈ Rn×n is an arbitrary matrix. Taking into account (1.183) and (1.184) for u i = 0, i ∈ Z+ it is easy to see that if x0 ∈ Im E¯ d E¯ dD , then x0 = E¯ d E¯ dD x0 . Thus, from (1.290) we have xi ≤ [ E¯ dD A¯ d + G d (In − E¯ d E¯ dD )]i x0 ≤ E¯ dD A¯ d + G d (In − E¯ d E¯ dD ) i x0 . (1.291) If the matrix E¯ dD A¯ d + G d (In − E¯ d E¯ dD ) is superstable (i.e. it satisfies the condition (1.264)), then from the inequality (1.291) we obtain xi ≤ E¯ dD A¯ d + G d (In − E¯ d E¯ dD ) i x0 ≤ σdi x0 , i ∈ Z+

(1.292)

and the norm of the state vector decreases monotonically to zero. Therefore, the following theorem has been proved. Theorem 1.42. The descriptor discrete-time linear system (1.182) (or equivalently (1.167)) is superstable if and only if there exists a matrix G d ∈ Rn×n such that the matrix E¯ dD A¯ d + G d (In − E¯ d E¯ dD ) satisfies the condition (1.264). The following theorem can also be used to test the superstability of descriptor discrete-time linear systems. Theorem 1.43. The descriptor discrete-time linear system (1.182) (or equivalently (1.167)) is superstable if and only if E¯ dD A¯ d < E¯ d E¯ dD .

(1.293)

1.2 Discrete-Time Systems

73

Proof. Using (1.27d) and (1.293) we can write ! !  ! !    ! ! ! ! ˆ !T E d 0 T −1 ! < !T In 1 0 T −1 ! , ! ! ! ! 0 0 0 0

(1.294)

where Eˆ d = Jd−1 A1d . Given that both matrices in the inequality (1.294) are transformed by the same similarity transformation, this condition does not depend on the matrix T . Consider T = In . Therefore, from (1.294) we have Eˆ d < In 1 = 1, which is similar to (1.264b). In the general case for any matrix T we obtain the formula (1.293).   Now let us consider the descriptor continuous-time linear system (1.24) and assume that it is superstable, i.e. there exists a matrix G ∈ Rn×n such that the matrix E¯ D A¯ + G(In − E¯ E¯ D ) satisfies the condition (1.133). Using (1.224) and (1.293) we obtain h E¯ D A¯ + E¯ E¯ D < E¯ E¯ D .

(1.295)

Theorem 1.44. If the descriptor continuous-time linear system (1.24) (or equivalently (1.1)) is superstable, i.e. there exists a matrix G ∈ Rn×n such that the matrix E¯ D A¯ + G(In − E¯ E¯ D ) satisfies the condition (1.133), then its discrete-time counterpart (1.182) (or equivalently (1.167)) is also superstable if and only if 0 2, 3) is less than 1 for p ∈ (0, 2), which follows from the proof of Theorem 1.40. Hence, the expression (1.298): 1) is equal to E¯ E¯ D for p = 2, 2) is greater than E¯ E¯ D for p > 2, 3) is less than E¯ E¯ D for p ∈ (0, 2). Therefore, the discrete-time equivalent of the superstable descriptor continuous-time linear system (1.24) is also superstable for h satisfying (1.296).   Example 1.27. (Continuation of Example 1.13) Consider the superstable descriptor continuous-time linear system (1.24) with the matrices E¯ and A¯ given by (1.155). We will search for such a range of the discretization period h, for which the discretetime equivalent (1.182) of the system is also superstable. Taking into account that ¯ = 4 and using (1.296) we obtain the range of the discretization E¯ E¯ D = 1, E¯ D A period 0 < h < 0.5. (1.301) Note that the conditions (1.287) and (1.301) are the same. It is easy to check that for h = 0.5 we have h E¯ D A¯ + E¯ E¯ D = E¯ E¯ D = 1, while for h satisfying (1.301) we obtain h E¯ D A¯ + E¯ E¯ D < E¯ E¯ D = 1. Therefore, the discrete-time equivalent (1.182) of the superstable continuous-time system (1.24) with (1.155) is also superstable if the condition (1.301) is satisfied. Consistent initial conditions of the system are determined by (1.150). The time plots of the state variables and the norm of the state vector for x0 = [ 2 3 0 ]T and two values of the discretization period: h = 0.3 satisfying (1.301) and h = 0.6 not satisfying (1.301) are presented in Figs. 1.6 and 1.7.

1.2.5.4

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient superstability conditions of the descriptor discrete-time linear system (1.195) (or equivalently (1.167)) will be established using the Weierstrass-Kronecker decomposition method. Considerations on the influence of the discretization period h on the superstability of the discrete-time equivalent (1.208) of the continuous-time model (1.55) will be presented.

1.2 Discrete-Time Systems

75

Note that every superstable system is also asymptotically stable. State vector of the subsystem (1.195b) x¯2,i = 0 for u i = 0, i ∈ Z+ . Using (1.193) and (1.197) we obtain     i x¯ A1d x¯10 xi = Q d x¯i = Q d 1,i = Q d x¯2,i 0  i   A1d 0 x¯10 = Qd (1.302) 0 0 Mdi  i  A1d 0 i Q −1 = Qd d x 0 = A Md x 0 , 0 Mdi where the matrix Md ∈ Rn 2 ×n 2 is arbitrary and  A Md = Q d

A1d 0 0 Md



Q −1 d .

(1.303)

From (1.302) we have xi = AiMd x0 ≤ AiMd x0 ≤ A Md i x0

(1.304)

If the matrix A Md is superstable (i.e. it satisfies the condition (1.264)), then from inequality (1.304) we obtain xi ≤ A Md i x0 ≤ σdi x0 , i ∈ Z+ ,

(1.305)

and the norm of the state vector decreases monotonically to zero. Therefore, the following theorem has been proved. Theorem 1.45. The descriptor discrete-time linear system (1.195) (or equivalently (1.167)) is superstable if and only if there exists a matrix Md ∈ Rn 2 ×n 2 such that the matrix A Md determined by (1.303) satisfies the condition (1.264). The following theorem can also be used to test the superstability of descriptor discrete-time linear systems. Theorem 1.46. Let  A Md,0 = Q d

   A1d 0 In 1 0 −1 Qd , Q I = Q Q −1 . 0 0 0 0

(1.306)

The descriptor discrete-time linear system (1.195) (or equivalently (1.167)) is superstable if and only if (1.307) A Md,0 < Q I . Proof. The proof is similar to the proof of Theorem 1.43.

 

Now let us consider the descriptor continuous-time linear system (1.55) and assume that it is superstable, i.e. there exists a matrix M ∈ Rn 2 ×n 2 such that the matrix A M

76

1 Descriptor Linear Systems

determined by (1.160) satisfies the condition (1.133). Using (1.206) and (1.209) we obtain     (A1 h + In 1 )i x˜10 x˜ xi = Q x˜i = Q 1,i = Q x˜2,i 0    x˜10 (A1 h + In 1 )i 0 (1.308) =Q 0 0 Mi   (A1 h + In 1 )i 0 Q −1 x0 = A¯ iM x0 , =Q 0 Mi where the matrix M ∈ Rn 2 ×n 2 is arbitrary and A¯ M = Q



A1 h + In 1 0 0 M



Q −1 .

(1.309)

Let A M,0 and A¯ M,0 be the matrices defined by (1.160) and (1.309) for M = 0, respectively. Thus, we have (1.310) A¯ M,0 = h A M,0 + Q I , where the matrix Q I is determined by (1.306). Using (1.307) and (1.310) we obtain h A M,0 + Q I < Q I .

(1.311)

Theorem 1.47. If the descriptor continuous-time linear system (1.55) (or equivalently (1.1)) is superstable, i.e. there exists a matrix M ∈ Rn 2 ×n 2 such that the matrix A M defined by (1.160) satisfies the condition (1.133), then its discrete-time counterpart (1.208) (or equivalently (1.167)) is also superstable if and only if 0 0

(2.1)

0

is called the Euler gamma function. The Euler gamma function can be also defined by (x) = lim

n→∞

n!n x , x ∈ C\{0, −1, −2, . . .}. x(x + 1) . . . (x + n)

(2.2)

We shall show that (x) satisfies the equality (x + 1) = x(x).

(2.3)

Proof. Using (2.1) we obtain ∞ (x + 1) =

x −t

t e dt = 0

∞ −t x e−t 0

∞ +x

t x−1 e−t dt = x(x).

(2.4)

0

  © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Kaczorek and K. Borawski, Descriptor Systems of Integer and Fractional Orders, Studies in Systems, Decision and Control 367, https://doi.org/10.1007/978-3-030-72480-1_2

79

80

2 Fractional Descriptor Linear Systems

Example 2.1. From (2.3) we have for x = 1 : (2) = 1 · (1) = 1 since (1) = x = 2 : (3) = 2 · (2) = 1 · 2 = 2!, x = 3 : (4) = 3 · (3) = 3 · 2 · (2) = 3!.

∞

e−t dt = 1,

0

In general case for x ∈ N we have (n + 1) = n(n) = n(n − 1)(n − 1) = n(n − 1)(n − 2) . . . (1) = n!. The gamma function is also well-defined for x being any real (complex) numbers. For example we have for: x = 1.5 : (2.5) = 1.5 · (1.5) = 1.5 · 0.5(0.5), x = −0.5 : (0.5) = −0.5 · (−0.5) = −0.5 · (−1.5)(−1.5).

2.1.2 Mittag-Leffler Function The Mittag-Leffler function is a generalization of the exponential function esi t and it plays important role in solution of the fractional differential equations. Definition 2.2. A function of the complex variable z defined by [57, 132, 135] E α (z) =

∞  k=0

zk (kα + 1)

(2.5)

is called the one parameter Mittag-Leffler function. Example 2.2. For α = 1 we obtain E 1 (z) =

∞  k=0

∞

 zk zk = = ez , (k + 1) k! k=0

(2.6)

i.e. the classical exponential function. An extension of the one parameter Mittag-Leffler function is the following two parameters function. Definition 2.3. A function of the complex variable z defined by [57, 132, 135] E α,β (z) =

∞  k=0

zk (αk + β)

(2.7)

2.1 Continuous-Time Systems

81

is called the two parameters Mittag-Leffler function. For β = 1 from (2.7) we obtain (2.5).

2.1.3 Caputo Definition of Fractional Derivative-Integral Definition 2.4. The function defined by [57, 132, 135] α 0 Dt

1 f (t) = (N − α)

t 0

f (N ) (τ ) d N f (τ ) (N ) dτ, f (τ ) = (t − τ )α+1−N dτ N

(2.8)

is called the Caputo fractional derivative-integral, where N − 1 ≤ α < N , N ∈ N. Theorem 2.1. The Caputo derivative-integral operator is linear satisfying the relation α α α (2.9) 0 Dt [λ f (t) + μg(t)] = λ 0 Dt f (t) + μ 0 Dt g(t), λ, μ ∈ R. Proof. From (2.8) and (2.9) we have α 0 Dt [λ f (t)

1 + μg(t)] = (N − α) =

λ (N − α)

+

μ (N − α)

=

λ 0 Dtα

t

dN dτ N

0

t 0

t 0

[λ f (τ ) + μg(τ )] dτ (t − τ )α+1−N

f (N ) (τ ) dτ (t − τ )α+1−N

(2.10)

g (N ) (τ ) dτ (t − τ )α+1−N

f (t) + μ 0 Dtα g(t).  

Theorem 2.2. The Laplace transform of the derivative-integral (2.8) for N − 1 < α < N has the form L



α 0 Dt

N   f (t) = s α F(s) − s α−k f (k−1) (0+ ).

(2.11)

k=1

Proof. Using Definitions 2.4, C.2 and Eqs. (C.8), (C.12) for N − 1 < α < N we obtain

82

2 Fractional Descriptor Linear Systems





⎡

1 L 0 Dtα f (t) = L ⎣ (N − α)

t

⎤ (t − τ ) N −α−1 f (N ) (τ )dτ ⎦

0

    1 L t N −α−1 L f (N ) (t) = (N − α)

(2.12) N  (N − α) N 1 N −k (k−1) + s f (0 ) = s F(s) − (N − α) s N −α k=1 = s α F(s) −

N 

s α−k f (k−1) (0+ ).

k=1

 

2.1.4 State Equations of Fractional Descriptor Continuous-Time Linear Systems Consider the continuous-time linear system described by state equations E 0 Dtα x(t) = Ax(t) + Bu(t), 0 < α < 1

(2.13a)

y(t) = C x(t) + Du(t),

(2.13b)

where x(t) ∈ Rn , u(t) ∈ Rm , y(t) ∈ R p are state, input and output vectors, 0 Dtα is the Caputo derivative-integral operator defined by (2.8) and E, A ∈ Rn×n , B ∈ Rn×m , C ∈ R p×n , D ∈ R p×m . We assume that detE = 0 and det[Eλ − A] = 0 for some λ ∈ C.

(2.14)

If the matrix E is nonsingular (detE = 0), then we can transform (2.13a) to the state equation of a fractional standard continuous-time linear system through premultiplication of (2.13a) by E −1 . The system (2.13) with the singular matrix E (detE = 0) will be called a fractional descriptor continuous-time linear system.

2.1.5 Solution to the State Equation of Fractional Descriptor Continuous-Time Linear Systems Let U be the set of admissible input vectors u(t) ∈ U ⊂ Rm and X 0 ⊂ Rn the set of consistent initial conditions x(0) = x0 ∈ X 0 for which the Eq. (2.13a) has the solution x(t) with u(t) ∈ U .

2.1 Continuous-Time Systems

2.1.5.1

83

Laurent Series Expansion Method

In this section the solution to the Eq. (2.13a) will be given using the Laurent series expansion method [64]. Theorem 2.3. Solution to the Eq. (2.13a) for x0 ∈ X 0 and u(t) ∈ U is given by the formula ⎫ ⎧ t ∞ ⎨ ⎬  Tk Et kα Tk (t − τ )(k+1)α−1 x0 + Bu(τ )dτ , (2.15) x(t) = ⎭ ⎩ (kα + 1) [(k + 1)α] k=−n 0

0

where n 0 is called the index of the system and the matrices Tk ∈ Rn×n can be found by the Laurent series expansion of the matrix [Es α − A]−1 . Proof. Using the Laplace transform and taking into account that for 0 < α < 1 L[0 Dtα ] = s α X (s) − s α−1 x0

(2.16)

Es α X (s) − Es α−1 x0 = AX (s) + BU (s).

(2.17)

we obtain for (2.13a)

From (2.17) we have X (s) = [Es α − A]−1 [Es α−1 x0 + BU (s)]. Let α

[Es − A]

−1

=

∞ 

(2.18)

Tk s −(k+1)α .

(2.19)

k=−n 0

The matrices Tk , k = −n 0 , −n 0 + 1, . . . , 0, 1, . . . can be found by the expansion (2.19). Substituting (2.19) into (2.18) we obtain X (s) =

∞ 

Tk s −(k+1)α [Es α−1 x0 + BU (s)]

k=−n 0

=

∞  k=−n 0

Tk Es

−(kα+1)

x0 +

∞ 

(2.20) Tk Bs

−(k+1)α

U (s).

k=−n 0

Applying to (2.20) the inverse Laplace transform and the convolution theorem we obtain (2.15) since L[t α ] = (α+1) .   s α+1 Remark 2.1. If E = In , then n 0 = 0 and Tk = Ak , k = 0, 1, . . .. In this case from (2.15) we have

84

2 Fractional Descriptor Linear Systems ∞ 

⎛

k kα ⎝ A t x0 + x(t) = (kα + 1) k=0

t 0

⎞ Ak (t − τ )(k+1)α−1 Bu(τ )dτ ⎠ [(k + 1)α] (2.21a)

t = 0 (t)x0 +

(t − τ )Bu(τ )dτ, 0

where 0 (t) = E α (At α ) =

∞  k=0

(t) =

Ak t kα , (kα + 1)

∞  Ak t (k+1)α−1 , [(k + 1)α] k=0

(2.21b)

(2.21c)

where E α (At α ) is the one parameter Mittag-Leffler function defined by (2.5). The formula (2.21a) is the solution to the state equation of a fractional standard continuoustime linear system [57]. Example 2.3. Consider the fractional descriptor continuous-time linear system (2.13a) with ⎡ ⎤ ⎡ ⎤ 0.5 −0.375 0 −1 2 0 0.25 0 ⎦ , A = ⎣ 0 −1 0 ⎦ E =⎣ 0 −0.5 −0.125 0 1 0 −2 ⎡ ⎤ (2.22) 1 0 B = ⎣ 0 1 ⎦. −1 −2 The matrix pencil (E, A) of (2.22) is regular since det[Es α − A] = 0.25(s α + 2)(s α + 4) = 0.

(2.23)

The matrix [Es α − A]−1 has the form ⎡

[Es α − A]−1

⎤−1 0.5s α + 1 −0.375s α − 2 0 0 0.25s α + 1 0 ⎦ =⎣ α −0.5s − 1 −0.125s α 2 ⎡ 2 ⎤ 3s α +16 0 s α +2 (s α +2)(s α +4) 4 =⎣ 0 0 ⎦. s α +4 0.5 1 0.5

Using (2.19) for n 0 = −1, the matrix (2.24) can be written in the form

(2.24)

2.1 Continuous-Time Systems

85

[Es α − A]−1 =

∞ 

Tk s −(k+1)α ,

(2.25a)

k=−1

where ⎤ ⎧⎡ 0 0 0 ⎪ ⎪ ⎪⎣ 0 0 0 ⎦ ⎪ for k = −1, ⎪ ⎪ ⎨ 0.5 1 0.5 Tk = ⎡ ⎤ k k k ⎪ 0 ⎪ ⎪ 2(−2) 5(−2) − 2(−4) ⎪ ⎣ 0 ⎪ 0 ⎦ for k = 0, 1, 2, . . . . 4(−4)k ⎪ ⎩ 0.5 1 0.5

(2.25b)

Using (2.22) and (2.25b) we have ⎤ ⎧⎡ 000 ⎪ ⎪ ⎪ ⎣0 0 0⎦ ⎪ for k = −1, ⎪ ⎪ ⎨ 000 Tk E = ⎡  ⎤  ⎪ (−2)k 0.5 (−2)k − (−4)k 0 ⎪ ⎪ ⎪ ⎣ 0 ⎪ 0 ⎦ for k = 0, 1, 2, . . . , (−4)k ⎪ ⎩ 0 0 0 ⎤ ⎧⎡ 00 ⎪ ⎪ ⎪⎣0 0⎦ ⎪ for k = −1, ⎪ ⎪ ⎨ 00 Tk B = ⎡ ⎤ k k k ⎪ ⎪ ⎪ 2(−2) 5(−2) − 2(−4) ⎪ ⎣ 0 ⎦ for k = 0, 1, 2, . . . . ⎪ 4(−4)k ⎪ ⎩ 0 0

(2.26a)

(2.26b)

Using (2.15), (2.22) and (2.26) we obtain ∞ 

⎛

t

(k+1)α−1

⎞

Tk (t − τ ) ⎝ Tk Et x0 + Bu(τ )dτ ⎠ (kα + 1) [(k + 1)α] k=0 0 ⎡ ⎤ E α (−2t α ) 0.5 [E α (−2t α ) − E α (−4t α )] 0 0 ⎦ x0 0 E α (−4t α ) =⎣ 0 0 0 ⎡ ⎤  t 2E α,α [−2(t − τ )α ] 5E α,α [−2(t − τ )α ] − 2E α,α [−4(t − τ )α ] ⎦ 0 4E α,α [−4(t − τ )α ] + ⎣ 0 0 0

x(t) =

kα

× (t − τ )α−1 u(τ )dτ, (2.27) where E α (z) and E α,β (z) are Mittag-Leffler functions defined by (2.5) and (2.7).

86

2 Fractional Descriptor Linear Systems

The equation above shows the solution (2.15) of the state equation of the fractional descriptor linear system (2.13) with (2.22).

2.1.5.2

Drazin Inverse Matrix Method

In this section the solution to the Eq. (2.13a) will be given using the Drazin inverse matrix method [61, 63, 90]. Consider the Eq. (2.13a) and assume that the condition (2.14) is met. There exists a number c ∈ C such that det[Ec − A] = 0. Premultiplying the Eq. (2.13a) by the matrix [Ec − A]−1 we obtain ¯ ¯ + Bu(t), E¯ 0 Dtα x(t) = Ax(t)

(2.28a)

where E¯ = [Ec − A]−1 E, A¯ = [Ec − A]−1 A, B¯ = [Ec − A]−1 B.

(2.28b)

Note that the Eqs. (2.13a) and (2.28a) have the same solution x(t). Basic definitions and properties concerning Drazin inverses are given in Sect. 1.1.2.2. Lemma 1.2 can also be used for the fractional descriptor continuous-time linear system (2.28). Therefore, the Eq. (2.28a) is equivalent to the following equations: α 0 Dt x 1 (t)

¯ 1 (t) + E¯ D Bu(t), ¯ = E¯ D Ax

(2.29a)

¯ n − E¯ E¯ D ) 0 Dtα x2 (t) = x2 (t) + (In − E¯ E¯ D ) A¯ D Bu(t), ¯ A¯ D E(I where

x1 (t) = E¯ E¯ D x(t), x2 (t) = (In − E¯ E¯ D )x(t).

(2.29b)

(2.29c)

Theorem 2.4. [63] Solution to the Eq. (2.28a) for x0 ∈ X 0 and u(t) ∈ U has the form t D D ¯ x(t) = 0 (t) E¯ E¯ x0 + E¯ (t − τ ) Bu(τ )dτ 0



(2.30a)

q−1

+ ( E¯ E¯ D − In )

¯ (kα) (t), ( E¯ A¯ D )k A¯ D Bu

k=0

where ¯ α) = 0 (t) = E α ( E¯ D At

∞  ¯ k t kα ( E¯ D A) k=0

(kα + 1)

,

(2.30b)

2.1 Continuous-Time Systems

(t) =

87 ∞  ¯ k t (k+1)α−1 ( E¯ D A) k=0

[(k + 1)α]

,

(2.30c)

¯ α ) is the one parameter Mittag-Leffler function u (kα) (t) = 0 Dtkα u(t), E α ( E¯ D At ¯ defined by (2.5) and q is the index of E. It is easy to check that the first two components of (2.30a) are the solution to the Eq. (2.29a) and the third component of (2.30a) is the solution to the Eq. (2.29b). From (2.30a) for t = 0 we have x0 = E¯ E¯ D v + ( E¯ E¯ D − In )

q−1 

¯ (kα) (0), ( E¯ A¯ D )k A¯ D Bu

(2.31)

k=0

where v ∈ Rn is an arbitrary vector. In the case when u(t) = 0 from (1.46) we have x0 = E¯ E¯ D v. A homogenous equation E 0 Dtα (t) = Ax(t) has the solution if and only if x0 ∈ Im E¯ E¯ D . ¯ E¯ D A, ¯ E¯ A¯ D , E¯ D B, ¯ A¯ D B, ¯ By Lemma 1.3 and Theorem 1.5 the matrices E¯ D E, ¯ the index q of the matrix E, the solution (2.30a) of the Eq. (2.13a) and the set of consistent initial conditions X 0 do not depend of the choice of c. Example 2.4. (Continuation of Example 2.3) Consider the fractional descriptor continuous-time linear system (2.13a) with (2.22). According to (2.23) the matrix pencil (E, A) is regular. We choose c = 0 and using (2.28b) we obtain ⎡

⎤ 0.5 0.125 0 E¯ = [−A]−1 E = ⎣ 0 0.25 0 ⎦ , 0 0 0 ⎡ ⎤ ⎡ −1 0 0 1 A¯ = [−A]−1 A = ⎣ 0 −1 0 ⎦ , B¯ = [−A]−1 B = ⎣ 0 0 0 −1 0

⎤ 2 1⎦. 0

(2.32)

It is easy to see that rank E¯ = rank E¯ 2 , so q = 1. The Drazin inverse of E¯ can   be  0.5 0.125 0 ¯ ¯ ¯ ¯ computed using (B.1) and (B.2) for E = E 1 , where E 11 = , E 12 = 0 0.25 0 and ⎡ ⎤   −1 −2 2 −1 0 E¯ 11 E¯ 11 E¯ 12 = ⎣0 4 0⎦. (2.33) E¯ D = 0 0 0 0 0

88

2 Fractional Descriptor Linear Systems

¯ Next, we compute the matrices Note that A¯ D = A¯ −1 = A. ⎡

⎤ ⎡ ⎤ 100 −2 1 0 E¯ E¯ D = ⎣ 0 1 0 ⎦ , E¯ D A¯ = ⎣ 0 −4 0 ⎦ , 000 0 0 0 ⎡ ⎤ ⎡ ⎤ 00 0 23 E¯ D B¯ = ⎣ 0 4 ⎦ , ( E¯ E¯ D − I3 ) = ⎣ 0 0 0 ⎦ , 0 0 −1 00 ⎡ ⎤ ⎡ ⎤ −0.5 −0.125 0 −1 −2 A¯ D E¯ = ⎣ 0 −0.25 0 ⎦ , A¯ D B¯ = ⎣ 0 −1 ⎦ 0 0 0 0 0 and 0 (t) =

∞  ¯ k t kα ( E¯ D A) k=0

⎡

(kα + 1)

⎤ E α (−2t α ) 0.5 [E α (−2t α ) − E α (−4t α )] 0 0⎦, 0 E α (−4t α ) =⎣ 0 0 0 0 (t) =

(2.34)

∞  ¯ k t (k+1)α−1 ( E¯ D A)

= t α−1

¯ k t kα ( E¯ D A) [(k + 1)α]

[(k + 1)α]   ⎤ ⎡ E α,α (−2t α ) 0.5 E α,α (−2t α ) − E α,α (−4t α ) 0 0⎦. 0 E α,α (−4t α ) = t α−1 ⎣ 0 0 0 k=0

(2.35a)

(2.35b)

Taking into account that q = 1 and using (2.30a), (2.34) and (2.35) we obtain ¯D

x(t) = 0 (t) E¯ E x0 + ⎡

t

¯ ¯ (t − τ ) E¯ D Bu(τ )dτ + ( E¯ E¯ D − I3 ) A¯ D Bu(t)

0

⎤ E α (−2t α ) 0.5 [E α (−2t α ) − E α (−4t α )] 0 0 ⎦ x0 0 E α (−4t α ) =⎣ 0 0 0 ⎡ ⎤  t 2E α,α [−2(t − τ )α ] 5E α,α [−2(t − τ )α ] − 2E α,α [−4(t − τ )α ] ⎦ 0 4E α,α [−4(t − τ )α ] + ⎣ 0 0 0 × (t − τ )α−1 u(τ )dτ, (2.36) where ( E¯ E¯ D − I3 ) A¯ D B¯ = 0 and E α (z), E α,β (z) are Mittag-Leffler functions defined by (2.5) and (2.7). The equation above shows the solution (2.30a) of the state equation of the fractional descriptor linear system (2.13a) with (2.22). Note that solutions (2.27) and (2.36) are equivalent.

2.1 Continuous-Time Systems

2.1.5.3

89

Weierstrass-Kronecker Decomposition Method

In this section the solution to the Eq. (2.13a) will be given using the WeierstrassKronecker decomposition method [57, 90]. It is well-known [57, 90] that if the condition (2.14) holds, then there exists a pair of nonsingular matrices P, Q ∈ Rn×n such that  0 In 1 s α − A1 , A1 ∈ Rn 1 ×n 1 , N ∈ Rn 2 ×n 2 , (2.37) P[Es − A]Q = 0 N s α − In 2 α



where n 1 = deg{det[Es α − A]}, n 2 = n − n 1 and N is a nilpotent matrix with the nilpotency index μ, i.e. N μ−1 = 0 and N μ = 0. The matrices P and Q can be computed using one of procedures given in [47, 48, 168]. Premultiplying (2.13a) by the matrix P and introducing the new state vector  x(t) ¯ = we obtain

 x¯1 (t) = Q −1 x(t), x¯1 (t) ∈ Rn 1 , x¯2 (t) ∈ Rn 2 , x¯2 (t)

(2.38)

P E Q Q −1 0 Dtα x(t) = P AQ Q −1 x(t) + P Bu(t),

(2.39a)

y(t) = C Q Q −1 x(t) + Du(t)

(2.39b)

and using (2.37) we get α 0 Dt x¯ 1 (t)

= A1 x¯1 (t) + B1 u(t),

(2.40a)

N 0 Dtα x¯2 (t) = x¯2 (t) + B2 u(t),

(2.40b)

y(t) = C1 x¯1 (t) + C2 x¯2 (t) + Du(t),

(2.40c)



where PB =

 B1 , B1 ∈ Rn 1 ×m , B2 ∈ Rn 2 ×m , B2

  C Q = C1 C2 , C1 ∈ R p×n 1 , C2 ∈ R p×n 2 .

(2.41a) (2.41b)

Theorem 2.5. [57] Solution to the Eq. (2.40a) for given initial condition x¯10 ∈ Rn 1 and input u(t) ∈ U has the form t x¯1 (t) = 10 (t)x¯10 +

11 (t − τ )B1 u(τ )dτ, 0

(2.42)

90

2 Fractional Descriptor Linear Systems

where 10 (t) = E α (A1 t α ) =

∞  k=0

11 (t) =

Ak1 t kα , (kα + 1)

∞  Ak1 t (k+1)α−1 , [(k + 1)α] k=0

(2.43)

(2.44)

where E α (A1 t α ) is the one parameter Mittag-Leffler function defined by (2.5). Theorem 2.6. Solution to the Eq. (2.40b) for given input u(t) ∈ U and x¯20 = 0 has the form μ−1  N k B2 u (kα) (t), (2.45) x¯2 (t) = − k=0

where u (kα) (t) = 0 Dtkα u(t) and the consistent initial conditions of the subsystem (2.40b) are given by the formula x¯20 = −

μ−1 

N k B2 u (kα) (0).

(2.46)

k=0

Proof. From (2.40b) and (2.45) we have N 0 Dtα x¯2 (t) = −

μ−1 

N k+1 B2 u (k+1)α (t) = −

k=0

μ−1 

N k B2 u (kα) (t) + B2 u(t)

k=0

(2.47)

= x¯2 (t) + B2 u(t) since N μ = 0. Substituting t = 0 into (2.45) we obtain (2.46).

 

Example 2.5. (Continuation of Examples 2.3 and 2.4) Consider the fractional descriptor continuous-time linear system (2.13a) with (2.22). According to (2.23) the matrix pencil (E, A) is regular. In this case ⎡

⎤ ⎡ ⎤ 200 1 0.75 0 0 ⎦ P = ⎣0 4 0⎦, Q = ⎣0 1 121 0 0 −0.5

(2.48)

2.1 Continuous-Time Systems

91

⎤ 100 = PEQ = ⎣0 1 0⎦, 000 ⎡ ⎤   −2 2.5 0 A1 0 = P AQ = ⎣ 0 −4 0 ⎦ , 0 In 2 0 0 1 ⎡ ⎤   20 B1 = PB = ⎣0 4⎦. B2 00

and



In 1 0 0 N



⎡

(2.49)

Next we compute the transition matrices 10 (t) =

∞  k=0

 =

Ak1 t kα (kα + 1)

 E α (−2t α ) 1.25 [E α (−2t α ) − E α (−4t α )] . 0 E α (−4t α )

∞  Ak1 t (k+1)α−1 Ak1 t kα = t α−1 [(k + 1)α] [(k + 1)α] k=0    α α α α−1 E α,α (−2t ) 1.25 E α,α (−2t ) − E α,α (−4t ) . =t 0 E α,α (−4t α )

(2.50a)

11 (t) =

(2.50b)

Using (2.42), (2.49) and (2.50) we obtain 

 E α (−2t α ) 1.25 [E α (−2t α ) − E α (−4t α )] x¯1 (t) = x¯10 0 E α (−4t α )   t  2E α,α [−2(t − τ )α ] 5 E α,α [−2(t − τ )α ] − E α [−4(t − τ )α ] + 0 4E α,α [−4(t − τ )α ] 0

× (t − τ )α−1 u(τ )dτ. (2.51) Taking into account that μ = 1 and using (2.45), (2.49) we obtain x¯2 (t) = −B2 u(t) = 0.

(2.52)

92

2 Fractional Descriptor Linear Systems

From (2.38), (2.51) and (2.52) we have ⎡

⎤ E α (−2t α ) 1.25 [E α (−2t α ) − E α (−4t α )] 0 0 ⎦ x¯0 0 E α (−4t α ) x(t) ¯ =⎣ 0 0 0  ⎤ ⎡ t α  2E α,α [−2(t − τ ) ] 5 E α,α [−2(t − τ )α ] − E α [−4(t − τ )α ] ⎦ 0 4E α,α [−4(t − τ )α ] + ⎣ 0 0 0 × (t − τ )α−1 u(τ )dτ, (2.53) where E α (z) and E α,β (z) are Mittag-Leffler functions defined by (2.5) and (2.7). Taking into account that x(t) ¯ = Q −1 x(t) we have to premultiply the Eq. (2.53) by −1 Q and substitute x¯0 = Q x0 . Thus we obtain the formula (2.27) (or (2.36)), which means that all considered methods of the fractional descriptor systems analysis give the same result. Remark 2.2. Comparing all methods of the fractional descriptor systems analysis we can notice that the numbers: n 0 in (2.15), q in (2.30) and μ in (2.45) are equivalent.

2.1.6 Positive Fractional Descriptor Continuous-Time Linear Systems In positive systems inputs, outputs and state variables take only nonnegative values [40, 49]. The following definition of the positivity is suitable for all methods of the fractional descriptor systems analysis. Definition 2.5. The fractional descriptor continuous-time linear system (2.13) is p called (internally) positive if x(t) ∈ Rn+ and y(t) ∈ R+ , t ∈ [0, +∞) for any consisn tent initial condition x0 ∈ X 0 ⊂ R+ and all admissible inputs u(t) ∈ U ⊂ Rm +, t ∈ (kα) , k = 1, . . . , l − 1, t ∈ [0, +∞), where u (t) = [0, +∞) such that u (kα) (t) ∈ Rm + kα D u(t) and l = n = q = μ. 0 t 0 2.1.6.1

Laurent Series Expansion Method

In this section sufficient positivity conditions of the fractional descriptor continuoustime linear system (2.13) will be established using the Laurent series expansion method. Theorem 2.7. The fractional descriptor continuous-time linear system (2.13) is (internally) positive if p×n

p×m

n×m Tk E ∈ Rn×n + , Tk B ∈ R+ , C ∈ R+ , D ∈ R+

(2.54)

2.1 Continuous-Time Systems

93

where k = −n 0 , −n 0 + 1, . . . , 0, 1, . . . and the matrices Tk ∈ Rn×n can be found by expansion (2.19) of the matrix [Es α − A]−1 . Proof. The proof can be accomplished in a similar way to the proof of Theorem 1.9.   Example 2.6. Consider the (2.13) with  1 E= 0

fractional descriptor continuous-time linear system      0 2 1 1 , A= , B= , 0 0 −1 0   C = 2 1 , D = 0.

(2.55)

The matrix pencil (E, A) of (2.55) is regular since  α   s − 2 −1    = s α − 2 = 0. det[Es − A] =  0 1  α

(2.56)

The matrix [Es α − A]−1 has the form [Es α − A]−1 =



s α − 2 −1 0 1

−1

 =

1 1 s α −2 s α −2

0

1

 .

(2.57)

Using (2.19) the matrix (1.78) can be written in the form [Es α − A]−1 =

∞ 

Tk s −(k+1)α ,

(2.58a)

k=−1

 ⎧ 00 ⎪ ⎪ for k = −1, ⎨ 01 Tk =  k k  ⎪ ⎪ ⎩ 2 2 for k = 0, 1, 2, . . . . 0 0

where

(2.58b)

Using (2.55) and (2.58b) we have  Tk E =

 k  2k 0 2 for k = 0, 1, 2, . . . . , Tk B = 0 0 0

(2.59)

2×1 1×2 It is easy to see that Tk E ∈ R2×2 + , Tk B ∈ R+ , k = 0, 1, 2 . . . and C ∈ R+ , D ∈ 1×1 R+ . Therefore, by Theorem 2.7 the fractional descriptor continuous-time linear system (2.13) with (2.55) is positive.

94

2 Fractional Descriptor Linear Systems

2.1.6.2

Drazin Inverse Matrix Method

In this section necessary and sufficient positivity conditions of the fractional descriptor continuous-time linear system (2.28) (or equivalently (2.13)) will be established using the Drazin inverse matrix method. Lemma 2.1. [57] Let E¯ D A¯ ∈ Mn and 0 < α < 1. Then 0 (t) =

∞  ¯ k t kα ( E¯ D A) k=0

(t) =

(kα + 1)

∈ Rn×n + , t ∈ [0, +∞).

∞  ¯ k t (k+1)α−1 ( E¯ D A) k=0

[(k + 1)α]

∈ Rn×n + , t ∈ [0, +∞).

(2.60a)

(2.60b)

Lemma 2.2. The matrices E¯ D A¯ and E¯ D A¯ + G(In − E¯ E¯ D ), where G ∈ Rn×n is arbitrary, satisfy the equalities ∞  ¯ k t kα ( E¯ D A) k=0

(kα + 1)

E¯ E¯ D =

∞  ¯ k t (k+1)α−1 ( E¯ D A) k=0

[(k + 1)α]

E¯ D =

∞  [ E¯ D A¯ + G(In − E¯ E¯ D )]k t kα ¯ ¯ D EE , (kα + 1) k=0

(2.61a)

∞  [ E¯ D A¯ + G(In − E¯ E¯ D )]k t (k+1)α−1 ¯ D E . (2.61b) [(k + 1)α] k=0

Proof. From (2.61a) we have ∞  [ E¯ D A¯ + G(In − E¯ E¯ D )]k t kα ¯ ¯ D EE (kα + 1) k=0   [ E¯ D A¯ + G(In − E¯ E¯ D )]2 t 2α [ E¯ D A¯ + G(In − E¯ E¯ D )]t α + + . . . E¯ E¯ D = In + (α + 1) (2α + 1)   ∞ D α D 2 2α  ( E¯ D A) ¯ ¯ t ¯ k t kα ( E¯ A) E¯ At + = In + E¯ E¯ D = E¯ E¯ D , (α + 1) (2α + 1) (kα + 1) k=0 (2.62) since (In − E¯ E¯ D ) E¯ E¯ D = 0. The equality (2.61b) can be proved in a similar way   taking into account that (In − E¯ E¯ D ) E¯ D = 0.

Theorem 2.8. Let ¯ k = 0, 1, . . . , q − 1. F¯k = ( E¯ A¯ D )k A¯ D B,

(2.63)

The fractional descriptor continuous-time linear system (2.28) (or equivalently (2.13)) is positive if and only if there exists a matrix G ∈ Rn×n [79, 141] such that

2.1 Continuous-Time Systems

95

E¯ D A¯ + G(In − E¯ E¯ D ) ∈ Mn and

p×n p×m , Im E¯ E¯ D ⊂ Rn+ , E¯ D B¯ ∈ Rn×m + , C ∈ R+ , D ∈ R+

( E¯ E¯ D − In ) F¯k ∈ Rn×m + , k = 0, 1, . . . , q − 1.

(2.64a)

(2.64b) (2.64c)

Proof. Using the superposition principle we will consider components of the solution (2.30a) independently: n ¯ ¯D such 1. If Im E¯ E¯ D ⊂ Rn×n + , then E E x 0 ⊂ R+ . Therefore, for the matrix G  ∞ D D that E¯ A¯ + G(In − E¯ E¯ ) ∈ Mn according to Lemma 2.1 we have k=0  D ¯ D k kα ¯ ¯ ¯ ∞ [ E A+G(In − E E )] t ¯ E¯ D ∈ Rn×n and using Lemma 2.2 we obtain E + k=0 (kα+1)  D ¯ D k kα D k kα ¯ ¯ ¯ ¯ ¯ ∞ ( E A) t [ E A+G(In − E E )] t D D n ¯ ¯ ¯ ¯ E E x0 = k=0 (kα+1) E E x0 ∈ R+ , t ∈ [0, +∞). (kα+1) t ¯ 2. We can write the second component in the form 0 (t − τ ) E¯ D Bu(τ )dτ . n×m D ¯ D D ¯ m ¯ ¯ ¯ ¯ For E A + G(In − E E ) ∈ Mn , E B ∈ R+ and u(t) ∈ U ⊂ R+ we have t ¯ )dτ ∈ Rn+ . E¯ D 0 (t − τ ) Bu(τ ¯ 3. If ( E¯ E¯ D − In ) F¯k ∈ Rn×m + , k = 0, 1, . . . , q − 1, where the matrices Fk are given , k = 0, 1, . . . , q − 1, t ∈ [0, +∞), then the last by (2.63) and u (kα) (t) ∈ Rm + component of the solution (2.30a) is also nonnegative.

¯D ¯ Therefore, we have x(t) ∈ Rn+ , t ∈ [0, +∞) if Im E¯ E¯ D ⊂ Rn×n + , E A + G(In − n×m n×m D D D E¯ E¯ ) ∈ Mn , E¯ B¯ ∈ R+ and ( E¯ E¯ − In ) F¯k ∈ R+ , k = 0, 1, . . . , q − 1. Subp p×n stituting (2.30a) into (2.13b) it is easy to see that y(t) ∈ R+ , t ∈ [0, +∞) if C ∈ R+ p×m and D ∈ R+ .   Remark 2.3. [79] Note that the fractional descriptor continuous-time linear system (2.28) can be positive even though the matrix E¯ D A¯ is not a Metzler matrix. In the special case for G = 0 from (2.64a) we obtain the positivity condition E¯ D A¯ ∈ Mn . Example 2.7. (Continuation of Example 2.6) Consider the fractional descriptor continuous-time linear system with (2.55). According to (2.56) the matrix pencil (E, A) is regular. We choose c = 0 and using (2.28b) we obtain     −0.5 0 −1 0 −1 −1 ¯ ¯ E = [−A] E = , A = [−A] A = , 0 0 0 −1   −0.5 . B¯ = [−A]−1 B = 0

(2.65)

From (B.1) we have E¯ = E¯ 1 , where E¯ 11 = −0.5, E¯ 12 = 0 and the index of E¯ is q = 1 since rank E¯ = rank E¯ 2 = 2. Using (B.2) we compute

96

2 Fractional Descriptor Linear Systems

E¯ D =



A¯ D = A¯ −1 and

  −2 0 = , 0 0   −1 0 = A¯ = 0 −1

−1 ¯ −2 ¯ E 11 E 12 E¯ 11 0 0



(2.66)

    10 v D 2×2 ¯ ¯ Im E E = Im ⊂ R+ for v = 1 ⊂ R2+ , 0 00     20 1 D ¯ D ¯ ¯ ¯ E A= ∈ M2 , E B = ∈ R2×1 + . 00 0

(2.67a)

(2.67b)

According to Lemma 1.3 the matrices (2.67) and the index of E¯ do not depend of the choice of c. From (2.63) we have   0.5 D 0 ¯D ¯ ¯ ¯ ¯ . F0 = ( E A ) A B = 0

(2.68)

From (2.67a) and (2.68) we have ( E¯ E¯ D − I2 ) F¯0 =

  0 ∈ R2×1 + . 0

(2.69)

Therefore, the fractional descriptor continuous-time linear system with (2.55) is positive.

