Solubility of matrix inequalities

SOLUBILITY OF MATRIX INEQUALITIES A. N. Churilov

An important part is played in the theory of control by special matrix equations [i], and also by the matrix inequalities associated with them. The solutions of these equations or inequalities allow us to construct Lyapunov functions for nonlinear systems of automatic regulation, and to solve problems of the synthesis of optimal control with quadratic quality criterion. Convenient and effectively verifiable conditions for the solubility of Lur'e's inequalities are given by the proposition known as the frequency theorem or the Yakubovich-Kalman Lemma, different variants of whose proof may be found in [2-8]. Beginning with Kalman's work [3], the formulation of the frequency theorem in the often-occurring singular case contains requirements of controllability and observability of the linear part of the system, Although for systems with one nonlinearity these requirements are quite natural, and their verification does not present any special difficulties, for systems with several nonlinearities this verification can be extremely complicated (see Sec. 1.2 of [6] and Sec. 5.6 of [9]). Therefore other mathematicians have tried to completely remove, alter or weaken the requirements of controllability and observability in the formulation of the frequency theorem in the degenerate case. The first and most important result is that of Meyer [i0, ii] (which refers to systems with stable linear part and one nonlinearity). A generalization is introduced (without proof) in [12] of a result of Meyer to the case of several nonlinearities. However an extra restriction on the coefficients of the system is introduced. Another close result connected with the removal of the observ~bility condition was obtained in [13]. In this article we show that the conditions of controllability and observability may in practice be omitted for systems with several nonlinearities and a linear part which is not necessarily stable, i.e. for systems of a substantially wider class than that considered in [10-13]. Let A, b, G, g, and F be complex matrices of dimensions n x n, n x m, n x n, n x m, m • m, respectively, and let the matrices G and F be Hermitian. In control theory we are interested in determining the conditions under which there exists a Hermitian n x n-matrix H satisfying the inequality%

I

HA + A*H Hb

We say that we are in the real case, if the matrices A, b, G, g and F are real. In practice these matrices themselves are not usually known, but instead the following matrix-function is:

n (z) = b*A~GAzb + g*A,b + b*A~g + F. Here Az = (zln -- A) -~, where In is the unit n x n-matrix. We have the following statement as a variant of the frequency theorem in the absence of controllability and observability. THEOREM. Let the matrix A have no purely imaginary eigenvalues and let G < 0. For all real numbers m let the inequality n ( g o ) ~ 0 be satisfied, where at least at one point ~o this inequality is strict. Then there exists a nonsingular Hermitian matrix H satisfying inequality (i), such that the number of positive and negative eigenvalues of the matrix H (counting multiplicities) is equal to the number of eigenvalues of the matrix A with negative and positive real parts respectively. In the real case we may choose the matrix H to be real. Remark i. If in the formulation of the Theorem we discard the condition G < 0, then it follows from the above proof that inequality (i) is soluble in the class of Hermitian %The asterisk denotes the Hermitian conjugate. Leningrad ShipbuildingInstitute. Translated from Matematicheskie Zametki, Vol. 36, No. 5, pp. 725-732, November, 1984. Original article submitted July 12, 1983.

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matrices. But in this case the statement of the matrix H does not hold.

of the Theorem on nonsingularity

Remark 2. If F > 0, then the inequality sufficiently large in absolute value.

~(im)

> 0 is satisfied

for

and the spectrum values

which

are

Before we proceed directly to the proof of the fundamental theorem, we prove two subsidiary statements. Let P, Q, and R be complex n x n-matrices and let Q and R be Hermitian. We introduce a Hermitian matrix W(H) = Q + HP + P*H + HRH depending on the Hez~itian n x nmatrix H. We consider the quadratic matrix equation

W(U) = o and the quadratic

matrix inequality

(2)

connected w i t h it:

w (g) ~< o with respect

to an unknown Hermitian

n • n-matrix

(3)

H.

