Sampled-Data Control for Periodic Objects 9783031019555, 9783031019562

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Efim N. Rosenwasser Torsten Jeinsch Wolfgang Drewelow

Sampled-Data Control for Periodic Objects

Sampled-Data Control for Periodic Objects

Efim N. Rosenwasser · Torsten Jeinsch · Wolfgang Drewelow

Sampled-Data Control for Periodic Objects

Efim N. Rosenwasser Department of Ship Automation Marine Technical University St. Petersburg Saint Petersburg, Russia

Torsten Jeinsch Lehrstuhl Regelungstechnik Universität Rostock Rostock, Mecklenburg-Vorpommern Germany

Wolfgang Drewelow Lehrstuhl Regelungstechnik Universität Rostock Rostock, Mecklenburg-Vorpommern Germany

ISBN 978-3-031-01955-5 ISBN 978-3-031-01956-2 (eBook) https://doi.org/10.1007/978-3-031-01956-2 Mathematics Subject Classification: 93C05, 93C35, 93C57, 93C80, 93C83, 93D15, 93E20 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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

We dedicate this book to the memory of our good friend and colleague Bernhard P. Lampe, who made major contributions to the underlying theory of computer-controlled systems.

Preface

One of the actual problems in modern control theory is the control of objects with periodically changing parameters. Many of these objects are described with a sufficient degree of adequacy by mathematical models in the form of systems of linear differential equations with periodic coefficients. In the literature, such objects are referred to as Finite-Dimensional Linear Continuous Periodic (FDLCP), also simply referred to as periodic below. An incomplete list and description of different types of such FDLCP objects can be found in [1]–[10] and the literature cited there. This monograph deals with the problem of digital control of FDLCP objects. The corresponding closed-loop system belongs to the class of sampled-data (SD) systems, because it is built of both continuous- and discrete-time components. The main mathematical apparatus used in this book is a method based on the concept of the parametric transfer function or Parametric Transfer Matrix (PTM), respectively. By definition, unlike the ordinary transfer function W (s) for linear timeinvariant systems, the parametric transfer function W (s, t) depends on the complex variable s and the time t as parameters. The concept of the parametric transfer function was introduced into control theory in the work of L. A. Zadeh [11], [12]. The idea of transferring this concept to sampled-data systems with continuous linear time-invariant (LTI) elements was put forward in the work of J. R. Ragazzini and L. A. Zadeh [13]. The possibility of transferring the parametric transfer function method to SD systems with a periodic non-stationary continuous object was set in [14]. A systematic description of the methods for the analysis and synthesis of singleinput single-output sampled-data systems with continuous LTI elements based on the parametric transfer function is given in the monographs [15]–[19]. This parametric transfer function approach has been generalized to multidimensional computerized systems with continuous linear time-invariant objects by introducing the concept of the PTM [20]. In monograph [21], this approach was extended to systems with delay. The main content of this monograph is divided into five parts.

vii

viii

Preface

Part I, comprising Chaps. 1–4, develops the results of [22] and is devoted to the formulation of the frequency approach for the mathematical description and study of FDLCP objects. In Chap. 1, discrete operational transformations of functions of the continuous argument t in the infinite interval −∞ < t < ∞, as well as a representation of this transformation by the two-sided Laplace transformation, are introduced. This leads to the definition of the discrete Laplace transformation of functions of continuous arguments and the corresponding transformation of the image. Besides, the first part contains a chapter devoted to the operator description of LTI objects. Chapter 2 provides necessary material from the theory of linear differential equations with periodic coefficients, which is an important basis for the further presentation. The methodology for calculating the response of LTI objects to periodic and exponential periodic input signals using the discrete Laplace transformation is introduced. Chapter 3 defines the parametric transfer matrix of an FDLCP object based on the content of the previous chapters and gives general relations defining this PTM. Chapter 4 introduces the concept of the Floquet–Lyapunov decomposition, which makes it possible to represent a FDLCP object in the form of a serial connection of an LTI element and modulation elements at its input and output, respectively. Part II consists of Chaps. 5–7. These chapters describe methods for constructing the PTMs of open and closed sampled-data systems with a periodic object and delay. The discussion includes both synchronous systems, in which the period of the FDLCP object is equal to the sampling period of the discrete controller, and asynchronous systems, where these periods differ by an integer factor. Chapter 5 deals with the open-loop sampled-data system without delay. Chapter 6 develops the parametric transfer functions for open-loop synchronous and asynchronous sampled-data systems in case of a delay. Chapter 7 provides general expressions that define PTMs for different types of closed-loop SD systems with FDLCP objects and delays. Part III includes Chaps. 8–12. In this part, the technique of solving the problems of modal control and stabilization of closed-loop SD systems of considered classes is developed. Chapters 8–9 contain the information necessary for understanding the subsequent presentation from the theory of polynomial and rational matrices. Chapter 10 describes the method of solving problems of modal control and stabilization of SD systems based on applying the mathematical apparatus of Determinant Polynomial Equations (DPEs). The peculiarity of the proposed approach is the use of discrete models presented in polynomial form described by the backward operator (reverse shift) ζ . Using this model form eliminates the problem of separation of the subset of causal controllers from the whole set of stabilizing controllers because all discrete stabilizing controllers for a strictly causal discrete backward object are causal. In Chap. 11, the tasks of synchronous control stabilization of FDLCP objects are considered based on the described approach. For this purpose, a discrete backward model is built for which the set of all causally stabilizing discrete controllers is generated using the DPE mathematical apparatus. In Chap. 12, a similar approach

Preface

ix

is used to solve the problem of stabilizing asynchronous closed-loop SD systems, including delay in the loop. Part IV, formed by Chaps. 13–19, is devoted to the construction of a quality functional for solving the H2 -optimization problem of the synchronous closed-loop SD system Sτ with FDLCP object and delayed control. Chapters 13–15 establish some additional relations for the PTM of the system Sτ required for the construction of the quality functional. Chapters 16 and 17 show that the set of transfer matrices of all causal discrete controllers that stabilize the system Sτ can be parameterized by a stable rational matrix θ (ζ ), which is called the system function, and a representation of the PTM of the system Sτ by the system function is constructed. In Chaps. 18, 19, the relations expressing the H2 -norm of the system Sτ by its PTM are given. Using these relations, an integral quality functional is constructed which depends on matrix Δ θ˜ (s) = θ (e−sT ) specified on the imaginary axis in the interval − ω2 ≤ Im(s) ≤ ω2 with ω = 2π . T Part V (Chaps. 20–25) is devoted to the method of H2 -optimization of the system Sτ , based on minimization of the quadratic quality functional, built in Chap. 19. Chapter 20 provides auxiliary information from the theory of scalar and matrix quasipolynomials, including an important factorization theorem from [23]. Chapter 21 deals with the algorithm of minimizing the quadratic functional on the unit circle, based on the results of [24], [25]. Chapters 22 and 23 describe practical ways of building the matrix η(s, t), which appears in the PTM representation through the system function and matrix C˜ T (s, t), which is included in the expression of the constructed quality functional. In Chap. 24, the generated quality functional is translated into a functional specified on the circle |ζ | = 1 by replacing the integration variable ζ = e−sT . It is shown that the coefficients in the integral expression of the transformed functional are quasi-polynomials, where two of them on the circle |ζ | = 1 are non-negative. The final chapter (Chap. 25) is devoted to solving the H2 -optimization problem for the system Sτ . A general algorithm for minimizing the constructed transformed quality functional is given. As an application example of the proposed method, the sequential solution of the H2 -optimization problem for the system Sτ with an FDLCP object of first order is demonstrated. The book is intended for researchers and engineers, involved in research and development of modern control systems, as well as for teachers, postgraduates, and undergraduates. The authors very gratefully acknowledge the financial support from the Deutsche Forschungsgemeinschaft. We would like to express our gratitude for the dedicated and careful work of Oliver Jackson of Springer, who helped us greatly in overcoming editorial problems. We thank Vladislav Rybinskii and Renate Ziegler for their help in finalizing the manuscript. Saint Petersburg, Russia Rostock, Germany Rostock, Germany

Efim N. Rosenwasser Torsten Jeinsch Wolfgang Drewelow

Contents

Part I 1

2

3

4

The Frequency Approach to the Mathematical Description of Linear Periodic Objects

Discrete Operational Transformations of Continuous Argument Functions and Operator Description of Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Discrete Laplace Transformation of Continuous Argument Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Discrete Laplace Transform of the Image . . . . . . . . . . . . . . . . . . . . 1.3 Operator Description of a Finite-Dimensional Linear Time-Invariant System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Transfer of an Exponentially Periodic Signal Through a Linear Time-Invariant System . . . . . . . . . . . . . . . . . . . . . . . . . . . . State-Space Analysis of Finite-Dimensional Linear Continuous Periodic Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 State-Space Description of Periodic Objects . . . . . . . . . . . . . . . . . . 2.2 Transfer of Periodic Signals Through Periodic Objects . . . . . . . . . 2.3 Transfer of Exponentially Periodic Signals Through Periodic Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Higher-Order Periodic Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 9 11 17 17 20 23 27

The Frequency Method in the Theory of Periodic Objects . . . . . . . . . 3.1 Frequency Description of Linear Periodic Operators . . . . . . . . . . . 3.2 Linear Periodic Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Operator Description of a Basic Periodic Object . . . . . . . . . . . . . . 3.4 Parametric Transfer Matrix of the Basic Periodic Object . . . . . . . 3.5 Parametric Transfer Matrix of a Complemented Periodic Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 39 43

The Floquet–Lyapunov Decomposition and Its Application . . . . . . . . 4.1 Floquet–Lyapunov Transformation . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51

45

xi

xii

Contents

4.2 4.3 4.4

Part II 5

6

7

Floquet–Lyapunov Decomposition and Its Parametric Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Periodic Object with Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-Frequency Exponentially Periodic Excitation of the Floquet–Lyapunov Decomposition . . . . . . . . . . . . . . . . . . . .

53 54 58

The Parametric Transfer Matrix Approach to Sampled-Data Systems with Periodic Objects

Open-Loop Sampled-Data System with Periodic Object . . . . . . . . . . . 5.1 Multivariable Zero-Order Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Linearized Model of the Digital Controller . . . . . . . . . . . . . . . . . . . 5.3 Open-Loop System with Time-Invariant Object . . . . . . . . . . . . . . . 5.4 Synchronous Open-Loop System with Periodic Object . . . . . . . . . 5.5 Asynchronous Rising Open-Loop System with Periodic Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Open-Loop System with Periodic Object and High-Frequency Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open-Loop Sampled-Data System with Periodic Object and Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Open-Loop System with Linear Time-Invariant Object and Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Synchronous Open-Loop System with Periodic Object and Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Asynchronous Rising Open-Loop SD System with Periodic Object and Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Open-Loop System with Periodic Object, High-Frequency Hold and Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closed-Loop Sampled-Data System with Periodic Object and Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Synchronous Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Asynchronous Rising Closed-Loop System . . . . . . . . . . . . . . . . . . 7.3 Closed-Loop System with Periodic Object, High-Frequency Hold and Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 65 70 72 75 78 83 83 89 94 98 101 101 106 110

Part III Determinant Polynomial Equations, Sampled-Data Modal Control and Stabilization of Periodic Objects 8

Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 General Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Polynomial Matrices Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Latent Equation and Latent Numbers . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Pairs of Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 117 118 119

Contents

9

Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 General Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 McMillan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Matrix Fraction Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Strictly Proper Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Polynomial Pairs and Transfer Matrices . . . . . . . . . . . . . . . . . . . . . 9.6 Polynomial Matrix Division and Reduction of the Degree of Polynomial Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Determinant Polynomial Equations, Causal Modal Control and Stabilization of Discrete Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 General Definitions and Problem Setting for Causal Modal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Solving Backward Modal Control and Stabilization Problems for Polynomial Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Solving Modal Control and Stabilization Problems for Backward Processes in Polynomial Matrix Description . . . . . 11 Synchronous Sampled-Data Stabilization of Periodic Objects . . . . . . 11.1 Synchronous Stabilization of a Single Periodic Object . . . . . . . . . 11.2 Synchronous Stabilization of a Periodic Object with Prefilter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Synchronous SD Stabilization of an FDLCP Object with Control Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Asynchronous Sampled-Data Stabilization of Periodic Objects . . . . 12.1 Low-Frequency Stabilization of a Periodic Object . . . . . . . . . . . . . 12.2 Low-Frequency Stabilization of a Periodic Object with Prefilter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Low-Frequency Stabilization of a Periodic Object with Control Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Stabilization of a Periodic Object Using a High-Frequency Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Stabilization of a Periodic Object with Control Delay Using a High-Frequency Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

123 123 124 126 127 129 130 133 133 136 138 145 145 150 152 157 157 161 162 164 167

Part IV Building the Quality Functional for the H2 -Optimization Task of the Sampled-Data System Sτ 13 General Parametric Transfer Matrix Properties of the Synchronous Open-Loop Sampled-Data System with Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 14 Parametric Transfer Matrix of the Closed-Loop Sampled-Data System with Delay as Function of Argument s . . . . . . . . . . . . . . . . . . . 181 15 Calculation of Matrices v0 (s), ξ0 (s), ψ0 (s) . . . . . . . . . . . . . . . . . . . . . . . 189

xiv

Contents

16 System Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 17 Representing the Parametric Transfer Matrix of System Sτ by the System Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 18

H2 -Norm of the Closed-Loop Sampled-Data System . . . . . . . . . . . . . . 205

19 Construction of the Quality Functional . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Part V

H2 -Optimization of the Closed-Loop Sampled-Data System

20 Scalar and Matrix Quasi-Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 21 Minimization of a Quadratic Functional on the Unit Circle . . . . . . . . 225 22 Construction of Matrix η(s, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 23 Construction of Matrix C˜ T (s, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 24 Transformation of the Quality Functional . . . . . . . . . . . . . . . . . . . . . . . 239 25

H2 -Optimization of the System Sτ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 25.1 General Solution of the H2 -Optimization Task . . . . . . . . . . . . . . . . 243 25.2 General Solution of the H2 -Optimization Task for a First-Order Periodic Object . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Part I

The Frequency Approach to the Mathematical Description of Linear Periodic Objects

Chapter 1

Discrete Operational Transformations of Continuous Argument Functions and Operator Description of Linear Time-Invariant Systems

In this chapter, based on the mathematical apparatus of the discrete Laplace transform of a continuous argument function, the properties of multidimensional linear timeinvariant (LTI) systems necessary for further representation are discussed.

1.1 Discrete Laplace Transformation of Continuous Argument Functions 1. With Ωα,β we denote the set of functions f (t) which have the following properties: (a) f (t) is defined for all −∞ < t < ∞. (b) The estimation  Meαt , t ≥ +0, | f (t)| < Meβt , t ≤ −0,

(1.1)

is valid for constants M, α, β, where α < β. (c) The function f (t) is of bounded variation on any finite interval. If f (t) is a matrix, we write f (t) ∈ Ωα,β if conditions (a), (b), and (c) are met for its elements. 2.

For f (t) ∈ Ωα,β there exists the two-sided Laplace transform [26] ∞ F(s) =

f (t)e−st dt, α < Re(s) < β,

(1.2)

−∞

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_1

3

4

1 Discrete Operational Transformations of Continuous Argument Functions …

where the integral converges absolutely and the inversion formula 1 2π j

c+  j∞

F(s)est ds =

f (t + 0) + f (t − 0) , 2

(1.3)

c− j∞

holds. The improper integral is understood here in the sense of Cauchy’s principal value. Hereby, in any strip α < α1 ≤ Re(s) ≤ β1 < β the estimation |F(s)|
α.

(1.6)

0

The Eqs. (1.3) and (1.4) remain valid under the convergence conditions c > α and Re(s) > α. 3. We define the discrete Laplace transform (DLT) of the continuous argument function f (t) by Δ D˜ f (T, s, t) =

∞ 

f (t + kT )e−ksT , −∞ < t < ∞.

(1.7)

k=−∞

As shown in [17], the series converges absolutely and uniformly for f (t) ∈ Ωα,β and α < Re(s) < β. The following basic properties can be derived for the DLT [17]: (a) The relation 2π D˜ f (T, s, t) = D˜ f (T, s + jω, t), ω = T

(1.8)

holds. (b) For any integer m applies D˜ f (T, s, t + mT ) = D˜ f (T, s, t)emsT .

(1.9)

1.2 Discrete Laplace Transform of the Image

5

(c) Besides, if, for the functions f 1 (t) and f 2 (t), the condition (1.1) holds, then for α < Re(s) < β D˜ f1 + f2 (T, s, t) = D˜ f1 (T, s, t) + D˜ f2 (T, s, t).

1.2 Discrete Laplace Transform of the Image 1.

For f (t) ∈ Ωα,β , we consider the function ϕ f (T, s, t) = D˜ f (T, s, t)e−st .

(1.10)

It follows from (1.7), that ϕ f (T, s, t) is the series ϕ f (T, s, t) =

∞ 

f (t + kT )e−s(t+kT ) , −∞ < t < ∞.

(1.11)

k=−∞

In the following, the sum of the series (1.11) is called the shifted pulse frequency response (DPFR) of the function f (t). It follows from (1.11) that ϕ f (T, s, t) = ϕ f (T, s, t + T ).

(1.12)

∗ the subset of the set Ωα,β of functions, for which DPFR ϕ f (T, s, t) Denote by Ωα,β has bounded variation with respect to t. In this case, as it is shown in [17] for α < Re(s) < β, the function ϕ f (T, s, t) can be represented by the Fourier series

ϕ F (T, s, t) =

∞ 1  F(s + k jω)ek jωt , −∞ < t < ∞, T k=−∞

(1.13)

which converges for all t. Here F(s) is the image of the function f (t) for α < Re(s) < β. From the properties of Fourier series for functions of bounded variation, it follows that, in the point of continuity ϕ f (T, s, t) according to t, we have ϕ F (T, s, t) = ϕ f (T, s, t).

(1.14)

However, if the function ϕ f (T, s, t) has a jump discontinuity at t = t0 , then ϕ F (T, s, t0 ) =

ϕ f (T, s, t0 + 0) + ϕ f (T, s, t0 − 0) . 2

(1.15)

6

2.

1 Discrete Operational Transformations of Continuous Argument Functions …

The function ∞ 1  Δ D˜ F (T, s, t) = ϕ F (T, s, t)est = F(s + k jω)e(s+k jω)t T k=−∞

(1.16)

is called the discrete Laplace transform of the image F(s). In analogy to the given relations, it follows that, in the continuity points of D˜ f (T, s, t) with respect to t applies (1.17) D˜ F (T, s, t) = D˜ f (T, s, t) and for existing jump points t0 D˜ f (T, s, t0 + 0) + D˜ f (T, s, t0 − 0) . D˜ F (T, s, t0 ) = 2

(1.18)

Since a function of bounded variation does not possess more than a countable set ∗ of jump points, so for f (t) ∈ Ωα,β , the functions D˜ f (T, s, t) and D˜ F (T, s, t) are equivalent, i.e., they differ from each other on a set of measure zero. Further on, the equivalent functions are considered as equal. Hereby from (1.16), we have 2π , D˜ F (T, s, t) = D˜ F (T, s + k jω, t), ω = T

(1.19)

and also for integer m D˜ F (T, s, t + mT ) = D˜ F (T, s, t)emsT .

(1.20)

3. Formulas (1.7) and (1.16) can be considered as functional transformations of the original f (t) and the corresponding image F(s). The corresponding inversion formulas have the form [17] ω  2

c+ j

f (t) =

T 2π j

c− j

T F(s) = 0

where α < c < β, ω =

2π . T

D˜ f (T, s, t)ds,

ω 2

D˜ F (T, s, t)e−st dt,

(1.21)

1.2 Discrete Laplace Transform of the Image

7

∗ 4. Below it is always supposed that f (t) ∈ Ωα,β . In this case, the series (1.7) and (1.16) coincide everywhere, except at the jump points of function D˜ f (T, s, t). That is why on any interval of continuity, the equation holds

D˜ f (T, s, t) = D˜ F (T, s, t).

(1.22)

The further relations should be considered on those time intervals, on which the equality (1.22) holds. Below the series (1.7) and (1.16) on the interval 0 < t < T will be denoted as D˜ f (T, s, t) and D˜ F (T, s, t). The use of the relations given above allows to build the DLT and the DPFR on the infinite internal −∞ < t < ∞. In particular, it follows from (1.20) that, for kT < t < (k + 1)T , we have D˜ F (T, s, t) = D˜ F (T, s, t − kT )eksT , ϕ F (T, s, t) = D˜ F (T, s, t − kT )e−s(t−kT ) .

(1.23)

5. For many important special cases, closed expressions for the function D F (T, s, t) can be constructed. If the image F(s) is a strictly proper fractional-rational function of the form F(s) =

νi   i=1 k=1

f ik , (s − si )k

(1.24)

then we have [17] D˜ F (T, s, t) =

νi   i=1 k=1

∂ k−1 f ik esi t . (k − 1)! ∂sik−1 1 − e(si −s)T

(1.25)

If the image function (matrix) F(s) has the form F(s) = (s Iχ − A)−1 ,

(1.26)

where A is a constant matrix χ × χ , then we have [20] D˜ F (T, s, t) = e At (Iχ − e−sT e AT )−1 = (Iχ − e−sT e AT )−1 e At .

(1.27)

In the more general case, if F(s) = C(s Iχ − A)−1 B,

(1.28)

where C and B are constant matrices, from (1.27), we find D˜ F (T, s, t) = Ce At (Iχ − e−sT e AT )−1 B = C(Iχ − e−sT e AT )−1 e At B.

(1.29)

8

1 Discrete Operational Transformations of Continuous Argument Functions …

6. Let F(s) be a function (matrix), for which DPFR ϕ F (T, s, t) and DLT D˜ F (T, s, t) are defined. Then we use the notation Δ

Fτ (s) = F(s)e−sτ ,

(1.30)

where τ is a positive constant, that allows a representation τ = mT + θ = (m + 1)T − γ ,

(1.31)

where m is non-negative integer, θ and γ are constants, such that 0 ≤ θ < T, 0 < γ ≤ T.

(1.32)

Obviously, with regard to (1.16), we have ∞ 1  ˜ Fτ (s + k jω)e(s+k jω)t = D Fτ (T, s, t) = T k=−∞

=

∞ 1  F(s + k jω)e(s+k jω)(t−τ ) = D˜ F (T, s, t − τ ). T k=−∞

(1.33)

Using (1.20) and (1.31), we obtain from here D˜ Fτ (T, s, t) = D˜ F (T, s, t − mT − θ ) = D˜ F (T, s, t − θ )e−msT . Let be 0 < t < θ . Then −T < t − θ < 0, and therefore, due to (1.23) D˜ Fτ (T, s, t) = D˜ F (T, s, t − θ )e−msT = D˜ F (T, s, t + γ )e−(m+1)sT .

(1.34)

If θ < t < T , then 0 < t − θ < T and D˜ Fτ (T, s, t) = D˜ F (T, s, t − θ )e−msT . Bringing together (1.34) and (1.35), we find  D˜ (T, s, t + γ )e−(m+1)sT , 0 < t < θ, D˜ Fτ (T, s, t) = ˜ F D F (T, s, t − θ )e−msT , θ < t < T.

(1.35)

(1.36)

In particular, if (1.28), (1.29) holds, we obtain

D˜ Fτ (T, s, t) =



Ce A(t+γ ) (Iχ − e−sT e AT )−1 Be−(m+1)sT , 0 < t < θ, Ce A(t−θ) (Iχ − e−sT e AT )−1 Be−msT , θ < t < T.

(1.37)

1.3 Operator Description of a Finite-Dimensional Linear Time-Invariant System

9

1.3 Operator Description of a Finite-Dimensional Linear Time-Invariant System First, we consider a finite-dimensional linear time-invariant system described by the vector differential equation a( p)y(t) = b( p)x(t), where a( p), b( p) are polynomial matrices of corresponding sizes and p = differential operator. Suppose that the rational matrix Δ

W ( p) = a −1 ( p)b( p)

(1.38) d dt

is the

(1.39)

is strictly proper. Matrix (1.39) is called transfer matrix of the system (1.38). Let p1 , . . . , p be the poles of the matrix (1.39) and q1 , , . . . , qm be the set of different real parts of the poles pi . By construction, we have m ≤ . Assume q0 = −∞, qm+1 = ∞ and consider the corresponding open intervals Ji , (i = 1, . . . , m + 1) on the complex p plane defined by qi−1 < Re( p) < qi , (i = 1, . . . , m + 1).

(1.40)

As it follows from [17, 22], in each interval Ji , the transfer matrix (1.39) is the two-sided Laplace transform of a corresponding matrix function gi (t) defined by the relation c+  j∞

1 gi (t) = 2π j

W ( p)e pt dp, qi−1 < c < qi , (i = 1, 2, . . . , m + 1).

(1.41)

c− j∞

Hereby  W ( p) =

∞ −∞

gi (t)e− pt dt, qi−1 < Re( p) < qi , (i = 1, 2, . . . , m + 1). (1.42)

The integral

∞ y(t) =

gi (t − ν)x(ν)dν, (i = 1, . . . , m + 1) −∞

(1.43)

10

1 Discrete Operational Transformations of Continuous Argument Functions …

in case of convergence defines a particular solution of the differential equation (1.38). The relation (1.43) can be considered as a linear stationary operator y(t) = Ui [x(t)] defined on the set of functions x(t), for which the integral (1.43) converges. Following [17, 22], we define a corresponding transfer function (matrix) Wi ( p) for the operator Ui by the formula Wi ( p) = e− pt Ui [e pt ], qi−1 < Re( p) < qi .

(1.44)

Using (1.43) and (1.42), we find Wi ( p) = e

− pt

∞

∞ gi (t − ν)e dν = pν

−∞

∞ =

gi (t − ν)e− p(t−ν) dν =

−∞

gi (μ)e− pμ dμ = W ( p), qi−1 < Re( p) < qi .

(1.45)

−∞

From (1.45), it follows that the operator Ui in the interval Ji has the transfer matrix W ( p) in (1.39). Thus, for Re( p) ∈ Ji , W ( p) defines the solution to the equation (1.38) as (1.46) x(t) = e pt , y(t) = W ( p)e pt . Obviously, there is only one unique solution for the case that the matrix a( p) has no eigenvalue in p. As follows from [17, 22], in the interval Jm+1 , we have gm+1 (t) = 0, t < 0. In this case

(1.47)

t ym+1 (t) =

gm+1 (t − ν)x(ν)dν

(1.48)

−∞

follows from (1.43), which determines a process at zero initial energy. From the given thoughts, it follows that the LTI system (1.38) corresponds to a family of linear stationary operators (1.43). Each has a transfer function (matrix) (1.45) defined in the corresponding interval (1.40). Hereby, the transfer functions Wi ( p) corresponding to the different operators of the family are an analytical continuation of each other and form a single transfer function (1.39) corresponding to the solution of (1.46).

1.4 Transfer of an Exponentially Periodic Signal Through a Linear Time-Invariant System

11

1.4 Transfer of an Exponentially Periodic Signal Through a Linear Time-Invariant System 1. A function (matrix) f (t) will be called exponentially periodic (EP) if it allows a representation of the form f (t) = eλt f T (t),

f T (t) = f T (t + T ),

(1.49)

where λ is a complex number and T is a real positive constant. Further on, the number λ is called the exponent of the EP function (1.49) and the number T its period. Below it is supposed that function f T (t) can be represented by the convergent Fourier series f T (t) =

∞ 

fk e

k jωt

,

k=−∞

1 fk = T

T

f T (ν)e−k jων dν,

(1.50)

0

where

2π . T

ω=

(1.51)

Using (1.50), the EP function f (t) can be represented by the convergent series f (t) =

∞ 

f k e(λ+k jω)t .

(1.52)

k=−∞

2. The following auxiliary task plays an important role in the further explanations. Let us assume that we use the EP function x(t) = eλt x T (t), x T (t) = x T (t + T ),

(1.53)

as input signal x(t) for the LTI system (1.38). We need to find all EP solutions for the output signal y(λ, t) in (1.38), i.e., such that y(λ, t) = eλt yT (λ, t),

yT (λ, t) = yT (λ, t + T ).

(1.54)

Let us have the series of the form (1.52): x(t) =

∞ 

xk e(λ+k jω)t ,

(1.55)

yk (λ)e(λ+k jω)t .

(1.56)

k=−∞

and also y(λ, t) =

∞  k=−∞

12

1 Discrete Operational Transformations of Continuous Argument Functions …

By inserting (1.55) and (1.56) in (1.38) and setting equal the coefficients with the same harmonics, we get for all integers k a(λ + k jω)yk (λ) = b(λ + k jω)xk .

(1.57)

Suppose that among the numbers λ + k jω, there are no eigenvalues of the matrix a(λ). Then for any integer k from (1.57), we get yk (λ) = a −1 (λ + k jω)b(λ + k jω)xk = W (λ + k jω)xk .

(1.58)

Inserting (1.58) in (1.56), we find y(λ, t) =

∞ 

yk (λ)e(λ+k jω)t =

k=−∞

∞ 

W (λ + k jω)xk e(λ+k jω)t .

(1.59)

k=−∞

Combined with (1.54), it immediately follows that ∞ 

yT (λ, t) =

W (λ + k jω)xk ek jωt .

(1.60)

k=−∞

This formula can be transformed to a form convenient for later use. To do this, note that T 1 xk = x T (ν)e−k jων dν. (1.61) T 0

From (1.60) and (1.61), it follows  ∞ 1  W (λ + k jω) x T (ν)e−k jων dν ek jωt . yT (λ, t) = T k=−∞ T

(1.62)

0

By changing the order of addition and integration, we find T yT (λ, t) =

ϕW (T, λ, t − ν)x T (ν)dν,

(1.63)

∞ 1  W (λ + k jω)ek jωt T k=−∞

(1.64)

0

where ϕW (T, λ, t) =

1.4 Transfer of an Exponentially Periodic Signal Through a Linear Time-Invariant System

13

is the DPFR of the transfer function (matrix) W (λ). Considering yT (λ, t) = e−λt y(λ, t), x T (t) = e−λt x(t),

(1.65)

we get for the (1.63) EP output T y(λ, t) =

D˜ W (T, λ, t − ν)x(ν)dν.

(1.66)

0

We want to rewrite (1.63) and (1.66) in a closed form. For this, we bring (1.66) into the partitioned form t y(λ, t) =

D˜ W (T, λ, t − ν)x(ν)dν +

T

D˜ W (T, λ, t − ν)x(ν)dν.

(1.67)

t

0

Since the function W (λ) is assumed to be strictly proper, it can be represented in the form of (1.28) W (λ) = C(λIχ − A)−1 B. (1.68) Suppose 0 < t < T and 0 < ν < t. Then we have 0 < t − ν < T , and therefore, due to (1.23), (1.29), we have D˜ W (T, λ, t − ν) = D˜ W (T, λ, t − ν) = Ce At (Iχ − e−λT e AT )−1 e−Aν B, 0 < t − ν < T.

If 0 < t < T and −T < t − ν < 0, then from (1.23) and (1.29), follows

(1.69)

D˜ W (T, λ, t − ν) = D˜ W (T, λ, t − ν + T )e−λT = Ce At (Iχ − e−λT e AT )−1 e−λT e AT e−Aν B.

(1.70)

Using (1.69) and (1.70), Equation (1.66) can be written in the form T y(λ, t) = C

e At G T (λ, t − ν)e−Aν Bx(ν)dν,

(1.71)

0

where  G T (λ, t − ν) =  = Since

(Iχ − e−λT e AT )−1 , 0 < t − ν < T, = (Iχ − e−λT e AT )−1 e−λT e AT , −T < t − ν < 0 (Iχ − e−λT e AT )−1 , 0 < t − ν < T, (Iχ − e−λT e AT )−1 − Iχ , −T < t − ν < 0.

D˜ W (T, λ, t − ν) = ϕW (T, λ, t − ν)eλ(t−ν) ,

(1.72)

(1.73)

14

1 Discrete Operational Transformations of Continuous Argument Functions …

from (1.66), we also obtain T yT (λ, t) =

ϕW (T, λ, t − ν)x T (ν)dν,

(1.74)

0

that for 0 < t < T can be written in the form T yT (λ, t) =

Ce At G T (λ, t − ν)e−Aν Be−λ(t−ν) x T (ν)dν.

(1.75)

0

3. The above constructions can be extended to a linear stationary system with delay, when instead of (1.38), we have the equation a( p)yτ (t) = b( p)x(t − τ ),

(1.76)

where the matrices a( p), b( p) are the same as in (1.38) and τ > 0 is constant, allowing the representation of (1.31) and (1.32) τ = mT + θ = (m + 1)T − γ , 0 ≤ θ < T, 0 < γ ≤ T.

(1.77)

Let x(t) be an EP function. Then function x(t − τ ) is also EP. Indeed, if x(t) = eλt x T (t), we have   Δ x(t − τ ) = xτ (t) = eλ(t−τ ) x T (t − τ ) = eλt e−λτ x(t − τ ) , i.e.,

where

x(t − τ ) = eλt x T τ (t),

(1.78)

x T τ (t) = e−λτ x T (t − τ ).

(1.79)

Assuming that the conditions of applicability of (1.74) for the periodic component of the EP output are met, we have in view of (1.78) and (1.79)

yT τ (λ, t) = e

−λτ

T ϕW (T, λ, t − ν)x T (ν − τ )dν. 0

(1.80)

1.4 Transfer of an Exponentially Periodic Signal Through a Linear Time-Invariant System

15

With the help of substitution of the integration variable ν =μ+τ

(1.81)

Equation (1.80) can replaced by

yT τ (λ, t) = e

−λτ

T −τ 

ϕW (T, λ, t − τ − μ)x T (μ)dμ.

(1.82)

−τ

Since the integrand is periodic with respect to μ with the period T , the last relation can be written in the equivalent form

yT τ (λ, t) = e

−λτ

T ϕW (T, λ, t − τ − μ)x T (μ)dμ.

(1.83)

0

Taking into account ϕWτ (T, λ, t) =

∞ 1  Wτ (λ + k jω)ek jωt = T k=−∞

∞ 1  W (λ + k jω)e−(λ+k jω)τ ek jωt = T k=−∞   ∞  1 = e−λτ W (λ + k jω)ek jω(t−τ ) = e−λτ ϕW (T, λ, t − τ ), T k=−∞

=

(1.84) relation (1.83) can be written in the form T yT τ (λ, t) =

ϕWτ (T, λ, t − μ)x T (μ)dμ.