2.1.6.3

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient positivity conditions of the fractional descriptor continuous-time linear system (2.40) (or equivalently (2.13)) will be established using the Weierstrass-Kronecker decomposition method with the assumption that the matrix Q determined by (2.37) is a monomial matrix [59, 84]. Let the decomposition (2.37) of the positive fractional descriptor continuous-time linear system (2.13) be possible for a monomial matrix Q ∈ Rn×n + . In this case x(t) = ¯ ∈ Rn+ . It is also well-known that premultiplying the Q x(t) ¯ ∈ Rn+ if and only if x(t) equation (2.13a) by a nonsingular matrix P does not change the solution x(t). Theorem 2.9. [19, 59, 84] Let the decomposition (2.37) of the system (2.13) be possible for a monomial matrix Q ∈ Rn×n + . The fractional descriptor continuous-time linear system (2.40) (or equivalently (2.13)) is positive if and only if A1 ∈ Mn 1 , B1 ∈ Rn+1 ×m , C ∈ R+ , D ∈ R+ p×n

− N B2 ∈ k

Rn+2 ×m ,

k = 0, 1, . . . , μ − 1.

p×m

,

(2.70)

2.1 Continuous-Time Systems

97

n Proof. For a monomial matrix Q ∈ Rn×n ¯ ∈ Rn+ . Accord+ we have x(t) ∈ R+ if x(t) n×n ing to Lemma 2.1 for A1 ∈ Mn 1 we have 10 (t) ∈ R+ and 11 (t) ∈ Rn×n + . If A1 ∈ Mn 1 and B1 ∈ Rn+1 ×m , then from (2.42) we obtain x¯1 ∈ Rn+1 since from Definition n 2 ×m k 2.5 x¯10 = Q −1 x10 ∈ Rn+1 and u(t) ∈ Rm , k = 0, 1, . . . , μ − 1, + . If −N B2 ∈ R+ n2 then from (2.45) we obtain x¯2 ∈ R+ since from Definition 2.5 u (kα) (t) ∈ Rm +, p×n p×m p k = 0, 1, . . . , μ − 1. From (2.39b) for C ∈ R+ and D ∈ R+ we have y(t) ∈ R+ since x(t) ∈ Rn+ and u(t) ∈ Rm   +.

Example 2.8. (Continuation of Example 2.6) Consider the fractional descriptor continuous-time linear system with (2.55). According to (2.56) the matrix pencil (E, A) is regular. In this case 

and

P=

   1 1 10 , Q= 0 −1 01





 10 , 00     A1 0 20 = P AQ = , 0 In 2 01     1 B1 = PB = . B2 0 In 1 0 0 N

(2.71)



= PEQ =

(2.72)

Hence, n 1 = 1 and n 2 = 1. The matrix Q ∈ R3×3 + in (2.71) is monomial. From (2.55) and (2.72) we have 1×1 A1 = 2 ∈ M1 , B1 = 1 ∈ R1×1 + , −B2 = 0 ∈ R+ ,   1×1 C = 2 1 ∈ R1×2 + , D = 0 ∈ R+ .

(2.73)

By Theorem 2.9 the fractional descriptor continuous-time linear system with (2.55) is positive.

2.1.7 Stability of Fractional Descriptor Continuous-Time Linear Systems In the following considerations it is assumed that det A = 0, i.e. the fractional descriptor continuous-time linear system (2.13) has exactly one equilibrium point [47].

98

2 Fractional Descriptor Linear Systems

2.1.7.1

Laurent Series Expansion Method

In this section necessary and sufficient stability conditions of the fractional descriptor continuous-time linear system (2.13) will be established using the Laurent series expansion method. Definition 2.6. The fractional descriptor continuous-time linear system (2.13) is called asymptotically stable if (2.74) lim x(t) = 0 t→∞

for all consistent initial conditions x0 ∈ X 0 and u(t) = 0. Let (t) be the matrix given by (t) =

∞  k=−n 0

Tk Et kα . (kα + 1)

(2.75)

Definition 2.7. The characteristic equation of the matrix pair (E, A) has the form p(E,A) (λ) = det[Eλ − A] = ar λr + ar −1 λr −1 + . . . + a1 λ + a0 = 0,

(2.76)

where λ = s α , r = rank(t) and p(E,A) (λ) is the characteristic polynomial of the matrix pair (E, A). Theorem 2.10. [57] The matrix pair (E, A) (or equivalently the system (2.13)) is asymptotically stable if and only if its eigenvalues λk , k = 1, . . . , r (roots of the characteristic equation) satisfy the condition π |argλk | > α , k = 1, . . . , r. 2

(2.77)

Theorem 2.11. The descriptor continuous-time linear system (1.1) is asymptotically stable if and only if lim (t) = 0. (2.78) t→∞

Proof. The proof can be accomplished in a similar way to the proof of Theorem 1.13.   From the above considerations we have the following theorem. Theorem 2.12. The descriptor continuous-time linear system (2.13) is asymptotically stable if and only if one of the following equivalent conditions is satisfied: 1. Roots of the characteristic Eq. (2.76) satisfy the condition (2.77). 2. The matrix (t) defined by (2.75) satisfies the condition (2.78).

2.1 Continuous-Time Systems

99

Example 2.9. Consider the fractional descriptor continuous-time linear system (2.13) with     −0.5 0 4 0 E= , A= . (2.79) 0 0 1 −2 The matrix pencil of (2.79) is regular since   −0.5s α − 4 det[Es α − A] =  −1

 0  = −s α − 8 = 0. 2

(2.80)

The matrix [Es α − A]−1 has the form α

[Es − A]

−1



−0.5s α − 4 0 = −1 2

 − s α2+8 0 . = − s α1+8 0.5 

−1

(2.81)

Using (2.19) for n 0 = 1 the matrix (2.81) can be written in the form ∞ 

[Es α − A]−1 =

Tk s −(k+1)α ,

(2.82a)

k=−1

where

 ⎧ 0 0 ⎪ ⎪ for k = −1, ⎨ 0 0.5 Tk =   k ⎪ ⎪ ⎩ −(−8) k 0 for k = 0, 1, 2, . . . . −2(−8) 0

(2.82b)

Using (1.107) and (1.110b) we have 

0.5(−8)k 0 Tk E = (−8)k 0 and



∞  Tk Et kα (t) = (kα + 1) k=0  0.5E α (−8t α ) = E α (−8t α )

for k = 0, 1, 2, . . .

=

∞ 



k=0



0.5(−8)k t kα (kα+1) (−8)k t kα (kα+1)

0

(2.83)

0

(2.84)

0 , 0

where E α (z) is the one parameter Mittag-Leffler function defined by (2.5). By the second condition of Theorem 2.12 the fractional descriptor system (2.13) with (2.79) is asymptotically stable for any α ∈ (0; 1) since limt→∞ E α (−8t α ) = 0. The characteristic equation of the system (2.79) is given by   −0.5s α − 4 p(E,A) (s α ) = det[Es α − A] =  −1

 0  = −s α − 8 = 0. 2

(2.85)

100

2 Fractional Descriptor Linear Systems

From (2.85) we can see that the pair (E, A) has one eigenvalue s1α = −8 and by Theorem 2.12 the descriptor system (2.13) with (2.79) is asymptotically stable for any α ∈ (0; 1) since |args1α | = π > α π2 . 2.1.7.2

Drazin Inverse Matrix Method

In this section necessary and sufficient stability conditions of the fractional descriptor continuous-time linear system (2.28) (or equivalently (2.13)) will be established using the Drazin inverse matrix method. Definition 2.6 of the asymptotic stability of the fractional descriptor continuoustime linear system is suitable for this method since the solutions of (2.13) and (2.28) are equivalent. ¯ A) ¯ is given by Definition 2.8. The characteristic equation of the matrix pair ( E, ¯ − A] ¯ = a¯ r λr + a¯ r −1 λr −1 + . . . + a¯ 1 λ + a¯ 0 = 0, p( E, ¯ A) ¯ (λ) = det[ Eλ

(2.86)

where λ = s α , r = rank E¯ D A¯ and p( E, ¯ A) ¯ (λ) is the characteristic polynomial of the ¯ A). ¯ matrix pair ( E, Lemma 2.3. [11] The characteristic polynomials of the matrix pairs (E, A) and ¯ A) ¯ are related by ( E, p(E,A) (λ) p( E, . (2.87) ¯ A) ¯ (λ) = det[Ec − A] Proof. The proof is similar to the proof of Lemma 1.7.

 

¯ A) ¯ Remark 2.4. The characteristic equations of the matrix pairs (E, A) and ( E, have the same form. Therefore, both pairs have the same set of eigenvalues and Theorem 2.10 can be used for the roots of the Eq. (2.86). Theorem 2.13. The fractional descriptor continuous-time linear system (2.13) is asymptotically stable if and only if the matrix E¯ D A¯ has r stable eigenvalues (satisfying the condition (2.77)) and n − r zero eigenvalues. Proof. The proof follows immediately from Lemma 1.8.

 

Based on the above considerations we have the following Theorem. Theorem 2.14. The fractional descriptor continuous-time linear system (2.13) is asymptotically stable if and only if one of the following equivalent conditions is satisfied: 1. Roots of the characteristic Eq. (2.86) satisfy the condition (2.77). 2. The matrix E¯ D A¯ has r stable eigenvalues and n − r zero eigenvalues.

2.1 Continuous-Time Systems

101

Example 2.10. (Continuation of Example 2.9) Consider the fractional descriptor continuous-time linear system (2.13) with (2.79). According to (2.80) the matrix pencil (E, A) is regular. We choose c = 0 and using (2.28b) we obtain E¯ = [−A]−1 E =



   0.125 0 −1 0 −1 ¯ , A = [−A] A = . 0.0625 0 0 −1

(2.88)

From (B.1) we have E¯ = E¯ 2 , where E¯ 21 = 0.125, E¯ 22 = 0.0625 and the index of E¯ is q = 1 since rank E¯ = rank E¯ 2 = 1. Using (B.2) we compute E¯ D =



     −1 E¯ 21 −1 0 0 80 D −1 ¯ ¯ ¯ . = , A =A =A= −2 0 −1 40 0 E¯ 22 E¯ 21

(2.89)

From (2.86) and (2.88) we have    0.125s α + 1 0  α   = 0.125s α + 1 = 0. ¯ ¯ p( E, ¯ A) ¯ (s ) = det[ Es − A] =  0.0625s α 1  α

(2.90)

¯ A) ¯ has one eigenvalue s1α = −8, which is equal to the eigenvalue of the The pair ( E, pair (E, A). By Theorem 2.14 the fractional descriptor system (2.13) with (2.79) is asymptotically stable for any α ∈ (0; 1) since |args1α | = π > α π2 . Using (2.89) and (1.117) we obtain   α s + 8 0  D ¯  = s α (s α + 8) = 0.  ¯ p E¯ D A¯ (s ) = det[I3 s − E A] =  4 sα  α

α

(2.91)

It is easy to see that r = rank E¯ D A¯ = 1. According to (2.91) the matrix E¯ D A¯ has ¯ A) ¯ equal to s1α = −8 and n − r = 1 zero eigenvalue r = 1 eigenvalue of the pair ( E, s2α = 0. Therefore, by Theorem 2.14 the considered system is asymptotically stable for any α ∈ (0; 1).

2.1.7.3

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient stability conditions of the fractional descriptor continuous-time linear system (2.40) (or equivalently (2.13)) using the WeierstrassKronecker decomposition method. Note that the state vector of the subsystem (2.40b) x¯2 (t) = 0 for u(t) = 0. Therefore, the stability of the system (2.40) (or equivalently (2.13)) depends entirely on the subsystem (2.40a). Definition 2.9. The fractional descriptor continuous-time linear system (2.40) (or equivalently (2.13)) is called asymptotically stable if

102

2 Fractional Descriptor Linear Systems

lim x¯1 (t) = 0

t→∞

(2.92)

for all initial conditions x¯10 ∈ Rn 1 and u(t) = 0. Definition 2.10. [57] The characteristic equation of the matrix A1 is given by p A1 (λ) = det[In 1 λ − A1 ] = λn 1 + a˜ n 1 −1 λn 1 −1 + . . . + a˜ 1 λ + a˜ 0 = 0,

(2.93)

where λ = s α , n 1 = deg{det[Eλ − A]} and p A1 (λ) is the characteristic polynomial of the matrix A1 . Lemma 2.4. [11] The characteristic polynomials of the matrix pair (E, A) and of the matrix A1 are related by p(E,A) (λ) =

(−1)n 2 p A (λ). det(P Q) 1

Proof. The proof is similar to the proof of Lemma 1.9.

(2.94)  

Remark 2.5. The characteristic equations of the matrix pair (E, A) and of the matrix A1 have the same form. Therefore, the matrix pair (E, A) and the matrix A1 have the same set of eigenvalues and Theorem 2.10 can be used for the roots of the Eq. (2.93). Theorem 2.15. [57] The matrix A1 (or equivalently the system (2.40)) is asymptotically stable if and only if its eigenvalues satisfy the condition (2.77). Example 2.11. (Continuation of Example 2.9) Consider the fractional descriptor continuous-time linear system (2.13) with (2.79). According to (2.80) the matrix pencil (E, A) is regular. In this case 

   −2 0 1 0 P= , Q= 0 1 0.5 −0.5 and

 10 = PEQ = , 00     A1 0 −8 0 = P AQ = . 0 In 2 0 1 

In 1 0 0 N



(2.95)



(2.96)

Hence, n 1 = 1 and n 2 = 1. Using (2.93) and (2.96) we obtain p A1 (s α ) = det[I1 s α − A1 ] = s α + 8 = 0.

(2.97)

Eigenvalue of the matrix A1 is s1α = −8 and is equal to the eigenvalue of the pair (E, A). By Theorem 2.15 the considered fractional descriptor system is asymptotically stable for any α ∈ (0; 1) since |args1α | = π > α π2 .

2.1 Continuous-Time Systems

103

2.1.8 Superstability of Fractional Descriptor Continuous-Time Linear Systems The asymptotic stability of a dynamical system ensures that its free response decreases to zero for t → ∞, however its value may increase significantly in the initial part of the state vector trajectory (it is so-called “peak effect”). In superstable systems the norm of the state vector decreases monotonically to zero for t → ∞, which prevents such undesirable effects [70, 137, 138].

2.1.8.1

Properties of Superstable Fractional Continuous-Time Linear Systems

The concept of superstable continuous-time linear systems is given in Sect. 1.1.5.1. Consider the standard fractional continuous-time linear system α 0 Dt x(t)

= Ax(t), 0 < α < 1,

(2.98)

where x(t) ∈ Rn is the state vector, 0 Dtα is the Caputo derivative-integral operator defined by (2.8) and A ∈ Rn×n . The solution to the Eq. (2.98) can be obtained using (2.21) for u i = 0, i ∈ Z+ , i.e. it has the form xi = 0 (t)x0 = E α (At α )x0 ,

(2.99)

where 0 (t) is given by (2.21b) and E α (At α ) is the one parameter Mittag-Leffler function defined by (2.5). Lemma 2.5. For a superstable matrix A (i.e. satisfying (1.133)) we have

E α (At α ) ≤ E α (−σ t α ),

(2.100)

where σ is the superstability degree of the matrix A. Proof. The proof can be accomplished in a similar way as it has been shown in [137] for the norm of matrix exponential. For a small t α = δt α we have E α (Aδt α ) ≈ In + and

Aδt α (α + 1)

(2.101)

104

2 Fractional Descriptor Linear Systems

⎞ ⎛   n α    ⎜ δt α ⎟ + 1 + aii δt ai j  ⎟

E α (Aδt α ) ≈ max ⎜   ⎝ 1≤i≤n (α + 1) (α + 1) ⎠ ⎡

j=1 j=i

⎞

⎛

⎢ ⎜ 1+⎜ = max ⎢ ⎣ ⎝aii + 1≤i≤n

n  j=1 j=i

 ⎟ ai j ⎟ ⎠

⎤

α δt α ⎥ ⎥ ≤ 1 − σ δt ⎦ (α + 1) (α + 1)

≤ E α (−σ δt α ). Hence, for an arbitrary t α = nδt α we obtain

E α (At α ) = E α (Anδt α ) ≤ E α (Aδt α ) n ≤ [E α (−σ δt α )]n = E α (−σ t α ).

(2.102)

(2.103)  

Theorem 2.16. If the fractional system (2.99) is superstable, then

x(t) ≤ x0 E α (−σ t α ), t ∈ [0, +∞). Proof. The proof follows immediately from Lemma 2.5.

(2.104)  

Therefore, from the above considerations it follows that if the matrix A satisfies the condition (1.133), then the norm of the state vector decreases monotonically to zero for t → ∞. Example 2.12. Consider the fractional continuous-time linear system (2.98) with α = 0.9 and   −2 1 A= . (2.105) 0 −2 It is easy to check that the considered system is asymptotically stable since the eigenvalues of (2.105) are s10.9 = s20.9 = −2 and the condition (2.77) is satisfied. From (1.133) we have σ = 1 > 0 and the system is also superstable. In Fig. 2.1 we present time plots of the state variables and the norm of the state vector for x0 = [ 1 1 ]T , from which we can see that the norm decreases monotonically to zero. Now let us consider the fractional continuous-time linear system (2.98) with α = 0.9 and   −2 4 A= . (2.106) 0 −2 Note that the eigenvalues of (2.105) and (2.106) are the same. In this case the system is not superstable since by the condition (1.133) we have σ = −2 < 0. In Fig. 2.2 we show time plots of the state variables and the norm of the state vector for x0 = [ 1 1 ]T , from which we can see that the norm does not decrease monotonically to zero.

2.1 Continuous-Time Systems

105

1

1 x 1 (t)

||x(t)||

x 2 (t)

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

5

10

15

0

5

t [s]

10

15

t [s]

Fig. 2.1 State variables (on the left) and norm of the state vector (on the right) of the fractional continuous-time linear system (2.98) with (2.105) and α = 0.9 for x0 = [ 1 1 ]T 1.2

1.2 x 1 (t)

||x(t)||

x 2 (t)

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

5

10

15

0

5

10

15

t [s]

t [s]

Fig. 2.2 State variables (on the left) and norm of the state vector (on the right) of the fractional continuous-time linear system (2.98) with (2.106) and α = 0.9 for x0 = [ 1 1 ]T

2.1.8.2

Laurent Series Expansion Method

In this section necessary and sufficient superstability conditions of the descriptor continuous-time linear system (2.13) will be established using the Laurent series expansion method. Consider the fractional descriptor continuous-time linear system (2.13). Solution to the Eq. (2.13a) for u(t) = 0 has the form x(t) = (t)x0 ,

(2.107)

where (t) is defined by (2.75). Theorem 2.17. The fractional descriptor continuous-time linear system (2.13) is superstable if and only if

106

2 Fractional Descriptor Linear Systems

(t1 ) > (t2 ) for any t1 < t2 , t1 , t2 ∈ [0, +∞).

(2.108)

Proof. The proof can be accomplished in a similar way to the proof of Theorem 1.19.   Example 2.13. Consider the fractional descriptor continuous-time linear system (2.13) with α = 0.8 and ⎡

⎤ ⎡ ⎤ 0 −1 −1 010 E = ⎣ −0.3333 −0.1667 −0.1667 ⎦ , A = ⎣ 1 0 0 ⎦ . 0 0.5 0.5 001

(2.109)

The matrix pencil (E, A) of (2.109) is regular since det[Es α − A] = 0.1667s 1.6 + 0.8333s 0.8 + 1 = 0.

(2.110)

The matrix [Es α − A]−1 has the form ⎡

[Es 0.8 − A]−1

⎤−1 0 −s 0.8 − 1 −s 0.8 = ⎣ −0.3333s 0.8 − 1 −0.1667s 0.8 −0.1667s 0.8 ⎦ 0 0.5s 0.8 0.5s 0.8 − 1 ⎤ ⎡ s 0.8 s 0.8 3 1.6 0.8 +6 − s 0.8 +3 s 1.6 +5s 0.8 +6 ⎥ ⎢ s s+5s 0.8 −2 2s 0.8 ⎥. =⎢ 0 0.8 0.8 s +2 ⎦ ⎣ s +2 0.8 0.8 s 2s +2 − s 0.8 0 − 0.8 +2 s +2

(2.111)

Using (2.19) for n 0 = −1, the matrix (2.111) can be written in the form [Es 0.8 − A]−1 =

∞ 

Tk s −0.8(k+1) ,

(2.112a)

k=−1

where ⎤ ⎧⎡ 0 0 0 ⎪ ⎪ ⎪⎣ 1 0 2 ⎦ ⎪ ⎪ ⎪ ⎨ −1 0 −2 Tk = ⎡ ⎤ k k ⎪ −3(−3)k 3(−3)k − 2(−2)k ⎪ ⎪ 3(−3) − 2(−2) ⎪ ⎣ ⎦ ⎪ −4(−2)k 0 −4(−2)k ⎪ ⎩ 2(−2)k 0 2(−2)k

for

k = −1,

for

k = 0, 1, . . . . (2.112b)

From (2.109) and (2.112b) we have

2.1 Continuous-Time Systems

107

⎤ ⎧⎡ 000 ⎪ ⎪ ⎪ ⎣0 0 0⎦ ⎪ for k = −1, ⎪ ⎪ ⎨ 000 Tk E = ⎡ ⎤ ⎪ (−3)k (−2)k − (−3)k (−2)k − (−3)k ⎪ ⎪ ⎪ ⎣ 0 ⎦ for k = 0, 1, 2, . . . ⎪ 2(−2)k 2(−2)k ⎪ ⎩ k k 0 −(−2) −(−2) and (t) =

=

∞  Tk Et kα (kα + 1) k=0 ⎡ (−3)k t 0.8k [(−2)k −(−3)k ]t 0.8k ∞  k=0

⎡

⎢ ⎢ ⎣

(0.8k+1)

0 0

(0.8k+1) 2(−2)k t 0.8k (0.8k+1) (−2)k t 0.8k − (0.8k+1)

[(−2)k −(−3)k ]t 0.8k (0.8k+1) 2(−2)k t 0.8k (0.8k+1) (−2)k t 0.8k − (0.8k+1)

(2.113)

⎤ ⎥ ⎥ ⎦

(2.114)

⎤ E 0.8 (−3t 0.8 ) ψ12 (t) ψ13 (t) 0 2E 0.8 (−2t 0.8 ) 2E 0.8 (−2t 0.8 ) ⎦ , =⎣ 0 −2E 0.8 (−2t 0.8 ) −2E 0.8 (−2t 0.8 ) where ψ12 (t) = ψ13 (t) = E 0.8 (−2t 0.8 ) − E 0.8 (−3t 0.8 ) and E α (z) is the one parameter Mittag-Leffler function defined by (2.5). Analyzing (2.114) we can see that $⎡ ⎤$ $ 1 0 0 $ $ $ ⎣ 0 2 2 ⎦$ = 4 max (t) = (0) = $ $ $ t $ 0 −2 −2 $

(2.115)

and (t1 ) > (t2 ) for any t1 < t2 , t1 , t2 ∈ [0, +∞). By Theorem 2.17 the fractional descriptor continuous-time linear system (2.13) with α = 0.8 and (2.109) is superstable. Using (2.15) we can determine consistent initial conditions of the system ⎤⎡ ⎤ ⎡ ⎤ v1 1 0 0 v1 x0 = T0 Ev = ⎣ 0 2 2 ⎦ ⎣ v2 ⎦ = ⎣ 2(v2 + v3 ) ⎦ , 0 −1 −1 v3 −(v2 − v3 ) ⎡

(2.116)

where v1 , v2 , v3 ∈ R are any numbers. In Fig. 2.3 we present time plots of the state variables and the norm of the state vector for x0 = [ 0.5 2 − 1 ]T , from which we can see that the norm of the state vector decreases monotonically to zero for t → ∞.

108

2 Fractional Descriptor Linear Systems 2

2 x 1 (t)

||x(t)||

x 2 (t)

1.5

x 3 (t)

1.5

1 1

0.5 0

0.5 -0.5 -1

0

5

10

15

0

0

5

t [s]

10

15

t [s]

Fig. 2.3 State variables (on the left) and norm of the state vector (on the right) of the fractional descriptor continuous-time linear system (2.13) with α = 0.8 and (2.109) for x0 = [ 0.5 2 − 1 ]T

2.1.8.3

Drazin Inverse Matrix Method

In this section necessary and sufficient superstability conditions of the fractional descriptor continuous-time linear system (2.28) (or equivalently (2.13)) will be established using the Drazin inverse matrix method. Consider the fractional descriptor continuous-time linear system (2.28). Solution to the Eq. (2.28a) for u(t) = 0 has the form ¯ α )x0 , x(t) = 0 (t) E¯ E¯ D x0 = E α ( E¯ D At

(2.117)

¯ α ) is the one parameter Mittag-Leffler where 0 (t) is given by (2.30b) and E α ( E¯ D At function defined by (2.5). Using Lemma 2.2 from (2.117) we have % & ¯ α ) E¯ E¯ D x0 = E α [ E¯ D A¯ + G(In − E¯ E¯ D )]t α E¯ E¯ D x0 , (2.118) x(t) = E α ( E¯ D At where G ∈ Rn×n is an arbitrary matrix. Taking into account (2.30) and (2.31) for u(t) = 0 it is easy to see that if x0 ∈ Im E¯ E¯ D , then x0 = E¯ E¯ D x0 . Therefore, from (2.118) we have $ % &$

x(t) ≤ $ E α [ E¯ D A¯ + G(In − E¯ E¯ D )]t α $ x0 .

(2.119)

If the matrix E¯ D A¯ + G(In − E¯ E¯ D ) is superstable $ %(i.e. it satisfies the condition &$ (1.133)), then by Lemma 2.5 we have $ E α [ E¯ D A¯ + G(In − E¯ E¯ D )]t α $ ≤ E α (−σ t α ). From the inequality (2.119) we obtain

x(t) ≤ x0 E α (−σ t α ),

(2.120)

2.1 Continuous-Time Systems

109

and the norm of the state vector decreases monotonically to zero. Therefore, the following theorem has been proved. Theorem 2.18. The fractional descriptor continuous-time linear system (2.28) (or equivalently (2.13)) is superstable if and only if there exists a matrix G ∈ Rn×n such that the matrix E¯ D A¯ + G(In − E¯ E¯ D ) satisfies the condition (1.133). Example 2.14. (Continuation of Example 2.13) Consider the fractional descriptor continuous-time linear system (2.13) with α = 0.8 and (2.109). According to (2.110) the matrix pencil (E, A) is regular. Using (2.28b) for c = 0 we obtain ⎡

⎤ 0.3333 0.1667 0.1667 1 1 ⎦, E¯ = [−A]−1 E = ⎣ 0 0 −0.5 −0.5 ⎡ ⎤ −1 0 0 A¯ = [−A]−1 A = ⎣ 0 −1 0 ⎦ . 0 0 −1

(2.121)

Next, from one of the methods presented in Appendix B we have ⎡

E¯ D

and

⎤ 3 −1 −1 = ⎣0 4 4 ⎦ 0 −2 −2

⎡ ⎤ ⎤ 1 0 0 −3 1 1 E¯ D A¯ = ⎣ 0 −4 −4 ⎦ , E¯ E¯ D = ⎣ 0 2 2 ⎦ . 0 −1 −1 0 2 2

(2.122)

⎡

(2.123)

The matrix E¯ D A¯ given by (2.123) does not satisfy the condition (1.133). However, using (2.123) and ⎡ ⎤ 000 G = ⎣0 0 2⎦ (2.124) 020 we obtain

⎡

⎤ −3 1 1 E¯ D A¯ + G(I3 − E¯ E¯ D ) = ⎣ 0 −2 0 ⎦ , 0 0 −2

(2.125)

which satisfies (1.133). Therefore, by Theorem 2.18 the considered system is superstable. The time plots of the state variables and the norm of the state vector for x0 = [ 0.5 2 − 1 ]T are presented in Fig. 2.3.

110

2.1.8.4

2 Fractional Descriptor Linear Systems

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient superstability conditions of the descriptor continuous-time linear system (2.40) (or equivalently (2.13)) using the WeierstrassKronecker decomposition method. Note that every superstable system is also asymptotically stable. State vector of the subsystem (2.40b) x¯2 = 0 for u(t) = 0. Using (2.38) and (2.42) we obtain    E α (A1 t α )x¯10 x¯1 (t) =Q Q x(t) ¯ =Q 0 x¯2 (t)    α x¯10 0 E α (A1 t ) Q 0 0 E α (Mt α )   α 0 E α (A1 t ) Q −1 x0 Q 0 E α (Mt α )     A1 0 −1 x0 = E α (A M t α )x0 , Eα Q Q 0 M 

x(t) = = = =

(2.126)

where E α (A M t α ) is the one parameter Mittag-Leffler function defined by (2.5), the matrix M ∈ Rn 2 ×n 2 is arbitrary and  AM = Q

A1 0 0 M



Q −1 .

(2.127)

From the Eq. (2.126) we have

x(t) = E α (A M t α )x0 ≤ E α (A M t α )

x0 .

(2.128)

If the matrix A M is superstable (i.e. it satisfies the condition (1.133)), then by Lemma 2.5 we have E α (A M t α ) ≤ E α (−σ t α ). From the inequality (2.128) we obtain

x(t) ≤ x0 E α (−σ t α ),

(2.129)

and the norm of the state vector decreases monotonically to zero. Therefore, the following theorem has been proved. Theorem 2.19. The fractional descriptor continuous-time linear system (2.40) (or equivalently (2.13)) is superstable if and only if there exists a matrix M ∈ Rn 2 ×n 2 such that the matrix A M defined by (2.127) satisfies the condition (1.135). Example 2.15. (Continuation of Example 2.13) Consider the fractional descriptor continuous-time linear system (2.13) with α = 0.8 and (2.109). According to (2.110) the matrix pencil (E, A) is regular. In this case

2.1 Continuous-Time Systems

111

⎡

⎤ ⎡ ⎤ −2 0 −2 0 1 0 P = ⎣ 1 −3 1 ⎦ , Q = ⎣ 2 0 −2 ⎦ 0.5 0 1 −1 0 2 ⎤ 100 = PEQ = ⎣0 1 0⎦, 000 ⎡ ⎤   −2 0 0 A1d 0 = P AQ = ⎣ 1 −3 0 ⎦ , 0 In 2 0 0 1

and



In 1 0 0 N

(2.130)

⎡



(2.131)

where n 1 = 2 and n 2 = 1. Using (2.127) for the matrices (2.130), (2.131) and M = [m] ∈ R1×1 we obtain ⎡

AM

0 =⎣ 2 −1 ⎡ −3 =⎣ 0 0

⎤⎡ ⎤⎡ ⎤ 1 0 −2 0 0 0 1 1 0 −2 ⎦ ⎣ 1 −3 0 ⎦ ⎣ 1 0 0 ⎦ 0 2 0 0 m 0 0.5 1 ⎤ 1 1 −4 − m −4 − 2m ⎦ . 2 + m 2 + 2m

(2.132)

Assuming m = −2 the matrix (2.132) satisfies the condition (1.133). Therefore, by Theorem 2.19 the considered system is superstable.

2.2 Discrete-Time Systems 2.2.1 Grünwald-Letnikov Fractional-Order Backward Difference Definition 2.11. A discrete-time function defined by

xi = n

n−1

xi −

n−1

xi−1

  n  k n xi−k , = (−1) k k=0

(2.133)

i = 1, 2, 3, . . . , n ∈ Z+ , xi ∈ R, where

  n n! n(n − 1) . . . (n − k + 1) = = , k k!(n − k)! k!

is called the n-order (backward) difference of the function xi .

(2.134)

112

2 Fractional Descriptor Linear Systems

Definition 2.12. The fractional n-order (backward) difference on the interval [0, k] of the function xi is defined as follows n 0 k x i =

  k  n xi− j . (−1) j j j=0

(2.135)

From (2.133) it follows that the n-order difference can be written as a linear combination of the values of discrete-time function in n + 1 points. The definitions are valid for n being natural numbers and integers. Note that (2.135) is also well defined for fractional and real numbers. In general case n can be also a complex number. Example 2.16. From (2.133) we have for: n = 1 : xi = xi − xi−1 , n = 2 : 2 xi = xi − xi−1 = xi − 2xi−1 + xi−2 , n = 3 : 3 xi = 2 xi − 2 xi−1 = xi − 3xi−1 + 3xi−2 − xi−3 . From (2.135) we obtain for: −1 0 k x k

=

n = −1 : =

  k  −1 xk− j = xk + xk−1 + . . . + x0 (−1) j j j=0 k 

xk− j ,

j=0 −2 0 k x k

n = −2 :

  k  j −2 xk− j = xk + . . . + (k + 1)x0 = (−1) j j=0 k  = ( j + 1)xk− j . j=0

Definition 2.13. The discrete-time function   k  j α

xk = xk− j , (−1) j j=0 α

(2.136)

where 0 < α < 1, α ∈ R and   ' 1 α = α(α−1)...(α− j+1) j j!

for j = 0, for j = 1, 2, 3, . . . ,

(2.137)

2.2 Discrete-Time Systems

113

is called the Grünwald-Letnikov fractional α-order (backward) difference of the function xk . Example 2.17. Using (2.137) for 0 < α < 1 we obtain for:   α = α < 0, k = 1 : (−1) 1   α α(α − 1) k = 2 : (−1)2 = < 0, 2! 2   α(α − 1)(α − 2) 3 α = < 0. k = 3 : (−1) 3 3! 1

2.2.2 State Equations of Descriptor Discrete-Time Linear Systems Consider the discrete-time linear system described by state equations E d α xi+1 = Ad xi + Bd u i , 0 < α < 1,

(2.138a)

yi = Cd xi + Dd u i , i ∈ Z+ ,

(2.138b)

where xi ∈ Rn , u i ∈ Rm , yi ∈ R p are state, input and output vectors and E d , Ad ∈ Rn×n , Bd ∈ Rn×m , Cd ∈ R p×n , Dd ∈ R p×m . We assume that detE d = 0 and det[E d λ − Ad ] = 0 for some λ ∈ C.

(2.139)

If the matrix E d is nonsingular (det E d = 0), then we can transform (2.138a) to the state equation of a fractional standard discrete-time linear system through premultiplication of (2.138a) by E d−1 . The system (2.138) with the singular matrix E d (detE d = 0) will be called a fractional descriptor discrete-time linear system. Substituting the definition of fractional difference (2.136) into (2.138a) we obtain E d xi+1 = Adα xi +

i 

E d c j xi− j + Bd u i , i ∈ Z+ ,

(2.140a)

j=1

where Adα = Ad + E d α. 

and c j = (−1)

j

 α . j +1

(2.140b)

(2.140c)

114

2 Fractional Descriptor Linear Systems

Remark 2.6. From (2.140a) it follows that the fractional system is equivalent to the system with increasing number of delays. Note that coefficients c j given by (2.140c) strongly decrease for increasing i. Hence, we can limit the upper bound of the sum by some natural number L, which is called the length of the practical implementation [53, 57, 88]. In this case the equations (2.140) can be written in the form E d xi+1 = Adα xi +

L 

E d c j xi− j + Bd u i ,

(2.141a)

j=1

yi = Cd xi + Dd u i

(2.141b)

with x−k = 0, k = 1, 2, . . . .

2.2.3 Solution to the State Equation of Fractional Descriptor Discrete-Time Linear Systems Let U be the set of admissible input vectors u i ∈ U ⊂ Rm and X 0 ⊂ Rn the set of consistent initial conditions x(0) = x0 ∈ X 0 for which the equation (2.138a) has the solution xi with u i ∈ U .

2.2.3.1

Laurent Series Expansion Method

In this section the solution to the Eq. (2.138a) will be given using the Laurent series expansion method. Theorem 2.20. Solution to the Eq. (2.138a) for x0 ∈ X 0 and u i ∈ U , i ∈ Z+ is given by the formula n 0 +i−1 Td,i−k−1 Bd u k , (2.142a) xi = Td,i E d x0 + k=0

where the matrices Td,k ∈ Rn×n are determined by ⎧ i  ⎪ ⎪ ⎪ E d Td,k − Adα Td,k−1 − c j E d Td,k− j−1 = In for k = 0, ⎪ ⎪ ⎨ j=1 i  ⎪ c j E d Td,k− j−1 = 0 for k = 0, E d Td,k − Adα Td,k−1 − ⎪ ⎪ ⎪ j=1 ⎪ ⎩ E d Td,k = 0 for k ≤ −n 0

(2.142b)

2.2 Discrete-Time Systems

115

and n 0 is called the index of the system. The matrices Td,k can also be found by the

−1 i  −j Ed c j z . Laurent series expansion of the matrix E d z − Adα − j=1

Proof. Applying the Z-transform to (2.140a) we obtain E d z X (z) − E d zx0 = Adα X (z) +

i 

E d c j z − j X (z) + Bd U (z).

(2.143)

j=1

From (2.143) we have ⎡ X (z) = ⎣ E d z − Adα −

i 

⎤−1 Ed c j z− j ⎦

[E d x0 z + BU (z)].

(2.144)

j=1

Let

⎡ ⎣ E d z − Adα −

i 

⎤−1 Ed c j z− j ⎦

=

j=1

∞ 

Td,k z −(k+1) .

(2.145)

k=−n 0

The matrices Td,k , k = −n 0 , −n 0 + 1, . . . , 0, 1, . . . can be found by the expansion (2.145). Substituting (2.145) into (2.144) we obtain X (z) =

∞  

 Td,k E d x0 z −k + Td,k Bd z −(k+1) U (z) .

(2.146)

k=−n 0

Applying to (2.146) the inverse Z-transform and the convolution theorem we obtain (2.142a).   From (2.142a) for i = 0 we have x0 = Td,0 Ev +

n 0 −1

Td,−k−1 Bd u k ,

(2.147)

k=0

where v ∈ Rn is an arbitrary vector. Remark 2.7. If the matrix E d = In , then n 0 = 0 and Td,i = d,i , where d,i+1 = d,i Adα +

i 

c j d,i− j , d,0 = In ,

(2.148)

j=1

Adα = Ad + In α. Therefore, from (2.142) and (2.148) we have

(2.149)

116

2 Fractional Descriptor Linear Systems

xi = d,i x0 +

i−1 

d,i−k−1 Bd u k .

(2.150)

k=0

The formula (2.150) is the solution to the state equation of a standard fractional discrete-time linear system [57]. Example 2.18. Consider the fractional descriptor discrete-time linear system (2.141a) with α = 0.5, L = 1 and ⎡

Adα

⎤ ⎡ ⎤ −0 −0.4 0 0 0.2 0 E d = ⎣ −1.6 0 0 ⎦ , Ad = ⎣ 0.8 0 0 ⎦ 0 0 0 0 0 0.2 ⎡ ⎤ ⎡ ⎤ 00 0 −0.2 0 = Ad + 0.5E d = ⎣ 0 0 0 ⎦ , Bd = ⎣ 0 −0.8 ⎦ . 0 0 0.2 0 0

(2.151)

The matrix pencil (E d , Ad ) of (2.151) is regular since det[E d λ − Ad ] = 0.032(2λ + 1)2 = 0.

(2.152)

The matrix [E d z − Adα − E d c1 z −1 ]−1 has the form ⎡ [E d z − Adα − 0.125E d z −1 ]

−1

0

= ⎣ − (8z20z 2 −1) 0

⎤ − (8z5z 0 2 −1) 0 0 ⎦. 0 −5

(2.153)

Using (2.145) for n 0 = 1, the matrix (2.153) can be written in the form −1 −1

[E d z − Adα − 0.125E d z ]

=

∞ 

Td,k z −(k+1) ,

(2.154a)

k=−1

where

Td,k

⎤ ⎧⎡ 00 0 ⎪ ⎪ ⎪ ⎣0 0 0 ⎦ ⎪ for k = −1, ⎪ ⎪ ⎪ ⎪ 0 0 −5 ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ 0 −0.625(0.125)0.5k 0 ⎨ 0 0 ⎦ for k = 0, 2, 4, . . . , = ⎣ −2.5(0.125)0.5k ⎪ ⎪ 0 0 0 ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ 000 ⎪ ⎪ ⎪ ⎣ ⎪ 0 0 0⎦ for k = 1, 3, 5, . . . . ⎪ ⎩ 000 (2.154b)

2.2 Discrete-Time Systems

117

Using (2.151) and (2.154b) we have ⎤ ⎧⎡ (0.125)0.5k 0 0 ⎪ ⎪ ⎪ ⎣ ⎪ 0 (0.125)0.5k 0 ⎦ for k = 0, 2, 4, . . . , ⎪ ⎪ ⎨ 0 0 0 Td,k E d = ⎡ ⎤ ⎪ 000 ⎪ ⎪ ⎪ ⎣0 0 0⎦ ⎪ for k = 1, 3, 5, . . . , ⎪ ⎩ 000 ⎤ ⎧⎡ 0 0.5(0.125)0.5k ⎪ ⎪ ⎪ ⎣ 0.5(0.125)0.5k ⎦ for k = 0, 2, 4, . . . , ⎪ 0 ⎪ ⎪ ⎨ 0 0 Td,k Bd = ⎡ ⎤ ⎪ 00 ⎪ ⎪ ⎪ ⎣ ⎦ ⎪ for k = 1, 3, 5, . . . . ⎪ ⎩ 00 00

(2.155a)

(2.155b)

Using (2.142a), (2.151), (2.155) and taking into account that n 0 = 1 we obtain xi = Td,i E d x0 + ⎡

i 

Td,i−k−1 Bd u k = Td,i E d x0 +

k=0 i

i−1 

Td,i−k−1 Bd u k + Td,−1 Bd u i

k=0

⎤ [(−1)i + 1] (0.0625)i 0 0 0 [(−1)i + 1]i (0.0625)i 0 ⎦ x0 =⎣ 0 0 0 ⎡ ¯ ¯ ¯⎤ k k i−1 0 0.5[(−1) + 1] (0.0625)k  ⎣ 0.5[(−1)k¯ + 1]k¯ (0.0625)k¯ ⎦ uk , + 0 k=0 0 0 (2.156) where k¯ = i − k − 1. The equation above shows the solution (2.142a) of the state equation of the fractional descriptor linear system (2.141a) with α = 0.5, L = 1 and (2.151).

