An Introduction to the Theory of Linear Systems [1, 1 ed.]

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An Introduction to the Theory of Linear Systems

An Introduction to the Theory of Linear Systems

R. Fratila U.S. Naval Electronic Sy stems Command, Washington, D.C.

For sale by the Superintendent of D ocuments, 1.:.s. Government Printing.Office Washington, D .C. 20402 - Price $1.60 Stock No . 008--050-0:1143-i

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Whoever undertakes to set himself up as judge in the field of Truth and Knowledge is shipwrecked by the laughter of the gods. Einstein

Acknowledgement

The suggestions and constructive comment offered by the individuals listed below helped make this volume more credible. Their efforts are greatly appreciated .

1. S. Lawson, Jr. M. I. Skolnik 1. C. Tieman L. Young L. A. Zadeh

Preface

Until the past decade the tenn "system" as used in the many disciplines of science and engineering conveyed a variety of meaning . Since the early 1960's theorists have been folding these varied impressions into the single, fonnidable conceptual framework of linear system theory. The discussions of this volume are intended to lend to this unification. Specifically. the intent is to draw on the conceptual ideas that are common to linear filters, linear time-invariant systems, Markov chains and quantum theory and present , in a non-rigorous manner. their respective similarities. Chapter 1 deals with the fundamental concepts upon which the definin·on of a system can be established. Such notions as ordered pairs , oriented abstract objects, etc. , serve as the basics of the definiti on. The system response is defined in terms of an input-output-state relationship which must satisfy certain consistency conditions. The system definition of Chapter 1 is built upon in Chapters 2. 3 and 4 . In Chapter 2 the conditions for consistency are re-cast in the form of demonstrating the separation property inherent in the input-output-state relationship.

vi

PREFACE

In Chapter 3 the state-space representation is formalized for bnear differential systems, and Chapter 4 highlights the canonical form of the state-space representation, i.e. x = ·Ax + Bu y =Cx+Du

In Chapter 5 solutions to the canonical equations are presented for a variety of systems. A natural fall-out of the state-space formulation are the theoretical concepts of controlability , observability and stability , which are addressed in Chapter 6. In the last two chapters the reader is reminded that the linear theory developed thus far has limited application to real world problems, such as treating large complex systems. The objective of Chapter 7 is to introduce the idea of an imprecisely defined (or probabilistic) system. The methods of extracting signals from noise , both of which are treated as random processes , are discussed as a means to augment the linear theory. Finally , Chapter 8, which deals with quantized systems, ends this volume by highlighting the perturbation method for treating state transition probabilities. It exemplifies the high degree of difficulty encountered in analyzing statistical systems.

Contents

CHAPTER 1

Preface

V

Fundamental Concepts and Definitions

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14

Introduction Time Functions Objects, Attributes and Terminal Relations Oriented Model Notion of State State-Input-Output Relations Condition I, M utuaJ-Consistency Condition II , First Self-Consistency Condition III , Second Self-Consistency Condition IV , Third Self-Consistency State of Time t Multiple Objects Interconnected Objects Tandem Combinations- Ini tially Free

l 2 4 5 9 9

IO 11 12 15 16 18 19 20 VII

""'

CHAPTER 2

CONTENTS

1. l5 Tandem Combinations- Constrained 1.16 Svstem

23 24

State Equations

25

2.1 2.2 2.3

CHAPTER 3

3.4 3.5 3.6 3.7

,_)

28

31

Introduction Time Invariance linearity Zero State or Impulse Response Zero In put Response and Basis Functions Input-Ou tpu t-State Relation and Basis Functions State Equations

4.2 4.3 4.4 4.5 4.6

5.2 5.3 5.4 5 .5 5.6

33 35 38 40 42

46 47 52

I.n troduction Reci procal Differential System Differential Operator S~'stem General Soluti l)n System Realiza tion and Equivalence Method of Partial fractions

54

57 61

68

Solutions to the Canonical Equations

5. 1

31 32

46

Canonical Formulation 4.1

CHAPTER 5

25

Time Invariance, Linearity and Basis Functions

3.1 3.2 3.3

CHAPTER 4

In trodu ction Separation Property State Equati ons

Fixed Con tinuous-Tirne System : Time Domain A.J13lysis Fixed Continu0us-Time Systems : Freque ncy Domain Analysis Fixed Discrete-Time Svstem : Time Doma in Analvsis Fixed CGntinuous-Time Systems: Discrete Inputs Fixed Discrete-Time Systems: ::: Domain Analysis Finite State .Marki.)\' Systems: Input -Free J

•

68 71

78 79

84 90

ix

CONTENTS

5.7 5.8

CHAPTER 6

Controllability, Observability and Stability 6.1 6.2 6.3 6.4 6.5 6.6 6.7

CHAPTER 7

Time-Varying Continuous-Time Systems: Time Domain Analysis Time-Varying Discrete-Time Systems : Time Domain Analysis

Notation Spectral Decomposition Controllability Observability Stability-Un[orced System Stability-Forced System Comment

10 l

103

103 106 107 1 10 11 2 116 120

Statistical Systems-Signals in Noise

122

Introduction Averages (Mean Value) and Correlation Fun ction Filtering Problem Optimum Linear Filter Spectral Density of Signal and Noise Power Spectral Density and Correlation Fun ction Correlation Time-Pass Band Product Transfer Functions and Linear Operators 7.9 Monochromatic Signals 7.10 Signals of Known Form in Noise 7.11 The Matched Filter

122 123 125 126 133 133 137 138 142 148

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

CHAPTER 8

97

Quantized Systems-Perturbation Theory and State Transitions

8.1 8.2 8.3 8.4 8.5 8.6

Introduction Fundamental Postulate s of Quantum Theory Energy Picture Time-Dependent Perturbations; State Transiti ons Constant Perturbation Exponenti al Decay

l5l

159

I 59

160 164 166 17 1 17 3

CONTENTS

X

APPENDIX A Dirac Delta Function and the Unit lmpube

177

APPENDIX B

Resolution of Continuous-Time Signals into Unit Impulses

179

APPENDIX C

Discrete-Time State Equations

180

APPENDIX D Z Transforms

182

APPENDIX E Analogous Quantities of Continuous-Time and DiscreteTime Systems

184

APPENDIX F Stochastic Processes

185

BIBLIOGRAPHY

191

INDEX

195

1 Fundamental Concepts and Definitions

1.1

INTRODUCTION

Linear system theory is intended as a discipline to provide a unified conceptual framework for system analysis . In establishing this framework we will introduce such notions as abstract objects , their measurable attributes. and the mathematical relations between attributes. These concepts will serve as the primitives of the theory , although they defy precise definition in unequivocal terms . Our goal will be to identify a small part of the physical world we intuitively understand and attempt to evolve a quantitative basis for analysis. The proof of this quantitative understanding will be manifest in our ability to better describe the behavior of linear differential systems. We think of a system (in vague terms) as being a collection of things or objects which somehow are united through interaction or interdependence. More precisely . we define 1 a system to be a partially interconnected set of abstract l. After Zadeh and Desoer , Linear Sy stem Th eo ry. McGraw-Hill , New York. 196 3, pp. 1-65. 1

