PERTURBATION ANALYSIS OF COUPLED MATRIX RICCATI EQUATIONS

Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain

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PERTURBATION ANALYSIS OF COUPLED MATRIX RICCATI EQUATIONS Mihail Konstantinov * Vera Angelov a**Petko Petko v*** Dawei Gu **** Vassilios Tsachouridis t * University of Architecture and Civil Enginering, 1 Hr. Smirnenski Blvd., 1421 Sofia, Bulgaria, e-mail: [email protected]. b g ** Institute of Information Technologies, Akad. G. Bonchev Str., B1. 2, 1113 Sofia, Bulgaria, e-mail: [email protected]. bg * * * Department of Automatics, Technic a1 University of Sofia, 1756 Sofia, Bulgaria, e-mail: [email protected] **** Department of Engineering, University of Leicester, L eicester LE1 7RH, England, e-mail:dag@le. ac.uk t Department of Engineering, University of Leicester, L eicester LEI 7RH, England, e-mail:[email protected]

Abstract: Local and non-local perturbation bounds for real continuous-time coupled algebraic matrix Riccati equations are derived using the technique of Lyapunov majorants and fixed point principles. Equations of this type arise in the robust analysis and design of linear control systems. Copyright © 2002 IFAC Keywords: Perturbation analysis, Riccati equations, Robust control

or the spectral (or 2-) norm in R m x n ;11 . [IF the Frobenius (or F-) norm in R m x n ;11 . 11 a replacement of either 11 . 112 or 11 . [IF; rad (A) the spectral radius of the square matrix A; det(A) the determinant of the square matrix A; lllPlll = [IlE111,. . . , llETll]' E R1; is the generalized norm of P , when P = ( E l , .. . , E T )is a collection of T matrices.

1. INTRODUCTION AND NOTA TION

~

~

In this paper we present a complete perturbation analysis of real contin uous-time coupled algebraic matrix Riccati equations (CAMRE) of the form Fi(Xl,Xz,Pi) = 0, i = 1 , 2 , where Fi are matrix quadratic functions in the unknown matrices X i , and Pi are collections of matrix coefficients.

~

~

Throughout the paper we use the following notation: R m x n the space of m x n real matrices; Rm = R m X 1R+ ; = [O,CQ); AT the transpose of the matrix A; 5 the component-wise order relation on R m x n ;vec(A) the column-wise vector representation of the matrix A; Mat(L) E R p ' J x m n the matrix representation of the linear matrix operator L : R m x n R p x q ; I , the unit n x n matrix; n,z the n2x n2 vec-permutation matrix such that vec(AT) = II,zvec(A) for all A E R n x n ; A 8 B the Kronecker product of the matrices A and B ; 11 . 112 the Euclidean norm in R" ~

~

~

~

~

~

~

~

~

307

We also denote R = R n X nand S = { A E R : A = AT} c R;S+ = { A E S : A 2 O}; Lin(L1,Lz) the space of linear operators C1 + C2, where C 1 , C2 are linear spaces. We also use the abbreviation Lin = Lin(R,R). ~

We identify the Cartesian product R m x nx R m X " , endow ed with the structure of a linear space, with , a m x nand Ram,. In an yof the spaces R m x a nR particular, the ordered pair ( A , B ) E R m x n x Rmxn and the matrix [A,B] E R m x z n are considered as iden tical objects. Finally, w e use

the same notation P for an ordered matrix rtuple ( E l , . . . ,E,) as well as for the collection

L(.) := F ~ ( xP, ) ( . )E L i n ( R 2 , R 2 ) , Li(.) := F i , x ( X ,Pi)(.)E L i n ( R 2 , R ) ,

{El,...,&}.

Lij(.) := Fi,x, ( X ,Pi)(.)E L i n ( R , R ) . Thus

2. PROBLEM STATEMENT Consider the system of real continuous-time CAMRE arising in the robust control of linear timeinvariant systems (see e.g. [I])

F2(Xl,X2,P2) := (A2+ X l B 2 ) X 2

+ X2(Az +

+ Cz

F x (X,P )( Y )= (L1 ( Y ) L2 , (Y)) = (Lll(Y1)

+ L12(Y2),LZl(Yl) + L22(Y2)).

Applying the vec operation to F x ( X ,P ) ( Y ) and using the equality (A 8 B)&z = II,z(B @ A ) we find the matrix representation of the operator L(.)

where

XzDzXz = 0, where Xi E R are the unknown matrices, Ai, Bi E R,Ci,Di E S , are given matrix coefficients and Pi := (Ai,Bi,Ci,D i ) E R4. -

Lll:=I,@G:+G:@In,

L12 := (In2 + rI,2)(In ~ 2 1 : =(1n2

We set

+

~

L22 :=I, @ Gz

=: (E1,EZ,E3,E4,E5,E6,E7,E8) E R8.