2.2.3.2

Drazin Inverse Matrix Method

In this section the solution to the Eq. (2.138a) will be given using the Drazin inverse matrix method [61, 63, 90]. Consider the Eq. (2.138a) and assume that the condition (2.139) is met. There exists a number c ∈ C such that det[E d c − Ad ] = 0. Premultiplying the Eq. (2.138a) by the matrix [E d c − Ad ]−1 we obtain E¯ d α xi+1 = A¯ d xi + B¯ d u i ,

(2.157a)

118

2 Fractional Descriptor Linear Systems

where E¯ d = [E d c − Ad ]−1 E d , A¯ d = [E d c − Ad ]−1 Ad , B¯ d = [E d c − Ad ]−1 Bd . (2.157b) Observe that the Eqs. (2.138a) and (2.157a) have the same solution xi . Basic definitions and properties concerning Drazin inverses are given in Sect. 1.1.2.2. Substituting the definition of fractional difference (2.136) into (2.157a) we obtain E¯ d xi+1 = A¯ dα xi +

i 

E¯ d c j xi− j + B¯ d u i ,

(2.158a)

j=1

where

A¯ dα = A¯ d + E¯ d α.

(2.159)

and the coefficients c j , j = 1, 2, . . . are given by (2.140c). Lemma 1.2 can also be used for the fractional descriptor discrete-time linear system (2.158). Therefore, the Eq. (2.158a) is equivalent to the following equations:

α x1,i+1 = E¯ dD A¯ d x1,i + E¯ dD B¯ d u i , A¯ dD E¯ d (In − E¯ d E¯ dD ) α x2,i+1 = x2,i + (In − E¯ d E¯ dD ) A¯ dD B¯ d u i , where

x1,i = E¯ E¯ D xi , x2,i = (In − E¯ E¯ D )xi .

(2.160a) (2.160b)

(2.160c)

Substitution of (2.136) into (2.160) yields x1,i+1 = E¯ dD A¯ dα x1,i +

i 

c j x1,i− j + E¯ dD B¯ d u i ,

(2.161a)

j=1

A¯ dD E¯ d (In − E¯ d E¯ dD )x2,i+1 = [In + α A¯ dD E¯ d (In − E¯ d E¯ dD )]x2,i +

i 

c j A¯ dD E¯ d (In − E¯ d E¯ dD )x2,i− j

(2.161b)

j=1

+ A¯ dD B¯ d (In − E¯ d E¯ dD )u i , Theorem 2.21. The solution to the equation (2.158) (or equivalently (2.157a)) for x0 ∈ X 0 and u i ∈ U is given by

2.2 Discrete-Time Systems

119

xi = d,i E¯ d E¯ dD x0 +

i−1 

d,i−k−1 E¯ dD B¯ d u k

k=0

(2.162a)

 − In ) ( E¯ d A¯ dD )k A¯ dD B¯ d ψi,k , q−1

+

( E¯ d E¯ dD

k=0

where q is the index of E¯ d and d,i+1 = d,i E¯ dD A¯ dα +

i 

c j d,i− j , d,0 = In ,

(2.162b)

j=1

ψi,0 = u i , ψi,1 = ψi+1,0 − αψi,0 −

i 

c j ψi− j,0 = u i+1 − αu i −

j=1

ψi,2 = ψi+1,1 − αψi,1 −

i 

i 

c j u i− j ,

j=1

c j ψi− j,1

j=1

= u i+2 − 2αu i+1 + α 2 u i + 2α

i 

c j u i− j −

j=1

−

i  j=1

c j u i− j+1 +

i 

cj

j=1

i− j 

i+1 

c j u i− j+1

(2.162c)

j=1

cl u i− j−l

l=1

.. . ψi,q−1 = ψi+1,q−2 − αψi,q−2 −

i 

c j ψi− j,q−2 .

j=1

Proof. We shall show that the solution (2.162) satisfies the equality (2.158a). Using (1.26), (1.27) and (2.162) we obtain

120

2 Fractional Descriptor Linear Systems

E¯ d xi+1 = E¯ d d,i+1 E¯ d E¯ dD x0 +

i 

E¯ d d,i−k E¯ dD B¯ d u k

k=0

+ E¯ d ( E¯ d E¯ dD − In )

q−1 

( E¯ d A¯ dD )k A¯ dD B¯ d ψi+1,k

k=0

⎛ = ⎝d,i A¯ dα +

i 

⎞

E¯ d c j d,i− j ⎠ E¯ d E¯ dD x0

j=1

+

i−1  k=0

+

i−1 

(2.163)

d,i−k−1 A¯ dα E¯ dD B¯ d u k + E¯ d E¯ dD B¯ d u i ⎛ ⎝

⎞

k 

k=0

c j d,k− j ⎠ E¯ d E¯ dD B¯ d u i−k−1

j=1

+ ( E¯ d E¯ dD − In )

q−1 

( E¯ d A¯ dD )k+1 B¯ d ψi+1,k ,

k=0 i  j=1

E¯ d c j xi− j =

i 

E¯ d c j d,i− j E¯ d E¯ dD v

j=1

⎛ ⎞ i−1 k   ⎝ c j d,k− j ⎠ E¯ d E¯ dD B¯ d u i−k−1 + k=0

(2.164)

j=1

+ ( E¯ d E¯ dD − In )

q−1 

( E¯ d A¯ dD )k+1 B¯ d ψi− j,k

k=0

and A¯ dα xi = d,i A¯ dα E¯ d E¯ dD x0 +

i−1 

d,i−k−1 A¯ dα E¯ dD B¯ d u k

k=0

 ( E¯ d A¯ dD )k+1 B¯ d ψi,k q−1

+ α( E¯ d E¯ dD − In )

(2.165)

k=0

+ ( E¯ d E¯ dD − In )

q−1 

( E¯ d A¯ dD )k B¯ d ψi,k .

k=0

From (2.163), (2.164) and (2.165) it follows that E¯ d xi+1 − A¯ dα xi −

i 

E¯ d c j xi− j = B¯ d u i ,

(2.166)

j=1

which completes the proof.

 

2.2 Discrete-Time Systems

121

It is easy to check that the first two components of (2.162a) are the solution to the Eq. (2.161a) and the third component of (2.162a) is the solution to the Eq. (2.161b). From (2.162a) for i = 0 we have x0 = E¯ d E¯ dD v + ( E¯ d E¯ dD − In )

q−1 

( E¯ d A¯ dD )k A¯ dD B¯ d ψ0,k .

(2.167)

k=0

where v ∈ Rn is an arbitrary vector and ψ0,k is defined by (2.162c). In the case when u i = 0, i ∈ Z+ from (2.167) we have x0 = E¯ d E¯ dD v. A homogenous equation E d α xi+1 = Ad xi has the solution if and only if x0 ∈ Im E¯ d E¯ dD . By Lemma 1.3 and Theorem 1.5 the matrices E¯ dD E¯ d , E¯ dD A¯ d , E¯ d A¯ dD , E¯ dD B¯ d , D ¯ ¯ Ad Bd , the index q of the matrix E¯ d , the solution (2.162a) of the Eq. (2.157a) and the set of consistent initial conditions X 0 do not depend of the choice of c. Example 2.19. (Continuation of Example 2.18) Consider the fractional descriptor discrete-time linear system (2.141a) with α = 0.5, L = 1 and (2.151). According to (2.152) the matrix pencil (E d , Ad ) is regular. We choose c = 0 and using (2.157b), (2.159) we obtain ⎡

⎤ ⎡ ⎤ 200 −1 0 0 E¯ d = [−Ad ]−1 E d = ⎣ 0 2 0 ⎦ , A¯ d = [−Ad ]−1 Ad = ⎣ 0 −1 0 ⎦ 000 0 0 −1 ⎡ ⎤ ⎡ ⎤ (2.168) 00 0 01 A¯ dα = A¯ d + 0.5 E¯ d = ⎣ 0 0 0 ⎦ , B¯ d = [−Ad ]−1 Bd = ⎣ 1 0 ⎦ . 0 0 −1 00 It is easy to see that rank E¯ d = rank E¯ d2 , so q = 1. The Drazin of E¯d can  inverse   be 2 0 0 computed using (B.1) and (B.2) for E¯ d = E¯ 1 , where E¯ 11 = , E¯ 12 = and 02 0 E¯ dD =



−1 ¯ −2 ¯ E 11 E 12 E¯ 11 0 0



⎡

⎤ 0.5 0 0 = ⎣ 0 0.5 0 ⎦ . 0 0 0

¯ Next, we compute the matrices Note that A¯ D = A¯ −1 = A.

(2.169)

122

2 Fractional Descriptor Linear Systems

⎡

⎤ ⎡ ⎤ 100 000 E¯ d E¯ dD = ⎣ 0 1 0 ⎦ , E¯ dD A¯ dα = ⎣ 0 0 0 ⎦ , 000 000 ⎡ ⎤ ⎡ ⎤ 0 0.5 00 0 E¯ dD B¯ d = ⎣ 0.5 0 ⎦ , ( E¯ d E¯ dD − I3 ) = ⎣ 0 0 0 ⎦ , 0 0 0 0 −1 ⎡ ⎤ ⎡ ⎤ −2 0 0 0 −1 A¯ dD E¯ d = ⎣ 0 −2 0 ⎦ , A¯ dD B¯ d = ⎣ −1 0 ⎦ 0 0 0 0 0

(2.170)

and d,i+1 = d,i E¯ dD A¯ dα +

L 

c j d,i− j = c1 d,i−1 = 0.125d,i−1 ,

(2.171)

j=1

where d,0 = I3 , d,−1 = 0. From (2.171) it follows that

d,i

⎤ ⎧⎡ (0.125)0.5i 0 0 ⎪ ⎪ ⎪ ⎣ ⎦ for i = 0, 2, 4, . . . , ⎪ 0 0 (0.125)0.5i ⎪ ⎪ 0.5i ⎨ 0 0 (0.125) = ⎡ ⎤ ⎪ 000 ⎪ ⎪ ⎪ ⎣0 0 0⎦ ⎪ for i = 1, 3, 5, . . . . ⎪ ⎩ 000

(2.172)

Using (2.162), (2.170) and (2.172) and taking into account that q = 1 we obtain xi = d,i E¯ d E¯ dD x0 + ⎡

i−1 

d,i−k−1 E¯ dD B¯ d u k + ( E¯ d E¯ dD − In ) A¯ dD B¯ d ψi,0

k=0

⎤ [(−1)i + 1]i (0.0625)i 0 0 0 [(−1)i + 1]i (0.0625)i 0 ⎦ x0 =⎣ 0 0 0 ⎡ ¯k ¯ ¯⎤ i−1 0 0.5[(−1) + 1]k (0.0625)k  ⎣ 0.5[(−1)k¯ + 1]k¯ (0.0625)k¯ ⎦ uk , + 0 k=0 0 0

(2.173)

where k¯ = i − k − 1. The equation above shows the solution (2.162a) of the state equation of the fractional descriptor linear system (2.141a) with α = 0.5, L = 1 and (2.151). Note that solutions (2.156) and (2.173) are equivalent.

2.2 Discrete-Time Systems

2.2.3.3

123

Weierstrass-Kronecker Decomposition Method

In this section the solution to the Eq. (2.138a) will be given using the WeierstrassKronecker decomposition method [57, 90]. If the conditon (2.139) holds, then there exists a pair of nonsingular matrices Pd , Q d ∈ Rn×n such that [57, 90] 

 In 1 λ − A1d 0 , A1d ∈ Rn 1 ×n 1 , N ∈ Rn 2 ×n 2 , 0 N λ − In 2 (2.174) where n 1 = deg{det[E d λ − Ad ]}, n 2 = n − n 1 and N is a nilpotent matrix with the nilpotency index μ, i.e. N μ−1 = 0 and N μ = 0. The matrices Pd and Q d can be computed using one of procedures given in [47, 48, 168]. Premultiplying (2.138a) by the matrix Pd and introducing the new state vector Pd [E d λ − Ad ]Q d =

 x¯i =

x¯1,i x¯2,i



n1 n2 = Q −1 d x i , x¯ 1,i ∈ R , x¯ 2,i ∈ R ,

(2.175)

we obtain −1 α Pd E d Q d Q −1 d x i+1 = Pd Ad Q d Q d x i + Pd Bd u i ,

(2.176a)

yi = Cd Q d Q −1 d x i + Dd u i

(2.176b)

α x¯1,i+1 = A1d x¯1,i + B1d u i ,

(2.177a)

N α x¯2,i+1 = x¯2,i + B2d u i ,

(2.177b)

yi = C1d x¯1,i + C2d x¯2,i + Dd u i ,

(2.177c)

and using (2.174) we get



where Pd Bd =

 B1d , B1d ∈ Rn 1 ×m , B2d ∈ Rn 2 ×m , B2d

  Cd Q d = C1d C2d , C1d ∈ R p×n 1 , C2d ∈ R p×n 2 .

(2.178a) (2.178b)

Substituting the definition of fractional difference (2.136) into (2.177) we obtain x¯1,i+1 = A1dα x¯1,i +

i  j=1

c j x¯1,i− j + B1d u i ,

(2.179a)

124

2 Fractional Descriptor Linear Systems i 

N x¯2,i+1 = (In 2 + N α)x¯2,i +

c j N x¯2,i− j + B2d u i ,

(2.179b)

j=1

where A1dα = A1d + In 1 α

(2.179c)

and the coefficients c j , j = 1, 2, . . . are given by (2.140c). Theorem 2.22. [57] Solution to the Eq. (2.177a) for given initial condition x¯10 ∈ Rn 1 and input u i ∈ U , i ∈ Z+ has the form x¯1,i = d,i x¯10 +

i−1 

d,i− j−1 B1d u k .

(2.180a)

k=0

where d,i+1 = d,i A1dα +

i 

c j d,i− j , d,0 = In 1 .

(2.180b)

j=1

Theorem 2.23. Solution to the Eq. (2.177b) for given input u i ∈ U , i ∈ Z+ has the form μ−1  N k B2d ψi,k , (2.181) x¯2,i = − k=0

and the consistent initial conditions of the subsystem (2.177b) are given by the formula μ−1  N k B2d ψ0,k , (2.182) x¯20 = − k=0

where ψi,k ∈ Rm is defined by (2.162c). Proof. Using (2.179b) and (2.181) we obtain N x¯2,i+1 = −

μ−1 

N k+1 B2d ψi+1,k ,

(2.183)

k=0

(In 2 + N α)x¯2,i +

i 

c j N x¯2,i− j = −

j=1

μ−1 

(N k + N k+1 α)B2d ψi,k

k=0

−

μ−1  i  k=0 j=1

(2.184) c j N k+1 B2d ψi− j,k

2.2 Discrete-Time Systems

125

Therefore, from (2.183), (2.184) and (2.162c) we have N x¯2,i+1 − (In 2 + N α)x¯2,i −

i 

c j N x¯2,i− j = B2d u i ,

(2.185)

j=1

 

which completes the proof.

Example 2.20. (Continuation of Example 2.18) Consider the fractional descriptor discrete-time linear system (2.141a) with α = 0.5, L = 1 and (2.151). According to (2.152) the matrix pencil (E d , Ad ) is regular. In this case ⎡

⎤ ⎡ ⎤ −2.5 0 0 010 P = ⎣ 0 −0.625 0 ⎦ , Q = ⎣ 1 0 0 ⎦ 0 0 1 005 ⎤ 100 In 1 0 = PEQ = ⎣0 1 0⎦, 0 N 000 ⎡ ⎤   −0.5 0 0 A1d 0 = P AQ = ⎣ 0 −0.5 0 ⎦ , 0 In 2 0 0 1 ⎡ ⎤   0.5 0 B1d = P B = ⎣ 0 0.5 ⎦ . B2d 0 0

and





(2.186)

⎡

Next we compute

 A1dα = A1d + 0.5I2 =

00 00

(2.187)

 (2.188)

and d,i+1 = d,i A1dα +

L 

c j d,i− j = c1 d,i−1 = 0.125d,i−1 ,

(2.189)

j=1

where d,0 = I2 , d,−1 = 0. From (2.189) it follows that

d,i

 ⎧ (0.125)0.5i 0 ⎪ ⎪ for i = 0, 2, 4, . . . , ⎨ 0 (0.125)0.5i =   ⎪ ⎪ ⎩ 00 for i = 1, 3, 5, . . . . 00

(2.190)

126

2 Fractional Descriptor Linear Systems

Using (2.180), (2.187) and (2.190) we obtain  0 [(−1)i + 1]i (0.0625)i x¯ = 0 [(−1)i + 1]i (0.0625)i 10

(2.191) i−1 ¯ ¯ ¯  0 0.5[(−1)k + 1]k (0.0625)k + ¯ ¯ ¯ uk , 0 0.5[(−1)k + 1]k (0.0625)k k=0 

x¯1,i

where k¯ = i − k − 1. Taking into account that μ = 1 and using (2.181), (2.187) we obtain x¯2,i = −B2d ψi,0 = −B2d u i = 0.

(2.192)

From (2.175), (2.191) and (2.192) we have ⎡

⎤ [(−1)i + 1]i (0.0625)i 0 0 0 [(−1)i + 1]i (0.0625)i 0 ⎦ x¯0 x¯i = ⎣ 0 0 0 ⎡ ⎤ ¯ ¯ ¯ k k k i−1 0.5[(−1) + 1] (0.0625) 0  ¯ ¯ ¯ ⎣ + 0 0.5[(−1)k + 1]k (0.0625)k ⎦ u k , k=0 0 0

(2.193)

where k¯ = i − k − 1. Taking into account that x¯i = Q −1 xi we have to premultiply the Eq. (2.193) by Q and substitute x¯0 = Q −1 x0 . Thus we obtain the formula (2.156) (or (2.173)), which means that all considered methods of the fractional descriptor systems analysis give the same result.

2.2.4 Positive Fractional Descriptor Discrete-Time Linear Systems In positive systems inputs, outputs and state variables take only nonnegative values [40, 49]. The following definition of the positivity is suitable for all methods of the fractional descriptor systems analysis. Definition 2.14. The fractional descriptor discrete-time linear system (2.138) is p called (internally) positive if xi ∈ Rn+ and yi ∈ R+ , i ∈ Z+ for any consistent initial condition x0 ∈ X 0 ⊂ Rn+ and all admissible inputs u i ∈ U ⊂ Rm + , i ∈ Z+ .

2.2 Discrete-Time Systems

2.2.4.1

127

Laurent Series Expansion Method

In this section necessary and sufficient positivity conditions of the fractional descriptor discrete-time linear system (2.138) will be established using the Laurent series expansion method. Theorem 2.24. The descriptor discrete-time linear system (2.138) is (internally) positive if and only if Td,k E d ∈ Rn×n + , k = 0, 1, . . . , n×m Td,k Bd ∈ R+ , k = −n 0 , −n 0 + 1, . . . , 0, 1, . . . Cd ∈

p×n R+ ,

Dd ∈

(2.194)

p×m R+ ,

where the matrices Td,k ∈ Rn×n are given by (2.142b). Proof. The proof can be accomplished in a similar way to the proof of Theorem 1.28.   Example 2.21. (Continuation of Example 2.18) Consider the fractional descriptor discrete-time linear system (2.141a) with (2.151), α = 0.5, L = 1 and     C d = 1 0 0 , Dd = 0 0 .

(2.195)

The matrices Td,k E d and Td,k Bd are given by (2.155). From (2.155) and (2.195) it 3×2 1×3 follows that Td,k E d ∈ R3×3 + , Td,k Bd ∈ R+ , k = 0, 1, 2, . . . and C d ∈ R+ , Dd ∈ 1×2 R+ . Therefore, by Theorem 2.24 the considered fractional descriptor system is positive.

2.2.4.2

Drazin Inverse Matrix Method

In this section necessary and sufficient positivity conditions of the fractional descriptor discrete-time linear system (2.157) (or equivalently (2.138)) will be established using the Drazin inverse matrix method. Theorem 2.25. Let F¯d,k = ( E¯ d A¯ dD )k A¯ dD B¯ d , k = 0, 1, . . . , q − 1.

(2.196)

The fractional descriptor discrete-time linear system (2.157) (or equivalently (2.138)) is positive if and only if there exists a matrix G d ∈ Rn×n [79, 141] such that E¯ dD A¯ dα + G d (In − E¯ d E¯ dD ) ∈ Rn×n + ,

(2.197a)

128

2 Fractional Descriptor Linear Systems

and p×n p×m Im E¯ d E¯ dD ⊂ Rn+ , E¯ dD B¯ d ∈ Rn×m + , C d ∈ R+ , Dd ∈ R+

( E¯ d E¯ dD − In ) F¯d,k ∈ Rn×m + , k = 0, 1, . . . , q − 1.

(2.197b) (2.197c)

Proof. Using the superposition principle we consider components of the solution (2.162a) independently: n ¯ ¯D 1) If Im E¯ d E¯ dD ⊂ Rn×n + , then E d E d x 0 ⊂ R+ . Therefore, for the matrix G d such that n×n D D E¯ d A¯ dα + G d (In − E¯ d E¯ d ) ∈ R+ we have [ E¯ dD A¯ dα + G d (In − E¯ d E¯ dD )]i ∈ ¯D ¯ ¯ ¯D i ¯ ¯D ¯D ¯ i Rn×n + . Taking into account that [ E d Adα + G d (In − E d E d )] E d E d = ( E d Adα ) D D D D D i i E¯ d E¯ d we obtain [ E¯ d A¯ dα + G d (In − E¯ d E¯ d )] E¯ d E¯ d x0 = ( E¯ d A¯ dα ) E¯ d E¯ dD x0 ∈ Rn+ , i ∈ Z+ . Taking into account that for α ∈ (0; 1) coefficients c j > 0, j = 1, 2, . . . we have d,i E¯ d E¯ dD x0 ∈ Rn×n + , i ∈ Z+ .  n×n D ¯ n ¯ ¯D ¯ 2) If 1) holds and E d Bd ∈ R+ , then i−1 k=0 d,i−k−1 E d Bd u k ∈ R+ for u i ∈ U ⊂ m R+ , i ∈ Z+ . ¯ 3) If ( E¯ d E¯ dD − In ) F¯d,k ∈ Rn×m + , k = 0, 1, . . . , q − 1, where the matrices Fd,k are given by (2.196), then the last component of the solution (2.162a) is also nonnegative.

Therefore, we have xi ∈ Rn+ , i ∈ Z+ if and only if the conditions (2.197) hold. p p×n Substituting (2.162a) into (2.138b) it is easy to see that yi ∈ R+ , i ∈ Z+ if Cd ∈ R+ p×m and Dd ∈ R+ .   Remark 2.8. Note that the fractional descriptor discrete-time linear system (2.157) can be positive even though the matrix E¯ dD A¯ dα contains negative entries. In the special case for G d = 0 from (2.197a) we obtain the positivity condition E¯ dD A¯ dα ∈ Rn×n + . Example 2.22. (Continuation of Example 2.19) Consider the fractional descriptor discrete-time linear system (2.141a) with α = 0.5, L = 1 and (2.151), (2.195). From (2.170) we have ⎡

Im E¯ d E¯ dD

⎤ ⎡ ⎤ 100 v1 3 ⎣ v = Im ⎣ 0 1 0 ⎦ ⊂ R3×3 for v = 2 ⎦ ⊂ R+ + 000 0

(2.198)

3×2 3×2 ¯D ¯ ¯ ¯D ¯D ¯ and E¯ dD A¯ dα ∈ R3×3 + , E d Bd ∈ R+ , ( E d E d − I3 ) Ad Bd ∈ R+ . From (2.195) it 1×3 1×2 follows that Cd ∈ R+ , Dd ∈ R+ . Therefore, by Theorem 2.25 the considered fractional descriptor system is positive.

2.2.4.3

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient positivity conditions of the fractional descriptor discrete-time linear system (2.177) (or equivalently (2.138)) will be established

2.2 Discrete-Time Systems

129

using the Weierstrass-Kronecker decomposition method with the assumption that the matrix Q d determined by (2.174) is a monomial matrix [59, 66]. Theorem 2.26. [19, 59, 84] Let the decomposition (2.174) of the system (2.138) be possible for a monomial matrix Q d ∈ Rn×n + . The fractional descriptor discrete-time linear system (2.177) (or equivalently (2.138)) is positive if and only if A1dα ∈ Rn+1 ×n 1 , B1d ∈ Rn+1 ×m , Cd ∈ R+ , Dd ∈ R+ p×n

− N k B2d ∈

Rn+2 ×m ,

k = 0, 1, . . . , μ − 1.

p×m

,

(2.199)

Proof. For a monomial matrix Q d ∈ Rn×n we have xi ∈ Rn+ , i ∈ Z+ if x¯i ∈ Rn+ , + i ∈ Z+ . Taking into account that for α ∈ (0; 1) coefficients c j > 0, j = 1, 2, . . . and A1dα ∈ Rn+1 ×n 1 , B1d ∈ Rn+1 ×m , then from (2.180a) we obtain x¯1,i ∈ Rn+1 , i ∈ Z+ since n 2 ×m k , by Definition 2.14 x¯10 = Q −1 x10 ∈ Rn+1 and u i ∈ Rm + , i ∈ Z+ . If −N B2d ∈ R+ n2 m k = 0, 1, . . . , μ − 1, then from (2.181) we obtain x¯2,i ∈ R+ since u i ∈ R+ , i ∈ Z+ . p×n p×m p From (2.176b) for Cd ∈ R+ and Dd ∈ R+ we have yi ∈ R+ , i ∈ Z+ since xi ∈ n m   R+ and u i ∈ R+ , i ∈ Z+ . Example 2.23. (Continuation of Example 2.20) Consider the fractional descriptor discrete-time linear system (2.141a) with α = 0.5, L = 1 and (2.151), (2.195). From 2×2 (2.187) and (2.188) we have A1dα ∈ R2×2 and −N 0 B2d = −B2d ∈ + , B1d ∈ R+ 1×2 1×3 1×2 R+ . From (2.195) it follows that Cd ∈ R+ , Dd ∈ R+ . Taking into account that the matrix Q ∈ R3×3 + is monomial, by Theorem 2.26 the considered fractional descriptor system is positive.

2.2.5 Stability of Fractional Descriptor Discrete-Time Linear Systems 2.2.5.1

Laurent Series Expansion Method

In this section necessary and sufficient stability conditions of the fractional descriptor discrete-time linear system (2.138a) will be established using the Laurent series expansion method. Definition 2.15. [53, 57, 88] The fractional descriptor discrete-time linear system (2.138a) is called practically stable for given length L of practical implementation if the fractional linear system (2.141a) is asymptotically stable. The fractional descriptor discrete-time linear system (2.138a) is called asymptotically stable if the fractional linear system (2.141a) is asymptotically stable for L → ∞. To test the stability of the system (2.138a) well-known methods for fractional discrete-time linear systems can be applied, see e.g. [25, 152].

130

2 Fractional Descriptor Linear Systems

Theorem 2.27. The fractional descriptor discrete-time linear system (2.138a) is practically stable for given length L of the practical implementation if and only if all roots of the characteristic equation ⎡ det ⎣ E d z − Adα −

L 

⎤ Ed c j z− j ⎦ = 0

(2.200)

j=1

lie inside the unit circle. Proof. The proof can be accomplished in a similar way as it has been shown in [152]. The characteristic Eq. (2.200) is obtained through application of the Z-transform to the Eq. (2.141a) for u i = 0, i ∈ Z+ .   Taking into account that [25] ∞ 

c j z − j = z − α − (z − 1)α z 1−α

(2.201)

j=1

we obtain the following theorem. Theorem 2.28. The fractional descriptor discrete-time linear system (2.138a) is asymptotically stable if and only if all roots of the characteristic equation   det E d (z − 1)α z 1−α − Ad = 0

(2.202)

lie inside the unit circle. Example 2.24. Consider the fractional descriptor discrete-time linear system (2.138a) with α = 0.4 and 

   1 0 0.2 0 Ed = , Ad = , 0.5 0 0.1 0.3   0.6 0 . Adα = Ad + 0.4E d = 0.3 0.3

(2.203)

The matrix pencil (E d , Ad ) of (2.203) is regular since det[E d λ − Ad ] = −0.3z + 0.06 = 0.

(2.204)

To test the stability of the system (2.138a) we have to consider the system (2.141a). The characteristic equation of (2.141a) with α = 0.4, L = 1 and (2.203) has the form det[E d z − Adα − 0.12E d z −1 ] = −0.3z + 0.036z −1 + 0.18 = 0.

(2.205)

2.2 Discrete-Time Systems

131

The system (2.141a) with L = 1 is asymptotically stable since roots of (2.205) are z 1 = −0.1583, z 2 = 0.7583. Therefore, by Theorem 2.27 the fractional descriptor discrete-time linear system (2.138a) is practically stable for length of the practical implementation L = 1. Now let us consider the system (2.141a) with α = 0.4, L → ∞ and (2.203). Using (2.202) we obtain det[E d (z − 1)0.4 z 0.6 − Ad ] = −0.3(z − 1)0.4 z 0.6 + 0.06 = 0.

(2.206)

The system (2.141a) with L → ∞ is unstable since the root of (2.206) is z 1 = 1.0174. Therefore, by Theorem 2.28 the fractional descriptor discrete-time linear system (2.138a) is unstable.

2.2.5.2

Drazin Inverse Matrix Method

In this section necessary and sufficient stability conditions of the fractional descriptor discrete-time linear system (2.157) (or equivalently (2.138)) will be established using the Drazin inverse matrix method. The upper bound of the sum in (2.158a) can be limited by some natural number L (length of practical implementation), like in the case of (2.141a), i.e. the state equation has the form E¯ d xi+1 = A¯ dα xi +

L 

A¯ d c j xi− j + B¯ d u i ,

(2.207)

j=1

where A¯ dα is given by (2.159) and x−k = 0, k = 1, 2, . . . . Definition 2.16. The fractional descriptor discrete-time linear system (2.157) (or equivalently (2.138)) is called practically stable for given length L of practical implementation if the fractional linear system (2.207) is asymptotically stable. The fractional descriptor discrete-time linear system v (or equivalently (2.138)) is called asymptotically stable if the fractional linear system (2.207) is asymptotically stable for L → ∞. Lemma 2.6. The characteristic equation ⎡ det ⎣ E¯ d z − A¯ dα −

L 

⎤ E¯ d c j z − j ⎦ = 0

(2.208)

j=1

of (2.207) has the same set of roots as the characteristic equation (2.200) of (2.141a).

132

2 Fractional Descriptor Linear Systems

Proof. From (2.157b) and (2.208) we have ⎡ det ⎣ E¯ d z − A¯ dα − ⎧ ⎨

L 

⎤ E¯ d c j z − j ⎦

j=1

⎛

= det [E d c − Ad ]−1 ⎝ E − Adα − ⎩ ⎡

L  j=1

⎞⎫ ⎬ Ed c j z− j ⎠ ⎭

= {det[E d c − Ad ]}−1 det ⎣ E d z − Adα −

L 

(2.209) ⎤

E d c j z − j ⎦ = 0,

j=1

 

which is equivalent to (2.200).

Therefore, by Lemma 2.6 and the approach given in Sect. 2.2.5.1 we obtain the following theorem. Theorem 2.29. The fractional descriptor discrete-time linear system (2.138a) is: 1) practically stable for given length L of the practical implementation if and only if all roots of the characteristic equation (2.208) lie inside the unit circle; 2) asymptotically stable if and only if all roots of the characteristic equation   det E¯ d (z − 1)α z 1−α − A¯ d = 0

(2.210)

lie inside the unit circle. Example 2.25. (Continuation of Example 2.24) Consider the fractional descriptor discrete-time linear system (2.138a) with (2.203). According to (2.204) the matrix pencil (E d , Ad ) is regular. We choose c = 0 and using (2.157b) we obtain     −5 0 −1 0 −1 −1 ¯ ¯ E d = [−Ad ] E d = , Ad = [−Ad ] Ad = , 0 0 0 −1   −3 0 . A¯ dα = A¯ d + 0.4 E¯ d = 0 −1

(2.211)

To test the stability of the system (2.157) (or equivalently (2.138a)) we have to consider the system (2.207). The characteristic equation of (2.207) with α = 0.4, L = 1 and (2.211) has the form det[ E¯ d z − A¯ dα − 0.12 E¯ d z −1 ] = −5z + 0.6z −1 + 3 = 0.

(2.212)

The system (2.207) with L = 1 is asymptotically stable since roots of (2.212) are z 1 = −0.1583, z 2 = 0.7583. Therefore, by Theorem 2.29 the fractional descriptor discrete-time linear system (2.157) (or equivalently (2.138a)) is practically stable for length of the practical implementation L = 1.

2.2 Discrete-Time Systems

133

Now let us consider the system (2.207) with α = 0.4, L → ∞ and (2.211). Using (2.210) we obtain det[ E¯ d (z − 1)0.4 z 0.6 − A¯ d ] = −5(z − 1)0.4 z 0.6 + 1 = 0.

(2.213)

The system (2.207) with L → ∞ is unstable since the root of (2.213) is z 1 = 1.0174. Therefore, by Theorem 2.29 the fractional descriptor discrete-time linear system (2.157) (or equivalently (2.138a)) is unstable.

2.2.5.3

Weierstrass-Kronecker Decomposition Method

In this section necessary and sufficient stability conditions of the fractional descriptor discrete-time linear system (2.177) (or equivalently (2.138)) using the WeierstrassKronecker decomposition method. Note that the state vector of the subsystem (2.177b) x¯2 (t) = 0 for u(t) = 0. Therefore, the stability of the system (2.177) (or equivalently (2.138)) depends entirely on the subsystem (2.177a). The upper bound of the sum in (2.179a) can be limited by some natural number L (length of practical implementation), like in the case of (2.141a), i.e. the state equation has the form x¯1,i+1 = A1dα x¯1,i +

L 

c j x¯1,i− j + B1d u i ,

(2.214)

j=1

where A1dα is given by (2.179c) and x¯1,−k = 0, k = 1, 2, . . . . Definition 2.17. The fractional descriptor discrete-time linear system (2.177) (or equivalently (2.138)) is called practically stable for given length L of practical implementation if the fractional linear system (2.214) is asymptotically stable. The fractional descriptor discrete-time linear system (2.177) (or equivalently (2.138)) is called asymptotically stable if the fractional linear system (2.214) is asymptotically stable for L → ∞. Lemma 2.7. The characteristic equation ⎡ det ⎣In 1 z − A1dα −

L 

⎤ In 1 c j z − j ⎦ = 0

(2.215)

j=1

of (2.214) has the same set of roots as the characteristic equation (2.200) of (2.141a).

134

2 Fractional Descriptor Linear Systems

Proof. Using (2.174) and (2.200) we obtain ⎧ ⎡ ⎤ ⎫ L ⎨ ⎬  det P ⎣ E d z − Adα − Ed c j z− j ⎦ Q ⎩ ⎭ j=1 ⎡ ⎤ L  −j In 1 c j z 0 ⎢ In 1 z − A1dα − ⎥ ⎢ ⎥ j=1 = det ⎢ ⎥ L  ⎣ ⎦ c j N z− j 0 N z − (In 2 + N α) − ⎡ = (−1)n 2 det ⎣In 1 z − A1dα −

L 

j=1

⎤ In 1 c j z − j ⎦ = 0

j=1

(2.216) since

⎡ det ⎣ N z − (In 2 + N α) −

L 

⎤ c j N z − j ⎦ = (−1)n 2 .

(2.217)

j=1

Therefore, from (2.216) it follows that the characteristic equations (2.200) and (2.215) have the same set of roots.   Using Lemma 2.7 and the approach given in Sect. 2.2.5.1 we obtain the following theorem. Theorem 2.30. The fractional descriptor discrete-time linear system (2.138a) is: 1) practically stable for given length L of the practical implementation if and only if all roots of the characteristic equation (2.215) lie inside the unit circle; 2) asymptotically stable if and only if all roots of the characteristic equation   det In 1 (z − 1)α z 1−α − A1d = 0

(2.218)

lie inside the unit circle. Example 2.26. (Continuation of Example 2.24) Consider the fractional descriptor discrete-time linear system (2.138a) with (2.203). According to (2.204) the matrix pencil (E d , Ad ) is regular. In this case  P= and



   1 0 1 0 , Q= −0.5 1 0 3.3333 



 10 = PEQ = , 00     0.2 0 A1d 0 = P AQ = , 0 In 2 0 1 In 1 0 0 N

(2.219)

(2.220a)

2.2 Discrete-Time Systems

135

A1dα = A1d + 0.4I1 = 0.6.

(2.220b)

To test the stability of the system (2.177) (or equivalently (2.138a)) we have to consider the system (2.214). The characteristic equation of (2.214) with α = 0.4, L = 1 and (2.220) has the form det[In 1 z − A¯ 1dα − 0.12In 1 z −1 ] = z − 0.12z −1 + 0.6 = 0.

(2.221)

The system (2.214) with L = 1 is asymptotically stable since roots of (2.221) are z 1 = −0.1583, z 2 = 0.7583. Therefore, by Theorem 2.30 the fractional descriptor discrete-time linear system (2.177) (or equivalently (2.138a)) is practically stable for length of the practical implementation L = 1. Now let us consider the system (2.214) with α = 0.4, L → ∞ and (2.220). Using (2.218) we obtain det[In 1 (z − 1)0.4 z 0.6 − A1d ] = (z − 1)0.4 z 0.6 − 0.2 = 0.

(2.222)

The system (2.214) with L → ∞ is unstable since the root of (2.222) is z 1 = 1.0174. Therefore, by Theorem 2.30 the fractional descriptor discrete-time linear system (2.177) (or equivalently (2.138a)) is unstable.

2.2.6 Superstability of Fractional Descriptor Discrete-Time Linear Systems The asymptotic stability of a dynamical system ensures that its free response decreases to zero for i → ∞, however its value may increase significantly in the initial part of the state vector trajectory (it is so-called “peak effect”). In superstable systems the norm of the state vector decreases monotonically to zero for i → ∞, which prevents such undesirable effects [70, 137, 138].

2.2.6.1

Properties of Superstable Fractional Discrete-Time Linear Systems

The concept of superstable discrete-time linear systems is given in Sect. 1.2.5.1. Consider the standard fractional discrete-time linear system

α xi+1 = Ad xi , i ∈ Z+ , 0 < α < 1,

(2.223)

136

2 Fractional Descriptor Linear Systems

where xi ∈ Rn is the state vector and Ad ∈ Rn×n . Substituting the definition of fractional difference (2.136) into (2.223) and limiting the upper bound of the sum by some natural number L we obtain xi+1 = Adα xi +

L 

c j xi− j ,

(2.224)

j=1

where Adα is given by (2.149) and x−k = 0, k = 1, 2, . . . . The solution to the equation (2.224) can be obtained using (2.150) for u i = 0, i ∈ Z+ , i.e. it has the form xi = d,i x0 ,

(2.225)

where the matrices d,i are determined by (2.148). Definition 2.18. The fractional discrete-time linear system (2.223) is called practically superstable for given length L of practical implementation if the fractional linear system (2.224) is superstable. The fractional discrete-time linear system (2.223) is called superstable if the fractional linear system (2.224) is superstable for L → ∞. Theorem 2.31. The fractional discrete-time linear system (2.224) is superstable if (

Adα ∈

d−

)

d 2 − 4c L d + ; 2

)

d 2 − 4c L 2

* for L ≥ 1

(2.226a)

or

Adα < 1 for L = 0, where d =1−

L−1 

cj.

(2.226b)

(2.226c)

j=1

Proof. According to Theorem 1.38 the fractional discrete-time linear system (2.223) will be superstable if xi+1 < xi and therefore

d,i+1

< 1, i ∈ Z+

d,i

(2.227)

since xi ≤ d,i

x0 . Given that $ $ $ L $ $ $ $ ¯

d,i+1 ≤ d,i

Adα + $ c j i− j $ $ $ j=1 $

(2.228)

2.2 Discrete-Time Systems

137

the inequality (2.227) can be rewritten in the form

d,i+1

≤ Adα +

d,i

$ $ $ $ $ $ L $ c j d,i− j $ $ $ j=1

d,i

$ ⎤$ ⎡ $ d,i−1 $ $ $ $ ⎥$ ⎢ $ ⎢ d,i−2 ⎥$ $ c1 In c2 In . . . c L In ⎢ .. ⎥$ $ ⎣ . ⎦$ $ $ $ d,i−L $ ⎤$ ⎡ ≤ Adα + $ $ d,i−1 $ $ $ $ ⎥$ ⎢ $ ⎢ d,i−2 ⎥$ $ Adα c1 In . . . c L−1 In ⎢ .. ⎥$ $ ⎣ . ⎦$ $ $ $ d,i−L $ L 

cj

j=1

≤ Adα +

(2.229)

Adα +

L−1 

< 1. cj

j=1

From (2.229) we have ⎛

Adα 2 + Adα ⎝

L−1 

⎞ c j − 1⎠ +

j=1

L 

cj −

j=1

L−1 

c j < 0.