2

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

objects , wltich are to is uniquely determined by x and u(to. t ] . This must be true fo r all to. To qualify as a st ate space of lt, the spa ce '¥ must have the property that. given any point x in '¥ and any input u( to. t] (defined over the input segment space), the output at time tis uniquel y dete rmined by x

AN INTRODUCTION TO TKE THEORY OF LINEAR SYSTEMS

12

and u. The point x at time to will be called the state of Ct at t9. The output is independent of u or y prior to time to- Th.is is a key property of the state space concept. An input-output-state relation of form (1.9) dearly satisfies the first selfconsistency condition-regardJess of the range of a. (For all to the ootput y at time tis uniquely determined by a and u. y is independent of u or y prior to to.) However. by changing the upper limit of the integral (1.9) from t tot+ 1, i.e. , if we have

y(t) = a +

l

t+l

u(t)d~

r > r0

to

we no longer satisfy the first self-consistency condition. Th-e output at time t cannot be determined without knowledge of u(~) between t and t + 1.

1.9 CONDITION Ill, SECOND SELF.CONSISTENCY

If the input-output pair {u[to,ttl , y[to ,td) satisfies {1.6), then (u[t,ti] , y[t.ti]) also satisfies (1.7). u[t.ti) and y[t,ti) are sections of u[to ,tt1 and y[to, td , respectively , and to~ t ~ t 1 . This must hold for all x in 'I', all u[to, ttl in the input segment space , and all to , t , t1- The purpose of this condition is to ensure that the state space 'Ir can include all possible initial

er.

conditions for To help clarify the meaning of the second self-consistency condition consider the input to Ct over ( to ,r 1 ] as consisting of two contiguous segments , u 0 (to t] follo~ved by u 1 {t,ti) , i.e. ,

(Note : u0 u 1 is not to be interpreted as the product of u0 and u 1 .) If the input -output pair (u 0 u 1 . v0 v1 ) satisfies (I .6), then the segment response to u0 u 1 beginning in state x(t o) = xo is

{1.10) To say that (u1 . y 1) also satisfies {1.6 ). as the second self-consistency condition does. means that there must be some values of x in + such that yl = F(x ;u 1)

for all arbitrary times t. where t o

< t ~ t 1.

(1.11)

Denoting these values of x in '¥ as

FUNDAMENTAL CONCEPTS ANO DEFINITIONS

13

the set A the second self-consistency condition requires that this set be nonempty. (Figure 1.4 graphically illustrates the idea.) To further illustrate the second self-consistency condition let Ct be characterized by the relation t1

J

y (~) = 0'. 2 +

u(~) dt

(1.12)

to

u and y are real-valued time functions and 'V includes the set of all real numbers, i.e ., O'. = x is any real number. Let's consider the value of y at the particular time ~ = t. For~= t equation (1.12) becomes

y(t) = 0'. 2

+

J,

ti

(1.13)

u(t ) dt

to

Let the input u(t) over the observation interval (0 ,5] be u(t) = - 3t 2 and let the initial conditions at time to = 0 be xo = O'.o = I . From (1.13) the response over [0 ,5] is

Thus , the input-output pair associated with (-3t 2 , I - t 3 ).

xo =

1 for the interval [0,5) 1s

Time

-, ----i---f,, ........

x0

I

}A

--

. . 1,.---

--1- --- -

I

I

fI I Time

Figure 1.4

Val ue s of x in "1 .

14

AN INTRODUCTION TO THE THEORY O-F LINEAR SYSTEMS

Now consider the intermediate time t = 2. The second self-consistency condition requires that if the input-0utput pair (u(0,5),y(0,5] ), which is (-3r 2 , I- r 3 ) , satisfies (1.6), then the same pair over the time interval (2 ,5] also must satisfy (I .6 ). Since Ct is characterized by (1.12), we have for (2 15) ,

y(t) =

o}

+

Ii

5

-3t 2dt

(1.14)

Also , from (1.12) we can write for the pair (-3t2, I- r 3 ) ·

(1.15)

The only way the pair (- 3t 2 , 1- t 3 ) can satisfy (I .14) is if

= -7

Clearly , a 2 = - 7 has no solution in \}I , where '11 consists of real numbers. Therefore A is empty and the second self-consistency condition is not satisfied by relation (1.12). On the other hand , if (1 were characterized by an input-output-state relation of the form y(t) = a +

J

t u(~)d~

(1.16)

to

instead of ( 1.12 ), then for any xo in'¥ and any~ in (to ,t1] , where tosu(nd~

-00

(5.7) Evaluating (5 .7) at t = to gives the initial state as rto

x(to) =

J-~ e- A(Ho>su(nd~

The refore, solution (5 .7) becomes

J t

x(t) = eAU - tolx(to) +

eA(r-t>su(nd~

(5 .8)

1(t

(5.9)

to

= (t - to)x(to)

f

+

- ~) Bu{nd~

to

For fixed systems it is convenient to establish (5.8) and (5.9) become

to = 0, in

which case equations

1 t

x(t ) =

eA 1x(O)

+

eA(l-t)Bu(nd~

(5.10)

0

=

(t) x(O) +

r

(t - nsu(~)d~

0

(5.1 I)

71

SOLUTIONS TO THE CANONICAL EQUATIONS

Finally , on substituting (5.11) into (2.27) the input-output-state relation for a continuous-time system is

(5.12)

Equations (5.11) and (5.12) are the system state equations. It is clear from (5 .11) that the system state at time t can be determined if the system state at some previous time to (to < t) is known and if the input u(t) is known. The manner in which the initial state is "transformed" is cha racte rized by the makeup of and how the input is applied , i.e .. by the matric operators A and B. The system output (5 .12) reflects a dependency on all four operators.

5.2

FIXED CONTINUOUS-TIME SYSTEMS: FREQUENCY DOMAIN ANALYSIS

For fixed continuous-time systems the diffe rential equations to be solved are x =Ax+ Bu y =

Cx + Du

where A . 8 C and D are constant matrices. From previous discussions the method of Laplace transforms was seen to be a convenient method for solving equations of this form. It was established in (4.48) and (4.49) that the Laplace transform of both sides of the above equations gives : respectively. 1

sX(s) - x(O) = AX(s) + BU(s) Y(s) = CX(s)

+ DU(s)

From (4.48) X(s) = (sl -

Ar 1 x(O)

=

=

Xn (s)

+ (s l -

Ar 1BU(s)

(5 .1 3)

(5. 14)

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

72

which, when substituted in ( 4.49), resulted in the input-output equation Y(s) = C{sl - Af 1x(0) + [C{sl - A)- 1B + D]U(s)

Yi (s) =

(5.15)

=

The zero-state response , i.e. , x(O) = 0 , identifies the transfer function matrix H{s) in (4.50) as H{s) = C(sl - Af 1B + D

(5.16)

(5.17)

=

H,z1 (s) Hn2(s) ... Hn,1 (s) Writing the zero-state response as

Y(s) = H(s)U(s)

(5.18)

the ith component of the transform vector Y(s) is (5.19)