(1.85)

0

Besides, considering yT τ (λ, t) = e−λt yτ (λ, t), x T (μ) = e−λμ x(μ),

(1.86)

16

1 Discrete Operational Transformations of Continuous Argument Functions …

from (1.85) we can obtain that the EP output yτ (λ, t) is determined by T yτ (λ, t) =

D˜ W (T, λ, t − τ − μ)x(μ)dμ.

(1.87)

0

In consideration of (1.78), the last expression may be substituted by

yτ (λ, t) = e

−mλT

T 0

D˜ W (T, λ, t − θ − μ)x(μ)dμ.

(1.88)

Chapter 2

State-Space Analysis of Finite-Dimensional Linear Continuous Periodic Objects

This chapter extends the results of Chap. 1 to finite-dimensional linear continuous periodic objects.

2.1 State-Space Description of Periodic Objects 1. An FDLCP object is a system in which the state vector v(t) of size χ × 1 is described by the vector differential equation dv(t) = A(t)v(t) + B1 (t)x(t) + B2 (t)u(t), dt

(2.1)

where x(t) is the input vector  × 1, u(t) is the control vector m × 1 and A(t) = A(t + T ), Bi (t) = Bi (t + T ), (i = 1, 2) are real continuous periodic matrices of sizes χ × χ , χ × , χ × m. It is assumed that matrices B1 (t) and B2 (t) are of bounded variation for +0 ≤ t ≤ T − 0. The output vector y(t) n × 1 of the FDLCP object is determined by y(t) = C(t)v(t), (2.2) where the n × χ matrix C(t) = C(t + T ) is continuous and of bounded variation for +0 ≤ t ≤ T − 0. 2. For further discussion, let us consider some properties of the homogeneous vector equation dv(t) = A(t)v(t) (2.3) dt and the corresponding matrix equation © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_2

17

18

2 State-Space Analysis of Finite-Dimensional Linear Continuous Periodic Objects

d V (t) = A(t)V (t), dt

(2.4)

where V (t) is a matrix χ × χ . The solution of the matrix equation (2.4) under the initial condition V (0) = Iχ is called matrizant of the FDLCP object (matrix A(t)) and is denoted by H (t). Under the given assumptions, the Floquet equation [27, 28] H (t) = L(t)e N t

(2.5)

can be applied for the matrizant H (t), where L(t) = L(t + T ) is a continuous differentiable periodic matrix χ × χ for which inf | det L(t)| > 0.

0≤t≤T

(2.6)

Moreover, in (2.5), N is a constant matrix χ × χ . In a number of cases, matrices L(t) and N can occur to be complex. Hereby, matrizant H (t) always remains real. Since e N 0 = Iχ it follows from (2.5) that

3.

The matrix

L(0) = Iχ .

(2.7)

M = H (T ) = L(T )e N T = e N T

(2.8)

Δ

we will call the monodromy matrix of the FDLCP object (2.1), (2.2). It directly follows from (2.8) that the monodromy matrix is non-singular, i.e., det M = det e N T = 0.

(2.9)

The eigenvalues of the monodromy matrix, i.e., the roots of det(z Iχ − M) = 0

(2.10)

are called multipliers of the FDLCP object, and roots of the equation det(Iχ − ζ M) = 0

(2.11)

its reverse multipliers. Let z 1 , . . . , z ρ be the set of different multipliers of the FDLCP object. From (2.9), it follows that z i = 0, (i = 1, . . . , ρ).

(2.12)

Then from (2.11) and (2.12), it follows that the sequence of reverse multipliers is defined by the relation ζi = z i−1 , (i = 1, . . . , ρ).

2.1 State-Space Description of Periodic Objects

19

As stated in [27, 28], the number z i is a multiplier if and only if there exists a solution of the vector equation (2.3), for which v(t + T ) = z i (t)

(2.13)

is valid. It follows from (2.13) that Equation (2.3) has a non-zero solution of period T , if and only if, at least one of the multipliers equals one. Below, the set of multipliers of the FDLCP object (2.1), (2.2) is denoted by Mz , and the corresponding set of reverse multipliers by Mζ . 4.

From (2.8), we get N=

1 ln M. T

(2.14)

As it is known from [27, 28], the logarithm of a constant square non-singular matrix always exists and is defined ambiguously. The set of matrices N , satisfying the relation (2.14), will be denoted by M N . Let s1 , . . . , sχ be the sequence of eigenvalues of an arbitrary matrix N ∈ M N , i.e., the roots of the equation det(s Iχ − N ) = 0.

(2.15)

Then the numbers sik = si + k

2π j (i = 1, . . . , χ ; k = 0, ±1, . . .) T

(2.16)

will be called characteristic indices of the FDLCP object. Below, the set of all characteristic indices is denoted by Ms . This set does not depend on the particular choice of the matrix N ∈ M N . It follows from (2.14) that the number z i = esik T (i = 1, . . . , χ ; k = 0, ±1, . . .)

(2.17)

is a multiplier and the number ζi = e−sik T (i = 1, . . . , χ ; k = 0, ±1, . . .)

(2.18)

a reverse multiplier. 5.

Substituting t by t + T in (2.5), we obtain H (t + T ) = L(t + T )e N (t+T ) = L(t)e N t e N T = H (t)M.

(2.19)

Consequently, for any integer , we have H (t + T ) = H (t)M  .

(2.20)

20

2 State-Space Analysis of Finite-Dimensional Linear Continuous Periodic Objects

It follows from (2.6) that matrix L(t) is invertible for all t. Therefore, matrix H (t) is also invertible for all t since Δ

G(t) = H −1 (t) = e−N t L −1 (t).

(2.21)

Let us show that the matrix G(t) is the solution of the matrix differential equation dG(t) = −G(t)A(t) dt

(2.22)

G(0) = Iχ .

(2.23)

under the initial condition

The last relation follows from the equality G(0) = H −1 (0) = Iχ .

(2.24)

For the proof of (2.22), we notice that by virtue of (2.21) dG(t) d H (t) d [G(t)H (t)] = H (t) + G(t) = 0χχ , dt dt dt

(2.25)

where 0χχ is the zero matrix χ × χ . Taking into account the relations given earlier, from (2.25), we find dG(t) H (t) + G(t)A(t)H (t) = 0χχ . dt

(2.26)

Since matrix H (t) is non-singular, then after reducing the equation (2.26) by H (t), we get that matrix G(t) = H −1 (t) is the solution of (2.22) under the initial condition (2.23). Note also that, by transposition, (2.22) can be brought into the form dG  (t) = −A (t)G  (t), dt

(2.27)

which is analogous to (2.4).

2.2 Transfer of Periodic Signals Through Periodic Objects 1. Suppose that in the equation (2.1), we have no control signal u(t) and the input signal x(t) = x T (t) is T-periodic, i.e., Δ

x(t) = x T (t) = x T (t + T ).

(2.28)

2.2 Transfer of Periodic Signals Through Periodic Objects

21

Then from (2.1), we come to the equation dv(t) = A(t)v + B1 (t)x T (t), dt

(2.29)

and the problem is to find non-zero solutions of (2.29), such that Δ

v(t) = vT (t) = vT (t + T ). 2.

(2.30)

Below, the notation Δ

v(kT ) = vk , (k = 0, ±1, . . .)

(2.31)

is used systematically. Thus, the general solution of the equation (2.29) can be written in the form t (2.32) v(t) = H (t)v0 + H (t)H −1 (ν)B1 (ν)x T (ν)dν. 0

Since A(t) = A(t + T ), B1 (t) = B1 (t + T ), the solution (2.32) will be T-periodic if and only if (2.33) v(0) = v0 = v1 = v(T ). For t = T from (2.32) taking into account (2.33) and (2.8), we obtain T v0 = Mv0 + M

H −1 (ν)B1 (ν)x T (ν)dν

(2.34)

H −1 (ν)B1 (ν)x T (ν)dν.

(2.35)

0

which is equivalent to T (Iχ − M)v0 = M 0

From here, it immediately follows that, if det(Iχ − M) = 0,

(2.36)

i.e., Eq. (2.3) has no multiplier equal to one, then there exists a unique vector of initial conditions that satisfy (2.33). This means that if (2.36) holds, the equation (2.29) has a unique non-zero T-periodic solution. For the vector of initial conditions using (2.35), we find

22

2 State-Space Analysis of Finite-Dimensional Linear Continuous Periodic Objects

v0 = (Iχ − M)

−1

T M

H −1 (ν)B1 (ν)x T (ν)dν.

(2.37)

0

This, given (2.32), leads to the expression for the periodic solution vT (t) = vT (t + T ) with vT (t) = H (t)(Iχ − M)−1 M

T

H −1 (ν)B1 (ν)x T (ν)dν +

0

t

H (t)H −1 (ν)B1 (ν)x T (ν)dν.

0

(2.38) We now show that the solution obtained is indeed T-periodic. For this, it is sufficient to show that relation (2.33) holds. For t = T from (2.38), we have   vT (T ) = M (Iχ − M)−1 M + Iχ

T

H −1 (ν)B1 (ν)x T (ν)dν =

0

= M(Iχ − M)−1

T

H −1 (ν)B1 (ν)x T (ν)dν = vT (0),

(2.39)

0

where it is used that (Iχ − M)−1 M + Iχ = (Iχ − M)−1 . 3.

(2.40)

For 0 < t < T , Eq. (2.38) can be written in integral form T vT (t) =

G T (t, ν)B1 (ν)x T (ν)dν,

(2.41)

0

where the matrix G T (t, ν) is defined by  G T (t, ν) =

  H (t) (Iχ − M)−1 M + Iχ H −1 (ν), ν < t, H (t)(Iχ − M)−1 M H −1 (ν), ν > t,

(2.42)

that with regard to (2.40) is equal to  G(t, ν) =

H (t)(Iχ − M)−1 H −1 (ν), ν < t, H (t)(Iχ − M)−1 M H −1 (ν), ν > t.

(2.43)

4. Let the condition (2.36) hold. Then the constructed T-periodic solution vT (t) is unique. Here, the general solution of (2.29) can be represented in the form v(t) = H (t)v0 + vT (t),

(2.44)

2.3 Transfer of Exponentially Periodic Signals Through Periodic Objects

23

where v0 is an arbitrary constant vector χ × 1. Using (2.44) and (2.2), we find the general expression for the output of the FDLCP object for a T-periodic input signal (2.28) (2.45) y(t) = C(t)v(t) = C(t)H (t)v0 + C(t)vT (t). For v0 = 0χ1 , the output y(t) is T-periodic and equal to y(t) = yT (t) = C(t)vT (t) = C(t + T )vT (t + T ). Theorem 2.1 Under condition (2.36), the periodic output (2.46) is unique.

(2.46) 

Proof We assume that there exist a constant non-zero vector v10 for which the output y1 (T ) is T-periodic. Then, using (2.45), we have y1 (t) − y1 (t + T ) = [C(t)H (t) − C(t + T )H (t + T )] v10 = 0n1 .

(2.47)

Since C(t) = C(t + T ), H (t + T ) = H (t)M, the latter expression can be written as (2.48) C(t)H (t)(Iχ − M)v10 = 0n1 . Since condition (2.36) is fulfilled, the vector Δ

v20 = (Iχ − M)v10

(2.49)

is also non-zero and we have from (2.48) C(t)H (t)v20 = 0n1 . It follows from (2.45) that y1 (t) = yT (t) = C(t)vT (t).

(2.50) 

2.3 Transfer of Exponentially Periodic Signals Through Periodic Objects 1. In this section, the results of Sect. 2.2 are generalized for the case that the input signal of the FDLCP object (2.29) is exponentially periodic (EP), i.e., has the form Δ

x(t) = x(λ, t) = eλt x T (t), x T (t) = x T (t + T ).

(2.51)

We will look for the EP solution of Eqs. (2.2), (2.29) in which v(λ, t) = eλt vT (λ, t), vT (λ, t) = vT (λ, t + T ), y(λ, t) = eλt yT (λ, t),

yT (λ, t) = yT (λ, t + T ).

(2.52)

24

2 State-Space Analysis of Finite-Dimensional Linear Continuous Periodic Objects

From these relations together with the system equations for the FDLCP object (2.29), (2.2), we get the T-periodic vectors vT (λ, t) and yT (λ, t) dvT (λ, t) = Aλ (t)vT (λ, t) + BT (t) dt

(2.53)

yT (λ, t) = C(t)vT (λ, t).

(2.54)

and Matrix Aλ (t) and vector BT (t) in (2.53) are Δ

Aλ (t) = A(t) − λIχ = Aλ (t + T ), Δ

BT (t) = B1 (t)x T (t) = BT (t + T ).

(2.55)

It is easy to check that the matrizant Hλ (t) of matrix Aλ (t) is defined by Hλ (t) = H (t)e−λt ,

(2.56)

where H (t) is the matrizant of the matrix A(t), and the corresponding monodromy matrix Mλ equals (2.57) Mλ = Hλ (T ) = Me−λT , where M is the monodromy matrix (2.8). Using the results of Sect. 2.2, it can be stated that, provided det(Iχ − Mλ ) = det(Iχ − e−λT M) = 0,

(2.58)

Equation (2.53) has a unique T-periodic solution vT (λ, t), which for 0 < t < T in accordance with (2.41) can be represented as T vT (λ, t) =

G T (λ, t, ν)B1 (ν)x T (ν)dν,

(2.59)

0

where due to (2.43)  G T (λ, t, ν) =

Hλ (t)(Iχ − Mλ )−1 Hλ−1 (ν), ν < t, Hλ (t)(Iχ − Mλ )−1 Mλ Hλ−1 (ν), t < ν.

Taking into account (2.56) and (2.57), we get from (2.60)

(2.60)

2.3 Transfer of Exponentially Periodic Signals Through Periodic Objects  G T (λ, t, ν) =  =

25

H (t)(Iχ − e−λT M)−1 H −1 (ν)e−λ(t−ν) , 0 < t − ν < T, = H (t)(Iχ − e−λT M)−1 e−λT M H −1 (ν)e−λ(t−ν) , −T < t − ν < 0 L(t)e N t (Iχ − e−λt M)−1 e−N ν L −1 (ν)e−λ(t−ν) , 0 < t − ν < T, L(t)e N ν (Iχ − e−λT M)−1 e−λT Me−N ν L −1 (ν)e−λ(t−ν) , −T < t − ν < 0.

(2.61) 2. Formula (2.59) can be converted into a form that defines the vector vT (λ, t) for the interval −∞ < t < ∞. For this, note that from the contents of Chap. 1, we have for the interval 0 < t < T ∞ 1  [(λ + k jω)Iχ − N ]e(λ+k jω)t = e N t (Iχ − e−λT e N T )−1 = D˜ N (T, λ, t) = T k=−∞

= (Iχ − e−λT M)−1 e N t = D˜ N (T, λ, t).

(2.62)

Since for kT < t < (k + 1)T , the equality D˜ N (T, λ, t) = D˜ N (T, λ, t − kT )ekλT

(2.63)

applies, it follows from (2.62) that D˜ N (T, λ, t − ν) =



D˜ N (T, λ, t − ν), 0 < t − ν < T, D˜ N (T, λ, t − ν − T )e−λT , −T < t − ν < T.

(2.64)

With the help of (2.62), we can obtain the explicit form of (2.64) D˜ N (T, λ, t − ν) =



e N t (Iχ − e−λT M)−1 e−N ν , 0 < t − ν < T, (2.65) e N t (Iχ − e−λT M)e−λT Me−N ν , −T < t − ν < 0.

From here, it follows that ∞ −1 k jω(t−ν) 1   ϕ N (T, λ, t − ν) = D˜ N (T, λ, t − ν)e−λ(t−ν) = e = (λ + k jω)Iχ − N T k=−∞  Nt e (Iχ − e−λT M)−1 e−N ν e−λ(t−ν) , 0 < t − ν < t, = (2.66) e N t (Iχ − e−λT M)−1 e−λT Me−N ν e−λ(t−ν) , −T < t − ν < 0.

Comparing (2.61) and (2.66), we come to the equality G T (λ, t, ν) = L(ν)ϕ N (T, λ, t − ν)L −1 (ν)

(2.67)

26

2 State-Space Analysis of Finite-Dimensional Linear Continuous Periodic Objects

which is valid for 0 < t < T, 0 < ν < T . As a result, we obtain that formula (2.59) can be represented in the form T vT (λ, t) =

L(t)ϕ N (T, λ, t − ν)L −1 (ν)B1 (ν)x T (ν)dν.

(2.68)

0

In contrast to (2.59), in the obtained expression, the integrand is T-periodic with respect to t. Therefore, unlike (2.59), formula (2.68) defines the vector vT (λ, t) for −∞ < t < ∞. 3.

The periodic solution (2.68) corresponds to the T-periodic output vector yT (λ, t) = C(t)vT (λ, t).

(2.69)

Repeating the proof of Theorem 2.1, instead of (2.50), we find a condition of the form (2.70) C(t)Hλ )(t)v20 (λ) = 0n1 , which indicates a unique periodic output for a fixed, non-specific value of λ. 4. Starting from (2.68), it is possible to get a corresponding relation for the EP output y(λ, t) in the interval −∞ < t < ∞. For this, we find with (2.68) and (2.69) T yT (λ, t) =

C(t)L(t)ϕ N (T, λ, t − ν)L −1 (ν)B1 (ν)x T (ν)dν,

(2.71)

0

from where, given that yT (t) = e−λt y(λ, t), x T (ν) = e−λν x(λ, ν),

(2.72)

we get the desired expression for the EP output T y(λ, t) =

C(t)L(t)ϕ N (T, λ, t − ν)eλ(t−ν) L −1 (ν)B1 (ν)x(λ, ν)dν =

0

T = 0

C(t)L(t) D˜ N (T, λ, t − ν)L −1 (ν)B1 (ν)x(λ, ν)dν.

(2.73)

2.4 Higher-Order Periodic Objects

27

2.4 Higher-Order Periodic Objects 1. In a number of applications, the functionality of an FDLCP object is described by a scalar linear differential equation of order χ d χ−1 y(t) d χ y(t) + a (t) + · · · + aχ (t)y(t) = x(t), 1 dt χ dt χ−1

(2.74)

where ai (t) = ai (t + T ) are real continuous functions. First we will transform this representation of the FDLCP object into the form (2.1), (2.2). For this, the vector v(t) χ × 1 is introduced using the relations ⎡

⎤ v1 (t) v(t) = ⎣ ... ⎦ , vχ (t) v1 (t) = y(t), v2 (t) =

dy(t) d χ−1 y(t) , . . . , vχ (t) = . dt dt χ−1

(2.75)

With this, the system of χ differential equations dv1 (t) = v2 (t), dt ... dvχ−1 (t) = vχ (t), dt dvχ (t) = −aχ (t)v1 (t) − aχ−1 (t)v2 (t) − · · · − a1 (t)vχ (t) + x(t), dt

(2.76)

is set up, which can be represented in the form (2.1) dv(t) = Ac (t)v(t) + B1 x(t). dt

(2.77)

Matrix Ac (t) χ × χ has the form ⎡

⎤ 0 1 0 ... 0 0 ⎢ 0 0 1 ... 0 0 ⎥ ⎢ ⎥ ⎢ ... ... ... ... ... ⎥ Ac (t) = ⎢ ... ⎥, ⎣ 0 0 0 ... 0 1 ⎦ −aχ (t) −aχ−1 (t) −aχ−2 (t) ... −a2 (t) −a1 (t)

(2.78)

and is called adjoint matrix for Equation (2.74). Besides, in (2.77) B1 is a constant vector χ × 1

28

2 State-Space Analysis of Finite-Dimensional Linear Continuous Periodic Objects

⎡ ⎤ 0 ⎢0⎥ ⎥ B1 = ⎢ ⎣...⎦ . 1

(2.79)

Together with the output equation y(t) = Cv(t),

(2.80)

where C is a constant vector 1 × χ   C = 1 0 ... 0 ,

(2.81)

Equations (2.77)–(2.81) represent the FDLCP object (2.74) in state space form (2.1), (2.2). 2. For the study of the FDLCP object under consideration, all theoretical considerations of Sects. 2.1–2.3 are applicable. Due to the specific type of matrix Ac (t) and vectors B and C, simplifications result in this case. In particular, the associated matrizant Hc (t) can be constructed as follows. We look at the homogeneous equation d χ−1 y(t) d χ y(t) + a (t) + · · · + aχ (t)y(t) = 0. 1 dt χ dt χ−1

(2.82)

Let us consider the system of particular solutions y1 (t), . . . , yχ (t) for the equation (2.82), which correspond to the initial conditions y1 (0) = 1, y2 (0) = 0,

yχ (0) = 0,

dy1 (t) d χ−1 y(t) = 0, . . . , = 0, dt |t=0 dt χ−1 |t=0 dy2 (t) d χ−1 y(t) = 1, . . . , = 0, dt |t=0 dt χ−1 |t=0 ... dyχ (t) dy χ−1 (t) = 0, . . . , = 1. dt |t=0 dt |t=0

(2.83)

Then matrizant Hc (t) is defined by ⎡ ⎢ Hc (t) = ⎢ ⎣

y1 (t) dy1 (t) dt

...

d χ−1 y1 (t) dt χ−1

⎤ y2 (t) ... yχ (t) dy (t) dy2 (t) χ ⎥ ... dt dt ⎥. ... ... ... ⎦ χ−1 d y (t) d χ−1 y2 (t) ... dt χ−1χ dt χ−1

(2.84)

2.4 Higher-Order Periodic Objects

3.

29

To determine the corresponding inverse matrizant G c (t) = Hc−1 (t),

(2.85)

note, that by transposing matrix Ac (t), we obtain matrix ⎡

0 ⎢−1 ⎢ − Ac (t) = ⎢ ⎢ ... ⎣0 0

0 0 ... 0 0

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

⎤ 0 aχ (t) 0 aχ−1 (t)⎥ ⎥ ... ... ⎥ ⎥ 0 a2 (t) ⎦ −1 a1 (t)

(2.86)

which corresponds to the system of differential equations dq1 (t) = aχ (t)qχ (t), dt dq2 (t) = −q1 (t) + aχ−1 (t)qχ (t), dt ... dqχ−1 (t) = −qχ−2 (t) + a2 (t)qχ (t), dt dqχ (t) = −qχ−1 (t) + a1 (t)qχ (t). dt

(2.87)

We form the set of solutions for Eqs. (2.87) ⎡

⎤ qi1 (t) Δ ⎢ qi2 (t) ⎥ ⎥ qi (t) = ⎢ ⎣ ... ⎦ , (i = 1, 2, . . . , χ ) qiχ (t)

(2.88)

corresponding to the initial conditions qii (0) = 1, qik = 0, (k = i).

(2.89)

⎤ q11 (t) q12 (t) ... q1χ (t) ⎢ q21 (t) q22 (t) ... q2χ (t)⎥ ⎥. G c (t) = ⎢ ⎣ ... ... ... ... ⎦ qχ1 (t) qχ2 (t) ... qχu (t)

(2.90)

Due to (2.85), we have ⎡

30

2 State-Space Analysis of Finite-Dimensional Linear Continuous Periodic Objects

4. For the FDLCP object (2.77)–(2.81), all statements made in the previous sections are valid. In particular, under the condition det(Iχ − Hc (T )) = 0 the T-periodic output of the FDLCP object is unique. 5. The above can be generalized to a multidimensional FDLCP object of order χ which is described by a vector equation of the form (2.74), in which y(t) and x(t) are vectors p × 1 and q × 1, and ai (t) and bi (t) are matrices p × p and p × q.

Chapter 3

The Frequency Method in the Theory of Periodic Objects

In this chapter, the concept of the parametric transfer matrix (PTM) is introduced and explained on the basis of the operator description of an FDLCP object.

3.1 Frequency Description of Linear Periodic Operators 1. In this section, frequency properties of linear periodic operators and systems are considered which are used in the following. Sufficient conditions for the applicability of the relations specified below are given in [17–19]. In the following explanations, it is supposed that these conditions are met. Let the linear operator y(t) = U [x(t)] (3.1) be defined on the set M X of vectors x(t) l × 1, where y(t) is a vector n × 1 that belongs to the set MY . Operator U is called stationary if from the relation (3.1) and for any real constant τ the equality U [x(t − τ )] = y(t − τ )

(3.2)

is true. If (3.2) does not hold for at least one value of τ , we will call this operator U non-stationary. If a constant T > 0 exists for a non-stationary operator U , such that from (3.1) the equality U [x(t − T )] = y(t − T ) (3.3) follows, then such an operator U will be called T-periodic and the number T its period [17, 29]. Obviously, if T > 0 is a period of operator U , then numbers Tk = kT , where k > 0 is integer, also will be a period of the operator. Further, if not specified © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_3

31

32

3 The Frequency Method in the Theory of Periodic Objects

otherwise, the period of the operator is understood to be the least value of T , for which (3.3) holds for any input x(t) ∈ M X . Notice that in the case if the input vector is T-periodic, i.e., x(t) = x(t + T ), the output vector y(t) is also T-periodic, since y(t) = U [x(t)] = U [x(t − T )] = y(t − T ). 2. Let s be a complex variable that varies in the domain Ms of the complex plane. We introduce vectors x1 (s, t), . . . , x (s, t)  × 1, defined by the relations ⎡ st ⎤ ⎡ ⎤ ⎡ ⎤ e 0 0 ⎢0⎥ ⎢est ⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ x1 (s, t) = ⎢ ⎢ ... ⎥ , x2 (s, t) = ⎢ 0 ⎥ , ..., x (s, t) = ⎢ ... ⎥ , ⎣ ... ⎦ ⎣ ... ⎦ ⎣0⎦ est 0 0

(3.4)

where s ∈ Ms . Suppose that for s ∈ Ms , we have xi (s, t) ∈ M X , (i = 1, . . . , ).

(3.5)

Then for all 1 ≤ i ≤ , the vectors n × 1 yi (s, t) = U [xi (s, t)]e−st , (i = 1, . . . , )

(3.6)

belonging to the set MY are defined. Hereby, the n ×  matrix W yx (s, t) defined by  W yx (s, t) = y1 (s, t) y2 (s, t) ... y (s, t)

(3.7)

will be called the parametric transfer matrix (PTM) of the operator U in the domain Ms . Generalizing operator U for matrix inputs, (3.7) can be written in the form  W yx (s, t) = U est I e−st .

(3.8)

3. As shown in [17, 29], if the operator U is stationary, i.e., satisfying (3.2), then the PTM W yx (s, t) does not depend on t and we get W yx (s, t) = W yx (s), s ∈ Ms .

(3.9)

If operator U is periodic, then we have W yx (s, t) = W yx (s, t + T ), s ∈ Ms .

(3.10)

Under assumptions that are usually fulfilled in applications [17, 29], condition (3.10) is not only necessary, but also sufficient for the T-periodicity of operator U . It is assumed that these assumptions are being fulfilled everywhere in the following.

3.2 Linear Periodic Integral Operators

33

In the problems discussed below, the PTM W yx (s, t) and, in particular, W yx (s) is an analytical function of the complex argument s in the region Ms and allows an analytical continuation to the whole complex plane. Below, by PTM W yx (s, t), we will understand such an analytical continuation on the whole complex s plane. At that, the complex s plane is divided into a finite number of non-overlapping regions Msi , in each of which the PTM W yx (λ, t) is the transfer matrix of some periodic operator Ui , so we have  W yx (s, t) = Ui est I e−st , s ∈ Msi .

(3.11)

The totality of operators Ui built in this way will be called the family of operators of the corresponding PTM W yx (s, t). We will denote this family of operators by MW . Furthermore, in addition to (3.1), the notation y(t) = W yx (s, t)x(t), s ∈ Msi

(3.12)

is used, which means that for x(t) = est I and s ∈ Msi the corresponding output y(s, t) of operator Ui is defined by y(s, t) = W yx (s, t)est , s ∈ Msi .

(3.13)

3.2 Linear Periodic Integral Operators 1.

In the following discussion, linear integral operators of the type

∞ h(t, ν)x(ν)dν, −∞ < t < ∞

y(t) = U [x(t)] =

(3.14)

−∞

play an important role where h(t, ν) is a given matrix n × , x(t) and y(t) are vectors  × 1, n × 1. Operator U is considered on the set M X of vectors x(t), for which integral (3.14) converges absolutely (according to the norm). Matrix h(t, ν) will be called Green’s function of the operator (3.14). We show that for the T-periodicity of this operator, it is sufficient that equation h(t − T, ν) = h(t, ν + T )

(3.15)

holds. Indeed, from (3.14), we have

∞ h(t − T, ν)x(ν)dν.

y(t − T ) = −∞

(3.16)

34

3 The Frequency Method in the Theory of Periodic Objects

Furthermore, from (3.14), it follows that

∞ U [x(t − T )] =

∞ h(t, ν)x(ν − T )dν =

−∞

h(t, ν + T )x(ν)dν.

(3.17)

−∞

Comparing expressions (3.16) and (3.17), we come to the conclusion that if (3.15) holds, equality also (3.3) holds, i.e., the operator (3.14) is T-periodic. Note also that (3.16) corresponds to the expression h(t, ν) = h(t + T, ν + T ),

(3.18)

which is obtained from (3.15) by replacing t by t + T . 2.

By substitution of the integration variable ν =t −μ

(3.19)

relation (3.14) leads to the form

∞ g(t, μ)x(t − μ)dμ,

y(t) = U [x(t)] =

(3.20)

−∞

where

Δ

g(t, μ) = h(t, t − μ).

(3.21)

Below g(t, μ) is referred to as the weight function of the operator (3.14). Note that the periodicity condition of operator (3.14), defined by the relations (3.15), (3.18), is expressed using the weight function as g(t, μ) = g(t + T, μ).

(3.22)

g(t + T, μ) = h(t + T, t − μ + T ).

(3.23)

Indeed, from (3.21), we have

From here, using (3.18), we find g(t + T, μ) = h(t, t − μ) = g(t, μ).

(3.24)

3.2 Linear Periodic Integral Operators

35

3. The parametric transfer matrix of operator (3.14), defined by (3.8), is written in the form

∞

∞ h(t, ν)esν dν = h(t, ν)e−s(t−ν) dν. (3.25) W yx (s, t) = e−st −∞

−∞

With the help of substitution (3.19), we get

∞ W yx (s, t) =

g(t, μ)e−sμ dμ.

(3.26)

−∞

It is evident that the PTM W yx (s, t) is the two-sided Laplace transformation of the weight matrix g(t, μ) for the argument μ (if such a transformation exists). Let the matrix g(t, μ) satisfy the conditions for the uniform convergence of the integral (3.26) relative to t, formulated in Chap. 1. Then formula (3.26) allows the inversion. The corresponding relation has the form 1 g(t, μ) = 2π j

c+

j∞

W yx (s, t)esμ ds, α < c < β.

(3.27)

c− j∞

Substituting here μ into t − ν, we find the expression for Green’s function 1 h(t, ν) = 2π j

c+

j∞

W yx (s, t)es(t−ν) ds, α < c < β.

(3.28)

c− j∞

It is essential that (3.27) and (3.28) are valid in the corresponding strip of the convergence α < c < β. 4.

As an example, we consider the PTM W yx (s, t) =

sin(t) , s−a

(3.29)

where a is constant. As follows from [22, 26], for c > Re(a) 1 2π j

c+

j∞

c− j∞

and for c < Re(a)

esμ ds = s−a



eaμ , μ > 0 , c > Re(a) 0, μ < 0

(3.30)

36

3 The Frequency Method in the Theory of Periodic Objects

1 2π j

c+

j∞

c− j∞

esμ ds = s−a



0, μ > 0 −eaμ , μ < 0, c < Re(a).

(3.31)

So, for c > Re(a), we get from (3.27) and (3.30) the expression for the weight function  sin t eaμ , μ > 0, g1 (t, μ) = (3.32) 0, μ < 0, and for c < Re(a) from (3.28), (3.31) follows  g2 (t, μ) =

0, μ > 0, − sin t eaμ , μ < 0.

(3.33)

Substituting in (3.32) and (3.33) t − ν for μ, we find Green’s functions  h 1 (t, ν) =  h 2 (t, ν) =

sin t ea(t−ν) , t > ν, 0, t < μ, 0, t > ν, . − sin t ea(t−ν) , t < ν

(3.34)

Green’s functions (3.34) correspond to periodic integral operators of the form (3.14)

t U1 [x(t)] =

sin t ea(t−ν) x(ν)dν, −∞

∞ U2 [x(t)] = −

sin t ea(t−ν) x(ν)dν.

(3.35)

t

Operators (3.35) form a family of linear periodic integral operators that correspond to PTM (3.29) in the two half-planes Re(s) > Re(a) and Re(s) < Re(a). The expressions of operators (3.35) in terms of weight functions define the formulas

∞ U1 [x(t)] =

sin t eaμ x(t − μ)dμ. 0

0 U2 [x(t)] = − −∞

sin t eaμ x(t − μ)dμ.

(3.36)

3.2 Linear Periodic Integral Operators

37

5. An important class of linear integral operators for applications are those for which the following condition is fulfilled h(t, ν) = 0n , ν > t

(3.37)

g(t, μ) = 0n , μ < 0.

(3.38)

or, what is equivalent

If (3.37) holds, from (3.14), we obtain

t h(t, ν)x(ν)dν,

(3.39)

g(t, μ)x(t − μ)dμ.

(3.40)

y(t) = U [x(t)] = −∞

and if (3.38) holds, from (3.20), we find

∞ y(t) = U [x(t)] = 0

Operators (3.39) and (3.40) are usually called causal. 6. Suppose that an operator U in (3.1) is defined on the set M X and there exists a set of values of parameter λ, for which the condition holds Δ

x(t) = x(λ, t) = eλt x1 (t),

y(t) = eλt y1 (λ, t).

(3.41)

In this case, the expression  Δ y1 (t) = e−λt U eλt x1 (t) = Uλ [x1 (t)]

(3.42)

defines a linear operator that we call shift operator.The PTM Wλ (s, t) of this shift operator in accordance with (3.8) is defined by   Wλ (s, t) = e−st Uλ est I = e−(s+λ) U e(s+λ)t I .

(3.43)

Comparing this expression with (3.8), we obtain that for the PTM of the shift operator, the formula (3.44) Wλ (s, t) = W (s + λ, t) applies in the corresponding domain of convergence. From (3.44), it follows that from the periodicity condition W (s, t) = W (s, t + T ), the relation Wλ (s, t) = Wλ (s, t + T ) follows.