(2.230)

j=1

Introducing (2.226c) and solving (2.230) with respect to Adα we obtain (2.226a). For a system with no delays (L = 0) from (2.230) we obtain the condition (2.226b).   Theorem 2.32. The fractional discrete-time linear system (2.224) is superstable for L → ∞ if

Adα < α. (2.231) Proof. In [88] it has been shown that ∞  j=1

c j = 1 − α.

(2.232)

138

2 Fractional Descriptor Linear Systems

From (2.232) and (2.230) for L → ∞ we have

Adα 2 − Adα α < 0 Solving (2.233) with respect to Adα we obtain (2.231).

(2.233)  

Therefore, the following theorem has been proved. Theorem 2.33. The fractional discrete-time linear system (2.223) is: 1) practically superstable for given length L of practical implementation if the condition (2.226) holds; 2) superstable if the condition (2.231) holds. Example 2.27. Consider the fractional discrete-time linear system (2.223) with α = 0.5 and   −0.45 0 Ad = . (2.234) 0.72 −0.4 In this case we have $ $ $ 0.05 0 $ $ = 0.82. $

Adα = Ad + I2 α = $ 0.72 0.1 $

(2.235)

From (2.226) and (2.231) we obtain the desired values of the norm of the matrix (2.235) for superstable systems:

Adα ∈ (0.1464; 0.8536)

Adα ∈ (0.0785; 0.7965) .. .

Adα ∈ (0; 0.5)

for L = 1, for L = 2, for L → ∞.

Therefore, from the above values and (2.235) we conclude that the considered system is practically superstable for the length of practical implementation L = 1. Taking into account that the presented approach gives only sufficient superstability conditions, we can not determine if the system is practically superstable for higher lengths of practical implementation (L > 1) or superstable (for L → ∞). In Fig. 2.4 we present time plots of the state variables and the norm of the state vector for x0 = [ 2 1 ]T and L = 1. We can see that the norm of the state vector decreases monotonically for i → ∞ and the system is practically superstable for L = 1. In Fig. 2.5 we show time plots of the norm of the state vector for x0 = [ 2 1 ]T and different values of the length of practical implementation L, from which we can conclude that the system is not practically superstable for L > 1 and therefore is also not superstable.

2.2 Discrete-Time Systems

139 2

2 x 1,i

||x i ||

x 2,i

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

4

0

5

0

1

2

3

4

5

i

i

Fig. 2.4 State variables (on the left) and norm of the state vector (on the right) of the fractional discrete-time linear system (2.223) with (2.234) and α = 0.5 for x0 = [ 2 1 ]T and L = 1 Fig. 2.5 Norm of the state vector of the fractional discrete-time linear system (2.223) with (2.234) and α = 0.5 for x0 = [ 2 1 ]T and different values of L

2 ||x i ||, L = 1 ||x i ||, L = 2 ||x i ||, L = 10

1.5

1

0.5

0 0

2

4

6

8

10

i

2.2.6.2

Laurent Series Expansion Method

In this section sufficient superstability conditions of the fractional descriptor discretetime linear system (2.138) will be established using the Laurent series expansion method. Consider the fractional descriptor discrete-time linear system (2.138). Solution to the Eq. (2.138a) for u i = 0, i ∈ Z+ is given by xi = Td,i E d x0 , where the matrices Td,k ∈ Rn×n are determined by (2.142b).

(2.236)

140

2 Fractional Descriptor Linear Systems

Definition 2.19. The fractional descriptor discrete-time linear system (2.138) is called practically superstable for given length L of practical implementation if the fractional linear system (2.141) is superstable. The fractional descriptor discrete-time linear system (2.138) is called superstable if the fractional linear system (2.141) is superstable for L → ∞. Theorem 2.34. The fractional descriptor discrete-time linear system (2.138) is: 1) practically superstable for given length L of practical implementation if

Td,i E > Td,i+1 E for i ∈ Z+ ;

(2.237)

Td,i E > Td,i+1 E for i ∈ Z+ and L → ∞.

(2.238)

2) superstable if

Proof. The proof can be accomplished in a similar way to the proof of Theorem 1.41.   Example 2.28. Consider the fractional descriptor discrete-time linear system (2.138) with α = 0.4, L = 1 and ⎡ ⎤ ⎡ ⎤ 0 −2 0 010 E d = ⎣ −3.3333 −5 0 ⎦ , Ad = ⎣ 1 0 0 ⎦ 0 −1 0 001 ⎡ ⎤ (2.239) 0 0.2 0 Adα = Ad + 0.4E d = ⎣ −0.3333 −2 0 ⎦ . 0 −0.4 1 The matrix pencil (E d , Ad ) of (2.239) is regular since det[E d λ − Ad ] = 0.3333(2λ + 1)(10λ + 3) = 0.

(2.240)

The matrix [E d z − Adα − E d c1 z −1 ]−1 has the form ⎡ ⎢ [E d z − Adα − 0.12E d z −1 ]−1 = ⎢ ⎣

1875z 3 −750z 2 −225z 15z 2500z 4 −625z 2 +36 −50z 2 +5z+6 − 50z 225z 0 +5z−6 25z 2 −10z−3 0 50z 2 +5z−6

0

⎤

⎥ 0⎥ ⎦.

(2.241)

1

Using (2.145) for n 0 = 1, the matrix (2.241) can be written in the form [E d z − Adα − 0.12E d z −1 ]−1 =

∞  k=−1

Td,k z −(k+1) ,

(2.242a)

2.2 Discrete-Time Systems

where

Td,k

⎤ ⎧⎡ ⎪ ⎪ 0 0 0 ⎪ ⎣ 0 0 0 ⎦ for k = −1, ⎪ ⎪ ⎪ ⎪ ⎨ 0.5 0 −1 ⎤ = ⎡ td,11 td,12 0 ⎪ ⎪ ⎪ ⎪ ⎣ td,21 0 0 ⎦ for k = 0, 1, 2, . . . , ⎪ ⎪ ⎪ ⎩ td,31 0 0

141

(2.242b)

td,11 = 0.4286(−0.4)k − 0.6429(0.4)k − 0.4821(−0.3)k + 0.8036(0.3)k , td,12 = 0.0857(0.4)k − 0.1286(−0.3)k , td,21 = −0.2857(−0.4)k − 0.2143(0.3)k , td,31 = −0.1429(−0.4)k − 0.1071(0.3)k . (2.242c) Using (2.239) and (2.242) we have ⎤ ⎧⎡ 000 ⎪ ⎪ ⎪ ⎣0 0 0⎦ ⎪ for k = −1, ⎪ ⎪ ⎪ ⎨ 000 ⎤ Td,k E d = ⎡ ¯ td,11 t¯d,12 0 ⎪ ⎪ ⎪⎣ ⎪ 0 t¯d,22 0 ⎦ for k = 0, 1, 2, . . . , ⎪ ⎪ ⎪ ⎩ 0 t¯d,32 0

(2.243a)

t¯d,11 = 0.2857(0.4)k + 0.4286(−0.3)k , t¯d,12 = 0.8572(0.4)k − 0.8572(−0.4)k + 1.6071(−0.3)k − 1.6071(0.3)k , t¯d,22 = 0.5714(−0.4)k + 0.4286(0.3)k ,

(2.243b)

t¯d,32 = 0.2857(−0.4)k + 0.2143(0.3)k .

From (2.243) it follows that the condition (2.237) is satisfied. Therefore, by Theorem 2.34 the considered system is practically superstable for the length of practical implementation L = 1. Performing similar analysis for L = 2, i.e. expanding in the Laurent series the matrix [E d z − Adα − c1 E d z −1 − c2 E d z −2 ]−1 it can be shown that the system is not practically superstable for L > 1 and therefore is also not superstable (the condition (2.238) is not satisfied). In Fig. 2.6 we present time plots of the state variables and the norm of the state vector for x0 = [ 1 2 1 ]T and L = 1. We can see that the norm of the state vector decreases monotonically for i → ∞ and the system is practically superstable for L = 1. In Fig. 2.7 we show time plots of the norm of the state vector for x0 = [ 1 2 1 ]T and different values of the length of practical implementation L, from which we can conclude that the system is not practically superstable for L > 1 and therefore is also not superstable.

142

2 Fractional Descriptor Linear Systems 2

2 x 1,i

||x i ||

x 2,i

1.5

x 3,i

1.5

1 1 0.5 0.5

0 -0.5

0

1

2

3

4

0

5

0

1

2

3

4

5

i

i

Fig. 2.6 State variables (on the left) and norm of the state vector (on the right) of the fractional descriptor discrete-time linear system (2.138) with (2.239) and α = 0.4 for x0 = [ 1 2 1 ]T and L=1 Fig. 2.7 Norm of the state vector of the fractional descriptor discrete-time linear system (2.138) with (2.239) and α = 0.4 for x0 = [ 1 2 1 ]T and different values of L

2 ||x i ||, L = 1 ||x i ||, L = 2 ||x i ||, L = 10

1.5

1

0.5

0 0

2

4

6

8

10

i

2.2.6.3

Drazin Inverse Matrix Method

In this section sufficient superstability conditions of the fractional descriptor discretetime linear system (2.157) (or equivalently (2.138)) will be established using the Drazin inverse matrix method. Consider the fractional descriptor discrete-time linear system (2.157). From (1.26b) and (2.162a) for u i = 0, i ∈ Z+ it follows that xi = E¯ d E¯ dD xi , i ∈ Z+ ,

(2.244)

2.2 Discrete-Time Systems

143

Premultiplying (2.207) by E¯ D and taking into account (2.244) we obtain xi+1

= E¯ dD A¯ dα xi +

L 

c j xi− j ,

(2.245)

j=1

where A¯ dα is given by (2.159). The Eq. (2.245) can also be written in the form xi+1 = H¯ d xi +

L 

c j xi− j ,

(2.246)

j=1

where

H¯ d = E¯ dD A¯ dα + G d (In − E¯ d E¯ dD )

(2.247)

since (2.244) holds and (In − E¯ d E¯ dD ) E¯ d E¯ dD = 0, G d ∈ Rn×n is an arbitrary matrix. Taking into consideration (2.162a) the solution to the equation (2.246) can be expressed by ¯ d,i x0 , (2.248a) xi =  where ¯ d,i+1 =  ¯ d,i H¯ d + 

L 

¯ d,i− j , cj

j=1

(2.248b)

¯ d,−k = 0, k = 1, . . . , h. ¯ d,0 = In ,  

Definition 2.20. The fractional descriptor discrete-time linear system (2.157) (or equivalently (2.138)) is called practically superstable for given length L of practical implementation if the fractional discrete-time linear system (2.246) is superstable. The fractional descriptor discrete-time linear system (2.157) (or equivalently (2.138)) is called superstable if the fractional linear system (2.246) is superstable for L → ∞. Theorem 2.35. The fractional discrete-time linear system (2.246) is superstable if there exists a matrix G d ∈ Rn×n such that (

H¯ d ∈ or

d−

* ) ) d 2 − 4c L d + d 2 − 4c L ; for L ≥ 1 2 2

H¯ d < 1 for L = 0,

(2.249a)

(2.249b)

where the coefficient d is given by (2.226c). Proof. The proof is similar to the proof of Theorem 2.31.

 

144

2 Fractional Descriptor Linear Systems

Theorem 2.36. The fractional discrete-time linear system (2.246) is superstable for L → ∞ if there exists a matrix G d ∈ Rn×n such that

H¯ d < α.

(2.250)

Proof. The proof can be accomplished in a similar way to the proof of Theorem 2.32.   Therefore, from the above considerations we have the following theorem. Theorem 2.37. The fractional descriptor discrete-time linear system (2.157) (or equivalently (2.138)) is: 1) practically superstable for given length L of practical implementation if there exists a matrix G d ∈ Rn×n such that the condition (2.249) holds; 2) superstable if there exists a matrix G d ∈ Rn×n such that the condition (2.250) holds. Example 2.29. (Continuation of Example 2.28) Consider the fractional descriptor discrete-time linear system (2.138) with α = 0.4 and (2.239). According to (2.240) the matrix pencil (E d , Ad ) is regular. Using (2.157b) for c = 0 we obtain ⎡

⎤ 3.3333 5 0 E¯ d = [−Ad ]−1 E d = ⎣ 0 2 0 ⎦ , 0 10 ⎡ ⎤ −1 0 0 A¯ d = [−Ad ]−1 Ad = ⎣ 0 −1 0 ⎦ , 0 0 −1 ⎡ ⎤ 0.3333 2 0 A¯ dα = A¯ d + 0.4 E¯ d = ⎣ 0 −0.2 0 ⎦ . 0 0.4 −1

(2.251)

Next, from (B.1) and (B.2) we have ⎡

E¯ dD

⎤ 0.3 −0.75 0 = ⎣ 0 0.5 0 ⎦ 0 0.25 0 ⎡

and E¯ dD A¯ dα

⎤ 0.1 0.75 0 = ⎣ 0 −0.1 0 ⎦ . 0 −0.05 0

(2.252)

Note that for G d = 0 the norm of the matrix

(2.253)

2.2 Discrete-Time Systems

H¯ d = E¯ dD A¯ dα

145

$⎡ ⎤$ $ 0.1 0.75 0 $ $ $ ⎣ ⎦$ + G d (I3 − E¯ d E¯ dD ) = $ $ 0 −0.1 0 $ = 0.85 $ 0 −0.05 0 $

(2.254)

takes its minimal value. From (2.249) and (2.250) we obtain the desired values of the norm of the matrix (2.254) for superstable systems

H¯ d ∈ (0.1394; 0.8606)

H¯ d ∈ (0.08; 0.8) .. .

H¯ d ∈ (0; 0.4)

for L = 1, for L = 2, for L → ∞.

Therefore, from the above values and (2.235) we conclude that the considered system is practically superstable for the length of practical implementation L = 1. Taking into account that the presented approach gives only sufficient superstability conditions, we can not determine if the system is practically superstable for higher lengths of practical implementation (L > 1) or superstable (for L → ∞). The time plots of the state variables and the norm of the state vector for x0 = [ 1 2 1 ]T and L = 1 are presented in Fig. 2.6. The time plots of the norm of the state vector for x0 = [ 1 2 1 ]T and different values of the length of practical implementation L are presented in Fig. 2.7. From these we can conclude that the considered system is not practically superstable for L > 1 and therefore is also not superstable.

2.2.6.4

Weierstrass-Kronecker Decomposition Method

In this section sufficient superstability conditions of the fractional descriptor discretetime linear system (2.177) (or equivalently (2.138)) will be established using the Weierstrass-Kronecker decomposition method. Note that every superstable system is also asymptotically stable. State vector of the subsystem (2.177b) x¯2,i = 0 for u i = 0, i ∈ Z+ . Using (2.175) and (2.180) we obtain     x¯ d,i x¯10 xi = Q d x¯i = Q d 1,i = Q d x¯2,i 0    d,i 0 x¯10 (2.255) = Qd 0 0  M,i   d,i 0 ¯ Q −1 = Qd d x 0 = d,i x 0 , 0  M,i where ¯ d,i+1 =  ¯ d,i A Md + 

L  j=1

¯ d,i− j , cj

(2.256a)

146

2 Fractional Descriptor Linear Systems

 A Md = Q d

A1dα 0 0 Md



Q −1 d

(2.256b)

and the matrix Md ∈ Rn 2 ×n 2 is arbitrary. Definition 2.21. The fractional descriptor discrete-time linear system (2.177) (or equivalently (2.138)) is called practically superstable for given length L of practical implementation if the fractional discrete-time linear system (2.214) is superstable. The fractional descriptor discrete-time linear system (2.177) (or equivalently (2.138)) is called superstable if the fractional linear system (2.214) is superstable for L → ∞. Taking into account (2.255), (2.256) and the approach given in Sect. 2.2.6.1 we obtain the following theorems. Theorem 2.38. The fractional discrete-time linear system (2.214) is superstable if there exists a matrix Md ∈ Rn×n such that (

A Md ∈

d−

)

d 2 − 4c L d + ; 2

* ) d 2 − 4c L for L ≥ 1 2

(2.257a)

or

A Md < 1 for L = 0,

(2.257b)

where the coefficient d is given by (2.226c). Theorem 2.39. The fractional discrete-time linear system (2.214) is superstable for L → ∞ if there exists a matrix Md ∈ Rn×n such that

A Md < α.

(2.258)

From Theorems 2.38 and 2.39 we have the resulting proposition. Theorem 2.40. The fractional descriptor discrete-time linear system (2.177) (or equivalently (2.138)) is: 1) practically superstable for given length L of practical implementation if there exists a matrix Md ∈ Rn×n such that the condition (2.257) holds; 2) superstable if there exists a matrix Md ∈ Rn×n such that the condition (2.258) holds. Example 2.30. (Continuation of Example 2.28) Consider the fractional descriptor discrete-time linear system (2.138) with α = 0.4 and (2.239). According to (2.240) the matrix pencil (E d , Ad ) is regular. In this case ⎡

⎤ ⎡ ⎤ −0.5 0 0 0 10 Pd = ⎣ 0.75 −0.3 0 ⎦ , Q d = ⎣ 1 0 0 ⎦ −0.5 0 1 0.5 0 1

(2.259)

2.2 Discrete-Time Systems

and

147

⎤ 100 = Pd E d Q d = ⎣ 0 1 0 ⎦ , 000 ⎡ ⎤   −0.5 0 0 A1d 0 = Pd Ad Q d = ⎣ 0.75 −0.3 0 ⎦ , 0 In 2 0 0 1 

In 1 0 0 N

⎡



(2.260)

where n 1 = 2 and n 2 = 1. Next we compute 

A1dα = A1d

 −0.1 0 + 0.4I2 = . 0.75 0.1

(2.261)

Using (2.256b) for (2.259), (2.261) and Md = [m d ] ∈ R1×1 we obtain ⎡

A Md

0 =⎣ 1 0.5 ⎡ 0.1 =⎣ 0 0

⎤⎡ ⎤⎡ 0 1 10 −0.1 0 0 0 0 ⎦ ⎣ 0.75 0.1 0 ⎦ ⎣ 1 0 0 −0.5 01 0 0 md ⎤ 0.75 0 −0.1 0 ⎦. −0.5m d − 0.05 m d

⎤ 0 0⎦ 1

(2.262)

From (2.257) and (2.258) we obtain the desired values of the norm of the matrix (2.262) for superstable systems

A Md ∈ (0.1394; 0.8606)

A Md ∈ (0.08; 0.8) .. .

A Md ∈ (0; 0.4)

for L = 1, for L = 2, for L → ∞.

From the above values for m d = 0 and A Md = 0.85 we conclude that the considered system is practically superstable for the length of practical implementation L = 1.

2.3 Concluding Remarks In this chapter fractional descriptor continuous-time and discrete-time linear systems have been analyzed. In the subsequent sections solutions of the state equation and analytical conditions of the positivity, stability and superstability have been established considering the Laurent series expansion method, Drazin inverse matrix method and Weierstrass-Kronecker decomposition method. The Caputo definition

148

2 Fractional Descriptor Linear Systems

of fractional derivative-integral and the Grünwald-Letnikov fractional (backward) difference operators have been used for continuous-time and discrete-time systems, respectively. Many of the conclusions given in Sect. 1.3 can also be applied here. The solution of the state equation does not depend on the choice of the method of the descriptor systems analysis. However, the considered methods differ in defining the analytical conditions for the positivity, stability and superstability. In the case of fractional continuous-time systems, the main difference compared to integer-order systems (which have been considered in Sect. 1.1) is the Mittag-Leffler function occurring in the solution of the state equation (Theorems 2.3–2.5). The Mittag-Leffler function is a generalization of the exponential function and it plays important role in the analysis of fractional differential equations. The conditions for the positivity (Theorems 2.7–2.9), stability (Theorems 2.10-2.15) and superstability (Theorems 2.17–2.19) are analogous to those presented in Sect. 1.1. We can see more differences in the case of fractional discrete-time systems. Applying the fractional difference operator we obtain the system with increasing number of delays in the state vector. The computational complexity of the Laurent series expansion method grows with the higher length of the practical implementation (i.e. with more delays in the state space model) and it is not recommended to use in the analysis of the fractional discrete-time linear systems, even though it is the only method for which necessary and sufficient conditions for the positivity (Theorem 2.24) and superstability (Theorem 2.34) have been established. Using the other two methods necessary and sufficient conditions for the positivity (Theorems 2.25 and 2.26) and sufficient conditions for superstability (Theorems 2.37 and 2.40) have been obtained. Note that for the Weierstrass-Kronecker decomposition method additional positivity condition is needed, i.e. the matrix Q (or Q d ) has to be a monomial matrix.

Chapter 3

Stability of Positive Descriptor Systems

In this chapter stability of positive descriptor systems will be analyzed. In the subsequent sections the following topics will be considered: stability tests for positive linear systems, stability of positive interval systems, stability of nonlinear systems with positive linear parts.

3.1 Stability Tests for Positive Linear Systems To test the stability of descriptor systems an approach based on the location of the roots of the characteristic equation can be used, which is discussed in Sects. 1.1.4, 1.2.4, 2.1.7, 2.2.5. However, if the system is a positive one, some other methods of stability analysis can be used [47, 57]. These methods are the subject of research in this section.

3.1.1 Continuous-Time Systems Consider the positive descriptor continuous-time linear system E x(t) ˙ = Ax(t),

(3.1)

where x(t) ∈ Rn+ is the state vector and E, A ∈ Rn×n . It is assumed that detE = 0 and det[Es − A] = 0 for some s ∈ C. Following the considerations presented in Sect. 1.1 the system (3.1) can be transformed into the following equivalent forms. 1) Using the Drazin inverse matrix method we obtain ¯ 1 (t), x˙1 (t) = E¯ D Ax x2 (t) = 0, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Kaczorek and K. Borawski, Descriptor Systems of Integer and Fractional Orders, Studies in Systems, Decision and Control 367, https://doi.org/10.1007/978-3-030-72480-1_3

(3.2) 149

150

3 Stability of Positive Descriptor Systems

¯ A¯ ∈ Rn×n have where x1 (t), x2 (t) ∈ Rn+ are given by (1.28) and the matrices E, the form (1.24b). Theorem 1.10 gives the positivity conditions of the system (3.2). 2) Using the Weierstrass-Kronecker decomposition method we have x¯˙1 (t) = A1 x¯1 (t), x¯2 (t) = 0,

(3.3)

where x¯1 (t) ∈ Rn+1 , x¯2 (t) ∈ Rn+2 are given by (1.53) and the matrix A1 ∈ Rn 1 ×n 1 is determined by (1.52). Theorem 1.11 gives the positivity conditions of the system (3.3). Theorem 3.1. The positive descriptor continuous-time linear system (3.1) is asymptotically stable if and only if all coefficients of the following characteristic equations 1) 2) 3) 4)

p(E,A) (s) = 0 given by (1.100), p( E, ¯ A) ¯ (s) = 0 given by (1.114), p E¯ D A¯ (s) = 0 given by (1.117), p A1 (s) = 0 given by (1.125).

are positive. Proof. First we shall show the proof for the equation (1.125). Necessity. The eigenvalues s1 , s2 , . . . , sn 1 of A1 are real or complex conjugate since the coefficients a˜ k , k = 1, . . . , n 1 of p A1 (s) are real. Hence if Resk < 0, k = 1, . . . , n 1 then all coefficients of the polynomial p A1 (s) = (s − s1 )(s − s2 ) . . . (s − sn 1 ) are positive, i.e. a˜ k > 0 for k = 1, . . . , n 1 . Sufficiency. This will be proved by contradiction. It is well-known [49] that if A1 is a Metzler matrix, then s¯ = max Resk is its eigenvalue and Resk < 0 if s¯ < 0. For k

a˜ k > 0, k = 1, . . . , n 1 and real s we have p A1 (s) = s n 1 + an 1 −1 s n 1 −1 + . . . + a1 s + a0 > 0 and A1 has no real nonnegative eigenvalue. Thus, we get the contradiction and s¯ < 0. By Lemmas 1.7–1.9 the characteristic polynomials p(E,A) (s), p( E, ¯ A) ¯ (s), p E¯ D A¯ (s) and p A1 (s) are related by (1.115), (1.117) and (1.126), respectively. Therefore, if one of the foregoing characteristic equations have only positive coefficients, then also every considered characteristic equation has only positive coefficients.   Theorem 3.2. The positive descriptor continuous-time linear system   (3.1) is asymptotically stable if and only if there exists a strictly positive vector λ¯ = λ¯ 1 . . . λ¯ n , λ¯ k > 0, k = 1, . . . , n such that [ E¯ D A¯ + G(In − E¯ E¯ D )]λ¯ < 0 for an arbitrary matrix G ∈ Rn×n

(3.4)

  or there exists a strictly positive vector λ = λ1 . . . λn 1 , λk > 0, k = 1, . . . , n 1 such that (3.5) A1 λ < 0.

3.1 Stability Tests for Positive Linear Systems

151

Proof. First we consider the condition (3.5). We shall show that if the system (3.3) is asymptotically stable then there exists a strictly positive vector λ ∈ Rn+1 satisfying (3.5). Integrating in the interval [0, ∞) the first equation of (3.3) we obtain ∞

x˙¯1 (t)dt = A1

∞

0

x¯1 (t)dt

(3.6)

0

and

∞ x¯1 (∞) − x¯1 (0) = A1

x¯1 (t)dt.

(3.7)

0

For an asymptotically stable positive system we have ∞ x¯1 (∞) = 0, x¯1 (0) > 0,

x¯1 (t)dt > 0.

(3.8)

0

Taking into account (3.7) and (3.8) we obtain (3.5) for ∞ λ=

x¯1 (t)dt.

(3.9)

0

It is well-known that the system (3.3) is asymptotically stable if and only if the corresponding transposed system x˙¯1 (t) = A1T x¯1 (t)

(3.10)

is asymptotically stable. As a Lyapunov function for the positive system (3.10) we choose the function (3.11) V (x¯1 (t)) = x¯1T (t)λ, which is positive for any nonzero x¯1 (t) ∈ Rn+ and V (x¯1 (t)) = 0 for x¯1 (t) = 0. Using (3.10) and (3.11) we obtain V˙ (x¯1 (t)) = x˙¯1T (t)λ = x¯1T (t)Aλ.

(3.12)

Therefore, if the condition (3.5) is satisfied then from (3.12) we have V˙ (x¯1 ) < 0 and the system (3.3) is asymptotically stable. The condition (3.4) can be proved in a similar way. The matrix G ∈ Rn×n and the term G(In − E¯ E¯ D ) eliminates from the matrix E¯ D A¯ unimportant entires that are further canceled through multiplication by x1 (t) ∈ Im E¯ E¯ D . Therefore, we have to find both: an arbitrary matrix G and strictly positive vector λ such that (3.4) holds.  

152

3 Stability of Positive Descriptor Systems

Example 3.1. Consider the descriptor continuous-time linear system (3.1) with ⎡

⎤ ⎡ ⎤ 0 10 0 −3 1 E = ⎣ 0.125 1 0 ⎦ , A = ⎣ −0.25 −2.75 1 ⎦ . 0 0 0.5 0 00

(3.13)

The matrix pencil (E, A) of (3.13) is regular since 0 s + 3 −1 det[Es − A] = 0.125s + 0.25s + 2.75 −1 0 0 −0.5

(3.14)

= 0.0625(s + 2)(s + 3)  = 0. The characteristic equation of (3.13) is given by p(E,A) (s) = det[Es − A] = 0.0625(s 2 + 5s + 6) = 0.

(3.15)

From (3.15) it follows that the pair (E, A) has two eigenvalues s1 = −2, s2 = −3 and by condition (1.101) the considered system is asymptotically stable. To check the positivity of the system we use two approaches as follows. 1) Drazin inverse matrix method case We choose c = 0 and using (1.24b) we obtain ⎡

⎤ 0.5 0.3333 0 E¯ = [−A]−1 E = ⎣ 0 0.3333 0 ⎦ , 0 0 0 ⎡ ⎤ −1 0 0 A¯ = [−A]−1 A = ⎣ 0 −1 0 ⎦ . 0 0 −1

(3.16)

Next, from (B.1) and (B.2) we have ⎡

E¯ D

and

⎤ 2 −2 0 = ⎣0 3 0⎦ 0 0 0

⎤ −2 2 0 E A¯ = ⎣ 0 −3 0 ⎦ . 0 0 0

(3.17)

⎡

¯D

(3.18)

By Theorem 1.10 the matrix E¯ D A¯ ∈ M3 and the system (3.13) is positive. Therefore, Theorems 3.1 and 3.2 can be used to test the stability of the system. From (3.16) and (3.18) we have

3.1 Stability Tests for Positive Linear Systems

153

¯ − A] ¯ = 0.1667(s 2 + 5s + 6) = 0, p( E, ¯ A) ¯ (s) = det[ Es ¯ = s(s 2 + 5s + 6) = 0. p E¯ D A¯ (s) = det[I3 s − E¯ D A]

(3.19)

Note that all coefficients of the characteristic equations p(E,A) (s), p( E, ¯ A) ¯ (s), p E¯ D A¯ (s) given by (3.15) and (3.19) are positive and by Theorem 3.1 the positive descriptor continuous-time linear system with (3.13) is asymptotically stable. We obtain the same result using Theorem 3.2 since for ⎡

⎤ 00 0 G = ⎣0 0 0 ⎦ 0 0 −1

(3.20)

and λ¯ = [ 2 1 1 ]T we have ⎡

⎤⎡ ⎤ −2 2 0 2 [ E¯ D A¯ + G(I3 − E¯ E¯ D )]λ¯ = ⎣ 0 −3 0 ⎦ ⎣ 1 ⎦ 0 0 −1 1 ⎡ ⎤ ⎡ ⎤ 2 0 = −⎣3⎦ < ⎣0⎦ 1 0

(3.21)

and the condition (3.4) is satisfied. 2) Weierstrass-Kronecker decomposition method case In this case

and

⎡

⎤ ⎡ ⎤ 1 0 −2 010 P = ⎣ −8 8 0 ⎦ , Q = ⎣ 1 0 0 ⎦ 0 0 1 002 ⎤ 100 In 1 0 = PEQ = ⎣0 1 0⎦, 0 N 000 ⎡ ⎤

 −3 0 0 A1 0 = P AQ = ⎣ 2 −2 0 ⎦ , 0 In 2 0 0 1



(3.22)

⎡

(3.23)

where n 1 = 2, n 2 = 1. By Theorem 1.11 the matrix A1 ∈ M2 and Q ∈ R3×3 + is monomial. Therefore, the system (3.13) is positive. From (3.23) we have p A1 (s) = det[I2 s − A1 ] = s 2 + 5s + 6 = 0.

(3.24)

Note that all coefficients of the characteristic equation p A1 (s) given by (3.24) are positive and by Theorem 3.1 the positive descriptor continuous-time linear system with (3.13) is asymptotically stable.

154

3 Stability of Positive Descriptor Systems

We obtain the same result using Theorem 3.2 since for λ = [ 1 2 ]T we have

−3 0 A1 λ = 2 −2

 

  1 3 0 =− < 2 2 0

(3.25)

and the condition (3.5) is satisfied.

3.1.2 Discrete-Time Systems Consider the positive descriptor discrete-time linear system E d xi+1 = Ad xi , i ∈ Z+ ,

(3.26)

where xi ∈ Rn+ is the state vector and E d , Ad ∈ Rn×n . It is assumed that detE d = 0 and det[E d z − Ad ] = 0 for some z ∈ C. Following the considerations presented in Sect. 1.2 the system (3.26) can be transformed into the following equivalent forms. 1) Using the Drazin inverse matrix method we obtain x1,i+1 = E¯ dD A¯ d x1,i , x2,i = 0,

(3.27)

¯ A¯ ∈ Rn×n have where x1,i , x2,i ∈ Rn+ are given by (2.160c) and the matrices E, the form (1.182b). Theorem 1.29 gives the positivity conditions of the system (3.27). 2) Using the Weierstrass-Kronecker decomposition method we have x¯1,i+1 = A1d x¯1,i , x¯2,i = 0,

(3.28)

where x¯1,i ∈ Rn+1 , x¯2,i ∈ Rn+2 are given by (1.193) and the matrix A1d ∈ Rn 1 ×n 1 is determined by (1.192). Theorem 1.31 gives the positivity conditions of the system (3.28). Theorem 3.3. The positive descriptor discrete-time linear system (3.26) is asymptotically stable if and only if all coefficients of the following characteristic equations 1) 2) 3) 4)

p(Ed ,Ad ) (z + 1) = p(Ed ,Ad −Ed ) (z) = det[E d (z + 1) − Ad ] = 0, p( E¯ d , A¯ d ) (z + 1) = p( E¯ d , A¯ d − E¯ d ) (z) = det[ E¯ d (z + 1) − A¯ d ] = 0, p E¯ dD A¯ d (z + 1) = (z + 1)n−r p( E¯ d , A¯ d ) (z + 1) = 0, p A1d (z + 1) = p A1d −In1 (z) = det[In 1 (z + 1) − A1d ] = 0

are positive, where r = rank E¯ dD A¯ d .

3.1 Stability Tests for Positive Linear Systems

155

Proof. First we shall show the proof for the fourth equation. From the equality p A1d −In1 (z) = det[In 1 z − A1d + In 1 ] = det[In 1 (z + 1) − A1d ] = p A1d (z + 1) (3.29) it follows that if z 1 , z 2 , . . . , z n 1 is a spectrum of A1d then z 1 − 1, z 2 − 1, . . . , z n 1 − 1 is the spectrum of A1d − In 1 . The real eigenvalues of A1d of the positive system (3.26) are located inside the unit circle if and only if the roots of the characteristic equation p A1d −In1 (z) = 0 have negative real parts. Therefore, the positive descriptor discretetime linear system (3.26) is asymptotically stable if and only if all coefficients of the equation p A1d −In1 (z) = 0 are positive. By Lemmas 1.12–1.14 the characteristic polynomials p(Ed ,Ad ) (z), p( E¯ d , A¯ d ) (z), p E¯ dD A¯ d (z) and p A1d (z) are related by (1.252), (1.253) and (1.257), respectively. Therefore, if one of the foregoing characteristic equations have only positive coefficients, then also every considered characteristic equation has only positive coefficients.   Theorem 3.4. The positive descriptor discrete-time linear system (3.26) is asymp totically stable if and only if there exists a strictly positive vector λ¯ = λ¯ 1 . . . λ¯ n , λ¯ k > 0, k = 1, . . . , n such that [ E¯ dD A¯ d + G d (In − E¯ d E¯ dD ) − In ]λ¯ < 0 for an arbitrary matrix G d ∈ Rn×n (3.30)   or there exists a strictly positive vector λ = λ1 . . . λn 1 , λk > 0, k = 1, . . . , n 1 such that (3.31) (A1d − In 1 )λ < 0. Proof. First we consider the condition (3.31). We shall show that if the system (3.28) is asymptotically stable then there exists a strictly positive vector λ ∈ Rn+1 satisfying (3.31). From (3.28) we have ∞

x¯1,i+1 = A1d

i=0

∞

x¯1,i .

(3.32)

i=0

For an asymptotically stable positive system we obtain x¯1,∞ = 0, x¯10 > 0,

∞

x¯1,i > 0.

(3.33)

i=0

Taking into account (3.32) and (3.33) we obtain (3.31) for λ=

∞ i=0

since from (3.32) and (3.34) we have

x¯1,i

(3.34)

156

3 Stability of Positive Descriptor Systems

λ − x¯10 = A1d λ.

(3.35)

It is well-known that the system (3.28) is asymptotically stable if and only if the corresponding transposed system T x¯1,i x¯1,i+1 = A1d

(3.36)

is asymptotically stable. As a Lyapunov function for the positive system (3.36) we choose the function T λ, (3.37) V (x¯1,i ) = x¯1,i which is positive for any nonzero x¯1,i ∈ Rn+ and V (x¯1,i ) = 0 for x¯1,i = 0. Using (3.36) and (3.37) we obtain T T T λ − x¯1,i λ = x¯1,i (A1d − In )λ. V (x¯1,i ) = V (x¯1,i+1 ) − V (x¯1,i ) = x¯1,i+1

(3.38)

Therefore, if the condition (3.31) is satisfied then from (3.38) we have V (x¯1,i ) < 0 and the system (3.28) is asymptotically stable. The condition (3.30) can be proved in a similar way. The matrix G d ∈ Rn×n and the term G d (In − E¯ d E¯ dD ) eliminates from the matrix E¯ dD A¯ d unimportant entires that are further canceled through multiplication by x1,i ∈ Im E¯ d E¯ dD . Therefore, we have to find both: an arbitrary matrix G d and strictly positive vector λ such that (3.30) holds.   Example 3.2. Consider the descriptor discrete-time linear system (3.26) with ⎡

⎤ ⎡ ⎤ 1 0 0 0.5 0.1 0 2 0 ⎦ , Ad = ⎣ 1 0.6 0 ⎦ . Ed = ⎣ 2 −0.8 −0.8 0 −0.4 −0.24 2

(3.39)

The matrix pencil (E d , Ad ) of (3.39) is regular since z − 0.5 −0.1 0 2z − 0.6 0 det[E d z − Ad ] = 2z − 1 −0.8z + 0.4 −0.8z + 0.24 −2

(3.40)

= −4z 2 + 2.8z − 0.4 = 0. The characteristic equation of (3.39) is given by p(Ed ,Ad ) (z) = det[E d z − Ad ] = −4z 2 + 2.8z − 0.4 = 0.

(3.41)

From (3.41) it follows that the pair (E d , Ad ) has two eigenvalues z 1 = 0.5, z 2 = 0.2 and by condition (1.239) the considered system is asymptotically stable. To check the positivity of the system we use two approaches as follows.

3.1 Stability Tests for Positive Linear Systems

157

1) Drazin inverse matrix method case Using (1.182b) for c = 0 we obtain ⎡

⎤ −2 1 0 E¯ d = [−Ad ]−1 E d = ⎣ 0 −5 0 ⎦ , 0 0 0 ⎡ ⎤ −1 0 0 A¯ d = [−Ad ]−1 Ad = ⎣ 0 −1 0 ⎦ . 0 0 −1

(3.42)

Next, from (B.1) and (B.2) we have ⎡

E¯ dD and

⎤ −0.5 −0.1 0 = ⎣ 0 −0.2 0 ⎦ 0 0 0

⎤ 0.5 0.1 0 E¯ dD A¯ d = ⎣ 0 0.2 0 ⎦ . 0 0 0

(3.43)

⎡

(3.44)

By Theorem 1.29 the matrix E¯ dD A¯ d ∈ R3×3 + and the system (3.39) is positive. Therefore, Theorems 3.3 and 3.4 can be used to test the stability of the system. From (3.39), (3.42) and (3.44) we have p(Ed ,Ad ) (z + 1) = det[E d (z + 1) − Ad ] = −4z 2 − 5.2z − 1.6 = 0 p( E¯ d , A¯ d ) (z + 1) = det[ E¯ d (z + 1) − A¯ d ] = 10z 2 + 13z + 4 = 0, p E¯ dD A¯ d (z + 1) = det[I3 (z + 1) − E¯ dD A¯ d ] = (z + 1)(z 2 + 1.3z + 0.4) = 0. (3.45) Note that all coefficients of the characteristic equations p( E¯ d , A¯ d ) (z + 1), p E¯ dD A¯ d (z + 1) given by (3.45) are positive. Although the characteristic equation p(Ed ,Ad ) (z + 1) has only negative coefficients, we can multiply it by (−1) to obtain positive coefficients. Therefore, by Theorem 3.3 the positive descriptor discrete-time linear system with (3.39) is asymptotically stable. We obtain the same result using Theorem 3.4 since for G d = 0 and λ¯ = [ 1 1 1 ]T we have ⎡ ⎤⎡ ⎤ −0.5 0.1 0 1 [ E¯ dD A¯ d + G d (I3 − E¯ d E¯ dD ) − I3 ]λ¯ = ⎣ 0 −0.8 0 ⎦ ⎣ 1 ⎦ 0 0 −1 1 ⎡ ⎤ ⎡ ⎤ (3.46) 0.4 0 = − ⎣ 0.8 ⎦ < ⎣ 0 ⎦ 1 0 and the condition (3.30) is satisfied.

158

3 Stability of Positive Descriptor Systems

2) Weierstrass-Kronecker decomposition method case In this case

and

⎡

⎤ ⎡ ⎤ 1 0 0 100 Pd = ⎣ −1 0.5 0 ⎦ , Q d = ⎣ 0 1 0 ⎦ 0 0.2 0.5 001 ⎤ 100 = Pd E d Q d = ⎣ 0 1 0 ⎦ , 000 ⎡ ⎤

 0.5 0.1 0 A1d 0 = Pd Ad Q d = ⎣ 0 0.2 0 ⎦ , 0 In 2 0 0 1

In 1 0 0 N



(3.47)

⎡

(3.48)

where n 1 = 2, n 2 = 1. By Theorem 1.31 the matrix A1d ∈ R2×2 and Q d ∈ R3×3 + + is monomial. Therefore, the system (3.39) is positive. From (3.48) we have p A1d (z + 1) = det[In 1 (z + 1) − A1d ] = z 2 + 1.3z + 0.4 = 0

(3.49)

Note that all coefficients of the characteristic equation p A1d (z + 1) given by (3.49) are positive and by Theorem 3.3 the positive descriptor discrete-time linear system with (3.39) is asymptotically stable. We obtain the same result using Theorem 3.4 since for λ = [ 1 1 ]T we have

(A1 − I2 )λ =

−0.5 0.1 0 −0.8

 

  1 0.4 0 =− < 1 0.8 0

(3.50)

and the condition (3.31) is satisfied.

3.1.3 Fractional Continuous-Time Systems Consider the fractional positive descriptor continuous-time linear system E 0 Dtα x(t) = Ax(t), 0 < α < 1,

(3.51)

where x(t) ∈ Rn+ , is the state vector, 0 Dtα is the Caputo derivative-integral operator defined by (2.8) and E, A ∈ Rn×n . It is assumed that detE = 0 and det[Es α − A] = 0 for some s α ∈ C. Following the considerations presented in Sect. 2.1 the system (3.51) can be transformed into the following equivalent forms.