The solution pair (XI, X2) E S2 is called stabilizing if the matrices G I := A1 B l X z - D l X l and G2 := A2 X1 B2 - X2 D2 are stable.

+

Denote by Pi +6Pi the perturbed collection Pi, in which each matrix 2 E Pi is replaced by 2 6 2 and let SP = (SPl,SP2). Then the perturbed version of equation (2) is

+

Note that Fi as defined by (1) are functions from x R x R4 = R6 to R. It will be convenient to write the system of CAMRE as one matrix equation. For this purpose we denote

R

F(X

Then the system (1) may be written as

Here F is considered as a mapping RIO R2,or equivalently, as a mapping Rnxanx R8 + Rnxan. We assume that system (1) has a solution X = (X1,Xz) E S2 such that the partial Frhchet derivative F x ( X ,P)(.)of F in X at the point (X,P) is invertible.

E R:, the vector where Si := [SA,, SB,,Sc,, of absolute Frobenius norm perturbations Sz := 116211~in the data matrices 2 E P . The perturbation problem for CAMRE (1) is to find bounds

A direct calculation gives

Sx, 5

F i , x , ( X , P i ) ( Z )=G:Z+ Z G i ,

Fz.x, (X, P2)(2)= X2B7T 2T

T

Bi

+ ZB2X2,

(4)

Denote bv

(2)

T

+ 6X, P + 6 P ) = 0.

Equation (4) has a unique isolated solution Y = X + SX E S2 in the neighbourhood of X if the perturbation SP is sufficiently small.

X := ( X , , X Z ) , F := (F1,Fz).

+

+ Gz @ I,.

+

In this work we are interested only in symmetric solutions of system (1).

F 1 , x ~ ( x , p l ) (= Z )X I B I Z

( () ~ 2 x 2 8) In) ~

Let the matrices from Pi be perturbed as Ai H Ai 6Ai, etc. We assume that the perturbations SCi and 6Di are symmetric in order to ensure that the perturbed equation also has a solution in S2. Symmetric perturbations in Ci and Di arise naturally in many applications.

The generalized n o r m of the matrix 8-tuple P is the vector IlIpIII := [IIElIIF,. . . , IIE8llFIT E R?.

F ( X ,P ) = 0.

(XlBl)),

Here Lij E Rn2 n2 is the matrix representation of the operator Lij (.), i , j = 1 , 2 .

p := (Pl, p z ) = ( A1 ,B1, C1, D1, A2, B2 , Cz ,D2 )

+

2

@

fi(S),

6 E R C R:, i = 1 , 2 ,

(5)

for the perturbations 6xi := ll6Xill~.Here R is a certain set and .fi are continuous functions, nondecreasing in each of their arguments and satisfying f i ( 0 ) = 0. The inclusion S E R guarantees that the perturbed CAMRE (4) has a unique solution Y = X SX in a neighbourhood of the

(3)

+

Further on we set

308

unperturbed solution X such that the elements of 6 x 1 , 6x2 are analytic functions of the elements of the matrices 62,2 E P , provided S is in the interior of 0.

The inverse operator

M ( . ) := L(.)-’ E L i n ( R 2 , R 2 ) of the operator L = F x ( X , P ) ( . )may be rep, for resented as L P 1 ( . )= ( M 1 ( . ) , M 2 ( . ) )where,

First order local bounds

SX;

I esti(6) + O(llS112),S + 0,

z:= (21,ZZ) E R2,

i = 1 , 2 , (6)

are first derived with esti(6) = O(llSll), 6 + 0, which are then incorporated in the non-local bounds (5).

M i ( 2 )= Mil(&) Hence

where Wi(Y,SPi) := CztpiF ~ , ~ ( S Z ) + H ~ ( 6Y ,~ ~ ) . In this way

3. LOCAL PERTURBATION ANALYSIS

2

SXi = - C M i j ( W j ( S X , S P j ) ) ,

Consider first the conditioning of the CAMRE (1).

j=1

which gives

The perturbed equations may be written as 2

Fi(X

+ M i z ( & ) , M i j ( . ) E Lin.

+ SXi, Pi + SPi) = C L i j ( S X j )

2

SXi = -

j=1

C C Mij

o

Fj,z(SZ)

(10)

j=1 Z t P ,

+ C Fi,Z(SZ)+ Hi(SX,SPi) = 0 ,

CMij(Hj(SX,SPj)).

