Sensitivity analysis of the differential matrix Riccati equation based on the associated linear differential system

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1 Advances in Computational Mathematics 7 (1997) Sensitivity analysis of the differential matrix Riccati equation based on the associated linear differential system Mihail Konstantinov a and Vera Angelova b a University of Architecture and Civil Engineering, 1 Hr. Smirnenski Blv., 1421 Sofia, Bulgaria mmk fte@bgace5.uacg.acad.bg b Institute of Information Technologies, Akad. G. Bontchev Str., Bl.2, 1113 Sofia, Bulgaria vera@bgcict.acad.bg Received February 1995; revised December 1996 Communicated by S. Hammarling Perturbation bounds are given for the solution of the nth order differential matrix Riccati equation using the associated linear 2nth order differential system. The new bounds are alternative to those existing in the literature and are sharper in some cases. Keywords: differential matrix Riccati equation, matrix exponential, sensitivity analysis, linear quadratic optimization, optimal filtering AMS subject classification: 93B35, 93B4 1. Introduction The solution of the differential matrix Riccati equation (DMRE) arising in the theory of linear quadratic optimization and optimal filtering is usually contaminated with rounding and parameter errors. This may lead to significant loss of accuracy and, in particular, to breaking of symmetry and divergence of the numerical procedure, carried out in floating-point computing environment. In turn, the actual error in the computed solution depends on the sensitivity of the solution of DMRE to perturbations in its matrix coefficients. Hence obtaining perturbation bounds for DMRE is important from both the theoretical and computational point of view. The sensitivity of the solution to DMRE is studied in several papers, see, e.g., [3] and [4]. However, these estimates may be pessimistic in some cases. In [6] a new approach is presented to estimate the sensitivity of the solution of DMRE. It is based on the sensitivity analysis of the associated linear differential system and exploits the existing sensitivity estimates [1,2,5,8,9] for the matrix exponential. This paper was partially supported by the National Scientific Research Foundation under Grants TH 44/94 and I 612/96. J.C. Baltzer AG, Science Publishers

2 296 M. Konstantinov, V. Angelova / Sensitivity analysis of differential matrix Riccati equation In this paper we propose an improved sensitivity estimate for DMRE in 2-norm, based on the approach from [6]. An experimental analysis is made to compare the effectiveness of the different approaches. It is shown that in some cases the new bound is superior relative to the known ones [3,4], i.e., all these bounds are complementary. In what follows we denote by R m n the space of real m n matrices, R + = [, ), is the 2- (or spectral) norm in R m n, A T R n m is the transpose of the matrix A R m n, I n is the unit n n matrix, and the symbol := stands for equal by definition. 2. Statement of the problem Consider the DMRE P (t) = P (t)sp(t) + A T P(t) + P(t)A + Q, t R +, P () = F, where A, S, Q, F, P (t) R n n and the matrices S, Q, F, P (t) are symmetric and nonnegative definite. The linear 2n-dimensional differential equation associated to (1) is Ψ (t) = MΨ(t), Ψ() = I 2n ; M := [ ] A S Q A T := (1) [ ] M11 M 12. (2) M 21 M 22 The solution of (2) is [ ] Ψ(t) = e Mt Ψ11 (t) Ψ := 12 (t). (3) Ψ 21 (t) Ψ 22 (t) If the inverse of the matrix J (t) :=Ψ 11 (t) + Ψ 12 (t)f exists, then the solution of the initial value problem (1) is represented as P (t) = ( Ψ 21 (t) + Ψ 22 (t)f )( Ψ 11 (t) + Ψ 12 (t)f ) 1 := H(t)J 1 (t). (4) Let the round-off and parameter errors accompanying the numerical computations be represented by equivalent perturbations in the data: A A = A + A, S S = S + S, Q Q = Q + Q, F F = F + F, where Z (Z = A, S, Q, F ) is a matrix perturbation in Z with Z Z. Usually only a bound Z on the norm of the perturbation Z is known. When Z is the equivalent perturbation caused by the implementation of a numerically stable method for the solution of (1), then Z eps ϕ(n) Z, where eps is the round-off unit of the machine arithmetic and ϕ(n) is a low order polynomial in n. Denote by P = P + P the solution of the perturbed equation (1) in which Z is replaced by Z (Z = A, S, Q, F ). Then knowing the sensitivity of Ψ(t)

3 M. Konstantinov, V. Angelova / Sensitivity analysis of differential matrix Riccati equation 297 to perturbations in A, S, Q, F we may estimate the sensitivity of P (t) which is measured by the quantity P (t). The main problems to be solved in this paper are: (i) Find an interval T = [, t ) such that for each t T the matrix J (t) + J (t) :=Ψ 11 (t) + Ψ 11 (t) + ( Ψ 12 (t) + Ψ 12 (t) ) (F + F ) is invertible and a representation of type (4) holds for the perturbed solution where P (t) + P (t) = ( H(t) + H(t) )( J (t) + J (t) ) 1, (5) H(t):= Ψ 21 (t) + Ψ 22 (t) F + Ψ 22 (t)f + Ψ 22 (t) F, J (t):= Ψ 11 (t) + Ψ 12 (t) F + Ψ 12 (t)f + Ψ 12 (t) F. (ii) Find a domain Ω R 4 +, Ω, and an estimate P (t) f(t, ), := [ A, S, Q, F ] T, (6) where the function f : T Ω R + is real analytic, non-decreasing in each component of and f(t,)=. 3. Main results From (3) we have that Ψ 11 () = I n, Ψ 12 () = and(ψ 11 ()+Ψ 12 ()F ) 1 = I n. Denote by σ 1 (t) σ 2 (t) σ n (t) the singular values of the matrix Ψ 11 (t) and let Ψ(t) be the perturbation in Ψ(t), which results from the data perturbations when solving the perturbed system (2), i.e., Ψ(t) = e (M+ M)t e Mt. Then the interval T = [, t ) may be chosen from t := sup { t: Ψ11 (t) + ( Ψ12 (t) + Ψ12 (t) )( F + F ) σn (t) }. (7) Using (5) we get P (t) = ( H(t) + H(t) )( J (t) + J (t) ) 1 H(t)J 1 (t) = ( H(t) P(t) J(t) ) (J(t)+ J(t) ) 1. (8) It is known [7] that for any matrices X, Y R n n such that X 1 exists and X 1 Y <1 the quantity (I n + X 1 Y ) 1 may be estimated as ( I n + X 1 Y ) X 1 Y.

4 298 M. Konstantinov, V. Angelova / Sensitivity analysis of differential matrix Riccati equation Hence (8) yields the following bound for the norm of P (t) ( ) P (t) H(t) + P (t) J (t) J 1 (t) 1 J 1 (t) J (t) provided J 1 (t) J (t) < 1. (9) To obtain bounds for J (t) and H(t) we use the fact that for M = [M ij ] R 2n 2n it is fulfilled M ij f M f ( M f := tr(m T M) is the Frobenius norm of M) and M M f r M, r := rank(m).hencewehave J(t) Ψ 11 (t) f + