Mathematical Problems of Control Theory: An Introduction

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Volume 4

Series A

Mathematical Problems of Control Theory An Introduction Gennady A. Leonov

forld Scientific

Mathematical Problems of Control Theory An Introduction

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Series Editors: A. Guran, A. Belyayev, H. Bremer, C. Christov & G. Stavroulakis

About the Series Rapid developments in system dynamics and control, areas related to many other topics in applied mathematics, call for comprehensive presentations of current topics. This series contains textbooks, monographs, treatises, conference proceedings and a collection of thematically organized research or pedagogical articles addressing dynamical systems and control. The material is ideal for a general scientific and engineering readership, and is also mathematically precise enough to be a useful reference for research-specialists in mechanics and control, nonlinear dynamics, and in applied mathematics and physics.

Selected Volumes in Series B Proceedings of the First International Congress on Dynamics and Control of Systems, Chateau Laurier, Ottawa, Canada, 5-7 August 1999 Editors: A. Guran, S. Biswas, L Cacetta, C. Robach, K. Teo, and T. Vincent Selected Topics in Structronics and Mechatronic Systems Editors: A. Belyayev and A. Guran Selected

Volumes in Series A

Vol. 1

Stability Theory of Elastic Rods Author: T. Atanackovic

Vol. 2

Stability of Gyroscopic Systems Authors: A. Guran, A. Bajaj, Y. Ishida, G. D'Eleuterio, N. Perkins, and C. Pierre

Vol. 3

Vibration Analysis of Plates by the Superposition Method Author: Daniel J. Gorman

Vol. 4

Asymptotic Methods in Buckling Theory of Elastic Shells Authors: P. E. Tovstik and A. L Smirinov

Vol. 5

Generalized Point Models in Structural Mechanics Author: I. V. Andronov

Vol. 6

Mathematical Problems of Control Theory: An Introduction Author: G. A. Leonov

Vol. 7

Vibrational Mechanics: Theory and Applications to the Problems of Nonlinear Dynamics Author: llya I. Blekhmam

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS ':-:v Series A c ;^;-r>

Volume 4

Series Editors: Ardeshir Guran & Daniel J Inman

Mathematical Problems of Control Theory An Introduction

Gennady A. Leonov Department of Mathematics and Mechanics St. Petersburg State University

V f e World Scientific wl

Sinaapore Singapore • New Jersey •London London•• Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Pubiication Data A catalogue record for this book is available from the British Library.

MATHEMATICAL PROBLEMS OF CONTROL THEORY: An Introduction Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4694-3

Printed in Singapore by Mainland Press

Preface

It is human to feel the attraction of concreteness. The necessity of concreteness is most pronounced in the mathematical activity. The famous mathematician of the 19th century Karl Weierstrass has pointed to the fact that progress in science is impossible without studying concrete problems. The abandonment of concrete problems in favor of more and more abstract research led to a crisis, which is often discussed now by the mathematicians. The mathematical control theory is no exception. The Wiener idea on a partitioning of a control system into the parts, consisting of sensors, actuators, and control algorithms being elaborated first by mathematicians and being realized then, using the comprehensive facilities of electronics, by engineers, was of crucial importance in the making of cybernetics. However at the same time it has been considered as a contributary factor for arising a tendency for an abandonment of concreteness. The notorious and very nontrivial thesis of LA. Vyshnegradsky: "there is no governor without friction", and the conclusion of V. Volterra that an average number of preys in the ecological predator-pray system turns out to be constant with respect to the initial data are a result of the solutions of concrete problems only. The study of concrete control systems allows us to evaluate both the remarkable simplicity of the construction of a damper winding for a synchronous machine, which acts as a stabilizing feedback, and the idea of television broadcasting of two-dimensional pictures via one-dimensional information channels by constructing a system of generators synchronization. In this book we try to show how the study of concrete control systems has become a motivation for development of the mathematical tools needed

