SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Volume 4
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Mathematical Problems of Control Theory An Introduction Gennady A. Leonov
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Mathematical Problems of Control Theory An Introduction
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Mathematical Problems of Control Theory: An Introduction Author: G. A. Leonov
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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
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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
2
The Watt governor and the mathematical
theory of stability of motion
the vertical by the angle (p, the arms set in motion the sleeve fitting on a shaft. T h e working agent (steam, water, diesel fuel) is piped. The pipe has a butterfly valve. The working agent creates a rotational moment of a shaft, which the W a t t governor is placed on. In the case of a steam turbine, for instance, the steam jet acts on turbine blades fitting on the shaft and creates a moment of force F. The equation, relating the moment of force F and the angular velocity of shaft u>(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
The Watt flyball governor and its
modifications
3
Here the rectangles may be regarded as some operators, which take each element of one functional space to elements of the other functional space. T h e operator "engine" takes each element of {ip} to elements of {to} and the operator "the W a t t governor" takes each element of {w} to elements of {
Engine
CO
Watt governor
Fig. 1.2 In the second half of 19th century it has become obvious t h a t the value ui(t) control cannot always be performed by means of the W a t t governor. T h e development of technology has led to new types of steam engines, for which the W a t t governors are practically useless. In studying the problems, related with the Watt governor operation, the first scholars of the day, such as I.A.Vyshnegradsky, J.Maxwell, a n d A.Stodola arrived at the conclusion being unexpected for engineers and nontrivial for mathematicians. T h e main ideas of these scholars are discussed below. For simplicity we consider a certain modification of the classic W a t t governor rather t h a n it himself. Consider one of these modifications, namely a turbine shaft control system, which is often used in a turbine construction. In Fig. 1.3 is shown a schematic diagram of such a system. T h e two equal governor flyballs of masses m, which are fitted on the bar, are fixed by springs and can slide along the bar. Here, unlike the previous case, a bar is used rather than hinges. T h e governor flyballs are also joined together by means of a membrane. T h e value of bending of the
4
The Watt governor and the mathematical
theory of stability of motion
membrane depends on a motion of flyballs along the bar. The special oil fills a chamber (which is above the membrane) of device. This device is called a servomechanism. The oil fulfills here two functions. It lubricates all moving details and moves the piston, rigidly connected with the valve. A constant forcing pressure induces an inflow of oil, which is assumed to be an incompressible fluid. The quantity of an outflowing oil depends on the bending of membrane.
o o
o
Forcing pressure
^>
^
Shaft
. Valve Engine
Working agent Pipeline
Fig. 1.3 Let us now consider a negative feedback. If the angular velocity of the shaft ui(t) decreases, then the centrifugal force, acting on the flyballs, also decreases and the springs tighten the flyballs to the center of shaft. In this case the bending of membrane increases and a size of the outlet decreases. As a result the amount of fluid, running out the outlet, decreases and its
The Watt flyball governor and its
modifications
5
volume increases. The fluid, remaining in the chamber of servomechanism, moves up the piston, opening the valve and increasing thus the flow of working agent. If the angular velocity w(i) increases, then the valve comes down and cuts off an excess of working agent. Beginning from 1952 such governors have been arranged on turbines at the Leningrad Metallic Factory [41]. Now we consider the second modification of the Watt governor. The functions of it are the same as before (see Fig. 1.1 and Fig. 1.3) but the mathematical description, from the methodological point of view, is much simpler than a description of the governors mentioned above. Such a governor is shown in Fig. 1.4.
The only difference between the operation of the classic Watt governor and the system, shown in Fig. 1.4, is as follows. In the case of the system mentioned above the centrifugal force / acts on the flyballs of masses m, which slide along a bar and are connected with the shaft by springs. At time t this force declines the masses from the stress-free position of spring by a value x(t). Thus the output of the modified Watt governor is a value x(t), which is transmitted into a device to move a valve by a distance u depending on x. Further we are interested in the fact of the existence of a functional dependence u{x) only, leaving aside the design of this device.
6
The Watt governor and the mathematical
theory of stability of motion
Recall t h a t x(t) and / satisfy the following equations f =/3m{x + r)oj2,
(1.2)
mx + ax + jx = f.
(1-3)
Here a, (3,7, r are positive numbers. T h e value —jx corresponds to an elastic force of the spring (the Hooke's law), the value — ax corresponds to a friction force. We also assume t h a t the viscous friction law is valid, i.e., the friction is proportional to the velocity x(t) and the number —a is a coefficient of proportionality. T h e other of the kinds of friction is neglected. T h e number r denotes the length of spring in the stress-free state. Under the assumption that G is constant and F is a function of u, namely F = F(u), by (1.1)—(1.3) we have the following system of equations for describing a value u>(t) control Ju=F(u(x))-G, mx + ax + -yx = (3m(x + r)w 2 . We see t h a t for the normal functioning of this system, the equations (1.4) must have a solution of the form U)(t) = L00,
xit) = X0.
(1.5)
Here wo is a required angular velocity of shaft and XQ is a number satisfying the following equations F(u(x0)) = G, fx0 = /3m(x0 + r)u>Q.
(1.6)
It follows t h a t equations (1.6) are necessary and sufficient conditions for the existence of the solution of system (1.4) in the form (1.5). Equations (1.6) can be obtained in the following way. From physical considerations we can conclude t h a t the first equation of (1.6) has a root XQ. T h e second equation is satisfied by the choice of a value of spring deflection rate and by using XQ, obtained previously. In this case a stability of trivial solution (1.5) is a main scientific and technical problem, a solving of which was far nontrivial and fully unexpected for engineers. Consider now system (1.4). Let us linearize it in a neighborhood of solutions (1.5) under the assumption that relations (1.6) are satisfied and
The Hermite—Mikhailov
criterion
7
the function F(u(x)) is sufficiently smooth. Denoting Ax(t) — x(t) — XQ and Aw(t) = w(t) — wo, we obtain the following linear system J ( A w ) * = F0Ax, m(Ax)"+a(Ax)'+-fAx
= f3mojlAx
+ 2j3rmJJo{x0 + r)A<jj.
^ ' '
Here Fo = F'(U(XQ))U'(XO), the higher order terms are omitted, and all the functions are replaced by their linear approximations. The linearizations of the type (1.7) are usually called equations of linear approximation. We now consider the equations of linear approximation (1.7) and then a relation between equations (1.7) and original equations (1.4). System (1.7) is equivalent to the following equation of the third order (Aw)
H
(Aw) H m where /o = 2{3moJo(xo + r ) . equation has the form
-Aw — A w = 0, (1.8) m Jm T h e characteristic polynomial of such an
3 , a 2 , 7 fimul f0F0 . +—p + P-—j—• I1-9) m m Jm Consider equation (1.8). From the elementary theory of integration it follows t h a t any solution Aw(<), corresponding to small initial d a t a Aw(0), (Aw(0))*, (Aw(0))**, remains small and tends to zero as t —> + o o if and only if all zeros of the polynomial Q(p) have negative real parts. In this case the solution of linear system is called asymptotically stable. A polynomial Q(p), all zeros of which have negative real parts, is often called a stable polynomial. n,i \ Q{p)=p
1.2
T h e Hermite—Mikhailov criterion
Does it make possible to say anything about a stability of the polynomial Q(p) without finding its zeros? (For mathematicians this question was raised in 60s of the 19th century in connection with the problem of studying the W a t t governor). For a polynomial of the second degree Q(p) = p 2 + ap + /3 the answer to this question is sufficiently simple. This is a necessary and sufficient condition for a and /? .to be positive. In this case for a polynomial of the third degree of the type of (1.9) the answer to the question is considerably complicated.
8
The Watt governor and the mathematical
theory of stability of motion
At present there are many various criteria of a stability of polynomials, namely the criteria of Hurwitz, Routh, Lienard. These criteria can be found in the remarkable book [16]. A criterion being the most simple and popular among engineers is a criterion of Hermite-Mikhailov. It will be discussed below. Consider a polynomial of degree n with the real coefficients f{p)=p"+an_1pn~1
+ ... + a0.
We state a simple fact, which is sometimes called the Stodola theorem. P r o p o s i t i o n 1.1. If all zeros of the polynomial parts, then all its coefficients a8- are positive.
f(p)
have negative real
P r o o f . Denote by ay real zeros of f(p) and by j3k the complex ones. Since a,- are real, the numbers fbk are also zeros of f(p). T h u s , the polynomial f{p) may be represented in the form of the following product
/(p)=no° - a ^ n ^ - M(P - &)= /-
j
Q
= II(P- ;) j
II(p2-2(Re^)p+l^l2)k
If for all j and k the conditions ctj < 0 and Re/?/. < 0 are satisfied, then the following products
Y[(p2-2(Re/3k)p+\pk\2)
Hip-aj), j
k
are polynomials with positive coefficients. Consequently, the polynomial f(p) has also positive coefficients. Denote by m the number of zeros of / ( p ) with the positive real p a r t s . Consider the values of polynomial f(p) on the imaginary axis: f(iw),
w e R1
A curve on the complex plane, {p — /(icj)|w £ M 1 }, is called a hodograph of polynomial f(p). Sometimes this hodograph is called the Mikhailov hodograph. Consider a function
Argf(iu).
The Hermite—Mikhailov
criterion
9
Here Arg z is a continuous branch of the many-valued function: a r g z + 2fc7r,
(1.10)
where k is an integer and arg z is an argument of z such t h a t — 7r < arg z < 7r. For definiteness, we assume t h a t <^(0) = 0. If the hodograph crosses a ray in the complex plane { I m z = 0, R e z < 0} at the point w = u>o, then we take a branch of function (1.10), which assures the continuity of the function
P r o p o s i t i o n 1.2. If f{iu) Mikhailov
of
Hermite-
&
/(p)=n^-Qi) n(p-&) j
k
and applying the well-known theorems on an argument of a product of complex numbers, we obtain -f oo
+ 0O
. j
-OO
(1.11)
*!
Now let us compute the values AAxg(iu-aj)\+™,
AArg(jw-/3*)|+~.
For this purpose we consider on the complex plane (Fig. 1.5) the numbers ctj, iu>, iu — ctj and the corresponding to t h e m vectors. W i t h increasing u from —oo to + o o the end of vector, corresponding to the number iu —ctj, monotonically slides along the line R e 2 = — Reofj and
10
The Watt governor and the mathematical
theory of stability of motion
rotates counterclockwise as it is shown in Fig. 1.5. Whence it follows that A A r g ( z w - a i ) | t ^ - ir. Consider now the numbers /3k, «w, iui — /3k on the complex plane (Fig. 1.6) and the vectors, corresponding to these numbers. i
Im z
J*
-Re 4
Rez
' im
ia-fi/ Fig. 1.5
Fig. 1.6
As ui increases from —oo to +oo the end of vector, corresponding to the number iui — /3k, monotonically slides along the line Rez = — Refik and rotates clockwise as it is shown in Fig. 1.6. Consequently,
AArg(iw -/3k)\tZ = ~nFrom decomposition (1.11) it follows that Ayj(o;)|_
= [n — m)n — mn = 7r(n — 2m).
The proof of Proposition 1.2 is completed. The following criterion results from Proposition 1.2. The Hermite-Mikhailov criterion. Let f(iu) / 0 , V u G l 1 . for a polynomial f(p) to be stable it is necessary and sufficient that A< u
p( )\-Z
= n7T
-
Then
(1.12)
The Hermitt—Mikhailov
criterion
11
T h e following remark allows us to simplify a verification of the test (1.12). Since the coefficients of polynomial f(p) are real, we have the following relations R e / H w ) = Re/(iw)
Im/(—ioj) =
—Imf(ioj).
This implies t h a t the hodograph f(p) is symmetric with respect to the real axis (see Fig. 1.7)
Fig. 1.7 Then in place of formula (1.12) it makes possible to consider the following condition
Note t h a t together with the proof of Proposition 1.2 we have also proved the fact t h a t for stable polynomials the function
0, going out of the point w = 0 on the positive semi-axis {Re z > 0, I m z = 0 } , sequentially crosses the semi-axes { R e z = 0, I m z > 0 ) , { I m z = 0, R e z < 0},... while passing n quadrants. If f(iui) ^ 0 Vw 6 M.n and with increasing u> the hodograph of f(p) for ui > 0, going out of a point on the semi-axis {Re z > 0, I m z = 0}, sequentially crosses the semi-axes { R e z = 0 , l m z > 0 } , { R e z < 0, I m z = 0}, ..., asymptotically tending to the n—th semi-axis in order, then f(p) is a stable polynomial (by formula 1.12).
12
The Watt governor and the mathematical
theory of stability of motion
Consider a polynomial of the third degree f(p) = p3 + ap2 + j3p + 7. By Proposition 1.1, a necessary condition for f(p) to be stable is that the conditions a > 0, /3 > 0, 7 > 0 be satisfied. Here f(iw) - (7 - aco2) + (P - w2)wi. Then the crosspoints of the hodograph of f(iw) for w > 0 with the semiaxes {Rez > 0, Imz = 0}, {Rez = 0, Imz > 0}, {Rez < 0, Imz = 0} correspond to the following points
In this case /(0) = 7, f(tU2) = {P — l/ot) \/j]a i, /(z'w3) = 7 - a/?. Whence it follows that a polynomial is stable if and only if /?-->0, a
7-a/3<0.
(1.14)
In addition, we have Reftioj) , • ~»0 1.15 T Im/(zw) as w —> +00. Thus, if (1.14) and (1.15) are satisfied, then with increasing w the hodograph of f(p) for u > 0, going out of a point on the semiaxis {Rez > 0, Imz = 0}, crosses the semi-axes {Rez = 0, Imz > 0}, {Rez < 0, Imz = 0} sequentially and exactly one time and asymptotically tends to the third semi-axis {Rez = 0, Imz < 0} and vice versa such a behavior of the hodograph is possible only if (1.14) is satisfied. Thus, for the polynomial to be stable it is necessary and sufficient for the following inequalities a >0,
/?>0,
7
>0,
a/3>7
(1.16)
to be satisfied. Sometimes these inequalities are called the condition of Vyshnegradsky.
Theorem on stability by the linear
1.3
approximation
13
Theorem on stability by the linear approximation
Consider a solvability of a matrix equation A*H + HA = G,
(1.17)
which is often called the Lyapunov equation. Here A and G are the given n x n-matrices and n x n-matrix if is a solution of (1.17). We consider the case that H and G are symmetric. Lemma 1.1. Suppose, all the eigenvalues of matrix A have negative real parts and G < 0. Then equation (1.17) has a unique solution f eA'tGeAtdt. o
H=-
(1.18)
Recall that the inequality G < 0 means that the corresponding quadratic form z*Gz is negative definite. We also note that from (1.18) it follows that H > 0. Really, the nondegeneracy of a matrix eAt implies that -(eAtYGeAt>0,
Vi>0.
Hence for any x £ K n we have ~x*{eAt)*GeAtx>0,
Vt > 0.
Then —
OC
x*Hx = - J x*(eAt)*GeAtx>Q. P r o o f of L e m m a 1.1. The assumptions on the eigenvalues of the matrix A imply that finite integral (1.18) exists. It is obvious that j ( e ^ ' G e A t ) = A*(eA'lGeAt)
+
(e^'Ge^A.
Integrating this identity from 0 to +oo and taking into account that lim eA'*GeAt
= 0,
t—>-+oo
we obtain equation (1.17) with H, satisfying (1.18).
14
The Watt governor and the mathematical
theory of stability of motion
Now we show the uniqueness of the solution of equation (1.17). Suppose the contrary, i.e., t h a t H\ and H2 are two solutions of equation (1.17). Then H = H\ — H2 satisfies an equation A*H + HA = 0.
(1.19)
Consider a vector function eAtx0,
x(t) = where XQ is a vector. By (1-19) we have j (x(t)*Hx(t))
= x(t)*(A*H
+ HA)x(t)
= 0.
Hence, x(t)*Hx(t)
=
x*0Hx0.
However x(t) —> 0 as t —> + 0 0 by virtue of the assumptions on eigenvalues of A. Thus, we have XQHX0 = 0, Vzo € ffi". Whence it follows that the symmetric matrix H is null. This completes the proof of lemma. Consider now a differential equation -£ = f(t,x),
teM\
i f l " ,
(1.20)
where f(t,x) is a continuous vector function: R ' X I ™ - } ! " , Further we assume t h a t all the solutions x(t,to, x0) with initial d a t a x(t0,to, x0) — x0 are defined on the interval (to, + 0 0 ) . D e f i n i t i o n 1.1. A solution x(t,t0,x0) of system (1.20) is said to be Lyapunov stable if for any number e > 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)\
<£,
Vt>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 = <^o(^o), S(e) = 5(e,to). If <5o and S(e) can be chosen independent of to, then the solution x(t,to, XQ) is called uniformly Lyapunov stable and uniformly asymptotically stable. T h e Lyapunov instability is a logical negation of the Lyapunov stability. Below the examples of stable and unstable solutions are given. Consider the equation of pendulum 6 + a6+
(g/l) sin6> = 0.
(1.22)
Here 6(t) is an angle of deviation of the pendulum from the vertical position, / is a length (Fig. 1.8), g is acceleration of gravity and a is a friction factor.
Fig. 1.8 Equation (1.22) can be given in the form (1.20):
T) =
—arj
sin#.
(1.23)
Schematic representation of the two-dimensional phase space, filled by trajectories of system (1.23), which is often called a phase portrait, is shown for the case a — 0 in Fig. 1.9 and t h a t for the case a > 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 for a > 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<<5}
(1.27)
The existence of such a 5 follows from the relation V(0) = 0 and a continuity of function V[x). We shall prove that for initial data XQ such that \XQ\ < S the inequality \x(t, to, XQ)\ < e, Vi > to is satisfied, i.e., the Lyapunov stability is valid. Suppose the contrary. Then by the continuity of the solution x(t, to, x0) there exists a number r > 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))
(1.28)
On the other hand, by (1.26) and (1.27) V{x{T,t0,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><0
(1-32)
is also satisfied. Inequalities (1.31), (1.32), continuity of the function W(x), and the inequality W{x) < 0, Va; (E {x\ 7 < |a;| < £0} imply the existence of a negative number as such that W(x{t,t0,Xo))
< ae.
It follows that t
V{x(t, t0, x0)) < V{x0) + J W{X{T, t0, x0)) dr < to
'
-)•
' t^t+co
-OO.
20
The Watt governor and the mathematical
theory of stability of motion
T h e last inequality contradicts the inequality V(x(t, to, XQ)) > (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) <0forx^Q. Then the zero solution of system (1.20) is Lyapunov unstable. P r o o f . Suppose the opposite, i.e., for e > 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))
<0
and, consequently, there exists a number 7 > 0 such t h a t \x(t,to,x0)\
> 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
+ se(t-t0)
->
-00,
to, xo)) dr
<
Theorem on stability by the linear approximation
21
which contradicts the assumption on the Lyapunov stability. The theorem is proved. Consider now system (1.20) represented in the following form -£=Ax
+ g{t,x).
(1.33)
Here A is a constant n x n-matrix, g(t, x) is a continuous vector-function: M1 x l " —> M1. Suppose that in some neighborhood of the point x — 0 the following inequality holds |s(t,a:)|
VieM1.
(1.34)
Here ae is some number. Assume that A does not have pure imaginary eigenvalues. Let us construct functions V(x) and W(x), which satisfy the assumptions of Theorem 1.1 or Theorem 1.2. At first, consider the case that all the eigenvalues of A have negative real parts. In this case for G = —I by Lemma 1.1 a matrix H > 0 can be found such that A*H + HA=-I.
(1.35)
Let us consider further the quadratic form V(x) — x*Hx, which is positive definite: V{x) = x*Hx>0,
Vi^O.
We observe that V(x) satisfies assumption 1) of the theorem on asymptotic stability. Equation (1.35) can be rewritten as 2x*HAx = — |ar|2. Therefore, taking into account (1.33) and (1.34), we obtain V(x) = 2x*H(Ax + g{t,x)) < -\x\2 + 2\x*H\ |z|ae. If 32 satisfies an inequality as < (4|tf I)" 1 ,
(1.36)
then assumption 2) of the theorem on asymptotic stability is also satisfied with W{x) = -\x\2/2. Thus, the following result can be stated.
22
The Watt governor and the mathematical
theory of stability of motion
C o r o l l a r y 1.1. If A is a stable matrix, i.e., all its eigenvalues have negative real parts and condition (1.36) holds, then the zero solution of system (1.33) is asymptotically stable. Consider the case t h a t the matrix A has no pure imaginary eigenvalues and m of its eigenvalues have positive real parts. W i t h o u t loss of generality we may assume t h a t the matrix A has the following block representation
where A\ is a stable [n — m) x (n — m)-matrix, A2 is a stable m x m-matrix. Applying again Lemma 1.1, we can prove the existence of symmetric matrices H1 > 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:|.
<
Then for
x<
(4(1^1+
\H2\))~1
(1.39)
Theorem on stability by the linear
approximation
23
assumption 2) of the instability theorem is satisfied with W(x) = —\x\2/2. We can state the following Corollary 1.2. If A does not have eigenvalues on the imaginary axis and is unstable (i.e., it has also eigenvalues with positive real parts), then under the assumption that inequality (1.39) is valid the zero solution is Lyapunov unstable. For the following autonomous system dx dt = / ( * ) ,
/(O) = o
xe
with continuously differentiate vector function f(x), 1.2 may be stated in terms of the Jacobi matrix
df(x) dx
dx\
dxn
dxi
dxn '
(1.40) Corollaries 1.1 and
dfn_.
which is given at the point x = 0: A =
dj_ dx
x=0
In particular, the following result is true. Theorem 1.3 (On stability by the linear approximation). Let A have no pure imaginary eigenvalues. If A is stable, then the zero solution of system (1.40) is asymptotically stable. If A is unstable, then the zero solution of system (1-40) is Lyapunov unstable. As an example, consider pendulum equation (1.23). At the point 8 = 0, r\ = 0 the Jacobi matrix is as follows 0
l
-g/i
-a
Its characteristic polynomial takes the form p +ap +
9
24
The Watt governor and the mathematical
theory of stability of motion
It is clear that for a > 0 the matrix A is stable, i.e., both eigenvalues have negative real parts. Consequently, the equilibrium considered is asymptotically stable. Now we consider the equilibrium 8 = n, rj — 0. To apply the theorem, make the change of variables 8 = 8 + TT, r\ = rj. Then we obtain the system
0 = v, rj = —arj — — sm{8 + TT), whose the Jacobi matrix at the point 8 = 0, rj = 0 takes the form A=
{g/i
-a)'
Its characteristic polynomial is given by 2 , 9 p + ap- -. Obviously, for 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
(z + ayy +
This function has properties similar to those of functions V(x) in the Lyapunov theorems on asymptotic stability and instability. Therefore the function V(w, y, z) can also be called the Lyapunov function. It is obvious that for any solution w(t),y(t),z(t) of system (1.44) the following equation holds V(co(t), y(t), z(t)) = (
(1.45)
Lemma 1.2. Let an inequality mj0 > a2
(1.46)
be satisfied and on the interval [0,wi], where u i is defined by / ip'(u)
(1.47)
be valid. Then for a solution of system (1.44) with initial data w(0) = 0, 2/(0) — —j^ XQ, Z(0) = 0 the following inclusion holds w(0€[0,wi],
Vi>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<0 it follows t h a t V(u(T),y(T),z(r))<0.
(1.49)
In case 1) we have 0 since w(t) = y(t) and, consequently, V(UI(T), y(r), Z(T)) > 0. The last inequality contradicts inequality (1.49). In case 2) we have 0 since wi > 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)\
Vt>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<
+oo
C /
u(r)dr.
28
The Watt governor and the mathematical
theory of stability of motion
This estimate and assumption 3) of the lemma results in a convergence of the following integral +00 /
U(T)U(T)CIT.
0
Then from (1.50) it follows t h a t there exists a limit lim u[t)2 = v. t-H-oo
Assumptions 2) and 3) imply t h a t v = 0. T h e l e m m a is proved. L e m m a 1.4. Let for a continuous on [0,+co) function u{t) the following assumptions hold: 1) for a number C we have \u(t)\
V i > 0,
lim u(t) = 0.
Then
lim it(t) — 0.
P r o o f . Suppose the opposite, i.e., there exists a sequence tk —> + 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;(<) = 0,
lim (Az(t))* = 0
(1.56)
are satisfied. Recall that here /o = 2/3mwo(a;o + r). P r o o f . Inclusion (1.55) follows directly from Lemma 1.1. Really, wj can be computed, using the equation - (Jmruf
- 70K0W1 = 0.
Condition (1.47) takes the form a7o —r> 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))<-ey(t)2,
V 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)))
V< > 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:
" ' i ) "*(*><•
(2.10)
Fig. 2.4 The operator M(jj-) acts first on the function £(<):
Then
Tt'"(t)
= /(t)
-
(2.11)
It is clear that er(t) is not determined by the function £(t) only. The initial data are the following /
^(0) &(0)
\
xo =
The vector XQ is called initial data of block L. So, the block L is the operator, described above, which acts on the direct product of sets L
{£(<)} x {a*} 4 Mo}.
(2.12)
The set of inputs {£(<)} is a s e t of functions, which the operator L can be defined on. In the case considered it is the functions that at any point t G [0, +oo) have m continuous derivatives. Then the function
Description
of linear blocks
37
f(t) = • / ^ ( ^ ) ^ ( 0 i s defined and continuous at any point t 6 [0,+00). The continuity of f(t) implies the existence of the solution a(t) of equation (2.11), which is determined for alii > 0 and has the n—th continuous derivative. The set of initial data is a subset of Euclidean space. Now we discuss another description of block L, which makes it possible to define the operator L on a set of continuous functions {£(<)} only. Using previous notation, the operator L is as follows
7,(0)
M dt
»? = * ,
a = A<|-
M
,
(2.13)
x0
We define a function rj(t) as a solution of the equation
*l
5
!=«
with initial data XQ and apply A f ( ^ ) to the function r)(t) :
It can readily be seen that if the block L is given by equations (2.13) and a function £(£) has the m—th continuous derivative, then it is possible to describe this block by equations (2.10). Really, we have
A M ^ l ^
= Mlj-t)M
j)M{jt
r) =
*)' = »{*)<•
Now we shall show that description (2.13) is a partial case of the following equations dx = Ax + b£, ~dl
x0 = x(0),
(2.14)
where A is a constant n x n-matrix, b and c are constant matrices of dimensions nxm and n x / respectively. The asterisk * denotes the transposition in the real case and Hermitian conjugation in the complex case.
3 8 Linear
electric
circuits.
Transfer
functions
and frequency
responses
of linear
blocks
For the sequel we are needed in the following notation ' xi j] = xi,
77 = 1 2 ,
...
,??
(n
1(
= X * n•*„ ,
X —
Then equations (2.13) may be written by ii — x2, &n — 1 — ^n?
AToxi + Z{t), m-^m + 1
+ Mm-lXm + . . . + Mr,Xx. Hence we have / A =
0
1
0
...
0
0
1
...
:
:
'••
'••
0
0
...
0
0
b=
0
1
\-Afo
c =
/°\
\
0
-Afn-J
M„
1(0)
.*n(0)/
W^W/
\
Xo
0
V 0
'si(O)'
/
Let us recall that a solution of equation (2.14) may be written in the following integral form (the Cauchy formula) t At
x(t) = e x0+
IeA^-Th^r)dr.
Therefore a{t) = c*eAtx0 + f c*eA^-Thi{T)
dr.
(2.15)
Description
of linear blocks
39
Given the description of block L in the form (2.15), the following generalization can be obtained t
r{t,T)Z{T)dT.
(2.16)
0
Here a(t) is a continuous /-dimension vector-function from a certain functional set {a(t)}. In the theory of integral operators a matrix function 7(2, T) of dimension / x m is called a kernel of integral operator t
f*f(t,T)Z{T)dT. 0
T h e description of blocks (2.14) can also be transformed to the form (2.16) if matrices A and b depend on t. Further we consider the case of a difference kernel only 7(<> r )
=l{t-r).
T h e following transformation t
j l{t-r)i{T)dr o is often called a convolution operator. T h u s , among the descriptions of L, operator (2.16) is most general. In the case of description (2.14) we obtain a(t) = c*eMx0,
l{t, T) =
c*eA^-Th.
Now we discuss the answer to the question: In what sense is there understood a linearity of all the above-mentioned descriptions of block i ? In the case of descriptions (2.10), (2.13), and (2.14) the block is linear since all the equations mentioned are linear. However it is more convenient to introduce another definition of linearity t h a t includes description (2.16). D e f i n i t i o n 2 . 1 . A block L is called linear if to any linear of arbitrary inputs £i(i) and £2(2) t*iti(t)
+
pi&{t)
combination
40 Linear electric circuits.
Transfer functions
and frequency responses of linear blocks
assigns the linear combination (ii(<Ti(t)-a1(t))+ii2(
Here o-,{t)=ai(t)+[l(t,T)ti{T)dT.
Since descriptions (2.10), (2.13), and (2.14) of the block L may be reduced to the form (2.16) and description (2.16) is obviously linear, the expressions (2.10), (2.13), and (2.14) are also linear.
2.2
Transfer functions and frequency responses of linear blocks
For the linear blocks, described by equations (2.10) or (2.13) we define a transfer function in the following way. Definition 2.2. A function
»<* = -%$.
<->
defined on a complex plane C, is called a transfer function W(p) of linear block. Definition 2.3. An I x m-matrix W(p)^c*{A-pI)-1b,
(2.18)
whose elements are the functions defined on a complex plane C, is called a transfer function (matrix function) of a linear block given by equations (2.14). It is clear that transfer function (2.17) is defined everywhere except for the points C that are zeros of polynomial Af(p). These singular points are poles of function W(p). The same proposition is valid for W(p) in the form (2.18). By using the rule of matrix inversion we conclude that elements of
Transfer functions
matrix (A — pi)
1
and frequency responses of linear blocks
41
take the form a
ij(p)
t=l,
a{p)
where a(p) is a characteristic polynomial of matrix A: a(p) = det(pl — A) and aij(p) are some polynomials of degree not greater than n — 1. In this case the elements of W(p) are the following functions A-j(p) a(p)
1,
J,
J
-,••
where Pij{p) are some polynomials of degree not greater than n — 1. The poles of these functions are zeros of polynomial a(p). In other words, the poles of W(p) coincide with the eigenvalues of matrix A. Show that functions (2.17) and (2.18) are well defined. It means that if we rewrite equations (2.10) and (2.13) in the form (2.14)., then the transfer functions from Definitions 2.2 and 2.3 coincide. In this case the matrices A, b, and c have the form
( °0
1
0
0
1
0
0
0
A= 0
0 1 -Mn
/
M0\
\-Mo
/°\ 0
\
M„ 0
(2.19)
V o / For the matrices A, b, and c the following formula c*{A-PI)-1b
=
M0 + Mip+ ... + det(pl - A)
Mmpm
(2.20)
is valid. Indeed the type of vector b implies that for the expression (A—pl)~lb to be computed it is necessary to find the last column of the matrix (A —pi)-1
42 Linear electric circuits.
Transfer functions
and frequency responses of linear blocks
only. Using the rule of matrix inversion we obtain that this column consists of algebraical complements of the last row of matrix, namely / p
-1
0
...
0
p
-1
0
0
\
0 0 VA/O
p 7V„_2
...
...
-1 {Nn-l+p)J
divided by det(pl — A). It is obvious that the required algebraical complements are equal to - 1 , -p,..., —p""1. Therefore / -ii (A-PI)-1b=-
1
\
det (pi - A) VP"- 1 /
Formula (2.20) follows directly from the last relation. Since det(pJ - A) = pn + Afn^ip"-1
+ ... + Af0,
we obtain for the matrices of the form (2.19) the following formula
^{A-piyH-.
M(P) M(p)
(2.21)
Note a very important property of a transfer matrix of system (2.14). Theorem 2.1. The transfer matrix W(p) is invariant with respect to nonsingular linear transformation x — Sy (det 5 ^ 0 ) . P r o o f . Having performed the linear nonsingular transformation x = Sy in system (2.14), we obtain a new system dy = S-1ASy dt
+ S~1bc:,
cr = c*Sy. Let us compute a transfer matrix of this system Wx{p) =
c*S{S-lAS-pI)-lS-H.
(2.22)
Transfer functions
and frequency responses of linear blocks
43
Since
{S~lAS - pi)'1 = (S^AS - pS-^)-1 = = {S-1(A-PI)S)-1 =S~1{A-pI)-1S, we have c*S{S~1AS
- p / ) - 1 ^ - 1 ^ = c*{A-
pl)-lb.
Thus, the transfer matrices W(p) of system (2.14) and W\{p) of system (2.22) coincide. The correctness of Definitions 2.2 and 2.3 follows directly from formula (2.21) and the invariance of transfer functions of system (2.14). Consider now a linear block of the form (2.16) with difference kernel 7(*»T)
=l(t-T).
To define the transfer function of such a block we need the Laplace transformation. Consider a set of real continuous on [0, +oo) functions {f{t)} satisfying the following inequalities \f{t)\
V*>0.
(2.23)
Here the numbers p can be different for different functions from the set {/(<)} and the numbers as are the same for the whole set {/(<)}• Definition 2.4. The Laplace transformation is an operator, defined on the set {f(t)}, such that it takes each element from this set to an element of the set of complex-valued functions {g(p)}, given on the set {p\
PeC,
Rep>ae},
(2.24)
by the following rule +oo
g(p)= J e-KfWdt.
(2.25)
o Inequality (2.23) implies, obviously, the convergence of integral (2.25). This definition may be extended in the natural way to the vector functions f(t). In this case on the left of inequality (2.23) Euclidean norm | • | must be taken rather than the absolute value | • | and g(p) are vector functions, whose dimensions are the same as that of vector functions f(t).
44 Linear electric circuits.
Transfer functions
and frequency responses of linear blocks
Let us now state two important properties of the Laplace operator, denoting this operator by L: {/(<)} —> {g(p)}Proposition 2.1. If a function f(t) from a set {/(it)} has a continuous derivative at any point t, then for f(t) the Laplace operator may be defined by the formula +00
L(f(i))=
e~vtf{t)dt
j
and the following relation L(f(t))=PL(f(t))-f(0)
(2.26)
is valid. The proof of this proposition is of a chain of relations +00 p
. +00
J e->«f(t) dt = e-r'fit)
+00 ' .
+p J e^ f{t) dt = - / ( 0 ) +
0
PL(f(t)).
0
Thus, the Laplace operator may be extended from the set {f(t)} to the set {/(<)} U {/(*)}. We recall also that the integral +00
J e-r'ftt),) d t does not always converge absolutely. Proposition 2.2. For functions fi(t) and /2(<) from the set {/(*)} the following formula t
L^J
valid.
/i(<-r)/ 2 (r)dr) = L(h(t)) L{f2{t))
(2.27)
Transfer functions
and frequency responses of linear blocks
45
The proof is of the following chain of relations t
t
+00
L(J
1
/i(*-r)/2(r)dr)= J
J h{t - r ) / 2 ( r ) di dt
e^
0
0 +00
| | e - P 7 i ( t - 7-) f2(r)dtdr
+ OO
J e-' f2{r)
e~^h{t
-
r)dtdT
J
e-^f1(t1)dt1dr
=
+ OO
+ OO
0
/2(r) J
+OO T
= J
= J
+00
T
e-P f2(r)dr
j
e ^ / i M ^ i
= L(f2(t))
L{h{t)).
0
Here t\ = t — T and the domain f2 has the form shown in Fig. 2.5.
t=t
a
Fig. 2.5 All the integrals in these relations converge absolutely by virtue of inequality (2.23). Propositions 2.1 and 2.2 are also true for the set of matrix functions or vector functions {/(<)}. In this case in formula (2.27) fi(t) is a k x m - m a t r i x function and f2(t) is an m x /-matrix function. Let us apply the tools, developed above, to the analysis of equations (2.13) and (2.14). Note t h a t if an input £(t) satisfies the inequality £(t)|< Pie*1{,
V*>0,
then for the vector function x(t) and the function rj(t), numbers ee2 and p2
46 Linear electric circuits.
Transfer functions
and frequency responses of linear blocks
can be found such that inequalities \X(t)\
\V^(t)\
Vi>0
(2.28)
are satisfied. Here i = 0 , 1 , . . . , n — 1. For proving this fact it is sufficient to show that the first inequality holds. The latter follows directly from estimates
)dr <
KOI < |e *(0)l +
< /3ext\x(0)\+
f \eA^-^\m(T)\dT o
<
t A
A
e^'^dr.
Here numbers f3 and A are such that eAt\
Vf>0.
It is obvious that estimates (2.28) imply the existence of p$ such that a(t)\
\/t>0.
(2.29)
Estimates (2.28) and (2.29) can be obtained by different methods, they can also be improved, and so on. In our consideration an important point is that estimates of this kind exist. In this case, choosing ae = max(asi,ae2) and assuming zero initial data: x(0) = 0 or 77(^(0) = 0, i = 0 , 1 , . . . , n — 1, by Proposition 2.1 we obtain the following relations
Transfer functions
and frequency responses of linear blocks
47
1) for equations (2.13)
L(tT(t)) =
L(M(±r,(t)
Af{p)L(T,(t)) = Lm)
\
,9„m
^6{i>
L{
L(
-W{p)L(t(t))
2) for equations (2.14) L(x(t)) = L{Ax{t) + b£(t)) \ L(a(t)) = L(c*x(t)) ] PL{x(t))
= AL(x(t)) + bL(t(t)) L(a(t))=c'L(x(t))
\ j
L(x(t))=-{A-pI)-1bL(S(t)) L(*(t))=c*L(x(t))
\ j
L(a(t)) = L(
(2 31)
'
-c*(A-pir1bL(t(t)), -W(p)L{Z(t)).
Thus, we obtain a very simple relation between the Laplace transforms of the input and output of linear block. If we now consider a description of linear block (2.16) with the difference kernel j(t, r) = j(t — r): a(t)=a(t)+
f 7(t-rK(7-)dr, Jo
(2.32)
assuming that the initial data of this block are such that a(t) = 0, then by Proposition 2.2 we obtain a relation similar to relations
48 Linear electric circuits.
Transfer functions
and frequency responses of linear blocks
(2.30) and (2.31) L(
(2-33)
In this case the following definition can be stated. D e f i n i t i o n 2 . 5 . The Laplace transform ofj(t), taken with the sign, W(p) = -Lfr(t))
(2.34)
is called a transfer function (matrix function) described by equations (2.32). For the function j(t) = c*eAtb we have
i ( 7 ( t ) ) = L{c*eAtb)
=
opposite
W(p)
of the linear
block,
e-ptc*eMbdt
f
+ 00
=
c*{A-pI)-1e^-pI^tb
=
-c*(A-pI)-1b.
o Here we take into account the following relation lim
t-y+oo
e(
A
- p / ) ' = 0,
which is valid under the assumption t h a t R e p > aa > max Re Xj (A), were i Xj{A) are eigenvalues of matrix A. Therefore function (2.34) is well defined. Let us now proceed to the definition of frequency response of linear block. Suppose t h a t as < 0 and, consequently, the transfer function (matrix function) is also defined on the imaginary axis. D e f i n i t i o n 2.6. A function W(iu) is called a frequency response of linear block. T h e Fourier transformation, together with the Laplace transformation, plays an important role in many fields of applied mathematics. In the control theory the Fourier transformation is often defined in such a way t h a t it coincides with the Laplace transformation on the imaginary axis + oo
F(f(t))= J e-^f(t)dt.
(2.35)
Transfer functions
and frequency responses of linear blocks
49
Such a definition, for example, can be found in [26]. In the calculus (see [14]) the Fourier transformation of the functions f(t), given on (—oo,+oo), is defined in the following way + 0O
Hf(t)) = ^
f eiwtf(t)dt.
(2.36)
— oo
We assume that f(t) = 0 on (—oo, 0). In this case the transforms (F(f))(iu>) and (^ 7 (/))(iw) of the same function /(£) are related by the formula (F(f))(iu) = V^(T(f))(-iu).
(2.37)
Such a difference in definitions is, as a rule, unessential and the use of one or the other of definition is a matter of convenience only. Further we shall use definition (2.35). From formulas (2.30), (2.31), and (2.33) it follows that for zero initial data of block we have F(a(t)) = -W(iw) F(£(t))-
(2-38)
Definition 2.7. A hodograph of frequency response W(iu) (Nyquist plot) is a set of all its values on the complex plane C. In Fig. 2.6 is shown an example of hodograph W(ioj). The arrows on the hodograph indicate the direction in which W(iu) changes with increasing
Fig. 2.6 Consider the case of a harmonic input £(i). We restrict ourselves by descriptions of blocks (2.13) and (2.14) for m = I = 1. We shall seek solutions 77(f) and x(t) in the form 77(f) = V{iu)eiwt, x{t) = U{iu)eiwt.
50 Linear electric circuits.
Transfer functions
and frequency responses of linear blocks
Here V{ioj) is a scalar a n d U(iuj) is a vector. Substituting these expressions into equations (2.13) a n d (2.14), we obtain V{iw)Af(iw)eiut =e*' w t , iut (A - iul) U{iu)e + beiujt = 0. These equations are satisfied in the case t h a t V
^ )
= TrT-^
'
t/ ? w
( ' ) = ~(A
- M!)'1*-
(2-39)
Al (iu>) Thus, there exist harmonic solutions r)(t), x(t) a n d the o u t p u t a(i) takes the form a(t) =
M(lL0) e
zajt
-c*(A-iuI)~1beiwt.
Introducing a frequency response W(iw), finally we obtain
(2.40)
Thus, for £(t) = e2Wt there exist initial d a t a of blocks such t h a t the o u t p u t a{t) is determined by formula (2.40). Further we shall consider stable blocks only, namely block (2.13) provided t h a t Af(p) is a stable polynomial a n d block (2.14) provided t h a t A is a stable m a t r i x (i.e., all its eigenvalues have negative real parts). In this case for two solutions Xi(t) a n d X2{t) with different initial d a t a »i(0) a n d £2(0) we have (Xl{t)
- x2{t))'
= A(Xl(t)
-
x2{t)).
Taking into account the properties of the matrix A, we obtain lim
(xUt) -x2(t))
= 0.
Whence it follows t h a t for the corresponding inputs cri(t) a n d
(2.41)
Since equations (2.13) are a special case of description (2.14), we see t h a t (2.41) is valid for block description (2.13).
Transfer functions
and frequency responses of linear blocks
51
From (2.40) and (2.41) it follows t h a t for stable blocks for £{t) = and arbitrary initial d a t a we have lim
(
= 0.
eiut
(2.42)
Formula (2.42) results in t h a t for stable blocks the frequency response may be found experimentally. For this purpose it is necessary to apply a harmonic input £(£) = elut and then to wait some time until there occurs asymptotically harmonic o u t p u t cr(t) =
A{iu)eiwt+a(i^i.
Here A(iui) is an amplitude of signal, a(iio) a phase shift. We have A(iu>) = \W(iu)\,
a(iu)
— n + arg W(ico).
These formulas determine uniquely a complex number W(iu>). By varying u> from — oo to + o o , we obtain a hodograph W(iui). Thus, to find in this way the frequency response, we do not need information on the coefficients of polynomials M{p) and ftf(p) or t h a t on the matrix A and vectors b and c. Since many results of the control theory are stated in terms of frequency responses, the descriptions of blocks by means of differential or integral equations are not necessary. In this case a curve on the complex plane, which is a hodograph of frequency response, is required only. In some cases a frequency response makes it possible to determine uniquely the equations in question. Let us assume, for example, t h a t , a priori, we know t h a t polynomials M{p) and Af(p) in description (2.13) do not have common zeros. Then, using W(ico), one can uniquely determine W(p) (by the principle of analytic continuation) and, using the function W(p), the polynomials M(p) and Af(p) can be found.
Chapter 3
Controllability, observability, stabilization
3.1
Controllability
In this chapter the study of linear systems will be continued. Consider linear blocks, described by the following equations
— = Ax + b£, m
<x =
(3.1) c*x.
Here A is a constant n x n-matrix, 6 is a constant n x m-matrix, c is a constant n x /-matrix, £(2) is an m-dimensional vector function, which is regarded as an input of block. Thus, an o u t p u t of the block 0 and any pair of vectors xo £ M.n, x\ £ 1R" there exists a vector function £(t) such that for a solution x(t) of system (3.1) with both £(t) and the initial data x(0) = XQ the relation x{T) = xx is satisfied. Thus, system (3.1) is controllable if its state vector x(t) can be transformed in a definite time T from an arbitrary initial d a t a xo into a final state x\ by the control £{t). T h e o r e m 3 . 1 . The following conditions are equivalent and each of them is a necessary and sufficient condition of a controllability of system (3.1) : 1) a rank of an n x nm-matrix (b, Ab,..., A" 1b) equals n: va,nk{b,Ab,...,An-1b) 53
= n,
(3.2)
Controllability,
54
observability,
stabilization
1') a relation z* Akb = 0, V k = 0 , . . . , n — 1, where z £ l " , implies that z = 0, 2) a linear space L, spanned by n-vectors being the columns of a matrix eAtb for i G R 1 , coincides with W1 : L{eAtb\t
e(-oo,+oo)}=Rn,
(3.3)
2') for arbitrary numbers rj < T? the following relation holds L{eAtb\te(r1,r2)}=Rn,
(3.4)
3) for arbitrary numbers T\ < TI the following matrix is symmetric and positively definite: K = (e-Atbb*e~A'tdt
> 0.
(3.5)
Before proving this theorem we recall that a linear space L{a\,... ,ak}, spanned by vectors a i , . . . , a&, is a set of all possible linear combinations of k
these vectors { ^ ajdAotj G M }. i=i
The columns o i , . . . , a^ of the matrix eAtb depend on t, namely ai(t),... ...,ak(t). Therefore relation (3.3) has the following geometrical interpretation: to each of vectors aj(t) assign a curve in M n . Relation (3.3) implies that all these curves cannot be placed on a linear subspace of dimension less than n. Let us also remark that a symmetric matrix K is said to be a positively definite, K > 0, if the corresponding quadratic form z*Kz is positively definite: z*Kz > 0 V z £ t " , z ^ 0 . P r o o f of T h e o r e m 3.1. At first we note that condition 1') is simply another formulation of condition 1). It can easily be seen that condition 2') yields condition 2). Suppose, condition 2') is not satisfied. Then there exists a vector z £ Mn, z ^ 0 such that z*eAtb = 0, Vi G (ri,r 2 ). Since vector function z*eAtb is the analytic one, by the principle of analytic continuation we have an identity z*eAtb = 0 , Vi G K1. Therefore condition 2) is not satisfied and the equivalence of 2) and 2') is proved. Let us show that from the property of controllability, formula (3.3) follows.
Controllability
55
Suppose, formula (3.3) is not valid. Then there exists a nonzero vector z e l " such that z*eAtb = 0, Vi G M1. Recall that the solution x(t) of system (3.1) can be given in the Cauchy form t
'(*)
e
At
(x[0)+
f e-ATbt{T)dr\
(3.6)
Let a number T > 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 = -
- . . . - 60z*Ab = 0.
