E L S E V I E R Systems & Control Letters 32 (1997) 235 239

$7/$T|MS fJ COMTROL L|TT|R$

On upper bounds for the unified algebraic Riccati equation

Adam Czornik*, Aleksander Nawrat Institute of Mathematics, Silesian Technical Universi~, Pl-44-100 Gliwice, ul. Kaszubska 23, Poland

Received 28 May 1996; received in revised form 31 October 1996 and 28 July 1997

Abstract

In recent years, several eigenvalues, norms and determinants bounds have been investigated separately for the solutions of continuous and discrete Riccati equations. In this paper, an upper bound for solution of the unified Riccati equation is presentec~. In the limiting cases, the result reduces to a new upper bound for the solution of continuous and discrete Riccati equation. ~'~ 1997 Published by Elsevier Science B.V.

Keywords. Unified Riccati equation; Upper bounds

1. Introduction

The cont inuous and discrete algebraic Riccati equat ion are impor tant in various areas of engineering system theory, particularly in control system theory and also in study of the stability of linear systems, see [1, 6]. Numerous papers, [3-5 , 9, 11, 13], have presented eigenvalues bounds for the solution of cont inuous and discrete Riccafi equat ion separately. A summary of results from this area is given in [9].

The objective of this paper is to present an upper bound for sums of the eigenvalues of the solution of the unified algebraic Riccati equation. Similar results for lower bounds were obtained in [10J. In the limiting cases, we shall simplify this bound for the discrete and cont inuous Riccati equation.

Let ;.i(X), i = 1 . . . . . n, be the ith eigenvalue of a matrix X E ~" x,. Everywhere in the paper only real eigenvalues appear so we can order them in the nonincreasing order

; ~ ( x ) >1 ... >1 ;~.(x).

The following results shall be used.

Theorem 1.1 (Pate] and Toda [11]). For any matrices S e ~,×r, T E ~ r x ,

(I, + S T ) - 1 = I , - S(Ir + T S ) - I T . (1)

Theorem 1.2. For any symmetr ic matrices X ~ R" x , and Y ~ R" ×"

k k k 2 , (X + Y)<~ Z 2,(X) + ~ 2,(Y), k = 1 . . . . . n

i = 1 i = 1 i = l

(2)

* Corresponding author. Tel.: + 48 32 372864; fax: + 48 32 372864; e-mail: olek@zeus.polsl.gliwice.pl.

0167-6911/97/$17.00 (¢ 1997 Published by Elsevier Science B.V. All rights reserved PII S01 67-69 1 1 (97)00071-6

2 3 6 A. Czornik, A. Nawra t ,' Sys tems & Control Letters" 32 (1997) 235 239

and if, in addition, X >1 O, Y >~ O, then

k k

2,(XY) <~ ~ 2,(X)2,(Y), k = 1,. . . , n, i = 1 i = 1

and if X > O, Y >~ O, then

k k 2i(X)

2 "~i((X ' q- y ) - l ) ~ 2 1 q- )q(X),).n_i+l(Y ) ' i = 1 i = 1

k = l , . . . , n .

Inequality (2) was presented in [2]; for Eq. (3) Eq. (4), see [4].

Theorem 1.3 (Marshall and Olkin [7]). For k = 1 . . . . . n, let

k k

Y x, Zy/ i = 1 i = 1

for real numbers xi, yi arranged in nonincreasing order. Then

k k

i = 1 i = l

where real numbers ui arranged in nonincreasing order, are nonnegattve.

(3)

(4)

(5)

(6)

2. Main result

Consider the unified-type algebraic Riccati equation

-- Q = A 'P + PA + A A ' P A - (I,, + AA)'PB(Ir + AB'PB) 1B'P(I, + AA) (7)

P I P = (I, + A A ) ' ~ ( , + AA) ~ (I, + AA)'PB(Ir + AB'PB) 1B'P(I, + AA), (8)

where A e ~, A ~> 0, A, P, Q ~ N . . . . , and Be N" ×r, with P, Q, BB' > 0. If(A, B) is stabilizable and (C, A), where Q = C'C, has no unobservable mode inside the stability boundary, then Eq. (7) has a unique positive definite solution, see [8].

Remark 1. From the unified Riccati equation, Eq. (7), we obtain in limiting cases continuous and discrete Riccati equations, see [8]. In continuous case (A = 0) and from Eq. (7) we obtain the continuous-type algebraic Riccati equation. In discrete case and for A = 1, we obtain from Eq. (8) the discrete-type algebraic Riccati equation by substituting A by A - I,.

The following theorem contains the main result of this note.