3.1 Stability Tests for Positive Linear Systems

159

1) Using the Drazin inverse matrix method we obtain ¯ 1 (t), = E¯ D Ax x2 (t) = 0,

α 0 Dt x 1 (t)

(3.52)

¯ A¯ ∈ Rn×n where x1 (t), x2 (t) ∈ Rn+ are given by (2.29c) and the matrices E, have the form (2.28b). Theorem 2.8 gives the positivity conditions of the system (3.52). 2) Using the Weierstrass-Kronecker decomposition method we have α 0 Dt x¯ 1 (t)

= A1 x¯1 (t), x¯2 (t) = 0,

(3.53)

where x¯1 (t) ∈ Rn+1 , x¯2 (t) ∈ Rn+2 are given by (2.38) and the matrix A1 ∈ Rn 1 ×n 1 is determined by (2.37). Theorem 2.9 gives the positivity conditions of the system (3.3). Theorem 3.5. The fractional positive descriptor continuous-time linear system (3.51) is asymptotically stable if and only if all coefficients of the following characteristic equations 1) 2) 3) 4)

p(E,A) (s α ) = 0 given by (2.76), α p( E, ¯ A) ¯ (s ) = 0 given by (2.86), α p E¯ D A¯ (s ) = 0 given by (1.117), p A1 (s α ) = 0 given by (2.93).

are positive. Proof. The proof can be accomplished in a similar way to the proof of Theorem 3.1. Consider the Eq. (2.93). By condition (2.77) the stability region of a fractional continuous-time system can be extended to complex eigenvalues with positive real parts for α < 1. However, if A1 is a Metzler matrix, then its rightmost eigenvalue is real [49]. Therefore, for a fractional positive system the stability region is the same as for an integer-order system (α = 1) and A1 has no real nonnegative eigenvalue. α α By Lemmas 2.3–2.4 the characteristic polynomials p(E,A) (s α ), p( E, ¯ A) ¯ (s ), p E¯ D A¯ (s ) α and p A1 (s ) are related by (2.87), (1.117) and (2.94), respectively. Therefore, if one of the foregoing characteristic equations have only positive coefficients, then also every considered characteristic equation has only positive coefficients.   Theorem 3.6. The fractional positive descriptor continuous-time linear system (3.51) stable if and only if there exists a strictly positive vector  is asymptotically  λ¯ = λ¯ 1 . . . λ¯ n , λ¯ k > 0, k = 1, . . . , n such that [ E¯ D A¯ + G(In − E¯ E¯ D )]λ¯ < 0 for an arbitrary matrix G ∈ Rn×n

(3.54)

  or there exists a strictly positive vector λ = λ1 . . . λn 1 , λk > 0, k = 1, . . . , n 1 such that (3.55) A1 λ < 0.

160

3 Stability of Positive Descriptor Systems

Proof. The proof can be accomplished in a similar way to the proof of Theorem 3.2. Consider the inequality (3.55). Applying to the first equation of (3.53) the fractional integral defined by [57, 132, 135] α 0 It

we obtain

1 f (t) = (α)

t

(t − τ )α−1 f (τ )dτ

(3.56)

0

x¯1 (∞) − x¯1 (0) = A1 0 Itα x¯1 (t).

(3.57)

For an asymptotically stable positive system we have x¯1 (∞) = 0, x¯1 (0) > 0,

α 0 It x¯ 1 (t)

> 0.

(3.58)

From (3.57) and (3.58) we obtain (3.55) for λ = 0 Itα x¯1 (t).

(3.59)

Taking into account that the system (3.53) is asymptotically stable if and only if the corresponding transposed system α 0 Dt x¯ 1 (t)

= A1T x¯1 (t)

(3.60)

is asymptotically stable, the Lyapunov function V (x¯1 (t)) = x¯1T (t)λ

(3.61)

is chosen, which is positive for any nonzero x¯1 (t) ∈ Rn+ and V (x¯1 (t)) = 0 for x¯1 (t) = 0. From (3.60) and (3.61) we have α 0 Dt V ( x¯ 1 (t))

= 0 Dtα x¯1T (t)λ = x¯1 (t)T Aλ.

(3.62)

Therefore, if the condition (3.55) is satisfied then from (3.62) we have 0 Dtα V (x¯1 (t)) < 0 and the system (3.53) is asymptotically stable. The condition (3.54) can be proved similarly.   Example 3.3. Consider the fractional descriptor continuous-time linear system (3.51) with ⎡ ⎤ ⎡ ⎤ −1 −0.75 −0.75 210 E = ⎣ −2 −0.25 −0.25 ⎦ , A = ⎣ 4 0 1 ⎦ . (3.63) 0 −0.75 −0.75 021 The matrix pencil (E, A) of (3.63) is regular since

3.1 Stability Tests for Positive Linear Systems

161

−s − 2 −0.75s − 1 −0.75s det[Es α − A] = −2s − 4 −0.25s −0.25s − 1 0 −0.75s − 2 −0.75 − 1

(3.64)

= s 2α + 6s α + 8 = 0. The characteristic equation of (3.63) is given by p(E,A) (s α ) = det[Es α − A] = s 2α + 6s α + 8 = 0.

(3.65)

From (3.15) it follows that the pair (E, A) has two eigenvalues s1α = −2, s2α = −4 and by condition (2.77) the considered system is asymptotically stable for any α ∈ (0; 1). To check the positivity of the system we use the Drazin inverse matrix method. From (2.28b) for c = 0 we have ⎡

⎤ 0.5 0.125 0.125 0.5 ⎦ , E¯ = [−A]−1 E = ⎣ 0 0.5 0 −0.25 −0.25 ⎡ ⎤ −1 0 0 A¯ = [−A]−1 A = ⎣ 0 −1 0 ⎦ . 0 0 −1

(3.66)

Next, using one of the methods presented in Appendix B we obtain ⎡

E¯ D

and

⎤ 2 −1 −1 = ⎣0 8 8 ⎦ 0 −4 −4

⎤ −2 1 1 E¯ D A¯ = ⎣ 0 −8 −8 ⎦ . 0 4 4

(3.67)

⎡

(3.68)

By Theorem 2.8 there exists a matrix ⎡

⎤ 0 0 0 G = ⎣ 0 −4 0 ⎦ 0 0 −3 such that

⎤ −2 1 1 E¯ D A¯ + G(I3 − E¯ E¯ D ) = ⎣ 0 −4 0 ⎦ ∈ M3 0 1 −2

(3.69)

⎡

(3.70)

and the system (3.63) is positive. Therefore, Theorems 3.5 and 3.6 can be used to test the stability of the system. From (3.66) and (3.68) we have

162

3 Stability of Positive Descriptor Systems α ¯ α − A] ¯ = 0.125s 2α + 0.75s α + 1 = 0, p( E, ¯ A) ¯ (s ) = det[ Es ¯ = s α (s 2α + 6s α + 8) = 0. p E¯ D A¯ (s α ) = det[I3 s α − E¯ D A]

(3.71)

α Note that all coefficients of the characteristic equations p(E,A) (s α ), p( E, ¯ A) ¯ (s ), α p E¯ D A¯ (s ) given by (3.65) and (3.71) are positive and by Theorem 3.5 the fractional positive descriptor continuous-time linear system with (3.63) is asymptotically stable for any α ∈ (0; 1). We obtain the same result using Theorem 3.6 since for G given by (3.69) and λ = [ 2 1 1 ]T we obtain

⎡

⎤⎡ ⎤ −2 1 1 2 [ E A¯ + G(I3 − E¯ E )]λ = ⎣ 0 −4 0 ⎦ ⎣ 1 ⎦ 0 1 −2 1 ⎡ ⎤ ⎡ ⎤ 2 0 = −⎣4⎦ < ⎣0⎦ 1 0 ¯D

¯D

(3.72)

and the condition (3.54) is satisfied.

3.1.4 Fractional Discrete-Time Systems Consider the fractional positive descriptor discrete-time linear system E d α xi+1 = Ad xi , i ∈ Z+ , 0 < α < 1,

(3.73)

where xi ∈ Rn+ , is the state vector, α is the Grünwald-Letnikov fractional difference operator defined by (2.136) and E d , Ad ∈ Rn×n . It is assumed that detE d = 0 and det[E z − A] = 0 for some z ∈ C. Substituting (2.136) into (3.73) we get E d xi+1 = Adα xi +

i

E d c j xi− j ,

(3.74)

j=1

where Adα is given by (2.140b) and the coefficients c j are determined by (2.140c). Following the considerations presented in Sect. 2.2 the system (3.73) can be transformed into the following equivalent forms. 1) Using the Drazin inverse matrix method we obtain α x1,i+1 = E¯ dD A¯ d x1,i , x2,i = 0,

(3.75)

3.1 Stability Tests for Positive Linear Systems

163

where x1,i , x2,i ∈ Rn+ are given by (2.160c) and the matrices E¯ d , A¯ d ∈ Rn×n have the form (2.157b). Substituting (2.136) into (3.75) we get x1,i+1

= E¯ dD A¯ dα x1,i +

L

c j x1,i− j ,

j=1

(3.76)

x2,i = 0, where A¯ dα is given by (2.159), L is the length of practical implementation and x1,−k = 0, k = 1, 2, . . .. Theorem 2.25 gives the positivity conditions of the system (3.76). 2) Using the Weierstrass-Kronecker decomposition method we have α x¯1,i+1 = A1d x¯1,i , x¯2,i = 0,

(3.77)

where x¯1,i ∈ Rn+1 , x¯2,i ∈ Rn+2 are given by (2.175) and the matrix A1d ∈ Rn 1 ×n 1 is determined by (2.174). Substituting (2.136) into (3.77) we get x¯1,i+1 = A1dα x¯1,i +

L

c j x¯1,i− j ,

j=1

(3.78)

x¯2,i = 0, where A1dα is given by (2.179c), L is the length of practical implementation and x¯1,−k = 0, k = 1, 2, . . .. Theorem 2.26 gives the positivity conditions of the system (3.78). Remark 3.1. Note that the positive fractional system (3.76) (or (3.78)) is a system with increasing number of delays. It is well-known [23] that the asymptotic stability of the positive discrete-time linear systems with delays is independent of the numbers and values of the delays and it depends only on the sum of state matrices of the system. From Remark 3.1 we obtain the following theorems [23, 57, 88]. Theorem 3.7. The positive fractional system (3.76) is asymptotically stable if and only if the positive system x1,i+1 = D¯ 1 x1,i , (3.79) x2,i = 0 with D¯ 1 = E¯ dD A¯ dα +

L j=1

is asymptotically stable.

c j In ∈ Rn×n

(3.80)

164

3 Stability of Positive Descriptor Systems

Theorem 3.8. The positive fractional system (3.78) is asymptotically stable if and only if the positive system x¯1,i+1 = D1 x1,i , (3.81) x¯2,i = 0 with D1 = A1dα +

L

c j In 1 ∈ Rn 1 ×n 1

(3.82)

j=1

is asymptotically stable. Theorem 3.9. The fractional positive descriptor discrete-time linear system (3.73) is practically stable for given length L of practical implementation if and only if all coefficients of the following characteristic equations 1) 2) 3) 4)

p(Ed ,D) (z + 1) = p(Ed ,D−Ed ) (z) = det[E d (z + 1) − D] = 0, ¯ d (z + 1) − D] ¯ = 0, p( E¯ d , D) ¯ (z + 1) = p( E¯ d , D− ¯ E¯ d ) (z) = det[ E (z + 1) = 0, p D¯ 1 (z + 1) = (z + 1)n−r p( E¯ d , D) ¯ p D1 (z + 1) = p D1 −In1 (z) = det[In 1 (z + 1) − D1 ] = 0

are positive, where D = Adα +

L

c j E d ∈ Rn×n ,

(3.83a)

c j E¯ d ∈ Rn×n ,

(3.83b)

j=1

D¯ = A¯ dα +

L j=1

D¯ 1 , D1 are given by (3.80) and (3.82), respectively and r = rank E¯ dD A¯ dα . Proof. The proof follows immediately from Theorem 3.3. The equivalence of the above characteristic equations can be shown using Lemmas 1.8, 2.6 and 2.7.   Theorem 3.10. The positive descriptor discrete-time linear system (3.26) is practically stable for given length L of practical implementation if and only if there exists  a strictly positive vector λ¯ = λ¯ 1 . . . λ¯ n , λ¯ k > 0, k = 1, . . . , n such that [ D¯ 1 + G d (In − E¯ d E¯ dD ) − In ]λ¯ < 0 for an arbitrary matrix G d ∈ Rn×n

(3.84)

  or there exists a strictly positive vector λ = λ1 . . . λn 1 , λk > 0, k = 1, . . . , n 1 such that (3.85) (D1 − In 1 )λ < 0, where D¯ 1 , D1 are given by (3.80) and (3.82), respectively. Proof. The proof follows immediately from Theorem 3.4.

 

3.1 Stability Tests for Positive Linear Systems

165

To test the asymptotic stability of the positive fractional system (3.73) we assume that L → ∞. Lemma 3.1. For L → ∞ the matrices (3.80), (3.82) and (3.83) have the forms D∞ = A d + E d , D¯ ∞ = A¯ d + E¯ d , D¯ 1,∞ = E¯ dD A¯ d + In − (In − E¯ d E¯ dD )α,

(3.86)

D1,∞ = A1d + In 1 . Proof. Taking into account (2.232) from (3.80), (3.82) and (3.83) for L → ∞ we have D∞ = Adα +

∞

c j E d = Ad + E d α + E d (1 − α) = Ad + E d ,

j=1

D¯ ∞ = A¯ dα +

∞

c j E¯ d = A¯ d + E¯ d α + E¯ d (1 − α) = A¯ d + E¯ d ,

j=1

D¯ 1,∞ = E¯ dD A¯ dα +

∞

(3.87) c j In = E¯ dD A¯ d + E¯ d E¯ dD α + In (1 − α),

j=1

D1,∞ = A1dα +

∞

c j In 1 = A1d + In 1 α + In 1 (1 − α) = A1d + In 1 ,

j=1

 

which is equivalent to (3.86).

Theorem 3.11. The fractional positive descriptor discrete-time linear system (3.73) is asymptotically stable if and only if all coefficients of the following characteristic equations 1) 2) 3) 4)

p(Ed ,D∞ ) (z + 1) = det[E d (z + 1) − D∞ ] = det[E d z − Ad ] = p(Ed ,Ad ) (z) = 0, p( E¯ d , D¯ ∞ ) (z + 1) = det[ E¯ d (z + 1) − D¯ ∞ ] = det[ E¯ d z − A¯ d ] = p( E¯ d , A¯ d ) (z) = 0, p D¯ 1,∞ (z + 1) = det[In (z + 1) − D¯ 1,∞ ] = β1 (z + α)n−r p( E¯ d , A¯ d ) (z) = 0, p D1,∞ (z + 1) = det[In 1 (z + 1) − D1 ] = det[In 1 z − A1d ] = p A1d (z) = 0

are positive, where D∞ , D¯ ∞ , D¯ 1,∞ , D1,∞ are given by (3.86), r = rank E¯ dD A¯ d and β ∈ R is some scalar. Proof. The proof for Eqs. (1), (2) and (4) follows immediately from Theorem 3.9 and Lemma 3.1. For the Eq. (3) we shall show that the equality p D¯ 1,∞ (z + 1) = det[In (z + 1) − D¯ 1,∞ ] = det[In z − E¯ dD A¯ d + (In − E¯ d E¯ dD )α] 1 = (z + α)n−r p( E¯ d , A¯ d ) (z) β

(3.88)

166

3 Stability of Positive Descriptor Systems

holds, where β ∈ R is some scalar. Using (1.27d) and (3.88) we obtain p D¯ 1,∞ (z + 1) = det[In (z + 1) − D¯ 1,∞ ] = det[In z − E¯ dD A¯ d + (In − E¯ d E¯ dD )α]  



0 In 1 z − Jd−1 A1d −1 T = det T 0 In 2 (z + α)

(3.89)

= (z + α)n 2 det[In 1 z − Jd−1 A1d ] and

p( E¯ d , A¯ d ) (z) = det[ E¯ d z − A¯ d ]  



0 Jd z − A1d T −1 = det T 0 N z − A2d

(3.90)

= det[Jd z − A1d ]det[N z − A2d ] = βdet[Jd z − A1d ] = βdet[In 1 z − Jd−1 A1d ] since detT detT −1 = In and det[N z − A2d ] = β. Taking into account that n 2 = n − r from (3.89) and (3.90) we have (3.88). This completes the proof.   Theorem 3.12. The positive descriptor discrete-time linear system (3.26) is asymp totically stable if and only if there exists a strictly positive vector λ¯ = λ¯ 1 . . . λ¯ n , λ¯ k > 0, k = 1, . . . , n such that [ E¯ dD A¯ d + (G d − In α)(In − E¯ d E¯ dD )]λ¯ < 0 for an arbitrary matrix G d ∈ Rn×n (3.91)   or there exists a strictly positive vector λ = λ1 . . . λn 1 , λk > 0, k = 1, . . . , n 1 such that (3.92) A1d λ < 0, Proof. The proof follows immediately from Theorem 3.10 and Lemma 3.1.

 

Example 3.4. Consider the fractional descriptor discrete-time linear system (3.73) with α = 0.4 and ⎡ ⎤ ⎡ ⎤ 0 −5 0 010 (3.93) E d = ⎣ −5 −10 0 ⎦ , Ad = ⎣ 1 0 0 ⎦ . 0 −7.5 0 001 The matrix pencil (E d , Ad ) of (3.93) is regular since 0 −5z − 1 0 det[E z − A] = −5z − 1 −10z 0 = 25z 2 + 10z + 1 = 0. 0 −7.5z −1

(3.94)

3.1 Stability Tests for Positive Linear Systems

167

To check the asymptotic stability of the system (3.93) we can use the approach given in Sect. 2.2.5. Therefore, by Theorem 2.28 we have to compute the characteristic equation given by   det E d (z − 1)0.4 z 0.6 − Ad = [5(z − 1)0.4 z 0.6 + 1]2 = 0.

(3.95)

The system (3.93) with α = 0.4 is asymptotically stable since roots of (3.95) are z 1 = z 2 = −0.0656. To check the positivity of the system we use the Drazin inverse matrix method. From (2.157b) for c = 0 we have ⎡

⎤ 5 10 0 E¯ d = [−Ad ]−1 E d = ⎣ 0 5 0 ⎦ , 0 7.5 0 ⎡ ⎤ −1 0 0 A¯ d = [−Ad ]−1 Ad = ⎣ 0 −1 0 ⎦ , 0 0 −1 ⎡ ⎤ 14 0 A¯ dα = A¯ d + 0.4 E¯ d = ⎣ 0 1 0 ⎦ . 0 3 −1

(3.96)

Next, using (B.1) and (B.2) we obtain ⎡

E¯ dD and

⎤ 0.2 −0.4 0 = ⎣ 0 0.2 0 ⎦ 0 0.3 0

⎤ ⎡ ⎤ −0.2 0.4 0 0.2 0.4 0 E¯ dD A¯ d = ⎣ 0 −0.2 0 ⎦ , E¯ dD A¯ dα = ⎣ 0 0.2 0 ⎦ . 0 −0.3 0 0 0.3 0

(3.97)

⎡

(3.98)

and the system (3.93) is By Theorem 2.25 there exists the matrix E¯ dD A¯ dα ∈ R3×3 + positive. Therefore, Theorems 3.11 and 3.12 can be used to test the stability of the system. From (3.86) we have ⎡

D∞ D¯ ∞ D¯ 1,∞

⎤ 0 −4 0 = Ad + E d = ⎣ −4 −10 0 ⎦ , 0 −7.5 1 ⎡ ⎤ 4 10 0 = A¯ d + E¯ d = ⎣ 0 4 0 ⎦ , 0 7.5 −1

(3.99) ⎡

⎤ 0.8 0.4 0 = E¯ dD A¯ d + I3 − 0.4(I3 − E¯ d E¯ dD ) = ⎣ 0 0.8 0 ⎦ . 0 0.3 0.6

168

3 Stability of Positive Descriptor Systems

Using (3.93), (3.66) and (3.99) we obtain p(Ed ,D∞ ) (z + 1) = det[E d (z + 1) − D∞ ] = 25z 2 + 10z + 1 = 0, p( E¯ , D¯ ) (z + 1) = det[ E¯ d (z + 1) − D¯ ∞ ] = 25z 2 + 10z + 1 = 0, d

∞

p D¯ 1,∞ (z + 1) = det[In (z + 1) − D¯ 1,∞ ] = 0.04(z + 0.4)(25z 2 + 10z + 1) = 0. (3.100) Note that all coefficients of the characteristic equations p(Ed ,D∞ ) (z + 1) = p(Ed ,Ad ) (z), p( E¯ d , D¯ ∞ ) (z + 1) = p( E¯ d , A¯ d ) (z), p D¯ 1,∞ (z + 1) = β1 (z + α) p( E¯ d , A¯ d ) (z) given by (3.100) are positive and by Theorem 3.11 the fractional positive descriptor discretetime linear system with α = 0.4 and (3.93) is asymptotically stable. We obtain the same result using Theorem 3.12 since for G d = 0 and λ¯ = [ 3 1 1 ]T we obtain ⎡

⎤⎡ ⎤ −0.2 0.4 0 3 [ E¯ dD A¯ d + (G d − In α)(In − E¯ d E¯ dD )]λ¯ = ⎣ 0 −0.2 0 ⎦ ⎣ 1 ⎦ 0 0.3 −0.4 1 ⎡ ⎤ ⎡ ⎤ 0.2 0 = − ⎣ 0.2 ⎦ < ⎣ 0 ⎦ 0.1 0

(3.101)

and the condition (3.91) is satisfied.

3.2 Stability of Positive Interval Systems In this section necessary and sufficient stability conditions for positive interval linear systems will be established and Kharitonov theorem [98] will be extended to positive descriptor interval linear systems. It will be shown that such systems are asymptotically stable if and only if the respective lower and upper bound systems are asymptotically stable [72–74].

3.2.1 Continuous-Time Systems Consider the positive descriptor continuous-time linear system (3.1) with the interval matrix A ∈ Rn×n defined by A L ≤ A ≤ AU or equivalently A ∈ [A L , AU ].

(3.102)

It is assumed that det[Es − A L ] = 0 and det[Es − AU ] = 0 for some s ∈ C.

(3.103)

3.2 Stability of Positive Interval Systems

169

Following the considerations presented in Sect. 3.1.1 the interval system (3.1) can be transformed into two equivalent forms: 1) Using Drazin inverse matrix method we obtain the interval system (3.2), where E¯ ∈ [ E¯ L , E¯ U ], A¯ ∈ [ A¯ L , A¯ U ] and E¯ L = [Ec − A L ]−1 E, E¯ U = [Ec − AU ]−1 E A¯ L = [Ec − A L ]−1 A L , A¯ U = [Ec − AU ]−1 AU .

(3.104)

The matrix E¯ D A¯ in (3.2) will be denoted as A¯ 1 . Therefore, we have A¯ 1 ∈ [ A¯ 1L , A¯ 1U ] and A¯ 1L = E¯ LD A¯ L , A¯ 1U = E¯ UD A¯ U . 2) Using Weierstrass-Kronecker decomposition method we obtain the interval system (3.3), where the matrix A1 ∈ [A1L , A1U ] is determined by

P1 A L Q 1 =

 

A1L 0 A1U 0 , P2 AU Q 2 = 0 In 2 0 In 2

(3.105)

for two pairs of nonsingular matrices (P1 , Q 1 ), (P2 , Q 2 ). Definition 3.1. The positive descriptor interval continuous-time linear system (3.1) is called asymptotically stable if the system is asymptotically stable for all matrices E, A, where A ∈ [A L , AU ]. Definition 3.2. [72–74] The matrix A = (1 − k)A L + k AU , 0 ≤ k ≤ 1

(3.106)

is called the convex linear combination of matrices A L and AU . According to Definition 3.2 we can also define convex linear combinations of matrices A¯ 1L , A¯ 1U and A1L , A1U , i.e. A¯ 1 = (1 − k) A¯ 1L + k A¯ 1U , 0 ≤ k ≤ 1,

(3.107a)

A1 = (1 − k)A1L + k A1U , 0 ≤ k ≤ 1.

(3.107b)

Theorem 3.13. The convex linear combinations (3.107) are asymptotically stable if and only if: 1) the matrices A¯ 1L and A¯ 1U are asymptotically stable; 2) the matrices A1L and A1U are asymptotically stable. Proof. If the matrices A¯ 1L and A¯ 1U are asymptotically stable, then by (3.4) there exist matrices G L ∈ Rn×n , G U ∈ Rn×n and strictly positive vector λ¯ ∈ Rn+ such that

170

and

3 Stability of Positive Descriptor Systems

[ A¯ 1L + G L (In − E¯ L E¯ LD )]λ¯ < 0

(3.108a)

[ A¯ 1U + G U (In − E¯ U E¯ UD )]λ¯ < 0.

(3.108b)

From (3.107a) and (3.108) we obtain [ A¯ 1 + G(In − E¯ E¯ D )]λ¯ = {(1 − k)[ A¯ 1L + G L (In − E¯ L E¯ LD )] + k[ A¯ 1U + G U (In − E¯ U E¯ UD )]}λ¯ = (1 − k)[ A¯ 1L + G L (In − E¯ L E¯ LD )]λ¯ + k[ A¯ 1U + G U (In − E¯ U E¯ UD )]λ¯ < 0

(3.109)

for 0 ≤ k ≤ 1. Therefore, if the matrices A¯ 1L and A¯ 1U are asymptotically stable, then the convex linear combination (3.107a) is also asymptotically stable. The proof for the convex linear combination (3.107b) can be performed in a similar way. If the matrices A1L and A1U are asymptotically stable, then by (3.5) there exist strictly positive vector λ ∈ Rn+1 such that A1L λ < 0 and A1U λ < 0.

(3.110)

From (3.107b) and (3.110) we obtain A1 λ = [(1 − k)A1L + k A1U ]λ = (1 − k)A1L λ + k A1U λ < 0

(3.111)

for 0 ≤ k ≤ 1. Therefore, if the matrices A1L and A1U are asymptotically stable, then the convex linear combination (3.107b) is also asymptotically stable. The necessity follows immediately from the fact that k can be equal to zero and one.   Theorem 3.14. The positive descriptor interval continuous-time linear system (3.1) with (3.102) is asymptotically stable if and only if: 1) (Drazin inverse matrix method case) the matrices A¯ 1L and A¯ 1U are asymptotically stable; 2) (Weierstrass-Kronecker decomposition method case) the matrices A1L and A1U are asymptotically stable. Proof. By (3.4) the matrices A¯ 1L and A¯ 1U are asymptotically stable if and only if there exist G L ∈ Rn×n , G U ∈ Rn×n and strictly positive vector λ¯ ∈ Rn+ such that (3.108) holds. The convex linear combination (3.107a) satisfies the condition [ A¯ 1 + G(In − E¯ E¯ D )]λ¯ < 0 if and only if (3.108) holds. Therefore, the positive descriptor interval continuous-time linear system (3.1) with (3.102) is asymptotically stable if and only if the matrices A1L and A1U are asymptotically stable. The proof for the Weierstrass-Kronecker decomposition method case can be performed in a similar way.  

3.2 Stability of Positive Interval Systems

171

To test the stability of an interval system the Kharitonov theorem can also be used. It is recalled below and then extended to positive descriptor interval continuous-time linear systems. Consider the set (family) of the n-th degree polynomials pn (s) = an s n + an−1 s n−1 + . . . + a1 s + a0

(3.112a)

with interval coefficients a i ≤ ai ≤ a i , i = 0, 1, . . . , n.

(3.112b)

Using (3.112a) we define the following four polynomials: p1n (s) = a 0 + a 1 s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5 + . . . , p2n (s) = a 0 + a 1 s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5 + . . . , p3n (s) = a 0 + a 1 s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5 + . . . ,

(3.113)

p4n (s) = a 0 + a 1 s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5 + . . . . Theorem 3.15 (Kharitonov theorem). The set of polynomials (3.112) is asymptotically stable if and only if the four polynomials (3.113) are asymptotically stable. Theorem 3.16. The positive descriptor interval continuous-time linear system (3.1) with the characteristic equations 1) 2) 3) 4)

p(E,A) (s) = 0 given by (1.100), p( E, ¯ A) ¯ (s) = 0 given by (1.114), p A¯ 1 (s) = 0 given by (1.117), p A1 (s) = 0 given by (1.125)

with interval coefficients a i ≤ ai ≤ a i is asymptotically stable if and only if the lower bounds a i , i = 0, 1, . . . , r are positive, where r = rank A¯ 1 = rank A1 . Proof. For each of the above characteristic equations we can define four characteristic equations as in (3.113). If the lower bounds a i are positive, then the upper bounds a i are also positive. Therefore, by Theorem 3.1 and Kharitonov theorem (Theorem 3.15) the positive descriptor interval continuous-time linear system (3.1) is asymptotically stable.   Example 3.5. Consider the descriptor interval continuous-time linear system (3.1) with A ∈ [A L , AU ] and

172

3 Stability of Positive Descriptor Systems

⎡

⎤ ⎡ ⎤ 0 10 0 −3 1 E = ⎣ 0.125 1 0 ⎦ , A L = ⎣ −0.25 −2.75 1 ⎦ , 0 00 0 0 0.5 ⎡ ⎤ 0 −4 1 AU = ⎣ −0.5 −3 1 ⎦ . 0 0 1

(3.114)

It is easy to check that the matrix pencils (E, A L ), (E, AU ) of (3.114) are regular. To check the stability of the interval system (3.1) with (3.114) we use two approaches as follows. 1) Drazin inverse matrix method case From (3.104) for c = 0 we have ⎡

⎤ ⎡ 0.5 0.3333 0 0.25 0.5 E¯ L = ⎣ 0 0.3333 0 ⎦ , E¯ U = ⎣ 0 0.25 0 0 0 0 0 ⎡ ⎤ −1 0 0 A¯ L = A¯ U = ⎣ 0 −1 0 ⎦ . 0 0 −1

⎤ 0 0⎦, 0

(3.115)

Next, from (B.1) and (B.2) we have ⎡

E¯ LD

⎤ ⎡ ⎤ 2 −2 0 4 −8 0 = ⎣ 0 3 0 ⎦ , E¯ UD = ⎣ 0 4 0 ⎦ 0 0 0 0 0 0 ⎡

and A¯ 1L

A¯ 1U

⎤ −2 2 0 = E¯ LD A¯ L = ⎣ 0 −3 0 ⎦ , 0 0 0 ⎡ ⎤ −4 8 0 D = E¯ U A¯ U = ⎣ 0 −4 0 ⎦ . 0 0 0

(3.116)

(3.117)

By Theorem 1.10 the matrices A¯ 1L ∈ M3 , A¯ 1U ∈ M3 and the interval system (3.1) with (3.114) is positive. The characteristic equations of (3.117) are given by p A¯ 1L (s) = s(s 2 + 5s + 6) = 0, p A¯ 1U (s) = s(s 2 + 8s + 16) = 0.

(3.118)

Therefore, by Theorem 3.1 both matrices (3.117) are asymptotically stable and by Theorem 3.14 the interval system (3.1) with (3.114) is asymptotically stable.

3.2 Stability of Positive Interval Systems

173

Also, by Theorem 3.16 the lower bounds of (3.118) are positive and the considered interval system is asymptotically stable. 2) Weierstrass-Kronecker decomposition method case ⎡

⎤ ⎡ ⎤ 1 0 −2 010 P1 = ⎣ −8 8 0 ⎦ , Q 1 = ⎣ 1 0 0 ⎦ , 0 0 1 002 ⎡ ⎤ ⎡ ⎤ 1 0 −1 010 P2 = ⎣ −8 8 0 ⎦ , Q 2 = ⎣ 1 0 0 ⎦ . 0 0 1 001

In this case

(3.119)

From (3.105) and (3.115) we obtain

A1L =

  −3 0 −4 0 , A1U = . 2 −2 8 −4

(3.120)

3×3 By Theorem 1.11 the matrices A1L ∈ M2 , A1U ∈ M2 and Q 1 ∈ R3×3 + , Q 2 ∈ R+ are monomial. Therefore, the interval system (3.1) with (3.114) is positive. The characteristic equations of (3.120) are given by

p A1L (s) = s 2 + 5s + 6 = 0, p A1U (s) = s 2 + 8s + 16 = 0.

(3.121)

Therefore, by Theorem 3.1 both matrices (3.120) are asymptotically stable and by Theorem 3.14 the interval system (3.1) with (3.114) is asymptotically stable. Also, by Theorem 3.16 the lower bounds of (3.121) are positive and the considered interval system is asymptotically stable.

3.2.2 Discrete-Time Systems Consider the positive descriptor discrete-time linear system (3.26) with the interval matrix Ad ∈ Rn×n defined by Ad L ≤ Ad ≤ AdU or equivalently Ad ∈ [Ad L , AdU ].

(3.122)

It is assumed that det[E d z − Ad L ] = 0 and det[E d z − AdU ] = 0 for some z ∈ C.

(3.123)

Following the considerations presented in Sect. 3.1.2 the interval system (3.26) can be transformed into two equivalent forms:

174

3 Stability of Positive Descriptor Systems

1) Using Drazin inverse matrix method we obtain the interval system (3.27), where E¯ d ∈ [ E¯ d L , E¯ dU ], A¯ d ∈ [ A¯ d L , A¯ dU ] and E¯ d L = [E d c − Ad L ]−1 E d , E¯ dU = [E d c − AdU ]−1 E d A¯ d L = [E d c − Ad L ]−1 Ad L , A¯ dU = [E d c − AdU ]−1 AdU .

(3.124)

The matrix E¯ dD A¯ d in (3.27) will be denoted as A¯ 1d . Therefore, we have A¯ 1d ∈ D ¯ [ A¯ 1d L , A¯ 1dU ] and A¯ 1d L = E¯ dDL A¯ d L , A¯ 1dU = E¯ dU AdU . 2) Using Weierstrass-Kronecker decomposition method we obtain the interval system (3.28), where the matrix A1d ∈ [A1d L , A1dU ] is determined by

P1d Ad L Q 1d =

 

A1d L 0 A1dU 0 , P2d AdU Q 2d = 0 In 2 0 In 2

(3.125)

for two pairs of nonsingular matrices (P1d , Q 1d ), (P2d , Q 2d ). Definition 3.3. The positive descriptor interval discrete-time linear system (3.26) is called asymptotically stable if the system is asymptotically stable for all matrices E d , Ad , where Ad ∈ [Ad L , AdU ]. According to Definition 3.2 we can define convex linear combinations of matrices A¯ 1d L , A¯ 1dU and A1d L , A1dU , i.e. A¯ 1d = (1 − k) A¯ 1d L + k A¯ 1dU , 0 ≤ k ≤ 1,

(3.126a)

A1d = (1 − k)A1d L + k A1dU , 0 ≤ k ≤ 1.

(3.126b)

Theorem 3.17. The convex linear combinations (3.126) are asymptotically stable if and only if: 1) the matrices A¯ 1d L and A¯ 1dU are asymptotically stable; 2) the matrices A1d L and A1dU are asymptotically stable. Proof. If the matrices A¯ 1d L and A¯ 1dU are asymptotically stable, then by (3.30) there exist matrices G d L ∈ Rn×n , G dU ∈ Rn×n and strictly positive vector λ¯ ∈ Rn+ such that (3.127a) [ A¯ 1d L + G d L (In − E¯ d L E¯ dDL ) − In ]λ¯ < 0 and

D ) − In ]λ¯ < 0. [ A¯ 1dU + G dU (In − E¯ dU E¯ dU

From (3.126a) and (3.127) we obtain

(3.127b)

3.2 Stability of Positive Interval Systems

175

[ A¯ 1d + G d (In − E¯ d E¯ dD ) − In ]λ¯

= {(1 − k)[ A¯ 1d L + G d L (In − E¯ d L E¯ dDL )] D +k[ A¯ 1dU + G dU (In − E¯ dU E¯ dU )] − In }λ¯ (3.128) = (1 − k)[ A¯ 1d L + G d L (In − E¯ d L E¯ dDL )]λ¯ D +k[ A¯ 1dU + G dU (In − E¯ dU E¯ dU )]λ¯ − In λ < 0

for 0 ≤ k ≤ 1. Therefore, if the matrices A¯ 1d L and A¯ 1dU are asymptotically stable, then the convex linear combination (3.126a) is also asymptotically stable. The proof for the convex linear combination (3.126b) can be performed in a similar way. If the matrices A1d L and A1dU are asymptotically stable, then by (3.31) there exist strictly positive vector λ ∈ Rn+1 such that (A1d L − In 1 )λ < 0 and (A1dU − In 1 )λ < 0.

(3.129)

From (3.107b) and (3.110) we obtain (A1d − In 1 )λ = [(1 − k)A1d L + k A1dU − In 1 ]λ = (1 − k)A1d L λ + k A1dU λ − In 1 λ < 0

(3.130)

for 0 ≤ k ≤ 1. Therefore, if the matrices A1d L and A1dU are asymptotically stable, then the convex linear combination (3.126b) is also asymptotically stable. The necessity follows immediately from the fact that k can be equal to zero and one.   Theorem 3.18. The positive descriptor interval discrete-time linear system (3.26) with (3.122) is asymptotically stable if and only if: 1) (Drazin inverse matrix method case) the matrices A¯ 1d L and A¯ 1dU are asymptotically stable; 2) (Weierstrass-Kronecker decomposition method case) the matrices A1d L and A1dU are asymptotically stable. Proof. By (3.30) the matrices A¯ 1d L and A¯ 1dU are asymptotically stable if and only if there exist G d L ∈ Rn×n , G dU ∈ Rn×n and strictly positive vector λ¯ ∈ Rn+ such that (3.127) holds. The convex linear combination (3.126a) satisfies the condition [ A¯ 1d + G d (In − E¯ d E¯ dD ) − In ]λ¯ < 0 if and only if (3.127) holds. Therefore, the positive descriptor interval discrete-time linear system (3.26) with (3.122) is asymptotically stable if and only if the matrices A1d L and A1dU are asymptotically stable. The proof for the Weierstrass-Kronecker decomposition method case can be performed in a similar way.   To test the stability of an interval system an extension of the Kharitonov theorem (Theorem 3.15) can also be used. Theorem 3.19. The positive descriptor interval discrete-time linear system (3.26) with the characteristic equations

176

1) 2) 3) 4)

3 Stability of Positive Descriptor Systems

p(Ed ,Ad ) (z + 1) = p(Ed ,Ad −Ed ) (z) = det[E d (z + 1) − Ad ] = 0, p( E¯ d , A¯ d ) (z + 1) = p( E¯ d , A¯ d − E¯ d ) (z) = det[ E¯ d (z + 1) − A¯ d ] = 0, p A¯ 1d (z + 1) = (z + 1)n−r p( E¯ d , A¯ d ) (z + 1) = 0, p A1d (z + 1) = p A1d −In1 (z) = det[In 1 (z + 1) − A1d ] = 0

with interval coefficients a d,i ≤ ad,i ≤ a d,i is asymptotically stable if and only if the lower bounds a d,i , i = 0, 1, . . . , r are positive, where r = rank A¯ 1d = rank A1d . Proof. Taking into account Theorem 3.3 and Kharitonov theorem (Theorem 3.15) the proof can be performed similarly to the proof of Theorem 3.16.   Example 3.6. Consider the descriptor interval discrete-time linear system (3.26) with Ad ∈ [Ad L , AdU ] and ⎡

⎤ ⎡ ⎤ 1 0 0 0.5 0.1 0 2 0 ⎦ , Ad L = ⎣ 1 0.6 0 ⎦ , Ed = ⎣ 2 −0.8 −0.8 0 −0.4 −0.24 2 ⎡ ⎤ 0.75 0.15 0 AdU = ⎣ 1.5 0.9 0 ⎦ . −0.6 −0.36 3

(3.131)

It is easy to check that the matrix pencils (E d , Ad L ), (E d , AdU ) of (3.131) are regular. To check the stability of the interval system (3.26) with (3.131) we use two approaches as follows. 1) Drazin inverse matrix method case From (3.124) for c = 0 we have ⎡

E¯ d L

⎤ ⎡ −2 1 0 −1.3333 0.6667 0 −3.3333 = ⎣ 0 −5 0 ⎦ , E¯ dU = ⎣ 0 0 0 0 0 ⎡ ⎤ −1 0 0 A¯ d L = A¯ dU = ⎣ 0 −1 0 ⎦ . 0 0 −1

⎤ 0 0⎦, 0

(3.132)

Next, from (B.1) and (B.2) we have ⎡

E¯ dDL and

⎤ ⎡ ⎤ −0.5 −0.1 0 −0.75 −0.15 0 D −0.3 0 ⎦ = ⎣ 0 −0.2 0 ⎦ , E¯ dU =⎣ 0 0 0 0 0 0 0

(3.133)

3.2 Stability of Positive Interval Systems

177

⎡

⎤ 0.5 0.1 0 A¯ 1d L = E¯ dDL A¯ d L = ⎣ 0 0.2 0 ⎦ , 0 0 0 ⎡ ⎤ 0.75 0.15 0 D ¯ AdU = ⎣ 0 0.3 0 ⎦ . A¯ 1dU = E¯ dU 0 0 0

(3.134)

3×3 ¯ By Theorem 1.29 the matrices A¯ 1d L ∈ R3×3 and the interval system + , A1dU ∈ R+ (3.26) with (3.131) is positive. The characteristic equations of (3.134) are given by

p A¯ 1L (z + 1) = (z + 1)(10z 2 + 13z + 4) = 0, p A¯ 1U (z + 1) = (z + 1)(40z 2 + 38z + 7) = 0.