(3.10)
56
Controllability,
observability,
stabilization
Repeating this procedure, we obtain that z*Akb = 0 for any natural k. Whence it follows that ^-"
t—1
k\
k=0
k\
k=0
It is obvious that condition 2) is not true. Thus, condition 2) implies condition 1). Now we prove that condition 1) results in condition 3). Assume that 3) is not valid. Hence there exists a nonzero vector z G K" such that
The last relation can be given in the following form T-2
\z*e-Atb\2dt
= 0.
Tl
Hence z*e~Atb = 0, V< G {n, r 2 ). In this case, as it was already noted at the beginning of the proof, we have z*eAtb = 0 for all t G M1. Differentiating the expression z*e b k times, we arrive at the following identities
z*AkeMb = a,
vtem1,
A = o, 1,...
Putting t = 0, we obtain z*b = 0, z*Ab - 0 , . . . , z*An~1b = 0. It means that condition 1) is not valid. Thus, condition 1) implies condition 3). Let us prove now that from 3) it follows the controllability. For this purpose we choose an input £(t) in the form b*e-A'^0,
£(t) =
where the vector £o will be determined below. By the Cauchy formula we obtain T
x{T) = e
AT
(x{Q) + f e- ylf 66*e- A *^o o
dt\.
Controllability
57
The last relation can be written as e-ATXl
-x0
= K£0.
(3.11)
Since det K ^ 0, equation (3.11) is always solvable and for ^o =
K~1(e~ATx1-x0),
we obtain the required input £(t) that in a time T transforms the solution x(t) from the state x(Q) = Xo into the state x(T) = x\. So, we have controllability =>• 2) => 1) =5> 3) =>• controllability. The proof of theorem is completed. Consider now several additional important properties of the controllability. Theorem 3.2. The following conditions are equivalent and each of them is necessary and sufficient for a controllability of system (3.1): 4) a linear space L, spanned by the complex-valued n-vectors to be composed the matrix (pi — A)~1b for p 6 C, p ^ Xj(A), coincides with an n-dimensional space C" : L{(pI-A)-1b\Pe£,
p^\j(A)}=Cn,
(3.12)
Here Xj(A) are the eigenvalues of A. 4') for any set ficC, having a limiting point different from Xj(A), the following relation holds L{(pI-A)-1b\PeQ}=Cl,
(3.13)
5) there does not exist a nonsingular n x n-matrix S such that matrices S_1AS and S~1b take the form
^ = ( t £)•
s
->=(o)'
6) rank(
+ ... + q0= (det(pi - A))[pi -
A)-\
7) there does not exist a vector z ^ 0 such that it satisfies the relations A*z = Xz, z*b = 0, where X is some number.
58
Controllability,
observability,
stabilization
P r o o f . We prove first that condition 4) is equivalent to condition 4'). For this purpose it is sufficient to prove that 4) implies 4'). Assuming the opposite, we observe that there exists a nonzero vector z £ C" such that z*{pl -A)~1b
= 0,
Vpefi.
Since z*(pl — A)~1b is analytic on fl, by the principle of analytic continuation we obtain the following identity z*(PI-A)-1b
= 0,
Vp#Ai(A),
i.e., condition 4) is not true. Thus, 4) yields 4'). Now we prove that 1') and 4) are the same. Suppose that condition 4') is not satisfied for fi = {p\ T-4 < l } . Then there exists a nonzero vector z G C n such that z*{PI-A)-1b
= 0,
VpGft.
(3.14)
Since for |p| > \A\ the following expansion holds
lz'(l-A.Y1b=*z'(l+A p V PJ P
\
+ ^ + ..)b, P P )
(3.15)
by (3.14) and (3.15) we obtain z* A2b z*b+ + — — + . . . = 0, Vpeft. (3.16) p p' This relation can be regarded as an expansion of zero into the Laurent series with the coefficients z*Akb. This expansion is unique and we have z*Akb=0, Vfc = 0 , l , . . . Thus, if 4) is not satisfied, then 4') with f2, introduced above, is not also satisfied and therefore 1') is not satisfied too. If 1') is not satisfied, then, as it was shown by means of the Cayley identity in the proof of Theorem 3.1, the relations z*Akb = 0 are satisfied not only for k = 0 , . . . , n — 1 but also for all natural k. Hence identity (3.16) follows. Then from expansion (3.15) we obtain z* Ah
z*{PI-A)-1b
= 0,
VpfEfi.
We see that if 1') is not satisfied, then 4') is not also satisfied. The equivalence of 1') and 4) is proved. Let us prove the equivalence of Properties 1) and 5).
Controllability
59
Assume that 5) does not valid. In this case, by introducing the following notation b=S~H,
A =
S~1AS,
we obtain Akb = S'1 ASS'1
AS...
S^AS5_16
= S " 1 A*6.
Therefore (6, Ab,...,
A"-1 b) = S~x{b, Ab,...,
An~lb).
(3.17)
From the assumptions imposed on the matrices A and b it follows directly that a matrix (b, Ab,..., A " ~x b) has the following structure
(6,l6,...,I"-16)=(^. Then rank(6, Ab,... ,An~1 b) < n. Using (3.17) and the nondegeneracy of matrix S, we obtain rank(6, Ab,...,
I " - 1 b) = rank(6, Ab,...,
An~lb).
Consequently, i&nk(b,Ab,...,An-lb)
< n.
So, condition 1) is not satisfied. We now assume that condition 1) is not satisfied and rank(6, Ab,... .. .,>l n-1 &) = r < n. In this case a nonsingular matrix S can be found such that S-1(b,Ab,...,An-lb)=
(®y
(3.18)
where Q is an r x nm-matrix and 0 is a null matrix of dimension (n — r) x nm. Consider matrices
where S is a nonsingular matrix satisfying relation (3.18), A21 is an (n — r) x r-matrix, and 62 is a n ( n — r) x m-matrix.
60
Controllability,
observability,
stabilization
Obviously, in this case relation (3.17) is satisfied. Hence by (3.18) we obtain (b,M,...,A"-1b)=
(®y
(3.19)
Consequently, b2 = 0 and
From (3.19) it follows that A2ibi = 0. Hence we obtain
A*b = A(Ab)=( and by (3.19)
^I^II&I
A A
*f\ )
= 0. In this case we have
Comparing this expression with (3.19), we find that A2iAl1bi Repeating this procedure, we obtain A2iAknb1
= 0,
V* = 0 , l , . . . , n - 2 .
= 0.
(3.20)
Since An is an r x r-matrix, by the Cayley identity A"^1 is a linear combination of matrices A"± , . . . , An, I. Hence by (3.20) A2iAn1-1b1
= 0.
(3.21)
Since, as it was proved above, we have
Q=
(h,Anbu...,An-%)
and rankQ = r, by (3.20) and (3.21) the matrix A21 annuls all vectors z 6lr: A21z = Q,
Vz£W.
Therefore, A2\ — 0. Finally, we have proved that if 1) is not satisfied, then 5) is not also satisfied. The proof of equivalence of 1) and 5) is completed.
Controllability
61
The equivalence of 4) and 6) is almost obvious. If 6) is not satisfied, then there exists a nonzero vector z e C " such that z*qj = 0,
Vj = 0 , . . . , n - l.
(3.22)
Hence we obtain z*{pI-A)~1b = 0,
Vp^XjiA).
(3.23)
If 4) is not satisfied, then there exists a nonzero vector z £ C ™ such that relation (3.23) is valid. In this case we have z*qn-1pn-1
+ ... + z*qo = 0,
VpGC.
Hence (3.22) is true and 6) is not satisfied. Now we prove the equivalence of 1) and 7). If there exists a nonzero vector z g T such that z*b = 0 and A* z = \z, then z*Ahb = \kz*b = 0,
Vfc=l,...
Hence 1') is not satisfied. Suppose that 1) is not satisfied. Denote by L a linear space spanned by column-vectors of the matrix (b,Ab,...,An~1b). From the Cayley identity it follows that AL = L and from the unsatisfiability of 1) it follows that d i m i < n. Denote by L1- a linear subspace orthogonal to L. It is obvious that dimL-1 > 0, A*LX = I x and z*b = 0, Vz G LL. In the sequel we make use of the fact that in an invariant linear subspace there exists at least one eigenvector z, namely A*z = Xz,
zeL1.
From z*b = 0, V2 G Lx it follows that 7) is not satisfied. The theorem is proved.
Controllability,
62
3.2
observability,
stabilization
Observability
Consider now the notion of observability [23]. Definition 3.2. System (3.1) is said to be observable if for any number T > 0 the following relations
e(*) = o,
vt£[o,T]
imply x{t) = 0, Vt 6 [0,7]. Let us clarify this definition. It is obvious that for £(t) = 0 on [0,7] we have a:(i) = 0, V< G [0,7] under the initial data z(O) = 0 and therefore a(t) = c*x(t) = 0, Vi 6 [0, 7]. However for the zero output 0 relations Ut)
= 6 (*),
V i e [0,7]
imply the identity Xl(t)=x2{t)
VfG [0,7].
Here xi(t) is a solution of system (3.1) with the vector function £i(t) such that <Ti(t) = c*x\{t). Similarly, xi{t) is a solution of system (3.1) with the vector function £2(£) such that ^ ( i ) = c*X2{t). P r o o f . To prove this theorem it is sufficient to show that the property, stated in this theorem, follows from the observability. Subtracting from the system dxi . ,, — - = Axi + 6£i, at
. <Ti = c xi
dx2 = Ax2 + b£2, ~dt
a2 = c*X2,
the system
63
Observability
we obtain d(xi — x2) V
df
.,
\
, /,
,x
' = A{Xl - x2) + b{t! - &),
(<7i - < 7 2 ) = C * ( X 1
-X2).
Since this system is observable, we can easily see that from the relations
6(*)-fc(*) = 0, a1(t)-a2(t)
= 0,
V
it follows that xi{t)-x2(t)
= 0,
Vi€[0,T].
Then it is clear that the identities
£i(0 = 6M.
*i(0 = *2(0,
v<e[o,T]
imply the following relation n W = *2(<)»
V
which completes the proof of theorem. As it was noted above, the property of observability can be stated in the following way: the relations £(t) = 0, a(t) = 0, Vt 6 [0,T] yield the relation x(Q) = 0. If we recall that t At
a{t) = c*e
(x(0) + f e~ATbZ{T) dr) , o
then for cr(t) = 0, £(t) = 0 we obtain c*eAtx{Q) = 0.
(3.24)
Thus, system (3.1) is observable if and only if from relation (3.24) it follows that z(0) = 0. Rewriting (3.24) in the following form z(0)*e A , t c = 0 and comparing it with condition 2) of controllability of Theorem 3.1, we obtain the following.
Controllability,
64
observability,
stabilization
Theorem 3.4. System (3.1) is observable if and only if L{eA'tc\
t e (-oo,+oo)} = M " .
(3.25)
This result makes it possible to restate properties l)-7) of the controllability of Theorems 3.1 and 3.2 in terms of observability. We also remark that from the above-mentioned results it follows that the property of controllability can be stated in terms of the matrices A and b only and the property of observability that in terms of A and c without using system (3.1). Therefore we can consider controllable pairs of matrices (A, b), assuming that property 1) is valid (or any other of equivalent condition 2)7)). We also consider observable pairs of matrices (A,c), assuming that relation (3.25) is valid (or another condition equivalent to it). In this case Theorem 3.4 implies the following result. Theorem 3.5 (The Kalman duality theorem). A pair (A, c) is observable if and only if a pair (A*,c) is controllable. Consider system (3.1) in the case that m = 1, i.e., 6 is a vector. The following criterion for a controllability and observability is important which can be stated in terms of the transfer function. Theorem 3.6. System (3.1) is observable and controllable if and only if the polynomials det(pl — A) and W(p) det(pi — A) have no common zeros. P r o o f . We assume that polynomials det (pi — A) and W(p) det(pl — A) have a common zero p = po- Then a polynomial q(p) = det(pl - A)(pi -
A)-H
satisfies a relation Aq(po) = Po?(po)If a pair (A, b) is not controllable, then the theorem is proved. Now suppose that the controllability exists. Then by virtue of condition 6) of controllability the coefficients qj of the polynomial q(p) are linear independent vectors and therefore q(po) ^ 0. In addition, c*q(Po) = det(p0I - A) c*(PoI - A)~1b = = -det(PoI-A) W(Po) = 0.
Observability
65
Thus, there exists a nonzero vector q(po) such that Aq{po) = Poq(po),
c*q(po) = 0.
By the Kalman duality theorem and condition 7) of controllability we conclude that a pair (A, c) is not observable. We show that if system (3.1) is not controllable, then there exists a corresponding common zero p — poSuppose, a pair (A, b) is not controllable. Then by virtue of condition 5) of controllability there exists a nonsingular matrix S such that
*=*-'"=ft1£). '=*-» =0;)We have
Since W(p) is invariant with respect to replacement of A, b, c by A, b, c respectively, we obtain
W(p) = c*(A-PI)-n=~c*
((An-pir'bA
=
= c*(A11-p/)-161. We can easily see that the denominator cJ(Au — pl)~1bi is det(p/ — A n ) and its degree less than n. Thus the polynomials det(p/ — A) and det(p/ — A) W(p) have a common zero. The case that a pair (A, c) is not observable can be considered in just the same way. Theorem 3.6 is proved. We now discuss the form of system (3.1) in the case that it is not controllable. In this case by condition 5) of controllability there exists a matrix S such that
Having performed a linear change of variables
x = Sy,
i/=fM,
(3.27)
66
Controllability,
observability,
stabilization
where the dimension of vector j/i is equal to the dimension of columns of the matrix b\, we obtain y = S~1ASy a = c* Sy
+
S-1bt,
and by (3.26) and (3.27) 2/i = A n 2/1 + A12y2 2/2 = A22J/2-
+h(,,
Thus, in a system, which is not controllable, the subsystem 2/2 = A22 2/2 can be separated out such that there are no control actions in it.
3.3
A special form of the systems with controllable pair ( A , b )
Suppose that b is a vector, i.e., m = 1. If the pair (A, b) is assumed to be controllable, then by Theorem 3.1 the vectors en e n _i e„_ 2 d
= b, =(A + anI)b, = [A2 + anA + an^I)b, = (A"" 1 + anAn~2
(3.28)
+ ... + a2I)b,
where the coefficients a.j being those of the characteristic polynomial det(p/— A) = pn + anpn~1 + ... + ai are linearly independent and form a basis ]Rn. The matrix A transforms the vectors of this basis in the following way Aen Aen-i Aei
= e n _i anen, = e n _2 — a,n-ient =
-a\tn.
/0
„„•.
Stabilization.
The Nyquist
criterion
67
Then for this basis we obtain / A =
0
1 0
0
0
0
1
:
:
'•.
'•.
...
0
0 \-d!
0 -a2
/°\
\
(3.30)
b=
0
0
1 -anJ
M/
Thus, if 6 and c are vectors and the pair (A, b) is controllable, then the system —- = Ax + b£, at
a = c*x
(3.31)
can be given in the following scalar form: Xi
-
%n— 1 Xn
a
X2,
(3.32)
— *^n) —
. -aixi
(XnXn —
+ £,
~ c\X\ + .,
System (3.32) is equivalent to an equation
"'i)'="Ui<-
(3.33)
where the polynomials N(p) and M(p) are the following N(p) =pn + anp"-1 + ... + au
M{p) = Cnp"'1 + ... +
Cl.
By system (3.31) and equation (3.33) the equivalence of description of controllable linear blocks is proved.
3.4
Stabilization. The Nyquist criterion
In Chapter 1 we have discussed various aspects of the phenomenon of instability. This raises a question of whether it is possible to stabilize system (3.1) (i.e., to make it stable) by choice of the input £ = S*x,
Controllability,
68
observability,
stabilization
where 5 is a constant m x n-matrix. For the answer to this question we prove an auxiliary statement. Let the following matrices, namely an n x n-matrix A , a n m x m-matrix D, an n x m-matrix B, and an m x n-matrix C be given. Lemma 3.1 (The Schur lemma). 7/det^4 ^ 0, then d e t
(c
D)
=detA
(r
D)
=det
det(D - CA^B).
(3.34)
If det D^O, then det
- ° ^{A-BD^C).
P r o o f . Suppose, Q = -A~1B.
Then
A B\ / / „
Q\
(A
C
Im)
\C
DJ\0
(3.35)
0
\
D-CA-xB)'
Formula (3.34) follows directly from the last relation. The proof of (3.35) is in the same way. Corollary 3.1. If K and M are n x m-matrices, then det(/„ + KM*) = det(/ m + M*K). P r o o f . By (3.34) we have det (_^.
f
J = det/„ det(/ m + M*K).
By (3.35) det f
1
^
f \ = det Im det(/„ + KM*).
Formula (3.36) follows from two last relations. If £ = S*a;, then
(3.36)
Stabilization.
The Nyquist
criterion
69
and by (3.36) the characteristic polynomial of the matrix A + bS* takes the form det(pl - A - bS*) = det ( p J - A) det(J m - S* (A - p/) _ 1 6).
(3.37)
The following theorem states that in the case m = 1 for a controllable system, by fitting S the stabilization is always possible. In addition, by choice of 5 the characteristic polynomial of matrix A + bS* can be made arbitrary. Consider now a polynomial of degree n: V»(P) =Pn + V>n-ip n-1 + • • • + oTheorem 3.7. If m = 1 and a pair (A,b) is controllable, then for any polynomial i>{p) there exists a vector S £ l " such that det(PI-A-bS*)=i>{p).
(3.38)
P r o o f . We recall the notation 9 (p)
= (det(pJ-A))
q(p) = Qn-xp"
-1
(PI-A)-\
+ . . . +go-
By (3.37) det(p/ -A-bS*)
= det(p/ -A)-
S*q(p).
For proving the theorem it is sufficient to show a solvability of the following system $3 = Sj - S*qj,
j = l,...,n.
(3.39)
Here Sj are coefficients of polynomial det (pi — A): det(p/ - A) = pn + ^ - i p " - 1 + ... + S0. The pair (A, b) is controllable. This implies that by condition 6) the vectors qj are linearly independent and, consequently, the required vector S can be determined as
Jn-lJ
\Sn-l
-V>n-1,
Controllability,
70
observability,
stabilization
This concludes the proof of theorem. In many cases the input £ is a linear function of output a
where // is some m x /-matrix (see Fig. 3.1). In this case the matrix S is of a special form S*
=fic\
Further we shall assume that the characteristic polynomial S(p) of the matrix A has no zeros on the imaginary axis. The number of its zeros to the right of the imaginary axis is denoted by k0. The number k0 is called a degree of instability of an open system. From the point of view of the control theory it is convenient to consider the case /Li = 0 as an open loop (see Fig. 3.1).
i
.
a
L
4-flCT
Fig. 3.1 The number of zeros of the polynomial det(p/ - A - 6yuc*), which are placed to the right of the imaginary axis, is called a degree of instability of the closed system and is denoted by kc. Applying the Hermit-Mikhailov formula (see Chapter 1) to both sides of (3.37) being rewritten as det(pJ - A - bfic*) = S(p) det (lm + /j W(p)), we obtain Tr(n — 2kc) = Tr(n — 2k0) + 2nki.
Stabilization.
The Nyquist
criterion
71
Here _ AArgdet(Jm+MW(^))|+~ kl
~
2^
•
Note that for the application of the Hermit-Mikhailov formula it is necessary for the following inequality d e t ( i w J - A-bfic*)
^ 0,
Vw G M1
to be satisfied. This inequality is satisfied if we take det (Im + p W(iu)) £ 0,
W6M 1 .
(3.40)
Finally, we arrive at the following result. Theorem 3.8. Let Vugl1
S(iu)^Q,
(3.41)
and inequality (3.40) be satisfied. Then the Nyquist formula kc-k0-
jfci
(3.42)
is valid. The Nyquist criterion Under assumptions (3.40) and (3.41) system (3.1) is stable if and only if k0 = hi. The Nyquist criterion holds much favor due to its geometric interpretation in the case m — I = 1. Recall that in place of the description of block by differential or integral operators the hodograph of its frequency response (Nyquist plot) on the complex plane is often given (Fig. 3.2).
Fig. 3.2
72
Controllability,
observability,
stabilization
In the case t h a t m = I — 1 the number k\ can be given by A A r g ^ + WM)!!" *
1 =
2^r
•
From this relation and Fig. 3.2 we can conclude t h a t ki is the number of counterclockwise rounds of a vector such t h a t its origin is the point — yTx and the end is at the point on the hodograph. Here w running from — oo to + o o . In the case, shown in Fig. 3.2, we have k\ = 1. Since k0 can be obtained by the Hermit-Mikhailov formula, the Nyquist criterion is effective for stabilization of linear systems. In conclusion we note that definitions of controllability and observability can also be introduced for more general systems than (3.1). In the nonlinear generalizations of these notions, in place of mathematical tools, operating on linear subspaces, some smooth manifolds can be introduced [20, 35] in the n a t u r a l way. T h e Nyquist criterion is a part of the linear theory. T h e latter can be found in many books on the control theory, for example, in [11, 12, 18, 34, 39].
3.5
T h e time-varying stabilization. The Brockett problem
In 1999 the book [36] has published in which many interesting and import a n t problems of the control theory have been formulated. In particular, R. Brockett has stated the following problem. "Given a triple constant matrices (A,b,c*) under what circumstances does there exist a time-dependent matrix K(t) such t h a t the system dx -- = Ax + bK(t)c*x,
i£M"
(3.43)
is asymptotically stable ?" Note t h a t for system (3.43) the problem of stabilization due to a constant matrix K is a classic problem in the control theory (see §3.4). From this point of view the Brockett problem can be restated in the following way. How much does the introduction of matrices K(t), depending on time t, enlarge the possibilities of a classic stabilization ? In studying the stabilization problems of mechanical systems it is sometimes necessary to consider another class of stabilizing matrices K{t). These
The time-varying
stabilization.
The Brockett
problem
73
matrices must be periodic and have a zero mean value on the period [0,T]: T
/ K { t ) d t = Q.
(3.44)
o Consider, for example, a linear approximation in a neighborhood of the equilibrium position of a pendulum with vertically oscillating pendulum pin 6 + a6 + (K(t)-wl)0
= 0,
(3.45)
where a and w0 are positive numbers. Here the functions K(t) being most frequently considered are of the form [15, 32] /3 sin ut and those of the following form [3, 4] K
W-\
[/? _p
for*€[0,T/2) for
^Ab)
For such functions K(t) the effect of stabilization of the upper equilibrium position for large u and small T is well known. Here the algorithms of constructing the periodic piecewise constant functions K(t), which make it possible to solve in some cases the Brockett problem and stabilization problem in the classes of functions satisfying condition (3.44), are obtained. For pendulum equation (3.45) with K(t) of the form (3.46) the possibility of low-frequency stabilization is proved. Suppose, there exist matrices Kj (j = 1 , 2 ) such that the systems ~
= (A + bKjC*)x,
i£l"
(3.47)
have stable linear manifolds Lj and invariant linear manifolds Mj. We assume that Mj P Lj = {0}, dim Mj + dimZy = n, and for the positive numbers \j, sej, aj, /3j the inequalities hold 1^(^)1 < a i e- A ^|a;(0)|, \x{t)\
Vx(0) G Lj,
(3.48)
Vx(0) £ M J .
(3.49)
Suppose also that there exists a matrix U(t) and a number r > 0 such that for the system ^
= (A + bU(t)c*)y
(3.50)
74
Controllability,
observability,
stabilization
the following inclusion Y{T)M1
C
L2
(3.51)
is valid. Here Y(t) is a fundamental matrix of system (3.50), Y(0) = / . Theorem 3.9. / / the inequality holds AiA2 > aeiae2,
(3.52)
then there exists a periodic matrix K(t) such that system (3.43) is asymptotically stable. P r o o f . Condition (3.52) implies the existence of positive numbers t\ and t2 satisfying the following inequalities -Ai
-\2t2
+ ae^i < - T .