Theorem 2.1. inequality:

k

2,(P) <~

The eigenvalues of the positive definite matrix solution P of Eq. (7), satisfies the following

i = 1

N + ,~/N 2 + 4M[1 + A2~(A + A' + AAA')] E~= 1 /~i(~_~)

fi)r k = 1 . . . . . n, where

k

M = y ~ i = 1

2[1 + A2~(A + A' + AAA')]

1 + A2~(A + A' + AAA' )

(9)

Z . _ i + I ( B B )

A. Czornik, A. N a w r a t / Sys tems & Control Let ters 32 (1997) 235 239 2 3 7

and

k

N = M2~(A + A' + AAA') + A[1 + A21(A + A' + AAA')] ~ 2i(Q). i = 1

Proof. Using the matrix identity (1), Eq. (7) can be t ransformed to

P = (I, + AA)'(P -~ + ABB')-I(I, + AA) + AQ. (10)

Use of Eq. (3) gives

k

;¢i((I. + AA)' (P -1 + ABB') -1 (I, + AA)) i = 1

k

<<, ~ 2~((I, + AA)(I, + AA)')2~((P 1 + ABB,)-~) ( l l ) i - 1

k 2,((I, + AA)(I. + AA)')2,(P) <~ ~ 1 + A2,(P))c,_i+I(BB' ) ' (12)

i = 1

where to obtain Eq. (12), Eq. (4) was identified with Eq. (5) and then Eq. (6) was used. The application of Eq. (2) in Eq. (10), using Eq. (12) gives

k k ~ ] t i ( ( i . + AA)(In + AA),)2i(p) 1 Z ;.,(P) <~ Z L ~ - + - A ~ , z . ~ + A;,,(Q)j. (13)

i = l i = 1

For the convex function f : [0, o~ ) ~ E , f (x ) = - x/(1 + x) and real numbers xie [0, 3c ), ~i > 0, i = 1 . . . . . k, we have

where S = ~2~= 1 :q, see [-12]. Application of Eq. (14) with xi = A2i(P)2.-i+ I(BB') and

2i((I. + A 4)(I. + AA)') 1 + A2~(A + A' + AAA') 3( i = =

A2.-i+I(BB') A2. i+I(BB') '

produces

k V2/((/" + AA)(In + AA)')2i(,)] = i [i + A2i(A + A ' + AAA') A_2i(P))., i+l(BB') 1

L T + Z~oY,-+ ~ j ,=, AAn_i+ I(BB' ) 1 + A2i(P)2, i+ I(BB' U i = l

M/A ~ = , (1 + A}.,(A + A' + AAA'))2i(P) <~ k t ~ • M/A + ~i=1 (1 + A2i(A + A' + AAA ))/.i(P)

Since the function q : [0, ~c ) --+ R, g(x) = (M/A)x/((M/A) + x) is increasing and

k k

(1 + A}.i(A + A' + AAA'))2~(P) <~ (1 + A;tjA + A' + AAA')) y' 2i(P), i = 1 i = 1

we can bound Eq. (13) as

k M/A (1 + A21(A + A' + AAA'))y~= 1 2~(P) k

, = 1 M/z~ +(1 + A)q(A + A' + AAA'))E~=~ 2,(P) + ,=1 ~ A2,(Q).

Solving Eq. (17), we get Eq. (9). [5]

The following two corollaries follow from Eq. (9) and Remark 1.

(14)

(15)

(16)

(17)

238 A. Czornik, A. Nawrat / Systems & Control Letters 32 (1997) 235 -239

Corollary 2.1. The eigenvalues o f the positive definite matrix solution P of continuous Riccati equation

- Q = A'P + PA - PBB'P, (18)

satisfies the following inequality:

4 k k )q(A + A') + ~/(21(A + A')) 2 + (E,=l (1/2. ~+I(BB'))) -~ y~=, 2i(Q) £~(p) ~< (19)

i = 1 2 (E~=I (1/2.-i+ ,(BB'))) -1 '

J b r k = 1 . . . . . n.

The following bound was obtained in [5]:

k ZI(A + A') + ~/().~(A + A')) z + 4,L,(BB') (Z~=l £,(Q)/k) (20) Z,(P) -%< 2Z.(BB')/k

i = 1

for cont inuous Riccati equation. Our bound, Eq. (19), is stronger, than Eq. (20), because

k 1 k 2 ,_ I(BB') <~ " (21) i= 1 i+ ;~,(BB')

Corollary 2.2. The eigenvalues o f the positive-definite matrix solution P of discrete Riccati equation

P = A ' P A - A'PB(I~ + B ' P B ) - 1 B ' P A + Q,

satisfies the following inequality

(22)

k R + x ~ 2 + 4).,(AA') E~=, ()q(AA')/Z,_,+ I(BB')) Y~=, ~i(Q) (23) Z £~(P) ~< 22,(AA')

i = l

for k = 1 . . . . , n, where

L ~. t k = 2i(AA') 1) + 21(AA') ~, 2,(Q).