38

3 The Frequency Method in the Theory of Periodic Objects

7. It should be emphasized that (3.44) has a general character and is applicable to any linear operators if the PTM exists. Let us, in particular, have a scalar linear stationary differential operator y(t) = Ud [x(t)] = d( p)x(t), where p =

d dt

(3.45)

is the differentiation operator and d( p) = p n + a1 p n−1 + · · · + an

(3.46)

is a given polynomial. The transfer function of operator (3.45), obtained by (3.8), shows that, in this case  W yx (s, t) = e−st Ud est = d(s). If

x(t) = eλt x1 (t),

(3.47)

y(t) = eλt y1 (t),

(3.48)

 Ud eλt x1 (t) = eλt d( p + λ)x1 (t).

(3.49)

then a direct calculation shows that

From here and with (3.47), we find that Wλ (s) = W yx (s + λ) = d(s + λ).

(3.50)

It is not difficult to check that the above remains true when we have y(t) = Ud1 [x(t)] = d( p, t)x(t)

(3.51)

instead of (3.45), where d( p, t) is a polynomial with variable coefficients d( p, t) = p n + a1 (t) p n−1 + · · · + an (t).

(3.52)

Using conversions similar to those above, we obtain that the transfer function of operator (3.52) has the form W yx (s, t) = d(s, t)

(3.53)

and for the shift operator U (λ), defined by (3.40), the transfer function Wλ (s, t) is defined (3.54) Wλ (s, t) = W yx (s + λ, t) = d(s + λ, t).

3.3 Operator Description of a Basic Periodic Object

39

3.3 Operator Description of a Basic Periodic Object 1.

Let us consider the basic FDLCP object dv(t) = A(t)v(t) + x(t) dt

(3.55)

H (t) = L(t)e N t .

(3.56)

with the matrizant

Let the indices of Eq. (3.55), i.e., the roots of the characteristic polynomial Δ

Δ(s) = det(s Iχ − N )

(3.57)

be the numbers s1 , . . . , sχ , among which can be equal ones. Let ri , (i = 1, . . . , q ≤ χ ) be the sequence of different real parts of indices sk , arranged in ascending order. In the following, we will refer to these numbers ri as Lyapunov indices. The open intervals (3.58) J0 = (−∞, r1 ), J1 = (r1 , r2 ), Jq = (rq , ∞) are called the regularity intervals of Eq. (3.55) (of matrix A(t)). 2.

Consider the rational matrix Δ

W N (s) = (s Iχ − N )−1

(3.59)

which can be represented as a fraction W N (s) =

B N (s) , Δ(s)

(3.60)

where B N (s) is a polynomial matrix χ × χ and Δ(s) is the characteristic polynomial (3.57). In general, the fraction on the right side of (3.60) is reducible. After all possible reductions, we get from (3.60) the irreducible representation [30] (s Iχ − N )−1 =

C(s) , ψ(s)

(3.61)

where C(s) is a polynomial matrix, ψ(s) is a monic polynomial that is called minimal. In this case, each root of the characteristic polynomial Δ(s) is a root of the minimal polynomial. Let the minimal polynomial have the form ψ(s) = (s − s˜1 )ν1 ...(s − s˜ p )ν p ,

(3.62)

40

3 The Frequency Method in the Theory of Periodic Objects

where all roots s˜i are different and Re(˜si ) ≤ Re(˜si+1 ). Then we have a decomposition into simple fractions [30] (s Iχ − N )−1 =

 p Z k1 C(s) Z k,νk Z k2 + · · · + , (3.63) = + ψ(s) (s − s˜k ) (s − s˜k )2 (s − s˜k )νk k=1

where Z ki are constant matrices χ × χ . Matrices Δ

Z k1 = Pk , (k = 1, . . . , p)

(3.64)

are commonly called components of the matrix N . We note a number of important properties of the components of the matrix N [22, 30]. (a) The number of components Pk is equal to the number of different roots of the characteristic polynomial Δ(s) (the minimal polynomial ψ(s)). (b) For any function f (N ) of the matrix N , we have Pk f (N ) = f (N )Pk .

(3.65)

(c) The relations apply p

Pk = Iχ ,

(3.66)

k=1

Pk2 = Pk ,

(3.67)

Pk P j = 0χχ , (k = j).

(3.68)

3. Denote by R ( = 1, . . . , q) the set of numbers of sequence elements s˜1 , . . . , s˜ p that have the same Lyapunov indices r . Introduce matrices Δ

Q =



Pk , ( = 1, . . . , q).

(3.69)

k∈R

From (3.66)–(3.68), analogous formulas for the matrices Q  follow q

Q  = Iχ ,

(3.70)

=1

Q 2 = Q  , ( = 1, . . . , q)

(3.71)

Q  Q m = 0χχ , ( = m).

(3.72)

3.3 Operator Description of a Basic Periodic Object

41

Assuming r0 = −∞, rq+1 = ∞, each regularity interval Ji , (i = 0, 1, . . . , q) can be linked to a pair of matrices Q i+ and Q i− , determined by the relations Q i+ =

i

Q,

Q i− =

=1

q

Q  = Iχ − Q i+ .

(3.73)

=i+1

In particular Q 0+ = 0χχ , Q q+ = Iχ ,

Q 0− = Iχ , Q q− = 0χχ .

(3.74)

For the matrices Q i+ and Q i− , the following equations apply Q i+ + Q i− = Iχ , 2 Q i+

2 = Q i+ , Q i− = Q i− ,

Q i+ Q i− = Q i− Q i+ = 0χχ .

(3.75)

Furthermore Q i+ e N t = e N t Q i+ ,

Q i− e N t = e N t Q i− .

(3.76)

4. Let H (t) be the matrizant of Equation (3.55). Then, using (3.73), we can define matrices  H (t)Q i+ H −1 (ν), t > ν, (3.77) h i (t, ν) = −H (t)Q i− H −1 (ν), t < ν. Let us show that the equalities h i (t, ν) = h i (t + T, ν + T ), (i = 0, 1, . . . , q)

(3.78)

are true. For this purpose, let us note that taking into account (3.56), the expression (3.77) can be represented as  h i (t, ν) =

L(t)e N t Q i+ e−N ν L −1 (ν), t > ν, L(t)e N t Q i− e−N ν L −1 (ν), t < ν,

(3.79)

which, given (3.76), leads to the expressions  h i (t, ν) =

L(t)e N (t−ν) Q i+ L −1 (ν), t > ν, −L(t)e N (t−ν) Q i− L −1 (ν), t < ν.

From here, (3.78) follows immediately, because L(t) = L(t + T ), L −1 (ν + T ).

(3.80) L −1 (ν) =

42

5.

3 The Frequency Method in the Theory of Periodic Objects

Using matrices (3.77), we construct a set of linear integral operators

∞ vi (t) = Ui [x(t)] =

h i (t, ν)x(ν)dν.

(3.81)

−∞

It follows from (3.78) that operators (3.81) are T-periodic and matrices h i (t, ν) are their Green’s functions. By substituting the integration variable ν = t − u,

(3.82)

(3.81) are brought to the form

∞ vi (t) = Ui [x(t)] =

gi (t, u)x(t − u)du,

(3.83)

−∞

where the weight functions gi (t, u) are defined by the formulas derived from (3.80) and (3.82)  L(t)Q i+ e N u L −1 (t − u), u > 0, (3.84) gi (t, u) = −L(t)Q i− e N u L −1 (t − u), u < 0. Below, the operators Ui [x(t)] will be called the basic operators of the object (3.55) corresponding to regularity intervals (3.58). We will call the basic operators U0 [x(t)] and Uq [x(t)] the extrema. With regard to (3.74), we have  h 0 (t, ν) =

0χχ , t > ν, −H (t)H −1 (ν), t < ν.

(3.85)

That is why operator U0 [x(t)] is given by the formula

∞ v0 (t) = −

H (t)H −1 (ν)x(ν)dν.

(3.86)

t

Analogously, with the help of (3.74), it is established that operator Uq [x(t)] is defined by the relation

t vq (t) = H (t)H −1 (ν)x(ν)dν, (3.87) −∞



since in this case h q (t, ν) =

H (t)H −1 (ν), t > ν, 0χχ , t < ν.

(3.88)

3.4 Parametric Transfer Matrix of the Basic Periodic Object

43

It follows from (3.88) that operator Uq is causal. The other operators Ui for i = q do not have the same property. 6. Each of the formulas (3.81) defines a solution of Equation (3.55) on the set of input signals x(t), at which the corresponding integral converges and allows differentiation by the argument t. Indeed, taking into account (3.75), relation (3.81) takes the form

t vi (t) =

H (t)Q i+ H −∞

−1

∞ (ν)x(ν)dν −

H (t)Q i− H −1 (ν)x(ν)dν.

(3.89)

t

The differentiation of this formula by t while fulfilling the above assumptions shows that dvi (t) = A(t)vi (t) + x(t), (i = 0, 1, . . . , q). (3.90) dt

3.4 Parametric Transfer Matrix of the Basic Periodic Object 1. As it follows from [22], relations (3.89) are applied when the input signal x(t) has the form (3.91) x(t) = eλt x1 (t), where Re(λ) ∈ Ji and vector x1 (t) is uniformly bounded on the axis −∞ < t < ∞, i.e., (3.92) x1 (t) < k = const, −∞ < t < ∞. In this case, the corresponding solution of the equation (3.90), calculated by the formula (3.89), can be written in the form vi (λ, t) = eλt v1i (λ, t),

(3.93)

where vector v1 (λ, t) for Re(λ) ∈ Ji is also uniformly bounded, i.e., v1i (λ, t) < d = const, Re(λ) ∈ Ji , −∞ < t < ∞. Hereby, if (3.91), (3.92) hold, the solution of (3.93), (3.94) is unique.

(3.94)

44

3 The Frequency Method in the Theory of Periodic Objects

2. It follows from (3.91) and (3.93) that the vector v1i (λ, t) for Re(λ) ∈ Ji defines the unique and bounded solution on the axis −∞ < t < ∞ of the displaced equation dv1i (λ, t) = (A(t) − λIχ )v1i (λ, t) + x1 (t). dt

(3.95)

On the other side, from (3.81), (3.91), and (3.93), we obtain

∞ v1i (λ, t) =

h i (λ, t, ν)x1 (ν)dν = Uiλ [x1 (t)],

(3.96)

−∞

where Uiλ [x1 (t)] is a linear integral operator with Green‘s function Δ

h i (λ, t, ν) = h i (t, ν)e−λ(t−ν) = h i (λ, t + T, ν + T ),

(3.97)

i.e., operator Uiλ [x1 (t)] is T-periodic, and therefore, from the condition x1 (t) = x1 (t + T ), it can be concluded that v1i (λ, t) = v1i (λ, t + T ).

(3.98)

It follows from the said that for Re(λ) ∈ Ji and for the input signal x(t) = eλt x1 (t), x1 (t) = x1 (t + T ),

(3.99)

where vector x1 (t) is bounded on the norm for 0 ≤ t ≤ T , operator (3.96) defines a unique solution of the displaced equation (3.95) of the form (3.93), where vector v1i (λ, t) is bounded and T-periodic, i.e., relation (3.98) is valid. 3. By extending Equation (3.55) and operators (3.81) for matrix inputs, on the base of the proven, we can assume that for the input signal x(t) = eλt Iχ , Re(λ) ∈ Ji

(3.100)

the corresponding operator defines a matrix χ × χ Δ

v1i (λ, t) = Wi (λ, t), Wi (λ, t) = Wi (λ, t + T ),

(3.101)

where Wi (λ, t) is the T-periodic solution of the matrix equation dWi (λ, t)  = A(t) − λIχ Wi (λ, t) + Iχ . dt

(3.102)

3.5 Parametric Transfer Matrix of a Complemented Periodic Object

45

Hereby, from (3.95) and (3.96), it follows that the matrix Wi (λ, t) is the PTM of the operator Ui [x(t)]. By construction, Wi (λ, t) is defined for Re(λ) ∈ Ji . 4. At the same time, from Sect. 2.3, it follows that for all λ not being roots of the characteristic function d(λ) = det(Iχ − e−λt M), Equation (3.102) has a unique T-periodic solution Wvx (λ, t) = Wvx (λ, t + T ), which due to (3.100), (3.59) for 0 ≤ t ≤ T has the form

T Wvx (λ, t) =

HT (λ, t, ν)e−λ(t−ν) dν,

(3.103)

0

where  HT (λ, t, ν) =

H (t)(Iχ − e−λT M)−1 H −1 (ν), t > ν, H (t)(Iχ − e−λT M)−1 e−λT M H −1 (ν), ν > t.

(3.104)

From (3.103) and (3.104), it follows that the matrix W yx (λ, t) can be represented in the form Rvx (λ, t) , (3.105) Wvx (λ, t) = d(λ) where the matrix Rvx (λ, t) for each t is an integer function of the argument λ. Further on, matrix Wvx (λ, t) is called parametric transfer matrix (PTM) of the equation (3.55). From (3.105), it follows that the PTM Wvx (λ, t) for each t is a meromorphic function of the argument λ, which is defined everywhere on the whole complex λ plane, except for the countable set of poles which belong to the indices set of equation Δ(λ) = det(Iχ − e−λT M) = 0. 5.

From the proven, it follows that for Re(λ) ∈ Ji , the equality is true Wi (λ, t) = Wvx (λ, t),

(3.106)

which means that the PTM Wi (λ, t) of operators (3.81) as function of the complex variable λ is an analytical continuation of each other. Hence follows the conclusion that the operators (3.81) form a family of linear periodic integral operators, generated by PTM Wvx (λ, t).

3.5 Parametric Transfer Matrix of a Complemented Periodic Object 1. The results of Sect. 3.4 for the basic FDLCP object (3.55) can be transferred without significant changes to the case of a complemented FDLCP object described by

46

3 The Frequency Method in the Theory of Periodic Objects

dv(t) = A(t)v(t) + B(t)x(t), dt

(3.107)

y(t) = C(t)v(t),

(3.108)

where the assumptions of Sect. 2.1 apply. Supposing x(t) = eλt x1 (t), v(t) = eλt v1 (t),

y(t) = eλt y1 (t),

we arrive at the equations dv1 (λ, t) = [A(t) − λIχ ]v1 (λ, t) + B(t)x1 (t) dt

(3.109)

y1 (t) = C(t)v1 (λ, t).

(3.110)

and

Let Equation (3.107) have regularity intervals Ji (3.58), each of which corresponds to a basic operator in (3.81). Then, by virtue of the proof in Sect. 3.4, for Re(λ) ∈ Ji , x1 (t) = x1 (t + T ), (i = 1, . . . , q)

(3.111)

there exists a unique T-periodic solution vT i (λ, t) = vT i (λ, t + T ) of Equation (3.109) and a unique T-periodic output yT i (λ, t), which are defined by

∞ vT i (λ, t) =

h i (t, ν)e−λ(t−ν) B(ν)x1 (ν)dν

−∞

∞ yT i (λ, t) =

C(t)h i (t, ν)e−λ(t−ν) B(ν)x1 (ν)dν

(3.112)

−∞

with matrices h i (t, ν) defined by (3.77) and (3.79). 2. If we extend Equations (3.107) and (3.108) to matrix inputs, assuming x1 (t) = I , we obtain from (3.112) T-periodic matrices (i) (λ, t) Wvx

=

(i) Wvx (λ, t

∞ + T) =

h i (t, ν)B(ν)e−λ(t−ν) dν

(3.113)

−∞

and

(i) (i) (i) W yx (λ, t) = W yx (λ, t + T ) = C(t)Wvx (λ, t),

(3.114)

3.5 Parametric Transfer Matrix of a Complemented Periodic Object

47

(i) which are PTMs of the operators (3.112). By construction, the PTM Wvx (λ, t) is a T-periodic solution of the displaced matrix equation

dv(λ, t) = [A(t) − λIχ ]v(λ, t) + B(t), dt

(3.115)

which we will call Zadeh’s equation. On the other hand, as it follows from Sect. 2.3, for all λ, which are not roots of the function d(λ) = det(Iχ − e−λT M), Zadeh’s equation has a unique T-periodic solution

T vT (λ, t) =

HT B (λ, t, ν)e−λ(t−ν) dν,

(3.116)

0

where the matrix HT B (λ, t, ν) is connected with the matrix HT (λ, t, ν) in (3.104) by (3.117) HT B (λ, t, ν) = HT (λ, t, ν)B(ν), or in the explicit form  HT B (λ, t, ν) =

H (t)(Iχ − e−λT M)−1 H −1 (ν)B(ν), t > ν, H (t)(Iχ − e−λT M)−1 e−λT M H −1 (ν)B(ν), v > t.

(3.118)

3. Further on, matrix vT (λ, t) (3.116) will be called the PTM of Equation (3.107), and matrix (3.119) yT (λ, t) = yT (λ, t + T ) = C(t)vT (λ, t) the PTM of the FDLCP object (3.107), (3.108). Here, as before, we will use the notation Δ Δ (3.120) vT (λ, t) = Wvx (λ, t), yT (λ, t) = W yx (λ, t). From (3.110)–(3.119), it follows that the PTM W yx (λ, t) for each t is a meromorphic function of the complex variable λ defined on the whole complex plane except for a countable set of poles, which belong to the set of roots of the function d(λ) = det(Iχ − e−λT M) and which are not dependent on t. For Re(λ) ∈ Ji , x1 (t) = I , we get from (3.120) the equations

∞

h i (t, ν)B(ν)e−λ(t−ν) dν,

(3.121)

C(t)h i (t, ν)B(ν)e−λ(t−ν) dν

(3.122)

Wvx (λ, t) = −∞

∞ W yx (λ, t) = −∞

48

3 The Frequency Method in the Theory of Periodic Objects

which are equivalent to (3.116) and (3.119). Replacing ν with t − u in (3.121) results in the expression

∞ gi (t, u)B(t − u)e−λu du, (3.123) Wvx (λ, t) = −∞

where as before gi (t, u) = h i (t, t − u)

(3.124)

is the corresponding weight function. As follows from [17, 26], the right part of the formula (3.123) for a fixed value of t is an absolutely convergent two-sided Laplace transform on the variable u. Therefore, using the general inversion formula, we come to c+

j∞ 1 Wvx (λ, t)eλu dλ, ri < c < ri+1 , (3.125) gi (t, u)B(t − u) = 2π j c− j∞

which after the substitution of t − ν for u leads to c+

j∞

1 h i (t, ν)B(ν) = 2π j

Wvx (λ, t)eλ(t−ν) dλ, ri < c < ri+1 .

(3.126)

c− j∞

After left multiplying the last formula by matrix C(t), we obtain c+

j∞

1 C(t)h i (t, ν)B(ν) = 2π j

W yx (λ, t)eλ(t−ν) dλ.

(3.127)

c− j∞

In particular, for c > rq , (3.127) with regard to (3.88) gives 1 2π j

c+

j∞

W yx (λ, t)eλ(t−ν) dλ =



C(t)H (t)H −1 (ν)B(ν), t > ν, 0n , t < ν.

(3.128)

c− j∞

4.

We now introduce the notation Δ

h Bi (t, ν) = h i (t, ν)B(ν), Δ

g Bi (t, u) = h Bi (t, t − u) = h i (t, t − u)B(t − u). Thus, Eq. (3.123) takes the form

(3.129)

3.5 Parametric Transfer Matrix of a Complemented Periodic Object

∞ Wvx (λ, t) =

g Bi (t, u)e−λu du, Re(λ) ∈ Ji .

49

(3.130)

−∞

Formula (3.130) denotes that for a fixed t = t ∗ , the PTM Wv (λ, t ∗ ) is the absolutely convergent two-sided Laplace transform of the matrix gi (t ∗ , u) in the interval Ji . 5.

Let the input signal x(t) ∈ Ωα,β in (3.107) have the two-sided Laplace transform

∞ X (λ) =

x(t)e−λt dt,

(3.131)

−∞

absolutely converging in the interval J¯i ⊂ Ji . Then for a fixed t = t ∗ the product V (λ, t ∗ ) = Wvx (λ, t ∗ )X (λ), Re(λ) ∈ J¯i

(3.132)

can be considered as convolution image of the corresponding originals, absolutely converging in the interval J¯i . Considering (3.130) and (3.131), using the inversion formula for convolution of originals [26] for Re(λ) ∈ Ji , we have 1 2π j

c+

jω

∗ (i) Wvx (λ, t ∗ )X (λ)eλt dλ

∞ =

g Bi (t ∗ , u)x(t ∗ − u)du =

−∞

c− jω

∞

∗

∗

∗

∞

h Bi (t , t − u)x(t − u)du =

= −∞

h Bi (t ∗ , ν)x(ν)dν.

(3.133)

−∞

This leads, with regard to (3.132), to the expression for the original v(t), corresponding to the image (3.132) 1 v(t) = 2π j

c+

j∞

(i) Wvx (λ, t)X (λ)eλt dλ =

c− j∞

∞ =

h Bi (t, ν)x(ν)dν, ri−1 ≤ r¯i−1 < c < r¯i ≤ ri ,

(3.134)

−∞

where r¯i−1 and r¯i are the boundaries of the interval J¯i . Based on the above calculations, it follows that the expressions in (3.134) define a particular solution of Equation (3.107). In particular, if r¯i > rq and r¯q+1 = ∞, then the corresponding solution (3.134) has the form

50

3 The Frequency Method in the Theory of Periodic Objects

t H (t)H

v(t) = −∞

−1

c+

j∞

1 (ν)B(ν)x(ν)dν = 2π j

Wvx (λ, t)X (λ)eλt dλ, c > r¯i .

c− j∞

(3.135) If we additionally have x(t) = 01 , t < 0,

(3.136)

then (3.135) defines the process at initial energy zero v0 (t) =

6.

⎧ t ⎨ ⎩0

H (t)H −1 (ν)B(ν)x(ν)dν, t ≥ 0,

(3.137)

0χ1 , t < 0.

By multiplying (3.134) from the left by matrix C(t), we obtain the equations 1 yo (t) = 2π j

c+

j∞

λt

∞

W yx (λ, t)X (λ)e dλ = c− j∞

C(t)h Bi (t, ν)x(ν)dν,

(3.138)

−∞

which specifies the determination of the output y(t) of an FDLCP object based on the operator method.

Chapter 4

The Floquet–Lyapunov Decomposition and Its Application

In this chapter, the concept of the Floquet–Lyapunov transformation is introduced, and based on this, the Floquet–Lyapunov decomposition is developed. The Floquet– Lyapunov decomposition, which is systematically used in the following, maps the structure of a multidimensional FDLCP object as a series connection of inertia-free periodic gain blocks and an LTI object.

4.1 Floquet–Lyapunov Transformation 1.

Let us have the equations of an FDLCP object dv(t) = A(t)v(t) + B(t)x(t), dt

(4.1)

y(t) = C(t)v(t)

(4.2)

and suppose that the assumptions of Sect. 2.1 hold. Let H (t) be the matrizant of the object, given by the Floquet equation H (t) = L(t)e N t .

(4.3)

We introduce into Eq. (4.1) the new state vector v L (t) using the Floquet–Lyapunov transformation (4.4) v(t) = L(t)v L (t) = H (t)e−N t v L (t). Differentiating the last relation, we find

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_4

51

52

4 The Floquet–Lyapunov Decomposition and Its Application

d L(t) dv(t) dv L (t) = v L (t) + L(t) . dt dt dt

(4.5)

Regarding that  d H (t) −N t d L(t) d  H (t)e−N t = − H (t)e−N t N = A(t)L(t) − L(t)N , = e dt dt dt (4.6) from (4.5), we can obtain dv(t) dv L (t) = A(t)L(t)v L (t) − L(t)N v L (t) + L(t) . dt dt

(4.7)

On the other hand, from (4.1) and (4.4), we have dv(t) = A(t)L(t)v L (t) + B(t)x(t). dt

(4.8)

Setting equal the right parts of Eqs. (4.7) and (4.8), we get L(t)

dv L (t) = L(t)N v L (t) + B(t)x(t). dt

(4.9)

Since matrix L(t) = L(t + T ) for all t is invertible, Eq. (4.9) is equal to dv L (t) = N v L (t) + L −1 (t)B(t)x(t). dt

(4.10)

By adding the expression of the output y(t) = C(t)L(t)v L (t),

(4.11)

we obtain together with (4.10) the equations of the transformed FDLCP object, in which, instead of the periodic matrix A(t), we have a constant matrix N . This fact significantly simplifies further calculations. 2.

Using the differential operator p = Δ

d , dt

we introduce the transfer matrix −1

W N ( p) = ( p Iχ − N ) .

(4.12)

Using this transfer matrix, (4.10) can be written in operator form  −1  v L (t) = W N ( p) L (t)B(t) x(t).

(4.13)

4.2 Floquet–Lyapunov Decomposition and Its Parametric Transfer Matrix

53

4.2 Floquet–Lyapunov Decomposition and Its Parametric Transfer Matrix Equation (4.13) together with Eq. (4.11) define an open-loop system with an LTI element and periodic modulation at the input and the output. The structure of this system is shown in Fig. 4.1. We will refer to this form of decomposition of the FDLCP object (4.1), (4.2) into three blocks as Floquet–Lyapunov (FL) decomposition. We want to show that the PTM W F L (λ, t) for the object (4.11) and (4.13) coincides with the earlier built PTM of the FDLCP object (4.1) and (4.2). For this, we assume that x(t) = eλt I and look for a solution of Equations (4.11) and (4.13) in the form y(t) = eλt W F L (λ, t), W F L (λ, t) = W F L (λ, t + T ).

(4.14)

From Fig. 4.1, it follows that the problem is reduced to the construction of the EP response of the system shown in Fig. 4.2 to the EP input signal Δ

−1

x N (t) = eλt L (t)B(t).

(4.15)

Using the results from Sect. 1.4, we get that, for this EP input signal, the corresponding EP output signal of the stationary part (4.12) for the EP input signal (4.15) has the form (4.16) v L (λ, t) = eλt W L x (λ, t), W L x (λ, t) = W L x (λ, t + T ), where the matrix W L x (λ, t) is defined by T W L x (λ, t) =

−1

ϕ N (T, λ, t − ν)L (ν)B(ν)dν.

(4.17)

0

Here applies for 0 < t < T , 0 < ν < T

x(t)

 L−1 (t)B(t)

xN (t)



WN (p)

vL (t)

 C(t)L(t)

Fig. 4.1 Floquet–Lyapunov decomposition of FDLCP object (4.11) and (4.13) xN (t)



WN (p)

vL (t)

 C(t)L(t)

Fig. 4.2 Substructure for calculation of PTM W F L (λ, t)

y(t)



y(t)



54

4 The Floquet–Lyapunov Decomposition and Its Application

 ϕ N (T, λ, t − ν) =

−1

e N t (Iχ − e−λT M) e−N ν e−λ(t−ν) , t > ν, −1 e N t (Iχ − e−λT M) e−λT Me−λ(t−ν) , t < ν.

(4.18)

Since from Fig. 4.1, follows W F L (λ, t) = C(t)L(t)W L x (λ, t),

(4.19)

we get from (4.17) and (4.18) T W F L (λ, t) =

−1

C(t)L(t)ϕ N (T, λ, t − ν)L (ν)B(ν)dν.

(4.20)

0

Given (2.71) for x T (t) = I , a corresponding equation for the PTM of the FDLCP object W yx (λ, t) follows, so that we get W F L (λ, t) = W yx (λ, t).

4.3 Periodic Object with Delay 1.

In this section, we consider an FDLCP object with input delay, described by dvτ (t) dt

= A(t)vτ (t) + B(t)x(t − τ ), yτ (t) = C(t)vτ (t),

(4.21)

where the matrices A(t) = A(t + T ), B(t) = B(t + T ), andC(t) = C(t + T ) are the same as in (2.1) and (2.2). Furthermore, the input delay τ is constant and allows the representation (1.77) τ = mT + θ = (m + 1)T − γ , 0 ≤ θ < T, 0 < γ ≤ T.

(4.22)

Equations (4.21) correspond to the open-loop system in Fig. 4.3 generated by the FL decomposition.

x(t)



e−pτ

x(t − τ)

 L−1 (t)B(t)

xNτ (t)

 WN (p)

vLτ (t)

Fig. 4.3 FL decomposition of the FDLCP object with input delay

 C(t)L(t)

yτ (t)



4.3 Periodic Object with Delay

55

2. We consider the problem of transferring an EP signal through the system shown in Fig. 4.3. For this, we suppose that Δ

x(t) = x(λ, t) = eλt x T (t), x T (t) = x T (t + T ).

(4.23)

x(λ, t − τ ) = eλt x T τ (λ, t),

(4.24)

x T τ (λ, t) = e−λτ x T (t − τ ).

(4.25)

Then

where

If (4.24) and (4.25) hold, the EP signal −1

x N τ (λ, t) = eλ(t−τ ) L (t)B(t)x T (t − τ )

(4.26)

acts on the input of the stationary element as shown in Fig. 4.3. The EP reaction of the stationary element on this EP input can be represented with the help of (1.66) in the form T v Lτ (λ, t) = D˜ N (T, λ, t − ν)x N τ (λ, ν)dν = =e

−λτ

T

0 −1 D˜ N (T, λ, t − ν)eλν L (ν)B(ν)x T (ν − τ )dν

(4.27)

0

and the corresponding output yτ (λ, t) is yτ (λ, t) = C(t)L(t)v Lτ (λ, t) =e

−λτ

T

−1 C(t)L(t) D˜ N (T, λ, t − ν)eλν L (ν)B(ν)x T (ν − τ )dν. (4.28)

0

Supposing here x T (t) = I ,

yτ (λ, t) = eλt W yτ x (λ, t),

(4.29)

where W yτ x (λ, t) is the PTM from input x(t) to output yτ (t), from (4.28), we get W yτ x (λ, t) = e−λτ

T

C(t)L(t) D˜ N (T, λ, t − ν)e−λ(t−ν) L (ν)B(ν)dν = −1

0

=e

−λτ

T

(4.30) −1

C(t)L(t)ϕ N (T, λ, t − ν)L (ν)B(ν)dν.

0

Comparing this equality to expression (4.20), we come to the equality W yτ x (λ, t) = e−λτ W yx (λ, t),

(4.31)

56

4 The Floquet–Lyapunov Decomposition and Its Application x(t)

 L−1 (t)B(t)

xN (t)

 WN (p)

vL (t)

 C(t)L(t)

y(t)



e−pτ

yτ (t)



Fig. 4.4 FL decomposition of the FDLCP object with output delay xN (t)



Wyx (p)

y(t)



yτ (t)



e−pτ

Fig. 4.5 Equivalent structure to Fig. 4.4

where W yx (λ, t) is the PTM for the FDLCP object without input delay. 3. In case of an existing τ conforming to (4.22) at the output of an FDLCP object, using the FL decomposition, we come to a structure shown in Fig. 4.4. In accordance with the general approach used in this book, we should suppose that x(t) = eλt I and equalities hold y(λ, t) = eλt W yx (λ, t), W yx (λ, t) = W yx (λ, t + T )

(4.32)

y τ (λ, t) = eλt W y τ x (λ, t), W y τ x (λ, t) = W y τ x (λ, t + T ),

(4.33)

where W y τ x (λ, t) is the desired PTM for the system in Fig. 4.4. With the help of (4.32), the structure in Fig. 4.4 can be substituted by the equivalent structure in Fig. 4.5. Hereby, from (4.32), we have y τ (λ, t) = y(λ, t − τ ) = eλ(t−τ ) W yx (λ, t − τ ). Comparing the last expression to (4.33), we find the PTM W y τ x (λ, t) W y τ x (λ, t) = e−λτ W yx (λ, t − τ ),

(4.34)

what with regard to the relation (4.20) T W yx (λ, t) =

−1

C(t)L(t)ϕ N (T, λ, t − ν)L (ν)B(ν)dν, 0

leads to the formula W y τ x (λ, t) = e

−λτ

T

−1

C(t − τ )L(t − τ )ϕ N (T, λ, t − τ − ν)L (ν)B(ν)dν. 0

(4.35)

4.3 Periodic Object with Delay

57

4. We want to show that (4.35) can be expressed by the matrizant H (t) of the FDLCP object. For this, regarding that ϕ N (T, λ, t − τ − ν) = D˜ N (T, λ, t − τ − ν)e−λ(t−τ −ν) ,

(4.36)

from (4.35) with (4.22), we obtain W y τ x (λ, t)

= e−mλT

T

C(t − θ )L(t − θ ) D˜ N (T, λ, t − θ − ν)L

−1

(ν)B(ν)e−λ(t−ν) dν,

0

(4.37)

where the relations are used C(t − τ ) = C(t − θ ), L(t − τ ) = L(t − θ ), D˜ N (T, λ, t − τ − ν) = D˜ N (T, λ, t − θ − ν)e−mλT .

(4.38)

We consider the square OABC: 0 ≤ t < T , 0 < ν < T on the t − ν plane. As follows from Fig. 4.6, the square OABC is divided by the two straight lines t − ν − θ = 0, t − ν − θ = −T into the three domains: R I , R I I , and R I I I , in which we have 0 < t − ν − θ < T, t, ν ∈ R I , −T < t − ν − θ < 0, t, ν ∈ R I I , −2T < t − ν − θ < −T, t, ν ∈ R I I I .

Fig. 4.6 The t − ν plane

t  T

(4.39)

t −ν −θ = 0 B

A RI

θ

t − ν − θ = −T RII

RIII O

T −θ

C  ν T

58

4 The Floquet–Lyapunov Decomposition and Its Application

For these domains, using (1.23) and (1.29), it follows that D˜ N (T, λ, t − ν − θ ) = D˜ N (T, λ, t − ν − θ ) = −1 = e N (t−θ) (Iχ − e−λT M) e−N ν , t, ν ∈ R I , D˜ N (T, λ, t − ν − θ ) = D˜ N (T, λ, t − ν − θ + T )e−λT = −1 = e N (t−θ) (Iχ − e−λT M) e−λT Me−N ν , t, ν ∈ R I I , D˜ N (T, λ, t − ν − θ ) = D˜ N (T, λ, t − ν − θ + 2T )e−2λT = −1 = e N (t−θ) (Iχ − e−λT M) e−2λT M 2 e−N ν , t, ν ∈ R I I I .