It is apparent from (5.19) that Hu(s) is the transfer function between input uj(t) and output Yi(!). In Section 3.4 hu(t) was identified as the response at the ith output terminal due to a unit impulse applied at the jth input terminal. We conclude. therefore, that (5.2 0)

Equa ti on (5 .10) gave the general time-domain solution fo r a fixed system as

x(t ) = (l)x(O) +

r

{t - OBu(~M

0

73

SOLUTIONS TO THE CANONICAL EQUATIONS

where , for convenience , to = 0. Applying the theorem for the Laplacee transform of a convolution the Laplace transform of the above equation is X(s) = ~s)x(O) + ~s)BX(s)

(5.21)

(k) x(O) +

L (Q) Bu(k - Q- I ) Q=O

(5.56)

or k-l

x(k) = (k) x(O) +

L (k-Q-I) Bu(k)

Q=O

(5.5 7)

Substituting (5.56) into (5 .5 1) the output equation becomes k-l

y(k) = C(k) x(O) +

L C(Q) Bu(k - Q- I )

Q=O

+ Du(k)

(5.58)

An alternate expression for the output is readily derived using the state vector

fom1 (5.57). Solutions (5.56) and (5.58) are the discrete-time analogs of(S. l I) and (5 .1 2), respectively. The corresponding formulation of the discre te-time state equat ions is given in Appendix C.

5.4

FIXED CONTINUOUS-TIME SYSTEMS : DISCRETE INPUTS

In many continuous-time systems the input is often a sam pled signal. thus characterizing the input as discrete. There are two kinds of sampled signals (and

80

AN INTRODlJCTION TO THE THEORY OF LINEAR SYSTEMS

associated discreteness) we wiU consider: (I) the piece-wise constant signals where the samplin g function is a pulse of constant amplitude over a time interval ( tk Jk+I ) , i.e. ,

(5.59) and (2) the impulse-modulated signal represented by a modulated delta-function series 00

(5.60)

Signals having the forms of (5 .59) and (5 .60) are respectively depicted in Figs. 5 .1 a and 5 .1 b. In generating u(tk) a zero order holding circuit of sorts is implied. The various lk are arbitrary but satisfy the condition tk+ 1 > tk. From previous discussions the response of a continuous-time system to sampled inputs can be readily determined. In fact , it can be determined ·.vith considerably more ease than would be required to determine the response for all time. The continuous-time system is governed by the dynamic equation

x=

+

Ax

Bu

wherein the solution is

x(t) = (t)x(to) +

I

t

(t -

nsumdt

to

~ I I

f---!

!'G

,....JII a. - - - - '

I

:

~

I

:

I

I

Ill

I

I

--u(t,1:J

I

~ I

I

I

I

I

I

I

I

I

I

I I

u(t)

b.

.___

__.___.__....__.......___

___._ _..___.__..............____.____._

,, t 2 .. . . . . . .. . . .

'* ... t,1: +1

Figure 5.1

Sampled signal .

__.__ _ _ _ _ _ t

81

SOLUTIONS TO THE CANONICAL EQUATIONS

Let the input be the sampled signal of (5.59). It is a signal of constant amplitude over a prescribed time interval. Then

J'

x(t) = (t)x(to) +

(t -

0 Bu(tk )d~

(5.61)

to

Now let the input consist of a series of constant vectors of form (5.59). Then at time tk + T. where O < T ~ fk+ I - fk for each time interval (tk Jk+ 1 ), we have

(5.62)

If the "staircase" input changes uniformly at intervals becomes

1

tk =

kT equation (5.62)

kT+r

x(kT + r) = (r)x(kD +

(kT +

T-

OBu(kDd~

(5.63)

kT

Finally i f or r

===

T

J

kT+T

x(kT+ T) = (T)x(kT) +

(kT+ T- OBu(kT)d~

(5.64)

kT

Clearly ( 5 .64) can be written as the difference equation x(k + I) = Ax(k) + Bu(k)

(5.65)

= {T) = eAt

(5.66)

where A

,..T

B =

J 4>(T- ~)d~ 0

Thus. the continuous-time system governed by

x=

Ax + Bu

(5.67)

82

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

having as the input a piece-wise constant signal is equivalent at sampling instants to the system governed by the difference equation (5 .65). To illustrate some of the theory developed consider a simple system governed by the differential equation 2

d + 5 !!_ + ( dt2 dt

4) y (t) = u(t)

Let the input to the system be a continuous-time signal in one case, and in another case let it be the sampled signal u(t) = 2k for kT < t ~ (k + I )T, where k = 0 , 1,2 , ... , 00 . From equation (4.19) the system matrices are

C =

[I]

D = 0

The state transition matrix (t) is the inverse Laplace Transform of (sl - A)- 1 , where

s

(sl - At 1

- [ -4 = [s+ 5

-4

I

]-l

s +5 1] 1 s (s+4Xs+l)

By (5.39) and (5.44) we can evaluate the inverse transform of each polynomial element of the above matrix giving

For the case of a continuous-time input the state vector is , from (5.10), 1 [ 4e-t - e-4t

e-t - e-4t ]

x(t) = -

3 - 4(e-t - e-4t) - e-t + 4e-4t 1 +3

f,

x(to)

t [ e- (t-t) _ e- 4(t- 0 ]

0

e- (t- 0 + 4e-4(t- 0

u(nd~

83

SOLUTIONS TO THE CANONICAL EQUATIONS

For the staircase input u(t) = 2k equation (5.64) specifies the state vector as the difference equation

1 [ 4e-T - e-4T

e-T - e-4T

x(kT+ T) = -

-e-T + 4e-4T

3 -4(e-T - e-47) 3

-

I 4 +-

Jx(kT)

I

- e-T + - e-4T

4

3

which is of the form

x(k +I) = Ax(k) + Bu(k) Hence, by (5.56) the state vector for the continuous-time system having a staircase input is k- l

L A B 2k-Q-l

x(k) = Akx(0) +

2

Q=O where

A and Bare determined by equations (S.66) and (S.67), respectively.

We next consider the continuous-time system wherein the input is the series of impulses represented by (S .60). Referring once again to state equation of a continuous-time system we have ... t

x(t) = (t)x(to) +

j

(t- ~)Bu(~)d~

,., to At time fk + r , where 0 above sta te equation

< r ~ tk+I -

tk.

we have on substituting (5.60) into the

(S.68) Equation (5.68) is the state equation of a continuous-time system where the input is a series of impulses occurrin g at times fk (k = 0 , 1, 2 .. .. If the impulses occur at uniformly spaced time intervals where tk+l - tk = T. equation (5.68) gives the state equa ti on for this condition as 00

x(kT+ r) = (r)x(kT) + (r)Bu(kT)

).

(5.69)

84

.