LEMMA I. Let the pair of matrices (P, R) be controllable [6, p. 43], let equation (2) be soluble. Then there exists a solution H of inequality matrix P + RH has only purely imaginary eigenvalues.

let R >~ 0 and (3) such that the

Proof. It was shown in [14, 15] that for the conditions of the Lemma, there exist solutions HI and Ha of equation (2), such that the eigenvalues of the matrices P + RHx and P + RH2 lie in the closed left half-plane and the closed right half-plane respectively. Moreover [14] H2 ~ H:. Write H = (H~ + H2)/2. We see that H is the required matrix. Clearly, W (H) : --(H~ - - H1) R (H 2 --.H~)/4. Since R ~

0, then H satisfies

inequality

(3).

It is easily

seen that

(H2 - - H1) (P --~ RH) + (P Jr RH)* (H~ - - H 0 = 0. Let there exist a number we see that

X and a nonzero vector x such that X* ( H 2 - - H i ) x R e ~

=

(4)

(P + RH)x = Xx.

Then from

(4)

0.

If x*(H2 -- H1)x = 0, then since H2 ~ H ~ it follows that Hax = H1x = H x . Therefore, ( P + R H I ) x (P + RHz)x = Xx. Since the common eigenvalues of the matrices P + RH~ and P + RH2 can only be purely imaginary, Re X = 0. LEMMA 2. Suppose that the matrix P has no purely imaginary eigenvalues. Then for any Hermitian matrix Q, there exists a nonsingular Hermitian solution H of the Lyapunov inequality

HP+P*H 0. Under to the quadratic matrix inequality (3), where

Q =gF-lg*--G,

P ==A --bF-Zg *,

R

=

bF"lb *.

this condition

in-

(6)

863

=

It is known (see, for example, p. 45 of [6]) that either the pair of matrices trollable, or there exists a nonsingular transformation U such that

(P, b) is con-

where the pair of matrices (P~, b,) is controllable (in the special case when b = O, the statement of the theorem can be obtained easily using Lemma 2). We note that the idea of using such a transformation to prove the frequency theorem belongs to K. Meyer [ii]. Set R, = b t F ~ b ~*. Without loss of generality, we may assume that the matrices P and b are already of the form of the matrices on the right hand sides in equations (7). We divide the remaining matrices into blocks according to dimension:

']:hen inequality (3) is equivalent to the inequality

I

where

W~

W~]

0

W* W~] '~ '

W1 (H*) = Qt + H*Pi + P~Ht + H,R,HI. W2 (HI. H.) = Q2 + H,P~ + (P, + RtH~)*H, + H,P3. W3 (g2, H3) : Q3 + H~P2 + P~g2 + H~R1H2 + H3P~ -+- P~Y~. I t i s e a s i l y seen t h a t the f r e q u e n c y i n e q u a l i t y

H (i~)~O

implies the frequency inequality

R -- R (P* + i~L~)-lQ (p -- i~In)-* R ~ 0 for all real numbers ~ for which det (P to the inequality

--i~l~)=/=O.

This inequality,

in turn, is equivalent

R, -- R, (P~ + ~I)-IQ, (p, _ i~l)-*Rt ~ 0 for all real numbers ~ for which det (P,--i~I)=/=O. Since the pair of matrices (P,, R,) is controllable, then from the traditional formulation of the frequency theorem (see, for example, [16]) we see that there exists a solution H, of the equation

w , (H,) = O. Then by Lemma i a solution H, of the inequality W~ (H,) < 0 may be chosen such that the matrix Pt + R,H, has only purely imaginary eigenvalues. Fix such a matrix H,. Since the matrix A has no purely imaginary eigenvalues and P = A -- bF-*g *, then it follows from (7) that the matrix P3 also has no purely imaginary eigenvalues. Therefore the matrices P~ and --(P, + R,H,)* have no common eigenvalues. Then the matrix equation

H2P8 + (P* + R*HI)*H~ = --HIP2 - - Q2, considered with respect to the unknown matrix Ha, has a unique solution (see [17], p. 207). Fix this solution Ha. Since the matrix Ps has no purely imaginary eigenvalues, then by Lemma 2 there exists a matrix H3 such that

tI3P3 + P ~ H ~ < - - Q 3 - - H ~ P 2 - - P ~ H 2 - - H ~ R I H 2. Thus the matrices H,, Ha, and H3 may be chosen so that W I ~ 0

, W2 = 0, W ~ < 0 .