(4.40)

Formulas in (4.40) can be written in the compact form D˜ N (T, λ, t − ν − θ ) = e N ()t−θ G T θ (λ, t − ν)e−N ν ,

(4.41)

where the matrix G T θ (λ, t − ν) is given by ⎧ −1 ⎨ (Iχ − e−λT M) , θ < t − ν < θ + T, −1 G T θ (λ, t − ν) = (Iχ − e−λT M) e−λT M, θ − T < t − ν < θ, ⎩ −1 (Iχ − e−λT M) e−2λT M 2 , θ − 2T < t − ν < θ − T. (4.42) Inserting (4.41) and (4.42) in (4.37), we obtain the desired expression

W y τ x (λ, t) = e

−mλT

T

−1

C(t − θ )H (t − θ )G T θ (λ, t − ν)H (ν)B(ν)e−λ(t−ν) dν.

0

In this case, for the EP of the output y τ (λ, t), we come to the relation

= e−mλT

T

(4.43)

y τ (λ, t) = eλt W y τ x (λ, t) = −1

C(t − θ )H (t − θ )G T θ (λ, t − ν)H (ν)B(ν)eλν dν

(4.44)

0

for 0 ≤ t ≤ T .

4.4 Low-Frequency Exponentially Periodic Excitation of the Floquet–Lyapunov Decomposition 1. We assume that the input to the decomposition shown in Fig. 4.1 is an EP signal. The period Tμ of this signal is connected with the period T of the FDLCP object by Δ

Tμ = μT,

(4.45)

4.4 Low-Frequency Exponentially Periodic Excitation of the Floquet–Lyapunov …

59

where μ ≥ 2 is an integer. In this case, we have Δ

x(t) = xμ (λ, t) = eλt xμT (t), xμT (t) = xμT (t + μT )

(4.46)

and therefore, on the input of the stationary element acts an EP signal Δ

Δ

−1

x N μ (λ, t) = eλt L (t)B(t)xμT (t) = eλt x N Tμ (t). −1

(4.47)

−1

Since L (t) = L (t + T ) and B(t) = B(t + T ) from (4.46) and (4.47), it follows that (4.48) x N Tμ (t) = x N Tμ (t + μT ). 2. From the relations of Sect. 1.4, it follows that for all λ, such that among the j for any integer k, there are no roots of the equation numbers λ = k 2π μT det(Iχ − e−μλT M μ ) = det(Iχ − e−μλT eμN T ) = 0,

(4.49)

there exists a unique EP output of the stationary element v Lμ (λ, t) = =

μT 

μT 

D˜ N (μT, λ, t − ν)x N μ (λ, ν)dν =

0

(4.50)

−1 D˜ N (μT, λ, t − ν)L (ν)B(ν)x N Tμ (ν)eλν dν.

0

Here D˜ N (μT, λ, t − ν) is the sum of series −1 ∞ 2kπ j k2π j 1 (λ + )Iχ − N e(λ+ μT )(t−ν) . D˜ N (μT, λ, t − ν) = μT k=−∞ μT

(4.51)

Since, according to (1.26) and (1.27), for 0 < t < μT , we have −1 −1 D˜ N (μT, λ, t) = D˜ N (μT, λ, t) = e N t (Iχ − e−μλT M μ ) = (Iχ − e−μλT M μ ) e N t , (4.52) for 0 ≤ t < μT and 0 < ν < μT , we obtain with regard to (1.9)

 D˜ N (μT, λ, t − ν) =

−1

e N t (Iχ − e−μλT M μ ) e−N ν , 0 < t − ν < μT, −1 e N t (Iχ − e−μλT M μ ) e−μλT M μ e−N ν , −μT < t − ν < 0.

(4.53) This formula can be written in the compact form D˜ N (μT, λ, t − ν) = e N t G T μ (λ, t − ν)e−N ν ,

(4.54)

60

4 The Floquet–Lyapunov Decomposition and Its Application

where  G T μ (λ, t − ν) =

−1

(Iχ − e−μλT M μ ) , 0 < t − ν < μT, −1 (Iχ − e−μλT M μ ) e−μλT M μ , −μT < t − ν < 0.

(4.55)

Inserting (4.46) and (4.54) in (4.50), we find μT v Lμ (λ, t) =

−1

e N t G T μ (λ, t − ν)e−N ν L (ν)B(ν)x T μ (ν)eλν dν.

(4.56)

0

With the help of (4.56), we find the EP output yμ (λ, t) of the FL decomposition (FDLCP object) for the output (4.46)

=

μT 

yμ (λ, t) = C(t)L(t)v Lμ (λ, t) = −1

C(t)L(t)e N t G T μ (λ, t − ν)e−N ν L (ν)B(ν)x T μ (ν)eλν dν =

0

=

μT 

(4.57)

−1

C(t)H (t)G T μ (λ, t − ν)H (ν)B(ν)x T μ (ν)eλν dν.

0

From here, supposing yμ (λ, t) = eλt yT μ (t),

yT μ (t) = yT μ (t + μT ),

(4.58)

we obtain that for 0 ≤ t ≤ T μT yT μ (λ, t) = 0

−1

C(t)H (t)G T μ (λ, t − ν)e−λ(t−ν) H (ν)B(ν)x T μ (ν)dν.

(4.59)

Part II

The Parametric Transfer Matrix Approach to Sampled-Data Systems with Periodic Objects

Chapter 5

Open-Loop Sampled-Data System with Periodic Object

The chapter is dedicated to the PTM calculation of open-loop SD systems with an FDLCP object. Both synchronous and asynchronous systems are considered.

5.1 Multivariable Zero-Order Hold 1. By a multivariable zero-order hold, we understand a linear system with input x(t) and output u(t), where x(t) and u(t) are vectors  × 1 and m × 1, defined by the relations (5.1) u(t) = h(t − kT )xk , kT < t < (k + 1)T, where xk = x(kT ) and h(t) is a matrix m × , defined on the interval 0 ≤ t ≤ T . It is supposed that the elements of the matrix h(t) are of bounded variation in the definition interval, and the input signal x(t) is continuous at the sampling points Δ tk = kT , where T > 0 is the sampling period. If we introduce the periodic matrix h T (t), defined by the relations h T (t) = h(t − kT ), kT < t < (k + 1)T,

(5.2)

Equation (5.1) can be written as u(t) = h T (t)xk = h T (t)x(kT ), kT < t < (k + 1)T.

(5.3)

2. Relations (5.1) and (5.3) determine the linear operator u(t) = L 0 [x(t)] that maps an arbitrary vector x(t), continuous at the sampling points, to the unique corresponding output vector u(t). It is easy to prove that the operator L 0 is T-periodic, which means that for any integer N the equality © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_5

63

64

5 Open-Loop Sampled-Data System with Periodic Object

L 0 [x(t − N T )] = u(t − N T )

(5.4)

follows from (5.3). To show this, we introduce x N (t) = x(t − N T ), x N k = x N (kT ) = x(kT − N T ).

(5.5)

It is obvious that x N k = xk−N and it follows from (5.3) that L 0 [x N (t)] = h T (t)x N k = h T (t)xk−N , kT < t < (k + 1)T.

(5.6)

Since by construction h T (t) = h T (t − N T ), from (5.1) and (5.6), we obtain u N (t) = L 0 [x(t − N T )] = u(t − N T ),

(5.7)

which is equivalent to (5.4). 3.

Let us find the PTM W0 (s, t) of the operator L 0 by the general formula   W0 (s, t) = L 0 est I e−st .

(5.8)

Using (5.1), we get for this operator   L 0 est I = h(t − kT )eksT , kT < t < (k + 1)T

(5.9)

and from (5.8), we obtain W0 (s, t) = h(t − kT )e−s(t−kT ) , kT < t < (k + 1)T

(5.10)

or its equivalent form W0 (s, t) = h(t)e−st , 0 < t < T, W0 (s, t) = W0 (s, t + T ).

(5.11)

4. Since the elements of matrix h(t) have a bounded variation for 0 ≤ t ≤ T , the matrix W0 (s, t) may be represented by a Fourier series convergent for all t W0 (s, t) =

∞  k=−∞

Wk (s)ek jωt , ω =

2π , T

(5.12)

5.2 Linearized Model of the Digital Controller

where 1 Wk (s) = T

T

65

W0 (s, t)e−k jωt dt.

(5.13)

0

Using (5.11), from (5.13), we obtain Wk (s) = where H¯ (s) =

1 ¯ H (s + k jω), T T

h(t)e−st dt

(5.14)

(5.15)

0

is the transfer matrix of the forming element [31]. As a result, formula (5.12) takes the form ∞ 1  ¯ W0 (s, t) = (5.16) H (s + k jω)ek jωt . T k=−∞ Taking into account (1.13), the latter relation can be represented as W0 (s, t) = ϕ H¯ (T, s, t),

(5.17)

where ϕ H¯ (T, s, t) is the DPFR of the matrix H¯ (s).

5.2 Linearized Model of the Digital Controller 1. Using x(t) for the input and u(t) for the output signal, where x(t) and u(t) are vectors  × 1 and m × 1, the linearized model of a digital controller (DC) is determined by the following relations: ξk = x(kT ), (k = 0, ±1, . . .),

(5.18)

α0 ψk + · · · + αρ ψk−ρ = β0 ξk + · · · + βρ ξk−ρ ,

(5.19)

u(t) = h(t − kT )ψk , kT < t < (k + 1)T.

(5.20)

In these equations, as before, T is the sampling period and h(t) is a matrix m × q, which appears in (5.1). Besides, ξk and ψk are vectors n × 1 and q × 1, and αi , βi are constant matrices q × q and q × n.

66

5 Open-Loop Sampled-Data System with Periodic Object

2. Equation (5.18) is the equation of the analogue-to-digital converter (ADC). Equation (5.19) represents the control program (CP) for which the causality condition det α0 = 0

(5.21)

is fulfilled. If we use the backward operator ζ [32], which takes one step back according to the relations ζ ψk = ψk−1 , ζ ξk = ξk−1 ,

(5.22)

the equation of the CP can be written in the polynomial form α(ζ )ψk = β(ζ )ξk ,

(5.23)

α(ζ ) = α0 + α1 ζ + · · · + αρ ζρ , β(ζ ) = β0 + β1 ζ + · · · + βρ ζρ

(5.24)

where

are polynomial matrices q × q and q × n. From now on, we will call the pair of the polynomial matrices α(ζ ), β(ζ ) discrete controller or simply controller. Under condition (5.21), matrix α(ζ ) is invertible, and there exists a rational matrix −1

Wd (ζ ) = α (ζ )β(ζ ),

(5.25)

which we will call transfer matrix of the discrete controller. In addition to the above, note that (5.20) is the equation of the zero-order hold where the matrix h(t) is the same as in the previous section. This zero-order hold corresponds to the functionality of a digital-to-analogue converter (DAC). The structure of the DC is shown in Fig. 5.1. 3. The union of relations (5.18)–(5.20) may be regarded as a linear periodic system [17, 19] which generates a set of linear periodic operators u(t) = L Ri [x(t)], (i = 1, . . . , μ).

(5.26)

For each of the operators L Ri in some domain of the complex plane G i , there exists a corresponding PTM W Ri (s, t), defined by the formula DC

ξk

x(t)



ADC



ψk CP

Fig. 5.1 Structure of the digital controller (DC)



u(t) DAC



5.2 Linearized Model of the Digital Controller

  W Ri (s, t) = L Ri est In e−st = W Ri (s, t + T ).

67

(5.27)

These matrices W Ri (s, t) as functions of the complex variable s are analytical extensions of each another and define in their totality the single function W R (s, t) of the complex variable s. This function is defined on the whole complex plane except for some special points (poles). In the following, we will call the matrix W R (s, t) = W R (s, t + T ) the PTM of the DC (5.18)–(5.20). 4.

To determine the PTM W R (s, t), we should, as shown in [20], assume x(t) = est I

(5.28)

and find a solution to Equations (5.18)–(5.20), in which u(t) = est W R (s, t), W R (s, t) = W R (s, t + T ).

(5.29)

As shown in [17, 20, 21], such a solution exists if the conditions ψk+1 = esT ψk

(5.30)

are met. Bearing in mind the fact that, in this case ξk = eksT I , (k = 0, ±1, . . .),

(5.31)

and substituting (5.30) and (5.31) into (5.19), we get ksT ˜ , α(s)ψ ˜ k = β(s)e

(5.32)

˜ are determined by the relations where matrices α(s) ˜ and β(s)  α(s) ˜ = α(ζ ) ζ =e−sT = α0 + α1 e−sT + · · · + αρ−ρsT , ˜ β(s) = β(ζ ) ζ =e−sT = β0 + β1 e−sT + · · · + βρ−ρsT .

(5.33)

From (5.21), it follows that matrix α(s) ˜ is invertible. So from (5.32), we get

where

ψk = W˜ d (s)eksT ,

(5.34)

 −1 ˜ W˜ d (s) = Wd (ζ ) ζ =e−sT = α˜ (s)β(s).

(5.35)

If (5.34) holds, from (5.20), we find u(t) = h(t − kT )eksT W˜ d (s), kT < t < (k + 1)T.

(5.36)

68

5 Open-Loop Sampled-Data System with Periodic Object

Multiplying the last expression by e−st , we obtain W R (s, t) = h(t − kT )e−s(t−kT ) W˜ d (s), kT < t < (k + 1)T,

(5.37)

which can be expressed in the equivalent form W R (s, t) = h(t)e−st W˜ d (s), 0 < t < T, W R (s, t) = W R (s, t + T ).

(5.38)

Taking into account (5.11), the last relations may be written as W R (s, t) = W0 (s, t)W˜ d (s),

(5.39)

where W0 (s, t) is the PTM of the zero-order hold. 5. Equations (5.18)–(5.20) correspond to the DC structure in Fig. 5.1. Using (5.39), we can get an equivalent structural representation of the DC, which is used from now on. This, however, requires an auxiliary formula. Let f (t) be an arbitrary vector function  × 1 of the real argument t, defined on −∞ < t < ∞. Let τ be an arbitrary real number and Δ (5.40) f τ (t) = f (t − τ ) be a shift of function f (t) by τ . Then the relation (5.40) may be regarded as a linear operator (5.41) f τ (t) = L τ [ f (t)], which we will name τ -shift operator. Obviously, the operator L τ is stationary and its transfer matrix Wτ ( p), defined by the formula

turns out to be equal to

  Wτ (s) = L τ est I e−st

(5.42)

Wτ (s) = e−sτ I .

(5.43)

In particular, in the scalar case for the value of the sampling period T, the transfer function WT (s) of the shift operator takes the form WT (s) = e−sT .

(5.44)

From (5.44), it follows that the matrix W˜ d (s) in (5.35) can be regarded as a transfer matrix of a linear stationary system which is composed of pure delay elements with a delay time value of the sampling period. In this case, according to the definition of a transfer matrix of a linear stationary system, we obtain the operator equation x1 (t) = W˜ d (s)x(t)

(5.45)

5.2 Linearized Model of the Digital Controller Fig. 5.2 PTM structure of the digital controller (DC)

69 DC

x(t)



with x(t) = est I , we get

W˜ d (s)

x1 (t)

 W0 (s,t)

x1 (t) = W˜ d (s)est .

u(t)



(5.46)

Using the above, let us consider the continuous sampled-data open-loop system shown in Fig. 5.2, where the matrix W˜ d (s) is understood in the sense of (5.42), (5.46), and W0 (s, t) is the PTM of the zero-order hold (5.11). By assuming x(t) = est I and using (5.40) and (5.11), it is easy to see that the PTM of the system in Fig. 5.2 is determined by (5.39). 6. For example, let us consider scalar equations (5.18)–(5.20) while the control program equation has the following form: ψk + 2ψk−2 = ξk−1 . Then

(5.47)

ζ 1 + 2ζ 2

(5.48)

e−sT . 1 + 2e−2sT

(5.49)

Wd (ζ ) = and after replacing ζ by e−sT , we obtain W˜ d (s) =

The right-hand side of this relation corresponds to the connection of pure delay elements shown in Fig. 5.3.

Fig. 5.3 Structure of delay elements for Equation (5.49)

x(t)

   

x1 (t) • 

e−sT

−2e−sT



70

5 Open-Loop Sampled-Data System with Periodic Object

5.3 Open-Loop System with Time-Invariant Object 1. In this section, we consider the open-loop SD system depicted in Fig. 5.4. Here, W0 (s, t) is the PTM of the zero-order hold (5.11) of dimensions m × q, and W (s) is the transfer matrix of the LTI object of dimensions n × m which is assumed to be rational and strictly proper. In total, we have a linear periodic system to which the PTM W yx1 (s, t) = W yx1 (s, t + T ) of dimension n × q corresponds. 2. For the determination of the PTM W yx1 (s, t) in accordance with the general approach described, we assume that x1 (t) = est Iq , u(t) = est Wux1 (s, t), where

y(t) = est W yx1 (s, t),

Wux1 (s, t) = Wux1 (s, t + T ), W yx1 (s, t) = W yx1 (s, t + T )

(5.50)

(5.51)

is the PTM from input x1 (t) to the outputs u(t) and y(t). It follows from (5.16) that u(t) = est W0 (s, t) =

∞ 1  ¯ H (s + k jω)e(s+k jω)t = D˜ H¯ (T, s, t), T k=−∞

where as before H¯ (s) =

T

h(ν)e−sν dν.

(5.52)

(5.53)

0

The exponentially periodic (EP) response of the LTI object to the EP input signal u(t) in (5.52) may be written as y(t) =

∞ 1  W (s + k jω) H¯ (s + k jω)e(s+k jω)t = D˜ W H¯ (T, s, t). T k=−∞

(5.54)

Multiplying (5.54) by e−st gives the desired PTM ∞ 1  W yx1 (s, t) = W (s + k jω) H¯ (s + k jω)ek jωt = ϕW H¯ (T, s, t). T k=−∞

Fig. 5.4 Series connection of zero-order hold and LTI system

(5.55)

Wyx1 (s,t) x1 (t)

 W0 (s,t)

u(t)



W (s)

y(t)



5.3 Open-Loop System with Time-Invariant Object

71

3. Using [17] allows us to rewrite the given relations in a closed form. Notice that (5.53) implies T H¯ (s + k jω) = h(t)e−(s+k jω)ν dν, (5.56) 0

and after we substitute this in (5.54), we get  ∞ 1  W (s + k jω) h(ν)e−(s+k jω)ν dνe(s+k jω)t . T k=−∞ T

y(t) =

(5.57)

0

By changing the order of addition and integration here, we come to the formula y(t) =

T 0

 1 T

W (s + k jω)e(s+k jω)(t−ν) h(ν)dν =

∞ k=−∞ T

D˜ W (T, s, t − ν)h(ν)dν,

=

(5.58)

0

where as before ∞ 1  D˜ W (T, s, t) = W (s + k jω)e(s+k jω)t . T k=−∞

(5.59)

Taking into account the equality D˜ W (T, s, t − ν) = ϕW (T, s, t − ν)es(t−ν)

(5.60)

and multiplying (5.58) by e−st , we find T W yx1 (s, t) =

ϕW (T, s, t − ν)e−sν h(ν)dν.

(5.61)

0

4. If we have a digital controller (DC) with PTM W R (s, t) (5.39) as pulse element in the open-loop system in Fig. 5.4, then we come to the structure shown in Fig. 5.5. In this case, in accordance with the general rules of transforming PTMs [17, 21], the corresponding RTM W yx (s, t) is equal to W yx (s, t) = W yx1 (s, t)W˜ d (s).

(5.62)

72

5 Open-Loop Sampled-Data System with Periodic Object Wyx (s,t) x(t)



W˜ d (s)

x1 (t)

 Wyx1 (s,t)

y(t)



Fig. 5.5 Series connection of DC and LTI system

5.4 Synchronous Open-Loop System with Periodic Object 1. In this section, we consider an open-loop SD system with an FDLCP object, whose structure is shown in Fig. 5.6, where W0 (s, t) is the PTM of the zero-order hold (5.11), W F L (s, t) is the PTM of the FDLCP object (4.2) and W˜ d (s) is the matrix (5.35). It is assumed that the period of the FDLCP object coincides with the sampling period T (synchronous mode). Let us denote the PTM of the system shown in Fig. 5.6 by W yx (s, t) and the PTM of the subsystem shown in Fig. 5.7 by W yx1 (s, t). It is clear that (5.63) W yx (s, t) = W yx1 (s, t)W˜ d (s). In the following, we bring the subsystem in Fig. 5.7 in a form that uses the Floquet– Lyapunov decomposition as shown in Fig. 4.1, which leads to the structure in Fig. 5.8, where according to (5.1) u(t) = h(t − kT )x(kT ), kT < t < (k + 1)T. Furthermore, we have W N (s) = (s Iχ − N )

(5.64)

−1

(5.65)

and L(t) = L(t + T ),

B(t) = B(t + T ), C(t) = C(t + T ).

(5.66)

Wyx (s,t) x(t)



W˜ d (s)

x1 (t)

 W0 (s,t)

u(t)

 WFL (s,t)

y(t)



Fig. 5.6 Open-loop SD system with an FDLCP object Fig. 5.7 Part of the system in Fig. 5.6

Wyx1 (s,t) x1 (t)

 W0 (s,t)

u(t)

 WFL (s,t)

y(t)



5.4 Synchronous Open-Loop System with Periodic Object

73 Wyx1 (s,t)

x1 (t)

u1 (t)

u(t)

 WN (s)

 L−1 (t)B(t)

 W0 (s,t)

vL (t)

 C(t)L(t)

y(t)



Fig. 5.8 FL decomposition of structure in Fig. 5.7 W01 (s,t) x1 (t)

 W0 (s,t)

u(t)

 L−1 (t)B(t)

u1 (t)



Fig. 5.9 First two blocks of Fig. 5.8

2. Let us consider the first two blocks occurring in Fig. 5.8. It follows from (5.64) that input x1 (t) and output u 1 (t) in Fig. 5.9 are related by −1

−1

u 1 (t) = L (t)B(t)u(t) = L (t)B(t)h(t − kT )x1 (kT ) = = L (t − kT )B(t − kT )h(t − kT )x1 (kT ), kT < t < (k + 1)T, −1

(5.67)

which can be written as u 1 (t) = h 1 (t − kT )x1 (kT ), kT < t < (k + 1)T, where

−1

h 1 (t) = L (t)B(t)h(t), 0 < t < T.

(5.68)

(5.69)

Equation (5.68) shows that the connection in Fig. 5.9 can be considered as a conventional zero-order hold with the modulated hold function h 1 (t) and the sampling period T . The PTM W01 (s, t) of this hold is determined by the equations arising from (5.11) W01 (s, t) = e−st h 1 (t), 0 < t < T, (5.70) W01 (s, t) = W01 (s, t + T ). The representation of PTM W01 (s, t) in the form of a Fourier series, similar to (5.16), has the form W01 (s, t) = ϕ H¯ 1 (T, s, t) = where

∞ 1  ¯ H1 (s + k jω)ek jωt , T k=−∞

(5.71)

74

5 Open-Loop Sampled-Data System with Periodic Object Wyx1 (s,t) u1 (t)

x1 (t)



 W01 (s,t)

WN (s)

vL (t)

 C(t)L(t)

y(t)



Fig. 5.10 Modified representation of Fig. 5.8

H¯ 1 (s) =

T

−1

L (t)B(t)h(ν)dν

(5.72)

0

and the matrix B(t) is assumed to have a bounded variation. From this, we can conclude that the structures shown in Figs. 5.8 and 5.10 are equivalent. According to (5.55), the PTM WvL x1 (s, t) from input x1 (t) to output v L (t) is equal to WvL x1 (s, t) = ϕ N H¯ 1 (T, s, t), where ϕ N H¯ 1 (T, s, t) =

∞ 1  W N (s + k jω) H¯ 1 (s + k jω)ek jωt . T k=−∞

(5.73)

(5.74)

Thus, the PTM W yx1 (s, t) from input x1 (t) to the output y(t) has the form W yx1 (s, t) = C(t)L(t)ϕ N H¯ 1 (T, s, t) = e−st C(t)L(t) D˜ N H¯ 1 (T, s, t),

(5.75)

and the PTM of the original open-loop system in Fig. 5.6 turns out to be equal to W yx (s, t) = C(t)L(t)ϕ N H¯ 1 (T, s, t)W˜ d (s).

(5.76)

3. Let us find closed expressions for the PTM W yx1 (s, t). For this purpose, note that from (5.61), with W (s) = N (s) and H¯ (s) = H¯ 1 (s), it follows that ϕ N H1 (T, s, t) = =

T

T

ϕ N (T, s, t − ν)e−sν h 1 (ν)dν =

0

(5.77) −1

ϕ N (T, s, t − ν)L (ν)B(ν)h(ν)e−sν dν.

0

Therefore D˜ N H¯ 1 (T, s, t) = est ϕ N H¯ 1 (T, s, t) =

T

−1 −1 D˜ N (T, s, t − ν) L (ν)B(ν)h(ν)dν.

0

(5.78)

5.5 Asynchronous Rising Open-Loop System with Periodic Object

75

From (1.69) and (1.70), we have under 0 < t < T, 0 < ν < T . D˜ N (T, s, t − ν) =



−1

e N t (Iχ − e−sT M) e−N ν , ν < t, −1 e N t (Iχ − e−sT M) e−sT Me−N ν , ν > t,

(5.79)

where it is taken into account that e N T = M. Since −1

−1

L(t)e N t = H (t), e−N ν L (ν) = H (ν),

(5.80)

where H (t) is the matrizant of the FDLCP object, from (5.79), we find that if 0 < t < T and 0 < ν < T , the following equalities hold Δ

L (t) D˜ N (T, s, t − ν)L (ν) = HT (s, t, ν) =  −1 −1 H (t)(Iχ − e−sT M) H (ν), ν < t, = −1 −1 H (t)(Iχ − e−sT M) e−sT M H (ν), ν > t −1

−1

(5.81)

where HT (s, t, ν) is the matrix (3.104). By means of the last equality, we find C(t)L(t) D˜ N H¯ 1 (T, s, t) = C(t)

T HT (s, t, ν)B(ν)h(ν)dν,

(5.82)

0

which, taking into account (5.75), leads to the formula W yx1 (s, t) = e

−st

T HT (s, t, ν)B(ν)h(ν)dν.

C(t)

(5.83)

0

5.5 Asynchronous Rising Open-Loop System with Periodic Object 1. In the previous section, an SD system was considered, containing a periodic pulse element and an FDLCP object, which we called synchronous, because the sampling period Td of the pulse element was equal to the period T of the FDLCP object. If, on the other hand, Td = T , the corresponding system will be called asynchronous. In the case of (5.84) Td = μT, where μ > 1 is an integer, the SD system will be termed rising, and if T = μTd , it will be termed declining.

(5.85)

76

5 Open-Loop Sampled-Data System with Periodic Object

2. This section deals with a rising open-loop SD system in which the DC, which is described by equations of the type (5.18)–(5.20), is used as pulse element ξk = x(kμT ), (k = 0, ±1, . . .)

(5.86)

α0 ψk + · · · + αρ ψk−ρ = β0 ξk + · · · + βρ ξk−ρ ,

(5.87)

u μ (t) = h μ (t − kμT )ψk , kμT < t < (k + 1)μT.

(5.88)

The equation (5.88) of the digital-to-analogue converter can be given in the form of (5.3) (5.89) u μ (t) = h T μ (t)ψk , kμT < t < (k + 1)μT, where the periodic matrix h T μ (t) = h T μ (t + μT ) is defined by the formula Δ

h T μ (t) = h μ (t − kμT ), kμT < t < (k + 1)μT.

(5.90)

With the equations of the DC (5.86)–(5.88), we can associate the structure shown in Fig. 5.11, where  W˜ dμ (s) = Wd (ζ ) ζ =e−μsT

(5.91)

and Wd (ζ ) is matrix (5.25). Besides, in Fig. 5.11 appears the PTM Wμ (s, t) = h T μ (t)e−st , 0 < t < μT, Wμ (s, t) = Wμ (s, t + μT ).

(5.92)

3. To describe the FDLCP object, we will use the FL decomposition shown in Fig. 4.1. The structure represented in Fig. 5.12 corresponds to the open-loop SD system for input x1 (t) and output yμ (t). The first two elements in Fig. 5.12 may be combined into a fictitious zero-order hold with PTM Wμ1 (s, t) = Wμ1 (s, t + μT ), which is described by −1

Wμ1 (s, t) = L (t)B(t)h T μ (t)e−st , 0 < t < μT, −1 Wμ1 (s, t) = L (t)B(t)h T μ (t)e−s(t−kμT ) , kμT < t < (k + 1)μT,

Fig. 5.11 DC subsystem x(t)



W˜ d µ (s)

x1 (t)

 Wµ (s,t)

(5.93)

uµ (t)



5.5 Asynchronous Rising Open-Loop System with Periodic Object

77

Wyμ x1 (s,t) x1 (t)

 Wμ (s,t)

uμ (t)

 L−1 (t)B(t)

uN (t)

 WN (p)

vL μ (t)

 C(t)L(t)

yμ (t)



Fig. 5.12 FL decomposed asynchronous rising SD system Wyμ x1 (s,t) x1 (t)

uN (t)

 Wμ1 (s,t)



WN (s)

vL μ (t)

 C(t)L(t)

yμ (t)



Fig. 5.13 Corresponding structure to Fig. 5.12 including the modified zero-order hold

−1

−1

where L (t) = L (t + μT ), B(t) = B(t + μT ), h T μ (t) = h T μ (t + μT ) has been taken into account. The corresponding structural scheme is shown in Fig. 5.13. 4. To determine the PTM W yμ x1 (s, t) of the system shown in Fig. 5.13, we assume as usual that (5.94) x1 (t) = est I. In this case, the EP signal u N (s, t) = est Wμ1 (s, t)

(5.95)

acts on the stationary element W N (s) of the system. Using (5.93), we obtain that for 0 < t < μT , we have −1 (5.96) u N (s, t) = L (t)B(t)h μ (t). Thus, it follows that the EP response v N μ (s, t) to the input u N (s, t) in accordance with (5.95) is μT v Lμ (s, t) =

−1 D˜ N (μT, s, t − ν)L (ν)B(ν)h μ (ν)dν,

(5.97)

0

where D˜ N (μT, s, t − ν) is the series sum

−1 ∞  ω ω 1  ˜ (s + k j )Iχ − N e(s+k j μ )(t−ν) . D N (μT, s, t − ν) = μT k=−∞ μ

(5.98)

The closed expressions for the series sum (5.98) can be obtained from (5.79) by substituting μT for T . As a result, we find for 0 < t, ν < μT

78

5 Open-Loop Sampled-Data System with Periodic Object

D˜ N (μT, s, t − ν) =



−1

e N t (Iχ − e−sμT M μ ) e−N ν , ν < t, e N t (Iχ − e−sμT M μ )e−sμT M μ e−N ν , ν > t,

(5.99)

by virtue of which it follows from (5.97) and (5.71) that −1 −1 Δ L (t) D˜ N (μT, s, t − ν)L (ν) = HμT (s, t, ν) =  −1 −1 H (t)(Iχ − e−μsT M μ ) H (ν), ν < t, . = −1 −1 H (t)(Iχ − e−μsT M μ ) e−μsT M μ H (ν), ν > t.

5.

(5.100)

Fig. 5.13 implies that for input x1 (t) = esT I , the EP output yμ (s, t) is equal to yμ (s, t) = C(t)L(t)v Lμ (s, t),

(5.101)

and the corresponding PTM W yμ x1 (s, t) is determined by the formula W yμ x1 (s, t) = e−st C(t)L(t)v Lμ (s, t).

(5.102)

Taking into account (5.97) and (5.98), from (5.102), we get

W yμ x1 (s, t) = e

−st

μT HμT (s, t, ν)B(ν)h(ν)dν.

C(t)

(5.103)

0

5.6 Open-Loop System with Periodic Object and High-Frequency Hold 1. In this section, we consider an open-loop system with an FDLCP object of period T in which the hold operates according to i −1 T − kT )ψi,k , (i = 1, . . . , η) η

(5.104)

i i −1 T < t < kT + T, (k = 1, ±1, . . .). η η

(5.105)

u(t) = h i (t − while kT +

In these relations, η > 1 is an integer, ψi,k , (i = 1, . . . , η) are the elements of vector control sequences of dimensions q × 1. Besides, in (5.104), h i (t) are matrices m × q, defined on the interval 0 ≤ t ≤ Tη and having bounded variation within it.

5.6 Open-Loop System with Periodic Object and High-Frequency Hold

2.

79

Let us introduce matrices h¯ i (t) defined on the interval 0 ≤ t ≤ T by the relations ⎧ i−1 ⎪ ⎨ 0mq , 0 < t < η T, T ), i−1 T < t < ηi T, h¯ i (t) = h i (t − i−1 η η ⎪ ⎩ 0 , i T < t < T. mq η

(5.106)

Then formulas (5.104) and (5.105) are equivalent to formula u(t) =

η 

h¯ i (t − kT )ψi,k , kT < t < (k + 1)T.

(5.107)

i=1

After introducing T -periodic matrices h¯ T i (t) = h¯ i (t − kT ), kT < t < (k + 1)T,

(5.108)

relation (5.107) can be replaced by u(t) =

η 

h¯ T i (t)ψi,k , kT < t < (k + 1)T.

(5.109)

i=1

3.

By introducing the block matrix m × ηq   h ∗T η (t) = h¯ T 1 (t) ... h¯ T η (t) ,

(5.110)

as well as the block vector ηq × 1 ⎡

⎤ ψ1k ψk = ⎣ ... ⎦ , ψηk Δ

(5.111)

formula (5.109) can be regarded as the equation of a multivariable zero-order hold u(t) = h ∗T η (t)ψk , kT < t < (k + 1)T

(5.112)

with period T . We suppose below that the control vector sequence (5.111) is constructed by relations similar to (5.18), (5.19) ξk = x(kT ), (k = 0, ±1, . . .),

(5.113)

α0 ψk + · · · + αρ ψk−ρ = β0 ξk + · · · + βρ ξk−ρ ,

(5.114)

where matrices αi , βi have dimensions ηq × ηq and ηq × n, while det α0 = 0.

80

5 Open-Loop Sampled-Data System with Periodic Object

4. In their totality, relations (5.112)–(5.114) may be considered as the equations of a certain DC with sampling period T . Therefore, to construct the PTM of an openloop system with an FDLCP object of period T and a pulse element (5.112), we can use the relations from Sects. 5.3 and 5.4 and the original structure from Fig. 5.6. Then matrix W˜ d (s) will be found from (5.35), and according to (5.11) and (5.112), the PTM W0 (s, t) is equal to W0∗ (s, t) = h ∗T η (t)e−st , 0 < t < T, W0∗ (s, t) = W0 (s, t + T ).