-

Z€Pi

j=1

where Fi,z(.) := F i , z ( X , P i ) ( . ) E Lin, 2 E Pi, are the Frkchet derivatives of F i ( X , P i ) in the matrix argument 2, evaluated at the point ( X ,Pi). The matrix expressions H i ( S X ,bpi) = O (II[SX,SPi]112),SX + 0, S P ~ + 0, contain second and higher order terms in S X , SPi. In fact, for Y = (Y1,Y2) E S 2 ,we have

Therefore 2

ax; I

C C Kij,zSz + O(llSll”, j=1

+ 0,

ZEP,

where the quantity Kij,z := IIMij o F ~ , z I I Lis~ ~ , the absolute condition number of the solution component X i with respect to the matrix coefficient 2 E Pj. Here 1 l . I I ~ iis~ the induced norm in the space L i n of linear operators R + R. If X i # 0, estimates in terms of relative perturbations are 2

EX,

IC

C

j=1 Z t P ,

kj,ZEZ

+ O(ll~ll”,

+ 0,

where the quantity kij,z := K i ljl Z l l,F ~ is ~the, relative condition number of the solution component X i with respect to the matrix coefficient 0 # E Pj.

z

The calculation of the condition numbers Kij,z is staightforward for the Frobenius. Denote by Li,z E R n z X nthe 2 matrix of the operator Fi,z E Lin. We have

309

be denoted as

M := M a t ( M ) = L p l :=

where

[ 2I:

. (13) esti(6) := min {~st!~)(6),est!~)(S)}

M . . E Rn2Xn2.

The local bounds are continuous, first order homogeneous, non-linear functions in 6.

23

Having in mind the expressions for Li,z the absolute condition numbers are calculated from

The bounds est!k) are majorants for the solution of a complicated optimization problem, defining the conditioning of the problem as follows. Set := vec(bXi), i = 1 , 2 and 6 := [&, . . . ,681’ := . . ,SD,]’ E R!. Then we have

Kij,z = ~ ~ M ~ j L j2, Ez Pj, ~ ~ i~, j, = 1,2. Rewrite equations (10) in vectorized form as

cc c 2

vec(6Xi) =

Ni,zvec(SZ)

8

(11)

j=1 Z€P, 2

Mijvec(Hj(dX,dPj)),

-

j=1

where

Ni,z := -MijLj,z E RnZXn2 , 2 E Pj.

k = 1,..., 8} is the exact upper bound for the first order term in the perturbation bound for the solution component Xi (note that Ki(S) is well defined, since the minimization in 77 is carried out over a compact set). The calculation of Ki(6) is a difficult task. Instead, one can use a bound above such as esti(d) 2 Ki(S).

The condition number based perturbation bounds may be derived as an immediate consequence of (11)7

Let y E R t be a given vector and let Xi # 0. Then the relative condition number of Xi with Relations (11) also give a second perturbation respect to y is ~ i ( y := ) Ki(y)/llXill~.If lllPlll is bound III) the generalized norm of P , then K ~ ( ~ ~ ~isP the dx, Iest12)(6) ~ ( l l ~ l l est12)(6) ~), := ll~~11~11611~relative norm-wise condition number of X i .

+

where 4. NON-LOCAL PERTURBATION ANALYSIS Local bounds of the type considered in Section 3 are valid only asymptoticaly, for S + 0. But in practice they are usually used simply neglecting terms of order O(llSl12).

The bounds estjl)(S) and est!2) (6) are alternative, i.e., which one is smaller depends on the particular value of 6. There is a third bound, which is always less or equal to estil)(S). Indeed, we have axi

Iest!3)(~)+ 0 ( l l ~ l l 2 ) ,

est!3)(~):= JSTG~S,

A

6 X = II(dX,dP), II = ( I I l , I I 2 ) ,

where Ni = [ni,pq] E R Y 8 is a matrtix with elements

ni,pq :=

~

~

~

&

~

p~ , q, = q ~ 1,..., ~ 28

,

where Ni,k, k = 1,.. . , 8 denote the n2 x n2 h

blocks of Ni,i.e., Ni

=

Ni,k E RnZXn2 , and @ i , ~ N. .- N . Ni,B1,.. . , 2,s .- z r D z . A

A

:= N ~ , A Ni,2 ~ , :=

h

where II(Y,SP) := -M(Fp(X, P)(SP)+H(Y,SP)).

Equation (14) comprizes two equations, namely

esti1’(6) and we have the overall estimates

dx;= esti(8) + O(llSll”,

(14)

Here H ( Y , S P ) := ( H 1 ( Y , S P I ) , H 2 ( Y , S P 2 )con) tains second and third order terms in Y and S P , see (71, (8).

. . ,Nii,~], A

[@i,~,@i,,,.