Preface

VI

for solving such problems. In many cases by this a p p a r a t u s far-reaching generalizations were made and its further development exerts a material effect on many fields of mathematics. A plan to write such a book has arisen from author's perusal of many remarkable books and papers of the general control theory and of special control systems in various domains of technology: energetics, shipbuilding, communications, aerotechnics, and computer technology. A preparation of the courses of lectures: " T h e control theory. Analysis" and "Introduction to the applied theory of dynamical systems" for students of the Faculty of Mathematics and Mechanics, making a speciality of "Applied m a t h e m a t i c s " was a further stimulus. An additional motivation has also become the following remark of o

K.J. Astrom [5]: " T h e introductory courses in control are often very similar to courses given twenty or thirty years ago even if the field itself has developed substantially. The only difference may be a sprinkling of M a t l a b exercises. We need to take a careful look at our knowledge base and explore how it can be weeded and streamlined. We should probably also pay more attention to the academic positioning of our field". T h e readers of this book are assumed to be familiar with algebra, calculus, and differential equations according with a program of two first years of m a t h e m a t i c a l departments. We have tried to make the presentation as near to being elementary as possible. T h e author hopes t h a t the book will be useful for specialists in the control theory, differential equations, dynamical systems, theoretical and applied mechanics. But first of all it is intended for the students and postgraduates, who begin to specialize in the above-mentioned scientific fields. T h e author thanks N.V. Kuznetsov, S.N. Pakshin, I.I. Ryzhakova, M.M. Shumafov, and Yu.K. Zotov for the patient cooperation during the final stages of preparation of this monograph. T h e author also thanks prof. A.L. Fradkov and A.S. Matveev who read the manuscript in detail and made many useful remarks. Finally, the author wishes to acknowledge his great indebtedness to Elmira A. Gurmuzova for her translating the manuscript from Russian into English. [email protected] St.Petersburg, April 2001.

Contents

Preface

v

Chapter 1 The Watt governor and the mathematical theory of stability of motion 1 1.1 The Watt flyball governor and its modifications 1 1.2 The Hermite—Mikhailov criterion 7 1.3 Theorem on stability by the linear approximation 13 1.4 The Watt governor transient processes 25 Chapter 2 Linear electric circuits. Transfer functions and frequency responses of linear blocks 33 2.1 Description of linear blocks 33 2.2 Transfer functions and frequency responses of linear blocks . . 40 Chapter 3 Controllability, observability, stabilization 3.1 Controllability 3.2 Observability 3.3 A special form of the systems with controllable pair (A,b) 3.4 Stabilization. The Nyquist criterion 3.5 The time-varying stabilization. The Brockett problem

53 53 62 66 67 72

Chapter 4 Two-dimensional control systems. Phase portraits 93 4.1 An autopilot and spacecraft orientation system 93 4.2 A synchronous electric machine control and phase locked loops 106 4.3 The mathematical theory of populations 126 vii

Vlll

Contents

Chapter 5 Discrete systems 5.1 Motivation 5.2 Linear discrete systems 5.3 The discrete phase locked loops for array processors

133 133 140 148

Chapter 6

155

The Aizerman conjecture. The Popov method

Bibliography

167

Index

171

Chapter 1

The Watt governor and the mathematical theory of stability of motion 1.1

T h e W a t t flyball g o v e r n o r a n d i t s m o d i f i c a t i o n s

T h e feedback principle has been used for centuries. An outstanding early example is the flyball governor, invented in 1788 by the Scottish engineer James W a t t to control the speed of the steam engine. The W a t t governor is used for keeping a constant angular velocity of the shaft of the engine (in a classic steam engine, in a steam turbine or hydroturbine, in a diesel device, and so on).