(5.115) η

5. Using the given relations and also the formula (5.63), we find the PTM W yx (s, t) from input x(t) to output y(t) with η W yx (s, t)

=e

−st

T C(t)

HT (s, t, ν)B(ν)h ∗T η (ν)dν W˜ d (s).

(5.116)

0

Since it follows from (5.110) that for 0 < t < T   h ∗T η (t) = h¯ 1 (t) ... h¯ η (t) ,

(5.117)

from (5.115), we obtain η (s, t) W yx

=e

−st

T C(t)

  HT (s, t, ν)B(ν) h¯ 1 (ν) ... h¯ η (ν) dν W˜ d (s).

(5.118)

0 η

If we represent matrix W˜ d (s) of dimensions nq ×  in the block form ⎤ ⎡ m W˜ d1 (s) q η W˜ d (s) = ⎣ ... ⎦ ... , W˜ dη (s) q

(5.119)

where the letters outside the matrix denote the corresponding block dimensions, relation (5.118) may be written as η  

T

η W yx (s, t)

=e

−st

C(t)

i=1 0

HT (s, t, ν)B(ν)h¯ i (ν)W˜ di (s)dν,

(5.120)

5.6 Open-Loop System with Periodic Object and High-Frequency Hold

81

which with (5.106) leads to the formula i

η (s, t) = e−st C(t) W yx

T η η  i=1 i−1 η

T

  i −1 T W˜ di (s)dν. (5.121) HT (s, ν)B(ν)h i ν − η

Chapter 6

Open-Loop Sampled-Data System with Periodic Object and Delay

At this point, the results from Chap. 5 are extended to open-loop SD systems with pure delay.

6.1 Open-Loop System with Linear Time-Invariant Object and Delay 1. In this section we consider an open-loop system, whose structure is shown in Fig. 6.1 where a pure delay is assumed at the input of the LTI object. Using the DC structural representation in Fig. 5.2 and combining the two LTI elements in Fig. 6.1, we arrive at the structure in Fig. 6.2, where Wτ (s) = W (s)e−sτ .

(6.1)

Here W (s) is a strictly proper rational matrix and τ > 0 is a real constant, allowing the representation τ = mT + θ = (m + 1)T − γ , (6.2) where m is a non-negative integer and 0 ≤ θ < T , 0 < γ = T − θ ≤ T . Furthermore, in Fig. 6.2 W0 (s, t) is the PTM of the zero-order hold (5.11) and W˜ d (s) is a matrix defined by formula (5.35).

τ (s, t) from input x1 (t) to output y(t). For this, in 2. Let us find the PTM W yx 1 accordance with the general approach used in this work, we assume x1 (t) = est Iq .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_6

83

84

6 Open-Loop Sampled-Data System with Periodic Object and Delay Wyxτ (s,t) u(t)

x(t)



DC



uτ (t)



e−sτ

W (s)

y(t)



Fig. 6.1 SD system with LTI object and delay τ (s,t) Wyx

x(t)



W˜ d (s)

x1 (t)

u(t)

 W0 (s,t)



Wτ (s)

y(t)



Fig. 6.2 Modified representation of Fig. 6.1

Then, by using (5.16) we can see that the input of the LTI element Wτ (s) in Fig. 6.2 is driven by the EP signal ∞ 1  ¯ H (s + k jω)e(s+k jω)t , T k=−∞

u(s, t) = est W0 (s, t) = where H¯ (s) =

T

h(ν)e−sν dν.

(6.3)

(6.4)

0

The EP response of the LTI element to the EP input (6.3) is equal to y(s, t) =

∞ 1  Wτ (s + k jω) H¯ (s + k jω)e(s+k jω)t . T k=−∞

(6.5)

Substituting here (6.1) and (6.4), gives T ∞ 1  −(s+k jω)τ y(s, t) = W (s + k jω)e h(ν)e−(s+k jω)ν dνe(s+k jω)t . T k=−∞

(6.6)

0

By changing the order of addition and integration, we obtain T y(s, t) = 0

D˜ W (T, s, t − τ − ν)h(ν)dν,

(6.7)

6.1 Open-Loop System with Linear Time-Invariant Object and Delay

where

∞ 1  D˜ W (T, s, t − τ − ν) = W (s + k jω)e(s+k jω)(t−τ −ν) . T k=−∞

85

(6.8)

From here, using (6.2) and (1.20), we have D˜ W (T, s, t − τ − ν) = e−(m+1)sT D˜ W (T, s, t + γ − ν),

(6.9)

and formula (6.7) takes on the form y(s, t) = e

−(m+1)sT

T

D˜ W (T, s, t + γ − ν)h(ν)dν.

(6.10)

0

3. Let us find closed expressions for the right-hand side of formula (6.10). First we will assume that 0 < t < θ, (6.11) then it is clear that γ < t + γ < T.

(6.12)

In this case we can rewrite formula (6.10) in the form ⎡ ⎢ y(s, t) = e−(m+1)sT ⎣

t+γ 

D˜ W (T, s, t + γ − ν)h(ν)dν +

T

⎤ ⎥ D˜ W (T, s, t + γ − ν)h(ν)dν ⎦ .

t+γ

0

(6.13) Formulas (6.11), (6.12) imply that for the first summand on the right-hand side of (6.13) we have 0 < t + γ − ν < T. (6.14) Therefore from (1.23) we obtain D˜ W (T, s, t + γ − ν) = D˜ W (T, s, t + γ − ν).

(6.15)

If we have a representation −1

W (s) = C(s Iχ − A) B,

(6.16)

for the LTI object, it follows from (1.29) that −1 −1 D˜ W (T, s, t) = Ce N t (Iχ − e−sT e AT ) B = C(I − e−sT e AT ) e N t B.

(6.17)

86

6 Open-Loop Sampled-Data System with Periodic Object and Delay

Therefore, if (6.11), (6.14) hold, we can get D˜ W (T, s, t + γ − ν) = −1 ˜ = DW (T, s, t + γ − ν) = Ce A(t+γ ) (Iχ − e−sT e AT ) e−Aν B.

(6.18)

In the second summand on the right-hand side of (6.13) for t + γ < ν < T the inequality − T < t + γ − ν < 0, (6.19) is valid, which means that

 D˜ W (T, s, t + γ − ν) = D˜ W T, s, (t + γ − ν + T ) − T = = e−sT D˜ W (T, s, t + γ − ν + T ) = e−sT D˜ W (T, s, t + γ − ν − T ).

(6.20)

It follows from this relation and (1.29), that if (6.19) holds, we have D˜ W (T, s, t + γ − ν) = e−sT D˜ W (T, s, t + γ − ν + T ) = −1 = Ce A(t+γ ) (Iχ − e−sT e AT ) e−sT e AT e−Aν B,

(6.21)

which by means of the identity previously used −1

(Iχ − e−sT e AT )e−sT e AT = (Iχ − e−sT e AT ) − Iχ ,

(6.22)

can be written as −1 D˜ W (T, s, t + γ − ν) = Ce A(t+γ ) (Iχ − e−sT e AT ) e−Aν B − Ce A(t+γ −ν) B. (6.23)

τ (s, t) If we substitute (6.18) and (6.23) into (6.13), then considering that the PTM W yx 1 from input x1 (t) to output y(t) is equal to τ (s, t) = e−st y(s, t), W yx 1

(6.24)

under the condition (6.11) we get = e−(m+1)sT Ce A(t+γ )

τ (s, t) = est W yx 1  T T

−1 Δ e−Aν Bh(ν)dν − e−Aν Bh(ν)dν = (Iχ − e−sT e AT ) 0

t+γ

Δ

= y1 (s, t). (6.25) 4.

For θ 1 is an integer, and is described by the equations ξk = y(kTμ ), (k = 0, ±1, . . .), α(ζ )ψk = β(ζ )ξk , u μ (t) = h μ (t − kTμ )ψk , kTμ < t < (k + 1)Tμ .

(7.27)

7.2 Asynchronous Rising Closed-Loop System

107

In their totality the relations (7.26), (7.27) define a periodic linear system with period Tμ . Using the FL decomposition, we can come up with a system of equations similar to (7.2) dv L (t) dt

−1

−1

= N v L (t) + L (t)B1 (t)x(t) + L (t)B(t)u μ (t − τ ), y(t) = C(t)L(t)v L (t), ξk = y(kTμ ) = C(0)v L (kTμ ), α(ζ )ψk = β(ζ )ξk , u μ (t) = h μ (t − kTμ )ψk , kTμ < t < (k + 1)Tμ .

(7.28)

The last equation in this system can also be written as u μ (t) = h Tμ (t)ψk , kTμ < t < (k + 1)Tμ ,

(7.29)

where the periodic matrix h Tμ (t) = h Tμ (t + Tμ ) is defined by the relation h T μ (t) = h μ (t − kTμ ), kTμ < t < (k + 1)Tμ .

(7.30)

The PTM Wμ (s, t) of hold (7.29) has the form Wμ (s, t) = h Tμ (t)e−s(t−kTμ ) , kTμ < t < (k + 1)Tμ . μ

(7.31)

μ

To determine the PTM W yx (s, t) = W yx (s, t + Tμ ) of system (7.28), as before, we should set x(t) = est I and find the EP solution where v L (s, t) = est WvμL x (s, t), WvμL x (s, t + Tμ ), μ μ μ y(s, t) = est W yx (s, t), W yx (s, t) = W yx (s, t + Tμ ), μ μ μ u μ (t) = est Wux (s, t), Wux (s, t) = Wux (s, t + Tμ ), ψk+1 = esTμ ψk , (k = 0, ±1, . . .).

(7.32)

2. The structural scheme in Fig. 7.7 is associated with the Eqs. (7.28) where Wμ (s, t) = Wμ (s, t + Tμ ) is PTM (7.31). Below we suppose that the representation τ = m 1 Tμ + θ1 = (m 1 + 1)Tμ − γ1 ,

Fig. 7.7 Asynchronous closed-loop SD system with FDLCP object and delay

(7.33)

108

7 Closed-Loop Sampled-Data System with Periodic Object and Delay

Fig. 7.8 Equivalent structures, formation of PTM Wμθ (s, t)

holds, in which m 1 is a non-negative number and Δ

0 ≤ θ1 < Tμ , 0 < Tμ − θ1 = γ1 ≤ Tμ .

(7.34)

Moreover, in Fig. 7.7 W˜ dμ (s) is a matrix of the form W˜ dμ (s) = Wd (ζ ) ζ =e−sTμ . Using the transformations applied in Chap. 6, it is not difficult to verify the equivalence of the two structures shown in Fig. 7.8, where Wμθ (s, t) is the PTM of the extended zero-order hold −1

u μθ (t) = L (t + θ1 )B(t + θ1 )h Tμ (t)ψk , kTμ < t < (k + 1)Tμ

(7.35)

with period Tμ . It follows from (7.35) that PTM Wμθ (s, t) can be represented by −1

Wμθ (s, t) = L (t + θ1 )B(t + θ1 )h Tμ (t)e−s(t−kTμ ) , kTμ < t < (k + 1)Tμ , (7.36) where it is obvious that Wμθ (s, t) = Wμθ (s, t + Tμ ). This allows us to move from the structure in Fig. 7.7 to the equivalent structure in Fig. 7.9.

Fig. 7.9 Equivalent structure to Fig. 7.7

7.2 Asynchronous Rising Closed-Loop System

109

Fig. 7.10 Equivalent structure to Fig. 7.9 due to the stroboscopic property

3. We assume that the mode of functioning corresponding to (7.32) is described by the system structure in 7.9. Then the feedback signal influencing the pulse element is μ (s, t). (7.37) y(s, t) = est W yx The signal is continuous and has period Tμ . Therefore, using the stroboscopic property of the pulse element, we can proceed to consider the open-loop system in Fig. 7.10. Let us determine the EP response of this open-loop system to the EP input −1 μ signals est W yx (s, 0) and est L (t)B1 (t). Using the results of Sect. 6.3, from Fig. 7.10 μ we find that the EP response of the open-loop system to input signal est W yx (s, 0) is μ (s, 0), yq (s, t) = est C(t)L(t)WvμL x1 (s, t)W˜ dμ (s)W yx

(7.38)

μ

where WvL x1 (s, t) is PTM (6.81). According to (1.66) the EP response r (s, t) (LTI −1 element) to input signal est L (t)B1 (t) has the form T r (s, t) =

−1 D˜ N (T, s, t − ν)L (ν)B1 (ν)esν dν.

(7.39)

0

By structure, we have

 μ (s, t) = yq (s, t) + r (s, t) e−st . W yx

(7.40)

By substituting here (7.38) and (7.39), after simplification, we get μ μ (s, 0) + C(t)L(t)r T (s, t) = W yx (s, t) C(t)L(t)WvμL x1 (s, t)W˜ dμ (s)W yx

with

T r T (s, t) =

−1

ϕ N (T, s, t − ν)L (ν)B1 (ν)dν.

(7.41)

(7.42)

0

The left and right parts of equality (7.41) are continuous with respect to t for −∞ < t < ∞, therefore equality (7.41) applies particularly for t = 0. So we get

110

7 Closed-Loop Sampled-Data System with Periodic Object and Delay μ μ C(0)WvμL x1 (s, 0)W˜ dμ (s)W yx (s, 0) + C(0)r T (s, 0) = W yx (s, 0),

(7.43)

where μ

WvL x1 (s, 0) = e−(m 1 +1)sTμ

Tμ 0

r T (s, 0) =

−1 D˜ N (Tμ , s, γ1 − ν)L (ν + θ1 )B(ν + θ1 )h μ (ν)dν,

T

−1

ϕ N (T, s, −ν)L (ν)B1 (ν)dν, 0 W˜ dμ (s) = Wd (ζ ) ζ =e−sTμ

(7.44) and the fact that L(0) = Iχ has been taken into account. Equation (7.43) may be rewritten as   μ In − C(0)WvμL x1 (s, 0)W˜ dμ (s) W yx (s, 0) = C(0)r T (s, 0), (7.45) from which follows that  −1 μ W yx (s, 0) = In − C(0)WvμL x1 (s, 0)W˜ dμ (s) C(0)r T (s, 0).

(7.46) μ

After substituting (7.46) into (7.41), we get the desired expression for PTM W yx (s, t) μ (s, t) = C(t)L(t)WvμL x1 (s, t) R˜ μ (s)C(0)r T (s, 0) + C(t)r T (s, t), W yx

where

 −1 R˜ μ (s) = W˜ dμ (s) In − C(0)WvμL x1 (s, 0)W˜ dμ (s) .

(7.47)

(7.48)

It follows from (7.47) that μ μ (s, t) = W yx (s, t + Tμ ), W yx

since

(7.49)

WvμL x1 (t) = WvμL x1 (s, t + Tμ ), C(t) = C(t + Tμ ), B(t) = B(t + Tμ ),

B1 (t) = B1 (t + Tμ ), r T (s, t) = r T (s, t + Tμ ).

7.3 Closed-Loop System with Periodic Object, High-Frequency Hold and Delay This section studies a closed-loop system in which the FDLCP object is defined by the standard equations

7.3 Closed-Loop System with Periodic Object, High-Frequency Hold and Delay dv(t) dt

= A(t)v(t) + B1 (t)x(t) + B(t)u(t − τ ), y(t) = C(t)v(t).

111

(7.50)

The functionality of the pulse element is described by equations ξk = y(kT ), (k = 0, ±1, . . .), α(ζ )ψk = β(ζ )ξ, η u(t) = i=1 h¯ i (t − kT )ψi,k , kT < t < (k + 1)T,

(7.51)

where η > 1 is an integer. In the last equation in (7.51) h¯ i (t) are matrices m × q, given by relations (5.106) ⎧ i−1 ⎪ ⎨ 0mq , 0 < t < η T, T ), i−1 T < t < ηi T, h¯ i (t) = h i (t − i−T η η ⎪ ⎩ 0 , iη < t < T, mq T where h i (t) are matrices m × q, defined in the interval 0 < t < based on (5.111), the vector ⎡ ⎤ ψ1,k ψk = ⎣ ... ⎦ ψη,k

(7.52)

T . η

If we introduce,

(7.53)

of dimensions ηq × 1 and the matrix m × ηq

 h ∗T η (t) = h¯ 1 (t) ... h¯ η (t) ,

(7.54)

of dimensions m × nq, then the last equation in (7.51) will take the form of a zeroorder hold (7.55) u(t) = h ∗T η (t)ψk , kT < t < (k + 1)T. Finally, the equations of the considered system can be written in the form of (7.1). η After that, to construct the PTM W yx (s, t) of system (7.51), we can use the formulas from Sect. 7.1.

Part III

Determinant Polynomial Equations, Sampled-Data Modal Control and Stabilization of Periodic Objects

Chapter 8

Polynomial Matrices

The chapter describes general properties of polynomial matrices that will be used in later chapters. The chapter contains mainly well-known material presented in a form adapted for the purposes of this book.

8.1 General Definitions and Properties 1.

The matrix

⎡

⎤ a11 (λ) ... a1m (λ) A(λ) = ⎣ ... ... ... ⎦ , an1 (λ) ... anm (λ)

(8.1)

where aik (λ) are polynomials, is called a polynomial matrix. Further, it is assumed that the polynomials aik (λ) have real coefficients and the variable λ is complex. We will denote the set of matrices of the type (8.1) by R nm [λ]. The matrix A(λ) can be represented as (8.2) A(λ) = A0 λq + A1 λq−1 + · · · + Aq , where Ai are real matrices n × m and q ≥ 0 is an integer. If n = m, the matrix A(λ) will be termed square, if n < m—horizontal, and if n > m—vertical. If the coefficient A0 = 0mn , the number q is called the degree of the matrix A(λ). This we denote by q = deg A(λ). (8.3) If m = n and det A0 = 0, the matrix (8.2) is called regular.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_8

115

116

2.

8 Polynomial Matrices

For m = n, there exists a polynomial Δ A (λ) = det A(λ),

(8.4)

which is called the characteristic polynomial of the matrix A(λ), and the equation Δ A (λ) = 0

(8.5)

is called the characteristic equation of the matrix A(λ). Its roots are the eigenvalues of the matrix A(λ). If (8.6) Δ A (λ) ≡ 0, the square matrix A(λ) is called non-singular. If n = m and det A(λ) ≡ 0, as well as if n = m, the matrix A(λ) is called singular. If A(λ) ∈ R nn [λ], det A(λ) = const = 0, the matrix A(λ) is usually called unimodular. The set of all eigenvalues of a nonsingular matrix A(λ) is called its spectrum. Two non-singular matrices A1 (λ), A2 (λ), possibly of different sizes, are called coprime if their spectra do not intersect. 3.

For a fixed λ = λ0 from (8.1) we get a constant matrix ⎡

⎤ a11 (λ0 ) ... a1m (λ0 ) A(λ0 ) = ⎣ ... ... ... ⎦ . an1 (λ0 ) ... anm (λ0 )

(8.7)

The rank of A(λ0 ) is commonly denoted by rank A(λ0 ). The maximum value of rank A(λ0 ) for all the complex values of λ0 is called the normal rank and is denoted by Rank A(λ). Observe that the normal rank has a number of general properties. (a) If A(λ) ∈ R nm [λ], we have rank A(λ) ≤ min{n, m}.

(8.8)

(b) If A(λ) ∈ R n [λ], B(λ) ∈ R m [λ], we have rank [A(λ)B(λ)] ≤ min{rank A(λ), rank B(λ)}

(8.9)

rank [A(λ)B(λ)] ≥ rank A(λ) + rank B(λ) − .

(8.10)

as well as

The last relation is usually called Sylvester rank inequality [33]. (c) Let a polynomial block matrix D(λ) have the form   D(λ) = A(λ) B(λ) .

(8.11)

8.2 Polynomial Matrices Equivalence

117

Then rank D(λ) ≤ rank A(λ) + rank B(λ).

(8.12)

(d) For A(λ) ∈ R nm [λ], B(λ) ∈ R nm (λ) we have rank [A(λ) + B(λ)] ≤ rank A(λ) + rank B(λ). 4. Definition 8.1 The matrix A(λ) ∈ R nm [λ] is called non-degenerated if the following equality holds rank A(λ) = min{n, m}. (8.13) 

8.2 Polynomial Matrices Equivalence 1. Definition 8.2 The two matrices A1 (λ) ∈ R nm [λ] and A2 (λ) ∈ R nm [λ] are called equivalent if (8.14) A2 (λ) = p(λ)A1 (λ)q(λ), where p(λ) and q(λ) are real unimodular matrices n × n and m × m. If p(λ) = In , the matrices A1 (λ) and A2 (λ) are called right-equivalent, and if q(λ) = Im leftequivalent.  Theorem 8.1 An arbitrary matrix A(λ) ∈ R nm (λ), rank A(λ) = ρ is equivalent to the matrix S A (λ) ∈ R nm [λ] of the type Sρ (λ) 0ρ,n−ρ S A (λ) = . 0n−ρ,ρ 0n−ρ,m−ρ 

(8.15)

Here Sρ (λ) is a diagonal matrix ρ × ρ Sρ (λ) = diag{a1 (λ), . . . , aρ (λ)},

(8.16)

where ai (λ) are real scalar polynomials with the leading coefficient for λ is equal to one (monic). Also, the polynomial ai (λ) divides the polynomial ai+1 (λ). The matrix  S A (λ) is wholly determined by the matrix A(λ). The matrix S A (λ) is commonly termed the Smith normal form.

118

8 Polynomial Matrices

2. The polynomials ai (λ) appearing in (8.16) are called invariant polynomials of the matrix A(λ). Theorem 8.2 Two polynomial matrices of the same size are equivalent if and only if their sets of invariant polynomials coincide.  3. Definition 8.3 Two polynomial matrices of different sizes will be called pseudoequivalent if the sets of their invariant polynomials other than one, coincide. 

8.3 Latent Equation and Latent Numbers 1.

Consider a non-degenerate horizontal matrix A(λ) ∈ R nm [λ], i.e., n < m, rank A(λ) = n.

(8.17)

Definition 8.4 If there exists a number λ0 , for which rank A(λ0 ) < n,

(8.18)

the matrix A(λ) is called latent, and the number λ0 is called its latent number. The matrix A(λ) which does not have latent numbers is called alatent.  2. According to Theorem 8.1, a non-degenerate horizontal matrix A(λ) ∈ R nm [λ] can be represented as   A(λ) = p(λ) diag{a1 (λ), . . . , an (λ1 )} 0n,m−n q(λ),

(8.19)

where p(λ) and q(λ) are unimodular matrices n × n, (n + m) × (n + m), and there is no zero polynomial among the invariant polynomials of ai (λ), (i = 1, . . . , n). Definition 8.5 The polynomial d A (λ) = a1 (λ)...an (λ)

(8.20)

is called the latent polynomial of the matrix A(λ), and the equation d A (λ) = 0

(8.21)

is called its latent equation. Its roots are called the latent numbers of the matrix A(λ). 

8.4 Pairs of Polynomial Matrices

119

3. Theorem 8.3 The set of latent numbers of the matrix A(λ) coincides with the set of its latent equation roots. Also, if λ0 is a latent number, rank A(λ0 ) = n − d0

(8.22)

holds true, where d0 is the number of invariant polynomials for which the number  λ0 is a root. By transposing, everything said in this section can by applied to vertical nondenerated polynomial matrices. 4. Representation (8.19) implies that the matrix A(λ) is alatent if and only if all its invariant polynomials are equal to one. Theorem 8.4 An arbitrary vertical or horizontal matrix composed of the rows or columns of a unimodular matrix is alatent. 

8.4 Pairs of Polynomial Matrices 1. Let us be given the matrices A(λ) ∈ R nn [λ] and B(λ) ∈ R nm [λ]. We will call the union of the matrices A(λ) and B(λ) a horizontal pair, and denote it (A(λ), B(λ)). Similarly, if A(λ) ∈ R nn [λ], C(λ) ∈ R mn [λ], we will call them a vertical pair [A(λ), C(λ)]. It is obvious that if the matrices A(λ), B(λ) form a horizontal pair, then the transposed matrices A (λ), B  (λ) form a vertical pair. Consequently, we can confine ourselves to considering the properties of horizontal pairs. 2.

An accompanying horizontal matrix N AB (λ) ∈ R n,n+m [λ] of the type   N AB (λ) = A(λ) B(λ)

(8.23)

can be associated with the horizonal pair (A(λ), B(λ)). Definition 8.6 The pair (A(λ), B(λ)) is called irreducible if the matrix N AB (λ) is alatent. If the matrix N AB (λ) is latent , the pair (A(λ), B(λ)) is called reducible. The latent numbers of the matrix N AB (λ) will also be called the latent numbers of the pair (A(λ), B(λ)).  3. Definition 8.7 Let the following relations be true for the pair (A(λ), B(λ)): A(λ) = g (λ)A1 (λ),

B(λ) = g (λ)B1 (λ),

(8.24)

where the matrix g (λ) ∈ R nn [λ] is non-singular and A1 (λ), B1 (λ) are polynomial matrices. Then the matrix g (λ) is called the common left divisor (CLD) of the pair

120

8 Polynomial Matrices

(A(λ), B(λ)). If under the condition (8.24) the pair (A1 (λ), B1 (λ)) is irreducible,  the CLD of g (λ) is called the greatest common left divisor (GCLD). It is known [34] that the GCLD of the pair (A(λ), B(λ)) is not uniquely determined. If g0 (λ) is one of the GCLDs, an arbitrary g (λ) can be found from the formula g (λ) = g0 (λ)χ (λ),

(8.25)

where χ (λ) is a unimodular matrix n × n. Formula (8.25) implies that all the GCLDs of the pair (A(λ), B(λ)) are right-equivalent. 4.

Similarly to the said above, an accompanying vertical matrix (n + m) × n  N AC (λ) =

A(λ) C(λ)

(8.26)

can be associated with a vertical pair [A(λ), C(λ)]. If we have A(λ) = A2 (λ)gr (λ), C(λ) = C1 (λ)gr (λ),

(8.27)

where gr (λ) ∈ R nn [λ], A2 (λ) ∈ R nn [λ], C1 (λ) ∈ R mn [λ], the non-singular matrix gr (λ) is called a common right divisor of the pair [A(λ), C(λ)] (CRD). If (8.27) is satisfied, from (8.26) we obtain A2 (λ) g (λ). N AC (λ) = C1 (λ) r 

(8.28)



If in (8.28) the pair [A2 (λ), C1 (λ)] is irreducible, i.e., the matrix A2 (λ) C1 (λ) is alatent, the matrix gr (λ) is called the greatest common right divisor (GCRD) of the pair [A(λ), C(λ)]. We can prove that if gr 0 (λ) is a GCRD of the pair [A(λ), C(λ)], an arbitrary GCRD gr (λ) is determined by the formula gr (λ) = χr (λ)gr 0 (λ),

(8.29)

where χr (λ) is an arbitrary unimodular matrix n × n, similar to the one from (8.25). 5. General criteria for the irreducibility of a horizontal pair (A(λ), B(λ)) are given by Theorem 8.5 A horizontal pair (A(λ), B(λ)) is irreducible if and only if one of the following conditions is fulfilled: (a) There exists a unimodular matrix q(λ) ∈ R n+m,n+m [λ] satisfying the equality   N AB (λ) = p1 (λ) 0nm q(λ), where p1 (λ) ∈ R nn [λ] is a unimodular matrix.

(8.30)

8.4 Pairs of Polynomial Matrices

121

(b) There exist matrices X (λ) ∈ R nn [λ] and Y (λ) ∈ R mn [λ] satisfying the equality A(λ)X (λ) + B(λ)Y (λ) = In .

(8.31)

(c) The conditions rank N AB (λi ) = rank



 A(λi ) B(λi ) = n

(8.32)

hold true for all λi , (i = 1, . . . , κ) which are all the different eigenvalues of the matrix A(λ). (d) There exist matrices α(λ) ∈ R mm [λ], β(λ) ∈ R mn [λ] such that the matrix Q  (λ) ∈ R n+m,n+m [λ] of the type  Q  (λ) =

A(λ) B(λ) β(λ) α(λ)

(8.33)

is unimodular.  Similar conditions for the irreducibility of vertical pairs can be obtained by transposing from Theorem 8.5. 6. Theorem 8.6 If the pair (A(λ), B(λ)) is reducible, the matrix p1 (λ) from (8.30) is not unimodular and possesses eigenvalues. In this case the matrix p1 (λ) determines one of the GCLDs of the pair (A(λ), B(λ)). 

Chapter 9

Rational Matrices

The chapter describes the general properties of rational matrices. Besides known results, the chapter contains additional information that is necessary for further presentation.

9.1 General Definitions and Properties 1.

A matrix L(λ) n × m of the form ⎡

⎤ 11 (λ) ... 1m (λ) L(λ) = ⎣ ... ... ... ⎦ n1 (λ) ... nm (λ) where ik (λ) =

m ik (λ) , dik (λ)

(9.1)

(9.2)

and m ik (λ), dik (λ) are polynomials from R 11 [λ], we will call rational. The set of these matrices is denoted by R nm (λ). 2. Let d(λ) be the least common multiple of the denominators dik (λ). Then the matrix L(λ) can be represented as L(λ) =

M(λ) , d(λ)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_9

(9.3)

123

124

9 Rational Matrices

where M(λ) ∈ R nm [λ]. Below we will call the matrix M(λ) the numerator and the polynomial d(λ) the denominator of the rational matrix L(λ) in the representation of type (9.3). Definition 9.1 If in representation (9.3) the polynomial d(λ) is of the lowest possible degree, the matrix L(λ) is called irreducible. If the matrix L(λ) is irreducible and the denominator d(λ) is monic, i.e., its leading coefficient equals one, the right-hand side of (9.3) is called the standard form of the matrix L(λ).  If in the standard form, we have d(λ) = (λ − λ1 )ν1 ...(λ − λr )νr

(9.4)

where all the numbers λi , (i = 1, . . . , r ) are different, these numbers are called the poles of the matrix L(λ) and the numbers νi , (i = 1, . . . , r ) are called their multiplicities.

9.2 McMillan Canonical Form 1. Let a horizontal matrix L(λ) ∈ R nm be represented in the standard form, the numerator M(λ) being a non-degenerated matrix. Then according to (8.19) we have   M(λ) = p(λ) diag{a1 (λ), . . . , an (λ)} 0n,m−n q(λ),

(9.5)

where ai (λ) are the corresponding invariant polynomials and p(λ), q(λ) are unimodular matrices of corresponding sizes. Substituting (9.5) into the right-hand side of (9.3) gives   (9.6) L(λ) = p(λ) K (λ) 0n,m−n q(λ), where the matrix K (λ) ∈ R nn (λ) has the form an (λ) a1 (λ) ,..., . K (λ) = diag d(λ) d(λ) 

(9.7)

After all possible cancellations in (9.7) we arrive at  K (λ) = diag

ξn (λ) ξ1 (λ) ξ2 (λ) , ,..., , ψ1 (λ) ψ2 (λ) ψn (λ)

(9.8)

where ξi (λ), (i = 1, . . . , n) and ψi (λ), (i = 1, . . . , n) are real monic polynomials, the polynomials ξi (λ) and ψi (λ) being coprime, the polynomial ξi (λ) dividing ξi+1 (λ) and the polynomial ψi+1 (λ) dividing ψi (λ).

9.2 McMillan Canonical Form

125

2. Definition 9.2 The matrix SL (λ) ∈ R nm (λ) of the type

 0n,m−n , . . . , ψξnn(λ) SL (λ) = diag ψξ11(λ) (λ) (λ) is called the McMillan canonical form of the matrix L(λ).

(9.9) 

The McMillan canonical form is uniquely determined by the original matrix L(λ). 3. Definition 9.3 The monic polynomial ψ L (λ) = ψ1 (λ)ψ2 (λ)...ψn (λ)

(9.10)

is called the McMillan denominator, and the monic polynomial ξ L (λ) = ξ1 (λ)ξ2 (λ)...ξn (λ)

(9.11) 

the McMillan numerator. Theorem 9.1 The quotient δ L (λ) =

ψ L (λ) d(λ)

(9.12) 

is a monic polynomial. If (9.4) holds true, Theorem 9.1 yields ψ L (λ) = (λ − λ1 )μ1 ...(λ − λr )μr ,

(9.13)

μi ≥ νi .

(9.14)

where Definition 9.4 The number μi is called the McMillan multiplicity of the pole λi .  Below, the multiplicity of the pole λi is denoted multλi , and the McMillan multiplicity is denoted Multλi . Definition 9.5 The number Mind L(λ) 

r  i=1

μi =

r 

Multλi = deg ψ L (λ)

(9.15)

i=1

is called the McMillan index of the matrix L(λ).



126

9 Rational Matrices

Theorem 9.2 Let the matrix L(λ) ∈ R nm (λ) and a (λ) ∈ R nn [λ] and ar (λ) ∈ R mm [λ] be non-singular matrices, the products a (λ)L(λ) and L(λ)ar (λ) being polynomial matrices. Then the following inequalities are true: deg det a (λ) ≥ Mind L(λ), deg det ar (λ) ≥ Mind L(λ).

(9.16) 

4. Theorem 9.3 Let the matrices L 1 (λ) ∈ R n (λ) and L 2 (λ) ∈ R m (λ) be given. Then the product L(λ) = L 1 (λ)L 2 (λ) ∈ R nm (λ) is defined and we have Mind L(λ) ≤ Mind L 1 (λ) + Mind L 2 (λ).

(9.17) 

Definition 9.6 If there is an equality in (9.17), then matrices L 1 (λ) and L 2 (λ) will be called independent.  It should be noted, that the independence of the two matrices depends, in the general case, on their multiplication order. This means that if the products L 1 (λ)L 2 (λ) and L 2 (λ)L 1 (λ) are defined, it may occur that Mind[L 1 (λ)L 2 (λ)] = Mind[L 2 (λ) L 1 (λ)].

9.3 Matrix Fraction Description 1. Definition 9.7 Let the matrix L(λ) ∈ R nm (λ) be represented by one of the two equations (9.18) L(λ) = a−1 (λ)b (λ), L(λ) = br (λ)ar−1 (λ), where a (λ) ∈ R nn [λ], ar (λ) ∈ R mm [λ], b (λ) ∈ R nm [λ], br (λ) ∈ R nm [λ]. Then we will say that the right-hand sides of (9.18) define matrix fraction descriptions (MFDs) of L(λ), the left one (LMFD) and the right one (RMFD) respectively.  Definition 9.8 If in (9.18) the polynomial matrices   N (λ) = a (λ) b (λ) ,

 Nr (λ) =

ar (λ) br (λ)

 (9.19)

are alatent, the corresponding MFDs are called irreducible (IMFDs). In this case, the first formula in (9.18) defines the irreducible left MFD (ILMFD) and the second the irreducible right MFD (IRMFD). 

