8
Ukrainian Mathematical Journal, Vol. 48, No. 8, 1996 LOCALIZATION OF SPECTRUM AND STABILITY OF CERTAIN CLASSES OF DYNAMICAL SYSTEMS A. G. Mazko UDC 517.512 We develop a method for the localization of spectra of multiparameter matrix pencils and matrix func- tions, which reduces the problem to the solution of linear matrix equations and inequalities. We formu- late algebraic conditions for the stability of linear systems of differential, difference, and difference-dif- ferential equations. 1. Introduction and Principal Results Algebraic methods based on the solution of linear matrix equations and inequalities are widely used in the theory of stability and stabilization of dynamical systems [ 1]. In particular, the problems of localization of spectra of linear systems are reduced to the investigation of Hermitian solutions of Lyapunov-type matrix equations [2--4]. The purpose of this paper is to develop the method of matrix equations and inequalities for the investigation of spectral problems for multiparameter matrix pencils and matrix functions, and to apply this method to problems of stability of linear dynamical systems. Let ~(z) = A o- z 1A 1 - ... - ZmAm be a multiparameter pencil of n x n matrices satisfying the regularity condition det~(z) ~ 0, z = Zl Zm C m. (1) The spectrum c(~) of a given matrix is defined as a geometric locus z such that det ~(z) = O. We pose the problem of localization of the spectrum cr(~), i.e., the problem of construction of vector sets ~ containing all points of the spectrum cy(q~). Theorem 1. Assume that Hermitian matrices X and Y satisfy the conditions m Z 7ijAiXA; = Y" i,j=O BXB* > 0, B = A11 i , Am (2) (3) Y+ ~(z)EdP(z)* > 0 (Vz~ ~), (4) where ~ij = ~t ji are scalar coefficients which form a matrix F, E > O, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhumal, Vol. 48, No. 8, pp. 1074-1079, August, 1996. Original article submitted October 24, 1996. 1214 0041-5995/96/4808-1214 $15.00 9 1997 Plenum Publishing Corporation

Localization of spectrum and stability of certain classes of dynamical systems

Embed Size (px)

Citation preview

Page 1: Localization of spectrum and stability of certain classes of dynamical systems

Ukrainian Mathematical Journal, Vol. 48, No. 8, 1996

LOCALIZATION OF SPECTRUM AND STABILITY OF CERTAIN CLASSES OF DYNAMICAL SYSTEMS

A. G. Mazko UDC 517.512

We develop a method for the localization of spectra of multiparameter matrix pencils and matrix func- tions, which reduces the problem to the solution of linear matrix equations and inequalities. We formu- late algebraic conditions for the stability of linear systems of differential, difference, and difference-dif- ferential equations.

1. Introduction and Principal Results

Algebraic methods based on the solution of linear matrix equations and inequalities are widely used in the

theory of stability and stabilization of dynamical systems [ 1 ]. In particular, the problems of localization of spectra

of linear systems are reduced to the investigation of Hermitian solutions of Lyapunov-type matrix equations [2--4].

The purpose of this paper is to develop the method of matrix equations and inequalities for the investigation of

spectral problems for multiparameter matrix pencils and matrix functions, and to apply this method to problems of

stability of linear dynamical systems.

Let ~ ( z ) = A o - z 1A 1 - ... - ZmAm be a multiparameter pencil of n x n matrices satisfying the regularity

condition

d e t ~ ( z ) ~ 0, z =

Zl

Zm

C m. (1)

The spectrum c ( ~ ) of a given matrix is defined as a geometric locus z such that det ~ ( z ) = O. We pose the

problem of localization of the spectrum cr(~), i.e., the problem of construction of vector sets ~ containing all

points of the spectrum cy(q~).

T h e o r e m 1. Assume that Hermitian matrices X and Y satisfy the conditions

m

Z 7 i jA iXA; = Y" i,j=O

BXB* > 0, B = A11 i ,

Am

(2)

(3)

Y + ~(z )EdP(z )* > 0 ( V z ~ ~ ) , (4)

where ~ij = ~t ji are scalar coefficients which form a matrix F, E > O,

Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhumal, Vol. 48, No. 8, pp. 1074-1079, August, 1996. Original article submitted October 24, 1996.

1 2 1 4 0041-5995/96/4808-1214 $15.00 �9 1997 Plenum Publishing Corporation

Page 2: Localization of spectrum and stability of certain classes of dynamical systems

LOCALIZATION OF SPECTRUM AND STABILITY OF CERTAIN CLASSES OF DYNAMICAL SYSTEMS 1215

= {z : rangA(z) + s ign~(z) > 2}, (5)

A(Z) = ZFZ*, Z = [z, Im ], and I m is the identity matrix o f order m. Then every point of the spectrum ~y(d~)

belongs to the set ~..

The vector sets ~ that localize the spectrum ~(~b) are described in terms of rank and signature of the Her-

mitian matrix A(z) and are determined only by the values of scalar coefficients in Eq. (2). The condition z ~

means that the matrix A(Z) has at least one positive eigenvalue, and the condition z ~ ~ is equivalent to the in-

equality A(z) = T00zz * + zg* + gz* + G < O, where

For example, if 7oo > 0 and 700 G < gg*, then the set ~ is located outside an m-dimensional ball ~ c f2 = {z :

II z - z0 II 2 > r }, where Zo = -Tg, r = 7e, 7 = 1/700, and e > 0 is the maximum eigenvalue of the matrix 7gg* - G.

For the investigation of the set (5), we can use the equalities [5]

r ankF = rankA(z) + rankS(z), s ignF = signA(z) + signS(z),

where S(z ) = F - FZ*A+ZF and A + is an arbitrary semiinverse matrix satisfying the equality A(z)A+A(z) = A(Z).

If the matrix F has only one positive eigenvalue, then the conditions z ~ ~ and S(z) < 0 are equivalent. Theorem 1 implies a method for the construction of domains in the complex plane that contain the spectrum of

matrix functions of the form

where z (k )

T h e o r e m 2. Suppose that Hermitian matrices

tion (4), where ~. ff A, A = { L : z (X)~ ~ } , and

function F ( L ) belongs to the domain A.

F ( ~ ) ~ ~ ( z ( ~ . ) ) = A O - Z l ( ~ . ) A I - . . . - Z m ( ~ . ) A m ,

is a given vector function with components zk(~.), k = 1, m.

X and Y satisfy relations (2)-(4) with z = z ( k ) in condi-

is a set of the form (5). Then the spectrum of the matrix

Remark 1. In order that conditions (3) ((4)) be satisfied, it is sufficient that X(Y) be a nonnegative (positive)

definite matrix.

If Y > 0, then restriction (4) is equivalent to the identity

r a n k [ ~ ( z ) , Y] = n ( V z ~ ~ ) ,

which is an analog of the conditions of controllability and stability in the Simon -Mi t t e r form for linear systems [6,

7]. Furthermore, to realize condition (4) it suffices to set

Y > CQC*, C = [ A 1 . . . . . Am].

Here and below, Q is an arbitrary positive definite matrix of order rim. In the scalar case where n = 1, in particu-

lar, for the polynomial F(X ) = a o + Xa I + ... + Xma, n, conditions (2)-(4) reduce to a single inequality a F a*> O,

Page 3: Localization of spectrum and stability of certain classes of dynamical systems

1216 A.G. MAZKO

where a = [ a 1 . . . . . a m ]

One can show that if zk(L) = - X m, k = 1, m,

XI" 1 - ~FI* + E ~ F 2 , where

then the matrix A in (5) is congruent to the matrix A = F 0 -

m m

r0 = Ilvolll, r l = [ l~ ' i - l~ [ l l , r '2 = IIv,-lj-111?

Moreover, the conditions 7~ < 0 and X ~ A are equivalent. In particular, if the inequality A < 0 is satisfied for

all X such that Re X > 0, then the domain A lies to the left of the imaginary axis.

2. Stability in the Domain of Localization of Spectra of Some Dynamical Systems

As corollaries of Theorem 2, we formulate algebraic conditions of stability of continuous and discrete dynami-

cal systems, which are most often used in applications.