/^N / ^ \

'eeve

*

1

c

>a>

A

~\

/////////////// Valve

Engine

Working agent Pipeline

Fie. 1.1 In Fig. 1.1 is shown a basic diagram of such a governor. T h e two identical arms with equal governor flyballs at their ends are hung at the top of a shaft. The arms are fastened by additional hinges and can leave the vertical by the angle (t), is as follows

Here J is a moment of inertia of rotating rigid body (in the case of a turbogenerator it consists of a shaft and a rotor of electric generator, which are rigidly connected to one another). For simplicity we neglect the masses of various parts of the W a t t governor since they are negligible compared to J. T h e moment of force G is a sum of a useful load and a drag torque. At power stations, for example, G forms an electrical network power. Equation (1.1) is well known in theoretical mechanics. The determination of a moment of inertia J is an applied problem of calculus. Recall t h a t the W a t t governor is employed for a given angular velocity to be kept constant, in which case co(t) = LOQ. T h e value UQ is determined by specific requirements of a concrete technology problem. For a turbogenerator, for example, a very important condition is that u>o must coincide with the frequency of electric current. T h e W a t t governor operates in the following way. If the value of w(t) is greater t h a n U>Q, then the centrifugal force (and therefore the angle 0 there exists a number 8(e) > 0 such that for all y0, satisfying the inequality \x0 - y0\ < 6(e), the following relation holds \x(t,t0,x0)

-x(t,t0,y0)\

t0.

(1.21)

Theorem on stability by the linear

approximation

15

D e f i n i t i o n 1.2. If a solution x{t,to,xo) is Lyapunov stable and there exists a number So such that for all yo, satisfying the inequality \xo — yo\ < So, the following equation lim

\x(t,t0,x0)

-

x(t,t0,y0)

0

- + + OO '

is valid, then the solution x(t,to,Xo)

is called asymptotically

stable.

Note t h a t , generally speaking, in Definitions 1.1 and 1.2 the numbers S(e) and S0 also depend on to'. So = 0 in Fig. 1.10. For

16

The Watt governor and the mathematical

theory of stability of motion

a = 0 system (1.23) has the following first integral V(0,77) = T?2 - ^

cos 9 = C,

(1.24)

where C is an arbitrary number.

Fig. 1.9 We see t h a t any solution 0(t), rj(t) of system (1.23) satisfies the following identity j

t

V(6(t),

V(t))

= 2r,(t) ( - f

sinfl(t)) + 2-f (8infl(t)) r,(t) = 0.

Thus, the trajectories in the phase space of system (1.23) are placed wholly on the level lines {6,T,\V(6,T,)

=

C}.

Whence it follows t h a t trajectories are closed in the neighborhood of a stationary solution 6(t) = 0. This implies t h a t the solution is Lyapunov stable but not asymptotically stable. Using the first integral (1.24) we see t h a t two trajectories tend to stationary solution 6(t) = 7T, rj(t) = 0 as t —> —00 and the same trajectories tend to equilibria 9{t) = —n, r)(t) = 0 and 6{t) = 37r, r)(t) = 0 as t —>• + 0 0 . Such trajectories are often called heteroclinic. Their existence proves t h a t the solution 6(t) = n, rj(t) = 0 is Lyapunov unstable. T h e first stationary solution, which is Lyapunov stable, corresponds to the lower equilibrium position of the pendulum. In some neighborhood of this position the closed trajectories correspond to the periodic oscillations of a pendulum in a neighborhood of the lower equilibrium position. T h e second stationary solution, which is Lyapunov unstable, corresponds to the upper equilibrium position. The latter exists theoretically

Theorem on stability by the linear

approximation

17

but we cannot observe it because of its instability. This fact is the same for many other physical, technical, biological, and economical systems, that is, the Lyapunov unstable equilibrium is nonrealizable. We consider an intuitive "mechanical" proof of asymptotic stability of the lower equilibrium position 6(t) = 0,?7( 0 only. If a > 0, then there exist friction forces, which assure the decay of oscillation near the lower equilibrium position. Thus, solutions in a neighborhood of the stationary point 9(t) = 0, r](t) = 0 tend to zero as t —>• +oo. It means that the stationary solution is asymptotically stable (Fig. 1.10).