(3.53)
Here T is an arbitrary positive number. We now define a periodic matrix K{t) in the following way: K(t)=Kx, V*€[(Mi), K(t) = U{t-h), V*6[
K(t) = K2,
\fte[ti
+ T),
(3.54)
+ T,h+t2 + T).
The period of the matrix K(t) is as follows: ti + t2 + T. We show that for sufficiently large T system (3.43) with such a matrix K(t) is asymptotically stable. For this purpose, consider nonsingular matrices Sj, reducing systems (3.47) to the canonical form:
dt ~ W j * " dwj
dim Zj = dim Lj,
(3.55)
„
dim Wj = dim Mj
-df = P>WHere Sji
:
U)'
(3.56)
and without loss of generality we may assume t h a t
MOI < e-^\zj(0)\, K (01 < c8e-*»|u;i(0)|.
(3.57)
The time-varying
stabilization.
The Brochett problem
75
From relations (3.54)-(3.56) it follows that (Zt1tT\)=S2Y{r)S^(Zl^\). \W2(tl+T)J V^l^l)/ Inclusion (3.51) implies that the matrix S2Y(T)SI SIY(T)SI
-[R2i{T)
(3.58)
has the form
„
) •
Therefore (3.53) and (3.57) result in the estimates \z2(h+h
+ r ) | < \Rn(T)\e-2T\Zl(0)\
\R12(r)\e~T\Wl(0)\,
+
M*i+*2 + r)| < l^iMle^MO)!. Whence it follows that for sufficiently large values of T for the initial values from the ball |;E(0)| < 1 we have \x(ti + t2 + r)\ < 1/2. This relation and the periodicity of the matrix K(t) imply the asymptotic stability of system (3.43). Suppose now that the matrix K(t) in equation (3.43) is a scalar function, A'i = K2 = Ko, Ai = A2 = A, a?i = ae2 = ae,
U(t)
= Uo, KQUO < 0,
the
function |y(<)l i s uniformly bounded on the interval (0,+oo), and there exists a sequence r- —> +oo such that Yir^Mx
C L2.
(3.59)
Theorem 3.10. Suppose, an inequality A > as is satisfied. Then there exists a T-periodic function K(t) such that (3.44) is satisfied and system (3.43) is asymptotically stable. For Theorem 3.10 to be proved it is sufficient to introduce the following function A'(^): K(t) = K0, ViG[0, \U0Tj/2K0\), K(t) = Uo, Vt£[\U0Tj/2Ko\, TJ + \UoTj/2K0\), K(t) = Ko, Vte[Tj + \U0TJ/2KO\, rj + \UQTJ/K0\).
(3.60)
The period of K(t) is equal to T = r , ( l + \U0/K0\). Here Tj is a sufficiently large number satisfying condition (3.59). The further proof of Theorem 3.10 is just in the same way as the proof of Theorem 3.9. We now apply Theorem 3.10 to equation (3.45) with the function K(t) of the form (3.46).
Controllability,
76
observability,
stabilization
Let a2<4(/?-wg).
(3.61)
W i t h o u t loss of generality we may assume t h a t f3 — WQ — a 2 / 4 = 1. P u t A'o = —/?, UQ = /?. From condition (3.61) it follows t h a t the characteristic polynomial of equation (3.45) with K(t) — UQ has complex roots, and, consequently, condition (3.59) with some T\ > 0 is satisfied. Then we have Tj = T\ + 2jn. It can easy be seen that the negativity of real parts of the roots of the characteristic polynomial implies the uniform boundedness of \Y(t)\ on (0,+oo). For K(t) = A'o = —/? the values of A and as can be calculated:
A=
I + \/T +
(/3+W )
° '
Thus, all the assumptions of Theorem 3.10 hold, and equation (3.45) with K(t) of the form (3.60) is asymptotically stable for sufficiently large j . This result can be stated in the following way. P r o p o s i t i o n 3 . 2 . / / (3.61) is satisfied, then for any number r there exists a number T > r such that equation (3.45) with the function K(t) of the form (3.46) is asymptotically stable. In particular, we see t h a t the stabilization of the upper position of pend u l u m is possible under the low-frequency vertical oscillation of the pend u l u m pin. It is natural t h a t in this case the oscillation amplitude a turns out to be large a = lT2f3/8, where Z is a pendulum length, /? is the absolute acceleration value divided by I. For high-frequency oscillation (for small T) the effect of stabilization is well known [3, 4]. For the effective testing of condition (3.51) sometimes the following lemmas are useful. Consider a system i = Qz,
z£l",
(3.62)
where Q is a constant nonsingular n x n-matrix, and a vector h G K n .
The time-varying
stabilization.
The Brockett problem
77
Lemma 3.2. Suppose, a solution z(t) of system (3.62) has the form z(t) = v(t)+w(t), where v(t) is a periodic vector function such that h*v(t) ^ 0, w(t) is a vector function such that + 0O
/
\w(T)\ dr < +oo,
lim
w(t) = 0.
o
Then there exist numbers T\ and TI such that the following inequalities h*z(n) > 0,
h*z{T2) < 0
(3.63)
are satisfied. P r o o f . Assuming the opposite, we obtain h*z{t) > 0, Vi > 0 or h*z(t) < 0, V< > 0. Let, for definiteness, h*z(t) > 0, Vt > 0. The inequality h*v(t) ^ 0 implies that t
lim
h*z(r)dr
= +oo.
(3.64)
o On the other hand, we have t
f h*z(r)dr
= h*Q~1(z{t) - 2(0)).
From'the above and the uniform boundedness of z(t) on (0, +oo) we have a uniform boundedness of the following function t
f h*z{r)dr, o which contradicts relation (3.64). This contradiction proofs the lemma. Lemma 3.3. Let n = 2 and a matrix Q have complex eigenvalues. Then for any pair of nonzero vectors / i £ l 2 , u £ R 2 there exist numbers T\ and Ti such that h*eQTlu>0,
h*eQT2u<0.
(3.65)
This obvious statement may be regarded as a corollary of Lemma 3.2.
78
Controllability,
observability,
stabilization
Lemma 3.4. Let a matrix Q have two complex eigenvalues Ao ± icog and other its eigenvalues Aj(Q) satisfy the condition ReAj(Q) < Ao. Let also for the vectors h E M.", u G Mn the following inequalities hold det(/i, Q*h,...,
( Q T _ 1 / i ) ± 0.
det(u,Qu,.-.,<2n_1u)#°-
(3-66) (3.67)
Then there exist numbers T\ and T2 such that h*eQTlu>Q,
h*eQr2u
(3.68)
Recall that conditions (3.67) and (3.66) are the conditions of controllability for a pair (Q, u) and those of observability for a pair (Q, h). To prove Lemma 3.4 it is sufficient to note that the solution z(t) = e^u can be represented as z(t) = ex°t{v(t) + w(t)), where v(t) and w(t) satisfy the assumptions of Lemma 3.2. The inequality h*v(t) ^ 0 follows from the observability of (Q,h) and controllability of (Q,u). Theorem 3.9 and Lemma 3.3 imply the following proposition. Theorem 3.11. Let n = 2 and there exist matrices KQ and UQ, satisfying the following conditions: 1) det bK0c* = 0 and TrbK0c* ^ 0, 2) the matrix A + bUoc* has complex eigenvalues. Then there exists a periodic matrix K(i) of the form (3.54) such that system (3.43) is asymptotically stable. For proving this theorem it is sufficient to put Ki = A'2 = yuA'o, where |//| is a sufficiently large number and Tr^&A'oe* < 0. It is obvious that in this case all the assumptions of Theorem 3.9 are true. Consider now the case that 6 and c are vectors, K(t) is a piecewise continuous function: M1 —^IR1. Consider a transfer function of system (3.43) W(p) — c*(A — pl)~xb, where p is a complex variable. We assume that W(p) is a nondegenerate function. This means that the pair [A, b) is controllable and the pair {A, c) is observable.
The time-varying
L e m m a 3.5. the system
stabilization.
If a hyperplane
The Brockett problem
{h*z = 0} is an invariant
x = (A + fibc*)x,
/z^O,
79
manifold
for
(3.69)
then a pair (A, h) is observable. P r o o f . Suppose t h a t (A,h) is not observable. In this case there exist both the vector q and the number 7 such t h a t h*q = 0, Aq = -yq, q ^ 0 (see Theorems 3.5 and 3.2). T h e observability of a pair (A, c) implies the inequality c*q 7^ 0. From the invariance of {h*z = 0} with respect to equation (3.69) it follows t h a t for all z £ {h* z = 0} we have h*(A + p.bc*)k z = 0, k = 1, 2 , . . . For z = q and k = 1 we obtain h*bc*q = 0. Hence h*b = 0. For z — q and k = 2, using the previous relation, we obtain h* Ab = 0. Continuing in the same way, we obtain further the equalities h*Ak~1b = 0. T h e controllability of the pair (A, 6) implies t h a t h — 0. Hence the assumption on the unobservability of the pair (A, h) is incorrect. The lemma is proved. L e m m a 3.6. If the straight line { a n } , u £ M.", a £ ffi1 is invariant with respect to system (3.69), then the pair (A,u) is controllable. P r o o f . T h e invariance implies t h a t (^4 + pbc*)ku = fkU, where k = 0 , 1 , . . . , jk are some numbers. From the observability of (A, c) it follows t h a t c*u ^ 0. Therefore for vectors z £ M n such t h a t z*u = 0, z*Au = 0, .. .,z*An~1u = 0 we obtain the following relations z*b = 0, z*Ab = 0 , . . . , z*An~1b — 0. From the controllability of the pair (A, b) it follows t h a t z = 0. T h u s , the relations z*u = 0 , . . . , z*An~1u = 0 imply t h a t 2 = 0. This proves the controllability of the pair (A, u). From Theorem 3.9 and Lemmas 3.4-3.6 the following result can be obtained: Theorem 3.12. Let b £ R", c £ M n , d i m M i = 1, d i m l 2 = n - I, inequality (3.52) be satisfied, for some number UQ 7^ Kj a matrix A + Uobc*i has complex eigenvalues XQ ± iuiQ and its other eigenvalues Xj satisfy a condition Re Xj < AoThen there exists a periodic function K(t) such that system (3.43) is asymptotically stable. To prove Theorem 3.12 it is sufficient to note t h a t by Lemma 3.5 the inequality UQ 7^ Kj, the controllability (A, b), and the observability (A, c) imply the observability of the pair (A+Uobc*, h), where ft is a normal vector
80
Controllability,
observability,
stabilization
of the subspace Li- By Lemma 3.6 the pair (A + Uobc*, u) is controllable. Here u ^ 0, u £ M\. By Lemma 3.4 there exists number T such that h* exp [{A + U0bc*)T}u = 0. Whence it follows that condition (3.51) is satisfied. Lemmas 3.2-3.6 have been stated for testing inclusion (3.51) in the case that the rotation of the subspace Mi is caused by the presence of complex eigenvalues. Another approach is a pulse action U(t) = fi with a large |/i| in a small time interval. In this case a velocity vector x is often close to a vector fb, where 7 is a number. We discuss this approach in more detail. Consider system (3.69) with large parameter /J: |/i| ^> 1. Lemma 3.7. Let c*b = 0 and for the vectors h,u the inequalities h*b 7^ 0, c*u ^ 0 be satisfied. Then there exist numbers /i and r(fi) > 0 such that h*x(r,u) = 0, lim r(fj.) = 0. P r o o f . Consider the following numbers h*u fih*bc*u ' n
(l + 2H|6||c|*0)M ~
l-(2\A\t0
+ 4\n\\A\\b\\c\tl)-
We choose the number ft such that to > 0 and 2|A|to + 4|Aip||&||c|fg
From equation (3.69) it follows that \x(t,u) — u — fibc*ut\ < <(2|i4|to + 4|/ip||&||c|tg) max te[o,2t 0 ]
\x(t,u)\.
The time-varying
stabilization.
The Brockett
problem
81
The estimation implies the following inequalities \x(t,u)\
V*e[0,2t o ],
\h*x(t, u) - h*u — fih*bc*ut\ < < {2\A\t0 + 4\n\\A\\b\\c\t20)R\h\,
V< G [0,2t0].
We can easily see that (2\A\t0 + 4\fi\\A\\b\\c\tl)R\h\ =
o(-
Then for large |/J| there exists r £ [0,2to] such that h*x(r,u) = 0. Lemma 3.8. Let c*b ^ 0 and for the vectors h and u the inequalities h*b^0, c * w ^ 0 h*uc*b h*bc*u be satisfied. Then there exist numbers fi and r(fi) > 0 such that h*x(r, u) = 0, lim TU) = 0. fl—fOO
P r o o f . Consider the numbers 1 , / t0 = — T In 1 /j,c*b \ R =
h*uc*b h*bc*ut
l + |6||c||c*6|- 1 (e 2 ^* 6t0 + l))|w| 1 - (2|A|t 0 + 2|A|H|c|fo|c*6|-1(e2ftc*6*° + 1)) '
We choose the number ^ such that to > 0 and 2\A\t0 + 2\A\\b\\c\t0\c*b\-1(e2»c'bt°
+ 1) < 1.
For t £ [0, 2to] we have \c*x(t,u) — fic*bc*x(t,u)\ < \A\\c\ max |x(f,u)|. te[o,2t 0 ]
Then for t £ [0,2i0] we obtain \c*x(t,u)-ef*c
,
i _
bt
c*u\<
ptpc'bta
—\A\\c\ max
\x(t,u)
Controllability,
82
observability,
stabilization
Then from (3.69) it follows that
x(t,u)-u-b-ff(e^'bt-l
<
< (2|A|t 0 + 2|A||6||c|to|c*6|- 1 (e 2 " c * 6 ' 0 + l)) max
\x{t,u)
t£[0,2£o]
The estimation results in the following inequalities \x{t,u)\
ViG[0,2i o ],
v(e"
c 6t
* - 1) < '
< (2|A|< 0 +2|A||6||c| ae. We also assume that the inequality holds lim v(ae P->ae
"*> < 1. — p)W(p) ' v '
The time-varying
stabilization.
The Brockett
problem
83
Then there exists a periodic function K{t) such that the system (3.43) is asymptotically stable. P r o o f . W i t h o u t loss of generality it can be assumed t h a t the matrix A and vectors b and c take the form
where A2 is an (n — 1) x (n — l ) - m a t r i x , b% € M n _ 1 , ci G Mn~1. In this case the vector h normal to subspace L\ = L2 and the vector u G M\ = M^ take the form
*-(!)• - ( i ) By L e m m a 3.8 there exist numbers p, and r(p) such t h a t for U(t) = p for system (3.50) condition (3.51) is satisfied in the case t h a t c*b cibi
1
It is obvious t h a t C1&1 ^ 0 by virtue of the controllability of (A, b) and the observability of (^4, c) and in addition we have C161 = lim (as — p)W(p). p—*-36
T h u s all the assumptions of Theorem 3.9 are valid and therefore system (3.43) is stabilizable. L e m m a 3 . 9 . Let an (n — 2)-dimensional linear subspace L invariant with respect to system (3.69) be placed in the hyperplane {h*z = 0 } . If the relations h*b = 0, c*b = 0 are satisfied, then a pair (A, h) is observable. P r o o f . From a controllability of a pair (A, b) and from an invariance of L with respect to system (3.69) the relation &GX follows. Then we see t h a t the linear subspace, spanned over b and L, coincides with the hyperplane {h*z = 0}. Let us assume t h a t a pair (A,h) is not observable. In this case (see Theorems 3.5 and 3.2) there exist both the vector q ^ 0 and the number 7 such t h a t h*q = 0, Aq — jq. T h e observability of a pair (A, c) implies the inequality c*q ^ 0. From the invariance of L it follows t h a t h*(A + pbc*)kz — 0, k = 0 , 1 , . . . , z G L. Arguing as above, we conclude t h a t for q there exist both the number v and the vector z £ L such t h a t q = z + vb. P u t v ^ 0. For k = 1 we obtain vh*Ab = 0. Then h*Ab = 0. Taking into account last relation and h*b = 0, c*b = 0, for k = 2 we obtain
84
Controllability,
observability,
stabilization
t h a t vh*A2b = 0. Continuing in the same way, we see t h a t for k = 3 , . . . the relation h*Akb = 0 is valid. By virtue of a controllability of (A, b) we obtain t h a t ft = 0, which contradicts the definition of vector ft. T h u s , for v ^ 0 the observability of (A, ft) is proved. For v — 0 we have the same line of reasoning as in proving Lemma 3.5. T h e o r e m 3 . 1 5 . Let b £ M", c £ M", c*6 = 0, d i m M x = 1, d i m i 2 = n — 2, the inequality (3.52) be satisfied and for a certain number UQ ^ Kj the assumption of Theorem 3.12 be valid. Then there exists a periodic function K(t) such that system (3.43) is asymptotically stable. P r o o f . Note t h a t the integral manifold Sl(fi), consisting of trajectories x(t, xo) of system (3.69) with initial d a t a XQ £ L2, tends to the hyperplane {h*x = 0} as p. —> 00. Here ft is a vector normal to linear subspace, spanned over £2 and b. This convergence is the same as described in proving Lemma 3.7. From L e m m a 3.9 it follows t h a t a pair ((A + Uobc*),h) is observable. From L e m m a 3.6 it follows t h a t a pair ([A + Uobc*), u) such t h a t u £ M\, u 7^ 0 is controllable. Then by Lemma 3.4 we obtain t h a t for system (3.50) with U(t) — Uo the function h*y(t, u) changes its sign for some values of t. Hence for sufficiently large \fi\ there exists a number 7o(/i) > 0 such t h a t y{r0(n),u) £ to{n). By using small perturbations of the right-hand side of system (3.43), one can obtain the inequality c*j/(r 0 (/i), u) ^ 0 (under a small perturbation of the right-hand sides of periodic systems the asymptotic stability is preserved). Further, letting in (3.50) U(t) = fi (or U(t) = —fj.) on (TO,T], we can reach a set L2, at time t = T. T h e sign of p, we choose in such a way t h a t T > TQ (see the proof of Lemma 3.7). We see t h a t y(r, u) £ Li and condition (3.51) is satisfied. Thus, all the assumptions of Theorem 3.9 are valid. In the case t h a t the transfer function W{p) of system (3.43) is nondegenerate we have
w{p)=c*{A-Pi)-H=
c l 1 n
f ~ t\';+cl
.
where CJ and a,j are real numbers, and system (3.43) may be written in the
The time-varying
stabilization.
The Brockett
problem
85
following scalar form (see § 3.3): x\ =
x2, (3.70)
*En— 1 — ^ n i
xn = ~(anxn + . . . + -K(t)(cnxn +... +
aixi) axi).
Then x1\
ci
*^n /
/°
1
0 0
\-Ol
...
0
\
0 •••
0 1 —an )
0 0 -a2
b =
0
Recall t h a t the nondegeneracy of the transfer function W(p) indicates t h a t the following polynomials c n p n _ 1 + . . . + a, pn + anpn~1 + .. , + a.i have no common zeros. Let cn ^ 0. We can assume, without loss of generality, t h a t cn = 1. T h e o r e m 3 . 1 6 . Suppose, the following conditions hold: l)/orn>2 ci < 0 , . . . , c „ _ 2 < 0, 2) ci(a„ - c „ _ i ) > ai, ci + (a„ - c n _ i ) c 2 > a 2 , Cn-2
c
n - l ) c n - l > On-1'
T/zen there does not exist a function asymptotically stable. P r o o f . Consider a set
K(t)
such that system
fi = {xi > 0 , . . . , £ „ _ i > 0, xn + c „ _ i x n _ ! + . . . + cixi
(3.43) is
> 0}.
We prove t h a t f2 is positively invariant, i.e., if x(to) £ Q, then x(t) £ fl, Vi >
Controllability,
86
observability,
stabilization
Note t h a t for j — 1 , . . . , n - 1 a n d for XJ{T)
— 0,
Xi(r) > 0, Vi ^ j , i
Zn(T-) + c „ - i x „ - i ( r ) + . . . + CXXX(T)
> 0.
the following inequality holds XS(T)
> 0.
(3.71)
Really, for j = 1 , . . . , n — 2 we have i
j{T)
= XJ+I(T)
> °.
for n = 2 £I(T") = X2(T) > -CIXI(T)
-
0,
and for n > 2 i „ _ i ( r ) = x n ( r ) > - c „ _ 2 a ; „ - 2 ( T ) - c^x^r)
> 0.
Note also t h a t (xn(r)
+ c„_i2r„_i(r) + . . . + C I Z I ( T ) ) * =
= (—a„_i + c n _ 2 + (a„ - c „ _ i ) c „ _ 1 ) x n _ 1 ( r ) + + . . . + ( - a 2 + ci + (a„ - c„_ 1)02)2:2(7-)+ + ( - a x + (a„ - c „ _ i ) c i ) a ; i ( r ) . Then by condition 2) of theorem we obtain inequality. (xn{r)
+ c n _ i x n _ ! ( r ) + . . . + W^T))'
> 0
(3.72)
for 2r n (r) + c n _ i i ; n _ i ( r ) + . . . + cia;i(r) = 0 a n d XJ{T) > 0, j = 1 , . . . , n- 1. From inequalities (3.71) a n d (3.72) it follows t h a t almost everywhere the boundary of set fi is a boundary without contact (transversal) with respect t o the vector field of system (3.70) a n d the solutions of system (3.70) intersect almost everywhere the boundary of set f2. In this case t h e continuous dependence of solutions of system (3.70) on initial d a t a implies t h a t t h e set Cl is positively invariant. T h e positive invariance of fi yields the lack of asymptotic stability of system (3.70). T h e proof of theorem is completed.
The time-varying
stabilization.
The Brockett
problem
87
It is well known [3, 4] another instability condition of system (3.43), namely V< e M1
Tv(A + bK(t)c*) > a > 0,
The results obtained are applied to the case that n = 2, 6, c are vectors, and K(t) is a scalar function. Consider a transfer function of system (3.43)
W(p)=c*(A-pI)-1b=
^ + \ R , p +ap + /3 2
2
where p is a complex variable. Put p ^ 0. Then without loss of generality we assume that p = 1. Suppose also that the function W(p) is nondegenerate, i.e., the inequality 7 2 — a~f + j3 ^ 0 is true. In this case system (3.43) may be written as (7 =
7/
(3.73) r) = —ar) — J3
By the constant K(t) = A'o a stabilization of system (3.73) is possible in the case that a + A'o > 0, j3 + 7A'o > 0. For the existence of a number A'o, satisfying these two inequalities, it is necessary and sufficient for either the condition 7 > 0 or the inequalities 7 < 0, aj < ft to be satisfied. Consider the case that by the constant K{t) =. A'o the stabilization is impossible: 7 < 0, a~f > j3. We apply Theorem 3.11. Condition 1) of Theorem 3.11 is satisfied since det&A'oc* = Aodet6c* = 0 and Tr&A'oc* = -A'o ± 0. Condition 2) of Theorem 3.11 is satisfied if for some Uo the polynomial p2 + ap + (3 + Uo{p + 7) has complex zeros. We see that for the existence of such Uo it is necessary and sufficient that the inequality holds 7 2 - a 7 + /0>O
(3.74).