(3.135)

Therefore, by Theorem 3.3 both matrices (3.134) are asymptotically stable and by Theorem 3.18 the interval system (3.26) with (3.131) is asymptotically stable. Also, by Theorem 3.19 the lower bounds of (3.135) are positive and the considered interval system is asymptotically stable. 2) Weierstrass-Kronecker decomposition method case In this case

⎡

⎤ ⎡ ⎤ 1 0 0 10 0 P1d = ⎣ −1 0.5 0 ⎦ , Q 1d = ⎣ 0 1 0 ⎦ , 0 0.4 1 0 0 0.5 ⎡ ⎤ ⎡ ⎤ 1 0 0 10 0 P2d = ⎣ −1 0.5 0 ⎦ , Q 2d = ⎣ 0 1 0 ⎦ . 0 0.4 1 0 0 0.3333

(3.136)

From (3.125) and (3.132) we obtain

A1d L



 0.5 0.1 0.75 0.15 = , A1dU = . 0 0.2 0 0.3

(3.137)

2×2 By Theorem 1.31 the matrices A1d L ∈ R2×2 and Q 1d ∈ R2×2 + , A1U ∈ R+ + , Q 2d ∈ 2×2 R+ are monomial. Therefore, the interval system (3.26) with (3.131) is positive. The characteristic equations of (3.137) are given by

p A1L (s) = 10z 2 + 13z + 4 = 0, p A1U (s) = 40z 2 + 38z + 7 = 0.

(3.138)

Therefore, by Theorem 3.3 both matrices (3.137) are asymptotically stable and by Theorem 3.18 the interval system (3.26) with (3.131) is asymptotically stable. Also, by Theorem 3.19 the lower bounds of (3.138) are positive and the considered interval system is asymptotically stable.

178

3 Stability of Positive Descriptor Systems

3.2.3 Fractional Continuous-Time Systems Consider the positive fractional descriptor continuous-time linear system (3.51) with the interval matrix A ∈ Rn×n defined by A L ≤ A ≤ AU or equivalently A ∈ [A L , AU ].

(3.139)

It is assumed that det[Es α − A L ] = 0 and det[Es α − AU ] = 0 for some s α ∈ C.

(3.140)

Following the considerations presented in Sect. 3.1.3 the fractional interval system (3.51) can be transformed into two equivalent forms: 1) Using Drazin inverse matrix method we obtain the fractional interval system (3.52), where E¯ ∈ [ E¯ L , E¯ U ], A¯ ∈ [ A¯ L , A¯ U ] and E¯ L , E¯ U , A¯ L , A¯ U are determined by (3.104). The matrix E¯ D A¯ in (3.52) will be denoted as A¯ 1 . Therefore, we have A¯ 1 ∈ [ A¯ 1L , A¯ 1U ] and A¯ 1L = E¯ LD A¯ L , A¯ 1U = E¯ UD A¯ U . 2) Using Weierstrass-Kronecker decomposition method we obtain the fractional interval system (3.53), where the matrix A1 ∈ [A1L , A1U ] is determined by (3.105) for two pairs of nonsingular matrices (P1 , Q 1 ), (P2 , Q 2 ). Definition 3.4. The positive fractional descriptor interval continuous-time linear system (3.51) is called asymptotically stable if the system is asymptotically stable for all matrices E, A, where A ∈ [A L , AU ]. According to Definition 3.2 we can define convex linear combinations of matrices A¯ 1L , A¯ 1U and A1L , A1U given by (3.107). Theorem 3.20. The positive fractional descriptor interval continuous-time linear system (3.51) with (3.139) is asymptotically stable if and only if: 1) (Drazin inverse matrix method case) the matrices A¯ 1L and A¯ 1U are asymptotically stable; 2) (Weierstrass-Kronecker decomposition method case) the matrices A1L and A1U are asymptotically stable. Proof. The proof is similar to the proof of Theorem 3.14.

 

To test the stability of a fractional interval system an extension of the Kharitonov theorem (Theorem 3.15) can also be used. Consider the set (family) of the n-th degree polynomials pn (s α ) = an s nα + an−1 s (n−1)α + . . . + a1 s α + a0 (3.141a) with interval coefficients a i ≤ ai ≤ a i , i = 0, 1, . . . , n.

(3.141b)

3.2 Stability of Positive Interval Systems

179

Using (3.141a) we define the following four polynomials: p1n (s α ) = a 0 + a 1 s α + a 2 s 2α + a 3 s 3α + a 4 s 4α + a 5 s 5α + . . . , p2n (s α ) = a 0 + a 1 s α + a 2 s 2α + a 3 s 3α + a 4 s 4α + a 5 s 5α + . . . , p3n (s α ) = a 0 + a 1 s α + a 2 s 2α + a 3 s 3α + a 4 s 4α + a 5 s 5α + . . . ,

(3.142)

p4n (s α ) = a 0 + a 1 s α + a 2 s 2α + a 3 s 3α + a 4 s 4α + a 5 s 5α + . . . . Theorem 3.21. The positive fractional descriptor interval continuous-time linear system (3.51) with the characteristic equations 1) 2) 3) 4)

p(E,A) (s α ) = 0 given by (2.76), α p( E, ¯ A) ¯ (s ) = 0 given by (2.86), α p A¯ 1 (s ) = 0 given by (1.117), p A1 (s α ) = 0 given by (2.93)

with interval coefficients a i ≤ ai ≤ a i is asymptotically stable if and only if the lower bounds a i , i = 0, 1, . . . , r are positive, where r = rank A¯ 1 = rank A1 . Proof. The proof is similar to the proof of Theorem 3.16.

 

Example 3.7. Consider the fractional descriptor interval continuous-time linear system (3.51) with A ∈ [A L , AU ] and ⎡

⎤ ⎡ ⎤ −1 −0.75 −0.75 210 E = ⎣ −2 −0.25 −0.25 ⎦ , A L = ⎣ 4 0 1 ⎦ . 0 −0.75 −0.75 021 ⎡ ⎤ 420 AU = ⎣ 8 0 2 ⎦ . 042

(3.143)

It is easy to check that the matrix pencils (E, A L ), (E, AU ) of (3.143) are regular. To check the stability of the fractional interval system (3.51) with (3.143) we use the Drazin inverse matrix method. From (3.104) for c = 0 we have ⎡

⎤ ⎡ ⎤ 0.5 0.125 0.125 0.25 0.625 0.625 0.5 ⎦ , E¯ U = ⎣ 0 0.25 0.25 ⎦ , E¯ L = ⎣ 0 0.5 0 −0.25 −0.25 0 −0.125 −0.125 ⎡ ⎤ −1 0 0 A¯ L = A¯ U = ⎣ 0 −1 0 ⎦ . 0 0 −1 Next, using one of the methods presented in Appendix B we obtain

(3.144)

180

3 Stability of Positive Descriptor Systems

⎡

E¯ LD and

⎤ ⎡ ⎤ 2 −1 −1 4 −2 −2 = ⎣ 0 8 8 ⎦ , E¯ UD = ⎣ 0 16 16 ⎦ 0 −4 −4 0 −8 −8 ⎤ −2 1 1 A¯ 1L = E¯ LD A¯ L = ⎣ 0 −8 −8 ⎦ , 0 4 4 ⎡ ⎤ −4 2 2 A¯ 1U = E¯ UD A¯ U = ⎣ 0 −16 −16 ⎦ . 0 8 8

(3.145)

⎡

(3.146)

By Theorem 2.8 there exist matrices ⎡

⎤ ⎡ ⎤ 0 0 0 0 0 0 G L = ⎣ 0 −4 0 ⎦ , G U = ⎣ 0 −8 0 ⎦ 0 0 −3 0 0 −6 such that

⎤ −2 1 1 A¯ 1L + G L (I3 − E¯ L E¯ LD ) = ⎣ 0 −4 0 ⎦ ∈ M3 , 0 1 −2 ⎡ ⎤ −4 2 2 A¯ 1U + G U (I3 − E¯ U E¯ UD ) = ⎣ 0 −8 0 ⎦ ∈ M3 0 2 −4

(3.147)

⎡

(3.148)

and the fractional interval system (3.1) with (3.114) is positive. The characteristic equations of (3.146) are given by p A¯ 1L (s α ) = s α (s 2α + 6s α + 8) = 0, p A¯ 1U (s) = s α (s 2α + 12s α + 32) = 0.

(3.149)

Therefore, by Theorem 3.5 both matrices (3.146) are asymptotically stable and by Theorem 3.20 the fractional interval system (3.51) with (3.143) is asymptotically stable. Also, by Theorem 3.21 the lower bounds of (3.149) are positive and the considered fractional interval system is asymptotically stable.

3.2.4 Fractional Discrete-Time Systems Consider the positive fractional descriptor discrete-time linear system (3.74) with the interval matrix Ad ∈ Rn×n defined by

3.2 Stability of Positive Interval Systems

181

Ad L ≤ Ad ≤ AdU or equivalently Ad ∈ [Ad L , AdU ]

(3.150)

and Adα ∈ [AdαL , AdαU ], where AdαL = Ad L + E d α, AdαU = AdU + E d α. It is assumed that det[E d z − Ad L ] = 0 and det[E d z − AdU ] = 0 for some z ∈ C.

(3.151)

Following the considerations presented in Sect. 3.1.4 the fractional interval system (3.74) can be transformed into two equivalent forms: 1) Using Drazin inverse matrix method we obtain the interval system (3.76), where E¯ d ∈ [ E¯ d L , E¯ dU ], A¯ d ∈ [ A¯ d L , A¯ dU ], A¯ dα ∈ [ A¯ dαL , A¯ dαU ], the matrices E¯ d L , E¯ dU , A¯ d L , A¯ dU are determined by (3.124) and A¯ dαL = A¯ d L + E¯ d L α, A¯ dαU = A¯ dU + E¯ dU α.

(3.152)

The matrix E¯ dD A¯ dα in (3.76) will be denoted as A¯ 1dα . Therefore, we have A¯ 1dα ∈ D ¯ [ A¯ 1dαL , A¯ 1dαU ] and A¯ 1dαL = E¯ dDL A¯ dαL , A¯ 1dU = E¯ dU AdαU . 2) Using Weierstrass-Kronecker decomposition method we obtain the fractional interval system (3.78), where the matrix A1d ∈ [A1d L , A1dU ] is determined by (3.125) and A1dα ∈ [A1dαL , A1dαU ], (3.153) A1dαL = A1d L + In 1 α, A1dαU = A1dU + In 1 α. Definition 3.5. The positive fractional descriptor interval discrete-time linear system (3.74) is called practically stable for given length L of practical implementation if the system is asymptotically stable for some finite L and all matrices E d , Ad , where Ad ∈ [Ad L , AdU ]. The positive fractional descriptor interval discrete-time linear system (3.74) is called asymptotically stable if the system is asymptotically stable for L → ∞ and all matrices E d , Ad , where Ad ∈ [Ad L , AdU ]. Taking into account Theorems 3.7 and 3.8, to test the stability of the fractional interval system we will use the matrices D¯ 1 and D1 defined by (3.80) and (3.82). From (3.80), (3.82) and (3.152), (3.153) we have D¯ 1 ∈ [ D¯ 1L , D¯ 1U ], D1 ∈ [D1L , D1U ], where D¯ 1L = A¯ 1dαL + D1L = A1dαL +

L

c j In , D¯ 1U = A¯ 1dαU +

L

j=1

j=1

L

L

j=1

c j In , D1U = A1dαU +

j=1

(3.154a)

c j In (3.154b) c j In .

182

3 Stability of Positive Descriptor Systems

According to Definition 3.2 we can define convex linear combinations of matrices D¯ 1L , D¯ 1U and D1L , D1U , i.e. D¯ 1 = (1 − k) D¯ 1L + k D¯ 1U , 0 ≤ k ≤ 1,

(3.155a)

D1 = (1 − k)D1L + k D1U , 0 ≤ k ≤ 1.

(3.155b)

Theorem 3.22. The positive fractional descriptor interval discrete-time linear system (3.74) with (3.150) is practically stable for given length L of practical implementation if and only if: 1) (Drazin inverse matrix method case) the matrices D¯ 1L and D¯ 1U are asymptotically stable; 2) (Weierstrass-Kronecker decomposition method case) the matrices D1L and D1U are asymptotically stable. Proof. The proof follows immediately from Theorems 3.10 and 3.18.

 

Theorem 3.23. The positive fractional descriptor interval discrete-time linear system (3.74) with (3.150) is asymptotically stable if and only if: 1) (Drazin inverse matrix method case) the matrices A¯ 1d L and A¯ 1dU are asymptotically stable; 2) (Weierstrass-Kronecker decomposition method case) the matrices A1d L and A1dU are asymptotically stable. Proof. The proof follows immediately from Theorems 3.12 and 3.18.

 

To test the stability of a fractional interval system an extension of the Kharitonov theorem (Theorem 3.15) can also be used. Theorem 3.24. The positive fractional descriptor interval continuous-time linear system (3.51) with the characteristic equations 1) 2) 3) 4)

p(Ed ,D) (z + 1) = p(Ed ,D−Ed ) (z) = det[E d (z + 1) − D] = 0, ¯ d (z + 1) − D] ¯ = 0, p( E¯ d , D) ¯ (z + 1) = p( E¯ d , D− ¯ E¯ d ) (z) = det[ E n−r p D¯ 1 (z + 1) = det[In (z + 1) − D¯ 1 ] = (z + 1) p( E¯ d , D) ¯ (z + 1) = 0, p D1 (z + 1) = p D1 −In1 (z) = det[In 1 (z + 1) − D1 ] = 0

with interval coefficients a i ≤ ai ≤ a i is practically stable for given length L of practical implementation if and only if the lower bounds a i , i = 0, 1, . . . , r are positive, where r = rank A¯ 1dα = rank A1dα and the matrices D, D¯ are defined by (3.83). Proof. Taking into account Theorem 3.9 and Kharitonov theorem (Theorem 3.15) the proof can be performed similarly to the proof of Theorem 3.16.  

3.2 Stability of Positive Interval Systems

183

Theorem 3.25. The positive fractional descriptor interval continuous-time linear system (3.51) with the characteristic equations 1) 2) 3) 4)

p(Ed ,D∞ ) (z + 1) = det[E d z − Ad ] = p(Ed ,Ad ) (z) = 0, p( E¯ d , D¯ ∞ ) (z + 1) = det[ E¯ d z − A¯ d ] = p( E¯ d , A¯ d ) (z) = 0, p D¯ 1,∞ (z + 1) = β1 (z + α)n−r p( E¯ d , A¯ d ) (z) = 0, p D1,∞ (z + 1) = det[In 1 z − A1d ] = p A1d (z) = 0

with interval coefficients a i ≤ ai ≤ a i is asymptotically stable if and only if the lower bounds a i , i = 0, 1, . . . , r are positive, where r = rank A¯ 1dα = rank A1dα . Proof. The proof follows immediately from Theorem 3.9, Lemma 3.1 and Kharitonov theorem (Theorem 3.15).   Example 3.8. Consider the fractional descriptor interval discrete-time linear system (3.26) with α = 0.4, Ad ∈ [Ad L , AdU ] and ⎡

⎤ ⎡ ⎤ 0 −5 0 010 E d = ⎣ −5 −10 0 ⎦ , Ad L = ⎣ 1 0 0 ⎦ , 0 −7.5 0 001 ⎡ ⎤ 0 1.5 0 AdU = ⎣ 2 0 0 ⎦ . 0 0 1.5

(3.156)

It is easy to check that the matrix pencils (E d , Ad L ), (E d , AdU ) of (3.156) are regular. To check the stability of the fractional interval system (3.26) with (3.131) we use the Drazin inverse matrix method. From (3.124) for c = 0 we have ⎡

E¯ d L

A¯ dαL

⎤ ⎡ ⎤ 5 10 0 2.5 5 0 = ⎣ 0 5 0 ⎦ , E¯ dU = ⎣ 0 3.3333 0 ⎦ , 0 7.5 0 0 5 0 ⎡ ⎤ −1 0 0 A¯ d L = A¯ dU = ⎣ 0 −1 0 ⎦ , 0 0 −1 ⎡ ⎤ ⎡ ⎤ 14 0 0 2 0 = ⎣ 0 1 0 ⎦ , A¯ dαU = ⎣ 0 0.3333 0 ⎦ , 0 3 −1 0 2 −1

(3.157)

Next, using (B.1) and (B.2) we obtain ⎡

E¯ dDL

⎤ ⎡ ⎤ 0.2 −0.4 0 0.4 −0.6 0 D = ⎣ 0 0.2 0 ⎦ , E¯ dU = ⎣ 0 0.3 0 ⎦ 0 0.3 0 0 0.45 0

(3.158)

184

and

3 Stability of Positive Descriptor Systems

⎡

⎤ −0.2 0.4 0 A¯ 1d L = E¯ dDL A¯ d L = ⎣ 0 −0.2 0 ⎦ , 0 −0.3 0 ⎡ ⎤ −0.4 0.6 0 D A¯ 1dU = E¯ dU A¯ dU = ⎣ 0 −0.3 0 ⎦ , 0 −0.45 0 ⎤ 0.2 0.4 0 A¯ 1dαL = E¯ dDL A¯ dαL = ⎣ 0 0.2 0 ⎦ , 0 0.3 0 ⎡ ⎤ 0 0.6 0 D ¯ AdαU = ⎣ 0 0.1 0 ⎦ . A¯ 1dαU = E¯ dU 0 0.15 0

(3.159)

⎡

(3.160)

3×3 ¯ By Theorem 1.29 the matrices A¯ 1dαL ∈ R3×3 + , A1dαU ∈ R+ and the fractional interval system (3.74) with (3.156) is positive. The characteristic equations of (3.159) are given by p A¯ 1d L (z) = z(25z 2 + 10z + 1) = 0, (3.161) p A¯ 1dU (z) = z(50z 2 + 35z + 6) = 0.

Therefore, by Theorem 3.9 both matrices (3.159) are asymptotically stable and by Theorem 3.22 the interval system (3.74) with (3.156) is asymptotically stable. Also, by Theorem 3.24 the lower bounds of (3.161) are positive and the considered interval system is asymptotically stable.

3.3 Stability of Nonlinear Systems with Positive Linear Parts In this section the stability of nonlinear feedback systems with positive linear parts will be analyzed [75–78, 80, 81, 91, 93]. Sufficient conditions for the global stability will be established using Lyapunov method (Sect. 3.3.1–3.3.4) and Kudrewicz theorem [104] will be extended to positive descriptor continuous-time nonlinear systems (Sect. 3.3.5).

3.3.1 Continuous-Time Systems Consider the continuous-time nonlinear feedback system shown in Fig. 3.1, which consists of a positive descriptor linear part, a nonlinear element with characteristic

3.3 Stability of Nonlinear Systems with Positive Linear Parts

185

Fig. 3.1 Continuous-time nonlinear feedback system

u(t) = f (e(t)) and a positive scalar gain feedback h. The linear part is described by the equations E x(t) ˙ = Ax(t) + Bu(t), y(t) = C x(t),

(3.162)

where x(t) ∈ Rn+ , u(t) ∈ R+ , y(t) ∈ R+ are state, input and output vectors and E, A ∈ Rn×n , B ∈ Rn×1 , C ∈ R1×n . It is assumed that detE = 0 and det[Es − A] = 0 for some s ∈ C. Following the considerations presented in Sect. 1.1 the system (3.162) can be transformed into the following equivalent forms. 1) Using the Drazin inverse matrix method we obtain x˙1 (t) = A¯ 1 x1 (t) + B¯ 1 u(t), N¯ x˙2 (t) = x2 (t) + B¯ 2 u(t),

(3.163)

y(t) = C x(t) = C[x1 (t) + x2 (t)], where x1 (t), x2 (t) ∈ Rn+ are given by (1.28) and ¯ A¯ 1 = E¯ D A, ¯ B¯ 2 = (In − E¯ E¯ D ) A¯ D B,

¯ B¯ 1 = E¯ D B, ¯ N¯ = (In − E¯ E¯ D ) A¯ D E.

(3.164)

¯ A¯ ∈ Rn×n and B¯ ∈ Rn×1 have the form (1.24b). Theorem 1.10 The matrices E, gives the positivity conditions of the system (3.163). 2) Using the Weierstrass-Kronecker decomposition method we have x˙¯1 (t) = A1 x¯1 (t) + B1 u(t), N x˙¯2 (t) = x¯2 (t) + B2 u(t), y(t) = C1 x¯1 (t) + C2 x¯2 (t),

(3.165)

where x¯1 (t) ∈ Rn+1 , x¯2 (t) ∈ Rn+2 are given by (1.53) and the matrices A1 ∈ Rn 1 ×n 1 , B1 ∈ Rn 1 ×1 , B2 ∈ Rn 2 ×1 , N ∈ Rn 2 ×n 2 , C1 ∈ R1×n 1 , C2 ∈ R1×n 2 are determined

186

3 Stability of Positive Descriptor Systems

by (1.52) and (1.56). Theorem 1.11 gives the positivity conditions of the system (3.165). The characteristic of the nonlinear element is shown in Fig. 3.2 and it satisfies the condition f (e(t)) 0≤ ≤ k < ∞. (3.166) e(t) It is assumed that the positive descriptor linear part is asymptotically stable. Definition 3.6. The positive descriptor continuous-time nonlinear system is called globally stable if it is asymptotically stable for all admissible nonnegative initial conditions x0 ∈ Rn+ . The following theorem gives sufficient conditions for the global stability of the positive descriptor continuous-time nonlinear system. Theorem 3.26. The continuous-time nonlinear system consisting of the positive and asymptotically stable descriptor linear part, the nonlinear element satisfying the condition (3.166) and the gain feedback h is globally stable if: 1) (Drazin inverse matrix method case) a) B¯ 2 = 0 and there exists a matrix G ∈ Rn×n such that A¯ C1 + G(In − E¯ E¯ D ) ∈ Mn

(3.167)

is asymptotically stable, where A¯ C1 = A¯ 1 + kh B¯ 1 C;

Fig. 3.2 Characteristic of the nonlinear element

(3.168)

3.3 Stability of Nonlinear Systems with Positive Linear Parts

187

b) C(In − E¯ E¯ D ) = 0 and there exists a matrix G ∈ Rn×n such that (3.167) is asymptotically stable and − N¯ j A¯ C2 A¯ C1 ∈ Rn×n + , j = 1, . . . , q − 1, j

(3.169)

where A¯ C1 is given by (3.168), A¯ C2 = kh B¯ 2 C

(3.170)

¯ and q is the index of E. 2) (Weierstrass-Kronecker decomposition method case) a) B2 = 0 and

AC1 ∈ Mn 1

(3.171)

AC1 = A1 + kh B1 C1 ;

(3.172)

is asymptotically stable, where

b) C2 = 0, (3.171) is asymptotically stable and − N j AC2 AC1 ∈ Rn+2 ×n 1 , j = 1, . . . , μ − 1, j

(3.173)

where AC1 is given by (3.172), AC2 = kh B2 C1

(3.174)

and μ is the nilpotency index of N . Proof. The proof will be accomplished for the Weierstrass-Kronecker decomposition method. Taking into account (3.165) for B2 = 0 we obtain x˙¯1 (t) = A1 x¯1 (t) + B1 u(t), N x˙¯2 (t) = x¯2 (t), y(t) = C1 x¯1 (t) + C2 x¯2 (t).

(3.175)

From the second equation of (3.175) it follows that x¯2 (t) = 0 since N is a nilpotent matrix. Assuming u(t) = ke(t) and taking into account that e(t) = hy(t) from (3.175) we obtain x˙¯1 (t) = (A1 + kh B1 C1 )x¯1 (t), (3.176) x¯2 (t) = 0

188

3 Stability of Positive Descriptor Systems

As the Lyapunov function V (x¯1 (t)) we choose V (x¯1 (t)) = λT x¯1 (t) ≥ 0 for x¯1 (t) ∈ Rn+1 ,

(3.177)

  where λ = λ1 . . . λn 1 is a strictly positive vector, i.e. λ j > 0, j = 1, . . . , n 1 . Using (3.175)–(3.177) we obtain V˙ (x¯1 (t)) = λT x˙¯1 (t) = λT [A1 x¯1 (t) + B1 u(t)] = λT [A1 x¯1 (t) + B1 h f (e(t))] ≤ λT [(A1 + kh B1 C1 )x¯1 (t)]

(3.178)

since u(t) = f (e(t)) ≤ ke(t) = kC1 x¯1 (t). From (3.178) it follows that V˙ (x¯1 (t)) < 0 if the condition (3.171) is satisfied with B2 = 0 and the nonlinear system is globally stable. Now let us consider (3.165) for C2 = 0. In this case we have x¯˙1 (t) = A1 x¯1 (t) + B1 u(t), N x¯˙2 (t) = x¯2 (t) + B2 u(t), y(t) = C1 x¯1 (t).

(3.179)

Assuming u(t) = ke(t) and taking into account that e(t) = hy(t) from (3.179) we obtain x˙¯1 (t) = (A1 + kh B1 C1 )x¯1 (t), (3.180) N x¯˙2 (t) = kh B2 C1 x¯1 (t) + x¯2 (t). The second equation of (3.180) can be solved in a similar way to the Eq. (1.55b). From (3.180) and (1.59) we have x¯2 (t) = −

μ−1

( j)

kh N j B2 C1 x¯1 (t),

(3.181)

j=0 ( j)

where μ is the nilpotency index of the matrix N and x¯1 (t) is the j-th time derivative of the vector x¯1 (t). Let AC1 = A1 + kh B1 C1 and AC2 = kh B2 C1 . Taking into account that x˙¯1 (t) = AC1 x¯1 (t), 2 x¨¯1 (t) = AC1 x˙¯1 (t) = AC1 x¯1 (t), (3.182) .. . ( j)

( j−1)

x¯1 (t) = AC1 x¯1

j

(t) = AC1 x¯1 (t)

from (3.181) we obtain x¯2 (t) = −

μ−1 j=0

j

N j AC2 AC1 x¯1 (t).

(3.183)

3.3 Stability of Nonlinear Systems with Positive Linear Parts

189

Choosing the Lyapunov function in the form (3.177) we get (3.178) and V˙ (x¯1 (t)) < 0 if the condition (3.171) is satisfied with C2 = 0. From (3.183) it follows that if lim x¯1 (t) = 0 then also lim x¯2 (t) = 0. The condition (3.173) ensures x¯2 (t) ∈ Rn+2 . t→∞ t→∞ Therefore, the nonlinear system is globally stable. The proof for the Drazin inverse matrix method can be accomplished similarly. Note that if C(In − E¯ E¯ D ) = 0, then C x2 (t) = 0 since x2 (t) = (In − E¯ E¯ D )x(t). The matrix G ∈ Rn×n and the term G(In − E¯ E¯ D ) eliminates from the matrix A¯ 1 = E¯ D A¯ unimportant entires that are further canceled through multiplication by x1 (t) ∈   Im E¯ E¯ D . Remark 3.2. Theorems 3.1 and 3.2 can be used to test the stability of (3.167) and (3.171). Example 3.9. Consider the nonlinear system consisting of the descriptor continuoustime linear part (3.162) with ⎡

⎤ ⎡ 0.5 0 0 −1 0 E = ⎣ 0 0.25 0 ⎦ , A = ⎣ 0.5 −0.5 0 0.25 0 0.5 −0.5   C= 210 ,

⎤ ⎡ ⎤ 0 1 0⎦, B = ⎣ 2 ⎦, 1 −1

(3.184)

the nonlinear element satisfying the condition (3.166) and the gain feedback h = 0.5. Find k satisfying (3.166) for which the nonlinear system is globally stable. The matrix pencil (E, A) of (3.184) is regular since 0.5s + 1 0 0 det[Es − A] = −0.5 0.25s + 0.5 0 = −0.125(s + 2)2 = 0. −0.5 0.25s + 0.5 −1

(3.185)

The characteristic equation of (3.184) is given by p(E,A) (s) = det[Es − A] = −0.125(s + 2)2 = 0.

(3.186)

From (3.186) it follows that the pair (E, A) has two eigenvalues s1 = s2 = −2 and by condition (1.101) the considered linear part is asymptotically stable. To check the positivity and global stability of the system we use two approaches as follows. 1) Drazin inverse matrix method case We choose c = 0 and using (1.24b) we obtain

190

3 Stability of Positive Descriptor Systems

⎡

⎤ 0.5 0 0 E¯ = [−A]−1 E = ⎣ 0.5 0.5 0 ⎦ , 0 0 0 ⎡ ⎤ ⎡ ⎤ −1 0 0 1 A¯ = [−A]−1 A = ⎣ 0 −1 0 ⎦ , B¯ = [−A]−1 B = ⎣ 5 ⎦ 0 0 −1 3

(3.187)

and q = 1 since rank E¯ = rank E¯ 2 . Next, from (B.1) and (B.2) we have ⎡

E¯ D

⎤ ⎡ ⎤ 2 00 −1 0 0 = ⎣ −2 2 0 ⎦ , A¯ D = A¯ −1 = A¯ = ⎣ 0 −1 0 ⎦ 0 00 0 0 −1

(3.188)

and ⎡

⎤ ⎡ ⎤ −2 0 0 2 A¯ 1 = E¯ D A¯ = ⎣ 2 −2 0 ⎦ , B¯ 1 = E¯ D B¯ = ⎣ 8 ⎦ , 0 0 0 0 ⎡ ⎤ ⎡ ⎤ 0 000 B¯ 2 = (I3 − E¯ E¯ D ) A¯ D B¯ = ⎣ 0 ⎦ , N¯ = (I3 − E¯ E¯ D ) A¯ D E¯ = ⎣ 0 0 0 ⎦ , −3 000   D ¯ ¯ C(I3 − E E ) = 0 0 0 . (3.189) By Theorem 1.10 the conditions (1.88) for the matrices (3.189) are satisfied and the descriptor linear part described by (3.184) is positive. Also, from (3.189) it follows that B¯ 2 = 0 and C(I3 − E¯ E¯ D ) = 0. Therefore, the case b) of Theorem 3.26 can be used to test the global stability of the nonlinear system. Taking into account (3.168), (3.170), (3.184), (3.189) and h = 0.5 we obtain ⎡

⎤ 2k − 2 k 0 A¯ C1 = A¯ 1 + 0.5k B¯ 1 C = ⎣ 8k + 2 4k − 2 0 ⎦ , 0 0 0 ⎡ ⎤ 0 0 0 0 0⎦. A¯ C2 = 0.5k B¯ 2 C = ⎣ 0 −3k −1.5k 0

(3.190)

The matrix A¯ C1 ∈ M3 for any k > 0 and by Theorem 3.1 it is asymptotically stable if and only if all coefficients of the characteristic equation p A¯ C1 (s) = det[I3 s − A¯ C1 ] = s[s 2 + (4 − 6k)s + 4 − 14k] = 0

(3.191)

are positive, i.e. if and only if 4 − 6k > 0 and 4 − 14k > 0, and this implies k < 0.2857. Taking into account (3.169) the matrix

3.3 Stability of Nonlinear Systems with Positive Linear Parts

⎡

0 − N¯ 0 A¯ C2 A¯ C1 = − A¯ C2

⎤ 0 0 0 = ⎣ 0 0 0 ⎦ ∈ R3×3 + 3k 1.5k 0

191

(3.192)

for any k > 0. Therefore, the positive nonlinear system is globally stable if the characteristic u(t) = f (e(t)) of the nonlinear element satisfies the condition (3.166) for k < 0.2857. 2) Weierstrass-Kronecker decomposition method case ⎡

⎤ ⎡ ⎤ 0 1 0 020 P = ⎣1 0 0⎦, Q = ⎣4 0 0⎦ 0 −1 1 001

In this case

and

⎤ 100 = PEQ = ⎣0 1 0⎦, 000 ⎡ ⎤

 −2 1 0 A1 0 = P AQ = ⎣ 0 −2 0 ⎦ , 0 In 2 0 0 1 ⎡ ⎤

 2     B1 = P B = ⎣ 1 ⎦ , C1 C2 = C Q = 4 4 0 B2 −3

In 1 0 0 N



(3.193)

⎡

(3.194)

where n 1 = 2, n 2 = 1. By Theorem 1.11 the conditions (1.94) for the matrices (3.194) are satisfied and Q ∈ R3×3 + is monomial. Thus, the descriptor linear part described by (3.184) is positive. Also, from (3.194) it follows that B2 = 0 and C2 = 0. Therefore, the case b) of Theorem 3.26 can be used to test the global stability of the nonlinear system. Taking into account (3.172), (3.174), (3.194) and h = 0.5 we obtain

AC1

 4k − 2 4k + 1 = A1 + 0.5k B1 C1 = , 2k 2k − 2   AC2 = 0.5k B2 C1 = −6k −6k .

(3.195)

The matrix AC1 ∈ M2 for any k > 0 and by Theorem 3.1 it is asymptotically stable if and only if all coefficients of the characteristic equation p AC1 (s) = det[I2 s − AC1 ] = s 2 + (4 − 6k)s + 4 − 14k = 0

(3.196)

are positive, i.e. if and only if 4 − 6k > 0 and 4 − 14k > 0, and this implies k < 0.2857. Taking into account (3.173) the matrix   0 = −AC2 = 6k 6k ∈ R1×2 − N 0 AC2 AC1 +

(3.197)

192

3 Stability of Positive Descriptor Systems

for any k > 0. Therefore, the positive nonlinear system is globally stable if the characteristic u(t) = f (e(t)) of the nonlinear element satisfies the condition (3.166) for k < 0.2857.

3.3.2 Discrete-Time Systems Consider the discrete-time nonlinear feedback system shown in Fig. 3.3, which consists of a positive descriptor linear part, a nonlinear element with characteristic u i = f (ei ) and a positive scalar gain feedback h. The linear part is described by the equations E d xi+1 = Ad xi + Bd u i , i ∈ Z+ , (3.198) yi = Cd xi , where xi ∈ Rn+ , u i ∈ R+ , yi ∈ R+ are state, input and output vectors and E d , Ad ∈ Rn×n , Bd ∈ Rn×1 , Cd ∈ R1×n . It is assumed that detE d = 0 and det[E d z − Ad ] = 0 for some z ∈ C. Following the considerations presented in Sect. 1.2 the system (3.198) can be transformed into the following equivalent forms. 1) Using the Drazin inverse matrix method we obtain x1,i+1 = A¯ 1d x1,i + B¯ 1d u i , N¯ d x2,i+1 = x2,i + B¯ 2d u i ,

(3.199)

yi = Cd xi = Cd [x1,i + x2,i ], where x1,i , x2,i ∈ Rn+ are given by (2.160c) and B¯ 2d

A¯ 1d = E¯ dD A¯ d , = (In − E¯ d E¯ dD ) A¯ dD B¯ d ,

B¯ 1d = E¯ dD B¯ d , N¯ d = (In − E¯ d E¯ dD ) A¯ dD E¯ d .

(3.200)

The matrices E¯ d , A¯ d ∈ Rn×n and B¯ d ∈ Rn×1 have the form (1.182b). Theorem 1.29 gives the positivity conditions of the system (3.199).

Fig. 3.3 Discrete-time nonlinear feedback system

3.3 Stability of Nonlinear Systems with Positive Linear Parts

193

2) Using the Weierstrass-Kronecker decomposition method we have x¯1,i+1 = A1d x¯1,i + B1d u i , Nd x¯2,i+1 = x2,i + B2d u i , yi = C1d x¯1,i + C2d x¯2,i ,

(3.201)

where x¯1,i ∈ Rn+1 , x¯2,i ∈ Rn+2 are given by (1.193) and the matrices A1d ∈ Rn 1 ×n 1 , B1d ∈ Rn 1 ×1 , B2d ∈ Rn 2 ×1 , Nd ∈ Rn 2 ×n 2 , C1d ∈ R1×n 1 , C2d ∈ R1×n 2 are determined by (1.192) and (1.196). Theorem 1.31 gives the positivity conditions of the system (3.201). The characteristic of the nonlinear element is shown in Fig. 3.2 and it satisfies the condition (3.166). It is assumed that the positive descriptor linear part is asymptotically stable. Definition 3.7. The positive descriptor discrete-time nonlinear system is called globally stable if it is asymptotically stable for all admissible nonnegative initial conditions x0 ∈ Rn+ . The following theorem gives sufficient conditions for the global stability of the positive descriptor discrete-time nonlinear system. Theorem 3.27. The discrete-time nonlinear system consisting of the positive and asymptotically stable descriptor linear part, the nonlinear element satisfying the condition (3.166) and the gain feedback h is globally stable if: 1) (Drazin inverse matrix method case) a) B¯ 2d = 0 and there exists a matrix G d ∈ Rn×n such that A¯ C1d + G d (In − E¯ d E¯ dD ) ∈ Rn×n +

(3.202)

is asymptotically stable, where A¯ C1d = A¯ 1d + kh B¯ 1d Cd ;

(3.203)

b) Cd (In − E¯ d E¯ dD ) = 0 and there exists a matrix G d ∈ Rn×n such that (3.202) is asymptotically stable and j j − N¯ d A¯ C2d A¯ C1d ∈ Rn×n + , j = 1, . . . , q − 1,

(3.204)

where A¯ C1d is given by (3.203), A¯ C2d = kh B¯ 2d Cd and q is the index of E¯ d .

(3.205)

194

3 Stability of Positive Descriptor Systems

2) (Weierstrass-Kronecker decomposition method case) a) B2d = 0 and

AC1d ∈ Rn+1 ×n 1

(3.206)

is asymptotically stable, where AC1d = A1d + kh B1d C1d ;

(3.207)

b) C2d = 0, (3.206) is asymptotically stable and − Nd AC2d AC1d ∈ Rn+2 ×n 1 , j = 1, . . . , μ − 1, j

j

(3.208)

where AC1d is given by (3.207), AC2d = kh B2d C1d

(3.209)

and μ is the nilpotency index of Nd . Proof. The proof will be accomplished for the Weierstrass-Kronecker decomposition method. Taking into account (3.201) for B2d = 0 we obtain x¯1,i+1 = A1d x¯1,i + B1d u i , Nd x¯2,i+1 = x¯2,i ,

(3.210)

yi = C1d x¯1,i + C2d x¯2,i . From the second equation of (3.210) it follows that x¯2,i = 0 since Nd is a nilpotent matrix. Assuming u i = kei and taking into account that ei = hyi from (3.210) we obtain x¯1,i+1 = (A1d + kh B1d C1d )x¯1,i , (3.211) x¯2,i = 0 As the Lyapunov function V (x¯1,i ) we choose V (x¯1,i ) = λT x¯1,i ≥ 0 for x¯1,i ∈ Rn+1 ,

(3.212)

  where λ = λ1 . . . λn 1 is a strictly positive vector, i.e. λ j > 0, j = 1, . . . , n 1 . Using (3.210)–(3.212) we obtain V (x¯1,i ) = V (x¯1,i+1 ) − V (x¯1,i ) = λT x¯1,i+1 − λT x¯1,i = λT [(A1d − In 1 )x¯1,i + B1d u i ] = λT [(A1d − In 1 )x¯1,i + B1d h f (ei )] ≤ λT [(A1d + kh B1d C1d − In 1 )x¯1,i ]

(3.213)

3.3 Stability of Nonlinear Systems with Positive Linear Parts

195

since u i = f (ei ) ≤ kei = kC1d x¯1,i . It is well-known [49] that elements on the main diagonal of the state matrix of a positive and asymptotically stable discrete-time system are less than 1. Therefore, from (3.213) it follows that V (x¯1,i ) < 0 if the condition (3.206) is satisfied with B2d = 0 and the nonlinear system is globally stable. Now let us consider (3.201) for C2d = 0. In this case we have x¯1,i+1 = A1d x¯1,i + B1d u i , Nd x¯2,i+1 = x¯2,i + B2d u i ,

(3.214)

yi = C1d x¯1,i . Assuming u i = kei and taking into account that ei = hyi from (3.214) we obtain x¯1,i+1 = (A1d + kh B1d C1d )x¯1,i , Nd x¯2,i+1 = kh B2d C1d x¯1,i + x¯2,i .

(3.215)

The second equation of (3.215) can be solved in a similar way to the Eq. (1.195b). From (3.215) and (1.198) we have x¯2,i = −

μ−1

kh N j B2d C1d x¯1,i+ j ,

(3.216)

j=0

where μ is the nilpotency index of the matrix Nd . Let AC1d = A1d + kh B1d C1d and AC2d = kh B2d C1d . Taking into account that x¯1,i+1 = AC1d x¯1,i , 2 x¯1,i+2 = AC1d x¯1,i+1 = AC1d x¯1,i , .. .

(3.217)

j

x¯1,i+ j = AC1d x¯1,i+ j−1 = AC1 x¯1,i from (3.216) we obtain x¯2,i = −

μ−1

j

j

Nd AC2d AC1d x¯1,i .

(3.218)

j=0

Choosing the Lyapunov function in the form (3.212) we get (3.213) and V (x¯1,i ) < 0 if the condition (3.206) is satisfied with C2d = 0. From (3.218) it follows that if lim x¯1,i = 0 then also lim x¯2,i = 0. The condition (3.208) ensures x¯2,i ∈ Rn+2 .

i→∞

i→∞

Therefore, the nonlinear system is globally stable. The proof for the Drazin inverse matrix method can be accomplished similarly. Note that if Cd (In − E¯ d E¯ dD ) = 0, then Cd x2,i = 0 since x2,i = (In − E¯ d E¯ dD )xi .

196

3 Stability of Positive Descriptor Systems

The matrix G d ∈ Rn×n and the term G d (In − E¯ d E¯ dD ) eliminates from the matrix A¯ 1d = E¯ dD A¯ d unimportant entires that are further canceled through multiplication   by x1,i ∈ Im E¯ d E¯ dD . Remark 3.3. Theorems 3.3 and 3.4 can be used to test the stability of (3.202) and (3.206). Example 3.10. Consider the nonlinear system consisting of the descriptor discretetime linear part (3.162) with ⎡

⎤ ⎡ ⎤ ⎡ ⎤ 1 2 0 0.4 1.2 0 3 E d = ⎣ 0 −1 0 ⎦ , Ad = ⎣ 0 −0.5 0 ⎦ , Bd = ⎣ −1 ⎦ , 0.5 1 0 0.2 0.6 −0.5 1.5   Cd = 1 2 1 ,

(3.219)

the nonlinear element satisfying the condition (3.166) and the gain feedback h = 0.5. Find k satisfying (3.166) for which the nonlinear system is globally stable. The matrix pencil (E d , Ad ) of (3.219) is regular since z − 0.4 2z − 1.2 0 0 −z + 0.5 0 = −0.05(2z − 1)(5z − 2) = 0. det[E d z − Ad ] = 0.5z − 0.2 z − 0.6 0.5 (3.220) The characteristic equation of (3.219) is given by p(Ed ,Ad ) (z) = det[E d z − Ad ] = −0.05(2z − 1)(5z − 2) = 0.