Fig. 1.10 A.M. Lyapunov has suggested a method of investigation of a solution stability, involving special functions, which are called now the Lyapunov functions. Consider the case that the solution x(t,to,xo) is the zero solution, namely x(t,to,xo) = 0. The general case may be reduced to it by the change of variable x=y+

x(t,t0,xQ).

Then ^=9(t,v),

(1-25)

where g(t, y) = f(t, y + x(t, t0, x0)) - f{t, x(t, t0, x0)). We see that equation (1.25) has the same structure as (1.20) and in addition we have g(t,0) = 0. However such a substitution is not always effective since in this case we must know the form of solution x(t,to,xo).

18

The Watt governor and the mathematical

theory of stability of motion

Consider a differentiable in some neighborhood of the point x = 0 function V(x) (V : Mn -> M1) such that V{0) = 0. We introduce the following notation

V(x):=(gvaAV(x)rf(t,x)

=

T~f,(t,x). '—' OXi

The expression V(x) is often called a derivative of the function V{x) with respect to system (1.20). Here x,- is the i—th component of the vector x and fi is the i—th component of the vector function / . It is clear that if we take the solution x(t,to,xo) rather than x, then by the differentiation rule the following relation holds — V(x{t,t0,x0))

=

(giadV(x(t,to,xo)))*f(t,x(t,t0,x0)).

Theorem 1.1 (On asymptotic stability). Suppose, there exist a differentiable function V(x) and a continuous function W(x) such that in some neighborhood of the point x = 0 the following conditions hold: 1) V[x) > 0 for x ± 0, V(0) = 0, 2) V{x) < W{x) < 0 forx^O. Then the zero solution of system (1.20) is asymptotically stable. P r o o f . Assume that the ball {x\ \x\ < e} is inside the neighborhood considered. Put a =

inf

V(x)

(1.26)

{x\ \x\=e]

Since the sphere is closed, assumption 1) implies that a > 0. Let us choose now a number d such that sup V{x) < a. W W to such that \X(T, to, XQ)\ — e and \x(t,to,xQ)\<e,

Vt£[t0,T}.

Theorem on stability by the linear approximation

19

Under the assumption 2) we obtain the following inequality V(x{T,to,x0))a>V(x0).

(1.29)

Since (1.28) and (1.29) are in contrast to each other, we have \x(t,tr,,xo)\ < e, Vt >t0. Let us now prove the asymptotic stability mentioned above. Holding a number £o fixed such that a ball {x\ \x\ < £Q} is placed wholly inside the neighborhood of the point x = 0, we choose So such that \x(t,t0,x0)\ <e0, V i > i o , Vx0 e {x\ \x\ < 60}. In this case assumption 2) yields that for any XQ from the ball {x\ \x\ < SQ] there exists a limit lim V{x(t,t0,x0))

=P

(1.30)

and V{x(t, t0, x0)) >/3,Vt>t0. This implies that j3 = 0. Assuming the opposite, that is, that /3 > 0, we obtain that the solution x(t,to,xo) is separated from zero, i.e., there exists a number 7 such that |a:(Mo,zo)|>7,

Vi>*0.

(1.31)

Recall that in addition to (1.31) the following condition \x{t,t0,x0)\<eo,

Vt> (3 > 0. Thus, we have f3 — 0. From relation (1.30) and the continuity of V{x) we conclude that lim

t—y+co

\x{t,t0,x0)\

= 0.

T h a t establishes the theorem. T h e o r e m 1.2 (On instability). Suppose that there exist a differentiable function V(x) and a continuous function W(x), for which in some neighborhood of the point x — 0 the following conditions hold: 1) V(0) — 0 and for a sequence Xk —> 0 as k —> oo the inequalities V{xk) < 0 are valid, 2) V(x) < W(x) 0 a number 5(e) can be found such t h a t \x{t,t0,x0)\

<e,

V

t>t0

for all xo € {z| \x\ < S(e)}. In this case by assumption 1) of the theorem we can choose xo such t h a t V(xo) < 0. Then from assumption 2) it follows that V{x{t,to,xo))

7,

V t > t0.