Moreover

P + R H = [P,-~R,H, P~+ R,H~] P~ J' where the matrix P, + R,H, has only purely imaginary eigenvalues, and the matrix P~ has no purely imaginary eigenvalues. Thus it obviously follows that the above-constructed matrix H satisfies the inequality

w (H) < o, and therefore it also satisfies inequality (l), and moreover the zero-space of the matrix W(H) does not contain any eigenvectors of the matrix P + RH, corresponding to eigenvalues with nonzero real parts. We note that in proving this fact we h a v e n o t used the assumption that G ~ 0 . 864

It follows from the definition of the matrix W(H) and the notation --N, where

N = --G + ( H b - - g )

r -~(b*H-g*)-

(6) that HA + A*H =

W(H).

It is clear from the condition of the Theorem and from the construction of the matrix H that N ~ 0. We see that the pair of matrices (A, N) is observable (see [6, p. 46]). Then it follows from Lemma 1.2.4 of [6] that the statement of the Lemma on the distribution of the eigenvalues of the matrix H is true. Suppose that there exist a nonzero vector x and a number % such that Ax = %x, Nx = O. Hence it follows that Gx = 0, W(H)x = 0, b*Hx = g*x. Since moreover (P + RH)x = Ax = %x, then in view of the above property of the zero-space of the matrix W(H), we see that Re% = 0. This contradicts our assumption that the matrix A has no purely imaginary eigenvalues. Thus by Theorem 1.2.3 of [6], the pair of matrices (A, N) is observable, and we have proved the existence of a matrix H with the~above-mentioned properties. We prove the Theorem in tNe general case.

Write

V = ( i ~ o [ , ~ - - A ) -1, where ~o is a real number the matrix

for which ~(i~o) > 0.

Multiply inequality

(I) on the right by

ImJ' and on the left by the Hermitian conjugate matrix.

We obtain an equivalent inequality

(see

[8]): t

(8)

where ~ = --V, ~ = V*GV, g = --V*g + V*GVb, ~ = ~(i~o). to an equivalent inequality of the same type,

Thus we have reduced inequality

for which r > 0.

(i)

Clearly tile inequality G d

0

implies the inequality G ~ O. Moreover, the number of eigenvalues of the matrix A lying to the left and the right of the imaginary axis is equal to the number of corresponding eigenvalues of the matrix A. From the coefficients of inequality (8) we make a matrix N(im) in the same way as the matrix H(im) was composed of the coefficients of the inequality (i). Moreover, as it was shown in [8], the inequality ~ ( i ~ ) ~ 0 implies the equality ~ ( i ~ ) ~ 0 . Thus we may apply all the arguments from the first part of the proof. Suppose that we are in the real case, and let the solution H of the inequality (I) (which in general is complex) be choseh as described above. Since the coefficients of inequality (i) are real, then the matrix H also satisfies this inequality. ~ Set Ho = (H + H)/2. The matrix Ho is real and satisfies the linear inequality (i). It follows from the above that we have a relation H A +

A*H =--N~, where the matrix N~ is defined from the coef-

ficients of inequality (8) in the same way as the matrix N was defined from the coefficients of inequality (I) for F > 0. Moreover N I ~ O and the pair of matrices (A, NI) Js observable. Since the matrix H satisfies (i), then it also satisfies (8), and therefore ~ + ~ * H = --N~, where N2 is some nonnegative definite matrix. Set No = (N~ + N2)/2. Clearly, from the inequalities N x > 0 , N 2 ~ 0 and the fact that the pair of matrices (A, NI) is observable, we see that the pair of matrices %

(A, No) is observable.

Then from the relation

%

HoA + A*Ho =--No we have the statement of the Theorem in the real case. LITERATURE CITED 1.

2. 3.