Thus, if inequality (3.74) is valid, then there exists a periodic function K(t) such that system (3.73) is asymptotically stable. Now we obtain the same result by means of Theorem 3.14. Without loss of generality it can be assumed that a > 0. For this purpose it is sufficient to choose an acceptable A'o in the expression — (a + A'o)r?— (/3 + 7A'o)cr— (K(t) — Ao)(r/ + 7
Controllability,
88
observability,
stabilization
a + A' 0 —»<*,/? + 7.K0 —> /?, A'(<) — A'o —• A'(i). From the inequality a > 0 it follows t h a t A > a?. Here c*6
_ ae + A
lim (ae —p)W(p) p-vaev
as + 7 '
T h u s all the assumptions of Theorem 3.14 are valid in the case t h a t (A - 7)(as + 7) = - 7 2 + 0:7 - (1 < 0. This inequality coincides with inequality (3.74). If the inequality 7
2
-a
7
+ /?<0
(3.75)
is satisfied, then the assumptions of Theorem 3.16 are obviously satisfied too. T h u s , we have the following result. T h e o r e m 3 . 1 7 [27]. / / inequality (3.74) is valid, then there exists a periodic function K(t) such that system (3.73) is asymptotically stable. If inequality (3.75) is valid, then there does not exist a function K{t) such that system (3.73) is asymptotically stable. For other class of the stabilizing functions K(t) of the form K(t) = (ko + kiw coscot), w » l , this result was also obtained in [33] by means of the averaging method. Consider now several systems of the third order with various transfer functions. ;
w
p3 + ap2 + /3p + 7
where a, /?, 7 are some numbers. If a > 0, ft > 0, then the stationary stabilization is possible. Let a > 0, P < 0. In this case the stationary stabilization is impossible. We now apply Theorem 3.15. Obviously, for sufficiently large UQ there are one negative and two complex roots Ao ± iwo, Ao > 0 of the polynomial p 3 + ap2 + ftp + UQ + 7. Take K\ such t h a t the relation p3 + ap2 +/3p + Id + f=(p-
se1)(p2 +
aip+Pi)
is valid for a i = a + aei, /3i = /J + ( a + aei)aei. For large sei the polynomial p2 + aip + pi has complex roots with the real part —(a + 33i)/2. Take A'2 such t h a t the relation p3 + ap2 + Pp + K2 + 7 = (p + A 2 )(p 2 + a2p + P2)
The time-varying
stabilization.
The Brockett
problem
89
is satisfied for 02 = a — ^2i P2 = P — if- — ^2)^2- Then for large A2 the polynomial p 2 + <*2P + Pi has complex roots with the real part (A2 — a ) / 2 . We obtain dim M i = dim £2 = 1, d i m M 2 = dimZ-i = 2, a + aei Ai = ^ — ,
A2 - a ** = — 3 - ,
AiA2 - a3iee2 = a(A 2 + aei) > 0. Thus, all the assumptions of Theorem 3.15 are satisfied. Since Tr (A + bK(t)c*) — —a, the asymptotic stability is impossible for a<0. Thus, we may state the following result. T h e o r e m 3 . 1 8 . For a > 0 the system is stabilizable. For a < 0 the stabilization is impossible. 2
) W(p) = -^ ;
^'
£—z
p3 + ap2 + /3p+~f
.
For a > 0, 7 > 0 the stationary stabilization is possible. Consider the case a > 0, 7 < 0 and apply Theorem 3.13 with K\ = K2, Xi = A2 = A, aei = 3B2 = as- Take A'i such t h a t the relation (p - ae)(p 2 + a i p + fa) = p3 + ap2 + (A'x + fap + 7 is satisfied with OL\ = a + as, fa — — 7/as. For small as the polynomial p 2 + a i p + Pi has complex zeros with the real part —(a + as)/2. Then Mi = M2, Li = L2, d i m M i = 1, d i m i 2 = 2, A = (a + ae)/2. It is clear t h a t for small as the inequality A > as holds. Since Tr (A + bK(t)c*) = —a for a < 0, the asymptotic stability is impossible. Therefore in this case we can state the following result. T h e o r e m 3 . 1 9 . Let a ^ O , 7 / O . Then for stabilization and sufficient that a > 0.
it is
necessary
P2
3) W(p) = -=— f—5 . ; vr/ p 3 + ap2 + Pp + 7 For P > 0, 7 > 0 the stationary stabilization is possible. In the case P < 0, 7 < 0 by Theorem 3.16 the stabilization is impossible. Consider the case P > 0, 7 < 0. We apply Theorem 3.9. P u t A'i = A'2, Ai = A2 = A, asi = 332 = as. Take A'i such t h a t the relation (p - as)(p 2 + aip + Pi) =p3+(a
+ A'i)p 2 + Pp + 7
90
Controllability,
observability,
stabilization
is valid. Here - ( 7 + ae/3) <*i =
5
7 i
Pi =
>
where as is a small number. Whence it follows that A can be determined in the following way _ - ( 7 +as/?) _ / ( 7 + aep')2 2as2 V 4ae4
1 as '
+
Obviously, inequality A > as is satisfied for b + as2 > 0. Thus, we can conclude that condition (3.52) is satisfied. Further we choose Uo such that the relation (p ~ V){P2 + <*2P + #2) = P3 + {a + U0)p2 + P> + 7 is valid. Here
~(7+^) a2 =
5
a ,
7
P2 = - - i
where v is a sufficiently large number. Without loss of generality we can assume that
Here Q is a 2 x 2-matrix, 62 G ffi , C2 6 ffi . Note that the following relations c*b = C161
-1 = lim(^ — p)VF(p)
1
|
aaiz + ft ^2
= 1
27 f3 ^ J^3 v2
are valid for [iv > —27. The matrix Q has complex eigenvalues. Therefore for nonzero vector u G Mi there exists a number T\ > 0 such that the vectors 6,
L j ,
[exp(A + [/0fec*)ri]M
belong to the same plane. By Lemma 3.8 inequality (3.76) implies that there exist numbers \x and T(/J,) such that 1\* J [exp(A + {U0 + n)bc*)T(ji)] [exp(A + £/ 0 6c*)n]« = 0.
Q
The time-varying
stabilization.
The Brockett problem
91
From the fact that planes L 2 and {x\ x* (J) = 0} intersect and a matrix Q has complex eigenvalues it follows that there exists a number r 2 > 0 such that [exp(A + Uobc*)r2] [exp(A + (£/0 + /*)6c*)r(/i)] [exp(A + [/o&c*)ri]u e £ 2 . This inclusion results in that condition (3.51) holds for the function U(t) = U0, U{t)=U0+p, tf (t) = 1/0,
ViGlO.n), V t G t n . n + r^)), V i e [Tl + T(H), n +
T{H)
+ r 2 ),
where r — T\ + r(/i) -+- r 2 . Finally, we may state the following result. Theorem 3.20. Let ft ^ 0, 7 < 0. Then for stabilization it is necessary and sufficient that /3 > 0. We hope that the mathematical tools developed here allow us to obtain in the sequel the other criteria of stabilization.
Chapter 4
Two-dimensional control systems. Phase portraits
In the case t h a t the dimension of a phase space (a state space) equals two, applying some auxiliary tools, we can demonstrate the partition of the phase plane into trajectories of differential equations, which correspond to one or the other of operating regimes of a control system. This allows us to describe qualitatively (and sometimes quantitatively) both the operating regimes of system and its transient processes.
4.1
A n autopilot and spacecraft orientation s y s t e m
Consider a system of angular orientation. A classic example of such a system is a two-positional autopilot. Let us obtain an equation of a ship rotation around the vertical axis passing through its center of gravity. (We neglect the lateral drift of a ship in the process of rotational motion and assume t h a t a ship moves with a constant velocity.)
Here 6(t) is a ship deviation from a given course, / is a ship moment of inertia with respect to its vertical axis, the value a6(t) corresponds to a moment of friction forces, a is a friction factor, M(V>) is a moment of forces of a rudder, ij) is a rudder blade angular deflection (Fig. 4.1). Note t h a t if the ship is uncontrollable (ifi = 0 and, consequently, M = 0), 93
Two-dimensional
94
control systems.
Phase
portraits
then we obtain the trajectories of the corresponding two-dimensional system v — x, X =
in the phase plane {x,6}. (4.2) an identity
(4.2)
a
——
X
In addition, for the solutions x(t), 0(t) of system
(x(t) + j 0(f))' = x{t) + j x{t) = 0 is satisfied. /Course at time t
Fig. 4.1 Therefore for any solution of system (4.2) we have x(t) + -0(t)
= const.
Whence it follows that each straight line x+jV=j consists wholly of three trajectories, one of which is an equilibrium x = 0, 6 = -yl/a and the two other of trajectories tending to this equilibrium (Fig. 4.2). Thus, the abscissa axis consists wholly of equilibria. To each ship movement with initial data 0(0) = 0O,
*(0) = 0(0) = x0
An autopilot and spacecraft orientation system
95
assigns the trajectory with the same initial data. Therefore the rotation of an uncontrolled ship decays with time and as t —>• +00 we have 0{t) -+0o+-
xQ.
Fig. 4.2 The purpose of control is to ensure conditions such that the following relation lim 0(t) = 0 is satisfied in a given region of initial data or we have lim 0(t) = 2kn t-y+oo
v
'
for almost all trajectories. Here the number k depends on the initial data k = k(60,x0)Consider a two-positional autopilot such that the rudder is in two positions only: ij> = i/'o and i\> = —V'o- In this case we have moments offerees equal in value but opposite in directions. At present there are various sensing devices (gyrocompasses), which measure quantities 0(t) and 0(t), and transmit a signal a(t) — 6(t)+b0(t) to an actuator, which in the ideal case handles instantly the rudder depending on (<x) = -V'o
for crG(0,7r),
i>(cr) — ip0 V>(CT + 2TT) = ip(a),
for <J G (-7T, 0), VcrGM1.
96
Two-dimensional
control systems.
Phase
portraits
If cr = 0 or IT = 7T, then the actuator is turned off and the rudder turns out to be in any position between —V>o and if>o, t h a t is, ip 6 [—^Oi V'o]We see t h a t the graph of a 27r-periodic function M(a) = M(ij>(cr)) has the form shown in Fig. 4.3. M
n
-7Z
27t
a
Fig. 4.3 T h e detailed description of technical realization of such a two-position autopilot can be found in [2]. Note also t h a t a similar problem arises in design of autopilot for an aeroplane. In this case an autopilot must control not one but three angles. T h e same problem occurs also in the spacecraft orientation systems. However in many cases such more complicated pilot-controlled motions can be regarded as a set of independent plane rotations, in which this case the angle controls is performed by three independent feedbacks. T h e peculiarity of a spacecraft orientation system is t h a t the forces of viscous resistance (a — 0) lack and jet engines can be used as an actuator rather than a rudder. In order to save a fuel the so-called dead zones are introduced, whose sizes are acceptable for the given orientation. In this case the graph of 27r-periodic function M(cr) is as follows (Fig. 4.4.) Here the interval (—A, A) is a dead zone in which the jet engine is put out. Thus, an autopilot and a spacecraft angular orientation system are described by the following equations I9 + a6 = M(a),
cr = 9 + b6,
where numbers a, A , b are such t h a t a > 0 , A > 0 , 6 > 0 . M(a) is shown in Fig. 4.4.
(4.3) T h e graph of
An autopilot and spacecraft orientation
system
97
Having performed the substitution r\ = 6 into system (4.3), we obtain 17 = -ar) + f(
f(<j) = M(a)/I,
/3 = 1 - ab. Put also /3 > 0.
M
A
-x
-A
K
2n-A
2n+A a
Fig. 4.4 Now we make a remark concerning the definition of solutions of system (4.4) on the lines of discontinuity r/ £ M1, a = A + 2irk, a = —A + 27rfc, 0;
2) Wv(t) + b J V - 0)] [Pv(t) + b !{
(4.5)
(4.6)
In the first case we assume that at time t a trajectory of system (4.4) "pass" from the semispace {a < a*} into that {cr > a*} (or vice versa) (see Fig. 4.5). Thus, we complete a definition of the solution of system (4.4) on a line of discontinuities by adding the limiting points of the trajectories Ti and T2 for t — 0 and t + 0 respectively. In the second case the velocity vectors Si and 52 of trajectories Ti and T2 for t — 0 and for t + 0 respectively are obtained in the following way (see Fig. 4.6).
98
Two-dimensional
control systems.
Phase
portraits
7-i
0=0*
Fig. 4.5
Fig. 4.6
We see that the trajectory Ti is not extensible by continuity into the semispace {a > cr*} and the trajectory F2 into that {cr < a*}. In this case the extended trajectory is also a continuation of trajectories Ti and F2 (here a solution of the Cauchy problem is not unique !). The solution slides along the line of discontinuity cr = a* until inequality (4.6) is violated. The velocity vector S3 of sliding solution is defined from Fig. 4.7. 7
• S ^ / S i
a*
a
Fig. 4.7 Thus, the end of vector 53 is a crosspoint of the line cr — cr* and a segment, connecting the ends of vectors Si and 52. The vector field so defined determines uniquely the trajectories, placed on the surfaces of discontinuity on the right-hand sides of differential equations. Such solutions are called sliding solutions. This method for supplementing a definition of solutions was suggested by A.F. Filippov and at present it is widely used in the control theory. For further details, concerning the theory of differential equations and differential inclusions with discontinuous right-hand sides, we refer an interested reader to the books [13, 17].
An autopilot and spacecraft orientation
system
99
Note t h a t in the case, considered above, for a sliding solution we obtain (r(i) — 0. Besides, for any solution r)(t),
/(*) = -§»>(*)
(4-7)
must be taken. Using the first equation of (4.4), we obtain very simple equations of a sliding regime cr(t)=cr\
f, = -(a+^jf,.
(4.8)
Note again t h a t t h e nonlinearity of / is determined in a sliding regime by formula (4.7), which depends on initial data. Such a supplementing of definition is due t o the rule of constructing the vector field on a line of discontinuity (see Fig. 4.7). In this case it is important t h a t for all sliding regimes inequality (4.6) is satisfied. From (4.7) it follows t h a t the values of f(t) always belong to t h e segment [/(cr* — 0), /(cr* + 0)] for /(cr* - 0) < /(cr* + 0) and t o the segment [f(cr* + 0), /(cr* - 0)] for /(<7*-0)>/(
V(r,,cr)=TJ + q j f(a)d<7,
(4.9)
0
where 2 ( 1 + ab) 9
~~(l-a6)2 •
We also consider the derivative of function V with respect to system (4.4), i.e., an expression ~ V(r,{t),
= M*)(-
<"?(*) + / ( * ( * ) ) ) +
(4 10)
+qf(*{t)){Pr)(t) + bf{
Two-dimensional
100
control systems.
Phase
portraits
Since for a sliding regime a(t) = a*, the following relation holds a(t)
a'
I
f(
0
0
f(
and the function / is completed by formula (4.7), we have jtV(V(t),tr(t))=-2(a+^jV(t)2.
(4.11)
By (4.10) 2ab = - 2 arj(t)2 + -—-1-ab)
d - V(r,(t),a(t)) dt , (l+a6)6 +
2
f{{))
JT^aW 2b
ab)2
f{a(t)) rj(t)+
- 2a H + (TT^)/^)2-
(4J2)
H*(t)f.
The function V with properties (4.11) and (4.12), constructed above, is the Lyapunov function (see Chapter 1). It is nonnegative for all r\ and a. A derivative of function V with respect to system (4.4) is nonpositive and satisfies relations (4.11) and (4.12) everywhere except for the points tk- Using this function, we show that any solution of system (4.4) tends to a certain equilibrium. For proving this fact we need in the following statement. Proposition 4.1. Ifrj(t), cr(t) is a solution of system (4.4), then rj(t), cr(t) + 2kn is also a solution of system (4.4). The proof is in substituting the solution n{t) a(t) + 2kn into system (4.4), taking into account a 27r-periodicity of function f(cr). It follows from Proposition 4.1 that if we have performed a shift of the band { f | £ l , a (E [— n, IT]} along the abscissa axis by 2nk, then the parts of trajectories, placed in this band, coincide with the corresponding parts of trajectories placed in the band {n 6 M1, cr £ [(2fc — 1)7T, (2k + 1)7r]}. We use this fact for a qualitative description of behavior of trajectories in the phase plane.
An autopilot and spacecraft orientation
system
101
Consider now the case of autopilot and sliding regimes in the phase plane (Fig. 4.8). These processes are described by equations (4.8). Using equations (4.7) and taking into account t h a t f(t) 6 [/(
where d = 6 |/(cr* - 0 ) | / | / 3 | . T h e analysis of the vector field makes it possible to describe the qualitative behavior of trajectories in small neighborhoods of sliding regimes (Fig. 4.9).
H
V
2%
i
II
H
2n
1
Fig. 4.8
Fig. 4.9
We see from this figure t h a t only two trajectories may enter the sliding regime on the line of discontinuity {?) £ I 1 , a — 7r}. Each point of the segment {a — n, r] £ [—d,d]} is a branch point of solutions. From this point the three trajectories are starting only. One of them enters into the right half-plane, the second into the left one and the third trajectory remains in a sliding regime. In other words, the sliding regime consists of three trajectories, namely of an equilibrium a = ir, r\ = 0 and of two trajectories, which tend to this equilibrium as t —>• oo. Each point on these trajectories is a branch point. Note t h a t the trajectory a(t),r](t), entering the sliding regime {rj £ [— d, d],a — 7r} in time t = to, has the following property. On the interval (—oo, to) the function |c(i) — n\ is the steadily decreasing one and \r](t)\ > d.
102
Two-dimensional
control systems.
Phase
portraits
This fact results from identity (4.12) and from the following relation a
q
f(a)da
< 0,
Vo-G ( - o o , + o o )
7T
(see Fig. 4.10).
Fig. 4.10 T h e segment {rj £ [—d, d], a = 0} of a sliding process is locally stable: in some e-neighborhood of this segment in a finite time all the trajectories reach this segment and then remain there, tending to zero as t —» + o o . Note t h a t a property to reach equilibrium in a finite time is a property of differential equations with discontinuous right-hand sides only. For the systems with smooth right-hand sides it is impossible. In this case the two trajectories reach a zero equilibrium in a finite time. Suppose now t h a t the trajectory considered does not tend to equilibrium. From the previous reasoning it is clear t h a t except for a countable set of crosspoints tk of the lines of discontinuity {77 £ M 1 , a = a*} on this trajectory the following inequality holds \f{l,
where I is a positive number (see Fig. 4.3). Identity (4.12) implies t h a t the
An autopilot and spacecraft orientation
system
103
following relation lim V(r){t),(r(t)) = - c o is valid. The last relation contradicts the boundedness from below of the function V(TJ,(T). The contradiction obtained makes it possible to prove that any solution of system (4.4) tends to equilibrium as t —> +00. Hence the following qualitative partition of the phase plane into the trajectories of system (4.4) (Fig. 4.10) can be obtained. In the control theory this qualitative picture of phase space, filled with trajectories, is called a phase portrait of system. We now discuss just in the same way as before the spacecraft orientation system. In this case the main distinction from the autopilot is in the presence of a dead zone and in the absence of a friction (a = 0). This fact implies the following changes in Fig. 4.8 and 4.9 (Fig. 4.11 and 4.12). 7
V / * ^
y
/*
((
V
Fig. 4.11
J\r K
^
s -*
Y
a
x
A
A
Fig. 4.12
From the configuration of trajectories in Fig. 4.12 we can conclude that the trajectory that does not tend to equilibrium as t —>• +00, as in the case of autopilot, satisfies relation (4.12) everywhere except for a countable set of crosspoints tk of the lines of discontinuity. Whence it follows that V(r,(t)Mt))<
V{r,(0),cj(Q)).
Therefore there exists a number T depending on initial data ?j(0), cr(0) and
Two-dimensional
104
control systems.
Phase
portraits
such that the following estimate holds \r)(t)\
Vt > 0.
(4.13)
We see t h a t for the trajectory considered there exists an infinite sequence tj —> + o o such t h a t t2i is an entry time of the trajectory rj(t):
a £ [ - A + 2nk,
A + 2nk]}
k
and ^2,-1 is an exit time of the trajectory from the set Q. Let us introduce the following notation L — m a x | / ( c ) | . From equations o
(4.4) it follows t h a t one of the inequalities hi-t
2
d i-i>j,
(4.14)
(4-15)
**-'«-i>iq^ is true. Really, we have either a(t2i) = cr(t2i-i) or the following relation \a{t2i)-a{t2i.1)\
= 2(7r-A).
(4.16)
In the first case we obtain \v(hi)-r){hi-i)\>d
(4.17)
(see Fig. 4.12). Therefore from the fact t h a t outside f2 U {cr = (2k — 1)TT} we have | / ( ( j ) | = L and from the first equation of system (4.4) (recall t h a t here a — 0) it follows t h a t \v{hi) - J?(*2i-i)| = L(hi
-hi-i)-
Then from (4.17) estimate (4.14) follows. In the second case from estimate (4.13) and the second equation of system (4.4) (recall t h a t here f3 = 1) we obtain an inequality \&{t)\
+ bL,
Relation (4.16) yields estimate (4.15).
Vt€[t2i-i,tn).
An autopilot and spacecraft orientation
system
105
Estimates (4.14), (4.15), and (4.12) imply the validity of one of inequalities V(rj(hi),(T{t2i))
< ^(r?(t2i_i),a(t2i-i)) -
V(r](t2i),a{t2i))
<
2bLd, r2
V(r]{t2i-1),<j(t2i-i))-2bL2
2(7T-A)
T + bL
Hence lim V(i](t-2i),cr(t2i))
= -oo.
This limit contradicts the boundedness from below of the function V(rj,a). T h e contradiction makes it possible to prove t h a t any trajectory of system (4.4) tends to equilibrium as t —> + o o . (Note t h a t a more universal method to use the Lyapunov functions for the analysis of stability of discontinuous systems can be found in [17]. However this generalization demands the developed theory of differential inclusions.) From the fact t h a t any solution of the system tends to equilibrium as t —> + o o and from the previous analysis of neighborhoods of sliding regimes we may obtain a qualitative partition of the phase plane into trajectories (see Fig. 4.13).
Fig. 4.13 Finally, note t h a t the equilibria r\ = 0, + o o except for special processes, tending
106
Two-dimensional
control systems.
Phase
portraits
to unstable equilibrium rj = 0,cr = (2k + 1)TT. However by virtue of instability this process is physically nonrealizable. T h e similar remarks can be made for the spacecraft orientation control. A systematic presentation of the spacecraft orientation control theory can be found in [38]. It should be noted t h a t the failure of spacecraft orientation system may cause a spacecraft rotation. As it can be seen from the phase portrait (Fig. 4.13) the initiation of the orientation control system may d a m p this rotation and orient the spacecraft in the proper direction. Such a situation arose at the space station "Mir" in 1987. A breakdown of a control system implies a complicated rotational motion of station. It had taken a certain time to recover the orientation system. After its initiation the orientation control system has d a m p e d the rotation of station.
Fig. 4.14 In conclusion we note t h a t in many cases the angular orientation control system must realize not constant direction but a very complicated maneuvering, which is usually called a programming motion. Consider, for example, a torpedoing of a moving ship. The torpedo is directed to not target but to a swirl. This trace can be detected by torpedo devices when the torpedo crosses it. Then the torpedo control system must perform the program motion of a torpedo roll-out and its second entry into the swirl (Fig. 4.14). When crossing a swirl the torpedo makes the second roll-out and so on until the torpedo will reach the target.
4.2
A synchronous electric machine control and phase locked loops
1. S y n c h r o n o u s m a c h i n e . Synchronous electric machines are widely spread as current generators. They are often used as electromotors. For example, as a motors for rolling
A synchronous
electric machine control and phase locked loops
107
mills. We discuss here a synchronous electromotor whose main elements are a stationary stator and rotating rotor (Fig. 4.15). In the figure the rotor slots are shown in which a rotor winding, the socalled direct current excitation winding, is placed. The winding is connected via brushes with a direct current source. On the stator there are also windings for alternating current, which creates an alternating magnetic field. These windings are such t h a t when passing an alternating current the magnetic-field vector is constant by absolute value and rotates with a constant angular velocity (Fig. 4.16).