Since W{x) is continuous, a negative number a? can be found such t h a t W(x) < a;, Mx G {x\ 7 < |a;| < e}. Therefore W(x(t,to,

xo)) < ae,

V t > to-

Hence t V(x(t,

t0, x0))

< V(x0)

+

/ W{X{T, to



-00,

to, xo)) dr


0 and H2 > 0 of dimension (n — m) x [n — m) and m x m respectively such t h a t the following relations hold AlK + HiAt A*2H2+H2A2 A function V(x) = x\H\Xi bility theorem. Here

— xlH2X2,

=-I, = -I.

[l 6l>

-

satisfies assumption 1) of the insta-

Hence, using (1.37), we obtain V(x) = -\x\2

+ 2x*1H1g1{t, x) - 2x*2H2g2(t, x),

(1.38)

where gi(t, x) and g2(t, x) satisfy the relation

\92{t,x)J Here gi(t, i j i l ' x l " ^ M " - " 1 , g2(t, x) : M1 x ffin -> M m . From (1.38) it follows that V < -\x\2 + 2\HlXl\ \gi{t,x)\ + 2\H2x2\ \g2{t,x)\ < -\x\2 + 2{\HlXl\ + |jy 2 ar 2 |) \g{t, x)\ < < -\x\2 + 2{\H1x1\ + |£T2a:2|)ae|a:|.


0 one of eigenvalues of A is positive and the other is negative. By the theorem, the solution is Lyapunov unstable. We again consider the Watt governor. The characteristic polynomial of the Jacobi matrix of system (1.4) has the form (1.9). By Theorem 1.3 and condition (1.16) we conclude that stationary solution (1.5) is asymptotically stable if the following inequalities FQ < 0, a(7-/3mwg)J>-F0/om

(1.41)

are satisfied and stationary solution (1.5) is Lyapunov unstable in the case that a{j - j3mu%)J < -Fofom.

(1.42)

This conclusion, which has been made by I.A.Vyshnegradsky in 1876, has impressed on his contemporaries. In the cases that the friction is small and (1.42) is satisfied, there occurs an effect being compared with the nonrealizability of the upper position of pendulum, in which case the required operating regime becomes nonrealizable due to its instability. For the conclusion to be cogitable for engineers, I.A.Vyshnegradsky has stated his notorious "thesis": The friction is a governing characteristic of a sensitive and correctly operating governor or shortly: "there is no governor without friction".

The Watt governor transient

processes

25

In the middle of 19th century an unstable operating regime of governors was explained by the fact t h a t with increasing a machine power the more heave valves were used and for their control the greater masses of balls, m, were necessary. In this case the improvement of a surface t r e a t m e n t led to the considerable decrease of friction factor a. In addition for the working speed of machines to be increased it was necessary to decrease the moments of inertia J of a shaft and the connected with it details. Notice t h a t since in the modern turbogenerators the value of J is large, inequalities (1.41) are always satisfied. Thus, we obtain the conditions, which ensure the operation of system: a machine - the W a t t governor. However in starting the system we must every time make a transition to the desired process from a given state of system. Such processes are called transient processes.

1.4

T h e Watt governor transient processes

Consider transient process for the W a t t governor, using main conceptions of the m e t h o d of Lyapunov functions. Suppose t h a t the function F(u(x)) is linear F{u(x))

-G

= F0Ax

= F0(x -

x0)

(see equations (1.4)-(1.7)). This assumption is natural in the case t h a t a spring constant 7 is sufficiently large and the changing of x(t) is sufficiently small. By the same argument the following approximation / = /3mru>2 + fimojQX is used. Thus, we have Jw =

F0Ax, (1.43)

m ( A x ) " + a(Ax)'

+ f0(Ax)

= Pmroj2 — foxo

with initial d a t a w(0) — 0, Aa;(0) = — XQ, (Aa;(0))* = 0, which correspond to an activation of system at time t = 0. Here 70 = 7 — (3mu>Q. We introduce the following notation y=—Ax,

z=—(Ax),

a = —,

6=—,

26

The Watt governor and the mathematical

^( w ) =

theory of stability of motion

I_ p \

r" {fimrui2 -

j0x0)-

mJ

Equations (1.43) in these notation are given by w = y,

y = z,

(1.44)

z = —az — by — ). We shall use further a function f V(u, y,z) =a

by2 0.

(1.48)

The

Watt governor

transient

27

processes

P r o o f . Obviously, for t = 0 inclusion (1.48) is true. We assume t h a t for some t > 0 inclusion (1-48) is not valid. Then the continuous differentiability of the function w(i) implies the existence of a number r > 0 such t h a t one of the following relations holds: 1) W(T) = 0, W(T) < 0, u{t) G [ 0 , « i ] , Vt G [ 0 , r ] , 2 ) « ( r ) = w i , « ( r ) > 0 , w(i) G [0,wi], V t e [ 0 , r ] . Let us remark t h a t in each of this cases by (1.47) we have the inequality V{u(t), y{t), z(t)) < 0, Vi £ [0, r ] . From the relations

VM0) > y(0),,(0))=-^^ a :S + ^^xS wo and therefore V(u(r),y(r),z(T)) > 0, which contradicts inequality (1.49). These contradictions prove the lemma. L e m m a 1.3. Let for a continuously differentiable on [0, +oo) function u(t) the following assumptions hold: 1) for a number C \ii(t)\0,

2) u{t) > 0, V< > 0, />+oo

3) /

u{t) dt < +oo.

Jo Then

lim u(t) = 0. t—>oo

P r o o f . Consider the following identity t 2

u(t)

= u(0)

2

+ 2 / U(T)U{T)

dr

(1.50)

o and obtain the estimation t

/ \u{T)u(T)\dr

t

< / \u(r)\u{T)dT
+ 0 0 such t h a t \u(th)\

>£.

In this case by 1) we obtain

HOI > I

(i-5i)

on the segments [t^, tj. + -^\. It follows from assumption 2) t h a t we may take t\ such t h a t

K^'-Iec'

v*i.

(1.52)

By (1.51) u (tk

+

ic)

>

3 e

2

- lec

T h e last inequality contradicts inequality (1.52). The proof of Lemma 1.4 is completed.

The Watt governor transient

processes

29

Theorem 1.4 (on transient process). Suppose, the parameters of governor satisfy the following assumptions m7o > a2,

(1.53)

a 7 o J > - A / 3 Fofom.

(1.54)

Tften /or a solution of equation (1.43) wiift initial data w(0) = 0, Ax(0) = —so; (Aa^O))* = 0 the following relations u(t)€

lim u ( ( ) = u 0 l

0, / ^ 1 V Pmr

[ 0 , >/3a,0],

=

(1.55)

J

lim Aa;( mz

2F0 pmrui mj

and is given by a-/oJ > - A / 3 F0m(f0 - 2/3mu>0x0). The last inequality follows directly from condition (1.54). Thus all the assumptions of Lemma 1.1 are satisfied and, consequently, inclusion (1.55) is valid. To prove (1.56), note that by using (1.45) and the inclusion (1.55) we obtain V(u(t),y(t),z(t)) 0,

(1.57)

where e is a sufficiently small positive number. From (1.55) it follows that the function V(u(t), y(t), z(t)) is uniformly bounded on [0,+oo). Then by

30

The Watt governor and the mathematical

theory of stability of motion

(1.57) we obtain the existence of a number C such that f y(r)2dr Jo

< - (V(u(0), y(0), z(0)) - V(u,(t),y(t), z(t))) 0.