The disadvantages of the local estimates may be overcome using the techniques of non-linear perturbation analysis. As a result, we find bounds (5). The estimate (5) is rigorous. The perturbed equation F ( X + S X , P+6P) = 0 may be rewritten as an operator equation for the perturbation 6 X

where the right-hand side of (15) is defined by relations (10). Setting

(12)

310

and

+ nn2)112, Vi2 := IlMi2(X2 8 In) (In2 + nn2)112. Vil

we obtain the vector operator equation

The function h : R$ x R: R$, constructed above, is a vector Lyapunov majorant for the operator equation (16), see [2].

in RZn2,which is reduced to two coupled vector equations

li = X i ( E , V ) ,

:= IlMil(L 8 X1) (In2

Consider the majorant system of two scalar quadratic equations

i = 1,2,

pi = hi(Pl,PZ,S),i = 1,2,

in R,', respectively. The vectorizations of the matrices Hi(Y, 6Pi) are

(19)

which may also be written in vector form as p = h ( p , 6 ) , where

Hence h(0,O) = 0, h,(O,O) = 0. Therefore, for 6 sufficiently small, the system (19) has a solution

and vec(Hz(Y,6P2)) = (X, @ I,) (In2

+ rI,2)

(18)

which is continuous, real analytic in 6 # 0 and satisfies p(0) = 0. The function f ( . ) is defined in a domain R c R t whose boundary dR may be obtained by excluding p from the system of equations

x vec (Y16B2 - Y26D2)

+ ( I , ~3 X I ) (I,* + n,s) vec(6B2Y2) vec(YZ(D2 + 6D2)Y2) + vec (SA2Y2+ Y2SA;) + vec ( Y ~ ( B Z + 6 ~ 2 + Y) ~~( B+~~SB,)Y,) ~ .

-

Let IIEIIF I p i , i = 1,2, where pi are nonnegative constants. Then it follows from (17), (18) that

The second equation in (20) is equivalent to

where

and

We can find a new Lyapunov majorant 9 , such that h(p,6) 5 g ( p , 6) and for which the equation P = 9(P, 4

311

(22)

has an explicit form solution. This can be done in many ways. Three of them are described below.

d = d ( 6 ) := (1- a1 - a2 -

+

Let

-

c i ( 6 ) := rnax{cli(S), czi(6)).

Hereinafter, in order to simplify the notation, we set aij := u i j ( 6 ) , ai := ui(S),b = b ( S ) , ci := c i ( S ) , ei := esti(b), e := est(6). Consider the function g with components 91 (P, 6) = &(P, 6) =e

+ alp1 + a2p2 + 2bp1p2 + ClPl + c2p;.

Now the majorant equation (22) has solutions with p1 = p2, where

e

-

(I - a1 - m ) p l

+ (2b + c1 + c2)pl = 0.

and j

4(Ul(b

+

ej))2

aj)ei

C2)

-

+ 4(b2

b(6) := max{bl(6), b 2 ( 6 ) } ,

- U2)2 -

-

-

+ c1))el 4 ( a ~ (+ b c1) + (1 al)(b + c2))e2

+ (1

:= max{ali(S), a2i(S)),

+ c1 + c2) ( a j e j + (1

cj(e1 - e2)’)

= (1 - U l

est(6) := max{estl(d), estZ(b)},

%(a)

4(2b

+ 2(b + c j ) ( e i

-

-

clcZ)(el

-

e2)2

# i . These bounds hold

provided d ( 6 ) 2 0.

5. CONCLUSIONS In this paper we have presented a complete local and non-local perturbation analysis of coupled continuous-time matrix Riccati equations, arising in the theory of H , control. The results obtained may be extended to other more general systems of matrix quadratic equations.

Hence, if

6. REFERENCES

6 E 0 := (6 E R: : a1 +a2

+

+

2 ~ e ( 2 b c1+

c2)

[l] P. Bernstein and W. Haddad, LQG control ‘ , performance bound: A Riccati with an H equation approach, IEEE Trans. Automat.

I I}

then

However, in this approach one of the bounds (23) is not asymptotically sharp unless el = e2. We next derive another explicit bound that is asymptotically sharp in the sense that its first order term is equal to esti(6).

Control 34 (1989) 293-305. [2] M. Konstantinov, P. Petkov, D. Gu, and I. Postlethwaite, Perturbation analysis in finite dimensional spaces, Technical Report 96-1 8 (Department of Engineering, Leicester University, Leicester, UK, June 1996).

Consider the function k with components

ki(6, P) := ei + alp1 + a2p2

+ 2bplp2 + clpf + czp;.

It is easy to see that k is again a Lyapunov majorant. Since h(p,6) 5 k(p,S) 5 g(p,S) the solution of the majorant system p = k(p,6) will majorize the solution of the system p = h(p, 6) thus producing less sharp bounds, but will give tighter bounds than these based on the majorant g. However, this solution is easily computable. Indeed, here we have p1 = p2 + el - e2. Substituting this expression in any of the equations pi = ki(p,S) we obtain quadratic equations for pi. Choosing the smaller solutions, we find the bounds

(24) where

312