Fig. 4.15
Fig. 4.16
It is clear t h a t this velocity coincides with the frequency of alternating current passing through the stator windings. Each loop of an excitation winding can be regarded as a loop with the direct current, placed in a magnetic field (Fig. 4.17). We consider the motion of the loop in the rotating coordinate system. In this coordinate system we have a loop with the direct current i, placed in a constant magnetic field. T h e force F acts on this loop (Fig. 4.18). It is well known t h a t F = /3iB, where B is a magnetic intensity, /? is a factor of proportionality. Whence it follows t h a t the value 0(t) satisfies the following equation 16 = -fiinlB
sin61 + M.
(4.18)
Here / is a moment of inertia, 21 is a distance between parallel parts of the loop, n is the number of loops, and M is a drag torque. Often, M is called an electromotor load.
108
Two-dimensional
control systems.
Phase
portraits
Let us consider equation (4.18) in the form 0 + &sin0 = 7.
(4.19)
Recall that for 7 = 0 we already considered this equation: it is an equation of a pendulum motion.
Fig. 4.17
Fig. 4.18
Equation (4.19) is equivalent to a system 0 = V, rj = —bsinO + 7,
(4.20)
which has the first integral V(rj,e) = ]-T]2
-bcos0-j6.
Since V(V(t),9(t)) = V(t)f,(t) + (bsine- y)6(t) = = r](t)[-bsm6 + j + bsin9 - -/] = 0, the trajectories of system (4.20) are wholly placed on the level curves of the function V(r],6) = C (Fig. 4.19). We consider here the most substantial case b > 7. For 6 < 7 the synchronous machine load is so large that in this case it is impossible to ensure the synchronous processes. Let us now determine what rotor motions correspond to one or other trajectory. The closed trajectories correspond to the motions that can be divided into two components, namely the rotations with a constant angular
A synchronous
electric machine control and phase locked loops
109
speed are equal to a magnetic vector speed and periodic swings of rotor with respect to this uniform rotation.
Fig. 4.19 T h e nonclosed trajectories correspond to the asynchronous motions of rotor. In course of time its angular speed becomes less than a magnetic field speed. Such processes are forbidden for a synchronous machine. Figure 4.19 show t h a t these two types of trajectories are separated by peculiar trajectories, which are also called homoclinic trajectories. As t —> + o o and t —> —oo these trajectories tend to the same stationary saddle point. Sometimes, to underline their separating role, they are also called separatrices. The objective of a synchronous machine control is to synchronize rotation of rotor and a rotation of magnetic field. For this purpose it was necessary to create the control devices, which could depress (damp) the above-mentioned swinging of rotor with respect to a rotating magnetic field and could expand regions, filled with the bounded solutions, (see Fig. 4.19). T h e first such devices were invented in the early 20th century. They were very simple by construction and consisted of two additional shortcircuited windings placed on the rotor. A motion of rotor with respect to the magnetic field induced a current in the windings, which, in turn, created forces acting on the rotor and damping its swings. T h e inventing of such control devices, which is brilliant by simplicity and technology, may be compared with the invention of the W a t t regulator. T h e complete mathematical model of the process of interaction of d a m p -
110
Two-dimensional
control systems.
Phase
portraits
ing windings (damper bar) with the machine is described by a system of differential equations of higher order. We refer an interested reader to the book [48]. There is also engineering argumentation, which we omit here, for a support of the assertion t h a t a moment of damping forces, caused by additional windings, equals —ad, where a is a constant parameter. In this case in place of equation (4.19) we obtain e + ae+bsm6
= j .
(4.21)
W i t h o u t loss of generality it can be assumed t h a t 6 = 1 . T h e equation of this type can be obtained from equation (4.21), using a change of variable T — t\J\/b. A qualitative study of equation (4.21) by the remarkable Italian mathematician Franchesko Tricomi in 1933 was primarily due to the problem of investigating a synchronous machine operating with the moment of damping force. We discuss the main results of Tricomi. Consider the following system equivalent to (4.21)
V=
(4.22)
-ctT)-
where 0 solution to some equilibrium as t —> + 0 0 .
of system
(4.22) tends
P r o o f . Consider a function e V(rj,e)
= ^n
2
+ j\(6)d6.
(4.23)
0
On the solutions rj(t), 6(t) of system (4.22) we have V(r,(t),6(t))
= -ar,(t)2.
(4.24)
Whence it follows t h a t the function V(r)(t),0(t)) is a monotonically decreasing function of t. By virtue of boundedness of n(t), 8(t) for t > 0 the function V(r)(t),6(t)) is also bounded. This fact implies the existence of a finite limit lim V(r](t),6{t))
= <£.
(4.25)
A synchronous
electric
machine
control
and phase
locked loops
111
Integrating both sides of equation (4.24) from 0 to t, we obtain t
a J n{r)2dT = Vfo(O), 6(0)) - V{f,{t),6(t)). Hence
/ o
\2, r](tydt < oo.
(4.26)
In addition, we have fo(t)2]' = 2r,{t)f,(t) = -2aV(t)2
- 2r,(t)
From the boundedness of the solution r](t), 0(t) it follows that there exists a number C such that |fo(*) 2 ]"|
V<>0.
(4.27)
By (4.26), (4.27), and Lemma 1.3 (Chapter 1) we obtain lim n(t) = 0.
(4.28)
Therefore, from relation (4.25) and expressions for the functions V(r], 9) and
= /3.
t—f + OO
Then for a certain number 6Q we have lim 6(t) - 60.
(4.29)
We see that the point TJ = 0, 6 = QQ is an equilibrium of system (4.22). The statement of Theorem 4.1 follows from relations (4.28) and (4.29). Theorem 4.1 implies that for 1 < 7, i.e., in the absence of equilibrium, all the solutions are unbounded (Fig. 4.20). Now we consider the case 1 > 7. Note that since function (4.23) satisfies inequality (4.24), the level curves of the function V(rj, 6) for rj ^ 0 are the curves without contact (transversal) for trajectories of system (4.22). In other words, these trajectories intersekt level curves from outside toward the interior (Fig. 4.21).
112
Two-dimensional
control systems.
Phase
portraits
Fig. 4.20
Fig. 4.21 Let us denote by 9Q a zero of function 0 and by 9\ a zero of function
lim 9(t) - 9X
and 77(f) < 0 for sufficiently large f. Let rj(t), 6{t) be a solution of system (4.22) such that lim n(f) - 0,
lim 0(f) = 6»i
A synchronous electric machine control and phase locked loops
113
a n d rj(t) > 0 for sufficiently large t (Fig. 4.22). From the fact t h a t a level curve of the function V(rj, 9), passing trough the point n = 0, 6 = 6\, is t h e level curve without contact it follows t h a t a trajectory rj(t), 9{t) for all f G l 1 lays below t h e curve {V(r],0) — V(0,9i), r) < 0} (Fig. 4.23). Since from relations V{n,9) = ^ ( 0 , ^ ) a n d 9 —)• + o o it follows t h a t n —> — oo, we can state t h e following lemma. Ik
MO,0(O \
V'
W)MO
Fig. 4.22 L e m m a 4 . 1 . The following
Fig. 4.23 relations
lim nit) — —oo, t-V-oo
lim 6(t) = + o o
v
i-f-oo
v
(4.30)
'
are valid. _ Obviously, there are three possibilities for t h e trajectory rj(t), 9{t): 1) a number r exists such t h a t ?y(r) = 0, 9{T) £ (9\ — 2ir, 9r,), rj(t) > 0, Vi G ( r , + o o ) (Fig. 4.24), •L>7
W)MO
Fig. 4.24 2) for all t we have
??(<) > 0, lim rj(t) = 0, and t—• — o o
(Fig. 4.25),
lim 9(t) =9X~ 2ir, £—>• — o o
114
Two-dimensional
control systems.
Phase
portraits
n(t)MO Fig. 4.25 3) for all t we have
?j(t) > 0 and
lim 6{t) = - o o (Fig. 4.26). t—y — oo
U<,t)Mt) Fig. 4.26 F. Tricomi has shown that in case 3) the following relation holds lim rj{t) = +oo.
(4.31)
It is a subtle theorem, the proof of which is omitted and can be found, for example, in [7]. In the works of scientists of the A. A. Andronov school the two-dimensional systems have recently been indicated such that in case 3) relation (4.31) is not valid [19]. From Theorem 4.1 and the properties of separatrices rj{t), 6{t) and rj(t), 0(t) of the saddle point rj — 0, 6 = 6\, we can conclude that for 1 > 7 three types of partition of the phase space into trajectories are possible: 1) The separatrices rj(t), 6(t) and rj(t), 0(t) are boundaries of a domain of attraction for a locally stable equilibrium 77 = 0, 6 = #0 (Fig. 4.27). When shifting along 6 by a value 2&7r these domains turn out to be domains of attraction for stationary solutions 77 = 0, 0 — 6Q + 2kir.
A synchronous
electric machine control and phase locked loops
115
Fig. 4.27 T h e trajectories outside these domains tend to infinity as t —>• + 0 0 . 2) A separatrix rj(t), 9(t) is heteroclinic, i.e., lim lj(t) t->-oo
lim n(i) = 0, C-> + oo
lim 0(t) =61t-f-00
V
'
2TT,
lim 6(t) = 6^. t-> + oo
y
'
In this case the domains of attraction of stable equilibria are also bounded by separatrices but in the half-plane {rj < 0} there do not exist unstable "corridors" (Fig. 4.28).
- • . . - '
Fig. 4.28 '
.'..
3) T h e whole phase space is partitioned into the domains of attraction of stable equilibria. The boundaries of these domains are separatrices of saddle equilibria (Fig. 4.29).
116
Two-dimensional
control systems.
Phase
portraits
Thus, the determination of domains of attraction of stable equilibria and the whole global analysis of system (4.22) are reduced to a computation or an estimation of one trajectory only, namely a separatrix rj(t), @(t). In case 1) it is also important t h a t a separatrix rj(t), 6{t) can be computed or estimated. F. Tricomi and his followers have obtained various analytic estimates of these separatrices. Case 2) in the parameter space { 0 , 7 } corresponds to interchanging the qualitative pictures of trajectory dispositions (i.e., corresponds to the replacement of case 1) by case 3) or vice versa). Such a qualitative change is called a bifurcation and the parameters a and 7, corresponding to case 2), are called parameters of bifurcation. T h e evaluations of separatrices lead to the evaluations of parameters of bifurcations. In the 60-70s of last century for the determination of separatrices and, consequently, of parameters of bifurcation the numerical methods were applied [40]. T h e computation of separatrices is divided by two parts. In some sufficiently small neighborhood of the saddle point the separatrix is approxim a t e d by a linear function with the coefficient
-2±VT-^
I )
-
Outside this neighborhood some numerical method for a solution of ordinary differential equations is used (the most often it is the R u n g e — K u t t a
A synchronous
electric machine control and phase locked loops
117
method). By using the numerical methods, mentioned above, the bifurcation curve is computed (see Fig. 4.30).
Fig. 4.30 The behavior of trajectories in the phase space, shown in Fig. 4.28, corresponds to the points on this curve. A structure, shown in Fig. 4.29, corresponds to the points, placed under this curve. A situation, shown in Fig. 4.27, corresponds to the points, placed above this curve and under the straight line {7 = 1}. Figure 4.20 corresponds to the points, placed above the straight line {7 = 1}. The above imply the following statement. Lemma 4.2. Any solution rj(t), 6(t) with initial data from the domain
{W
{0i-2ir,6i)
tends to equilibrium r) = 0, 6 = 8Q as t —>• +00. Now we discuss the problem on a limit load of synchronous machine. The limit load problem. Consider a synchronous electromotor, which rotates the rolls of hot rolling mill (Fig. 4.31). Till a red-hot bar moves along lower rolls only, we may assume that a synchronous machine load is equal to zero and the machine operates in a synchronous regime, i.e., 6(t) = 0, 0(t) = 0. At time t = r the bar enters the part of rolling mill, in which a rolling occurs. The bar moves forward and decreases in thickness due to
118
Two-dimensional
control systems.
Phase
portraits
the upper and lower rolls, rotating in opposite directions. Obviously, at time t = T a load-on occurs.
ooooooooo) (o o~c7b o~o o o o o o o o o o ) Fig. 4.31 Thus, for t < T the process is described by the following equation 6 + a9 + sin 6 = 0
(4.32)
and corresponds to the solution 9(t) = 0, 9(t) = 0. For O r w e have 6> + a0 + sin0 = 7
(4.33)
and the process corresponds to a solution with initial data 9(T) = 0, 0(r) = 0. We observe that such a solution of equation (4.33) is not equilibrium and the problem lies in the fact that after a load-on the synchronous machine remained synchronous as before. In other words, it is necessary for solution of equation (4.33) with initial data 6(r) = 0, 6(T) = 0 to be in a domain of attraction of a new stable equilibrium 6(t) = 6Q, 6(t) = 0. To solve this problem we apply Lemma 4.2, which can be restated in the following way. Theorem 4.2. An admissible load-on 7 is a value 7 satisfying the following inequality 0
[{sinG--/)
d6 <0.
(4.34)
Theorem 4.2 implies that the limit load-on 7 = T can be obtained from the equality of areas shown in Fig. 4.32
A synchronous
electric machine control and phase locked loops
119
Fig. 4.32
Fig. 4.33 fn the engineering practice such a method to determine the limit load-on is called a method of areas. Now we discuss what advantage has an algorithm of a slow loading of a synchronous machine. Suppose, we have m steps of loading. At the k—th step at time Tfc > T^-I the load increases instantly from 7^-1 to j ^ . Let a difference 77. — Tf,-\ be sufficiently large, namely such that a solution with initial data ^(r^), 9(TI,) (the previous equilibrium) is in a sufficiently
120
Two-dimensional
control systems.
Phase
portraits
small neighborhood of a new equilibrium. In this case by Lemma 4.2 the initial d a t a #(Tfc), 0(Tfe) belong to a domain of attraction of a new stable equilibrium if the upper shaded area in Fig. 4.33 is greater t h a n the lower shaded area. By choosing jk-i and 7^ sufficiently close together, we can always satisfy this condition. Thus, a synchronous machine can be slow loaded up to any value 7 < 1. Note t h a t the start of synchronous generators and their loading are complicated dynamic processes taking sometimes several tens of minutes. Note also t h a t at present there are many synchronous machine control systems, which make use of an exciting winding. T h e values 9(t) and 6(t) are transformed into an additional voltage, which is delivered to an exciting winding. This voltage creates a controlled force moment acting on a rotor and refining the dynamic properties of synchronous machine. We make mention here of the book of M.M. Botvinnik [8] in which such a control is considered. M.M. Botvinnik is also far-famed as a multiple world chess champion. P h a s e l o c k e d l o o p s . The synchronization principle is the central one in transmitting video pictures. We discuss a basic principle of black-andwhite display transmission. At broadcast station, on a photomatrix there is a picture of an object. An electronic ray passes sequentially along the lines of this matrix and, closing a circuit at time t, generates a voltage u(t), depending on brightness of a matrix element in which at time t the electronic ray occurs (Fig. 4.34).
r
Mt)
'1
h
Fig. 4.34 A scanning of the photomatrix takes the time interval [ i i , ^ ] - Then a high-frequency modulation of the signal u(t) occurs which is transmitted via a space to a television receiver. Here a demodulation arises and the signal is transmitted to a driver circuit of an electronic ray in a television electron-beam tube. T h e greater is the value of u(t), the greater is the intensity of electronic ray and, consequently, the greater is the brightness
A synchronous
electric machine control and phase locked loops
121
of the corresponding point of the display, at which the electronic ray occurs. The ray, passing through a deflector, scans sequentially display lines and creates a picture on the display. It is clear that in order to avoid the transmission errors, the generators of deflection of electronic rays at the television station and in the television receiver must work synchronously. In other words, the frequencies of both generators must coincide. This is an objective of the control systems being phase locked loops. These systems work in the following way. Information on the frequency wo of generator at the television station is transmitted to each television receiver by means of a sync-pulse signal, which is a part of television signal (Fig. 4.35). Sync-pulse signal
Brightness signal
Fig. 4.35 In television receivers this signal is separated and information on uio enters into the phase locked loops, shown in Fig. 4.36.
UL}AA/VV o\
LPF
SG
Fig. 4.36 In the case of open loop when the steering gear (SG) does not act on a voltage controlled generator (VCG), the latter (VCG) works in such a way that frequency wi is kept constant. We also assume that the frequency u>o of a TV station generator is constant.
122
Two-dimensional
control systems.
Phase
portraits
T h e objective of a control is to eliminate the frequency misalignment r = wo — wi. After initiating the control system, shown in Fig. 4.36 at time t = 0, the value LJ is not constant: u> = w(i), w(0) = wi. The objective of a control system is the following synchronization: u(t) —>• CJO as t —> + o o . T h e functions t
60{t) = 0O(O) + w 0 t,
0(i) = 0(0) + / * « ( r ) d r
(4.35)
o are called phases of generators. The numbers 6>o(0) and 6(0) are the values of these phases at time t = 0. T h e phases #o(t) and 6(t) are inputs of the block V, which is called a phase detector. There are many different electronic schemes being phase detectors. Their description can be found in [31, 40] It is significant t h a t in all these systems the output of block V has the form ai = F(60-6).
(4.36)
Here the function F(x) is 27r-periodic. Typical functions F(x) are sin a; and the functions whose graphs are shown in Fig. 4.37 and 4.38.
Fig. 4.37
Fig. 4.38
A synchronous
electric machine control and phase locked loops
123
The signal
Wl
+ *3{t),
*3{t) =S*2{t),
(4.37)
where 5 is a number. The action of the most of steering gears (SG) is based on the change of reactivity of the contour of a voltage controlled generator. The outlines of such devices can be found in the above-mentioned books [31, 40]. In the simplest case that a low pass filter does not enter into system (4.35)—(4.37), we have the following differential equation of the first order, describing the phase locked loops,
{Ba-ey
=
u0-ui-SF(80-0).
Using notation cr = 9Q — Q and recalling that r = wo — wj, we obtain finally the equation
J + SF(a)=r.
(4.38)
Consider now the equations of the phase locked loop with RC-filter. Since for RC-circuit the signals
* c ^ + *a = f(*).
(4 . 39)
Using a relation u>o — wi = T and substituting the expression for
124
Two-dimensional
control systems.
Phase
portraits
we obtain
RC + d +SF
^
i ^ =^
<4-4°)
where cr = Or, — 0. If a characteristic of the phase detector is given by F(
•-(:)•
-=(i)-
We see that if x(t) is a solution of the system, then x(t) + d is also solution of it. Consider a discrete group T={x
= kd\
k€%}.
Here 7L is a set of integer numbers. Let us consider now a factor-group Rn/T classes [x] £ M n /F, which are defined by
whose elements are residue
[x] = {x + u \ ue T}. We introduce the so-called plane metric p([x], [y]) = inf \z - tf |. Here, as previously, | • | is Euclidean norm in M".
A synchronous
electric machine control and phase locked loops
125
It follows from Proposition 4.1 §1 that [x(t)] is a solution and the metric space K " / r is a phase space. It is partitioned into nonintersecting trajectories [x{t)], i G l 1 . We can find the following diffeomorphism between M n /T and the surface of cylinder M1 x C. Here C is a circle of radius 1. Consider a set Q, = {x\ j] 6 M1,
Fig. 4.40
Fig. 4.41
Sometimes such a space is more convenient than M . Really, in cylindrical phase space there does not exist a multiplicity: all the values a + 2kir correspond to one state of system only. Besides in this case there are two types of the closed trajectories, which correspond to periodic processes of control systems. The closed trajectories of the first type can homotopically be collapsed onto a point (Fig. 4.40). This closedness is preserved in passing to M . The closed trajectories of the second type, which encircle the
126
Two-dimensional
control systems.
Phase
portraits
cylinder, cannot be collapsed homotopically onto a point. Such a closedness vanishes when passing to ffi2 (Fig. 4.41).
4.3
The mathematical theory of populations
A biological population, having an environment facilities such as a plenty of food, an absence of enemies, and a large livingspace, reproduces themselves in direct proportion to its strength. Let the number of persons be sufficiently large. Keeping in mind a certain idealization, assume that n(t) is a smooth function of t. This reproduction is described by the following differential equation
Here a constant positive number A is a birthrate of population. Equation (4.41) can also be found from the analysis of economic growth, assuming that on large time intervals the growth coefficient A is constant. Note that on a large time interval a bank deposit rate A is also described by equation (4.41). We consider the solution of this equation in the following form n(t) = e At n(0).
(4.42)
Under the ideal conditions the populations obey the law of exponential increase (4.42), which was experimentally observed in the case of the bacteria reproduction. However in ecological systems we see a strong competition of populations, concerning with space and food, and predators annihilating preys. Besides the human interference may also depress some populations. Thus, in the ecological systems there exist many feedbacks and therefore the human attempts to control an ecological system (for example, by means of shooting predators) may have also objectionable results. The study of the mathematical models of population interactions was begun in 20th of the previous century and in 1931 the notorious book of Vito Volterra [47], devoted to this problem, has been published. We consider the Lotka-Volterra equations, which describe the interaction of two species: predators and preys. This mathematical model, its various generalizations, and the other of interesting population models are
The mathematical
theory of populations
127
discussed in the book [47]. The preys (hare, for example) are assumed to be reproduced themselves under the absence of predators (wolves, for example) according to mathematical model (4.41). In the case that the preys (a food of predators) are absent the predators die out, what corresponds to a negative growth of the coefficient A. Assuming the coexistence of these two species, the predators and preys can encounter to one another. The number of these encounters is in direct proportion to strength of predators and preys an\(t)n2{t). These encounters decrease a population of preys (with a coefficient —j3\) and increase a population of predators (with a coefficient /32). Arguing as above, we arrive at the following differential equations for the strengths of the prey population ni(t) and the predators population n2(t) hi = Aini — /3icmi7i2,
(4.43)
h2 = —A2n2 + /32anin2
Here n\ and n2 are nonnegative. In this case system (4.43) has the following solutions ni(<) = 0, n2{t) = 0, m{t) = n2(t) = 0.
n2{t) = e- A2 *n 2 (0), ni(i) = e A l t n 2 (0),
Thus, the phase space of system (4.43) is the first quadrant of the plane {m > 0, n 2 > 0} (Fig. 4.42) kn2
-»-
n
\
Fig. 4.42 Now we show that for nj > 0, n2 > 0 system (4.43) has the first integral V(ni, n2) = A 2 lnni — afani + Ai lnn 2 — aP\n2.
Two-dimensional
128
control systems.
Phase
portraits
Really, V(ni(t),n2(t))
= A2
a/?2«i + Ai rii
a/3ih2 = n2
= A2(Ai - a/3in2) - a/32(Xini - a/3in x n 2 )+ +Ai(-A 2 + a/32ni) - a/3i(-\2n2 + a(32nin2) — 0. Whence it follows that the function W{n1,n2)
ev^n^=g{nl)h{n2),
=
where g(ni) = n^e-a^n\
h{n2) = nx2l
a n
e~
^\
is also the first integral of system (4.43). The functions g and h have the maximums at the points "i
A2 ap2
Ai
n2
a/3i
(4.44)
respectively. The graphs of these functions are shown in Fig. 4.43.
Fig. 4.43 The function W(rii, n 2 ) has a property W(nun2)
<W
A2
Ai
a/V aPi
Whence it follows that the level curves of function W(n\, n2) are the closed curves encircling a point given by (4.44). Thus the trajectories of system (4.43) are placed on the closed level curves and point given by (4.44) is an equilibrium state (see Fig. 4.44).