The uniform boundedness of V(u(t),y(t),z(t)) and inclusion (1.55) result in the fact that on [0, +oo) the functions y(t) and z(t) are uniformly bounded. This implies that the function ^ y(t)2 = 2y(t)z(t) is also uniformly bounded. Thus, all the assumptions of Lemma 1.3 are satisfied and, consequently, we have lim y(t) = 0.

(1.58)

t—++oo

Let us also remark that

y{t) =

-az{t)-by(t)-^{t))

and, as it was proved above, z(t),y(t),u>(t) are uniformly bounded on [0,+oo). From (1.58) by Lemma 1.4 we conclude that for z(t) = y(t) the following relation lim z(t) = 0

(1.59)

is satisfied. By (1.57)-(1.59) and using the special form of function V we obtain the existence of the following limits lim V(u(t),y(t),z(t)), —y+oo

lim t-++oo

/ ip(x) dx. J 0

Hence lim oj(t) = Ljn. t->+oo v ' This completes the proof of the theorem. Now we compare non-local conditions (1.53) and (1.54) of the transition from the initial state w = Q,x = 0 , i = 0 to the operating regime u — WQ,X = XQ,X = 0 with conditions (1.41) of asymptotic stability of the operating regime. Conditions (1.41) and (1.54) are similar in form. However, there is a slight difference: the expression on the right-hand side of inequality (1.54) includes a factor \ / 3 .

The Watt governor transient

processes

31

Condition (1.53) is an additional condition on the spring constant. Unlike conditions (1.41), the violation of which implies the physical nonrealizability of an operating regime (compare with the instability condition (1.42)), conditions (1.53) and (1.54) are sufficient conditions only. However in the engineering practice it is often impossible to pinpoint all parameters and the mathematical model is always a certain idealization. Therefore, in many cases the information, obtained by means of the sufficient conditions (1.53), (1.54), turns out to be quite sufficient. • y

n

/^ ~^x pa>o ffl

I

o

m

Fig. 1.11 The schematic representation of transient process is given in Fig. 1.11.

Chapter 2

Linear electric circuits. Transfer functions and frequency responses of linear blocks

2.1

D e s c r i p t i o n of l i n e a r b l o c k s

In the previous chapter a nonlinear mathematical model was considered. T h e linearization was an substantial part of investigation. In the present chapter we show t h a t the widespread electric circuits, containing resistors, capacitors, and inductors, can be described by linear differential equations. We consider first the simplest electrical circuit, i?C-circuit, which is often used in radiotechnology as a low-pass filter (Fig. 2.1). Here R is a resistance, C is a capacitance, U\{t) and U2(t) are voltages.

R u2

Fig. 2.1

Fig. 2.2

Let us find the relation between u\(t)

and ui(t),

Ri(t)-ui{t)-u2{t).

using of the O h m law (2.1)

Here i(t) is a current intensity. Recall t h a t dq{t) « • ( < )

=

dt

'

(2.2)

where q(t) is a quantity of electricity. We can consider this quantity of electricity on the capacitor plates with the capacity C . Since the plate-to33

34 Linear electric circuits.

Transfer functions

and frequency responses of linear blocks

plate voltage is v,2(t), from the property of capacity we have q(t)=Cu2(t).

(2.3)

Putting (2.3) in (2.2) and (2.2) in (2.1), we obtain RC^-

+ u2 = u1.

(2.4)

Consider now iilC-circuit (Fig. 2.2). In this case an electromotive force of self-induction e(t) is added to the voltage drop ui(t) — U2(t). For e(t) the following formula e{t) = -L

di(t) dt

(2.5)

is well known. Therefore in place of relation (2.1) we can take Ul{t)

- u2(t) + e(t) = Ri(t)

(2.6)

or, using (2.5), the following equation U!{t) - u3{t) - L^jp-

=

Ri(t).