In the case m = 1, the domain of localization of the spectrum of the linear pencil F ( X ) = A 0 + ~.A I is de-

scribed in Theorem 2 as

A = {X" 7ooXk-~,oxX-~,loX +7~i >0} .

As its boundary, we can take an arbitrary straight line or a circle. In particular, for matrices of coefficients of the

form

01 I~ ; l io 1 0 < r < l ,

we obtain the following algebraic conditions of stability of differential and difference systems unsolved with respect

to derivatives and iterations:

AoX(t)+ A 1 &(t) = O, (6)

AoXk+AlXk+ 1 = 0, k = 0 , 1 . . . . . (7)

Corollary 1. I f the matrix inequality

AoXA 1 + A1XA 0 + 2~A1XA 1 = Y > A1Q A ~

has a solution X such that A IXA1 >_ O, then the system of differential equations (6) is asymptotically stable

and its spectrum lies in the half plane A = {X" R e X < 7}- System (6) is asymptotically stable iff, for some Her- mitian matrices X and Y, the relations

* :tl

AoXA 1 + A I X A o = Y,

A1XA1 >_ O,

*

Y + (Ao+)~A1)(Ao+)~A 1 > 0 ('V'X: R e X > O )

Page 4: Localization of spectrum and stability of certain classes of dynamical systems

LOCALIZATION OF SPECTRUM AND STABILITY OF CERTAIN CLASSES OF DYNAMICAL SYSTEMS

are valid.

Corollary 2. I f the matrix inequality

r2A1XA ~ - A O X A 0 = Y >_ A 1 Q A 1

has a solution

its spectrum lies inside the circle matrices X and Y, the relations

are valid.

1217

X such that A 1XA1 > O, then the system of difference equations (7) is asymptotically stable and

A = { ~. : [ ~ ] < r }. System (7) is asymptotically stable iff, for some Hermitian

A I X A ~ - A o X A 0 = Y,

A 1 X A 1 > O,

Y + ( a o + ~ . a l ) ( a o + ~ . a l ) * > 0 ( V L " ]~.1>_1)

The first assertions of Corollaries 1 and 2 were established in [8] on the basis of the canonical representation of a regular matrix pencil. In the same paper, the conditions of solvability and the structure of solutions of the cor- responding matrix equations and inequalities were studied, which gave a possibility to formulate criteria of asymp-

totic stability of systems (6) and (7). In particular, unknown Hermitian matrices can be defined in the form X =

T ff T*, Y = A 1T t 'TA~, where T is a solution of the algebraic system [2]

with maximum rank.

AoTA I = A I T A o, T = T A I T

Theorem 2 can be used for the investigation of conditions of stability for dynamical systems more complicated than systems (6) and (7). As an example, consider the following classes of systems:

Aox( t ) + A 1 5c(t) + A 2 x ( t - " Q = O,

Aox( t ) + A 1Jc(t) + A 2 2 ( t ) = 0,

Aox k + AlXk+ 1 + A2Xk+ 2 = O.

(8)

(9)

(10)

The spectrum of system (8) with constant delay z > 0 is formed by the eigenvalues of the matrix function F(7~) =

A o + XA 1 + e-X~A2, and the spectra of systems (9) and (10) consist of the eigenvatues of the quadratic pencil

F (L) =Ao + ~.A 1 +~.2A 2. Let us introduce the matrices

[A1] , C = [AI, A2].

B = A2

Corollary 3. I f the matrix inequality

Page 5: Localization of spectrum and stability of certain classes of dynamical systems

1218 A.G. MAZKO

AoXA 1 + A I X A ; - czA1XA I - ~A2XA 2 = Y > CQC*,

where (z > 1 /6 > O, has a solution X such that BXB*>O, the system of the difference-differential equa- tions (8) is asymptotically stable for any constant values of the parameter "c > O. Moreover, its spectrum is lo-

cated in the domain A = { X: Re X< ~}, where ~ = ~('c) < 0 is the unique root of the transcendental equation

213~ + ~x13 = e-2*~.