(3.221)

From (3.221) it follows that the pair (E d , Ad ) has two eigenvalues z 1 = 0.5, z 2 = 0.4 and by condition (1.239) the considered linear part is asymptotically stable. To check the positivity and global stability of the system we use two approaches as follows. 1) Drazin inverse matrix method case We choose c = 0 and using (1.182b) we obtain ⎡

⎤ −2.5 1 0 E¯ d = [−Ad ]−1 E d = ⎣ 0 −2 0 ⎦ , 0 0 0 ⎡ ⎤ ⎡ ⎤ (3.222) −1 0 0 −1.5 A¯ d = [−Ad ]−1 Ad = ⎣ 0 −1 0 ⎦ , B¯ d = [−Ad ]−1 Bd = ⎣ −2 ⎦ 0 0 −1 0 and q = 1 since rank E¯ = rank E¯ 2 . Next, from (B.1) and (B.2) we have

3.3 Stability of Nonlinear Systems with Positive Linear Parts

⎡

E¯ dD

⎤ ⎡ ⎤ −0.4 −0.2 0 −1 0 0 ¯ ⎣ 0 −1 0 ⎦ = ⎣ 0 −0.5 0 ⎦ , A¯ dD = A¯ −1 d = Ad = 0 0 0 0 0 −1

197

(3.223)

and ⎡

B¯ 2d

⎤ ⎡ ⎤ 0.4 0.2 0 1 A¯ 1d = E¯ dD A¯ d = ⎣ 0 0.5 0 ⎦ , B¯ 1d = E¯ dD B¯ d = ⎣ 1 ⎦ , 0 0 0 0 ⎡ ⎤ 0   = (I3 − E¯ d E¯ dD ) A¯ dD B¯ d = ⎣ 0 ⎦ , Cd (I3 − E¯ E¯ D ) = 0 0 1 0 ⎡ ⎤ 000 N¯ d = (I3 − E¯ d E¯ dD ) A¯ dD E¯ d = ⎣ 0 0 0 ⎦ . 000

(3.224)

By Theorem 1.29 the conditions (1.223) for the matrices (3.224) are satisfied and the descriptor linear part described by (3.219) is positive. Also, from (3.224) it follows that B¯ 2d = 0 and Cd (I3 − E¯ E¯ D ) = 0. Therefore, the case a) of Theorem 3.27 can be used to test the global stability of the nonlinear system. Taking into account (3.203), (3.219), (3.224) and h = 0.5 we obtain ⎡

A¯ C1d = A¯ 1d

⎤ 0.5k + 0.4 k + 0.2 0.5k k + 0.5 0.5k ⎦ . + 0.5k B¯ 1d Cd = ⎣ 0.5k 0 0 0

(3.225)

The matrix A¯ C1d ∈ R3×3 + for any k > 0 and by Theorem 3.3 it is asymptotically stable if and only if all coefficients of the characteristic equation p A¯ C1d (z + 1) = det[I3 (z + 1) − A¯ C1d ] = (z + 1)[z 2 + (1.1 − 1.5k)z + 0.3 − 0.95k] = 0

(3.226)

are positive, i.e. if and only if 1.1 − 1.5k > 0 and 0.3 − 0.95k > 0, and this implies k < 0.3158. Therefore, the positive nonlinear system is globally stable if the characteristic u(t) = f (e(t)) of the nonlinear element satisfies the condition (3.166) for k < 0.3158. 2) Weierstrass-Kronecker decomposition method case In this case

⎡

⎤ ⎡ ⎤ 1 2 0 100 Pd = ⎣ 0 −1 0 ⎦ , Q d = ⎣ 0 1 0 ⎦ 1 0 −2 001

(3.227)

198

and

3 Stability of Positive Descriptor Systems

⎤ 100 = Pd E d Q d = ⎣ 0 1 0 ⎦ , 000 ⎡ ⎤

 0.4 0.2 0 A1d 0 = Pd Ad Q d = ⎣ 0 0.5 0 ⎦ , 0 In 2 0 0 1 ⎡ ⎤

 1     B1d = Pd Bd = ⎣ 1 ⎦ , C1d C2d = Cd Q d = 1 2 1 B2d 0

In 1 0 0 Nd

⎡



(3.228)

where n 1 = 2, n 2 = 1. By Theorem 1.31 the conditions (1.231) for the matrices is monomial. Thus, the descriptor linear part (3.228) are satisfied and Q d ∈ R3×3 + described by (3.219) is positive. Also, from (3.228) it follows that B2 = 0 and C2 = 0. Therefore, the case a) of Theorem 3.27 can be used to test the global stability of the nonlinear system. Taking into account (3.207), (3.228) and h = 0.5 we obtain

AC1d = A1d + 0.5k B1d C1d

 0.5k + 0.4 k + 0.2 = . 0.5k k + 0.5

(3.229)

The matrix AC1d ∈ R2×2 + for any k > 0 and by Theorem 3.3 it is asymptotically stable if and only if all coefficients of the characteristic equation p AC1d (z + 1) = det[I2 (z + 1) − AC1 ] = z 2 + (1.1 − 1.5k)z + 0.3 − 0.95k = 0 (3.230) are positive, i.e. if and only if 1.1 − 1.5k > 0 and 0.3 − 0.95k > 0, and this implies k < 0.3158. Therefore, the positive nonlinear system is globally stable if the characteristic u(t) = f (e(t)) of the nonlinear element satisfies the condition (3.166) for k < 0.3158.

3.3.3 Fractional Continuous-Time Systems Consider the fractional continuous-time nonlinear feedback system shown in Fig. 3.4, which consists of a fractional positive descriptor linear part, a nonlinear element with characteristic u(t) = f (e(t)) and a positive scalar gain feedback h. The linear part is described by the equations E 0 Dtα x(t) = Ax(t) + Bu(t), 0 < α < 1, y(t) = C x(t),

(3.231)

where x(t) ∈ Rn+ , u(t) ∈ R+ , y(t) ∈ R+ are state, input and output vectors, 0 Dtα is the Caputo derivative-integral operator defined by (2.8) and E, A ∈ Rn×n , B ∈ Rn×1 , C ∈ R1×n . It is assumed that detE = 0 and det[Es α − A] = 0 for some s α ∈ C.

3.3 Stability of Nonlinear Systems with Positive Linear Parts

199

Fig. 3.4 Fractional continuous-time nonlinear feedback system

Following the considerations presented in Sect. 2.1 the system (3.231) can be transformed into the following equivalent forms. 1) Using the Drazin inverse matrix method we obtain = A¯ 1 x1 (t) + B¯ 1 u(t), N¯ 0 Dtα x2 (t) = x2 (t) + B¯ 2 u(t), α 0 Dt x 1 (t)

(3.232)

y(t) = C x(t) = C[x1 (t) + x2 (t)], where x1 (t), x2 (t) ∈ Rn+ are given by (2.29c) and ¯ A¯ 1 = E¯ D A, ¯ B¯ 2 = (In − E¯ E¯ D ) A¯ D B,

¯ B¯ 1 = E¯ D B, ¯ N¯ = (In − E¯ E¯ D ) A¯ D E.

(3.233)

¯ A¯ ∈ Rn×n and B¯ ∈ Rn×1 have the form (2.28b). Theorem 2.8 The matrices E, gives the positivity conditions of the system (3.232). 2) Using the Weierstrass-Kronecker decomposition method we have

N

α 0 Dt x¯ 1 (t) α 0 Dt x¯ 2 (t)

= A1 x¯1 (t) + B1 u(t),

= x¯2 (t) + B2 u(t), y(t) = C1 x¯1 (t) + C2 x¯2 (t),

(3.234)

where x¯1 (t) ∈ Rn+1 , x¯2 (t) ∈ Rn+2 are given by (2.38) and the matrices A1 ∈ Rn 1 ×n 1 , B1 ∈ Rn 1 ×1 , B2 ∈ Rn 2 ×1 , N ∈ Rn 2 ×n 2 , C1 ∈ R1×n 1 , C2 ∈ R1×n 2 are determined by (2.37) and (2.41). Theorem 2.9 gives the positivity conditions of the system (3.234). The characteristic of the nonlinear element is shown in Fig. 3.2 and it satisfies the condition (3.166). It is assumed that the fractional positive descriptor linear part is asymptotically stable. Definition 3.8. The fractional positive descriptor continuous-time nonlinear system is called globally stable if it is asymptotically stable for all admissible nonnegative initial conditions x0 ∈ Rn+ .

200

3 Stability of Positive Descriptor Systems

The following theorem gives sufficient conditions for the global stability of the fractional positive descriptor continuous-time nonlinear system. Theorem 3.28. The fractional continuous-time nonlinear system consisting of the positive and asymptotically stable fractional descriptor linear part, the nonlinear element satisfying the condition (3.166) and the gain feedback h is globally stable if: 1) (Drazin inverse matrix method case) a) B¯ 2 = 0 and there exists a matrix G ∈ Rn×n such that A¯ C1 + G(In − E¯ E¯ D ) ∈ Mn

(3.235)

is asymptotically stable, where A¯ C1 = A¯ 1 + kh B¯ 1 C;

(3.236)

b) C(In − E¯ E¯ D ) = 0 and there exists a matrix G ∈ Rn×n such that (3.235) is asymptotically stable and − N¯ j A¯ C2 A¯ C1 ∈ Rn×n + , j = 1, . . . , q − 1, j

(3.237)

where A¯ C1 is given by (3.236), A¯ C2 = kh B¯ 2 C

(3.238)

¯ and q is the index of E. 2) (Weierstrass-Kronecker decomposition method case) a) B2 = 0 and

AC1 ∈ Mn 1

(3.239)

AC1 = A1 + kh B1 C1 ;

(3.240)

is asymptotically stable, where

b) C2 = 0, (3.239) is asymptotically stable and − N j AC2 AC1 ∈ Rn+2 ×n 1 , j = 1, . . . , μ − 1, j

(3.241)

where AC1 is given by (3.172), AC2 = kh B2 C1 and μ is the nilpotency index of N .

(3.242)

3.3 Stability of Nonlinear Systems with Positive Linear Parts

201

Proof. The proof can be accomplished in a similar way as for Theorem 3.26.

 

Remark 3.4. Theorems 3.5 and 3.6 can be used to test the stability of (3.235) and (3.239). Example 3.11. Consider the nonlinear system consisting of the fractional descriptor continuous-time linear part (3.231) with ⎡

⎤ ⎡ −1 −0.5 −0.5 21 E = ⎣ −2 −0.5 −0.5 ⎦ , A = ⎣ 4 0 0 −0.5 −0.5 02   C= 312 ,

⎤ ⎡ ⎤ 0 −4 1 ⎦ , B = ⎣ −4 ⎦ , 1 −4

(3.243)

the nonlinear element satisfying the condition (3.166) and the gain feedback h = 0.5. Find k satisfying (3.166) for which the fractional nonlinear system is globally stable. The matrix pencil (E, A) of (3.243) is regular since −s α − 2 −0.5s α − 1 −0.5s α det[Es α − A] = −2s α − 4 −0.5s α −0.5s α − 1 0 −0.5s α − 2 −0.5s α − 1

(3.244)

= s 2α + 6s α + 8 = 0. The characteristic equation of (3.243) is given by p(E,A) (s α ) = det[Es α − A] = s 2α + 6s α + 8 = 0.

(3.245)

From (3.245) it follows that the pair (E, A) has two eigenvalues s1α = −4, s2α = −2 and by condition (2.77) the considered fractional linear part is asymptotically stable for any α ∈ (0; 1). To check the positivity and global stability of the system we use the Drazin inverse matrix method. From (2.28b) for c = 0 we have ⎡

⎤ 0.5 0.125 0.125 E¯ = [−A]−1 E = ⎣ 0 0.25 0.25 ⎦ , 0 0 0 ⎡ ⎤ ⎡ ⎤ −1 0 0 1 A¯ = [−A]−1 A = ⎣ 0 −1 0 ⎦ , B¯ = [−A]−1 B = ⎣ 2 ⎦ 0 0 −1 0

(3.246)

and q = 1 since rank E¯ = rank E¯ 2 . Next, from (B.1) and (B.2) we have ⎡

E¯ D

⎤ ⎡ ⎤ 2 −1 −1 −1 0 0 = ⎣ 0 4 4 ⎦ , A¯ D = A¯ −1 = A¯ = ⎣ 0 −1 0 ⎦ 0 0 0 0 0 −1

(3.247)

202

3 Stability of Positive Descriptor Systems

and ⎡

⎤ ⎡ ⎤ −2 1 1 0 A¯ 1 = E¯ D A¯ = ⎣ 0 −4 −4 ⎦ , B¯ 1 = E¯ D B¯ = ⎣ 8 ⎦ , 0 0 0 0 ⎡ ⎤ ⎡ ⎤ 0 000 (3.248) D D D D B¯ 2 = (I3 − E¯ E¯ ) A¯ B¯ = ⎣ 0 ⎦ , N¯ = (I3 − E¯ E¯ ) A¯ E¯ = ⎣ 0 0 0 ⎦ , 0 000   C(I3 − E¯ E¯ D ) = 0 0 1 . By Theorem 2.8 there exists a matrix ⎡

⎤ 0 0 −1 G = ⎣ 0 −4 0 ⎦ 0 0 −1 ⎤ −2 1 0 A¯ 1 + G(I3 − E¯ E¯ D ) = ⎣ 0 −4 0 ⎦ 0 0 −1

(3.249)

⎡

such that

(3.250)

and the rest of the conditions (2.64) for the matrices (3.248) are satisfied. Thus, the fractional descriptor linear part described by (3.243) is positive. Also, from (3.248) it follows that B¯ 2 = 0 and C(I3 − E¯ E¯ D ) = 0. Therefore, the case a) of Theorem 3.28 can be used to test the global stability of the fractional nonlinear system. Taking into account (3.236), (3.243), (3.248) and h = 0.5 we obtain ⎡

A¯ C1

⎤ −2 1 1 = A¯ 1 + 0.5k B¯ 1 C = ⎣ 12k 4k − 4 8k − 4 ⎦ . 0 0 0

(3.251)

From (3.235), (3.249) and (3.251) we have ⎡

⎤ −2 1 0 A¯ C1 + G(I3 − E¯ E¯ D ) = ⎣ 12k 4k − 4 8k ⎦ ∈ M3 0 0 −1

(3.252)

for any k > 0. By Theorem 3.5 the matrix A¯ C1 is asymptotically stable if and only if all coefficients of the characteristic equation p A¯ C1 (s α ) = det[I3 s α − A¯ C1 ] = s α [s 2α + (6 − 4k)s α + 8 − 20k] = 0

(3.253)

are positive, i.e. if and only if 6 − 4k > 0 and 8 − 20k > 0, and this implies k < 0.4. Therefore, the positive fractional nonlinear system is globally stable if the characteristic u(t) = f (e(t)) of the nonlinear element satisfies the condition (3.166) for k < 0.4.

3.3 Stability of Nonlinear Systems with Positive Linear Parts

203

3.3.4 Fractional Discrete-Time Systems Consider the fractional discrete-time nonlinear feedback system shown in Fig. 3.5, which consists of a fractional positive descriptor linear part, a nonlinear element with characteristic u i = f (ei ) and a positive scalar gain feedback h. The linear part is described by the equations E d α xi+1 = Ad xi + Bd u i , i ∈ Z+ , 0 < α < 1, yi = Cd xi ,

(3.254)

where xi ∈ Rn+ , u i ∈ R+ , yi ∈ R+ are state, input and output vectors, α is the Grünwald-Letnikov fractional difference operator defined by (2.136) and E d , Ad ∈ Rn×n , Bd ∈ Rn×1 , Cd ∈ R1×n . It is assumed that detE d = 0 and det[E d z − Ad ] = 0 for some z ∈ C. Substituting (2.136) into (3.254) we get E d xi+1 = Adα xi +

i

E d c j xi− j + Bd u i ,

j=1

(3.255)

yi = Cd xi , where Adα is given by (2.140b) and the coefficients c j are determined by (2.140c). Following the considerations presented in Sect. 2.2 the system (3.255) can be transformed into the following equivalent forms. 1) Using the Drazin inverse matrix method we obtain α x1,i+1 = A¯ 1d x1,i + B¯ 1d u i , N¯ d α x2,i+1 = x2,i + B¯ 2d u i , yi = Cd xi = Cd [x1,i + x2,i ],

Fig. 3.5 Fractional discrete-time nonlinear feedback system

(3.256)

204

3 Stability of Positive Descriptor Systems

where x1,i , x2,i ∈ Rn+ are given by (2.160c) and B¯ 2d

A¯ 1d = E¯ dD A¯ d , B¯ 1d = E¯ dD B¯ d , = (In − E¯ d E¯ dD ) A¯ dD B¯ d , N¯ d = (In − E¯ d E¯ dD ) A¯ dD E¯ d .

(3.257)

The matrices E¯ d , A¯ d ∈ Rn×n and B¯ d ∈ Rn×1 have the form (2.157b). Substituting (2.136) into (3.256) we get x1,i+1 = E¯ dD A¯ dα x1,i +

L

c j x1,i− j + B¯ 1d u i ,

j=1

N¯ d x2,i+1 = (In 2 + N¯ d α)x2,i +

L

c j N¯ d x2,i− j + B¯ 2d u i ,

(3.258)

j=1

yi = Cd xi = Cd [x1,i + x2,i ], where A¯ dα is given by (2.159), L is the length of practical implementation and x1,−k = 0, x2,−k = 0, k = 1, 2, . . .. Theorem 2.25 gives the positivity conditions of the system (3.258). 2) Using the Weierstrass-Kronecker decomposition method we have α x¯1,i+1 = A1d x¯1,i + B1d u i , Nd α x¯2,i+1 = x2,i + B2d u i , yi = C1d x¯1,i + C2d x¯2,i ,

(3.259)

where x¯1,i ∈ Rn+1 , x¯2,i ∈ Rn+2 are given by (1.193) and the matrices A1d ∈ Rn 1 ×n 1 , B1d ∈ Rn 1 ×1 , B2d ∈ Rn 2 ×1 , Nd ∈ Rn 2 ×n 2 , C1d ∈ R1×n 1 , C2d ∈ R1×n 2 are determined by (1.192) and (1.196). Substituting (2.136) into (3.259) we get x¯1,i+1 = A1dα x¯1,i +

L

c j x¯1,i− j + B1d u i ,

j=1

Nd x¯2,i+1 = (In 2 + Nd α)x¯2,i +

L

c j Nd x¯2,i− j + B2d u i ,

(3.260)

j=1

yi = C1d x¯1,i + C2d x¯2,i , where A1dα is given by (2.179c), L is the length of practical implementation and x¯1,−k = 0, x¯2,−k = 0, k = 1, 2, . . .. Theorem 2.26 gives the positivity conditions of the system (3.260). The characteristic of the nonlinear element is shown in Fig. 3.2 and it satisfies the condition (3.166). It is assumed that the fractional positive descriptor linear part is asymptotically stable.

3.3 Stability of Nonlinear Systems with Positive Linear Parts

205

Definition 3.9. The fractional positive descriptor discrete-time nonlinear system is called: 1) globally stable for given length of practical implementation L if it is asymptotically stable for all admissible nonnegative initial conditions x0 ∈ Rn+ and some finite L; 2) globally stable independent of the length of practical implementation L if it is asymptotically stable for all admissible nonnegative initial conditions x0 ∈ Rn+ and L → ∞. The following theorems give sufficient conditions for the global stability of the fractional positive descriptor discrete-time nonlinear system. Theorem 3.29. The fractional discrete-time nonlinear system consisting of the positive and asymptotically stable fractional descriptor linear part, the nonlinear element satisfying the condition (3.166) and the gain feedback h is globally stable for given length of practical implementation L if: 1) (Drazin inverse matrix method case) a) B¯ 2d = 0 and there exists a matrix G d ∈ Rn×n such that A¯ C1d + E¯ E¯ D α +

L

c j In + G d (In − E¯ d E¯ dD ) ∈ Rn×n +

(3.261)

j=1

is asymptotically stable, where A¯ C1d = A¯ 1d + kh B¯ 1d Cd ;

(3.262)

b) Cd (In − E¯ d E¯ dD ) = 0 and there exists a matrix G d ∈ Rn×n such that (3.261) is asymptotically stable and j j − N¯ d A¯ C2d A¯ C1d ∈ Rn×n + , j = 1, . . . , q − 1,

(3.263)

where A¯ C1d is given by (3.262), A¯ C2d = kh B¯ 2d Cd

(3.264)

and q is the index of E¯ d . 2) (Weierstrass-Kronecker decomposition method case) a) B2d = 0 and

⎛ AC1d + In 1 ⎝α +

L j=1

⎞ c j ⎠ ∈ Rn+1 ×n 1

(3.265)

206

3 Stability of Positive Descriptor Systems

is asymptotically stable, where AC1d = A1d + kh B1d C1d ;

(3.266)

b) C2d = 0, (3.265) is asymptotically stable and − Nd AC2d AC1d ∈ Rn+2 ×n 1 , j = 1, . . . , μ − 1, j

j

(3.267)

where AC1d is given by (3.266), AC2d = kh B2d C1d

(3.268)

and μ is the nilpotency index of Nd . Proof. The proof will be accomplished for the Weierstrass-Kronecker decomposition method. Taking into account (3.260) for B2d = 0 we obtain x¯1,i+1 = A1dα x¯1,i +

L

c j x¯1,i− j + B1d u i ,

j=1

Nd x¯2,i+1 = (In 2 + Nd α)x¯2,i +

L

c j Nd x¯2,i− j ,

(3.269)

j=1

yi = C1d x¯1,i + C2d x¯2,i , From the second equation of (3.210) it follows that x¯2,i = 0 since Nd is a nilpotent matrix. Assuming u i = kei and taking into account that ei = hyi from (3.269) we obtain L c j x¯1,i− j , x¯1,i+1 = (A1dα + kh B1 C1 )x¯1,i + (3.270) j=1 x¯2,i = 0 By Remark 3.1 the positive fractional system (3.270) is asymptotically stable if and only if the positive system ⎡

⎛

x¯1,i+1 = ⎣ AC1d + In 1 ⎝α +

L j=1

⎞⎤ c j ⎠⎦ x¯1,i ,

(3.271)

x¯2,i = 0 is asymptotically stable, where AC1d is given by (3.266). As the Lyapunov function V (x¯1,i ) we choose (3.272) V (x¯1,i ) = λT x¯1,i ≥ 0 for x¯1,i ∈ Rn+1 ,

3.3 Stability of Nonlinear Systems with Positive Linear Parts

207

  where λ = λ1 . . . λn 1 is a strictly positive vector, i.e. λ j > 0, j = 1, . . . , n 1 . Using (3.269)–(3.272) we obtain V (x¯1,i ) = V (x¯1,i+1 ) − V (x¯1,i ) = λT x¯1,i+1 − λT x¯1,i ⎡⎛ ⎞ ⎤ L = λT ⎣⎝ A1dα + c j In 1 − In 1 ⎠ x¯1,i + B1d u i ⎦ j=1

⎡⎛ = λT ⎣⎝ A1dα +

⎞

L

⎤

c j In 1 − In 1 ⎠ x¯1,i + B1d h f (ei )⎦

(3.273)

j=1

⎧⎡ ⎫ ⎛ ⎞ ⎤ L ⎨ ⎬ ≤ λT ⎣ AC1d + In 1 ⎝α + c j ⎠ − In 1 ⎦ x¯1,i ⎩ ⎭ j=1

since u i = f (ei ) ≤ kei = kC1d x¯1,i . It is well-known [49] that elements on the main diagonal of the state matrix of a positive and asymptotically stable discrete-time system are less than 1. Therefore, from (3.273) it follows that V (x¯1,i ) < 0 if the condition (3.265) is satisfied with B2d = 0 and the nonlinear system is globally stable. Now let us consider (3.260) for C2d = 0. In this case we have x¯1,i+1 = A1dα x¯1,i +

L

c j x¯1,i− j + B1d u i ,

j=1

Nd x¯2,i+1 = (In 2 + Nd α)x¯2,i +

L

c j Nd x¯2,i− j + B2d u i ,

(3.274)

j=1

yi = C1d x¯1,i , Assuming u i = kei and taking into account that ei = hyi from (3.274) we obtain x¯1,i+1 = (AC1d + In 1 α)x¯1,i +

L

c j x¯1,i− j ,

j=1

Nd x¯2,i = kh B2d C1d x¯1,i + (In 2 + Nd α)x¯2,i +

L

(3.275) c j Nd x¯2,i− j ,

j=1

where AC1d is given by (3.266). The second equation of (3.275) can be solved in a similar way to the Eq. (2.179b). From (3.275) and (2.181) we have x¯2,i = −

μ−1 j=0

N j AC2d θi, j ,

(3.276)

208

3 Stability of Positive Descriptor Systems

where μ is the nilpotency index of the matrix Nd , AC2d is given by (3.268) and θi,0 = x¯1,i , θi,1 = θi+1,0 − αθi,0 −

L

c j θi− j,0 = x¯1,i+1 − α x¯1,i −

j=1

θi,2 = θi+1,1 − αθi,1 −

L

L

c j x¯1,i− j ,

j=1

c j θi− j,1

j=1

= x¯1,i+2 − 2α x¯1,i+1 + α 2 x¯1,i + 2α

L

c j x¯1,i− j −

j=1

−

L

c j x¯1,i− j+1 +

j=1

L

cj

j=1

L− j

L+1

c j x¯1,i− j+1 (3.277)

j=1

cl x¯1,i− j−l

l=1

.. . θi,μ−1 = θi+1,μ−2 − αθi,μ−2 −

L

c j θi− j,μ−2 .

j=1

Taking into account that x¯1,i+1 = (AC1d + In 1 α)x¯1,i +

L

c j x¯1,i− j ,

j=1

x¯1,i+2 = (AC1d + In 1 α)x¯1,i+1 +

L+1

c j x¯1,i− j+1 ,

j=1

= (AC1d + In 1 α)2 x¯1,i + (AC1d + In 1 α)

L

c j x¯1,i− j +

j=1

L+1

c j x¯1,i− j+1

j=1

.. . x¯1,i+l = (AC1d + In 1 α)x¯1,i+l−1 +

L+l−1

c j x¯1,i− j+l−1

j=1

(3.278)

3.3 Stability of Nonlinear Systems with Positive Linear Parts

209

from (3.277) and (3.278) we obtain θi,0 = x¯1,i , θi,1 = AC1d x¯1,i , 2 θi,2 = AC1d x¯1,i , .. .

(3.279)

μ−1

θi,μ−1 = AC1d x¯1,i . Using (3.276) and (3.279) we have x¯2,i = −

μ−1

j

j

Nd AC2d AC1d x¯1,i .

(3.280)

j=0

Choosing the Lyapunov function in the form (3.272) we get (3.273) and V (x¯1,i ) < 0 if the condition (3.265) is satisfied with C2d = 0. From (3.280) it follows that if lim x¯1,i = 0 then also lim x¯2,i = 0. The condition (3.267) ensures x¯2,i ∈ Rn+2 . i→∞

i→∞

Therefore, the nonlinear system is globally stable. The proof for the Drazin inverse matrix method can be accomplished similarly. Note that if Cd (In − E¯ d E¯ dD ) = 0, then Cd x2,i = 0 since x2,i = (In − E¯ d E¯ dD )xi . The matrix G d ∈ Rn×n and the term G d (In − E¯ d E¯ dD ) eliminates from the matrix A¯ 1d = E¯ dD A¯ d unimportant entires that are further canceled through multiplication by x1,i ∈   Im E¯ d E¯ dD . Theorem 3.30. The fractional discrete-time nonlinear system consisting of the positive and asymptotically stable fractional descriptor linear part, the nonlinear element satisfying the condition (3.166) and the gain feedback h is globally stable independent of the length of practical implementation L if: 1) (Drazin inverse matrix method case) a) B¯ 2d = 0 and there exists a matrix G d ∈ Rn×n such that A¯ C1d + In + (G d − In α)(In − E¯ d E¯ dD ) ∈ Rn×n +

(3.281)

is asymptotically stable, where A¯ C1d is given by (3.262); b) Cd (In − E¯ d E¯ dD ) = 0 and there exists a matrix G d ∈ Rn×n such that (3.281) is asymptotically stable and j j − N¯ d A¯ C2d A¯ C1d ∈ Rn×n + , j = 1, . . . , q − 1,

(3.282)

where A¯ C1d and A¯ C2d are given by (3.262) and (3.264), respectively and q is the index of E¯ d .

210

3 Stability of Positive Descriptor Systems

2) (Weierstrass-Kronecker decomposition method case) a) B2d = 0 and

AC1d + In 1 ∈ Rn+1 ×n 1

(3.283)

is asymptotically stable, where A¯ C1d is given by (3.262); b) C2d = 0, (3.283) is asymptotically stable and − Nd AC2d AC1d ∈ Rn+2 ×n 1 , j = 1, . . . , μ − 1, j

j

(3.284)

where A¯ C1d and A¯ C2d are given by (3.262) and (3.264), respectively and μ is the nilpotency index of Nd . Proof. The proof follows immediately from Theorem 3.29 and (2.232).

 

Remark 3.5. Theorems 3.9 and 3.10 can be used to test the stability of (3.261) and (3.265). Theorems 3.11 and 3.12 can be used to test the stability of (3.281) and (3.283). Example 3.12. Consider the nonlinear system consisting of the fractional descriptor discrete-time linear part (3.254) with α = 0.3 and ⎡

⎤ ⎡ ⎤ ⎡ ⎤ −10 −30 0 110 −1 E d = ⎣ 0 −10 0 ⎦ , Ad = ⎣ 0 1 0 ⎦ , Bd = ⎣ 0 ⎦ , 0 −20 0 001 −1   Cd = 2 2 0 ,

(3.285)

the nonlinear element satisfying the condition (3.166) and the gain feedback h = 0.5. Find k satisfying (3.166) for which the fractional nonlinear system is globally stable independent of the length of practical implementation L. The matrix pencil (E d , Ad ) of (3.285) is regular since −10z − 1 −30z − 1 0 0 −10z − 1 0 = (−10z + 1)2 = 0. det[E d z − Ad ] = 0 −20z −1

(3.286)

To check the asymptotic stability of the fractional linear part (3.285) we can use the approach given in Sect. 2.2.5. Therefore, by Theorem 2.28 we have to compute the characteristic equation given by det[E d (z − 1)0.3 z 0.7 − Ad ] = −[10(z − 1)0.3 z 0.7 + 1]2 = 0.

(3.287)

The fractional linear part (3.285) with α = 0.3 is asymptotically stable since roots of (3.287) are z 1 = z 2 = −0.0367. To check the positivity and global stability of the system we use the Drazin inverse matrix method. From (2.157b) for c = 0 we have

3.3 Stability of Nonlinear Systems with Positive Linear Parts

211

⎡

⎤ ⎡ ⎤ 10 20 0 −1 0 0 E¯ d = [−Ad ]−1 E d = ⎣ 0 10 0 ⎦ , A¯ d = [−Ad ]−1 Ad = ⎣ 0 −1 0 ⎦ , 0 20 0 0 0 −1 ⎡ ⎤ ⎡ ⎤ 26 0 1 A¯ dα = A¯ d + 0.4 E¯ d = ⎣ 0 2 0 ⎦ B¯ d = [−Ad ]−1 Bd = ⎣ 0 ⎦ 0 6 −1 1 (3.288) and q = 1 since rank E¯ d = rank E¯ d2 . Next, from (B.1) and (B.2) we have ⎡

E¯ dD

⎤ ⎡ ⎤ 0.1 −0.2 0 −1 0 0 ¯ ⎣ 0 −1 0 ⎦ = ⎣ 0 0.1 0 ⎦ , A¯ dD = A¯ −1 d = Ad = 0 0.2 0 0 0 −1

(3.289)

and ⎡

⎤ ⎡ ⎤ −0.1 0.2 0 0.2 0.2 0 A¯ 1d = E¯ dD A¯ d = ⎣ 0 −0.1 0 ⎦ , E¯ dD A¯ dα = ⎣ 0 0.2 0 ⎦ 0 −0.2 0 0 0.4 0 ⎡ ⎤ ⎡ ⎤ 0.1 0 B¯ 1d = E¯ dD B¯ d = ⎣ 0 ⎦ , B¯ 2d = (I3 − E¯ d E¯ dD ) A¯ dD B¯ d = ⎣ 0 ⎦ , 0 −1 ⎡ ⎤ 000   N¯ d = (I3 − E¯ d E¯ dD ) A¯ dD E¯ d = ⎣ 0 0 0 ⎦ , Cd (I3 − E¯ E¯ D ) = 0 0 0 000

(3.290)

By Theorem 2.25 the conditions (2.197) for the matrices (3.290) are satisfied and the fractional descriptor linear part described by (3.285) is positive. Also, from (3.290) it follows that B¯ 2d = 0 and Cd (I3 − E¯ E¯ D ) = 0. Therefore, the case b) of Theorem 3.30 can be used to test the global stability independent of the length of practical implementation L for the considered system. Taking into account (3.262), (3.264), (3.285), (3.290) and h = 0.5 we obtain ⎡

⎤ 2k − 0.1 2k + 0.2 0 0 −0.1 0 ⎦ , A¯ C1d = A¯ 1d + 0.5k B¯ 1d Cd = ⎣ 0 −0.2 0 ⎡ ⎤ 0 0 0 0 0⎦. A¯ C2d = A¯ 1d + 0.5k B¯ 1d Cd = ⎣ 0 −10k −10k 0

(3.291)

212

3 Stability of Positive Descriptor Systems

From (3.281) for G d = 0 and (3.291) we have ⎡

⎤ 2k + 0.9 2k + 0.2 0 0 0.9 0 ⎦ ∈ R3×3 D¯ 1,∞ = A¯ C1d + I3 − α(I3 − E¯ E¯ D ) = ⎣ + 0 0.4 0.7 (3.292) for any k > 0. By Theorem 3.11 the matrix D¯ 1,∞ is asymptotically stable if and only if all coefficients of the characteristic equation p D¯ 1,∞ (z + 1) = det[I3 (z + 1) − D¯ 1,∞ ] = 0.1(z + 0.3)[10z 2 + (2 − 20k)z + 0.1 − 2k] = 0

(3.293)

are positive, i.e. if and only if 2 − 20k > 0 and 0.1 − 2k > 0, and this implies k < 0.05. Taking into account (3.282) the matrix ⎡

0 − N¯ d0 A¯ C2d A¯ C1d = − A¯ C2d

⎤ 0 0 0 = ⎣ 0 0 0 ⎦ ∈ R3×3 + 10k 10k 0

(3.294)

for any k > 0. Therefore, the positive fractional nonlinear system is globally stable independent of the length of practical implementation L if the characteristic u(t) = f (e(t)) of the nonlinear element satisfies the condition (3.166) for k < 0.05.

3.3.5 Analysis of Global Stability of Descriptor Continuous-Time Nonlinear Feedback Systems by the Use of Nyquist Plots The following analysis presents an extension of Kudrewicz theorem [104] to positive descriptor continuous-time nonlinear systems (see also [78, 80]). Consider the continuous-time nonlinear feedback system shown in Fig. 3.6, which consists of a positive descriptor linear part and a nonlinear element with characteristic u(t) = f (e(t)). The linear part is described by (3.162). It is assumed that the linear part with transfer function T (s) is positive, but not necessarily asymptotically stable. The characteristic of the nonlinear element is shown

Fig. 3.6 Continuous-time nonlinear feedback system

3.3 Stability of Nonlinear Systems with Positive Linear Parts

213

in Fig. 3.2 and it satisfies the condition (3.166). It is well-known [47] that the transfer function of a descriptor system consists of two parts, i.e. T (s) = C[Es − A]−1 B = Tsp (s) + T p (s),

(3.295)

where Tsp (s) is a strictly proper part and T p (s) is a polynomial part. The following equivalent forms of transfer matrices can be obtained using Drazin inverse matrix method and Weierstrass-Kronecker decomposition method. Theorem 3.31. The transfer matrix of the descriptor continuous-time linear system (3.162) has the form: 1) for the Drazin inverse matrix method T (s) = C[In s − A¯ 1 ]−1 B¯ 1 + C[ N¯ s − In ]−1 B¯ 2 = Tsp (s) + T p (s);

(3.296)

2) for the Weierstrass-Kronecker decomposition method T (s) = C1 [In 1 s − A1 ]−1 B1 + C2 [N s − In 2 ]−1 B2 = Tsp (s) + T p (s), (3.297) where the matrices A¯ 1 , B¯ 1 , B¯ 2 , N¯ are given by (3.164) and the matrices A1 , B1 , B2 , C1 , C2 , N are determined by (1.52) and (1.56). Proof. Applying the Laplace transform to (3.163) with zero initial conditions we obtain s X 1 (s) = A¯ 1 X 1 (s) + B¯ 1 U (s), (3.298) N¯ s X 2 (s) = X 2 (s) + B¯ 2 U (s), Y (s) = C[X 1 (s) + X 2 (s)], and

X 1 (s) = [In s − A¯ 1 ]−1 B¯ 1 U (s), X 2 (s) = [ N¯ s − In ]−1 B¯ 2 U (s),

(3.299)

Y (s) = C[X 1 (s) + X 2 (s)]. From (3.299) we have T (s) =

Y (s) = C[In s − A¯ 1 ]−1 B¯ 1 + C[ N¯ s − In ]−1 B¯ 2 , U (s)

(3.300)

which is equal to (3.296). The proof for the transfer matrix (3.297) can be performed similarly applying the Laplace transform to (3.165) with zero initial conditions.   In many cases, if the linear part is unstable, then by a suitable choice of the gain k1 we may obtain (Fig. 3.7) an asymptotically stable positive linear part with the transfer function T (s) (3.301) T1 (s) = 1 + k1 T (s)

214

3 Stability of Positive Descriptor Systems

Fig. 3.7 Continuous-time nonlinear feedback system with the gain k1 Fig. 3.8 Characteristic of the nonlinear element with the gain k1

and a nonlinear element with the characteristic f 1 (e(t)) = f (e(t)) − k1 e(t) (Fig. 3.8) satisfying the condition k1 ≤

f (e(t)) ≤ k2 , k2 < ∞. e(t)

(3.302)

According to Definition 3.6 the nonlinear continuous-time system is called globally stable if it is asymptotically stable for all admissible nonnegative initial conditions x0 ∈ Rn+ . Definition with center in the point   3.10. The circle in the plane (P(ω), Q(ω)) k1 +k2 k2 −k1 1 1 − 2k1 k2 , 0 and radius 2k1 k2 is called the − k1 , − k2 circle (see Fig. 3.9). Theorem 3.32. The continuous-time nonlinear system consisting of the positive and asymptotically stable descriptor linear part with the transfer function T1 (s) and the nonlinear element satisfying the condition (3.302) is globally stable if

3.3 Stability of Nonlinear Systems with Positive Linear Parts

215

Fig. 3.9 Illustration of Definition 3.10 and Theorem 3.32

1) (Drazin inverse matrix method case) B¯ 2 = 0 or C(In − E¯ E¯ D ) = 0;

(3.303)

2) (Weierstrass-Kronecker decomposition method case) B2 = 0 or C2 = 0

(3.304)

and the Nyquist plot of T1 ( jω)  = P(ω)+ j Q(ω) of the linear part is located on the right-hand side of the circle − k11 , − k12 . Proof. The proof will be accomplished for the Weierstrass-Kronecker decomposition method. From (3.295) and (3.301) we have T1 (s) =

Tsp (s) + T p (s) T (s) = , 1 + k1 T (s) 1 + k1 [Tsp (s) + T p (s)]

(3.305)

where Tsp (s) = C1 [In 1 s − A1 ]−1 B1 and T p (s) = C2 [N s − In 2 ]−1 B2 . If the condition (3.304) is satisfied, then from (3.305) we obtain T1 (s) =

Tsp (s) . 1 + k1 Tsp (s)

(3.306)

From (3.306) we have ReT1 ( jω) +

1 >0 k

for ω ≥ 0 and k = k2 − k1 > 0. Taking into account that

(3.307)

216

3 Stability of Positive Descriptor Systems

ReT1 ( jω) +



1 1 T ( jω) + = Re k2 − k1 1 + k1 T ( jω) k2 − k1 

1 1 + k2 T ( jω) = Re k2 − k1 1 + k1 T ( jω)

(3.308)

and that the border of asymptotic stability is the jω axis, we obtain jω =

1 + k2 [P(ω) + j Q(ω)] 1 + k1 [P(ω) + j Q(ω)]

(3.309)

or jω{1 + k1 [P(ω) + j Q(ω)]} = 1 + k2 [P(ω) + j Q(ω)].

(3.310)

From (3.310) we have −ωk1 Q(ω) = 1 + k2 P(ω), ω[1 + k1 P(ω)] = k2 Q(ω)

(3.311)

and after elimination of ω we obtain

or

[1 + k1 P(ω)][1 + k2 P(ω)] + k1 k2 Q 2 (ω) = 0

(3.312)

k1 + k2 1 + P(ω) + P 2 (ω) + Q 2 (ω) = 0. k1 k2 k1 k2

(3.313)

Note that (3.313) can be rewritten in the form of the equation

P(ω) +

k1 + k2 2k1 k2

2

 + Q 2 (ω) =

k2 − k1 2k1 k2

2 ,

(3.314)

  which describes the circle − k11 , − k12 (Fig. 3.9). This completes the proof.

 

Example 3.13. (Continuation of Example 3.9) Consider the nonlinear system consisting of the positive descriptor continuous-time linear part (3.162) with (3.184) and the nonlinear element with the characteristic u(t) = f (e(t)) satisfying the condition (3.302). Using (3.184) and (3.295) we obtain the transfer function T (s) = C[Es − A]−1 B = where Tsp (s) = T (s) =

12s + 28 , s 2 + 4s + 4

12s + 28 , T p (s) = 0. + 4s + 4

s2

(3.315a)

(3.315b)

To check the global stability of the system we use two approaches as follows.