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

where 0 < r ~ T. Lastly , for r = T we have as the discrete-time state equation for a continuous-time system

x(kT + T)

cf>(_T)x(kT) + cf>(_T)Bu(kT)

=

(5.70)

5.5 FIXED DISCRETE-TIME SYSTEMS: z DOMAIN ANALYSIS

Sampled data systems can be considered as continuous-time systems operating on discrete-time functions. The Z transfom1 addre~ing discrete-time systems is introduced for the same reason that makes the Laplace transform useful in the study of continuous-time systems. Procedures similar to analyzing continuoustime systems in the frequency domain will be followed to analyze discrete-time systems in the z domain ( where z is a complex variable). The Z transform of a djscrete-time function f{k) is a power series in z- 1 . The coefficients f{k) are the an1plitudes of the discrete-time signal. We have 00

Z {f(k)}

=

L

f(k)z-k

(5.71)

k= - oo

where Z{f(k)} is the Z transform of f{k). For discrete-time systems those values of f{k) where k < 0 are of less interest than where k > 0. The discussion to follow will , therefore , center around situations of positive values of k. Accordingly , our consideration of(5.71) will be bound by the values k = 0 , l ,2 , ... , 00 . From Appendix D the Z transform of the first forward difference equation is

3 { M(k)} = Z { f(k + 1) - f(k)} =

(z - l) S{ f(k) } - zf(0)

(5.7 2)

where 00

Z {f(k)}

=

L f(k )z-k

= f(0)

+ f(I )z- l + f (2)z- 2 + ...

k=O

S { f (k+ I)} = f(l ) + f (2)z- 1 + f(3)z- 2 + ... = z(Z{ f(k)} +f (O)] (5.73) The characteriza tion of a linear discrete-time system of fixed forrn was specified earlier by equations (5 .50) and (5 .5 1):

SOLUTIONS TO THE CANONICAL EQUATIONS

85

x(k + I ) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) Taking the Z transform of both sides of the above normal equations we have , using property (5.73), zX(z) - zx(O) = AX(z) + BU(z)

(5 .74)

Y(z ) = CX(z) + DU(z)

(5 .75)

where

U(z) = S{ u(k) }

X(z) = S{ x(k)}

Y(z)

=

S{ y(k) }

Solving (5 .74) for X(z) we have

(z I - A)X(z) = zx(O) + BU(z) X(z) = (z l - Ar 1zx(O) + (z l - A)- 1BU(z)

(5 .76)

which , when substituted in (5.75), gives the discrete-time output as

Y(z)

=

C(zl - At 1zx(O) + (C(z l - A)- 18 + D] X(z)

(5.77)

Refe rring to state equations (5 .56) and (5 .58) the S transfom1 of t he output equation (5 .58) gives

Y(z) = C is an n X n state transition probability matrix. Oearly, from (5.92) and (5.93) 4> is of the form

(k)

(5.94)

=

where '{)ii must satisfy the following probability conditions: ~

,l'l .•

Yj l

L IPJI··

. 1 ;=

0

= I

for all i .j

(5.95)

for all i

(5.96)

If the elements r,.pii are fixed then (5.93) becomes p(k+l) = p(k)

(5.97)

The derivation of (5 .97) was based solely on probabilistic reasoning ; independent of the linear , time-invariant conditions of Chapter 3. However , f onnula (5.97) which describes a zero-input probabilistic system , is of the same form as state equation (5.50) , whjch describes a zero-input, linear. time-invariant system: x(k + I) = x(k)

(5.50)

Moreover , the system described by (5 .97) need only be a finite-state Markov system. The likeness of (5 .50) and (5 .96) suggests that a class of nonlinear systems , where the states are represented in terms of probabilities. is governed by linear equations. The refore , the solution of (5 .97) could be obtained by following the methods of Sections 3 and 5. By direct re cursion (5.97) becomes (5.98)

Hence. for input-free Markov systems the state probabilities are co mpletely "determined" by knowing the transiti on probability matrix and the initial state probability vector. The evaluation of (5.98) is best carried-out in the z domain , wherein the S transfonn gives zP(z) - zp(O) =

= A lx> + Biu )

Iv>=

Clx) + D lu )

(6.19)

1

CONTROLLABILITY, OBSERVABILITY ANO STABILITY

107

Their respective time-dependent solutions are

Ix) = (t) ~x(O)) +

l'

e-At Blu(~))d~]

ly) = C(t{ix(O)) + ~'e-Ate lu(n )d~] + Dlu(t))

(6.20)

(6.21)

By applying identities (6 .15) and (6.18) to solutions (6.20) and (6.21) we obtain the foil owing spectral decompositions: 1

(6.22)

where

st

is the conjugate transpose of B. The scalar product

applies. From (6.22) and (6.23) two observations are readily apparent: (1) the vectors (BTe; I may be regarded as "weights" which determine the magnitude of the effect of the input on the change in state in the Ie;) "direction ," and (2) the vector-valued , time-dependent quantities e"; r lei ) can be interpreted as the system modes.

6.3 CONTROLLABILITY

The system theoretic concept of controllability arises fr om the following twopoint boundary valve problem. Given an initial state Ix(O)) at time ze ro and a finaJ state Ix ) i= lx(O) ). determine whether it is possible to find a time rand an input l u ) which take the sys tem fr om its initial sta te at time zero to its final

l . No tatio nally x == ix ) .

108

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

state at time t , where tis finite. To fonnalize the definition of controllability we establish for the system representation

Ix> = Alx> + Blu> ly) = Clx) + Dlu)

(6.24)

and stipulate that at some initial time zero the system is in an allowable initial state Ix(O)). The system representation (6 .24) is defined 1 to be completely con rrollable 1f there exists a finite time t > 0 and a real input ~u) defined on the time interval (O,t) such that

TI1e necessary and sufficient conditions under which the system representation is controllable are readily deducible from (6.22), the spectral decomposition of the state. These conditions are presented below (without proof) in the form of the following theorems: 2 Theorem 6. /. Let the matn·x A in (6.24) have distinct eigenvalues. Representation (6.24) is completely controllable if and only 1f for all n vectors of (6.22) the condition (i=l,2 , .... n)

(6.25)

is satisfied. (Bei l = 0 for any i implies that the mode e1 cannot be excited. Hence the representation cannot be completely controlled. Theorem 6.2. Let A and B in (6.24) be n X n and n X m matn·ces, respectively. Representation (6.24) is completely controllable if and only if the ro ws of the matrix

are linearly independent on the time interval (0. t).

l. See Zadeh and Po lak. Syste m Theo ry , p. 244 , McGraw-Hill. New York, 1969 . 1

op. cit.

109

CONTROLLABILITY, OBSERVABILITY AND STABILITY

Theorem 6.3. Let A and Bin (6.24) be n X n and n X m matrices, respectively.

Representation (6.24) is completely controllable if and only if the n X nm controllability matrix Cle- , where

(6.26) has rank n. As a test of the above conditions consider the simple system of Fig. 6.1. We desire to establish the controllability of the system representation. By inspection we have the simultaneous equations

which can be written in matrix form as

[~1] x2

4 = [

I 0

y = [I

j[x1 lJ [llu +

5

-I

1

x2

i[:: ]

+ u

Thus lhe system matrices are identified as

A=[~:] B=[:]

C =

Figure6.1

[I

-1)

D = [I]

110

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

From (6.26) the controllability matrix

0 there exists a 8 > 0, dependent on€ and to, such that I x(O) I


(t,to)I ~ K

(for all t ~ to)

lim l(t,to) I = O

(for all to)

(6.32)

f-+oo

Theorem 6.8. The system is stable in the sense of Lyapunov implies (and is implied by) (a) All the eigenvalues of the constant matrix A have non-real parts, and (b) those eigenvalues of A that lie on the imaginary axis are simple zeros of the minimum polynomial of A .