,= 1 2,_~+ ~(BB') (2a(AA') - i = 1

Our bound, Eq. (23) is stronger than

4 " , k k a + ~/a 2 + kz , (BB)• i=l 2i(Q) )"(P) <~ 2£ , (BB ' )

i = 1

where a = k ( ) q ( A A ' ) - 1 + Z I (Q)) . , (BB ' ) ) given in [4]. This follows directly from Eq. (21) and

)ti(AA' ) <~ )q(AA'), i = 1 . . . . . n.

(24)

{25)

The results of Eqs. (19) and (22) specialize for k = 1 to bounds for the extreme eigenvalues of P and for k = n to bounds for the trace of P.

Corollary 2.3. The extreme eigenvalue 21(P) and the trace tr(P) of the positive-definite matrix solution P of continuous Riccati equation, Eq. (18), satisfy the following inequalities:

21(A + A') + ~/()q(A + A')) 2 + 4£,(BB')A1(Q) 2~(P) -%< , (26)

22,(BB')

4 " tr(P) ~< £1(A + A') + ~/(21(A + A')) 2 + (Yi=l (1/2i(BB'))) ~ tr(Q) (27) 2 (3~7- ~ (1/£,(BB')))- '

A. Czornik, A. Nawrat / Systems & Control Letters 32 (1997) 235 239 239

Corollary 2.4. The extreme eigenvalue 21(P) and the trace tr(P) of the positive-definite matrix solution P of discrete Riccati equation, Eq. (22) satisfies the following inequality:

) q ( A A ' t - 1 + 21(Q) + x/-()q(AA') - 1 + )q(Q)2 + 4),,(BB'))q(Q) 2~(P) <<. 22.(BB') , (28)

]~ 4- N/]~ 2 -}- 4)q(AA') ~ : 1 ()~i(AA')/~in-i+ I(BB')) tr(O) tr(P) ~< , (29)

2)q(AA')

where

2i(AA, ) = , . . . . (21(AA') - 1) + 21(AA') tr(Q).

i = 1 ~n-i+ l[ l~lyJ ]

B o u n d s g iven by Eqs. (26) and (28) were p r e s e n t e d in [4, 9], b o u n d s of Eqs. (27) and (29) are new.

3. Conclusions

U p p e r b o u n d for the sums of the e igenva lues of the so lu t ion of un i f i ed- type Ricca t i e q u a t i o n is p r e sen t ed in

this note . S o m e new b o u n d s for the d iscre te and c o n t i n u o u s Ricca t i e q u a t i o n are o b t a i n e d us ing this uni f ied

a p p r o a c h .

References

[1] S. Barnett, C. Storey, Matrix Methods in Stability Theory, Barness and Nobel, New York, 1970. [2] K. Fan, On a theo:em of Weyl concerning eigenvalues of linear transformations I, Proc. Nat. Acad. Sci. USA 35 (1949) 652 655. [3] J. Garloff, Bounds For the eigenvalues of the solution of the discrete Riccati and Lyapunov equation and the continuous Lyapunov

equation, Internat. J. Control 37 (1986) 423431. [4] N. Komaroff, Upper bounds for the solution of the discrete Riccati equation, IEEE Trans. Automat. Control 37 (1992) 1370 1372. [5] N. Komaroff, Diverse bounds for the eigenvalues of the continuous algebraic Riccati equation, IEEE Trans. Automat. Control 39

(1994) 532 534. [6] P. Lancaster, L. Rodman, Algebraic Riccati Equation, Oxford Univ. Press, Oxford, 1995. [7] A.W. Marshal, I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979. [8] R.H. Middleton, G.C. Goodwin, Digital Control and Estimation A Unified Approach, Prentice-Hall, Englewood Cliffs, N J, 1990. [9] T. Mori, A. Derese, A brief summary of the bounds on the solution of the algebraic matrix equations in control theory, Internat. J.

Control 39 (1984) 247 256. [10] M. Mrabti, M. Benseddik, Bounds for the eigenvalues of the solution of the unified algebraic Riccati matrix equation, Systems

Control Lett. 24 (! 995) 345-349. [1 l~ R.V. Patel, M. Toda, On norm bounds for algebraic Riccati and Lyapunov equations, IEEE Trans. Automat. Control 23 (1978)

87 88. [12] L. Schwartz, Analyse Math6matique, Hermann, Paris, 1967. [ 13] S. Wang, Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equations, IEEE Trans. Automat. Control 31

(1986) 654-656.