3.3 Stability of Nonlinear Systems with Positive Linear Parts

217

1) Drazin inverse matrix method case From (3.189) it follows that C(I3 − E¯ E¯ D ) = 0. Using (3.184), (3.189) and (3.296) we have 12s + 28 (3.316) T (s) = Tsp (s) = C[I3 s − A¯ 1 ]−1 B¯ 1 = 2 s + 4s + 4 and T p (s) = 0, which is equal to (3.315). The Nyquist plot of T ( jω) =

12 jω + 28 ( jω)2 + 4( jω) + 4

(3.317)

is shown in Fig. 3.10 and it is located in the fourth quarter of the plane (P(ω), Q(ω)). Therefore, by Theorem 3.32 the nonlinear system is globally stable for all nonlinear elements with characteristic u(t) = f (e(t)) located in the first and third quarter (Fig. 3.8) since for any positive k2> k1 ≥ 0 the Nyquist plot is located on the righthand side of the circle − k11 , − k12 . 2) Weierstrass-Kronecker decomposition method case From (3.194) it follows that C2 = 0. Using (3.194) and (3.297) we have T (s) = Tsp (s) = C1 [I2 s − A1 ]−1 B1 =

12s + 28 s 2 + 4s + 4

(3.318)

and T p (s) = 0, which is equal to (3.315) and (3.316). The conclusions are the same as for the Drazin inverse matrix method case.

Fig. 3.10 Nyquist plot of (3.317)

218

3 Stability of Positive Descriptor Systems

Remark 3.6. Similar analysis can be performed for fractional positive descriptor continuous-time nonlinear systems with the linear part described by the transfer function (3.319) T (s α ) = C[Es α − A]−1 B = Tsp (s α ) + T p (s α ), where Tsp (s α ) is a strictly proper part and T p (s α ) is a polynomial part.

3.4 Concluding Remarks In this chapter stability of positive descriptor systems have been investigated. In the subsequent sections stability tests for positive linear systems, stability of positive interval systems and stability of nonlinear systems with positive linear parts have been analyzed. In positive descriptor systems some other methods of stability analysis can be applied, in contrast to nonpositive descriptor systems, where an approach based on the location of the roots of the characteristic equation is mostly used. To make use of the first of the presented methods we have to check whether the coefficients of the corresponding characteristic equations are positive (Theorems 3.1, 3.3, 3.5, 3.9 and 3.11). The second of the methods is based on the search for some strictly positive vector and it results from the Lyapunov method (Theorems 3.2, 3.4, 3.6, 3.10 and 3.12). To investigate the stability of positive descriptor interval systems the approach based on the convex linear combination of matrices can be applied (Theorems 3.14, 3.18, 3.20, 3.22 and 3.23). Also, an extension of the Kharitonov theorem (Theorem 3.15) can be used, from which it follows that positive descriptor interval systems are asymptotically stable if and only if the lower bounds of corresponding characteristic equations are positive (Theorems 3.16, 3.19, 3.21, 3.24 and 3.25). The positive descriptor nonlinear system is called globally stable if it is asymptotically stable for all admissible nonnegative initial conditions. Sufficient conditions for the global stability of such systems can be established using Lyapunov method (Theorems 3.26, 3.27, 3.28, 3.29 and 3.30). Analysis of global stability of descriptor nonlinear feedback systems can be performed using Nyquist plots, i.e. using an extension of Kudrewicz theorem (Theorem 3.32). The results presented in Sect. 3.3.5 can be easily extended to fractional and interval systems using the approaches given in the previous sections of this chapter.

Appendix A

Extensions of the Cayley-Hamilton Theorem for Descriptor Linear Systems

A.1 Cayley-Hamilton Theorem for Descriptor Linear Systems with Commuting Matrices Consider the descriptor linear system (1.1a) with the commuting matrices E A = AE.

(A.1)

Theorem A.1. Let the condition (1.2) be satisfied and det[Eλ − A] =

r 

a k λk .

(A.2)

k=0

Then the matrices E, A satisfy the equation r 

ak Ak E n−k = 0.

(A.3)

k=0

Proof. Let [Eλ − A]ad = Bn−1 λn−1 + . . . + B1 λ + B0

(A.4)

be the adjoint matrix of the matrix [Eλ − A]. From the definition of the inverse matrix and (A.2), (A.4) we have [Eλ − A][Bn−1 λn−1 + . . . + B1 λ + B0 ] = In (ar λr + ar −1 λr −1 + . . . + a1 λ + a0 ). (A.5) © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Kaczorek and K. Borawski, Descriptor Systems of Integer and Fractional Orders, Studies in Systems, Decision and Control 367, https://doi.org/10.1007/978-3-030-72480-1

219

220

Appendix A: Extensions of the Cayley-Hamilton Theorem for Descriptor Linear Systems

Comparing the coefficients at the same powers of λ in (A.5) we obtain ⎡

E ⎢ −A ⎢ ⎢ 0 ⎢ ⎢ .. ⎢ . ⎢ ⎢ 0 ⎢ ⎣ 0 0

⎤ ⎡ 0 ⎤ Bn−1 . ⎥ ⎥ ⎢ Bn−2 ⎥ ⎢ ⎥⎢ ⎥ ⎢ .. ⎥ ⎥ ⎥ ⎢ Bn−3 ⎥ ⎢ ⎥⎢ ⎥ ⎢ 0 ⎥ ⎥ ⎥ ⎢ .. ⎥ ⎢ ar In ⎥ ⎥⎢ . ⎥ = ⎢ ⎥. ⎥⎢ ⎥ ⎢ ⎥ . ⎢ B2 ⎥ ⎢ . 0 0 . . . −A E 0 ⎥ ⎥⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ 0 0 . . . 0 −A E ⎦ ⎣ B1 ⎦ ⎣ a1 In ⎦ 0 0 . . . 0 0 −A B0 a0 In

0 E −A .. .

0 0 E .. .

... ... ... .. .

0 0 0 .. .

0 0 0 .. .

0 0 0 .. .

⎤⎡

(A.6)

Premultiplying (A.6) by the matrix

An An−1 E An−2 E . . . AE n−1 E n



and using (A.1) we obtain the Eq. (A.3).

(A.7)  

Example A.1. Consider the pair of matrices ⎡

⎤ ⎡ ⎤ 010 a00 E = ⎣ 0 0 1 ⎦ , A = ⎣ 0 a 0 ⎦ , a = 0, 010 00a

(A.8)

which satisfies the condition (A.1) for any value of a. The characteristic polynomial of the pair (A.8) has the form    −a λ 0    det[Eλ − A] =  0 −a λ  = a(λ2 − a 2 )  0 λ −a 

(A.9)

and its characteristic equation is λ2 − a 2 = 0.

(A.10)

r = 2 and a2 = 1, a1 = 0, a0 = −a 2 .

(A.11)

In this case we have

Taking into account that ⎡

⎤ ⎡ ⎤ 010 0 a2 0 E 3 = ⎣ 0 0 1 ⎦ , A2 E = ⎣ 0 0 a 2 ⎦ 010 0 a2 0

(A.12)

Appendix A: Extensions of the Cayley-Hamilton Theorem for Descriptor Linear Systems

221

and using (A.11) we obtain 2  k=0

ak Ak E n−k = −a 2 E 3 + A2 E ⎡

⎤ ⎡ ⎤ 010 0 a2 0 = −a 2 ⎣ 0 0 1 ⎦ + ⎣ 0 0 a 2 ⎦ 010 0 a2 0 ⎡ ⎤ 000 = ⎣0 0 0⎦. 000

(A.13)

Therefore, the matrices (A.8) satisfy their characteristic Eq. (A.10).

A.2 Cayley-Hamilton Theorem for Descriptor Linear Systems with Noncommuting Matrices Consider the descriptor linear system (1.1a) with the noncommuting matrices E A = AE.

(A.14)

Let the condition (1.2) hold and a constant c be chosen such that det[Ec − A] = 0. Following Lemma 1.1 the matrices

satisfy the condition

E¯ = [Ec − A]−1 E, A¯ = [Ec − A]−1 A

(A.15)

¯ E¯ A¯ = A¯ E.

(A.16)

Theorem A.2. Let (A.2) be the characteristic polynomial of the pair satisfying the condition (A.16). Then r  bk E¯ n−k = 0 (A.17) k=0

or

n 

ck A¯ k = 0,

(A.18)

k=0

where bk = (−1)k

r  j j−k c aj k j=k

(A.19)

222

Appendix A: Extensions of the Cayley-Hamilton Theorem for Descriptor Linear Systems

and ck =

n  j=n−k

an− j j , ak = 0 for k > r. n − k cj

(A.20)

¯ A) ¯ we have Proof. By Theorem A.1 for the pair ( E, r 

ak A¯ k E¯ n−k = 0.

(A.21)

k=0

From (A.15) we get ¯ − A¯ = [Ec − A]−1 [Ec − A] = In , Ec

(A.22)

¯ − In . A¯ = Ec

(A.23)

and

Substitution of (A.23) into (A.21) yields r 

¯ − In )k E¯ n−k = ak ( Ec

k=0

r 

bk E¯ n−k = 0,

(A.24)

k=0

where the coefficients bk are defined by (A.19). Similarly, substituting

1 ¯ A + In E¯ = c

(A.25)

(obtained from (A.23)) into (A.21) we obtain   n   n−k  k 1 ¯ ¯ A + In ak A = ck A¯ k = 0, c k=0 k=0

r 

where the coefficients ck are defined by (A.20).

(A.26)  

Example A.2. Consider the pair of matrices ⎡

⎤ ⎡ ⎤ 100 0 −1 0 E = ⎣ 0 1 0 ⎦ , A = ⎣ 0 −1 −2 ⎦ , 000 0 0 −1

(A.27)

which satisfies the condition (A.14) since ⎡

⎤ ⎡ ⎤ 0 −1 0 0 −1 0 E A = ⎣ 0 −1 −2 ⎦ , AE = ⎣ 0 −1 0 ⎦ . 0 0 0 0 0 0

(A.28)

Appendix A: Extensions of the Cayley-Hamilton Theorem for Descriptor Linear Systems

223

Choosing in (A.15) c = 1 we obtain ⎡

⎤ 1 −0.5 0 E¯ = [E − A]−1 E = ⎣ 0 0.5 0 ⎦ 0 0 0 ⎤ 0 −0.5 0 A¯ = [E − A]−1 A = ⎣ 0 −0.5 0 ⎦ . 0 0 −1

(A.29)

⎡

and

(A.30)

¯ A) ¯ has the form The characteristic polynomial of the pair ( E,   λ −0.5λ + 0.5  ¯ − A] ¯ =  0 0.5λ + 0.5 det[ Eλ  0 0

 0  0  = 0.5λ2 + 0.5λ 1

(A.31)

and a2 = 0.5, a1 = 0.5, a0 = 0.

(A.32)

Using (A.21), (A.32) and (A.29), (A.30) we obtain 2  k=0

ak A¯ k E¯ 3−k = a1 A¯ E¯ 2 + a2 A¯ 2 E¯ ⎤⎡ ⎤ 0 −0.5 0 1 −0.75 0 = 0.5 ⎣ 0 −0.5 0 ⎦ ⎣ 0 0.25 0 ⎦ 0 0 −1 0 0 0 ⎡ ⎤⎡ ⎤ 0 0.25 0 1 −0.5 0 + 0.5 ⎣ 0 0.25 0 ⎦ ⎣ 0 0.5 0 ⎦ 0 0 −1 0 0 0 ⎡ ⎤ 000 = ⎣0 0 0⎦. 000 ⎡

(A.33)

Using (A.32) and (A.19) we obtain b0 = a1 + a2 = 1, b1 = −(a1 + 2a2 ) = −1.5, b2 = a2 = 0.5

(A.34)

224

Appendix A: Extensions of the Cayley-Hamilton Theorem for Descriptor Linear Systems

and from (A.17), (A.34) we have 2  k=0

bk E¯ 3−k = b0 E¯ 3 + b1 E¯ 2 + b2 E¯ ⎡ ⎤2 ⎤3 1 −0.5 0 1 −0.5 0 = ⎣ 0 0.5 0 ⎦ − 1.5 ⎣ 0 0.5 0 ⎦ 0 0 0 0 0 0 ⎤ ⎡ ⎤ ⎡ 1 −0.5 0 000 + 0.5 ⎣ 0 0.5 0 ⎦ = ⎣ 0 0 0 ⎦ . 0 0 0 000 ⎡

(A.35)

Similarly, using (A.20) and (A.32) we obtain c0 = 0, c1 = a1 = 0.5, c2 = 2a1 + a2 = 1.5, c3 = a1 + a2 = 1

(A.36)

and from (A.18), (A.36) we have 3  k=0

ck A¯ k = c1 A¯ + c2 A¯ 2 + c3 A¯ 3 ⎡

⎤ ⎡ ⎤2 0 −0.5 0 0 −0.5 0 = 0.5 ⎣ 0 −0.5 0 ⎦ + 1.5 ⎣ 0 −0.5 0 ⎦ 0 0 −1 0 0 −1 ⎡ ⎤3 ⎡ ⎤ 0 −0.5 0 000 + ⎣ 0 −0.5 0 ⎦ = ⎣ 0 0 0 ⎦ . 0 0 −1 000

(A.37)

Therefore, the matrices (A.29) and (A.30) satisfy their characteristic equations.

A.3 Cayley-Hamilton Theorem for Drazin Inverse Matrices Theorem A.3. If

then

det[In λ − E] = λn + an−1 λn−1 + . . . + a1 λ,

(A.38)

 2  n−1  D n + E = 0, a1 E D + a2 E D + . . . + an−1 E D

(A.39)

where E D ∈ Rn×n is the Drazin inverse of the matrix E.

Appendix A: Extensions of the Cayley-Hamilton Theorem for Descriptor Linear Systems

225

Proof. Using (A.38) and the classical Cayley-Hamilton theorem we obtain E n + an−1 E n−1 + . . . + a2 E 2 + a1 E = 0.

(A.40)

Premultiplying and postmultiplying (A.40) by the Drazin inverse matrix E D we obtain E D E n E D + an−1 E D E n−1 E D + . . . + a2 E D E 2 E D + a1 E D E E D = 0

(A.41)

and using (1.26a), (1.26b) we have E D E n−1 + an−1 E D E n−2 + . . . + a2 E D E + a1 E D = 0

(A.42)

E D E k E D = E D E E D E k−1 = E D E k−1 for k = 1, 2, . . . , n.

(A.43)

since

Postmultiplying (A.42) by E D and using (A.43) we obtain  2 E D E n−2 + an−1 E D E n−3 + . . . + a2 E D + a1 E D = 0. Repeating n − 2 times this procedure we obtain (A.39).

(A.44)  

Example A.3. The Drazin inverse of the singular matrix ⎡

⎤ 1 0 −1 E = ⎣0 1 0 ⎦ 0 −1 0 ⎡

has the form ED

⎤ 1 −2 −1 = ⎣0 1 0 ⎦. 0 −1 0

(A.45)

(A.46)

The characteristic polynomial of (A.45) is given by  λ − 1 0  det[I3 λ − E] =  0 λ − 1  0 1

 1  0  = λ3 − 2λ2 + λ. s

(A.47)

226

Appendix A: Extensions of the Cayley-Hamilton Theorem for Descriptor Linear Systems

From the classical Cayley-Hamilton theorem we have ⎡

⎤3 ⎡ ⎤2 1 0 −1 1 0 −1 E 3 − 2E 2 + E = ⎣ 0 1 0 ⎦ − 2 ⎣ 0 1 0 ⎦ 0 −1 0 0 −1 0 ⎡ ⎤ ⎡ ⎤ 1 0 −1 000 + ⎣0 1 0 ⎦ = ⎣0 0 0⎦. 0 −1 0 000

(A.48)

Applying Theorem A.3 to (A.46) we obtain 

ED − 2 E

 D 2



+ E

 D 3

⎡ ⎤2 ⎤ 1 −2 −1 1 −2 −1 = ⎣0 1 0 ⎦ − 2⎣0 1 0 ⎦ 0 −1 0 0 −1 0 ⎡ ⎤ ⎡ ⎤ 1 −2 −1 000 + ⎣0 1 0 ⎦ = ⎣0 0 0⎦. 0 −1 0 000 ⎡

(A.49)

 k Postmultiplying (A.39) by E D , k = 1 = 2, . . . we obtain the following remark. Remark A.1. If (A.38) is the characteristic polynomial of E, then  k+1  k+2  n+k−1  D n+k + a2 E D + . . . + an−1 E D + E =0 a1 E D for k = 1, 2, . . ..

(A.50)

Appendix B

Computation of the Drazin Inverse

B.1 Method 1 Theorem B.1. If the matrix E ∈ Rn×n has one of the following forms 

   E 11 E 12 E 21 0 E1 = , E2 = , 0 0 E 22 0     0 0 0 E 41 , E4 = , E3 = 0 E 42 E 31 E 32

(B.1)

where E 11 , E 21 ∈ Rn 1 ×n 1 and E 32 , E 42 ∈ Rn 2 ×n 2 are nonsingular and E 12 , E 41 ∈ Rn 1 ×n 2 , E 22 , E 31 ∈ Rn 2 ×n 1 , n = n 1 + n 2 , then its Drazin inverse E D ∈ Rn×n has one of the forms, respectively: 

   −1 −1 −2 0 E 21 E 11 E 12 E 11 D = , , E2 = −2 0 0 0 E 22 E 21     −2 0 0 0 E 41 E 42 D D . E3 = −2 −1 , E 4 = −1 E 31 E 32 E 32 0 E 42

E 1D

(B.2)

Proof. Using Definition 1.2 we will show that the matrices (B.1) and (B.2) satisfy the conditions (1.26). Consider the matrix E 1 and its Drazin inverse E 1D . From (1.26a) we have   −1 −2     −1 E 11 E 12 E 11 E 11 E 12 E 12 In 1 E 11 D = (B.3a) E1 E1 = 0 0 0 0 0 0 

and E 1D E 1 =

−1 −2 E 11 E 12 E 11 0 0



E 11 E 12 0 0



 =

 −1 E 12 In 1 E 11 . 0 0

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Kaczorek and K. Borawski, Descriptor Systems of Integer and Fractional Orders, Studies in Systems, Decision and Control 367, https://doi.org/10.1007/978-3-030-72480-1

(B.3b)

227

228

Appendix B: Computation of the Drazin Inverse

Therefore, E 1D E 1 = E 1 E 1D . Next, from (1.26b) we have 

E 1D E 1 E 1D

  −1 −2  −1 E 12 In 1 E 11 E 11 E 11 E 12 = 0 0 0 0   −1 −2 E 11 E 11 E 12 = E 1D . = 0 0 

We will show that j

E1 =

j

j−1

E 11 E 11 E 12 0 0

(B.4)

 (B.5)

for j = 1, 2, . . .. For example, consider the matrix (B.5) for j = 3. Then     E 11 E 12 E 11 E 12 E 11 E 12 E 13 = E 1 E 1 E 1 = 0 0 0 0 0 0    3 2   2 E 11 E 12 E 11 E 11 E 12 E 11 E 11 E 12 = . = 0 0 0 0 0 0

(B.6)

j

It is easy to check that the matrix E 1 , j = 1, 2, . . . have the form (B.5). Finally, from (1.26c) we have  q+1 E 1D E 1

=  =

−1 −2 E 11 E 12 E 11 0 0 q

q−1

E 11 E 11 E 12 0 0

 

q+1

E 11 0

=

q

E 11 E 12 0

 (B.7)

q E1 .

Therefore, E 1D is the Drazin inverse of E 1 . In a similar way we can show that E kD ,   k = 2, 3, 4 are the Drazin inverses of E k , k = 2, 3, 4, respectively. Now consider the singular matrix E in a different form than (B.1). Let Pk ∈ Rn×n be a nonsingular matrix such that E k = Pk E Pk−1 , k ∈ {1, 2, 3, 4},

(B.8)

i.e. transforms E to the one of the forms (B.1). If for a given matrix E there exists the similarity transformation (B.8), then we can use the following approach. Theorem B.2. The Drazin inverse of (B.8) has the form E kD = Pk E D Pk−1 , k ∈ {1, 2, 3, 4}, where E D is the Drazin inverse of E.

(B.9)

Appendix B: Computation of the Drazin Inverse

229

Proof. Using Definition 1.2 we will show that the matrices (B.8) and (B.9) satisfy the conditions (1.26). From (1.26a) we have E k E kD = Pk E Pk−1 Pk E D Pk−1 = Pk E E D Pk−1

(B.10a)

and E kD E k = Pk E D Pk−1 Pk E Pk−1 = Pk E D E Pk−1 = Pk E E D Pk−1 .

(B.10b)

Therefore, E k E kD = E kD E k . Next, from (1.26b) we have E kD E k E kD = Pk E D Pk−1 Pk E Pk−1 Pk E D Pk−1 = Pk E D E E D Pk−1 = Pk E D Pk−1 = E kD .

(B.11)

Finally, from (1.26c) we get q+1

E kD E k

= Pk E D Pk−1 Pk E q+1 Pk−1 = Pk E D E q+1 Pk−1 = P E q Pk−1 = E k , q

since E k = Pk E j Pk−1 , j = 1, 2, . . .. j

(B.12)  

From the considerations above we have the following theorem. Theorem B.3. The Drazin inverse of E has the form E D = Pk−1 E kD Pk , k ∈ {1, 2, 3, 4},

(B.13)

where E kD is one of the matrices (B.2) obtained from the similarity transformation. Proof. The proof follows immediately from the equation (B.9). To compute the Drazin inverse E of the matrix E ∈ R can be used. D

n×n

 

the following procedure

Procedure B.1. Step 1. According to (B.8) find the nonsingular matrix Pk , k ∈ {1, 2, 3, 4} such that it transforms the matrix E to one of the forms (B.1). Step 2. Compute the Drazin inverse E kD of the transformed matrix E k , k ∈ {1, 2, 3, 4} using (B.2). Step 3. Compute the Drazin inverse E D of the matrix E using (B.13). Example B.1. Consider the matrix E in the form ⎡

⎤ 121 E = ⎣0 2 1⎦. 100

(B.14)

230

Appendix B: Computation of the Drazin Inverse

It is easy to check that detE = 0. Step 1. According to (B.8) we find the matrix ⎡

⎤ 1 00 P1 = ⎣ 0 1 0 ⎦ , −1 1 1

(B.15)

which transforms (B.14) to the form E 1 from (B.1). Thus ⎡

E 1 = P1 E P1−1

⎤⎡ ⎤⎡ ⎤ 1 00 121 1 0 0 = ⎣ 0 1 0⎦⎣0 2 1⎦⎣0 1 0⎦ −1 1 1 100 1 −1 1 ⎡ ⎤ 211 = ⎣1 1 1⎦. 000

(B.16)



   21 1 Step 2. From (B.16) we have E 11 = , detE 11 = 0 and E 12 = . Using (B.2) 11 1 we obtain ⎡ ⎤  −1 −2  1 −1 −1 E 11 E 11 E 12 E 1D = = ⎣ −1 2 2 ⎦ . (B.17) 0 0 0 0 0 Step 3. The Drazin inverse of (B.14) can be computed using (B.13): ⎡

ED

⎤⎡ ⎤⎡ ⎤ 1 0 0 1 −1 −1 1 00 = P1−1 E 1D P1 = ⎣ 0 1 0 ⎦ ⎣ −1 2 2 ⎦ ⎣ 0 1 0 ⎦ 1 −1 1 0 0 0 −1 1 1 ⎡ ⎤ 2 −2 −1 = ⎣ −3 4 2 ⎦ . 5 −6 −3

(B.18)

To avoid the calculation of negative matrix powers in (B.2) we can use the CayleyHamilton theorem as follows. Consider the matrix E in the form E 1 . From (A.38), (A.40) and (B.1) we have

and

det[In 1 λ − E 11 ] = s n 1 + an 1 −1 s n 1 −1 + . . . + a1 s + a0

(B.19)

n1 n 1 −1 + an 1 −1 E 11 + . . . + a1 E 11 + a0 In 1 = 0. E 11

(B.20)

−1 we obtain Multiplying the equation (B.20) by E 11 n 1 −1 n 1 −2 −1 E 11 + an 1 −1 E 11 + . . . + a1 In 1 + a0 E 11 =0

(B.21)

Appendix B: Computation of the Drazin Inverse

and −1 =− E 11

231

 1  n 1 −1 n 1 −2 E 11 + an 1 −1 E 11 + . . . + a1 In 1 a0

(B.22)

−1 since a0 = detE 11 = 0. Similarly, multiplying the Eq. (B.21) by E 11 we obtain n 1 −2 n 1 −3 −1 −2 E 11 + an 1 −1 E 11 + . . . + a1 E 11 + a0 E 11 =0

and −2 =− E 11

 1  n 1 −2 n 1 −3 −1 E 11 + an 1 −1 E 11 . + . . . + a1 E 11 a0

(B.23)

(B.24)

Substitution of (B.22) into (B.24) yields −2 E 11 =

where

 1  n 1 −1 n 1 −2 b0 E 11 + b1 E 11 + . . . + bn 1 −1 In 1 , 2 a0

bk = a1 an 1 −k − a0 an 1 −k+1 , k = 0, . . . , n 1 − 1, an 1 = 1, an 1 +1 = 0.

(B.25a)

(B.25b)

Similar analysis can be performed for the matrix E in the form E k , k = 2, 3, 4 from (B.1). Example B.2. (Continuation of Example B.1) Consider the matrix E 1 in the form (B.16), where     21 1 E 11 = , E 12 = . (B.26) 11 1 From (B.19), (B.25b) and (B.26) we have det[I2 λ − E 11 ] = λ2 − 3λ + 1

(B.27)

a1 = −3, a0 = 1, b1 = 8, b0 = −3.

(B.28)

and

Using (B.22), (B.25), (B.26) and (B.28) we have

−2 E 11



 1 −1 , −1 2   2 −3 = −3E 11 + 8I2 = . −3 5

−1 = −E 11 + 3I2 = E 11

(B.29)

232

Appendix B: Computation of the Drazin Inverse

Therefore, the matrix E 1D has the form  E 1D =

−1 E 11

−2 E 11 E 12

0

0



⎡

⎤ 1 −1 −1 = ⎣ −1 2 2 ⎦ , 0 0 0

(B.30)

which is equivalent to (B.17).

B.2 Method 2 Consider a singular matrix E ∈ Rn×n with the index q satisfying the condition q ≤n−1

(B.31)

and the characteristic polynomial (A.38). From (1.26a), (1.26c) and (A.42) we have E D E n−1 + an−1 E D E n−2 + . . . + a2 E D E + a1 E D = E n−2 + an−1 E n−3 + . . . + aq+2 E q + (aq+1 E q + . . . + a2 E + a1 In )E D = −A + B E D = 0, (B.32) where A = −E n−2 − an−1 E n−3 − . . . − aq+2 E q ,

(B.33)

B = aq+1 E q + . . . + a2 E + a1 In .

(B.34)

det B = 0,

(B.35)

E D = B −1 A.

(B.36)

If

then from (B.32) we have

Therefore, the following theorem has been proved. Theorem B.4. If the conditions (B.31) and (B.35) are satisfied, then the Drazin inverse E D of a singular matrix E is given by (B.36). According to Theorem B.4 the Drazin inverse can be computed using the following procedure. Procedure B.2. Step 1. Using (1.25) compute the index q of E and check the condition (B.31). Step 2. Compute the coefficients ak , k = 1, . . . , n − 1 of the characteristic polynomial (A.38).

Appendix B: Computation of the Drazin Inverse

233

Step 3. Compute the matrix B given by (B.34) and check the condition (B.35). Step 4. Compute the matrix B −1 and using (B.33) compute the matrix A. Step 5. Using (B.36) compute the Drazin inverse matrix E D . Example B.3. Compute the Drazin inverse of the matrix ⎡

⎤ 121 E = ⎣0 2 1⎦. 100

(B.37)

Step 1. The index of the matrix (B.37) is q = 1 since rank E = rank E 2 = 2. Step 2. The characteristic polynomial of (B.37) is det[I3 λ − E] = λ3 − 3λ2 + λ

(B.38)

a2 = −3, a1 = 1.

(B.39)

and

Step 3. In this case we have ⎡

⎤ −2 −6 −3 B = −3E + I3 = ⎣ 0 −5 −3 ⎦ −3 0 1

(B.40)

and the condition (B.35) is satisfied since det B = 1. Step 4. Using (B.33) we compute ⎡

⎤ −1 −2 −1 A = −E = ⎣ 0 −2 −1 ⎦ −1 0 0

(B.41)

and from (B.40) we have ⎡

B −1

⎤ −5 6 3 = ⎣ 9 −11 −6 ⎦ . −15 18 10

(B.42)

Step 5. Using (B.36), (B.41) and (B.42) we obtain ⎡

ED

⎤ 2 −2 −1 = B −1 A = ⎣ −3 4 2 ⎦ . 5 −6 −3

(B.43)

234

Appendix B: Computation of the Drazin Inverse

Similarly to Method 1, to avoid calculating the inverse of the matrix B we can compute it as follows. Let det[In λ − B] = λn + bn−1 λn−1 + . . . + b1 λ + b0 ,

(B.44)

then by Cayley-Hamilton theorem we have B n + bn−1 B n−1 + . . . + b1 B + b0 = 0 and B −1 = −

 1  n−1 B + bn−1 B n−2 + . . . + b2 B + b1 In b0

(B.45)

(B.46)

since b0 = det B = 0. Knowing the matrix B and the coefficients bk , k = 0, 1, . . . , n − 1 we may compute the inverse matrix B −1 . Example B.4. (Continuation of Example B.3) The characteristic polynomial of the matrix (B.40) has the form det[I3 λ − B] = λ3 + 6λ2 − 6λ − 1

(B.47)

b2 = 6, b1 = −6, b0 = −1.

(B.48)

and

From (B.40), (B.46) and (B.48) we have ⎡

B −1

⎤ −5 6 3 = B 2 + 6B − 6I3 = ⎣ 9 −11 −6 ⎦ , −15 18 10

(B.49)

which is equivalent to (B.42).

B.3 Method 3 To compute the Drazin inverse E D of a singular matrix E ∈ Rn×n the following procedure can be used.

Appendix B: Computation of the Drazin Inverse

235

Procedure B.3. Step 1. Find the pair of matrices V ∈ Rn×r , W ∈ Rr ×n such that E = V W, rankV = rankW = rank E = r.

(B.50)

As the r columns (rows) of the matrix V (W ) the r linearly independent columns (rows) of the matrix E can be chosen. Step 2. Compute the nonsingular matrix W E V ∈ Rr ×r .

(B.51)

Step 3. The desired Drazin inverse matrix is given by E D = V [W E V ]−1 W.

(B.52)

Proof. It will be shown that the matrix (B.52) satisfies the conditions (1.26). Taking into account that detW V = 0 and (B.50) we obtain [W E V ]−1 = [W V W V ]−1 = [W V ]−1 [W V ]−1 .

(B.53)

Using (1.26a), (B.50) and (B.53) we obtain E E D = V W V [W E V ]−1 W = V W V [W V ]−1 [W V ]−1 W = V [W V ]−1 W (B.54) and E D E = V [W E V ]−1 W V W = V [W V ]−1 [W V ]−1 W V W = V [W V ]−1 W. (B.55) Therefore, the condition (1.26a) is satisfied. To check the condition (1.26b) we compute E D E E D = V [W E V ]−1 W V W V [W E V ]−1 W = V [W V W V ]−1 W V W V [W E V ]−1 W −1

(B.56)

= V [W E V ] W = E . D

Therefore, the condition (1.26b) is also satisfied. Using (1.26c), (B.50), (B.52) and (B.53) we obtain E D E q+1 = V [W E V ]−1 W (V W )q+1 = V [W V ]−1 [W V ]−1 W V W (V W )q = V [W V ]−1 W (V W )q −1

(B.57)

= V [W V ] W V W (V W ) = (V W )q = E q ,

q−1

236

Appendix B: Computation of the Drazin Inverse

where q is the index of E. Therefore, the condition (1.26c) is also satisfied.

 

Example B.5. Compute the Drazin inverse of the matrix ⎡

⎤ 121 E = ⎣0 2 1⎦. 100

(B.58)

Step 1. We find the pair of matrices ⎡

⎤ −0.7427 0.3393 V = ⎣ −0.6652 −0.4735 ⎦ , −0.0775 0.8128   −0.8202 −2.8157 −1.4078 W = , 1.1521 −0.2685 −0.1342

(B.59)

which satisfy the conditions (B.50). Step 2. Using (B.51) we compute  W EV =

 6.7733 −0.2679 . −1.9999 0.2267

(B.60)

Step 3. The Drazin inverse of the matrix (B.58) has the form E D = V [W E V ]−1 W ⎡ ⎤  −0.7427 0.3393  0.2267 0.2679 = ⎣ −0.6652 −0.4735 ⎦ 1.9999 6.7733 −0.0775 0.8128   −0.8202 −2.8157 −1.4078 × 1.1521 −0.2685 −0.1342 ⎡ ⎤ 2 −2 −1 = ⎣ −3 4 2 ⎦ . 5 −6 −3

(B.61)

Appendix C

Laplace Transforms of Continuous-Time Functions and Z -Transforms of Discrete-Time Functions

C.1 Convolutions of Continuous-Time and Discrete-Time Functions and Their Transforms Definition C.1. The Laplace transform of a continuous-time function f (t) is defined by ∞ L[ f (t)] = f (t)e−st dt = F(s), (C.1) 0

where f (t) = 0 for t < 0. Definition C.2. The continuous-time function f (t) defined by t f 1 (t) ∗ f 2 (t) =

f 1 (t − τ ) f 2 (τ )dτ

(C.2)

0

is called the convolution of the continuous-time functions f 1 (t) and f 2 (t). Theorem C.1. If F1 (s) = L[ f 1 (t)], F2 (s) = L[ f 2 (t)], then

⎡ L⎣

t

⎤ f 1 (t − τ ) f 2 (τ )dτ ⎦ = F1 (s)F2 (s).

(C.3)

0

Proof. Taking into account that f 1 (t) = 0, f 2 (t) = 0 for t < 0 and using (C.1) we have © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Kaczorek and K. Borawski, Descriptor Systems of Integer and Fractional Orders, Studies in Systems, Decision and Control 367, https://doi.org/10.1007/978-3-030-72480-1

237

238

Appendix C: Laplace Transforms of Continuous-Time Functions

⎡ L⎣

t

⎤

⎡

f 1 (t − τ ) f 2 (τ )dτ ⎦ = L ⎣

0

∞

⎤ f 1 (t − τ ) f 2 (τ )dτ ⎦

0

⎤ ⎡ ∞ ∞ = ⎣ f 1 (t − τ ) f 2 (τ )⎦ e−st dτ dt 0

0

t f 1 (u)e

=

−su

0

t du

f 2 (τ )e−sτ dτ = F1 (s)F2 (s).

0

(C.4)   Definition C.3. The Z-transform of a discrete-time function f i is defined by Z[ f i ] =

∞ 

f k z −k = F(z),

(C.5)

k=0

where f i = 0 for i < 0. Definition C.4. The discrete-time function defined by f 1,i ∗ f 2,i =

i 

f 1,i−k f 2,k

(C.6)

k=0

is called the convolution of the discrete-time functions f 1,i , f 2,i . Theorem C.2. If F1 (z) = Z[ f 1,i ], F2 (z) = Z[ f 2,i ], 

then Z

i 

 f 1,i−k f 2,k = F1 (z)F2 (z).

(C.7)

k=0

Proof. The proof is similar to the proof of Theorem C.1.

 

C.2 Laplace Transforms of Derivative-Integrals Theorem C.3. The Laplace transform of the function t α has the form L[t α ] = for α ∈ R, α > −1.

(α + 1) s α+1

(C.8)

Appendix C: Laplace Transforms of Continuous-Time Functions

239

Proof. Using (C.1) and (2.1) for α > −1 we obtain ∞

α

L[t ] =

α −st

t e

∞ dt =

0

xα s α+1

e

−x

dx =

0

1

∞

s α+1

x α e−x d x =

(α + 1) . s α+1

(C.9)

0

  Theorem C.4. The Laplace transform of the first order derivative of the function f (t) has the form   d L f (t) = s F(s) − f (0+ ). (C.10) dt Proof. Using (C.1) we have  ∞ ∞ −st ∞ d d −st f (t) = f (t)e dt = e f 0 + s f (t)e−st dt L dt dt 

0

0

(C.11)

= s F(s) − f (0+ ).   Generalizing (C.10) for n-order derivative we obtain  L

where f (k) (0+ ) =

 n  dn n f (t) = s F(s) − s n−k f (k−1) (0+ ), dt n k=1

(C.12)

 d f (t)  . dt t=0

Theorem C.5. The Laplace transform of the fractional α-order integral has the form ⎡ L[ 0 Itα f (t)] = L ⎣

1 (α)

t

⎤ (t − τ )α−1 f (τ )dτ ⎦ =

F(s) . sα

(C.13)

0

Proof. Using (3.56), (C.8), Theorem C.1 and taking into account that α > 0 we obtain ⎡ ⎤ t 1 L[ 0 Itα f (t)] = L ⎣ (t − τ )α−1 f (τ )dτ ⎦ (α) (C.14) 0 α−1

1 F(s) L t = L[ f (t)] = α . (α) s  

240

Appendix C: Laplace Transforms of Continuous-Time Functions

Theorem C.6. The Laplace transform of the fractional α-order integral of n-order derivative of the function f (t) has the form L



n 

α (n) n−α I f (t) = s F(s) − s n−k−α f (k−1) (0+ ). 0 t

(C.15)

k=1

Proof. Using (3.56), (C.8) and Theorem C.1 we obtain L



α 0 It

⎡



f (n) (t) = L ⎣

1 (α)

t

⎤ (t − τ )α−1 f (n) (τ )dτ ⎦

0



1 = L t α−1 L f (n) (t) (α)   n  1 (α) n n−k (k−1) + = s f (0 ) s F(s) − (α) s α k=1 = s n−α F(s) −

n 

(C.16)

s n−k−α f (k−1) (0+ ).

k=1

  Theorem C.7. [90] The inverse Laplace transform of the expression s α F(s) has the form −1

α

L [s F(s)] =

α 0 D It

f (t) +

n−1  k=0

t k−α f (k) (0+ ), (k − α + 1)

(C.17a)

where n − 1 < α < n, n ∈ N and  α 0 D It

f (t) =

α 0 Dt

f (t) for α > 0,

α 0 It

f (t) for α < 0.

(C.17b)

C.3 Z-Transforms of Discrete-Time Functions Theorem C.8. If Z[xi ] =

∞  i=0

xi z −i ,

(C.18)

Appendix C: Laplace Transforms of Continuous-Time Functions

241

then Z[xi+1 ] = z X (z) − zx0 , Z[xi− p ] = z − p X (z) + z − p

(C.19a)

−p



xjz j.

(C.19b)

j=−1

Proof. Using (C.18) we obtain Z[xi+1 ] =

∞ 

xi+1 z −i =

i=0

∞ 

x j z −( j−1) = z

j=1

∞ 

x j z − j − zx0

j=0

= z X (z) − zx0 ,

Z[xi− p ] =

∞  i=0

= z− p

∞ 

xi− p z −i =

x j z −( j+ p) = z − p

j=− p ∞ 

x j z− j + z− p

j=0

= z − p X (z) + z − p

∞ 

x j z− j

(C.20)

j=− p −p 

x j z− j

j=−1 −p



x j z− j .

j=−1

 

Appendix D

Nilpotent Matrices

Definition D.1. A real matrix A ∈ Rn×n is called nilpotent if there exists a natural number μ ≤ n such that Aμ−1 = 0 and Aμ = 0. The natural number μ is called the nilpotency index of the matrix A. Lemma D.1. Matrices of the form ⎡ 0 a12 ⎢0 0 ⎢ ⎢ A = ⎢ ... ... ⎢ ⎣0 0 0 an,2

. . . a1,n−1 ... 0 . .. . .. ... 0 . . . an,n−1

⎤ 0 0⎥ ⎥ .. ⎥ ∈ Rn×n .⎥ ⎥ 0⎦ 0

(D.1)

have the nilpotency index μ = 2 for any values of the entries a12 , . . . , a1,n−1 , an,2 , . . . , an,n−1 and the characteristic polynomials of the form det[In λ − A] = λn .

(D.2)

Proof. Using (D.1) it is easy to check that A2 = 0 and   λ −a12  0 λ   det[In λ − A] =  ... ...  0 0   0 −an,2

. . . −a1,n−1 ... 0 .. .. . . ... λ . . . −an,n−1

 0  0  ..  = λn . .  0  λ

(D.3)

 

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Kaczorek and K. Borawski, Descriptor Systems of Integer and Fractional Orders, Studies in Systems, Decision and Control 367, https://doi.org/10.1007/978-3-030-72480-1

243

244

Appendix D: Nilpotent Matrices

Lemma D.2. Matrices of the form   0 A12 ∈ R2n×2n A= 0 0

(D.4)

have the nilpotency index μ = 2 and the characteristic polynomials of the form det[I2n λ − A] = λ2n

(D.5)

for any submatrices A12 ∈ Rn×n . Proof. Using (D.4) it is easy to verify that A2 = 0 and    In λ −A12   = λ2n .  det[I2n λ − A] =  0 In 

(D.6)



From the well-known property of the transposition of the matrix A, i.e. A  T k A for k = 1, 2, . . . we have the following remark.

 k T

  =

Remark D.1. The transpose matrix A T has a nilpotency index μ if and only if the matrix A has the same nilpotency index μ. Lemma D.3. A diagonal matrix A with at least one nonzero entry is not the nilpotent matrix. Proof. This follows immediately from the relation Ak = (diag[a1 , . . . , an ])k = diag[a1k , . . . , ank ] = 0 for k = 1, 2, . . . if at least one from the entries a1 , . . . , an is nonzero.

(D.7)  

Lemma D.4. A nonnegative matrix A ∈ Rn×n with at least one nonzero diagonal entry is not nilpotent matrix. Proof. Let decompose the matrix A as the sum of the diagonal matrix D and the nonnegative matrix B with zero diagonal entries. Let assume that D and B are commuting matrices, i.e. D B = D B. Then Ak = (D + B)k = D k + B D k−1 + . . . + B k for k = 1, 2, . . . .

(D.8)

If the matrix A has at least one nonzero diagonal entry, then D = 0 and by Lemma D.3 D k = 0 for k = 1, 2, . . .. From (D.8) we have Ak = 0 for k = 1, 2, . . . since   D k = 0 and the remaining entries are nonnegative.

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