Theorem 6. 9. The system is asymptotically stable if and only if all the eigenvalues of A have negative real parts. By theorems 6.7 and 6.9 above we assert that for fixed systems desctjbed

by (6.30) asymptotic stability implies that

lx(to , x(O), t)I ➔ 0 in an exponential manner. We therefore specify that

Ix(to , x(O), t) I ~ ce-'At

(6.33)

where C is a constant and X is a positive number. We can be more specific regarding the above definitions and theorems through a simple example. Let the matrix operator A for the unforced system governed by (6.30) have distinct eigenvalues X1 , X2 , ... , X,, , where each eigenvalue has the complex form X1 = 01

+

jw1

X2 = 02

+

/W2

An =

+

jwn

.. .. .. . . . . .. . On

CONTAOLLABI LITY, OBSE AVABI LITY AND STABILITY

115

The real parts of each eigenvalue can be ordered as 01 > 02 > ... >On. From equation (5.46) the state transition matrix can be written as n

=

r-1

mp

L eA; t r=l L, A,; ( t I ) , r·

i=l

where the A,; are matrix coefficients of the polynomial expansion and mp is the root multiplicity or degeneracy of the eigenvalues. For the unforced system we consider three situations: (1) a; < 0, (2) a; > 0, and (3) a; = 0. In each case we refer to equation (5 .46). Case (I): a; < 0. We see that for t ~ to the absolute value of remains finite. The condition of Theorem 6.6 is satisfied. Further, lim ( to , t) = 0 (-+oo

thereby also satisfying the conditions of Theorem 6. 7. It necessarily fol1ows that Theorem 6.8 and 6.9 are also satisfied. Clearly then for the case where the eigenvalues have negative real parts the state vector I x ) ➔ 0 as t ➔ 00 . The system representation is asy mptotically stable. Case (2): a; > 0 . The time limiting value of the transition matrix in this case is

lirn ( t O•t) =

oo

(-+oo

Therefore the state vector Ix) ➔ 00 as t ➔ 00 • Hence . for the case where the system eigenvalues have positive real parts the system represe ntation is unstable. Case (3): a; = 0. When the system eigenvalues are purely imaginary two situations arise: (a) The system has simple roots , in which case the A,; are constan t ma tri ces and remains finite. Thus in accordance with The orem 6.6 the system representation is stable. (b) The system has multiple roots. in which case the A,; are polynomia ls in t with matrix coefficients. Clearly

Jim (tn. t) = !

00

-+oo

Hence , the system representation is unstable .

116

AN INTRODUCTION TO THE THEORY OF LINEAR SYST6..

6.6 STABILITY-FORCED SYSTEM

Recalling the state dynamical equation and its solution we have for the forced ~ system

Ix>

=

Alx> + Blu>

+

Ix) = (t)[lx(O))

la'

(OBlu(O )d~]

For discussion purposes we specify that the system is initially at rest. Let the initial state at time zero be the zero-state , i.e., lx(O)) = 0 , after which time the system is perturbed by an impulse. We examine the system response as it relates to the concept of stability. For this situation the solution to the dynamical equation becomes

Ix) =

f

1(t

- ~)Blu(O)d~

0

=

i

t

(6.34)

Glu(O)d~

0

where (6.35)

G=(t-nB

The elements of the matrix G are of the form fl

g1•1· =

L l{)·k bk.

k= 1

1

(6.36)

1

Since the vector lu ) is a column matrix the relat ionsrup between the elements of the integrand and the state vector is established as

X·

1

= --

L. g--ulJ / I

= L L l{)-kbk ·U · j k t I I

(i= 1, 2, ... ,n)

(6.37)

CONTROLLABILITY, OBSERVABILITY ANO STABILITY

117

Expressing the elemental input as an impulse

U·l

= 0

for i =I= j

the components of the state vector become (i=l , 2, ... ,n) Thus G can be interpreted as the matrix of impulse responses of the state. For such a system we define stability as follows: A forced system is stable relative to a set of bounded inputs if and only if the state is bounded for all inputs lu)(t) in the set for all t ~to. Accordingly we require that

lx(t) I ~ K




= eAt =

(At)2 I + At + ! + ... 2

We desire to examine in closed form the time dependency of the fundamental matrix . This can readily be accomplished for simple systems by converting to the frequency domain then back to the time domain. Thus

..C{ 4>(t)} = [sl - Ar 1 -4

]-1

s- 5

=

I

4

s- I

(s- IXs-5) I

0

(s- 5)

Converting back to the time domain the fundamental matrix becomes et

4>(t) = [ 0

e' ] e5t

e5t -

Hence the product G = 4>B is G =

[e5t] e5t

It is readily apparent that for the system of Fig. 6.1 the positive nature of the elements of 4> and B will cause all the non-zero terms to grow beyond bounds as t becomes infinite . Therefore , the system representation for the example chosen, under both forced and unforced conditions , is unstable. This conclusion should be no surprise. Earlier it was shown that the representation for the system of Fig. 6.l was not completely controllable and not completely observable. However , with slight modifications we were able to render the representation as completely controllable and completely observable. We now logically inquire as

119

CONTROLLABILITY , OBSERVABILITY AND STABILITY

to whether or not sirrlilar modifications can be made m the case of system stability-under forced and unforced conditions. In the case of the unforced system the conditions for stability are manifest in equations (6.31) through (6.33). Equations (6.38) and (6.39) represent the conditions for stability of the forced system. All of these conditions can be satisfied in a variety of ways; the objective being to maintain a finite state vector for bounded inputs for all t ~to. An unstable representation of an unforced system does not necessarily mean an unstable representation of that same system when an input is applied. The terms giving rise to the instability may not be excited by the input. Nor does it necessarily follow that when the conditions for stability are met then the system is completely observable and/o r completely controllable. Similarly , a stable representation of an unforced system does not necessarily imply a stable representatio n of the forced system. We demonstrate these important facts through a simple example. Referring to the system of Fig . 6.1 , we render the system stable by changing the representation to that shown in Fig. 6.4. Essentially , the changes are in the magnitudes of signal amplification. The modified system ma trices become 1

A =[

~

C = [I

_'J

_\]

B = [

I]

D = [I]

The corresponding state transition matrix and impulse response matrix G are , respectively ,

et

I 2 (et - e-t )

0

e-t

=

)'

Figure 6.4

Syste m wherein the input does not excite the unstable mode.