The mathematical
theory of populations
129
Fig. 4.44 The strengths ni(t) and n2(2) of preys and predators respectively, as it can be seen from the phase portrait in Fig. 4.44, are periodic functions. Such an oscillating character of the change of a strength of population one can observe in various ecological systems and the Lotka-Volterra equations describe qualitatively the interaction of populations of predators and preys to one another in a sufficiently adequate manner. Consider system (4.43) in the form (In n i ) ' = Ai - / ? i a n 2 , (lnn 2 )* = —A2 + /? 2 ani and integrate the left-hand and right-hand sides of these equations from 0 to T, where T is a period of solution ni(t), n 2 (i). Then we obtain T
~fn2{t)dt = \ ,
(4.45)
0 T
ni{t)dt
fl
= ^k'
0
Recall that a magnitude T
^Jx(t)dt 0
(4 46)
'
Two-dimensional
130
control systems.
Phase
portraits
is called a mean value of a T-periodic function x(t). Thus, we proved the following T h e o r e m 4 . 3 . The mean values of functions ni(t) and n2(t) do not depend on the initial data rci(O) and n2(Q) respectively and coincide with the stationary values (4.44). Consider now the case t h a t strengths of populations are controlled by means of an annihilation of persons of each species. This annihilation is assumed t o b e in the direct proportion t o the strength of population. In this case the equations of interaction of populations have the form " i = (Ai - 7 i ) " i - Pian1n2, n2 = - ( A 2 + 72)7*2 + /32an1n2.
.
.
Here 71 and 72 are intensity coefficients of annihilation. We consider first the case Ai < 71 (it means t h a t the annihilation of preys is more intensive than their reproduction in the absence of predators). In this case both populations die out. Really, consider a function V(rii, n2) = /? 2 ni +/?i"2- For solutions rii(t) and n2(t) of equation (4.47) we have V(ni(<), n2(t)) = (Ai - 7 i ) / ? 2 n i ( 0 - (A2 + 7 a ) / W 0 < <-eV(ni{t),n3(t)), where e = min(7i — Ai,A 2 + 72). following form
u dR, l4,48j
Considering inequality (4.48) in t h e
(K(m(f),n2(t))e£t)* < 0
V<>0
(4.49)
and integrating both sides of inequality (4.49) from 0 t o t, we obtain a n estimate < e - £ V ( n ! ( 0 ) , n 2 (0)).
V{ni(t),n2(t))
(4.50)
By (4.50) lim nAt) = 0, J-+ + 0O
V
'
lim n2(t) = 0. t-t + OO
^ '
From the last relations a complete annihilation of preys and an extinction of predators follows. Now we consider the most interesting case Aj > 7 1 .
The mathematical
theory of populations
131
In this case in place of formulas (4.45) and (4.46) we obtain
0 T
o These relations can be restated in the following assertion. Theorem 4.4 / / two species are exterminated in direct proportion to the number of persons, then the mean value of preys increases and the mean value of predators decreases. A nontrivial fact is that, shooting hare and do not hunting wolves, we do not affect a mean strength of hare. In this case the mean strength of wolves decreases.
Chapter 5
Discrete systems
5.1
Motivation
1. Mathematical motivation. A generator of Cantor set. In the previous chapter a wide variety of different trajectories in twodimensional phase spaces was demonstrated. These are trajectories, tending to equilibrium or to infinity, the trajectories that are closed curves corresponding to periodic solutions or are closed in a two-dimensional cylindrical phase space. In a one-dimensional phase space the trajectories of a differential equation ^
= /(*).
^ »
!
(5-1)
with the continuous right-hand side have no such a variety. From the fact that trajectories of equation (5.1) either fill the whole intervals in M1 or turn out to be stationary points, coinciding with the zeroes of the function f(x), we can conclude that any solution of equation (5.1) as t —> +oo tends either to stationary point or to infinity (Fig. 5.1). x Fig. 5.1 A situation, which is rather different from the above, occurs in the case of one-dimensional discrete equation xk = f(xk-i). 133
(5.2)
Discrete
134
systems
This equation is a recurrent formula to determine a sequence xk, k = 1,2, To find uniquely the sequence Xk the initial data XQ must be given. This case is similar to that of solving the Cauchy problem for differential equations.
Fig. 5.2 Let the function f(x) in equation (5.2) be given in the following form (Fig. 5.2)
/(*) =
Zx 3(1 - x)
for for
x < 1/2, x > 1/2.
(5.3)
Then equation (5.2) has exactly two stationary solutions xk = 0,
xk = 3/4.
Lemma 5.1. For any a?o€T[0,1] the solution xk tends to —oo as k
oo.
P r o o f . For XQ < 0 from (5.3) it follows that xk < 0 for all k = 0 , 1 , . . . and Xk = 3 fc 2!o.
(5.4)
For xo > 1 we obtain x± < 0. Consequently, xk = 3 fe_1 a;i.
(5.5)
The statement of Lemma 5.1 follows directly from relations (5.4) and (5.5). Now we discuss the map / of the segment [0,1]. Let us assign the segment [0,1] to each of two segments [0,1/3] and [2/3,1]. The interval (1, 3/2) is assigned to the interval (1/3, 2/3). By Lemma 5.1 all the solutions Xk of equation (5.2) with initial data from interval (1/3, 2/3) tend to infinity as k —> oo. Thus, interval (1/3, 2/3) is excluded from the segment [0,1].
Motivation
135
T h e m a p / ( / ( • ) ) takes each of the segments [0,1/9], [ 2 / 9 , 3 / 9 ] , [ 6 / 9 , 7 / 9 ] , [8/9,1] to the segment [0,1]. T h e solutions Xfc with initial d a t a from intervals ( 1 / 9 , 2 / 9 ) , ( 1 / 3 , 2 / 3 ) , ( 7 / 9 , 8 / 9 ) are excluded from the segment [0,1] and by L e m m a 5.1 it tend to infinity as k —\ + o o . Solutions XJV belong to the segment [0,1] if and only if the initial d a t a are in the segments
1 ' 0, jjF
'2 5
31
3TV<
3w
6
)
•iN~\
7
N
W' z _
i • ••i
[ Z
N
l
' J
T h e process of eliminating the medial parts of remaining segments is usually illustrated by the following figure (Fig. 5.3).
Fig. 5.3 T h e part of the segment [0,1] t h a t remains after infinitely many of procedures of deleting the medial parts of remaining segments is called a Cantor set. It is a very "holey" set. In any neighborhood of a point of the set there are holes being small intervals, which do not belong to this set. Can one measure this set? We shall answer to this question in the frame of the measure theory. At first we make use of the Lebesgue measure. In this case it is sufficient to measure the final lengths of segments after the N—th iteration, to summarize them, and to tend N to infinity. This implies t h a t
1N 3=0
^ 3 ^ N-
Thus, the Lebesgue measure of a Cantor set equals zero. In 1916 F. Hausdorff has introduced another definition of measure (to be precise, of an outer measure). Let us consider the Hausdorff measure of the Cantor set. We measure the set by not linear meters m, square meters m 2 or cubic meters m 3 but by m d , where d is any positive number.
136
Discrete
systems
In this case for each segment, obtained after N iterations, we find the following measure
\w) • We see that if this segment is assumed to be measured in square meters, then in terms of outer measurement (outer measure), covering the segment by the square with side 1/3 N (Fig- 5.4), we obtain the following "outer" value (l/3N)2. Extending this expression to arbitrary d, we obtain a value (l/3 JV ) rf . Summing over the rest of segments, we find
This value tends neither to zero nor to infinity in the only case log 3 Thus, the Hausdorff dimension of a Cantor set is equal to log 2/log 3 and its Hausdorff measure is one. The existence of sets similar to the Cantor ones stimulated the development of the modern theory of measure and a metric dimension of sets. 1/3*
1/3*
Fig. 5.4 Note that if the initial point XQ is in a Cantor set A', then the corresponding solution Xk belongs to K for all k. The set K, possessing such a property, is called invariant (or positively invariant). In the modern theory of dynamical systems the invariant sets of a noninteger Hausdorff dimension are called strange. At present the continuous and discrete systems, having such invariant sets, are intensively studied. In discrete systems such sets exist in the simplest one-dimensional models.
Motivation
137
For continuous system such sets are found due to computer experiments for phase spaces of dimension greater t h a n two. (See [42].) T h e generator considered above tends to infinity all the points from M1 \ K. We consider now the discrete system (5.2) with the phase space [—1, 3/2] and with the continuous function f(x) satisfying the following conditions , > _ f 3x \ 3(1 - a:)
for for
/ ( X j -
-l
for for
x G [0,1/2], x G [1/2,1], a;G [1,3/2], i£[-l,0],
where a is a number from the interval (0,1). L e m m a 5 . 2 . For any XQ G [—1)0] U [1,3/2] the solution zero as k —} + o o .
Xk tends to
P r o o f . P u t x0 G [1,3/2]. Then xx G [ - 1 , 0 ] . T h e set [ - 1 , 0 ] is positively invariant, i.e., for xo G [—1,0] we have Xk G [—1,0] V I , 2 , . . . , in which case xk > ak~1xi
-> 0
as k —>• + o o . This completes the proof of L e m m a 5.2. Using Lemma 5.2 and arguing as in the case of constructing the Cantor set, we see t h a t in addition to the invariance of A' with respect to the m a p / there exists a property of a global attraction, namely any solution Xk in the phase space [—1, 3/2] tends to set K as k —> + o o : p{xk,K)
= inf \z — Xk\ -> 0
(5.6)
z€K
as k —> + 0 0 . T h e invariant bounded sets, possessing attracting property (5.6), are called attractors. T h e attractors, having the noninteger Hausdorff dimensions, are called strange attractors. In some works an attractor is said to be "strange" if it has a property of inner instability, which is in the following: for small changes of initial d a t a xo G K, the difference between the solutions, corresponding to these initial d a t a , is not small for sufficiently large values of k. For the Cantor set generator such a sensitivity with respect to initial d a t a from K occurs.
Discrete
138
systems
Note that the numerical methods for solving differential equations, which are based on the idea of discretization, lead to discrete equations (5.2) with an n-dimension phase space M n . 2. Examples from technology. We now pass from the Cantor set to the analysis of a work of production storage. Consider a storage for n different component parts of a product. At the end of each day one makes the report on the numbers of the component parts. These reports can be regarded as discrete times: t = 0 , 1 , 2 , . . . , k. The components of vector Xk are the numbers of component parts of the similar type, which are stocked at the end of previous day. Each of components of vector Uk is a number of component parts of a certain type, which are delivered at the storage for a day on account of external supplies. Suppose that for the products to be fabricated it is necessary to take Vxk component parts from the storage for a day. Here V is a diagonal matrix. It follows that industrial chain: a storage - an assembly plant can be described by the following discrete equations Xk + l = Xk—
T>Xk + Uk,
(5.7)
C*Xk-
Here the values a^ correspond to the finished goods in a day. The components of vector c coincide with the diagonal elements of matrix V. Input
Output
Water temperature
D
^ D
Time
D
~ —,
CP
—*•
••
Programm Switch i
L
i Fine
Colour
1
Heating element
Water pump
Engine
White Ul
Fig. 5.5
'
S
Motivation
139
As in Chapter 2, equation (5.7) can be regarded as a linear discrete block with the input Uk and the output <Jk- Introduction to the theory of such systems is presented in the next section.
Fig. 5.6 The use of computers in the control systems permits us to describe the whole system or its part by discrete equations. As an example we consider the use of microcomputers in a control system of a washing machine [10]. The block diagram of such a system is shown in Fig. 5.5.
Discrete
140
systems
T h e system controls three values: a temperature of water, a water level, and a washing time. By a switching element one of programs is started, which yields three parameters: a water level, a water temperature, and a washing time. By means of the sensors of a water level, a temperature, and a timer, prescribing the work time of electromotor, the current information enters the input in the form of continuous electric signals. A digitizer (D) transforms an analog information for a central processor (CP) in the discrete form. The signals of C P control heating elements, water p u m p s , and an electromotor. The signals of C P depend on a washing program. A flowchart of one of such programs is shown in Fig. 5.6. Obviously, the system, described above, changes its states at discrete times, except for the input signals, which remain to be continuous until they are transformed by the digitizer.
5.2
Linear discrete s y s t e m s
In this section we discuss a discrete analog of the linear theory, which was developed in Chapters 2 and 3 for continuous systems. We consider first a linear discrete homogeneous system with a constant matrix Xk = Axk-i,
A; = 0 , 1 , . . .
(5.8)
Here A is a constant n x n-matrix, x is an element of an n-dimensional vector space. In this case a solution of system (5.8) is given by xk = Akx0.
(5.9)
As in the theory of differential equations, we shall say t h a t XQ are initial data. Sometimes it is useful to consider a similar matrix B A =
S-XBS.
Then formula (5.9) takes the form xk = S-1BSS~1BSS-1B...BSj k
x0 = S~lBkSx0.
(5.10)
Linear discrete
systems
141
We recall that if the matrix B is a Jordan matrix, i.e.
M
o
\o
J rr
B = where Jj are Jordan blocks, then the matrix Bk has a very simple form •J?
0
0
Jkm
B" where for t h e diagonal m a t r i x Jj we obtain
'XI k
J
0
-
(5.11)
If a J o r d a n block is as follows
(\ Jj
1 (5.12)
=
1 Xj/
\0 then
l\«.
kX k-1
k(k-i)\kf2
Vo
k...(k-l
+ 2)\k-'
^
+1
\
/
Here Aj are the eigenvalues of the matrix A. We assume further that x is an element of Euclidean vector space with the norm | • |. This implies the following Theorem 5.1. All the solutions of system (5.8) tend to zero as k —> +oo if and only if all the eigenvalues Xj of the matrix A satisfy the inequality
M < i-
(5.13)
For all the solutions of system (5.8) to be bounded it is necessary and sufficient for the inequalities \Xj\ < 1 to be satisfied for the eigenvalues of the
Discrete
142
systems
matrix A, corresponding to Jordan blocks of the form (5.11), and the inequalities \\j\ < 1 are satisfied for the eigenvalues, corresponding to Jordan blocks of the form (5.12). D e f i n i t i o n 5 . 1 . We shall say that a zero solution of system (5.8) is Lyapunov stable if for any 8 > 0 there exists a number e > 0 such that from the inequality \XQ\ < 8 the relation \xk\ < e, V k = 0 , 1 , . . . follows. D e f i n i t i o n 5 . 2 . A zero solution of system (5.8) is said to be asymptotically stable if it is Lyapunov stable and all the solutions tend to zero as k —> + o o . Let us remark t h a t in this case by virtue of a linearity the two properties are equivalent: the property t h a t the solutions with initial d a t a from some small neighborhood of zero tend to zero and the property t h a t all the solutions tend to zero as k —>• + o o . Thus, condition (5.13) is necessary a n d sufficient for the asymptotic stability of a zero solution of system (5.8). We recall t h a t for a system of differential equations dx — = Ax dt the necessary and sufficient condition of asymptotic stability is t h a t the following inequality holds Re Aj < 0 for all the eigenvalues Aj of the matrix A. We show now t h a t , sometimes, for solving system (5.8) the change from formula (5.9) to (5.10) can be useful. As an example, consider a discrete system of the form (5.8), which was probably the first system of this kind, studied by mathematicians. Early in the 13th century Fibonacci considered a discrete equation fk = fk-i
+ fk-2,
k > 1,
/o = / i = 1.
(5.14)
This equation generates the so-called Fibonacci numbers fk. Let us find the explicit expressions for fk, making use of the approach, described above. We introduce the following notation
Linear discrete
systems
143
Then equation (5.14) can be written as system (5.8) with a m a t r i x
It is obvious t h a t the Jordan form of matrix A is a matrix B
>i
- ' o
0 P2
where
1 +V5
1 - V5
To determine the m a t r i x 5 , consider an equation BS = SA,
(5.15) S~XBS.
which is a corollary of the equation A —
W i t h o u t loss of generality we can represent the matrix 5 in the following form 1
S12
S21
S22
By (5.15) Su =Pi, P1S12 - 1 + 512, P2S21 — S22, P2S22 = £22'+ 521 • These relations hold for 5i2 = pi, Therefore
S21 = 1, 622
1 1
=
P2-
Pi P2
By the computation rule of an inverse matrix we have
P2-P1
\-l
1
Discrete
144
systems
By formula (5.10) xk = S
if
Pi 0
0 k P 2
J_(P2
-Pi\(Pi
-V5\-l _L(-P2
^\
OWpi + 1
1 J V0 P%) \P2 + l o
PI\(P1
-\) Vo Pk2
i
-P2P\+2
+ PlPk2 + 2
Pkl+l-p\+1pkl+2-pk2+2.
p\+2-p\+2
This yields a formula for the Fibonacci numbers fk
'l + V5V fk =
k+l
1-V5'
k + l'
VE
A careful reader can easily see here a direct analogy with the integration of linear differential equations by means of reducing the system to the J o r d a n form with the subsequent returning into the initial phase space. For a linear nonhomogeneous discrete system xk — Axk-\
+
(5.16)
fk-i
we have the following representation of solutions k-l
k
Y,Ak~J~lfy
xk = A x0 +
(5.17)
3=0
T h e proof of this formula can be obtained by substitution of the right-hand side of expression (5.17) into the right-hand and left-hand sides of equation (5.16). Notice t h a t , unlike the theory of differential equations, in this case the answer to the question of the existence and uniqueness of solutions of discrete systems Xk
F{xk-Uk-1)
0,1,.
Linear discrete
systems
145
are positive. Consider now a system xk+1 = Axk + b£k, o-k=c*xk, where A is a constant n x n-matrix, 6 and c are constant n x m and n x /matrices respectively. We consider £k as an input of a linear block, ak as an output, and xk as a state vector of block at discrete times k = 0,1,... Definition 5.3. System (5.18) is said to be controllable if for any pair of vectors y 6 M n , z E IR" there exist a natural number N < n and a vector sequence £k such that a solution xk of system (5.18) with the sequence £k and the initial data XQ = y satisfies a relation xjy = z. Theorem 5.2. System (5.18) is controllable if and only if the pair (A, b) is controllable. P r o o f . Let us recall that the controllability of the pair (A,b) is equivalent to the following relation v&nk(b,Ab,...,A"-1b)=n.
(5.19)
Put N = n and apply formula (5.17). Then we obtain n-l
xn = Anx0 + J2Ak~l~J^j-
(5.20)
Now we rewrite this equation in the following way 6£„_i + Abtn-2 + ... + An-lbZ0 = z - Any.
(5.21)
Equation (5.21) can be given in the form (b,Ab,...,An-1b)\
:
Ui
\=z-Any.
(5.22)
The solvability of this linear equation with respect to the unknown matrix
\ follows from condition (5.19).
&
Discrete
146
systems
T h u s , if the pair (A,b) is controllable, then from (5.22) one can find a sequence of inputs £o> • • v ^ n - i i which take the state vector x from the state XQ = y to the state xn = z. Suppose t h a t (^4, b) is not controllable. Then by Theorem 3.2 of Chapter 3 there exists a nonsingular matrix S such t h a t
Let us perform a change of variables xk — Syk in equations (5.18): yk+1
= S~1ASyk
+ S~1b£k,
(5.23)
Equations for yk can be given in the following way „(1) _
Vk
A
J1)
- Anyk
j . 4
,/2)
+ A12yk
_L
A
+bil;k,
„( 2 ) _
yk
4
„( 2 )
- A22yk
,
where
(yilh Vk = \
(2)
We see t h a t by the input £k it is impossible to transfer the vector yk ' from an arbitrary point y^ = z[ ' to t h a t yN — z\ . T h u s , the fact t h a t the pair (A, b) is not controllable results in t h a t system (5.18) is not controllable too. As for the continuous system, system (5.18) is said to be observable if a sequence ak determines uniquely a sequence xk. Here the relation between ak and xk are given by a0 — c*x0, <Ti - c*Ax0 + c*b£0,
3=0
By the above equations the initial d a t a of the block are uniquely determined if and only if
rank
Linear discrete
systems
147
Thus we establish the following Theorem 5.3. The observability of system (5.18) is equivalent to the observability of a pair (A, c). For discrete systems we have the theorems similar to those on linear stabilization. Consider one of them. Let £fc = s*xk. Then by (5.18)
xk+1 = (A + bs*)xk. We recall that under the assumption that m = 1 (i.e., b is an n-vector) and (A, b) is controllable the following theorem was obtained, which together with the theorem on stability of a linear homogeneous discrete system allows us to answer the question of stabilization, using the linear feedback of the form £ = s*x. Theorem 5.4. For any polynomial
there exists a vector s £ Mn such that det{pl - A - bs*) = V>(p) For discrete systems a ^-transformation is often used. This transformation is similar to the Laplace transformation for continuous systems. Consider a sequence fk, which increases as k —> oo not faster than dk, where d is any number. We define a Z-transformation of sequence fk into a function of complex variable z in the following way: 00
Z{(fk)} = F(z) = Y,z-khfc=0
Here z lies outside a circle {z\ \z\ > d}. Note that in the various fields of mathematics there are often used the generating functions of the sequences G{z) = £ r = o **/*• We have G(z) = F(z~l). We see that fk may be a vector sequence of any dimension. We apply Z-transformation to both sides of equations (5.18) under the assumption ZQ = 0: oo
oo
k
oo
]T z~ xk+1 = AJ2 ~ k + bJ2z~kZk, k=0
z kx
k=0
k=0
Discrete
148
systems
oo
oo
k=0
k=0
These equations can be rewritten as oo
oo
k
Y,z~ °k = -c*{A- zI)-HY,z-kZk.
(5.24)
fc=0 fe=0
As in the case of continuous system, we call the function W(z) = c* (A — zl)~xb a transfer function of system (5.18). A transfer function relates Z-transforms of the input and output of a linear discrete block like that a similar function W(z) relates the Laplace transforms of the input and output for continuous system. 5.3
The discrete phase locked loops for array processors
In modern computers the problems of synchronization the work of processors arise. Since in the array processors the clock skew may be significant [25], it may lead to an incorrect work of parallel algorithms. The problem of a clock skew in high-speed systems is so much important that a modern VLSI are often supplied by several phase locked loops, placed on one chip [45]. In this case for creating a distributed system of generators [9] the phase locked loops can be used. We consider the simplest discrete phase locked loop without filters. The main principles of an operation of a continuous phase locked loop was already considered and the equations, describing this work, was obtained in (§4.2). Here the analogue of equation (4.38) is an equation
ek+1-ek
+ aF{ek) = r,
(5.25)
where a is some number. We consider further the sinusoidal characteristics of a phase detector, F{6) = sin 61. The purpose of control is an elimination of a clock skew for almost all initial data 6Q\ lim 6k = 2j7r. k —*-+oo
(5.26)
The discrete phase locked loops for array processors
149
From relations (5.26) and (5.25) it follows t h a t T = 0. We arrive a t an equation 9k+1-9k+asm9k=0.
(5.27)
W i t h o u t loss of generality it can be assumed that a > 0. T h e following solutions 9 = 2jn,
(5.28)
9 = (2j + 1)TT
(5.29)
are equilibria of this equation. Now we consider the linearized form of equation (5.27) in a neighborhood of these equilibria 9k+1 ~9k + a{9k - 2j7r) = 0,
(5.30)
9k+1 - 9k - a{9k - (2j + l)n) = 0.
(5.31)
From Theorem 5.1 it follows t h a t solutions (5.28) of equation (5.30) are asymptotically stable if a < 2.