(2.7)

Formulas (2.2) and (2.3) are obviously valid. Substituting (2.3) into (2.2) and (2.2) in (2.7), we finally obtain L

C

d?U2 ^ +

dv.2 -dT +

RC

U2

=

Ul

-

(2.8)

Thus, equations (2.4) and (2.8) are those relating ui and W2 for the RC and i?CL-circuits respectively. It is also useful to regard the values ui(t) and U2(t) as the input and output of the block respectively. These blocks are described by equations (2.4) or (2.8) (Fig. 2.3).

Fig. 2.3 Let us remark that from the formal point of view in both cases the input and output can be interchanged: it is possible to supply voltage u\{t) on the input and then to observe voltage U2{t) on the output and vice

Description

of linear blocks

35

versa. However u-zft) as the o u t p u t of block L turns out to be a solution of equations (2.4) with initial d a t a 1*2(0) or with initial d a t a «2(0), "2(0) while the o u t p u t ui(t) is uniquely described by formulas (2.4) and (2.8) only. Note also t h a t in an engineering practice the cases t h a t a linear block is a s u m of operators of differentiation are rare in occurrence. In this case to an input 142(f) + A sinwi (A is small and w is large) assign the o u t p u t u\(t) = RCv,2{t) + « 2 ( 0 + RCAu

cos Lot +

Asmut.

Since in this case the value RCAOJ is not small, the signal U2(t) passes through the block L with a large distortion. Further we shall show t h a t in the case t h a t an input is ui(t) there occurs an inverse effect and highfrequency distortions of the kind Asinut are depressed. Since formulas (2.1)—(2.3) and (2.5), (2.6) are also valid for the other of electric circuits, containing conductors, resistors, capacitors, and inductors, these circuits are described by linear differential equations with constant coefficients. In this case we are interested in the answer to the question of how the signal « i ( i ) is changed when passing via linear circuit L, i.e., how the input Ui(t) and the output u-2(t) of linear block L are related to one another. To answer this question in the framework of the linear circuits theory, at the end of the 19th century and early in the 20s the fundamental concepts of the control theory such as input, output, transfer function, and frequency response were stated. We see t h a t equations (2.4) and (2.8) admit a natural generalization:

Here A/"(-4) and A4 (4-) are the following differential operators

M (^)

u := Afnu^

M ( j \

u := Mmu{m)

+ Un^n~^ + Mm-iu(m'1]

+ ... + Af1ii + Afou, + ••• + Miii

+

M0u,

where A/i and Mi are some numbers. T h e previous remark on the noise immunity of blocks (2.4) and (2.8) implies the following restriction: m < n. W i t h o u t loss of generality it can be assumed t h a t Afn = 1.

36 Linear electric circuits.

Transfer functions

and frequency responses of linear blocks

The following notation of inputs and outputs: 0 be fixed and in the definition of controllability we assume that x(T) = x\ = 0. Then identity (3.6) has the form 1 T

f e-ATb£(T)dr o

= 0.

(3.7)

Now we multiply both sides of this identity by the vector z* T1

z~ z*x_0 + f z*e-ATbZ{r)

dr = 0.

(3.8)

0

Since z*eAtb = 0, V< G M1, by identity (3.8) we obtain that z*x0 = 0. However the last relation cannot be true since Xo is an arbitrary vector fromIR". Thus, if formula (3.3) is not valid, then system (3.1) is not controllable, i.e., the controllability implies formula (3.3). Now we show that condition 2) yields condition 1). Suppose, condition 1) is not true. Then condition 1') is not also true. Hence there exists a nonzero vector z £ M.n such that z*Akb = 0, Vfc = 0 , . . . , n - 1. We also prove that z*Anb= 0. By the Cayley identity we have An + S^xA"-1

+ .. . + 6iA + 60I = Q,

(3.9)

where Sj are coefficients of characteristic polynomial of the matrix A. By (3.9) z*Anb = -=(o)'

6) rank(