1. After Schwarz and Friedland. Linear Systems , p. 38 1, McGraw-Hill . New York. 1965.

120

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

Due to the po&itive nature of the elen1ents 'Pi 1 and 'Pi 2 we have li1n I I =

(X)

{-+oo

Thus , the representation for the system of Fig . 6.4 is unstable when no input i:s applied. On the other hand the impulse response contains the diminishing terms e- 1 in both elements g 1 and g 2 . The refore the integrals and

both remain finite for all finite t ~ to. Consequently , the representation of the system of Fig. 6 .4 is stable when an input is applied. This apparent paradox is resolved when one examines the controllability and observability matrices for the system , under both forced and unforced conditions: Un forced system u

B ~

=0 =0 =0

Clo =

[: :]

Forced system

Qc = [

I

-I]

-2

2

Oo .:.. - [ I 1

l ] 0

Conclusions: I. unstable 2. not completely controllable 3. completely observable

I . stable 2. not completely controllable 3. completely observable

Thus. to resolve the above paradox it is concluded that the system input does not excite the unstable mode. Both modes , however. are observable. The unstable mode is observed when no input is applied. The stable mode is observed when the input is applied.

6.7 COMMENT

The central idea of the past discussion has been system representation in statespace. The algebra of rati onal funct ions (polynomials), linear vector spaces , and

CONTROLLABILITY , OBSERVABILITY ANO STABILITY

121

the methods of Fourier and Laplace transforms were the priocipal analytical tools . It was assumed that knowledge of the system matrix operators and va.riables was complete. However , it must be recognized that for large systems this assumption is short-lived . For large systems complete identification of all the variables and their operator elements is not practical , and therein lies the problem of applying the linear theory to practical problems dealing with systems of significant size. Consequently , a deterministic analysis for such systems gives way to the more realizable probabilistic or statistical analysis. Accordingly , the reader is referred to the methods of Markov chains, diakoptics, sparse matricesJ fuzzy sets , statistical mechanics , etc .. which bear on this problem.

7 Statistical Systems-Signals in Noise

7.1

INTRODUCTION

It is important to recognize the practical limitations of the theory developed this far. Certainly. a deterministic analysis is possible for sin1ple differential systems , where all the descriptors satisfying the input-output-state relations are known. However. fo r large complex systems all the sta te variables. the elements of the fundamental matrix, etc. , are not known nor can they even be defined in some cases. The theory addresses a small part of the real (nonlinear) world. Consequently. a deterministic analysis of many real systems is either not a simple task or not feasible using the linear theory developed thus far. As a meth od for augmenting the theory we turn to statistical or probabilisti c analysis. The input-output relations are treated as random processes. We will look at the statistical properties of each process. where the relations between th ese properties wil l form the basis of the analysis. l11e most important of these are the mean value and con-elation function. Our attention will specifi cally foc us on th e filterin g problem and stochastic processes. i.e .. on signals ~(t) that 122

123

STATISTICAL SYSTEMS - Sf.GNALS IN NOISE

are described by their avera~ rather than signals u(t) that are described by thcir point properties. Appendix F contains a short summary of some of the probabilistic expressions which will be used in the sequel.

7.2 AVERAGES (MEAN VALUE) AND CORRELATION FUNCTION

As a prelude to the discussion to follow we consider two different ways to represent an average. They are the time average and ensemble average. Both lead to the same results. The time average (u(t) ) of the signal u(t ) is defined as

IT

IT u(t )dt T- OQ 2 -T

( u(t) > = lim

(7. l )

where T is the time interval (-t .t ). This Umit is a number associated with the functional u(t ). On the other hand the average of a random signal is interpreted differently. Assume that the set {~(t) } is a stocha stic process as defin ed in Appendix F . For a given time t . the sample function ~(t) is a random process resulting in the random variable ~. The expected value of the ra ndom variable will be denoted as £ {u(t) }; it is the ensemble average of the random process {u(t) }. We have E {~(t) } = E {~(r ,r) }

= ~(t) -

=

j

(7 .2) (7 .3)

00

af(a; t ) dt

(7.4)

- CC

where f (a ;r) is the probabili ty density funct ion associated wi th the value a of the random variable. and f(a :t )da is the probab il ity that a \:vill be found in a± da. In conjunction wi th the above it foll ows that

j -

-

0,:

f(a:r)da ; l

( 7. 5)

0,:

In general of HO are known and satisfy the relation

(8.20)

where the corresponding energy eigenvalues Eio are discrete. If the initial state at time to is expanded in tenn s of the unperturbed eigenstates we have fr om (8. 17) - iE;ro / h.1x-o> c-e 1x(to) > = L)"' . l l l

ln the absence of any disturbance. we have fo r all time

I x(r) )

=

L .

I

.f .o ,

-J .1 £t h 0) CI r.! 1X · I

168

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS I

However , in th·e presence of V(t) (8 .17) is no longer the correct solution to the Schrodinger equation. To obtain the proper solution we must expand lx(t)) at every instant of time with the amplitude coefficients Ci depending on time:

Ix(t))

(8.2 I)

The probability amplitude of finding the system in the kth unperturbed state is EO /h ck(t) = (xk0 I x( t)) e 1 kt .

(8.22)

To readily determine the manner in which state transitions occur we examine the time dependency of Ck- Substituting (8.21) into the equation of motion jh

:t

Ix)

= Hix) = (HO+ V)lx)

gives

Forming the scalar product of the above with ( xk0 I results in

+

L c;(t)e-iE;°r/h

(8.23)

i

Since the

lxP> form an orthononnal set Eq. (8.20) gives

which redu ces (8.23) to

=

~ ci(t)< xf IV Ixio)e- f(E;°-Ef)r/h l

= )"' 0

l

('· )u

- j(/:,';° - t]) r/ h

Ci l "kie

(8.24)

QUANTIZED SYSTEMS-PERTURBATION THEORY AND STATE TRANSITIONS

169

where (8.25) Equation (8.24) is a system of simultaneous linear homogeneous differential equations. It expresses the equation of motion in terms of the eigenvectors of the unperturbed Hamiltonian H 0 . In matrix form (8.24) can be written as:

V11 . t:. d Ju dt

c2

=

V21 e-i(E i° -E2°)t/ h

As yet no approximations have been made in arriving at (8.24). It is in the solution of this complicated set of equations that approximations are invoked. The solutions to (8 .24) depend highly on the initial conditions. For simplicity it is assumed that at the initial time to = - 00 the system is definitely in one of the stationary states of the unperturbed Han1iltonian, say the ith state. We want to examine the probability of the system transitioning to state k. We assume that HO has discrete energy levels . Thus the initial conditions for the time-dependent probability amplitudes become (8.26) We begin our solution to (8.24) through successive approximations. Substituting in (8.24) the initial values of the coefficients ci we have at time to= - 00

(8.2 7)

Equation (8.27) is only valid fort such that (k

* i)

We again make use of the initial conditions (8 .~6) t o integrate (8.27): .