(5.32)
Solutions (5.29) of equations (5.31) are unstable. These properties of solutions (5.28) and (5.29) are also valid for system (5.27) under the condition a £ (0,2). We denote b y a i a root of the equation V a 2 — 1 = 7r + arccos —. a We show now t h a t any solution of equation (5.27) with initial d a t a $o 7^ (2j + 1)TT under condition (5.32) tends to one of stable equilibria. P r o p o s i t i o n 5 . 1 . If a < a i and #o S [—IT,IT], then 9k(k) £ [—TT, IT] for all k = 1 , 2 , . . . . P r o o f . T h e function g (9) = 9—a sin 9 satisfies the following inequalities g(9)>6>-Tr,
V0e[-7r,O],
g(9) < m a x g(0) = \/a2 - 1 - (arccos - ) < n, [-JT.0]
V
a
J
V0 £ [-7r, 0],
Discrete
150
systems
g{0)<0
g(6) > ming(6)
V0e[O,7r],
= ( arccos - j - \Ja2
- 1 > -TT,
V6» G [0, TT].
From the above inequalities and from the inclusion 6Q 6 [— TT, TT] the inclusion 6k G [—TT, TT] follows. Proposition 5.1 gives a sharp estimate of mapping the segment [—TT, TT] into itself. Really, arguing as in the proof of Proposition 5.1 we obtain t h a t for a > <*i there exists a point 6Q € [ — f) TT] such t h a t \6o — « sin #o| >
T.
Obviously, Qi > 2. Therefore the proposition is valid under condition (5.32). If 6 G ( - f, f ) , then for a G (0, 2) we have \{0-a
sin 0)'\ < 1.
(5.33)
On the set [7r/2,7r] the following inequality 6-asin6>
- |
(5.34)
is satisfied and on the set [—IT, — 7r/2] an estimate holds 6»-asin6><|.
(5.35)
From inequalities (5.34) and (5.35) it follows t h a t for any solution 6k of equation (5.27) with initial d a t a 6Q G (0, TT) either for all k the estimate 0 < 6k+i < 6k holds or for some k the inclusion /
*
TT
e (--
TT\ ¥
)
(5-36)
is valid. From the inequalities 0 < 6k+i < 6k for all k it follows directly that lim 6k = 0. Inclusion (5.36) and estimate (5.33) imply also relation (5.37). The case #0 G (—TT, 0) may be considered in the same way. T h u s the following result is proved.
(5.37)
The discrete phase locked loops for array processors
151
Proposition 5.2. For any solution of equation (5.27) with a 6 (0, 2) under the initial data 6Q ^ (2j' + 1)TT there exists an integer number N such that lim 6k = 2Nn. k-t+oo
Below the results of a computer simulation for equation (5.27) for a 6 (2, Qi) [30] are given. For a E (2, 3] there exists an asymptotically stable limit cycle, which is symmetric with respect to 6 = 0, with period 2 and with a domain of attraction, namely (-7T,0)U(0,JT).
For all initial data 0O = - 3 ; - 2 . 9 ; . . . - 0.1; 0.1; 0.2;... 2.9; 3 after 100 iterations for a = 2.2; 2 . 3 ; . . . ; 3 such limit cycles were obtained. They are given in Table 1. Table 1 a
2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3
value of cycle 0.7489866426 -0.7489866426 0.9028834041 -0.9028834041 -1.026738291 1.026738291 -1.131102585 1.131102585 -1.221496214 1.221496214 1.301256148 -1.301256148 1.372589846 -1.372589846 1.437050573 -1.437050573 -1.495781568 1.495781568
Thus, for a = 2 the first bifurcation occurs. The globally asymptotic stability of a stationary set vanishes and a unique on [—ir, 7r] globally asymptotically stable cycle with period 2 is generated. The amplitude of this cycle increases with a. In some neighborhood of the parameter a = 3.1 the second bifurcation occurs: the cycle loses its stability and two locally stable nonsymmetrical cycles of period 2 occur. In the computer simulation for a — 3.1 the instability occurs in the following way.
Discrete
152
systems
Here, as in the previous cases, after 100 iterations for the initial values #o = —3; —2.9;...; —0.1; 0 . 1 ; . . . 3 we observe the transition into a certain neighborhood 0k G (-1.5496 - e, -1.5496 + e), 0k+i 6 (1.5496-e, 1.5496+ e), or 6k e (1.5496-e, 1.5496+ e), 0k+i e (-1.5496 - e, -1.5496 + e) where k > 100, 0 < e < 0.0002. For example, for 90 = 2.5 we have the following sequence #101 #102 #103 #104 #105 #106
= = = = = =
-1.549501645 1.549795512 -1.549520909 1.549775220 -1.549537730 1.549761807.
Here the irregularity in the last significant digit is typical and it is a result of the instability of cycle. For a = 3.2 there exists a transition (depending on initial data #o = — 3 ; . . . ; —0.1; 0 . 1 ; . . . ; 3) to one of nonsymmetrical locally stable cycles of period 2. The values of these cycles are the following -1.379442769;
1.762149884
-1.762149884;
1.379442769.
and
We obtain the similar result for a = 3.3. Here the values of locally stable cycles are the following -1.259697452;
1.881895201
-1.881895201;
1.259697452
and
The discrete phase locked loops for array processors
153
For a = 3.4 the instability occurs. Here we can observe the following values: -1.178465;
1.963186142
and -1.963186142;
1.178465.
For a — 3.5 we have locally stable cycles with period 4: -0.9994532; -1.3135694;
1.9446608; 2.0712776
and 0.9994532; 1.3135694;
-1.9446608; -2.0712776.
For a — 3.52 we see the transient processes, which yields the locally stable cycle 6k of period 8: 1.003657629 1.284421617 0.960000216 1.379695669
-1.965256723 -2.092223694 -1.923554541 -2.076225446
or a cycle to be symmetric to it: 6k = —6\. '. For a — 3.7 we see chaotic dynamics without periodic or quasiperiodic trajectories. Here we observe a local instability. Thus, there exists the transition to chaos via a sequence of period doubling bifurcations [42]. Unlike the classic period doubling bifurcations in logistic mappings [42] the interesting property of the system considered is the fact that before the period doubling bifurcation, from one globally asymptotically stable cycle of period two there occur two locally stable cycles of period two. Various aspects, concerning the development of the theory of discrete dynamical systems and their applications, can be found in [6, 21, 22, 24, 43, 44, 46].
Chapter 6
The Aizerman conjecture. The Popov method
In 1949 M.A. Aizerman, the notorious specialist in the control theory, has stated a conjecture [1], which stimulated the development of new mathematical methods in studying nonlinear differential equations. At present these methods go far beyond the control theory and are used in solving many applied problems. Let us consider a system — = Ax + b
(6.1)
where A is a constant n x n-matrix, b and c are constant n-dimensional vectors,
(6.2)
a = cx
and suppose that for a < fi <
ft
(6.3)
all the solutions of systems (6.2) tend to zero as t —> +oo. The question arises whether the zero solution of system (6.1) is globally 155
156
The Aizerman
conjecture.
The Popov method
asymptotically stable if the following condition a <
(6.4)
is satisfied for all a ^ 0. We say t h a t the zero solution of system (6.1) is globally asymptotically stable (or system (6.1) is globally stable) if this zero solution is Lyapunov stable and any solution of system (6.1) tends to zero as t —> + o o . M.A. Aizerman conjectured t h a t the answer to this question is always positive. In the papers of N.N. Krasovskii, V.A. Pliss, E. Noldus, and G.A. Leonov the classes of nonlinear systems were given for which the Aizerm a n conjecture is not true. T h a t is, inequality (6.4) is satisfied, all the linear systems (6.2) with p, satisfying (6.3), are stable but system (6.1) has the solutions t h a t do not tend to zero as t —> + o o . These results are considered in [28,29]. Now we demonstrate a method to obtain the criteria of global stability. T h e method was suggested by V.M. Popov [37] in 1958. T h e investigations of V.M. Popov were stimulated in part due to the Aizerman problem. Consider system (6.1) under the additional assumption t h a t A is a stable matrix, i.e., all its eigenvalues have negative real parts. Now we change slightly condition (6.4). Suppose, the following inequality holds Q <
Vo-^0,
(6.5)
where k is any number. As in Chapter 2, consider a transfer function of the linear part of system (6.1) W{P)=c*{A-pI)-1b,
peC.
T h e P o p o v t h e o r e m . Suppose that there exists a number 9 such that for all real u inequalities hold i + R e [ ( l + 0tw)W(«w)] > 0 , lim W-++0O
Then system
n - + Re\(l
i + 6iw)W(iu)]
k
(6.1) is globally asymptotically
stable.
(6-6) > 0.
157
Before proving the theorem, let us recall the definition of the Fourier transformation and some of its properties. In Chapter 2 two different definitions of the Fourier transformation were cited. For an absolutely integrable and piecewise continuous on (—oo, +00) (and for F on (0, +00')) function f(t) they are given by the following way +00
T{f(t))
= -±= j
+00
e^f(t)
dt,
F(f(t))
= J e-iutf(t)
<
For the Fourier transformation the following inversion formula is well known +00
/(t) =
+00
( /'« M T _ t ) /M^)^,
i / —00
(6.7)
—00
which is proved in the second volume of the text-book [14]. Here the exterior integral is understood in the sense of the principal value, i.e., as the following limit M
lim
/
(...)du.
M1—zoo J -M
By (6.7) for the absolutely integrable product f(t)g(t) of the absolutely integrable and piecewise continuous functions f(t) and g(t), the following relations hold +00
+00
j f(t)g(t)dt=
J
+00
—-[J
+00
( / — OO
-foo
e<w(T-{)/(r)dr)dW)dt
—CO
+00
+00
Hence we have +00
j f(t)g(t) dt=
+00
j
T{f{t))
F(g(t)) dw.
(6.8)
The Aizerman
158
conjecture.
The Popov
method
Relation (6.8) demonstrates a very important property of the Fourier transformation, namely the transformation is a unitary operator in the space of square summable functions i2(—oo,+00). In other words, if for a set of square summable functions / and g a scalar product +00
(/,
+00
J f(t)g(t) dt=±j 0
F(f(t))FW)J do:
(6.9)
-00
is satisfied. We also recall that in Chapter 2 the following properties F(/(t))=iWF(/(t))-/(0)>
(6.10)
t
F^Jf(t-T)g(r)dT^=F(f(t))
F(g(t))
(6.11)
0
were proved (Propositions 2.1 and 2.2). Now we can proceed to the proof of the Popov theorem. Note that if inequalities (6.6) are valid, then there exists a number e > 0 such that l - £ + Re[{l + 6ioj)W{ioj)] > 0,
VueR1.
(6.12)
We notice also that, having performed the change of nonlinearity ip(a) — kcr — ip(a) and using in place of system (6.1) the following equations x = {A + bc*k)x-brl>(
er = c*x,
(6.13)
it is possible to reduce condition (6.6) to a formally more restrictive condition (6.6) with 6 > 0. Really, in this case system (6.13) have a transfer function
u(p)-
~w{p)
1 + kW{p)
159
and ip(cr) satisfied condition (6.5): Q
Vo-^0.
Inequality (6.6) for system (6.13) takes the form i+Re[(l+0xw)£/(*w)] = (l+kW{iw)-kdiuW(iu)-k2eiu\W{kj) Re k\l+kW(iu>)\2 1 7
+ Re((l-0»w)W(tw)) \l +
>0.
kW(iu)\2
Thus, without loss of generality we can assume that 6 > 0. L e m m a 6.1. If inequality (6.12) is satisfied, then for any number T > 0 the inequality T
(6.14) i
+8
is valid. P r o o f . Consider functions
0, t
T)
dr.
Here T is a certain positive number, a{t) = c*eAtx0, Sometimes the function V?T(0
ls
7(t) = c*eAt6.
called a cut-off function of
The Aizerman
160
conjecture.
The Popov method
Obviously, the following relations hold T
/
+sv(
)
+ OO
= j
frit) [ r(t)\w{t)-\
0
(6
+00
= / <pr{t) ji{t-T)ipT(T)dr+ 0
f£-Ij^T(i)+
0
t
+00
+OjiJl{t-T)
dt+ f
0
0
By (6.9)-(6.11) we obtain t
+ OO
J frit)
l{t-r)
+6
! ( / * - >
+
dt =
+ OO
F(7(i))%T(f))+(£--)^T(t))+
+eiuF{i{t))F{<pr(t))
du =
+ OO
1 2^
-{l+8iu)W(iu)+
[e-r
\F(9T(t))\2d".
161
We see that if inequality (6.12) is satisfied, then the quantity t
+00
/
l(t-T)
+6
(/?(*-7>T(r)dT
al
is negative. Hence relations (6.15) yields inequality (6.14). Lemma 6.2. If inequality (6.12) is valid for 9 > 0, then there exists a number q such that +00
J tp(
(6.16)
P r o o f . From condition (6.5) and 9 > 0 by Lemma 6.1 it follows that T
a(T) 2
e I
T
[ ip{cr) da + f
By using the inequality uv < - « 2 + — v2,
Vw, Vv
and the fact that
I ifi(a) da > 0,
V
(the last inequality results from condition (6.5)), we obtain the following
The Aizerman
162
conjecture.
The Popov method
estimate T
a(0) 2
f
0
< fc|c|2|z0|" + Y
+ Oc*AeAtx0f'dt
<
0
[(c*eAtxo o
+
0c*AeAtxoydt.
From the above and stability of the matrix A the statement of Lemma 6.2 follows. Consider now a linear system x£Rn,
x = Ax + f{t),
(6.17)
where A is a constant stable matrix, f(t) is a continuous vector function. Lemma 6.3. Suppose, for some number a the following inequality holds + oo
f \f(t)\2dt
(6.18)
o Then there exists a number /3 such that the inequality \x(t)\2<(3(\x(0)\2+a)
(6.19)
is satisfied. In addition, the following relations hold +co
/ \x(t)\2dt < +oo, o
lirr^ a;(*) = 0.
(6.20)
P r o o f . Recall (see Chapter 1) that there exists a positively definite matrix H such that
A*H +
HA=-I.
Therefore for the function W(t) = x(t)*Hx(t) mate W(x(t))'
= -\x{t)\2
+ 2x*Hf{t)
we have the following esti-
< (6.21)
< - ^ ( t ) ! 2 + \2H\ \x(t)\ | / ( 0 | < - ^ Ht)\2 + 2|ff| 2 |/(<)| 2 -
163
Hence < x(0)*Hx(Q) + 2a\H\2.
x{t)*Hx(t)
(6.22)
From the last inequality and the positive definiteness of matrix H the existence of a number /3 follows such that inequality (6.19) is satisfied. Integrating both sides of inequality (6.21) from 0 to t, we obtain t
W(x{t)) - W{x(0)) + ]- f \x{r)\2dT < 2a\H\2. o By (6.22) we have + oo
I |a;(<)|2^
(6.23)
o Now we consider an integral t
jx(Tyx(r)dr=1-\x(t)\2-l-\x(0)\2.
(6.24)
o It is obvious that the following relations hold t
t
t
j |a:(r)i(r)|dT< f \x{r)\2dr+
t
f \x{r)\2dr =
t 2
f \x(r)\ dT+
t
J\AX(T)
+ f(r)\2dt\
<
t 2
\x(r)\+\f(r)\)2d?j
< (j\x(T)\ dT+J(\A\
t 2
t 2
< ((1 + 2|A| ) J \x(r)\ dT +
2J\f(r)\2dt
<
164
The Aizerman
conjecture.
The Popov method
By (6.18) and (6.23) the integral
/,(,)•*(„* t
o converges absolutely. It means that there exists a limit t lim / X(T)*X(T) v t-++oo J ' ' 0
dr.
Then by (6.24) there exists a limit lim
\x{t)\.
t—f+OO
From (6.23) it follows that lim |a;(t)| = 0. The proof of Lemma 6.3 is completed. Lemmas 6.2 and 6.3 imply that system (6.1) is globally asymptotically stable. In conclusion we note that in the case 6 < 0 it is necessary to make use of the change of variables, mentioned above, and to pass then to the case 0>O. A geometric interpretation of the Popov theorem. We consider a hodograph of the modified frequency response X(u>) = Re W(iw), Y(u) = wlmW(Ju) (Fig. 6.1). If through the point X — — 1/fc, Y = 0 it is possible to draw the line X — 6Y = —1/k such that the total hodograph {X(u),Y(u>)} is placed to the right of this straight line, then assumption (6.6) of the Popov theorem is satisfied. It is interesting to compare this result with the Nyquist criterion, which is a necessary and sufficient condition of stability of linear systems. For linear systems (6.1) with ) to be not intersect the "forbidden ray" {X, Y | X e (-oo, -l/k),Y = 0}.
165
The Popov frequency condition is more rigorous. A "forbidden zone" is a half-plane to the left of the line X — QY = — l/k, where 6 is a running parameter. In this case for two-dimensional systems it is possible to show that if a maximal stability sector, obtained by means of the Nyquist criterion, takes the form [0, ko), then for any k < ko there exists a parameter 6 such that inequalities (6.6) are satisfied. y.,
Fig. 6.1 Now we shall show the way in which the Popov theorem can be applied to the study of the Aizerman conjecture. Consider the two-dimensional systems (1) in the case that the linear stability sector (6.3) is finite. This implies that for fi = a or /i = fi the matrix A + fibc* has eigenvalues on the imaginary axis. In this case, at least, one of the matrices A + abc* and A + fibc* is nondegenerate. Show that the Aizerman conjecture has a positive solution for the following class of nonlinearities a + e<
W ^ 0.
Here e is any positive number. Without loss of generality we may consider the case that 0 <
VCT^O
and W(p) =
PV + v p2 + 8p + 7 '
(6.25)
166
The Aizerman
conjecture.
The Popov method
where 7 and 8 are positive numbers, 8 is small, p > 0. In the sequel, we assume that p > 0. Suppose also that in (6.6) we have PI In this case we obtain - + 0(8) < Re [(1 + 0iu)W(iu)] <- + 0{8). 7 7 Thus, if v > 0, then for small J inequalities (6.6) are satisfied for all K > 0. If 1/ = 0, then inequalities (6.6) are satisfied for K = K(8). In this case we have K(8) —• +00 as 8 —} 0. If v < 0, then inequalities (6.6) holds for K < -j/v
+ 0{8).
We can easily see that for v < 0 and 8 = 0 a sector 0 < // < - 7 / ^ is a maximal sector of linear stability (6.3). The Popov theorem implies that the Aizerman conjecture is valid in the class of functions (6.25), where a = 0,
j3 = —f/v.
Globally stable systems (6.1) with nonlinearities, satisfying condition (6.5), are often called absolutely stable. An additional information on the theory of absolute stability and various applications of the Popov method an interested reader can find in the books [17, 28, 29, 37].
Bibliography
1] M.A. Aiserman, (1949) On one problem concerning global stability of dynamical systems [in russian]// Uspehi matematicheskih nauk, Vol.4 No. 4, pp. 187-188. \2] A.A. Andronov, A.A. Witt, and S.E. Khaikin, (1965) Theory of Oscillation. Pergamon Press. 3] V.I.Arnold, (1991) Gewohnliche Differentialgleichungen. Deutscher Verlag der Wissenschaften. 4] V.I. Arnold, (1978) Mathematical methods of the classical mechanics. Springer. o
5] K.J. Astrom, (1999) Automatic control — the hidden technology// Adv. Control Highlights of ECC'99. Springer. o
6] K.J. Astrom, B. Wittenmark, (1984) Computer controlled systems. Theory and design. Prentice-Hall. 7] E.A. Barabashin and V.A. Tabueva, (1969) Dynamical systems with cylindrical phase space [in russian], nauka. 8] M.M. Botvinnik, (1950) Excitation regulators and static stability of synchronous machine [in russian], Gosenergoizdat. [9] T.H.Cormen, C.E.Leiserson and R.L.Rivest, (1990) Introduction to algoritms. MIT Press. [10] A.J. Dirksen, (1979) Microcomputers. Kluwer. [11] J.J. D'Azzo , C.H. Houpis, (1995) Linear control systems. Analysis and design, Mc Graw-Hill. [12] M. Driels, (1996) Linear control systems engineering, Mc Graw-Hill. [13] A.F. Filippov, (1985) Differential equations with discontinuous right hand side [in russian], Nauka. [14] G.M. Fichtenholz, (1964) Calculus. Vol. I, II [in russian], Nauka. [15] A.L.Fradkov, A.Yu.Pogromskiy, (1999) Introduction to control of oscillations and chaos. World Scientific. [16] F.R. Gantmacher, (1959) Theory of matrices. Chelsea. [17] A.Kh. Gelig, G.A. Leonov and V.A. Jakubovich, (1978) Stability of nonlinear 167
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Index
digitizer, 140 discrete group, 124 domains of attraction, 114
actuator, 95 Aizerman conjecture, 165 asymptotically stable, 7, 15 asynchronous motion, 109 attractor, 137
equation of pendulum, 15 Euclidean norm, 43 excitation winding, 107
bifurcation, 116 block, 22 Brockett problem, 72
factor-group, 124 Fibonacci numbers, 142 formula of Hermite-Mikhailov, 9 Fourier transformation, 48 frequency response, 35, 48 friction force, 6 functional space, 3 fundamental matrix, 74
canonical form, 74 Cantor set, 135 Cauchy formula, 38 Cayley identity, 55 centrifugal force, 2 characteristic polynomial, 7 condition of Vyshnegradsky, 12 controllability, 53, 54 convolution operator, 39 cut-off function, 159 cylindrical phase space, 124
generator of Cantor set, 133 globally asymptotic stability, 151 Hausdorff dimension, 136 Hermite—Mikhailov criterion, 7 Hermitian conjugation, 37 heteroclinic trajectories, 16, 115 high-frequency oscillation, 76 hodograph of frequency response, 49 hodograph of polynomial, 8 homoclinic trajectories, 109 Hooke's law, 6
damper bar, 110 damping force, 110 damping winding, 110 dead zone, 96 degree of instability, 70 difTeomorphism, 125 difference kernel, 39 differential inclusion, 98 171
172
Index
initial data, 3 input, 3 invariant linear manifold, 73
Popov theorem, 156 positive definite matrix, 54 processor, 140
Jacobi matrix, 23 Kalman duality theorem, 64
ftC-circuit, 33 _RLC-circuit, 34 rotating coordinate system, 107
Laplace transformation, 43 Laurent series, 58 law of mean values, 130 Lebesgue measure, 135 limit cycle, 151 limit load problem, 117 limiting point, 97 line of discontinuities, 97 linear approximations, 7 linear block, 123 Lotka-Volterra equations, 126 low-frequency stabilization, 73 low-pass filter, 33 Lyapunov equation, 13 Lyapunov functions, 17 Lyapunov stable, 14
Schur lemma, 68 separatrices, 109 servomechanism, 4 sliding regime, 99 sliding solution, 98 spacecraft orientation system, 96 stabilization, 67 stable linear manifold, 73 stable matrix, 22 stable polynomial, 7 steering gear, 121 Stodola theorem, 8 strange attractor, 136 sync-pulse signal, 121 synchronous machine, 109 synchronous machine load, 108
magnetic vector , 109 moment of force, 2 moment of inertia, 2 negative feedback, 2 Nyquist criterion, 67 Nyquist plot, 49 observability, 62 Ohm law, 33 open loop, 70 operating regime, 31 output, 3 period doubling bifurcation, 153 phase detector, 122 phase locked loops, 120 phase portrait, 15 plane metric, 124
transfer function, 35 transient process, 25 two-positional autopilot, 93 uniformly asymptotically stable, 15 uniformly Lyapunov stable, 15 unitary operator, 158 vertically oscillating pendulum, 73 voltage, 33 voltage controlled generator, 121, 123 Z-transformation, 147