9.4 Strictly Proper Rational Matrices

2.

127

Note some important properties of the IMFD [20, 34] in

Theorem 9.4 (a) If the IMFD of L(λ) are defined by the right-hand sides of formulas (9.18), we have det a (λ) ∼ det ar (λ) ∼ ψ L (λ), (9.20) as well as deg det a (λ) = deg det ar (λ) = Mind L(λ).

(9.21)

(b) The matrices a (λ) and ar (λ) are pseudo-equivalent and the matrices b (λ) and br (λ) are equivalent. (c) Let the right-hand sides of formulas (9.18) be the IMFD and non-singular matrices a¯  ∈ R nn [λ] and a¯ r (λ) ∈ R mm [λ] be such that the products a¯  (λ)L(λ) and L(λ)a¯ r (λ) are polynomial matrices. Then we have the equalities of the type a¯  (λ) = a1 (λ)a (λ), a¯ r (λ) = ar (λ)ar 1 (λ), where the matrices a1 (λ) ∈ R nn [λ], ar 1 (λ) ∈ R mm [λ] are non-singular. (d) The right-hand sides of formulas (9.18) define the IMFD if and only if the pairs (a (λ), b (λ)) and [ar (λ), br (λ)] are irreducible. (e) Let the matrix L(λ) ∈ R nm (λ) have the form L(λ) = c(λ)a −1 (λ)b(λ),

(9.22)

where c(λ) ∈ R np [λ], a(λ) ∈ R pp [λ], b(λ) ∈ R pm [λ], and the pair (a(λ, b(λ)) is irreducible. Let us have the ILMFD c(λ)a −1 (λ) = a−1 (λ)c (λ).

(9.23)

Then the following relation holds true L(λ) = a−1 (λ)[c (λ)b(λ)],

(9.24)

where the right-hand side is the ILMFD. 

9.4 Strictly Proper Rational Matrices 1. Definition 9.9 Given a matrix L(λ) ∈ R nm (λ), the integer excL(λ) is called the excess of L(λ), if there exists a finite limit

128

9 Rational Matrices

lim L(λ)λexcL(λ) = L ∞ = 0nm .

λ→∞

(9.25) 

If the matrix L(λ) has the form (9.3), then excL(λ) = deg d(λ) − deg M(λ).

(9.26)

Definition 9.10 If excL(λ) = 0 and if excL(λ) > 0, the matrix L(λ) is called proper and strictly proper respectively. If excL(λ) ≥ 0 the matrix L(λ) is called proper at least. If excL(λ) < 0 the matrix L(λ) is called improper.  2. It is known [34] that for any strictly proper matrix L(λ) ∈ R nm (λ) there exists a set of constant real matrices A, B, C of dimensions p × p, p × m and n × p satisfying the equality (9.27) L(λ) = C(λI p − A)−1 B.

Definition 9.11 A triple of the matrices A, B, C for which equality (9.27) is fulfilled is called a realization of the matrix L(λ), and the integer p its order.  3. A realization for which the order has the lowest possible value p = p0 is called minimal. Theorem 9.5 The following assertions hold true: (a) A realization A, B, C is minimal if and only if the pair (A, B) is controllable and the pair [A, C] is observable. (b) Two minimal realizations A, B, C and A1 , B1 , C1 are related by C1 = C Q −1 , A1 = Q AQ −1 , B = Q B,

(9.28)

where Q is an arbitrary non-singular real matrix p0 × p0 . (c) The order p0 of an arbitrary minimal realization of the matrix L(λ) is determined by the formula (9.29) p0 = Mind L(λ). 

9.5 Polynomial Pairs and Transfer Matrices

129

9.5 Polynomial Pairs and Transfer Matrices 1. The pairs (a (λ), b (λ)), and [ar (λ), br (λ)] are called non-singular if the matrices a (λ) and ar (λ) are non-singular. For non-singular pairs, there exist transfer matrices (9.30) W (λ) = a−1 (λ)b (λ), Wr (λ) = br (λ)ar−1 (λ). The right-hand sides of formulas (9.30) can be considered as MFDs of the matrices W (λ) and Wr (λ). From the properties of MFDs described above it follows that there are pairs (a (λ), b (λ)) and [ar (λ), br (λ)] having the same transfer matrix. We will call such pairs related. 2. In the following discussion we shall consider preferably horizontal pairs, as appropriate statements for vertical pairs may be obtained by transposing. Theorem 9.6 The following assertions hold true: (a) An LMFD (9.30) for the transfer matrix W (λ) is an ILMFD if and only if the pair (a (λ), b (λ)) is irreducible, i.e., the associated matrix   Nab (λ) = a (λ) b (λ)

(9.31)

is alatent. (b) If the pair (a (λ), b (λ)) is irreducible, the set of all the associated pairs (a¯  (λ), b¯ (λ)) is determined by the relations a¯  (λ) = g (λ)a (λ), b¯ (λ) = g (λ)b (λ),

(9.32)

where g (λ) is an arbitrary non-singular matrix of the corresponding dimensions. (c) The right-hand sides of formulas (9.30) determine the IMFD if and only if

or equally

det a (λ) ∼ ψW (λ), det ar (λ) ∼ ψWr (λ),

(9.33)

deg det a (λ) = MindW (λ), deg det ar (λ) = MindWr (λ).

(9.34) 

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9 Rational Matrices

9.6 Polynomial Matrix Division and Reduction of the Degree of Polynomial Pairs 1. Definition 9.12 The degree of a polynomial pair (a (λ), b (λ)) is the degree of the corresponding associated matrix (9.31), i.e., by definition   deg(a (λ), b (λ))  deg a (λ) b (λ) .

(9.35) 

Definition 9.13 The pairs (a1 (λ), b1 (λ)) and (a2 (λ), b2 (λ)) where the matrices a1 (ζ ) and a2 (ζ ) as well as the matrices b1 (ζ ) and b2 (ζ ) are of the same dimensions are called equivalent if the corresponding associated matrices Na1 b1 (λ) and Na2 b2 (λ) are equivalent.  2. The problem of constructing equivalent pairs of reduced degree plays an important role in theoretical considerations and practical tasks. One of the methods to solve this problem is to use the properties of polynomial matrix division. 3. Theorem 9.7 Given a regular matrix a(λ) ∈ R nn [λ] and an arbitrary matrix b(λ) ∈ R nm [λ], there exist unique matrices p(λ) ∈ R nm [λ] and q(λ) ∈ R nm [λ] such that b(λ) = a(λ)q(λ) + p(λ)

(9.36)

deg p(λ) < deg a(λ).

(9.37)

with  Definition 9.14 Whenever (9.36) and (9.37) are fulfilled, we will say that q(λ) is a left quotient, and p(λ) is a left remainderas the matrix a(λ) divides the matrix b(λ) from the left.  4. Theorem 9.8 The following assertions hold true [24, 30]: (a) Given a(λ) = λIn − A, b(λ) = B0 λd + B1 λd−1 + · · · + Bd , where A, Bi are constant matrices, the matrix

(9.38)

9.6 Polynomial Matrix Division and Reduction of the Degree of Polynomial Pairs

p(λ) = Ad B0 + Ad−1 B1 + · · · + Bd  b (A)

131

(9.39)

from representation (9.36) is constant. (b) Given a1 (λ) = In − λA, b(λ) = B0 λd + B1 λd−1 + · · · + Bd ,

(9.40)

where the constant matrix A is non-singular, p(λ) = A−d B0 + A−d+1 B1 + · · · + Bd  b (A−1 )

(9.41)

is a constant matrix.  5. Theorem 9.9 Let equality (9.36) hold for matrices a(λ) ∈ R nn [λ], b(λ) ∈ R nm [λ]. Then the irreducibility of the pair (a(λ), b(λ)) implies the irreducibility of the pair (a(λ), p(λ)).  Proof When (9.40) holds, we have 

   a(λ) b(λ) = a(λ) a(λ)q(λ) +p(λ) =    In q(λ) = a(λ) p(λ) . 0mn Im

(9.42)

  Since  the last matrix  on the right-hand side is unimodular, the matrices a(λ) b(λ) and a(λ) p(λ) are equivalent and from the irreducibility of the pair (a(λ), b(λ)) follows the irreducibility of the pair (a(λ), p(λ)).  6. Theorem 9.10 For the irreducibility of the pair a1 (λ), b1 (λ) from (9.40), where matrix A is non-singular, it is necessary and sufficient that the pair of constant  matrices A1 b (A−1 ) is controllable. Proof It follows from Theorems 10.8 and 10.9 that the reducibility of the pair a1 (λ), b(λ) implies the irreducibility of the pair In − λA, b (A−1 ) and vice versa. Moreover, from the equality 

   In − λA b (A−1 ) = −A A−1 − λIn −A−1 b (A)−1

(9.43)

we obtain by the PBH test [34] that the pair A−1 , A−1 b (A−1 ) is controllable. Accord ing to [20] this is equivalent to the controllability of the A, b (A−1 ).

Chapter 10

Determinant Polynomial Equations, Causal Modal Control and Stabilization of Discrete Systems

The chapter describes the mathematical apparatus of determinant polynomial equations and discusses their application to solve the problems of pole placement and stabilization of multidimensional discrete systems.

10.1 General Definitions and Problem Setting for Causal Modal Control 1. Many technical objects can be described by discrete models in the form of difference equations a0 yk + a1 yk−1 + · · · + aρ yk−ρ = b0 u k + b1 u k−1 + · · · + bρ u k−ρ ,

(10.1)

where yk , u k are vectors n × 1, m × 1 and ai , bi are constant matrices n × n and n × m. Introducing the backward shift operator g [32], from the conditions gyk = yk−1 , gu k = u k−1

(10.2)

we can rewrite Equation (10.1) as a (g)yk = b (g)u k ,

(10.3)

where a (g), b (g) are polynomial matrices in the form a (g) = a0 + a1 g + · · · + aρ g ρ , b (g) = b0 + b1 g + · · · + bρ g ρ .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_10

(10.4)

133

134

10 Determinant Polynomial Equations, Causal Modal …

In the further discussion, the operator g will be identified with the complex variable ζ . Then we can associate a polynomial pair (a (ζ ), b (ζ )) with the Eq. (10.3) where a (ζ ) = a (g)| g=ζ , b (ζ ) = b (g)| g=ζ .

(10.5)

2. Definition 10.1 A polynomial pair (a (ζ ), b (ζ )) (3.5) is called a (left) backward object or, in short, a B-object. A B-object is called non-singular if the pair  (a (ζ ), b (ζ )) is non-singular. A transfer matrix

W (ζ ) = a−1 (ζ )b (ζ )

(10.6)

can be associated with a non-singular B-object. Definition 10.2 A non-singular B-object is called causal if its transfer matrix (10.6) has no pole in the point ζ = 0.  Definition 10.3 A causal B-object is called strictly causal if W (0) = 0nm .

(10.7) 

Since in (10.7) the matrix W (ζ ) has no pole in the point ζ = 0, equality (10.7) implies that the transfer matrix of a strictly causal B-object can be written as W (ζ ) =

ζ M(ζ ) , d(ζ )

(10.8)

where d(ζ ) is a scalar polynomial with no zero roots. 3. The right-hand side of (10.6) can be regarded as the LMFD of the transfer matrix W (ζ ). Theoretically, there also exists a RMFD W (ζ ) = br (ζ )ar−1 (ζ ).

(10.9)

Some useful properties of the MFDs from (10.6) and (10.9) are determined by Theorem 10.1 If a B-object is irreducible, i.e., the right-hand sides in (10.6) and (10.9) are IMFDs, the following assertions hold true: (a) If the B-object is causal and the expressions in (10.6), (10.9) are IMFDs, then det a (0) = 0, det ar (0) = 0.

(10.10)

10.1 General Definitions and Problem Setting for Causal Modal Control

135

(b) If the conditions (i) are satisfied and the object is strictly causal, besides the relations (10.10) we have b (ζ ) = ζ b1 (ζ ), br (ζ ) = ζ br 1 (ζ ),

(10.11)

where the matrices b1 (ζ ) and br 1 (ζ ) ∈ R nm [ζ ].  Definition 10.4 An arbitrary left B-object (α (ζ ), β (ζ )) with α (ζ ) ∈ R mm [ζ ],  β (ζ ) ∈ R mn [ζ ] is called a left B-controller. Given a B-object and a left B-controller, we can construct a matrix Q  (ζ, α , β ) 



n m  a (ζ ) −b (ζ ) n . −β (ζ ) α (ζ ) m

(10.12)

Definition 10.5 Matrix (10.12) is called a (left) characteristic matrix of the closedloop system. The determinant Δ Q (ζ ) = det Q  (ζ, α , β )

(10.13)

is called the characteristic polynomial of the closed-loop system. The roots of Δ Q (ζ ) are called the closed-loop system modes.  4. For the theory and numerous applications, the task of modal control is of fundamental importance which regarding B-objects is formulated as follows: Given a B-object (a (ζ ), b (ζ )) and a scalar monic polynomial Δ(ζ ). Find the set of B-controllers (α (ζ ), β (ζ )), such that  det Q  (ζ ) = det

 a (ζ ) −b (ζ ) ∼ Δ(ζ ), −β (ζ ) α (ζ )

(10.14)

where the symbol ∼ stands for the scalar polynomials equivalence. For given matrices a (ζ ), b (ζ ) and polynomial Δ(ζ ) relation (10.14) can be regarded as an equation with respect to the matrices α (ζ ) and β (ζ ). As in [20, 21], we will call the equations of this type determinant polynomial equations (DPEs). Definition 10.6 A B-object (a (ζ ), b (ζ )) is called modal controllable if DPE (10.14) has solutions for any polynomial Δ(ζ ). 

136

10 Determinant Polynomial Equations, Causal Modal …

5. Definition 10.7 A matrix Q  (ζ ) ∈ R nn [ζ ] is called stable if all its eigenvalues ζi satisfy the condition (10.15) |ζi | > 1. In particular, a scalar polynomial Δ(ζ ) is stable if its roots satisfy condition (10.15).  Definition 10.8 A B-object (a (ζ ), b (ζ )) is called stabilizable if there exists a stable polynomial Δ S (ζ ) for which DPE (10.14) is solvable. Controllers (α (ζ ), β (ζ )) which determine the solutions of DPE (10.14) for stable polynomials Δ(ζ ) = Δ S (ζ ) are called stabilizing. The problem of constructing the set of causal stabilizing controllers for a given object is called the stabilization problem. 

10.2 Solving Backward Modal Control and Stabilization Problems for Polynomial Pairs 1. Consider an irreducible B-object (a (ζ ), b (ζ )). Then, as part d) of Theorem 8.5 implies, there is a set of controllers (α0 (ζ ), β0 (ζ )) such that the matrix 

a (ζ ) −b (ζ ) Q  (ζ ) = −β0 (ζ ) α0 (ζ )

 (10.16)

is unimodular. Definition 10.9 An arbitrary controller (α0 (ζ ), β0 (ζ )) for which matrix (3.16) is unimodular is called basic.  Theorem 10.2 The following assertions hold true: (a) All the basic controllers (α0 (ζ ), β0 (ζ )) are irreducible. (b) If an irreducible object (a (ζ ), b (ζ )) is strictly causal, all the basic controllers (α0 (ζ ), β0 (ζ )) are causal.  2. The following assertion provides a general method of constructing the set of all the left basic controllers for a given irreducible B-object. Theorem 10.3 ([20, 35]) Let α¯ 0 (ζ ), β¯0 (ζ ) be a particular basic controller for an irreducible object (a (ζ ), b (ζ )). Then the set of all left basic controllers is determined by the formulas α0 (ζ ) = D0 (ζ )α¯ 0 (ζ ) − M(ζ )b (ζ ), β0 (ζ ) = D0 (ζ )β¯0 (ζ ) − M(ζ )a (ζ )

(10.17)

10.2 Solving Backward Modal Control and Stabilization Problems for Polynomial Pairs

137

where the matrix M(ζ ) ∈ R mn [ζ ] is arbitrary and D0 (ζ ) ∈ R mm [ζ ] is an arbitrary unimodular matrix.  Practical methods of constructing the particular controller α¯ 0 (ζ ), β¯0 (ζ ) appearing in (10.17) can be found in [20]. 3.

General results related to solving DPE (10.14) are given by

Theorem 10.4 The following assertions hold true: (a) DPE (10.14) is solvable for any polynomial Δ(ζ ) if and only if the object (a (ζ ), b (ζ )) is irreducible. (b) If the object (a (ζ ), b (ζ )) is irreducible, strictly causal, and the condition Δ(0) = 0,

(10.18)

is fulfilled, all the solutions of DPE (10.14) are causal. (c) For a given irreducible object and known initial basic controller α¯ 0 (ζ ), β¯0 (ζ ) the set of DPE (3.14) solutions is found by the formulas α(ζ ) = D(ζ )α¯ 0 (ζ ) − M(ζ )b (ζ ), β(ζ ) = D(ζ )β¯0 (ζ ) − M(ζ )a (ζ ).

(10.19)

Here the matrix M(ζ ) ∈ R mn [ζ ] is arbitrary, and the matrix D(ζ ) ∈ R mm [ζ ] satisfies the condition det D(ζ ) ∼ Δ(ζ ). (10.20) (d) The pair (α(ζ ), β(ζ )) from (10.19) is irreducible if and only if the pair (D(ζ ), M(ζ )) is irreducible.  4. A method for solving DPE (10.14) for the case when the pair (a (ζ ), b (ζ )) is reducible is given by Theorem 10.5 Let the pair (a (ζ ), b (ζ )) be reducible and let ψ(ζ ) be the GCLD of the matrices a (ζ ), b (ζ ). Then holds 

   a (ζ ) b (ζ ) = ψ(ζ ) a1 (ζ ) b1 (ζ ) ,

(10.21)

where the pair (a1 (ζ ), b1 (ζ )) is irreducible. In this case, the DPE (10.14) is solvable if and only if the polynomial Δ(ζ ) in (10.14) admits the representation Δ(ζ ) = det ψ(ζ )Δ1 (ζ ),

(10.22)

where Δ1 (ζ ) is a polynomial. If (10.21) and (10.22) are fulfilled, the solution set of (10.14) coincides with that of the DPE

138

10 Determinant Polynomial Equations, Causal Modal …

 det

 a1 (ζ ) −b1 (ζ ) ∼ Δ1 (ζ ). −α(ζ ) β(ζ )

This set can be constructed by means of Theorem 10.4.

(10.23) 

5. On the basis of Theorems 10.4 and 10.5 we can obtain a solution for the stabilization problem. The corresponding result is offered by Theorem 10.6 The following assertions hold true: (a) If the object (a (ζ ), b (ζ )) is irreducible, it is stabilizable. (b) If the object is reducible, it is stabilizable if and only if the GCLD ψ(ζ ), appearing in (10.21), is stable. (c) In the case when the GCLD ψ(ζ ) is stable and condition (10.22) is fulfilled with a stable polynomial Δ1 (ζ ), the set of stabilizing controllers coincides with the DPE (10.23) solution set.  6. Theorem 10.7 If a strictly causal object (a (ζ ), ζ b (ζ )) is irreducible, then all stabilizing controllers (α (ζ ), β (ζ )) are causal. In this case, the set of transfer matrices Wd (ζ ) of all stabilizing controllers is determined by the relation   Wd (ζ ) = [α¯ 0 − ζ θ (ζ )b (ζ )]−1 β¯0 (ζ )(ζ ) − θ (ζ )a (ζ ) .

(10.24)

Here θ (ζ ) is an arbitrary stable rational matrix (whose poles lie outside the unit circle). 

10.3 Solving Modal Control and Stabilization Problems for Backward Processes in Polynomial Matrix Description 1. This section discusses modal control and stabilization problems for discrete B-objects described by equations of the type a(ζ )xk = b(ζ )u k , yk = c(ζ )xk + d(ζ )u k .

(10.25)

Here xk , yk , u k are elements of discrete vector sequences p × 1, n × 1, m × 1. Besides, a(ζ ), b(ζ ), c(ζ ), d(ζ ) are polynomial matrices of corresponding sizes, whose union P(ζ ) = (a(ζ ), b(ζ ), c(ζ ), d(ζ )) (10.26)

10.3 Solving Modal Control and Stabilization Problems for Backward …

139

is called a polynomial matrix description (PMD) of a discrete process [34]. If d(ζ ) = 0nm , the PMD (10.25), (10.26) is called homogeneous. 2. As above, the arbitrary pair (α(ζ ), β(ζ )) where α(ζ ) ∈ R mm [ζ ], β(ζ ) ∈ R mn [ζ ] is called discrete controller, and the difference equation α(ζ )u k = β(ζ )yk

(10.27)

the discrete controller equation. Together, Eqs. (10.25) and (10.27) define a closedloop system having the form a(ζ )xk − b(ζ )u k = 0 p1 , yk − c(ζ )xk − d(ζ )u k = 0n1 , α(ζ )u k − β(ζ )yk = 0m1 .

(10.28)

3. Definition 10.10 The polynomial matrix Q p (ζ, α, β) of the type ⎡

⎤ a(ζ ) 0 pn −b(ζ ) Q p (ζ, α, β) = ⎣ −c(ζ ) In −d(ζ ) ⎦ 0mp −β(ζ ) α(ζ )

(10.29)

is called the characteristic matrix of the closed-loop system, and the polynomial ⎡

⎤ a(ζ ) 0 pq −b(ζ ) det Q p (ζ, α, β) = det ⎣ −c(ζ ) In −d(ζ ) ⎦ = Δ p (ζ ) 0mp −β(ζ ) α(ζ ) its characteristic polynomial.

(10.30) 

4. For the given PMD (10.26) the modal control problem is reduced to the solution of the DPE ⎡ ⎤ a(ζ ) 0 pn −b(ζ ) det ⎣ −c(ζ ) In −d(ζ ) ⎦ ∼ Δ(ζ ), (10.31) 0mp −β(ζ ) α(ζ ) where Δ(ζ ) is a given monic polynomial and α(ζ ), β(ζ ) are the polynomial matrices to be determined. If we introduce the polynomial matrices    a(ζ ) 0 pn ¯ ) = −b(ζ ) , , b(ζ −c(ζ −d(ζ )  ) In  ¯ ) = 0mp β(ζ ) , α(ζ β(ζ ¯ ) = α(ζ ), 

a(ζ ¯ )=

the DPE (11.31) can be represented in the form analogous to (11.31)

(10.32)

140

10 Determinant Polynomial Equations, Causal Modal …

 det

 ¯ ) a(ζ ¯ ) −b(ζ ∼ Δ(ζ ). ¯ ) α(ζ −β(ζ ¯ )

(10.33)

However, the general method of solving DPE (10.14) considered above is not appli¯ ) has a special structure. cable directly to the DPE (10.33), since the matrix β(ζ Therefore, the task of solving the DPE (11.31) requires a separate consideration. 5. Definition 10.11 PMD (10.26) is called non-singular if det a(ζ ) = 0,

(10.34)

and minimal if the pairs (a(ζ ), b(ζ )) and [a(ζ ), c(ζ )] are irreducible, i.e., for all ζ we have     a(ζ ) a(ζ ) b(ζ ) rank = p, rank = p. (10.35) c(ζ )  For a non-singular PMD (10.26), a transfer matrix W p (ζ ) ∈ R W p (ζ ) = c(ζ )a −1 (ζ )b(ζ ) + d(ζ ).

nm

(ζ ) is defined by (10.36)

Definition 10.12 A PMD (10.26) is called causal if the transfer matrix W p (ζ ) has no poles within the circle |ζ | ≤ 1. We will call the causal PMD (10.26) strictly causal if the equality (10.37) W p (0) = 0nm 

is fulfilled. Obviously, equality (10.37) holds if we have b(ζ ) = ζ b1 (ζ ), d(ζ ) = ζ d1 (ζ ),

(10.38)

where b1 (ζ ) and d1 (ζ ) are polynomial matrices. Below we will confine ourselves only to the consideration of strictly causal PMDs, for which we have (10.38). In this case, instead of (10.26) we use the notation P(ζ ) = (a(ζ ), ζ b1 (ζ ), c(ζ ), ζ d1 (ζ )).

(10.39)

With regard to (10.38), DPE (11.31) now takes the form ⎡

⎤ a(ζ ) 0 pn −ζ b1 (ζ ) det ⎣ −c(ζ ) In −ζ d1 (ζ ) ⎦ ∼ Δ(ζ ). 0mp −β(ζ ) α(ζ )

(10.40)

10.3 Solving Modal Control and Stabilization Problems for Backward …

141

6. Theorem 10.8 The following assertions hold true: (a) Let PMD (10.26) be non-singular. Then this PMD is modal controllable, i.e., DPE (11.31) is solvable for any polynomial Δ(ζ ) if and only if this PMD is minimal. (b) Let us have the ILMFD a1−1 (ζ )c1 (ζ ) = c(ζ )a −1 (ζ )

(10.41)

and the conditions from a) are satisfied. Then the set of solutions for DPE (11.31) coincides with the set of solutions for the DPE of type (10.14) 

where

 a1 (ζ ) −b˜1 (ζ ) det ∼ Δ(ζ ) −β(ζ ) α(ζ )

(10.42)

b˜1 (ζ ) = c1 (ζ )b(ζ ) + a1 (ζ )d(ζ )

(10.43)

and the pair (a1 (ζ ), b˜1 (ζ )) is irreducible. (c) Let the conditions in a) be satisfied and, in addition, PMD P(ζ ) be strictly causal and have the form (10.39). Then under the condition Δ(0) = 0

(10.44)

all solutions of DPE (10.40) are causal.  7. Definition 10.13 If there exists a stable polynomial Δ(ζ ), for which DPE (11.31) is solvable, the corresponding PMD (10.26) is called stabilizable and the corresponding solutions α(ζ ), β(ζ ) are called stabilizing controllers.  The following Theorem contains general results related to the stabilization of PMD process (10.26). Theorem 10.9 The following assertions hold true: (a) If PMD (10.26) is non-singular and minimal, it is stabilizable. (b) Let the non-singular PMD (10.26) be not minimal. Suppose that we have 

   a(ζ ) −b(ζ ) = ψ (ζ ) a1 (ζ ) −b¯1 (ζ ) ,     a(ζ ) a2 (ζ ) ψr (ζ ), = −c1 (ζ ) −c(ζ )

(10.45)

142

10 Determinant Polynomial Equations, Causal Modal …

where ψ (ζ ) and ψr (ζ ) are the GCLD and GCRD, respectively. Then for the stabilizability of PMD (10.26) is necessary and sufficient that the matrices ψ (ζ ) and ψr (ζ ) are stable. (c) If PMD (10.26) is strictly causal, all stabilizing controllers are causal.  Another form of stabilizability conditions for PMD (10.26) gives Theorem 10.10 Let the polynomial matrix a1 (ζ ) satisfy the relation a(ζ ) = ψ (ζ )a1 (ζ )

(10.46)

which follows from the first equality in (10.45). Let us also have 

   a1 (ζ ) a3 (ζ ) ψ1 (ζ ), = −c(ζ ) −c2 (ζ )

(10.47)

where ψ1 (ζ ) is the GCRD of the pair [a1 (ζ ), −c(ζ )]. Then PMD (10.26) is stabilizable if and only if the matrices ψ (ζ ) and ψ1 (ζ ) are stable. In this case, the stabilization problem for DPE (11.31) is solvable if and only if we have Δ(ζ ) ∼ det ψ (ζ ) det ψ1 (ζ )Δ1 (ζ ),

(10.48)

where Δ1 (ζ ) is an arbitrary stable polynomial.



8. A practical method to solve the DPE (11.31) under the conditions of Theorem 10.10 is as follows: using the first relation in (10.45) as well as (10.47), the matrix (10.29) can be represented as ⎤ a3 (ζ ) 0 pn −b¯1 (ζ ) Q p (ζ, α, β) = diag{ψ (ζ ), In , Im } ⎣ −c2 (ζ ) 0 −d(ζ ) ⎦ diag{ψ1 (ζ ), In , Im }. 0mp −β(ζ ) α(ζ ) 



⎡

(10.49) Calculating the determinants on the right-hand side of the last equality, after reduction we arrive, taking into account (10.48), to the DPE det Q 2 (ζ, α, β) ∼ Δ1 (ζ )

(10.50)

where the matrix Q 2 (ζ, α, β) has the form ⎡

⎤ a3 (ζ ) 0 pn −b¯1 (ζ ) Q 2 (ζ, α, β) = ⎣ −c2 (ζ ) In −d(ζ ) ⎦ . 0mp −β(ζ ) α(ζ )

(10.51)

Let us show that here the pairs (a3 (ζ ), −b¯1 (ζ )) and [a3 (ζ ), −c2 (ζ )] are irreducible. Indeed, the pair [a3 (ζ ), −c2 (ζ )] is irreducible by construction. In addition, since

10.3 Solving Modal Control and Stabilization Problems for Backward …

143

the pair (a1 (ζ ), −b¯1 (ζ )) is also irreducible by construction, there exist polynomial matrices X (ζ ) and Y (ζ ) such that a1 (ζ )X (ζ ) + b¯1 (ζ )Y (ζ ) = I p .

(10.52)

Since it follows from (10.47) that a1 (ζ ) = a3 (ζ )ψ1 (ζ ),

(10.53)

equality (10.52) can be written in the form a3 (ζ )X 1 (ζ ) − b¯1 (ζ )Y (ζ ) = I p ,

(10.54)

X 1 (ζ ) = ψ1 (ζ )X (ζ )

(10.55)

where

is a polynomial matrix. From (10.54) follows the irreducibility of the pair (a3 (ζ ), −b¯1 (ζ )). The proved above implies that DPE (10.50) is solvable for any polynomial Δ1 (ζ ). Also, if the MFD P(ζ ) is strictly causal and Δ1 (0) = 0, all the solutions of DPE (10.50) are causal.

Chapter 11

Synchronous Sampled-Data Stabilization of Periodic Objects

This chapter addresses the problem of synchronous SD stabilization of FDLCP objects. In addition to the simple case, an SD system with LTI prefilter and a system with control delay are considered.

11.1 Synchronous Stabilization of a Single Periodic Object 1. In this section, we consider the SD system ST in which the FDLCP object is described by the state equations dv(t) dt

= A(t)v(t) + B(t)u(t), y(t) = C(t)v(t),

(11.1)

where y(t), v(t), u(t) are vectors n × 1, χ × 1, m × 1. Besides, in (11.1), A(t) = A(t + T ), B(t) = B(t + T ), C(t) = C(t + T ) are continuous matrices of corresponding dimensions. 2. It is assumed that the object (11.1) is controlled by an SD controller which is described by the system of equations ξk = y(kT ), (k = 0, ±1, . . .), α0 ψk + α1 ψk−1 + · · · + αρ ψk−ρ = β0 ξk + β1 ξk−1 + · · · + βρ ξk−ρ , u(t) = h(t − kT )ψk , kT < t < (k + 1)T,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_11

(11.2) (11.3) (11.4)

145

146

11 Synchronous Sampled-Data Stabilization of Periodic Objects

where the notation of Sect. 5.1 is used and the assumptions made there are assumed to be fulfilled. 3.

As before, the pair of polynomial matrices (α(ζ ), β(ζ )) of the type α(ζ ) = α0 + α1 ζ + · · · + αρ ζ ρ , det α0 = 0, β(ζ ) = β0 + β1 ζ + · · · + βρ ζ ρ

(11.5)

is called the discrete controller. Then the stabilization problem for the system ST is reduced to find the set of causal discrete controllers (11.5) for which the system ST is asymptotically stable. Integrating (4.1) yields v(t) = H (t)H

−1

t (kT )vk +

H (t)H −1 (ν)B(ν)u(ν)dν

(11.6)

kT

where the notation vk = v(kT )

(11.7)

is used. For t = (k + 1)T from (11.6), we obtain kT  +T

vk+1 = Mvk + M

k+1

H −1 (ν)B(ν)u(ν)dν,

(11.8)

kT

where M is the monodromy matrix. In deriving (11.8) we have used the formulas M = H (T ),

M k = H (kT ).

(11.9)

By the use of (11.4) the integral on the right-hand side of (11.8) can be represented as kT  +T

H

−1

kT

kT  +T

(ν)B(ν)u(ν)dν =

H −1 (ν)B(ν)h(ν − kT )dνψk .

(11.10)

kT

The change of the integration variable μ = ν − kT, in (11.10) gives kT  +T

H kT

−1

(ν)B(ν)h(ν − kT )dν = M

−k

T 0

H −1 (μ)B(μ)h(μ)dμ,

(11.11)

11.1 Synchronous Stabilization of a Single Periodic Object

147

where we have taken into account that H −1 (μ + kT ) = M −k H −1 (μ),

B(μ + kT ) = B(μ).

(11.12)

Substituting (11.11) into (11.8) yields vk+1 = Mvk + M Dψk ,

(11.13)

where D is a constant matrix χ × q determined by the formula T D

H −1 (μ)B(μ)h(μ)dμ.

(11.14)

0

Replacing k by k − 1 in (11.13) and considering that ξk = y(kT ) = C(kT )v(kT ) = C(0)vk

(11.15)

with the equations of the FDLCP object we can associate the discrete backward model vk = Mvk−1 + M Dψk−1 , (11.16) ξk = C(0)vk . Using the backward shift operator ζ , we can assign to this model the PMD system a(ζ )vk = b(ζ )ψk , ξk = c(ζ )vk , α(ζ )ψk = β(ζ )ξk ,

(11.17)

a(ζ ) = Iχ − ζ M, b(ζ ) = ζ M D, c(ζ ) = C(0).

(11.18)

where

It is obvious that this PMD is homogeneous, strictly causal, and non-singular. 4. By employing the reasoning of [20], used in solving the SD stabilization problem for a continuous LTI object, it can be shown that the stabilization problem for the FDLCP object (11.1) is equivalent to the stabilization problem for PMD (11.17), which leads to the solution of the DPE problem ⎡

⎤ Iχ − ζ M 0χn −ζ M D 0nm ⎦ ∼ Δ(ζ ), In det ⎣ −C(0) 0mχ −β(ζ ) α(ζ ) where Δ(ζ ) is a stable polynomial.