Analogous stability conditions were established for system (8) in the case A 1 = In by using the Lyapunov-

Krasovskii functional [9-11].

For the quadratic pencil F().), we set

Z = [Z(~.), /2] = _X2 0 ~. 1 0 --TV "

In this case, domains of localization of the spectrum or(F) in Theorem 2 can be described as follows:

A = A o U A 1, A 0 = { ~ , : t r s A1 = { ) v : d e t A = m * r ' m < 0 },

where A = 2 F Z * , co* = [ 1, ~, ~2], f- is the adjoint matrix formed of the cofactors of elements of the matrix

F. For example, for the matrix

_! 0 1" = --82 8

5 --1

( 0 < 8 < 1 , - 1 < e < 1 - 2 8 2 ) ,

A 0 = 0 and A 1 is a domain located to the left of the imaginary axis. Moreover, in the case 0 = 1 - 2 ~5 2, the do-

main A 1 degenerates into an open left half plane.

Corollary 4. If the matrix inequality

- A o X A 0 - A 2 X A 2 - 5 A 1 X A I + e(AoXA2 + A 2 X A O)

+ 5(AoXA ~ + A I X A 0 + A I X A 2 + A2XAI ) = Y >_ CQC*

has a solution X such that BXB * > O, the system of differential equations of the second order (9) is asymptoti-

cally stable and its spectrum is located in the domain

A = { s r l 2 ( c - ~ ) > ~(1 + d ~ + ~ 2 ) } ,

where

1 - e 1 - e ~ , = ~ + i ~ ] , c = 8 2 8 ' d 8 + 25

Corollary 5. If the matrix inequality

Page 6: Localization of spectrum and stability of certain classes of dynamical systems

LOCALIZATION OF SPECTRUM AND STABILITY OF CERTAIN CLASSES OF DYNAMICAL SYSTEMS 1219

- A o X A o - A I X A 1 + 2 9 A z X A 2 = Y >_ CQC*,

where 0 < 9 < 1/4, has a solution X such that BXB*>O, the system of difference equations (10) is asymp-

totically stable and its spectrum is located inside the circle

Corollary 6. I f the matrix inequality

* * - 1 A 1 X A 1 - 4 a 2 A 2 X A * 2 = Y > CQC*, - A ~ - A 2 X A ~ 2

where e > O, has a solution X such that B X B *> O, then the spectrum of the quadratic pencil F (~.) = A 0 +

LA 1 + ~'2A2 is located in the domain A = { ),." IRe ),.1 > ~ }.

For E = 0, this corollary expresses the dichotomy conditions for system (9), i.e., system (9) has no imaginary

eigenvalues. By using the transformation X = ig, we can formulate conditions under which all eigenvalues of sys-

tem (9) are complex. In conclusion, note that the consequences of Theorems 1 and 2 formulated above are useful for applications

and, moreover, new assertions of this sort can be obtained. Note that the Hermitian matrices 1" corresponding to

certain given properties of the set ~ or domain A are not uniquely defined. To construct new stability conditions

for systems (9) and (10), one can use Corollaries 4 and 5 and the bilinear transformation of the spectral parameter

)~-- ( g + 1 ) / ( g - 1). Consider a generalization of system (8) of the form

A o x ( t ) + A l k ( t ) + A 2 x ( t - ' C l ) + ... + ArnX(t-'Cm_i) = O.

Assume that the conditions

F = 0 g* G = 1 , g* = [ 1 , 0 . . . . . 0], g G ' H

m-1

h* = [h 1 . . . . . hm_l], 711Y> ~ ( l + l h j l ) 2, n<0, j= l

where y < 0 is the maximum eigenvalue of the btock H, are satisfied. One can show that the domain A localizing

the spectrum of the matrix quasipolynomial

F(L) = A o + LA 1 + e-X~tA2 + ... + e-~'~m-lArn

in Theorem 2 is located to the left of the imaginary axis.. Moreover, relations (2)-(4) guarantee the asymptotic sta-

bility of the given system for any constant values of the delay parameters z 1 > 0 . . . . . "urn_ I > 0.