Ck(!) ~ - ~

.. f

J Vk;e- j(tjO·· t/)1 / 1, dt -

00

(k

* i)

(8. 28)

170

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEM&

For small perturbations the ck(t) may remain small throughout. After the per• turbations have stopped the system settles down to constant values of ci evaluated at t = -too:

Ck(t) = -

~

1-

VJ,; e-j(t./-Eko)t/h dt

(8.29)

- oo

From formula (8.29) it is seen that a system influenced by a time-dependent perturbation makes transitions to other energy eigenstates of Ho. The quantity lck((X))i 2 defines the transition probability from state i to k. It is proportional to the square of the absolute value of the Fourier component of the perturbation matrix element Vk; , evaluated at the transition frequency wk; ~i.e. ,

where Wk; is deduced from the relation Wki =

2rrf

If the system is initially in the higher energy state k the transition probability to the lower state i is , from (8.29),

It can be shown that V is a Hermitian perturbation operator , and that wk;= -wik· Thus it readily follows that

(8.30) The two transition probabilities are equal. Property (8.30) is the condition for detailed balancing. The energy difference E = hwk; is transferred to the radiation field of the system.

QUANTIZED SYSTEMS-PERTURBATION THEORY AND STATE TRANSITIONS

171

8.5 CONSTANT PERTURBATION

As a matter of practical interest we consider the case where V is constant or

varies slowly over the period 1/ wki· The system is in an initial unperturbed eigenstate Ixi0 ) at time to = 0 and then subjected to a weak perturbation which persists at some near constant value. In keeping with our previous discussion the time development of a system with Hamiltonian H = HO+ V can be conveniently described in terms of transitions between eigenstates of the unperturbed Hamiltonian Ho. The approximate equations (8.27) apply :

Treating V as a constant and specifying the initial and final discrete states as i and k , respectively , integration of (8.27) gives

(8.3 1)

where c;(O) = 1.

k

*

l

The probability that the system. in the initial state i at time final unperturbed eigenstate k (k i) at time tis

*

to= 0. will be in the

(8.3 2)

Equation (8.32) is illustrated graphically in Figure 8.1 . It is a periodic function o ft with a period equal to rr / wki and a peak at Wki = 0. The e.\pressi on is valid only as long as c;(t) can be approximat ed as c1(r) ~ I. du ring which time th e transition probability to states \\.·here t} =I= remains smJII for we ak pe rturbations . The probability of findin g the sys tc- m in sta te k is small unlc-ss the ene rgy of the kth state is cl ose to the energ:v. of tht? initi al state. Howeve r. transitions to

Ei°

'-

172

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

£? Figure 8.1

states where Ef ~ E,-0 have an important property. When (8.32) can be approximated as

Ef ~ E,-0

equation

(8.33) Thus the trans1t1on probability to the kth unperturbed eigenstate increases quadratically with time. This has special in1portance when the states in the neighborhood of the initial energy are very closely spaced and constitute a near continuum. It is not physically possible to measure lck 12 for a single value of k. The classical measurement is of the total probability that the system made a transition from an initial to a finaJ state. We define the total transition probability to all possible final states as Transition probability where the summation extends over all final states under consideration . For a quasi-continuum of energy states per unit energy level we introduce the density of final unperturbed states denoted by p(E). The quantity p(E)dE measures the number of final states in the interval dE containing energy £. The total transition probability into these states is determined by multiplying (8.32) by p(Ek0 )dEk0 and integrating with respect to dEko:

(8.34)

where

173

QUANTIZED SYSTEMS-PERTURBATION THEORY AND STATE TRANSITIONS

The time rate of change of the total transition probability , w, is (8.3 5) In analyzing (8.35) Vki and p are reasonably constant over a small energy range oE;° near E;0 . However , sin wkitl wki oscillates rapidly in this same energy interval for all t satisfying the relation (8.36)

£?.

and has a pronounced peak at Ef = Clearly those transitions which tend to conserve the unperturbed energy are dominant. Also oE;O is usually comparable to Ez9. Thus h/oE;O is a very short time. Hence there is a large range oft where (8.36) is fulfilled yet the initial state i is not appreciably depleted. During this time (8.35) can be approximated as

Under the conditions stated the transition probability per unit time becomes (8.38) Equation (8.38) shows a constant rate of transition. This result com es about because we summed over transitions which conserve the unperturbed energy (El= Ei 0 ) and transitions that violate this conservation. From (8.33) it is seen that the transition probabilities of the former type increase quadratically with time , whereas from (8.32) it is seen that transitions of the latter type are periodic. The result is a compromise between these two and the transition rate is constant. Result (8.38) has been termed by Fermi as the golden rnle of timedependent perturbation theory . 8.6

EXPONENTIAL DECAY

Our discussions on perturbation the ory was centered around the approximate solutions to equations (8.24):

L clr) vki eiw )df i

174

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

From tJ1ese equations it was seen that if Vk; =!- 0 transitions from an initial state to various available final states occur. The probability that the system will make a transition in time interval between t and t + dt is equal to wdt. By the condition for detailed balancing , (8.30), these final states contribute to the probability amplitudes of the initial states through a feedback process. As the amplitudes of the final states grow, they do so at the expense of the initial states, since probability is conserved. Because of the different frequencies wk; of the feedback process the contributions made by the amplitudes ck to amplitudes c; are all of different phases. Thus if there are many available states k , forming a near-continuum, the contributions made by tliese states tend to cancel. This destructive interfere nee in tlie probability amplitude is interpreted as a gradual (exponential) depletion of the initial state. It is inferred above that tJ1e probability of finding tlie system at time t still in the initial state is proportional to exp(-wt). This is the exponential decay law. To derive the exponential decay law we no longer consider c;(t) on the righthand side in (8.24) as a constant. However, in our approximation we will continue to neglect all otJ1er contributions to the change in ck(t). Equation (8.24) becomes

k =I= i

.

ck(t) = -

~ Vk;

l

t

c1{t') eiwkit'dt'

(8.39)

0

where Ck(O) = 0 fork =I= i. The prime identifies t witli Ck· However , by reciprocal manipulation of (8.24) it can be shown that tlie equation of motion for c;(t) is rigorously

(8.40)

Substituting (8.39) into (8.40) gives

d

c ·(t) dt l

(8.41)

Assuming that the pertinent final states are in a near-continuum with a density

QUANTIZED SYSTEMS-PERTURBATION THEORY AND STATE TRANSITIONS

175

of states p(El) the sum can be replaced by an integral extending over all possible transition frequencies. Equ4tion (8.41) becomes

Equation (8.42) is a differential equation of the Volterra type. It can be approximated by the simple equation 1

o) c·(t)

i -dc1-(t) = ( - -w - ~Edt 2 h 1

(8.43)

1

8Et

where is the shift of the unperturbed energy level perturbation. 2 With clO) = I equation (8.43) yields w

J

Et due to second-order

O\

c·(t) = exp - - - -8£. 1 ( 2 l h l J

(8.44)

which describes the exponential decay . We next examine the probability that the system has decayed into state k. Substituting (8.44) into (8.39) and integrating:

~ r'] dt'

0 0 0 Ck ( t ) = - j_ h Vik·1 j"'texp [- !_ h £.1 + 8£.1 - £ k - 1·h 2 0

Eo - (£.o + 8£.o) + 1·h~ k l l ., '-

The probability lck 12 that the system has decayed into state k becomes I -

=

0 0 r ) (£+ 8£. 2 exp ( t cos h, 1

Ef)

+ exp(- ft)

t 2 1Vk; 12 - - - - - - - - -- - -- -- - _ £.o _ 8 £.0)2 + ( £0 k 1 1

2 4

r

(8.45)

l. See V. F. Weisskopf and E. P. Wigner. Z. Physik . 63 , 54 (19 30). 2. The reader is reminded that our discussio n on perturbation addre ssed only fust-order terms . An exceptjon is made in the derivatio n of equatjon (8.43).