(11.19)

148

11 Synchronous Sampled-Data Stabilization of Periodic Objects

Applying Theorems 10.8 and 10.10, along with (11.17), (11.18), leads to the results formulated in Theorem 11.1 The following assertions hold true: (a) If the pairs (Iχ − ζ M, −ζ M D) and [Iχ − ζ M, −C(0)] are irreducible, i.e., for all ζ   rank Iχ − ζ M −ζ M

D = χ, Iχ − ζ M (11.20) rank = χ, −C(0) the DPE (11.19) is solvable for any polynomial Δ(ζ ) and, consequently, the FDLCP object (11.1) is stabilizable. (b) All stabilizing controllers (α(ζ ), β(ζ )) are causal. (c) Given a stable polynomial Δ(ζ ) under the condition that the equality (11.20) is fulfilled, the set of stabilizing discrete controllers coincides with the solution set of the DPE of type (10.14)

a1 (ζ ) −ζ b1 (ζ ) det ∼ Δ(ζ ) −β(ζ ) α(ζ )

(11.21)

where the matrix a1 (ζ ) ∈ R nn [ζ ] is found from the ILMFD C(0)(Iχ − ζ M)−1 = a1−1 (ζ )c1 (ζ )

(11.22)

and the matrix b1 (ζ ) has the form b1 (ζ ) = c1 (ζ )M D.

(11.23) 

Note 11.1 Since the monodromy matrix M is non-singular, relations (11.20) are equivalent to the requirement of controllability of the pair M, D and observability of the pair M, C(0), which follows from Theorem 10.9. Note 11.2 If the relations (11.20) are not satisfied, then the solution to the stabilization problem can be obtained by applying Theorem 10.10. 5. Example 11.1 Given an FDLCP object of first order with the period 2π dv(t) dt

 = λ+

cos t 2−sin t

v(t) + sin tu(t)

y(t) = cos tv(t)

(11.24)

11.1 Synchronous Stabilization of a Single Periodic Object

149

where λ is a real constant. Object (11.24) is controlled by an digital controller with the period 2π ξk = y(kT ) = vk , α(ζ )ψk = β(ζ )ξk , (11.25) u(t) = ψk , 2π k < t < 2π(k + 1). It may be checked directly that in this example H (t) =

2eλt , 2 − sin t

H −1 (ν) = 0, 5e−λν (2 − sin ν)

(11.26)

and besides 2π M =e

2πλ

,

D = 0, 5

e−λν (2 − sin ν) sin νdν, C(0) = 1, h(t) = 1.

0

(11.27) Then (11.24) yields C(0) = 1, b1 (ζ ) = M D.

(11.28)

With D = 0 in the given example the conditions of Theorem 10.4 are fulfilled and the set of stabilizing controllers α(ζ ), β(ζ ) is found as solution of the DPE of the type (10.14) which in this case has the form det

1 − ζ M −ζ M D −β(ζ ) α(ζ )

∼ Δ(ζ ).

(11.29)

The initial basic controller (α0 (ζ ), β0 (ζ )) can be found as solution of the polynomial equation (11.30) (1 − ζ M)α0 (ζ ) − ζ M Dβ0 (ζ ) = 1, for whose particular solution we can take α0 (ζ ) = 1, β0 (ζ ) = −D −1 .

(11.31)

In this case, using formulas (10.17), we find the set of all stabilizing controllers for the object (11.24) determined by α(ζ ) = Δ(ζ ) − ζ M (ζ )M D, β(ζ ) = −Δ(ζ )D −1 − M (ζ )(1 − ζ M),

(11.32)

where Δ(ζ ) is an arbitrary stable polynomial. All stabilizing controllers (11.32) are causal.

150

11 Synchronous Sampled-Data Stabilization of Periodic Objects

11.2 Synchronous Stabilization of a Periodic Object with Prefilter 1. The structural scheme of the SD system ST 1 studied here is represented in Fig. 11.1. Here FDLCP represents the linear periodic object, DC is the digital controller specified by Equations (11.2)–(11.4). Besides, W (s) is an LTI prefilter object described by the state equations dq(t) = Pq(t) + Qu(t), dt (11.33) f (t) = Rq(t), where q(t) is a vector r × 1 and P, Q, R are constant matrices of corresponding dimensions. 2.

Let us assume that the FDLCP object is defined by equations of type (11.1) = A(t)v(t) + B(t) f (t), y(t) = C(t)v(t).

dv(t) dt

(11.34)

If we introduce the augmented state vector

q(t) v(t) ¯ = , v(t)

(11.35)

then the set of equations (11.33) and (11.34) can be written as equations of the extended FDLCP object ¯ v(t) ¯ = A(t) ¯ + B(t)u(t), ¯ v(t), y(t) = C(t) ¯

d v(t) ¯ dt

(11.36)

¯ ¯ ¯ where the matrices A(t), B(t), C(t) are found from the relations ¯ = A(t)



P 0r χ , B(t)R A(t)



¯ B(t) =

u(t) DC





  Q ¯ , C(t) = 0nr C(t) . 0χm

W (s)

f (t)

 FDLCP

Fig. 11.1 Closed-loop SD system ST 1 with FDLCP and LTI prefilter

y(t)



(11.37)

11.2 Synchronous Stabilization of a Periodic Object with Prefilter

151

3. To solve the stabilization problem for the system ST 1 we can utilize the results of Sect. 11.1, replacing the FDLCP object (11.1) by the extended object (11.36), (11.37). Theorem 11.2 If H (t) is the matrizant of the FDLCP object (11.34), the matrizant H¯ (t) of the augmented object (11.36), (11.37) is determined by ⎡ H¯ (t) = ⎣

H (t)

t

e Pt

0r χ

H −1 (ν)B(ν)Re Pν dν H (t)

⎤ ⎦.

(11.38)

0

 Proof By the definition of the matrizant, H¯ (t) is a solution of the homogeneous matrix equation d H¯ (t) ¯ H¯ (t) = A(t) (11.39) dt with the initial condition

H¯ (0) = Ir +χ .

(11.40)

Condition (11.40) for matrix (11.38) is obviously satisfied. Let us show that the matrix H¯ (t) satisfies relation (11.39). To do this we denote t H (t)

H −1 (ν)B(ν)Re Pν dν  X (t).

(11.41)

0

Differentiating both left-hand and right-hand sides of this equality yields d X (t) = A(t)X (t) + B(t)Re Pt . dt

(11.42)

Then, bearing in mind (11.42) and differentiating the right-hand side of (11.38), we conclude that condition (11.39) is fulfilled for the matrix H¯ (t).  4. Theorem 11.2 implies that the monodromy matrix of the augmented FDLCP object (11.36) has the form ⎡ M¯ = H¯ (T ) = ⎣

M

T 0

ePT

0r χ

H −1 (ν)B(ν)Re Pν dν M

⎤ ⎦.

(11.43)

152

11 Synchronous Sampled-Data Stabilization of Periodic Objects

Besides, if follows from (11.38) that ⎡ ⎢ H¯ −1 (μ) = ⎢ ⎣

e−Pμ

μ −

H

−1

⎤

0r χ

(ν)B(ν)Re dνe Pν

−Pμ

H

−1

⎥ ⎥ (μ) ⎦

(11.44)

0

and taking into account (11.37), we obtain ⎡ ⎢ ¯ H −1 (μ) B(μ) =⎢ ⎣

μ −

⎡ ⎢ =⎢ ⎣

e−Pμ H −1 (ν)B(ν)Re Pν dνe−Pμ

0

H

−1

(ν)B(ν)Re

⎤

⎥ Q ⎥ = H −1 (μ) ⎦ 0χm ⎤

e−Pμ

μ −

0r χ

−Pν

dνe

(11.45)

⎥ ⎥ Q,

−Pμ ⎦

0

which makes it possible to construct the corresponding matrix (11.14) of the type

D¯ =

T 0

⎡ ⎢ ⎢ ⎣−

μ

e−Pμ

⎤

⎥ ⎥ Qh(μ)dμ. H −1 (ν)B(ν)Re Pν dνe−Pμ ⎦

(11.46)

0

The remaining part of solving the stabilization problem is based on (11.36), (11.43), and (11.46) similarly to the solution method described in Sect. 11.1.

11.3 Synchronous SD Stabilization of an FDLCP Object with Control Delay 1. We consider the system ST τ , whose structure is shown in Fig. 11.2, where, in addition to the notation used before, τ is a positive constant. The equations of the FDLCP object are assumed to have the form

Fig. 11.2 Closed-loop SD system ST τ with FDLCP and delay

11.3 Synchronous SD Stabilization of an FDLCP Object with Control Delay

dv(t) = A(t)v(t) + B(t)u(t − τ ), dt y(t) = C(t)v(t),

153

(11.47)

and the SD controller is described by (11.2)–(11.4). The constant τ is assumed to satisfy the relations τ = mT + θ = (m + 1)T − γ , (11.48) where m ≥ 0 is an integer, θ and γ = T − θ are constants with 0 ≤ θ < T and 0 < γ ≤ T . It follows from (11.48) and (11.4) that the controlling signal at the input of the FDLCP object is determined by the relations  u(t − τ ) =

h(t − kT + γ )ψk−m−1 , kT < t < kT + θ, h(t − kT − θ )ψk−m , kT + θ < t < kT + T.

(11.49)

2. Integrating the first equation in (11.47) with the initial condition v(kT ) = vk , and bearing in mind (11.49), for kT ≤ t ≤ kT + θ , we get v(t) = H (t)H

−1

t (kT )vk + H (t)

H −1 (ν)B(ν)h(ν − kT + γ )dνψk−m−1 .

kT

(11.50) Similarly, utilizing (11.49), for kT + θ ≤ t ≤ kT + T we get v(t) = H (t)H

−1

kT  +θ

(kT )vk + H (t)

H −1 (ν)B(ν)h(ν − kT + γ )dνψk−m−1 +

kT

t

H −1 (ν)h(ν − kT − θ )B(ν)h(ν − kT − θ )dνψk−m .

+H (t) kT +θ

(11.51) For t = kT + T it follows from (11.51) that kT  +θ

vk+1 = Mvk + M

k+1

kT  +T

+M k+1

H −1 (ν)B(ν)h(ν − kT + γ )dνψk−m−1 +

kT

(11.52)

H −1 (ν)B(ν)h(ν − kT − θ )dνψk−m .

kT +θ

By the change of variable ν − kT + γ = μ

(11.53)

154

11 Synchronous Sampled-Data Stabilization of Periodic Objects

we establish that kT  +θ

H −1 (ν)B(ν)h(ν − kT + γ )dν =

T

H −1 (μ + kT − γ )B(μ + kT − γ )h(μ)dμ.

γ

kT

(11.54) Similarly, assuming that ν − kT − θ = μ,

(11.55)

we can obtain kT  +T

H −1 (ν)B(ν)h(t − kT − θ )dν =

kT +θ

γ

H −1 (μ + kT + θ)B(μ + kT + θ)h(μ)dμ.

0

(11.56)

Since we have H −1 (μ + kT − γ ) = H −1 (μ + kT − T + θ ) = M −k+1 H −1 (μ + θ ), H −1 (μ + kT + θ ) = M −k H −1 (μ + θ ), B(μ + kT − γ ) = B(μ + kT − T + θ ) = B(μ + θ ),

(11.57)

equalities (11.54) and (11.56) yield kT  +θ

M

k+1

H kT kT  +T

M k+1

−1

T (ν)B(ν)h(ν − kT + γ )dν = M

2

H −1 (μ + θ )B(μ + θ )h(μ)dμ,

γ

H −1 (ν)B(ν)h(ν − kT − θ )dν = M

kT +θ

γ

H −1 (μ + θ )B(μ + θ )h(μ)dμ.

0

(11.58) Substituting expressions (11.58) into (11.52) and changing k into k − 1, we obtain vk = Mvk−1 + M 2 Γ2 (θ )ψk−m−2 + MΓ1 (θ )ψk−m−1 ,

(11.59)

where we have denoted T Γ2 (θ ) =

H −1 (μ + θ )B(μ + θ )h(μ)dμ,

γ

γ Γ1 (θ ) =

(11.60) H −1 (μ + θ )B(μ + θ )h(μ)dμ.

0

Joining the discrete model of the output y(t) as well as the discrete controller equation to (11.59), we obtain the discrete model of the system ST τ

11.3 Synchronous SD Stabilization of an FDLCP Object with Control Delay

155

vk = Mvk−1 + M 2 Γ2 (θ )ψk−m−2 + MΓ1 (θ )ψk−m−1 , ξk = C(0)vk , α0 ψk + α1 ψk−1 + · · · + αρ ψk−ρ = β0 ξk + β1 ξk−1 + · · · + βρ ξk−ρ , det α0 = 0. (11.61) 3. Using the backward shift operator, the system of difference equations (11.61) can be represented as a(ζ )vk = ζ m+1 Mbτ (ζ )ψk , ξk = C(0)vk , (11.62) α(ζ )ψk = β(ζ )ξk , where

a(ζ ) = I − ζ M, bτ (ζ ) = ζ MΓ2 (θ ) + Γ1 (θ ), α(ζ ) = α0 + α1 ζ + · · · + αρ ζ ρ , β(ζ ) = β0 + β1 ζ + · · · + βρ ζ ρ .

(11.63)

Thus the stabilization problem for the system ST τ reduces to solving a task for the PMD (11.62), (11.63), for which we can use the results of Theorem 10.9. However, a supporting statement is necessary here. Lemma 11.1 The pair (a(ζ ), ζ m+1 Mbτ (ζ )) from (11.63) is irreducible if and only if the pair (M, Γ (θ )) with T Γ (θ ) =

H −1 (μ + θ )B(μ + θ )h(μ)dμ

(11.64)

0



is controllable.

Proof By virtue of Theorem 9.9 the pair (a(ζ ), ζ m+1 M(ζ MΓ2 (θ ) + Γ1 (θ )) is irreducible if and only if the pair (M, M −m Γ (θ )) is controllable. At the same time the equality   rank M −m Γ (θ) M −m+1 Γ (θ ) ...  = (11.65) = rank M −m Γ (θ ) MΓ (θ ) ... implies that the pair (M, M −m Γ (θ )) is controllable if and only if the pair (M, Γ (θ )) is controllable.  Theorem 11.3 The following assertions hold true: (a) All stabilizing discrete controllers for the system ST τ are causal. (b) If the pair (M, Γ (θ )) is controllable and the pair [M, C(0)] is observable, then the system ST τ is stabilizable. Moreover, the set of all stabilizing discrete controllers coincides with the solution set of the DPE of the form

a1 (ζ ) −bT τ (ζ ) ∼ Δ(ζ ). det −β(ζ ) α(ζ )

(11.66)

156

11 Synchronous Sampled-Data Stabilization of Periodic Objects

Here the matrix a1 (ζ ) is found from the ILMFD C(0)(I − ζ M)−1 = a1−1 (ζ )b1 (ζ ),

(11.67)

the matrix bT τ (ζ ) is given by the formula bT τ (ζ ) = ζ m+1 b1 (ζ )M(ζ MΓ2 (θ ) + Γ1 (θ ))

(11.68)

and Δ(ζ ) is an arbitrary stable polynomial. (c) If assumptions (b) are not satisfied, the solution to the stabilization problem can be obtained using Theorem 10.10. 

Chapter 12

Asynchronous Sampled-Data Stabilization of Periodic Objects

In this chapter, the results of the previous chapter are extended to different variants of asynchronous SD stabilization.

12.1 Low-Frequency Stabilization of a Periodic Object 1. We consider the stabilization problem for an FDLCP object with the period T , described, as before, by the state equations dv(t) dt

= A(t)v(t) + B(t)u(t), y(t) = C(t)v(t)

(12.1)

under the assumptions introduced above. It is assumed that the object (12.1) is controlled by SD controller working with the period Tμ = μT where d > 1 is an integer. The equations of this SD controller are as follows: ξk = y(kTμ ), (k = 0, ±1, . . .), α0 ψk + α1 ψk−1 + · · · + αρ ψk−ρ = β0 ξk + β1 ξk−1 + · · · + βρ ξk−ρ , u μ (t) = h μ (t − kTμ )ψk , kTμ < t < (k + 1)Tμ .

(12.2) (12.3) (12.4)

In their totality relations (12.1)–(12.4) define the linear periodic system Sμ with a period Tμ = μT .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_12

157

158

12 Asynchronous Sampled-Data Stabilization of Periodic Objects

2. To solve the stabilization problem let us construct a discrete backward model of the system Sμ . Integrating the first equation in (12.1) with the initial condition v(kTμ ) = vk , yields v(t) = H (t)H

−1

t (kTμ )vk +

(12.5)

H (t)H −1 (ν)B(ν)u μ (ν)dν.

(12.6)

kTμ

For t = kTμ + Tμ this implies that v(kTμ + Tμ ) = H (kTμ + Tμ )H −1 (kTμ ) + H (kTμ + Tμ )

kT μ +Tμ

H −1 (ν)B(ν)u μ (ν)dν.

kTμ

(12.7) If we take into account that v(kTμ + Tμ ) = vk+1 , H (kTμ ) = H (kμT ) = M kμ H −1 (kTμ ) = H −1 (kμT ) = M −kμ ,

(12.8)

and formula (12.4), relation (12.7) takes the form vk+1 = M μ vk + M (k+1)μ

kTμ +Tμ

H −1 (ν)B(ν)h μ (ν − kTμ )dνψk .

(12.9)

kTμ

By a change of the integration variable ν − kTμ = λ

(12.10)

we can establish that kTμ +Tμ

H kTμ

−1

Tμ (ν)B(ν)h μ (ν − kTμ )dν =

H −1 (λ + kTμ )B(λ + kTμ )h μ (λ)dλ.

0

(12.11) Utilizing (12.8) as well as the periodicity of the matrix B(t), the last relation can be represented as kTμ +Tμ

kTμ

H −1 (ν)B(ν)h μ (ν − kTμ )dν = M −kμ

Tμ 0

H −1 (λ)B(λ)h μ (λ)dλ.

(12.12)

12.1 Low-Frequency Stabilization of a Periodic Object

159

Substituting (12.12) into (12.9) and replacing k by k − 1, we find vk = M μ vk−1 + M μ Dμ ψk−1 ,

(12.13)

where Dμ is a constant matrix specified by the formula Tμ

H −1 (λ)B(λ)h μ (λ)dλ.

(12.14)

ξk = C(kTμ )v(kTμ ) = C(0)vk ,

(12.15)

Dμ = 0

Joining to (12.13) the equation

we obtain the system of difference equations which determines the discrete backward model of FDLCP object (12.1) which corresponds to the period Tμ . By adding discrete controller equation (12.3) to (12.13) and (12.15), we obtain a discrete model of system Sμ vk = M μ vk−1 + M μ Dμ ψk−1 , ξk = C(0)v)k, α0 ψ1 + α1 ψk−1 + · · · + αρ ψk−ρ = β0 ξk + β1 ξk−1 + · · · + βρ ξk−ρ .

(12.16)

3. Using the polynomial notation, we come to a description of the system Sμ using the equations aμ (ζ )vk = ζ M Dμ ψk , ξk = C(0)vk , (12.17) α(ζ )ψk = β(ζ )ξk , where

aμ (ζ ) = I − ζ M μ .

(12.18)

Equations (12.17) is obtained from (11.17), (11.18) by replacing the matrix M with the matrix M μ , the matrix D (11.14) with the matrix Dμ (12.14) as well as the period T with the period Tμ = μT . Thus, the stabilization problem solution for the system Sμ can be obtained from the results of Sect. 11.1 by an appropriate replacement. The case that is essential for applications is treated by Theorem 12.1 The following assertions hold true: (a) If the pair (M μ , Dμ ) is controllable and the pair [M μ , C(0)] is observable, the system Sμ is stabilizable. In this case all stabilizing discrete controllers are causal.

160

12 Asynchronous Sampled-Data Stabilization of Periodic Objects

(b) If we have an ILMFD −1 (ζ )cμ1 (ζ ), C(0)(I − ζ M μ )−1 = aμ1

(12.19)

the stabilizing controller set for the system Sμ coincides with the solution set of the DPE   aμ1 (ζ ) −ζ cμ1 (ζ )M Dμ ∼ Δ(ζ ), (12.20) det −β(ζ ) α(ζ ) where Δ(ζ ) is an arbitrary stable polynomial.  4. Example 12.1 Consider an FDLCP object of the period T = 2π given by Equations (11.24) and the SD controller equations having the form (12.2)-(12.4) for Tμ = 2π μ, μ > 1. Then, utilizing the results of Example 11.1 we get μ

Tμ

M =e

2πλμ

,

Dμ = 0, 5

e−λν (2 − sin ν) sin νdν.

(12.21)

0

In this case DPE (12.20) can be obtained as follows  det

1 − ζ M μ −ζ M μ Dμ −β(ζ ) α(ζ )

 ∼ Δ(ζ ).

(12.22)

The initial basic controller (α0 (ζ ), β0 (ζ )) can be found as a solution of the diophantic polynomial equation (1 − ζ M μ )α0 (ζ ) − ζ M μ Dμ β0 (ζ ) = 1. Thus we can take

(12.23)

α0 (ζ ) = 1, β0 (ζ ) = −Dμ−1 .

as initial solution. Here the stabilizing controllers set for the system Sd is found from the relations α(ζ ) = Δ(ζ ) − M (ζ )M μ Dμ , (12.24) β(ζ ) = −Δ(ζ )Dμ−1 − M (ζ )(1 − ζ M μ ), where M (ζ ) is an arbitrary polynomial and Δ(ζ ) is an arbitrary stable polynomial.

12.2 Low-Frequency Stabilization of a Periodic Object with Prefilter

161

12.2 Low-Frequency Stabilization of a Periodic Object with Prefilter In this section we consider the problem of stabilizing the extended FDLCP object (11.36), (11.37) using a slow sampling SD controller (12.2)–(12.4) with the period Tμ = μT . To solve the stabilization problem we can use the formulas in Sect. 12.1, taking into account the notation of Sect. 11.2. In particular, Eq. (12.13) takes the form (12.25) v¯k = M¯ μ v¯k−1 + M¯ μ D¯ μ ψk−1 . To calculate the matrices M¯ μ and D¯ μ appearing here, notice that by definition M¯ = H¯ (t)|t=Tμ ,

(12.26)

where H¯ (t) is the matrix from (11.38). Since by virtue of the monodromy matrix properties (12.27) M¯ μ = H¯ (Tμ ) = H¯ (μT ), for t = Tμ from (11.38) we find ⎡

e P Tμ

0r χ

⎤

⎥ ⎢ μT ⎥ M¯ μ = ⎢ ⎣ M μ H −1 (ν)B(ν)Re Pν dν M μ ⎦ .

(12.28)

0

Besides, formula (12.14) yields D¯ μ =

Tμ

¯ H¯ −1 (λ) B(λ)h μ (λ)dλ.

(12.29)

0

Using (11.44) and (11.37), from (12.29) we obtain

D¯ μ =

Tμ 0

⎡

e−Pλ

⎤

⎢ λ ⎥ ⎢ ⎥ ⎣ − H −1 (ν)B(ν)Re Pν dνe−Pμ ⎦ Qh μ (λ)dλ.

(12.30)

0

Joining to (12.2) the discrete model of the output ¯ v¯k , ξ¯k = C(0) where according to (11.37)

(12.31)

162

12 Asynchronous Sampled-Data Stabilization of Periodic Objects

 ¯ C(0) = 0nr C(0) ,

(12.32)

as well as the discrete controller equation, we come to the stabilization problem for the PMD (Ir +χ − ζ M¯ μ )v¯k = ζ M¯ D¯ μ ψk , ¯ v¯k , (12.33) ξ¯k = C(0) α(ζ )ψk = β(ζ )ξ¯k . This problem can be solved on the basis of the results from Sect. 10.3.

12.3 Low-Frequency Stabilization of a Periodic Object with Control Delay 1. We are considering here the system Sμτ . The equations of the FDLCP object with period T are similar to those in (4.21) dv(t) = A(t)v(t) + B(t)u μ (t − τμ ), y(t) = C(t)v(t), dt

(12.34)

where the delay time τμ is a positive constant that can be represented by τμ = m μ Tμ + θμ = (m μ + 1)Tμ − γμ ,

(12.35)

where m μ is a non-negative integer, θμ and γμ are constants satisfying the relations 0 ≤ θμ < Tμ , 0 < γμ = Tμ − θμ ≤ Tμ .

(12.36)

The FDLCP object (12.34) is assumed to be controlled by an SD controller with the period Tμ = μT described by Eqs. (12.2)–(12.4). 2. According to the general approach employed above, let us construct a discrete model of the FDLCP object (12.34) corresponding to the sampling period Tμ = μT . To do this note that similarly to (11.49) we have

u μ (t − τμ ) =

h μ (t − kTμ + γμ )ψk−m μ −1 , kTμ < t < kTμ + θμ , (12.37) h μ (t − kTμ − θμ )ψk−m μ , kTμ + θμ < t < (k + 1)Tμ .

Below we use the notation vμk = v(kTμ ),

yμk = ξk = y(kTμ ).

(12.38)

Moreover, using transformations similar to those presented in Sect. 11.3, we can obtain a discrete backward model of FDLCP object (12.34) corresponding to the

12.3 Low-Frequency Stabilization of a Periodic Object with Control Delay

163

period Tμ , and having the form vμk = M μ vμ,k−1 + M 2μ Γμ2 (θμ )ψk−m μ −2 + M μ Γμ1 (θμ )ψk−m μ −1 , ξk = C(0)vμk ,

(12.39)

where the notation Tμ Γμ2 (θμ ) = γμ γμ

Γμ1 (θμ ) =

H −1 (λ + θμ )B(λ + θμ )h μ (λ)dλ, (12.40) H −1 (λ + θμ )B(λ + θμ )h μ (λ)dλ

0

is used. Attaching the discrete controller equation to (12.39), we obtain a discrete model of the system Sμτ : vμk = M μ vμ,k−1 + M 2μ Γμ2 (θμ )ψk−m μ −2 + M μ Γμ1 (θμ )ψk−m μ −1 , ξk = C(0)vdk , α0 ψk + α1 ψk−1 + · · · + αρ ψk−ρ = β0 ξk + β1 ξk−1 + · · · + βρ ξk−ρ .

(12.41)

3. Passing from (12.41) to a polynomial notation, we conclude that the stabilization problem for the system Sμτ reduces to the stabilization task for the nonsingular strictly causal PMD aμ (ζ )vμk = ζ m μ +1 M μ bμτ (ζ )ψk , ξk = C(0)vμk , (12.42) α(ζ )ψk = β(ζ )ξk where aμ (ζ ) = I − ζ M μ , bμτ (ζ ) = ζ M μ Γμ2 (θμ ) + Γμ1 (θμ ), α(ζ ) = α0 + α1 ζ + · · · + αρ ζ ρ , β(ζ ) = β0 + β1 ζ + · · · + βρ ζ ρ .

(12.43)

Comparing relations (11.62), (11.63) and (12.42), (12.43) leads to the conclusion that the stabilization problem solution for the system Sμτ can be obtained from the stabilization problem solution for the system ST τ , described in Sect. 11.3, by replacing T by Tμ , τ by τμ , θ by θμ , γ by γμ , M by M μ . Applying this replacement in Theorem 11.3 leads to Theorem 12.2 The following assertions hold true: (a) All the stabilizing controllers for the system Sμτ are causal. (b) If the pair (M μ , Dμ (θμ )) where Dμ (θμ ) is a constant matrix determined by the relation

164

12 Asynchronous Sampled-Data Stabilization of Periodic Objects

Tμ Dμ (θμ ) =

H −1 (λ + θμ )B(λ + θμ )h μ (λ)dλ

(12.44)

0

is controllable, and the pair [M μ , C(0)] is observable, then the system Sμτ is stabilizable. Here the set of all the stabilizing discrete controllers (α(ζ ), β(ζ )) coincides with the solution set of the DPE  aμ1 (ζ ) −b¯μ1 (ζ ) ∼ Δ(ζ ), det −β(ζ ) α(ζ ) 

(12.45)

where Δ(ζ ) is an arbitrary stable polynomial, the matrix aμ1 (ζ ) is found from the ILMFD −1 (ζ )bμ1 (ζ ), (12.46) C(0)(I − ζ M μ )−1 = aμ1 and the matrix b¯μ1 (ζ ) has the form b¯μ1 (ζ ) = ζ m μ +1 bμ1 (ζ )M μ (ζ M μ Γμ2 (θμ ) + Γμ1 (θμ )).

(12.47)

(c) If the conditions from (b) are not satisfied, the stabilization problem solution can be constructed on the basis of Theorem 10.10. 

12.4 Stabilization of a Periodic Object Using a High-Frequency Hold 1.

The present section considers the SD stabilization problem for an FDLCP object dv(t) dt

= A(t)v(t) + B(t)u η (t), y(t) = C(t)v(t)

(12.48)

with period T where y(t), v(t), u η (t) are vectors n × 1, χ × 1 and m × 1 and A(t) = A(t + T ), B(t) = B(t + T ), C(t) = C(t + T ) are continuous matrices of appropriate dimensions. 2. The control vector u(t) is formed by an SD controller which generates controlling pulses with a period T (12.49) Tη = η where η > 1 is an integer. The ADC operates with the period T and is described by the equation

12.4 Stabilization of a Periodic Object Using a High-Frequency Hold

165

ξk = y(kT ), (k = 0, ±1, . . .).

(12.50)

The discrete controller with the period T produces a discrete control sequence ψ¯ k of dimensions qη × 1 as solution of the difference equation α0 ψ¯ k + α1 ψ¯ k−1 + · · · + αρ ψ¯ k−ρ = β0 ξk + β1 ξk−1 + · · · + βρ ξk−ρ .

(12.51)

⎡

⎤ ψ1k ψ¯ k = ⎣ ... ⎦ , ψηk

We assume below that

(12.52)

where ψik , (i = 1, . . . , η) are vector sequences q × 1, αi , βi are constant matrices ηq × ηq and ηq × n. For the constructed vector ψ¯ k , the control vector u η (t) is defined on the interval kT < t < (k + 1)T by the formula   (i − 1)T (i − 1)T iT u η (t) = h i t − kT − ψik , kT + < t < kT + , (i = 1, . . . , η), η η η

(12.53)

where h i (t) are matrices m × q, defined on the interval 0 < t
appears instead of ≥, then we will call the quasi-polynomial L(λ) positive on the unit circle. Here we use the notation Δ

x ∗ = x¯  where the bar above, as before, means complex conjugation.

(20.13) 

5. Theorem 20.2 About factorization. Let the real square quasi-polynomial L(λ) be non-negative on the unit circle. Then there exists a real polynomial δ(λ) such that for all λ = 0 we have   1 . det L(λ) = δ(λ)δ λ

(20.14)

At the same time there exists a real square polynomial matrix D(λ) m × m with a factorization    1 L(λ) = D D(λ) (20.15) λ where det D(λ) = δ(λ).

(20.16) 

Chapter 21

Minimization of a Quadratic Functional on the Unit Circle

The chapter describes a well-known procedure for minimizing a quadratic functional on the unit circle. 1. Theorem 21.1 ([24, 25]) Let us have a functional of the form 

  ˆ ) − C(ζ )Θ(ζ ) dζ ˆ )Π(ζ ˆ )Π (ζ )Θ(ζ )Γ (ζ ) − Θ(ζ ˆ )C(ζ tr Γˆ (ζ )Θ(ζ ζ (21.1) where integration is carried out along the unit circle |ζ | = 1 in positive direction (counterclockwise), tr denotes the trace of the matrix and the notation Δ

L =

1 2π j

−1 Δ ˆ )= F  (ζ ). F(ζ

(21.2)

is used for the matrix F(ζ ). Furthermore, in (21.1) Γ (ζ ), Π (ζ ), C(ζ ) are rational matrices that do not have poles on the circle |ζ | = 1. It is assumed that matrices Γ (ζ ) and Π (ζ ) are stable together with their inverses. Under the above conditions, there exists a stable rational matrix Θ 0 (ζ ) for which integral (21.1) takes a minimal value. This matrix Θ 0 (ζ ) can be constructed using the following algorithm. (a) Build the rational matrix −1 Δ ˆ )Γˆ −1 (ζ ). R(ζ ) = Πˆ (ζ )C(ζ

(21.3)

R(ζ ) = R+ (ζ ) + R− (ζ )

(21.4)

(b) Perform the separation

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_21

225

226

21 Minimization of a Quadratic Functional on the Unit Circle

where R− (ζ ) is a rational matrix that is strictly proper and whose set of poles coincides with the set of unstable poles of the matrix R(ζ ). (c) The optimal matrix Θ 0 (ζ ) is determined by the formula −1

−1

Θ 0 (ζ ) = Π (ζ )R+ (ζ )Γ (ζ ).

(21.5) 

Proof Given that   −1 tr [C(ζ )Θ(ζ )] = tr C(ζ )Θ(ζ )Γ (ζ )Γ (ζ ) =  −1  = tr Γ (ζ )C(ζ )Θ(ζ )Γ (ζ )

(21.6)

we rewrite (21.1) in the form L=

where

 ˆ )Π(ζ ˆ )Π (ζ )Θ(ζ )Γ (ζ )− tr Γˆ (ζ )Θ(ζ  ˆ )(ζ ˆ ) − (ζ )Θ(ζ )Γ (ζ ) dζ Γˆ (ζ )Θ(ζ ζ 1 2π j



Δ

−1

(ζ ) = Γ (ζ )C(ζ ).