Page 7: Localization of spectrum and stability of certain classes of dynamical systems

1220 A.G. MAZKO

3. Proofs of Theorems 1 and 2

Let us represent the matrix equation (2) in the form

A(F| Y, A = [A 0 . . . . . Am], (11)

where | denotes the Kronecker multiplication. Let v* ~: 0 be the left eigenvector of the matrix ~b (z) correspond-

ing to a point z a G ( ~ ) of the spectrum. Then the relation

v*A = v*[Ao, C ] = v * C ( [ z , I m ] | In) = v * C [ Z | In]

is true. Moreover, v * C r 0. Otherwise, we obtain the inequality rank A < n, which contradicts condition (1).

Let z ~ ~ . Multiplying (11) from the left (right) by v* (v) and taking (3), (4), and the properties of the Kro- necker product into account, we get

v * C ( A ( z ) | = t r ( A ( z ) W r) = v * Y v > O,

where

W = VBXB*V* > O, V =

!" ... O ]

" ' , i .

, . . 2)*

For Y > O, the inequalities v * Y v > 0 and v * Y~: 0 are equivalent. By using the decomposition of the nonnega-

tive definite matrix w T = RR* > 0 and rearranging factors under the trace sign, we arrive at the inequality

tr(R* (z)R) > O.

In view of the law of inertia, this inequality implies that the matrix A (z) cannot be negative semidefinite, i.e.,

z ~ ~ . This contradicts the assumption that z ~ ~ . Hence, G ( ~ ) c ~. In particular, in Theorem 2, the inclusion

G(F) c A holds. Theorems 1 and 2 are proved.

This work was financially supported by the Ukrainian State Committee on Science and Technology.

REFERENCES

1. S. Boyd, L. El Chaoui, E. Feron, and V. Balakrishman, "Linear matrix inequalities in system and control theory," Stud. AppL Math.,

15, 193 (1994). 2. A.G. Mazko, "Construction of analogs of the Lyapunov equation for a matrix polynomial," Ukr. Mat. Zh., 47, No. 3, 337-343

(1995). 3. A.G. Mazko, Generalized Lyapunov Equation and Its Application to Problems of Stability and Localization of Spectra [in Russian],

Author's Abstract of the Doctoral Degree Thesis (Physics and Mathematics), Kiev (1995). 4. S. Gutman and F. Chojnowski, "Root-clustering criteria (II); Linear matrix equations," IMA J. Math Contr. lnf., 6, 269-300 (1989). 5. A.G. Mazko, "Semiinversion and properties of matrix invariants," Ukr. Mat. Zh., 40, No. 4, 525-528 (1988). 6. T.R. Crosslej and R. Porter, "Simple proof of the Simon-Mitter controllability theorem," Electron Lett., 9, No. 3, 51-52 (1973).

Page 8: Localization of spectrum and stability of certain classes of dynamical systems

LOCALIZATION OF SPECTRUM AND STABILITY OF CERTAIN CLASSES OF DYNAMICAL SYSTEMS 1221

7. E.N. Khasina, "On control over degenerate linear dynamical systems," Avtom. Telemekh_, No. 4, 30-37 (1982).

8. A.G. Mazko, "'Distribution of roots of matrix polynomials with respect to two-dimensional curves," in: Numerical-Analytic Methods for the Investigation of Dynamics and Stability of Complicated Systems [in Russian], Institute of Mathematics, Ukrainian Academy

of Sciences, Kiev (1984), pp. 90-96. 9. D.G. Korenevskii and A. G. Mazko, "Compact form of the algebraic criterion of absolute (with respect to delay) stability of solutions

of linear difference-differential equations," Ukr. Mat. Zh., 41, No. 2, 278-282 (t989).

10. A.L. Zelentsovskii, "Stability with probability one of solutions of systems of linear stochastic difference-differential equations,"

Ukr. Mat. Zh., 43, No. 2, 147-151 (1991). 11. D. Ya. Khusainov, "On a method for the construction of Lyapunov-Krasovskii functionals for linear systems with delay." Ukr.

Mat. Zh., 41, No. 3, 382-387 (1989).