176

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

where r = hw. For t long compared to the state lifetime J/r th~ transition probability is approximated as (8.46)

Equation (8.46) is a bell-shaped curve. It has a pronounced peak at those final state energy 1evels Ef equal to + 8Ei0 . The width of the curve is equal to hw.

Et°

Appendixes

Appendix A

DIRAC DEL TA FUNCTIONS

The delta function o(~ - a) is a mathematically improper function having the followi ng properties (in one dimension): o(~-a) = 0

for ~

*a

(A .I )

if region of integration includes ~ = a

(A .2)

if region of integrat ion does not include ~ = a

(A.3)

177

178

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

Higher order delta functions are defined as the derivative of b(t - a):

(A.4) The delta function of another function can be transformed according to the rule

}r o(~ - ~o)

5([(~)) =

(A.5)

d~

where fl~o) = 0. For an aribtrary function f(~) the sifting property of delta functions provides for the integral equations

f

00

f(nO(~ - a)dt = f(a)

(A.6)

-00

rmW(~-a)d~ = (-If!"(a) -

(A.7)

-00

It is difficult to physically imagine the delta function or the unit impulse as it is also referred-to. Qualitatively , it can be thought of as a small pulse of high magnitude and infinitesimally small duration (see Fig. A-I). We require that as the peak of the curve gets higher , the width gets narrower in such a way that the area under the curve remains constant (unity). Thus the unit impulse 5(t) can be regarded the limit as A~ ➔ 0 of the pulse p(t) having width A~ and height I/ A~.

b(l)

I

--7

~

I I I I I

I

Figure A-1

Unit Impulse.

APPENDIXES

179

Appendix B

RESOLUTION OF CONTINUOUS-TIME SIGNALS INTO UNIT IMPULSES

An arbitrary signal u(t) can be approximated in any finite time interval -T ~ t ~ T by a finite number of unit pulses of width ~~ occurring at times t= kt where k= ±I , ±2, ... ,±N = T/~~- Fig. 8-l illustrates the idea. Since the height of the unit pulse is I/ ~~ , the pulse at t = k~ is multiplied by u(k~~)~~ thereby resulting in the amplitude u(t). The approximation of u(t) can be

written as N

u(t) =

L

u(k~~)~~ p(t - k~)

(B. l)

lim

L u(k~~)~~ p(t - k~~)

(8.2)

k=-N

=

A(-0 N~ 00

where p(t) is defined in Appendix A. In the limit as ~~ ➔ 0 and N ➔ 00 the pulses become impulses and the summation becomes an integral. Thus the approximation becomes exact. We have

J

T

u(t) =

u(~)8(t -T

~M

(B.3)

over the finite time interval (-T. T). Extending the integral ove r tl1e entire time domain defining u(t), i.e. , letting T ➔ 00 ,

r

00

u(t) =

11(08(1 -

nd~

(BA)

..., _ 00

l11us, any continuous-time signa l u(r) can be resolved int o a co ntinu um of unit pulses. It follows that the response of a linear sys tem to excitati on u(t) can be readily fou nd if the response to the unit im pul se is known. Hence the unit impulse response complet ely ck1r;1c teri1.es the sys tem.

180

AN INTRODUCTION TO THE THE ORY OF LINEAR SYSTEMS

Figure B-1

Approximation of u(t) by unit impulses.

Appendix C DISCRETE-TIME STATE EQUATIONS

From (2 .1) the standard form of the nth order difference equation can be written as a0 y(k) + a 1y(k - 1) + ... + any(k- n) = b 1u(k - 1) + ... + b5 u(k - s)

(C.I)

(a 0 + a 1A-l + ... + a11 A-n)y(k) = (b 1A-l + ... + b5 A-5 )u(k)

(C.2)

where A is the linear advance operator defined for any interger v as Avf(k) = f(k + v)

(C.3)

Introducing the variable v(k) where (C.4)

and (C.5)

181

APPENDIXES

equation (C.2) gives v(k) = -

1

[u(k)- anv(k - n) - an_ 1v(k - n + 1) - ... - a 1v(k - I)]

ao

(C.6)

Specifying the elements of x(k) as x 1(k) = v(k- n) x2(k) = v(k-n +l)

(C.7)

x,.z(k) = v(k- 1)

we can write

x 1(k + 1) = x 2 (k)

x 2 (k + I) = x 3 (k)

xn _1(k + 1) = xn(k)

x,.z(k + 1) = -

1

ao

[u(k) - anx 1(k) - ... - a 1xn(k)]

(C.8)

[n matrix form (C.8) becomes

x(k +I) = Ax(k) + Bu(k)

(C.9)

where

0 0

0 0

0

0

0

0 B =

A =

(C.10)

an-I

From (C.5) we have (C. i I )

182

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

Substituting (C .7) into (C. 11 ) gives

(C.12) or , in matrix form ,

y(k) = Cx(k) + Du(k)

(C.13)

where D = 0

(C.14)

Matrices (C .10) and (C.14) are not unique . They are one of a variety o f ways to represent (C .9) and (C. 13).

Appendix D TRANSFORMS

Listed below are cal functions:

Z transforms of some of the more commonly used mathemati00

Z [f(k) ]

= F(z) = [:

k=O

s [c/J = S [eiwk]

=

Z [cos wk] =

S (sin wk ]

f (k)z-k

=

k~O

I

l - az- 1

1 I - eiw 2 - 1 I - z- 1cos w I - 2z-l cos w + z- 2

z- 1 sin w I - 2z- 1cos w + z- 2

lzl

> la 1

lz l

>1

lz l

>I

lz l

>

I

183

APPENDIXES

Z [k"f(k))

(-z ~ tF(z)

=

Z [k]

z-1

=

lz l > 1

z-1(1 - z-1)

lzl > 1

(1 _ z-1)3

k

z U,]

= elfz

~

IZI

1, I zl > 1

>0

S [f(k + 1)] = z [F(z) - f(O)] S [f(k-m)] = z-m F(z)

S transforms involving difference operat ors can be written according to the rule

S [~f(k)] = (z - l ) F(z ) - zf(O)

:S [-~f(k) ]

= (1 - z- l )F(z)

where ~f(k) is the forward difference operator defined as ~f(k) = f(k + 1) - f(k) and -~f(k) is the backward difference operator defined as -6f(k) = f(k) - f(k - l)

TI1e inve rsion of the S transform ca n be accomplished in a variety of

AN INTRODUCTION TO THE THEORY OF LINEAR SYSTEMS

184

ways. Listed below are a few elementary transform terms useful in expansion by partial fractions: Time sequence

F(z) l . F(z) converges lzl > a 1

z-a z (z - a) 2

2. F(z) converges lzl < a

Ik l

-kak-l

Ik