(21.7)

(21.8)

We use the identity ˆ )Π(ζ ˆ )Π (ζ )Θ(ζ )Γ (ζ ) − Γˆ (ζ )Θ(ζ ˆ )(ζ ˆ ) − (ζ )Θ(ζ )Γ (ζ ) = Γˆ (ζ )Θ(ζ  ˆ )R(ζ ), = Π (ζ )Θ(ζ )Γ (ζ ) − R(ζ ) [Π (ζ )Θ(ζ )Γ (ζ ) − R(ζ )] − R(ζ (21.9) which is verified immediately based on (21.3). With the help of separation (21.4) we can write (21.9) as   ˆ )R(ζ ) = Π (ζ )Θ(ζ )Γ (ζ ) − R(ζ ) [Π (ζ )Θ(ζ )Γ (ζ ) − R(ζ )] − R(ζ    = Π (ζ )Θ(ζ )Γ (ζ ) − R+ (ζ ) − R− (ζ ) Π (ζ )Θ(ζ )Γ (ζ ) − R+ (ζ ) − ˆ )R(ζ ) = − R(ζ    = Π (ζ )Θ(ζ  )Γ (ζ ) − R+ (ζ ) Π (ζ )Θ(ζ )Γ (ζ ) − R+ (ζ ) −     − Π (ζ )Θ(ζ  )Γ (ζ ) − R+ (ζ ) R− (ζ ) − Rˆ − (ζ ) Π (ζ )Θ(ζ )Γ (ζ ) − R+ (ζ ) + ˆ )R(ζ ). + Rˆ − (ζ )R− (ζ ) − R(ζ (21.10) Inserting (21.10) into (21.7), the functional L is written as sum L = L1 + L2 + L3 + L4 where

(21.11)

21 Minimization of a Quadratic Functional on the Unit Circle

227



  Π (ζ )Θ(ζ  )Γ (ζ ) − R+ (ζ ) Π (ζ )Θ(ζ )Γ (ζ ) − R+ (ζ ) dζ , ζ     dζ Δ 1 L 2 = − 2π j tr Rˆ − (ζ ) Π (ζ )Θ(ζ )Γ (ζ ) − R+ (ζ ) ζ ,    Δ L 3 = − 2π1 j tr Π (ζ )Θ(ζ  )Γ (ζ ) − R+ (ζ ) R− (ζ ) dζ , ζ   Δ 1  dζ ˆ ˆ L 4 = 2π j tr R− (ζ )R− (ζ ) − R(ζ )R(ζ ) ζ . (21.12) The integral L 4 does not depend on the system function Θ(ζ ), i.e., does not depend on the choice of the control algorithm. We consider integral L 2 . In this integral the matrix Π (ζ )Θ(ζ )Γ (ζ ) − R+ (ζ ) is stable per construction, i.e., its poles are located outside the unit circle |ζ | = 1. Due to separation (21.4) matrix R− (ζ ) has only poles inside the unit circle. Therefore, the −1 poles of matrix Rˆ − (ζ ) = R  (ζ ) are located outside the circle. Since matrix R− (ζ ) is strictly proper matrix Rˆ − (ζ ) allows an irreducible representation of the form Δ

L1 =

1 2π j



tr

ζ N (ζ ) Rˆ − (ζ ) = d(ζ )

(21.13)

where N (ζ ) is polynomial matrix and d(ζ ) is a scalar polynomial with d(0) = 0. From (21.13) follows that Rˆ − (0) = 0.

(21.14)

Substituting (21.13) into the expression for integral L 2 in (21.12), we conclude that the corresponding integrand is similar in the circle |ζ | ≤ 1. Therefore, according to the residue theorem, it follows that L 2 = 0. By moving to the integration variable −1 z = ζ in L 3 it is also established that L 3 = 0. Besides, integral L 1 is non-negative.  The minimal value of L 1 = 0 is achieved using (21.5). Corollary 21.1 The minimal value of integral (21.1) is determined by L min = L 4 .

(21.15) 

Chapter 22

Construction of Matrix η(s, t)

In this chapter, techniques of preparatory calculations for the construction of the quality functional are discussed. 1. In this section we give additional relations that are used to solve the H2 optimization problem for the system Sτ . The approach under consideration and the solution of the H2 -optimization problem are based on representation (17.21) of the PTM (22.1) W yx (s, t) = ψ(s, t)(s)ξ(s) + η(s, t) by the system function (ζ ). Matrices ψ(s, t), ξ(s), η(s, t) are determined by (17.22). According to Theorem 17.2, these matrices for all t are integral functions of the argument s. For the matrices ψ(s, t) and ξ(s) this fact is proved directly. For matrix η(s, t) this fact is established on the basis of indirect considerations. Equation (17.22) for determining matrix η(s, t) contains matrix L˜N (s, t) which has poles. It follows from Theorem 17.2 that the poles of this matrix should be shortened when calculating η(s, t). When solving practical problems, it is difficult to capture these shortenings due to inevitable calculation errors. Therefore, this section provides a formulation for matrix η(s, t) in which the poles of the specified matrix L˜N (s, t) do not appear. For this purpose we use the fact proved in Chap. 17 that the matrix η(s, t) is the PTM W yx (s, t) in case the basic controller α 0 (ζ ), β 0 (ζ ) is used for control. With this choice of the discrete controller the matrix est W yx (s, t) is an exponentially periodic solution of the system of matrix differential-difference equations dv(s,t) = A(t)v(s, t) + B1 (t)est + B(t)u(s, t − τ ), dt y(s, t) = C(t)v(s, t), (22.2) ξk (s) = C(0)v(s, kT ), α 0 (ζ )ψk (s) = β 0 (ζ )ξk (s), u(s, t) = h(t − kT )ψk (s), kT < t < (k + 1)T. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_22

229

22 Construction of Matrix η(s, t)

230

These equations can be associated with the discrete model (14.24) vk (s) = Mvk−1 (s) + M 2 2 (θ )ψk−m−2 (s) + M1 (θ )ψk−m−1 (s)+ +e(k−1)sT M G(s), ξk (s) = C(0)vk (s), α 0 (ζ )ψk (s) = β 0 (ζ )ξk (s)

(22.3)

where G(s) is matrix (14.23). 2. As it follows from Chap. 14, the matrix est W yx (s, t) is the solution to (22.2) for which the sequences

vk (s) = v(s, kT ), ξk (s), ψk (s)

(22.4)

satisfy the conditions vk (s) = eksT v0 (s), ξk (s) = eksT ξ0 (s), ψk (s) = eksT ψ0 (s).

(22.5)

After substituting relations (22.5) into (22.3), we obtain a system of linear equations defining the matrices v0 (s), ψ0 (s) and ξ0 (s) of the form (14.28), which in the case under consideration has the form (14.28) Q˜ τ (s, α˜ 0 , β˜ 0 )Z (s) = e−sT G(s)

(22.6)

˜ ˜ = α˜ 0 (s), β(s) = β˜ 0 (s) and Z (s) where Q˜ τ (s, α˜ 0 , β˜ 0 ) is matrix (14.12) for α(s) ˜ ˜ is matrix (14.30). In expanded form the matrix Q τ (s, α˜ 0 , β 0 ) looks like ⎤ ˜ Iχ − e−sT M 0χn −e−(m+1)sT M b(s) ⎦ 0nq In Q˜ τ (s, α˜ 0 , β˜ 0 ) = ⎣ −C(0) 0qχ −β˜ 0 (s) α˜ 0 (s) ⎡

(22.7)

˜ has the form (14.10) for ζ = e−sT . Replacing in (22.7) e−sT with where matrix b(s) ζ , we obtain the polynomial matrix ⎡

⎤ Iχ − ζ M 0χn −ζ m+1 Mb(ζ ) ⎦. 0nq In Q τ (ζ, α 0 , β 0 ) = ⎣ −C(0) 0qχ −β 0 (ζ ) α 0 (ζ ) Let be

−1

WT (ζ ) = ζ m+1 C(0)(Iχ − ζ M) Mb(ζ )

(22.8)

(22.9)

the transfer matrix of the discrete model of the LP object. If we have the ILMFD C(0)(Iχ − ζ M)

−1

−1

= a (ζ )b (ζ ),

(22.10)

22 Construction of Matrix η(s, t)

231

then, under the condition of modal controllability of the system Sτ , by virtue of what was proved earlier, the polynomial matrix 

α (ζ ) −ζ m+1 b (ζ )Mb(ζ ) Q τ1 (ζ, α 0 , β 0 ) = α 0 (ζ ) −β 0 (ζ )

 (22.11)

is unimodular since the controller α 0 (ζ ), β 0 (ζ ) is the basic one, i.e., we have

det Q τ1 (ζ, α 0 , β 0 ) = d = const = 0.

(22.12)

By what was proved earlier, from (22.12), under suppositions made, the matrix (22.8) is also unimodular, i.e.,

det Q τ (ζ, α 0 , β 0 ) = d1 = const = 0.

(22.13)

Substituting in (22.12) argument e−sT for ζ , we obtain

det Q˜ τ (s, α˜ 0 , β˜ 0 ) = dτ = const = 0.

(22.14)

It follows from (22.14) that matrix Q˜ τ (s, α˜ 0 , β˜ 0 ) is invertible and elements of the −1 matrix Q˜ τ (s, α˜ 0 , β˜ 0 ) are polynomials in the variable ζ = e−sT . 3.

Given the inverse of the matrix Q˜ τ (s, α˜ 0 , β˜ 0 ) from (14.32) we get −1 ¯ Z (s) = e−sT Q˜ τ (s, α˜ 0 , β˜ 0 )G(s),

where

⎤ ¯ M G(s) G(s) = ⎣ 0n ⎦ . 0ql

(22.15)

⎡

(22.16)

Denote by Q˜ 1τ (s) the matrix composed of the first χ columns of the matrix −1 Q˜ τ (s, α˜ 0 , β˜ 0 ). Let be χ ⎤ ⎡ K˜ v (s) χ (22.17) Q˜ 1τ (s) = ⎣ K˜ ξ (s) ⎦ n K˜ ψ (s) q where the letters outside the matrix indicate the size of the corresponding blocks. Then, using (22.16) and (22.17) from (22.15), we can get v0 (s) = e−sT K˜ v (s)M G(s), ξ0 (s) = e−sT K˜ ξ (s)M G(s), ψ0 (s) = e−sT K˜ ψ (s)M G(s).

(22.18)

22 Construction of Matrix η(s, t)

232

4. We construct a solution for v(s, t) from the first equation in (22.2) under the

initial condition v(s, 0) = v0 (s). This solution has the form t v(s, t) = H (t)v0 (s) + H (t)

−1

H (ν)B1 (ν)esν dν+ 0

t

(22.19)

−1

H (ν)B(ν)u(ν − τ )dν.

+H (t) 0

Further, to simplify the calculations, we will assume that τ = θ , i.e., m = 0. Then, by virtue of what was established earlier, for 0 < t < T we have u(t − τ ) =

h(t + γ )ψ−1 (s), 0 < t < θ, h(t − θ )ψ0 (s), θ < t < T.

(22.20)

By inserting (22.20) into (22.19), we find for 0 ≤ t ≤ θ . t v(s, t) = H (t)v0 (s) + H (t)

−1

H (ν)B1 (ν)esν dν+ 0

t

(22.21)

−1

H (ν)B(ν)h(ν + γ )dνψ−1 (s).

+H (t) 0

Using (22.20), we get for the interval θ ≤ t ≤ T t v(s, t) = H (t)v0 (s) + H (t) 0

θ +H (t)

−1

H (ν)B1 (ν)esν dν+

−1

H (ν)B1 (ν)h(ν + γ )dνψ−1 (s)+ 0

t

(22.22)

−1

H (ν)B(ν)h(ν − θ )dνψ0 (s).

+H (t) θ

Given that

ψ−1 (s) = e−sT ψ0 (s),

we can combine (22.21) and (22.22) in the form

(22.23)

22 Construction of Matrix η(s, t)

233

t v(s, t) = H (t)v0 (s) + H (t)

−1

−1

H (ν)B1 (ν)esν dν+

(22.24)

0

+H (t) R˜ ψ (s, t)ψ0 (s) where

t

R˜ ψ (s, t) =

H (ν)B(ν)h˜ τ (s, ν)dν −1

(22.25)

0

and

h˜ τ (s, t) =



h(t + γ )e−sT , 0 < t < θ, h(t − θ ), θ < t < T.

(22.26)

Below we use the notation

t

G(s, t) = 0

−1

H (ν)B1 (ν)esν dν, T

G(s, T ) = G(s) =

(22.27) −1

H (ν)B1 (ν)e dν, sν

0

with the help of which, using (22.18), (22.24) can be represented as

where

v(s, t) = H (t)G(s, t) + H (t) R˜ G (s, t)G(s, T )

(22.28)



R˜ G (s, t) = e−sT K˜ v (s) + R˜ ψ (s, t) K˜ ψ (s) M.

(22.29)

By construction we have R˜ G (s, t) = R˜ G (s + k jω, t).

(22.30)

Equation (22.28) determines for 0 ≤ t ≤ T the exponentially periodic solution v(s, t) of the differential equation of the system (22.2). The corresponding output y(s, t) has the form y(s, t) = C(t)v(s, t) = C(t)H (t)G(s, t) + C(t)H (t) R˜ G (s, t)G(s, T ). (22.31) This is the EP reaction of the system Sτ to the input signal x(t) = est I when using the discrete basic controller α 0 (ζ ), β 0 (ζ ). Therefore, from (22.31) follows η(s, t) = e−st C(t)H (t)G(s, t) + e−st C(t)H (t) R˜ G (s, t)G(s, T ).

(22.32)

Chapter 23

Construction of Matrix C˜ T (s, t)

In this chapter, further preparatory calculations for the construction of the quality function are presented. 1.

Following (19.14), matrix C˜ T (s, t) is determined by ∞ 1  C˜ T (s, t) = Bξ η ψ (s + k jω, t) T k=−∞

(23.1)

where, in accordance with (19.11) Bξ η ψ (s + k jω, t) = ξ(s)η (−s, t)ψ(s, t).

(23.2)

Thereby we have, as follows from the first two equations in (17.22) and from (14.3) for m = 0, ψ(s, t) = e−st e−sT C(t)L(t)L˜N (s, t)a˜ r (s), ξ(s) = a˜  (s)C(0)G ξ (s) where 

T

G ξ (s) =

−1 D˜ N (T, s, −ν)L (ν)B1 (ν)esν dν.

(23.3)

(23.4)

0

Below we want to express ξ(s) in an explicit form. For the construction of such a form we use the obvious equation D˜ N (T, s, −ν) = D˜ N (T, s, T − ν)e−sT = D˜ N (T, s, T − ν)e−sT

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_23

(23.5)

235

23 Construction of Matrix C˜ T (s, t)

236

which is true for 0 < ν < T . With −1 D˜ N (T, s, T − ν) = (Iχ − e−sT M) e N (T −ν) ,

(23.6)

inserting (23.6) and (23.5) into (23.4), we find

G ξ (s) = e

−sT

(Iχ − e

−sT

T

−1

−1

H (ν)B1 (ν)esν dν =

M) M

(23.7)

0

−1

= e−sT (Iχ − e−sT M) M G(s, T ) where (14.23) and the relations −1

−1

e N T = M, e−N ν L (ν) = H (ν)

(23.8)

have been used. With the help of (23.3) and (23.7) we find −1

ξ(s) = e−sT a˜  (s)C(0)(Iχ − e−sT M) M G(s, T ).

(23.9)

Using the ILMFD (17.5) C(0)(Iχ − ζ M)

−1

−1

= a (ζ )b (ζ ),

we obtain after the substitution of e−sT for ζ a˜  (s)C(0)(Iχ − e−sT M)

−1



= b˜ (s) = b (ζ )

ζ =e−sT

,

(23.10)

which with the help of (23.9) leads to ξ(s) = e−sT b˜ (s)M G(s, T ).

2.

(23.11)

From (22.32) we come to

η (−s, t) = est G  (−s, t)H  (t)C  (t) + est G  (−s, t) R˜ G (−s, T )H  (t)C  (t). (23.12) By inserting (23.3), (23.11) and (23.12) into (23.2) we obtain 

  Bξ η ψ (s, t) = ξ(s)η (−s, t)ψ(s, t) =  = e−2sT b˜ (s)M (G(s, T )G  (−s, t) + G(s, T )G  (−s, T ) R˜ G (−s, t) × (23.13) × H  (t)C  (t)ψ˜ 1 (s, t)

23 Construction of Matrix C˜ T (s, t)

with

237

 ψ˜ 1 (s, t) = −C(t)L(t)L˜N (s, t)a˜ r (s).

If we insert (23.13) into (23.1) and regard that matrices marked by a tilde are periodic with respect to s with a period of jω, we get   C˜ T (s, t) = e−2sT b˜ (s)M σ (s, t) + σ (s, T ) R˜ G (−s, t) H  (t)C  (t)ψ˜ 1 (s, t) (23.14) where σ (s, t) and σ (s, T ) are the series sums  1 T

∞ 

σ (s, t) =

 1 T

σ (s, T ) =

G(s + k jω, T )G  (−s − k jω, t),

k=−∞ ∞ 

G(s + k jω, T )G  (−s − k jω, T )

(23.15)

k=−∞

and matrices G(s, t) and G(s, T ) are defined by (22.27). Closed formulas for the sums of series (23.15) gives Theorem 23.1 (a) Matrix σ (s, t) does not depend on s and for 0 < t < T is given by t    −1  −1  H (ν)B1 (ν) H (ν)B1 (ν) dν. σ (s, t) = σ (t) = (23.16) 0

(b) The equality is true 

σ (s, T ) = σ (T ) =

T 

−1

H (ν)B1 (ν)



 −1 H (ν)B1 (ν) dν.

(23.17)

0

 Proof Let us have the matrices 

T

F(s) =

f (ν)e

−sν

dν,

0



T

L(s) =

(ν)e−sν ,

(23.18)

0

for which all conditions of Lemma 19.3 are met. Then from (19.33) follows D˜ F L  (T, s, 0) =

T f (ν)(ν)dν = const. 0

(23.19)

23 Construction of Matrix C˜ T (s, t)

238

We introduce the notation



−1

f (ν) = H (ν)B1 (ν),  f (ν), 0 < ν < t,  (ν) = 0, t < ν < t.

(23.20) (23.21)

Moreover, taking into account (22.27) and (23.1), we come to (23.16). If we suppose that t = T in (23.16), we come to (23.17).  3. As follows from Theorem 23.1, matrices σ (s, t) = σ (t), σ (s, T ) = σ (T ) do not depend on the argument s. Thus, (23.14) can be written in the form   C˜ T (s, t) = e−2sT b˜ (s)M σ (t) + σ (T ) R˜ G (−s, t) H  (t)C  (t)ψ˜ 1 (s, t).

(23.22)

Chapter 24

Transformation of the Quality Functional

In this chapter the quality function to be minimized is reduced into the form of an integral quadratic functional on the unit circle. 1. In Chap. 19 it is shown that the H2 -optimization problem for the system Sτ leads to the minimization of the functional (19.60) jω

2 JT =

1 2π j

 ˜ A˜ ξ (s)− tr Θ˜  (−s) A˜ ψ (s)Θ(s)

− jω 2

(24.1)



˜ ds − Θ  (−s)C˜ T (−s) − C˜ T (s)Θ(s) with regard to the system function ˜ Θ(s) = Θ(ζ )|ζ =e−sT

(24.2)

where Θ(ζ ) is an arbitrary stable rational matrix. Matrices A˜ ψ (s), A˜ ξ (s), C˜ T (s) involved in (24.1) are given by (19.15), (19.18), (19.52) and (19.61) A˜ ψ (s) = A˜ ξ (s) =

α  k=−α β  k=−β

ak e−ksT , a−k = ak , bk e−ksT , b−k = bk ,

C˜ T (s) =

δ 

(24.3)

ck e−ksT .

k=−γ

Here ak , bk , ck are real matrices, α, β, γ , δ are non-negative integers. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_24

239

240

2.

24 Transformation of the Quality Functional

We introduce in (24.1) the new integration variable

Obviously, we have

ζ = e−sT .

(24.4)

dζ = −T e−sT ds

(24.5)

and, therefore, ds = −

1 dζ . T ζ

(24.6)

When s changes over the interval − jω ≤ (s) < jω , the variable ζ moves in the 2 2 complex plane on the unit circle |ζ | = 1 in clockwise direction. Therefore, inserting (24.4) and (24.6) into (24.1), we come to    dζ ˆ )Cˆ T (ζ ) − C T (ζ )Θ(ζ ) (24.7) T JT = tr Θ(ζ )Aψ (ζ )Θ(ζ )Aξ (ζ ) − Θ(ζ ζ where the contour integral must be calculated over the circle |ζ | = 1 in positive direction (counterclockwise). The matrices Aψ (ζ ), Aξ (ζ ), C T (ζ ) appearing in (24.7) are quasi-polynomials and according to (24.4), (24.3) have the form Aψ (ζ ) =

α  k=−α

ak ζ , k

Aξ (ζ ) =

β 

bk ζ , C T (ζ ) = k

k=−β

δ 

ck ζ k .

(24.8)

k=−γ

3. Theorem 24.1 The quasi-polynomials Aψ (ζ ) and Aξ (ζ ) are symmetric and nonnegative on the circle |ζ | = 1.  Proof The symmetry of quasi-polynomials Aψ (ζ ) and Aξ (ζ ) immediately follows from (24.3). For the proof of non-negativity of Aψ (ζ ) on the circle |ζ | = 1 we note that due to (19.61) T 1 (24.9) A˜ ψ (s, t)dt A˜ ψ (s) = T 0

where the matrix A˜ ψ (s, t) is determined by (19.10) A˜ ψ (s, t) = a˜ r (−s)L˜N (−s, t)L  (t)C  (t)C(t)L(t)L˜N (s, t)a˜ r (s).

24 Transformation of the Quality Functional

241

After performing the substitution of the variable (24.4), from the last expression we find Aψ (ζ, t) = A˜ ψ (s, t)|e−sT =ζ =   (24.10)  = C(t)L(t)L N (ζ, t)ar (ζ ) [C(t)L(t)L N (ζ, t)ar (ζ )] .

From this and Theorem 20.2 follows that matrix Aψ (ζ, t) = A˜ ψ (s, t)|e−sT =ζ is a matrix quasi-polynomial, non-negative for |ζ | = 1. Since from (24.9) for ζ = e−sT we have T 1 Aψ (ζ ) = Aψ (ζ, t)dt, (24.11) T 0

the matrix Aψ (ζ ) is a symmetric matrix quasi-polynomial, non-negative for |ζ | = 1. It remains to prove the statement of Theorem 24.1 for the matrix Aξ (ζ ). For this we use formula (23.11) ξ(s) = e−sT b˜ (s)M G(s, T ). Hence,

and

ξ  (−s) = esT G(−s, T )M  b˜ (−s)

(24.12)

ξ(s)ξ  (−s) = b˜ (s)M G(s, T )G  (−s, T )M  b˜ (−s).

(24.13)

∞ 1  A˜ ξ (s) = ξ(s + k jω)ξ  (−s − k jω), T k=−∞

(24.14)

Since

then, using (23.11) and (23.17), we get A˜ ξ (s) = b˜ (s)Mσ (T )M  b˜ (−s).

(24.15)

When replacing e−sT with ζ , this yields −1

Aξ (ζ ) = b (ζ )Mσ (T )M  b (ζ ).

(24.16)

It immediately follows from (24.16) that matrix Aξ (ζ ) is a symmetric quasipolynomial. Given (23.17), we can represent (24.16) in the form T    −1 b (ζ )M H (ν)B1 (ν) b (ζ )M Aξ (ζ ) = H −1 (ν)B1 (ν) dν. 0

(24.17)

242

24 Transformation of the Quality Functional

From here we can be conclude that the quasi-polynomial Aξ (ζ ) is non-negative on the circle |ζ | = 1. Moreover, if det σ (T ) = 0, then the quasi-polynomial Aξ (ζ ) is positive on this circle. 

Chapter 25

H2 -Optimization of the System Sτ

The chapter describes the general procedure for the H2 -optimization of the system Sτ and provides an example of solving the H2 -optimization problem for a first-order FDLCP object.

25.1 General Solution of the H2 -Optimization Task 1. Theorem 25.1 Let the system Sτ be modal controllable and the quasi-polynomials Aψ (ζ ) and Aξ (ζ ) positive on the circle |ζ | = 1. Then the following statements can be made: (a) There exist factorizations ˆ )Π (ζ ), Aξ (ζ ) = Γ (ζ )Γˆ (ζ ) Aψ (ζ ) = Π(ζ

(25.1)

where Π (ζ ) and Γ (ζ ) are stable polynomial matrices. (b) Functional (24.7) can be represented in the form  T JT =

  dζ ˆ )Π(ζ ˆ )Π (ζ )Θ(ζ )Γ (ζ ) − Θ(ζ ˆ )Cˆ T (ζ ) − C T (ζ )Θ(ζ ˆ ) . tr Γˆ (ζ )Θ(ζ ζ

(25.2) (c) There is an optimal stable system function Θ 0 (ζ ) that minimizes the functional (25.2). This matrix can be constructed using the algorithm of Theorem 21.1, which is described by the formulas (21.3)–(21.5).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 E. N. Rosenwasser et al., Sampled-Data Control for Periodic Objects, https://doi.org/10.1007/978-3-031-01956-2_25

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25 H2 -Optimization of the System Sτ

244

(d) If we have a valid ILMFD −1

Θ 0 = D0 (ζ )M0 (ζ )

(25.3)

for the optimal matrix Θ 0 (ζ ), then the characteristic polynomial Δ0 (ζ ) of the H2 -optimal closed-loop system satisfies Δ0 (ζ ) ∼ det D0 (ζ ).

(25.4)

Proof (a) The existence of factorizations (25.1) follows from Theorem 20.2. (b) Relation (25.2) is obtained from (24.7) using equality     ˆ )Π(ζ ˆ )Π (ζ )Θ(ζ )Γ (ζ ) . (25.5) ˆ )Aψ (ζ )Θ(ζ )Aξ (ζ ) = tr Γˆ (ζ )Θ(ζ tr Θ(ζ (c) This statement follows from Theorem 21.1, since the functionals (21.1) and (25.2) coincide in form and all the assumptions of Theorem 21.1 are fulfilled. (d) This statement follows from the general relations of Chap. 16, since if (25.3) holds and we use the basic controller α0 (ζ ), β0 (ζ ), the H2 -optimized discrete controller α0 (ζ ), β0 (ζ ) is given by α0 = D0 (ζ )α0 (ζ ) − ζ m+1 M0 (ζ )b (ζ )Mb(ζ ), 0 (ζ ) = D0 (ζ )β0 (ζ ) − M0 (ζ )a (ζ ) where matrices a (ζ ) and b (ζ ) are defined from ILMFD C(0)(Iχ − ζ M)

−1

−1

= a (ζ )b (ζ )

and matrix b(ζ ) is defined by (11.63).

25.2 General Solution of the H2 -Optimization Task for a First-Order Periodic Object 1. In this section, as an example of applying the given theory, we consider the H2 optimization problem for the closed-loop system Sτ , in which the FDLCP object is described by scalar state equations dv(t) dt

= a(t)v(t) + b1 (t)x(t) + b(t)u(t − τ ) y(t) = c(t)v(t)

(25.6)

25.2 General Solution of the H2 -Optimization Task for a First-Order Periodic Object

245

with periodic continuous functions a(t) = a(t + T ), b1 (t) = b1 (t + T ), b(t) = b(t + T ), c(t) = c(t + T ). In addition, the function b1 (t) is of bounded variation in the interval 0 ≤ t ≤ T . To simplify the calculations it is also assumed that m = 0, i.e., τ = θ = T − γ , 0 ≤ θ < T, 0 < γ ≥ T. (25.7) For an FDLCP object of first order we have H (t) = (t)e N t ,

−1

H (t) =

e−N t , (t)

(25.8)

where (t) is a continuously differentiable real function, such that |(t)| ≥ const > 0 and N is real number. Using the Floquet–Lyapunov transformation v(t) = (t)v L (t),

(25.9)

from (25.6) we can obtain corresponding state equations dv L (t) dt

1 (t) = N v(t) + b(t) x(t) + b(t) u(t − τ ), (t) y(t) = c(t)(t)v L (t).

(25.10)

2. The equations of the closed-loop system Sτ containing the object (25.10) according to (7.6) have the form dv L (t) dt

1 (t) = N v L (t) + b(t) x(t) + u θ (t − θ ), y(t) = c(t)(t)v L (t), ξk = c(0)v L (kT ), (k = 0, ±1, ...), α(ζ )ψk = β(ζ )ξk , u θ (t) = h θ (t − kT )ψk

where due to (7.4) u θ (t) =

b(t + θ ) h(t), 0 < t < T. (t + θ )

(25.11)

(25.12)

3. Using (11.62) we find that under the taken suppositions, the discrete model for the system Sτ (25.11) has the following form a(ζ )vk = ζ Mb(ζ )ψk , ξk = C(0)vk , α(ζ )ψk = β(ζ )ξk where

(25.13)

25 H2 -Optimization of the System Sτ

246

a(ζ ) = (1 − ζ M), b(ζ ) = ζ MΓ2 (θ ) + Γ1 (θ ),

M = eN T

(25.14)

and it has been taken into account that (0) = 1.

(25.15)

The functions Γ1 (θ ) and Γ2 (θ ) appearing in (25.14) are determined by (11.60) γ

Γ1 (θ ) =

−1

H (μ + θ )b(μ + θ )h(μ)dμ,

0

T

Γ2 (θ ) =

γ

(25.16)

−1

H (μ + θ )b(μ + θ )h(μ)dμ.

4. According to (14.11) the characteristic matrix of the discrete model (25.13) has the form ⎡ ⎤ 1 − ζ M 0 −ζ Mb(ζ ) ⎦. 1 0 (25.17) Q τ (ζ, α, β) = ⎣ −c(0) 0 −β(ζ ) α(ζ ) For the modal controllability of the system Sτ in the considered case it is necessary and sufficient that the polynomial pairs (1 − ζ M, ζ Mb(ζ )), [1 − ζ M, c(0)] are irreducible. Since by assumption c(0) = 0, the pair [1 − ζ M, c(0)] is irreducible. In addition, for the irreducibility of the pair (1 − ζ M, ζ Mb(ζ )) it is necessary and sufficient that the pair (M, Γ (θ )) of constant matrices with

T

−1

H (ν + θ )b(ν + θ )h(ν)dν

Γ (θ ) =

(25.18)

0

is controllable. It is not difficult to show that this occurs if and only if we have Γ (θ ) = 0.

(25.19)

Below it is always supposed that the system under study Sτ is modal controllable. 5.

In the considered case the FDLCP object has the discrete transfer function WT (ζ ) =

ζ c(0)Mb(ζ ) . 1−ζM

(25.20)

Under conditions of modal controllability, the function WT (ζ ) is irreducible. Therefore, for the scalar case under consideration we have

25.2 General Solution of the H2 -Optimization Task for a First-Order Periodic Object

det Q τ (ζ, α, β) = det Q τ 1 (ζ, α, β) 

where Q τ 1 (ζ, α, β) =

1 − ζ M −ζ C(0)Mb(ζ ) . −β(ζ ) α(ζ )

247

(25.21)

(25.22)

We find a left basic controller α0 (ζ ), β0 (ζ ) from the condition det Q τ 1 (ζ, α0 , β0 ) = 1

(25.23)

which is equivalent to the Diophantine polynomial equation (1 − ζ M)α0 (ζ ) − ζ c(0)M(ζ MΓ2 (θ ) + Γ1 (ζ )) = 1.

(25.24)

This equation is solvable due to the irreducibility of fraction (25.20). The solution of (25.24) with minimal degree is Δ

α0 (ζ ) = α 0 + α 1 ζ, β0 (ζ ) = const = β0

(25.25)

where α 0 , α 1 , β0 are constants to be determined. Inserting (25.25) into (25.24), we come to the equality (1 − ζ M)(α 0 + ζ α 1 ) − ζ c(0)M [ζ MΓ2 (θ ) + Γ1 (θ )] β0 = 1

(25.26)

that should hold for all ζ . Setting ζ = 0, we immediately obtain α 0 = 1. Assuming ζ = M

−1

(25.27)

and considering that Γ1 (θ ) + Γ2 (θ ) = Γ (θ ), we get that β0 = −

1 . c(0)Γ (θ )

(25.28)

Besides, from (25.26) and (25.28) we obtain easily α1 =

MΓ2 (θ ) . Γ (θ )

(25.29)

When we choose the left basic controller in the form (25.25), (25.27)–(25.29), we have with (25.22) 

1 − ζ M −ζ c(0)Mb(ζ ) α0 (ζ ) −β0 (ζ )

−1



α0 (ζ ) ζ c(0)Mb(ζ ) = . 1−ζM β0

(25.30)

25 H2 -Optimization of the System Sτ

248

From here we get the equalities a (ζ ) = ar (ζ ) = aθ (ζ ) = 1 − ζ M 1 βr 0 (ζ ) = −β0 (ζ ) = C(0)Γ (θ) αr 0 (ζ ) = α0 (ζ ).

(25.31)

Moreover, when using the basic controller (25.25), (25.27)–(25.27), from (25.23) it follows that −1 (25.32) det Q τ (ζ, αr 0 , βr 0 ) = 1.

6. Using the above relations, we proceed to the calculation of the quasi-polynomials Aψ (ζ ), Aξ (ζ ), C T (ζ ) that appear in the quality functional (24.7). To calculate the quasi-polynomial Aψ (ζ ), we note that from (24.9) follows Aψ (ζ ) =

1 T

T

A˜ ψ (s, t)dt| e−sT =ζ

(25.33)

0

where due to (19.10) we have for the scalar case A˜ ψ (s, t) = a˜ r (−s)L˜N (−s, t)L˜N (s, t)ar (s)c2 (t)2 (t). Since from (13.6), (13.9) follows ⎧   Γ (θ) ⎨ Me N t − Γ (t, θ ) , 0 pairs of polynomial matrices, 119 regular, 115 singular, 116 spectrum, 116 square, 115 Sylvester rank inequality, 116 unimodular, 116 vertical, 115 Polynomial matrix description (PMD), 138 causal, 140 homogeneous, 139 non-singular, 140 stabilizable, 141 strictly causal, 140 Polynomial pairs, 129 degree, 130 equivalence, 130 nonsingular, 129 related, 129

R Rational matrices, 123 denominator, 124 excess, 127 improper, 128 irreduible, 124 McMillan canonical form, 125 independence, 126 McMillan denominator, 125 McMillan index, 125 McMillan multiplicity, 125 McMillan numerator, 125 multiplicities, 124 numerator, 124 poles, 124

Index proper, 128 proper at least, 128 realization of a strictly proper rational matrix, 128 minimal realization, 128 order, 128 standard form, 124 strictly proper, 128

S SD system asynchronous declining mode, 75 asynchronous mode, 75

263 asynchronous rising mode, 75 synchronous mode, 72, 75 Stabilization problem, 136 Stroboscopic property, 104, 109, 181

T T-periodic signal, 20, 22 Transfer matrix, 134

Z Zadeh’s equation, 47 Zero-order hold, 63