A. I. Lur'e, Some Nonlinear Problems in the Theory of Automatic Control [in Russian], Gostekhizdat, Moscow (1951). V. A. Yakubovich, "The solution of some m a t r i x inequalities occurring in the theory of automatic regulation," Dokl. Akad. Nauk SSSR, 143, No. 3, 1304-1307 (1962). R. E. Kalman, "Lyapunov functions for the problem of Lur'e in automatic control," Proc. Nat. Acad. Sci. USA, 49, No. 2, 20]-205 (1963).

#An upper bar denotes the elementwise complex conjugate. 865

4. 5. 6. 7~ 8. 9. I0. ii. 12. 13. 14. 15. 16.

17.

V. M. Popov, Hyperstability of Automatic Systems [in Russian], Nauka, Moscow (1970). V. A. Yakubovich, "The frequency theorem in control theory," Sib. Mat. Zh., 14, No. 2, 384-420 (1973). A. Kh. Gelig, G. A. Leonov, and V. A. Yakubovich, The Stability of Nonlinear Systems with Nonunique Equilibrium State [in Russian], Nauka, Moscow (1978). A. N. Churilov, "The frequency theorem and LurWe's equation," Sib. Mat. Zh., 20, No. 4, 854-868 (1979). A. N. Churilov, "The solubility of certain matrix inequalities," Vestn. Mosk. Gos. Univ., Ser. Mat., Mekh., Astron., No. 7, 51-55 (1980). A. A. Voronov, Stability, Controllability and Observability [in Russian], Nauka, Moscow (1979). K. R. Meyer, "Lyapunov functions for the problems of Lur~e, '' Proc. Nat. Acad. Sci. USA, 53, No. 3, 501-503 (1965). K. R. Meyer, "On the existence of Lyapunov functions for the problem of Lur'e," SIAM J. Control, Ser. A, ~, No. 3, 373-383 (1965). M. D, Srinath, M. A. L. Thathacher, and H. K. Ramapriyan, "Absolute stability of systems with multiple nonlinearities," Int. J. Control, ~, No. 4, 365-375 (1968). A. V. Zharkov, "The problem of the existence of a positive solution of Lur'e's equation," Vestn. Mosk. Gos. Univ., Set. Mat., Mekh., Astron., No. 13, 134-135 (1978). W. A. Coppel, "Matrix quadratic equations," Bull. Austral. Math. Soc., iO, No. 3, 377401 (1974). P. Lancaster and L. Rodman, "Existence and uniqueness theorems for the algebraic Riccati equation," Int. J. Control, 32, No. 2, 285-310 (1980). A. N. Churilov, "On solutions of a quadratic matrix equation," in: Nonlinear Oscillations and Control Theory [in Russian], Vol. 2, Udmurtsk Univ., Izhevsk (1978), pp. 2433. F. R. Gantmakher, The Theory of Matrices, Chelsea Publ.

~-SIMILARITY OF LINEAR OPERATORS IN A FINITE-DIMENSIONAL SPACE Nguen Khong Tkhai

In this note we introduce a new equivalence relation for linear operators in a finitedimensional vector space, namely, the relation of ~-slmilarity, which contains, as particular cases, the ordinary similarity [i] and Erugin's similarity relation [2]. As an application, we propose a new algebraic proof of Erugin's theorem [2] on the equivalence of linear differential systems. i, ~-Similar Operators. Let H be a finite-dimensional vector space over the field of complex numbers C; A a n d B be elements of L(H), which is the space of all linear operators in H; and G(H) be the group of all invertible operators with the identity operator I as the identity. e Definition i. Two operators A and B are said to be ~-similar ( A c ~ B ) , if

where [A, B] is the linear operator that acts in L(H) and is defined by the equality [A, B] X = A X

-- X B ,

X ~ L (H),

being the abelian group in C with respect to the usual addition of complex numbers. it is obvious that this definition turns into the definition of the ordinary similarity of linear operators for ~ = {0}. The cases where ~ coincides with the imaginary axis iR and with C will be considered in See. 4. Belorussian State University. Translated from Matematicheskie Zametki, Vol. 36, No. 5, pp. 733-741, November, 1984. Original article submitted January 24, 1984.

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0001-4346/84/3656-0866508.50

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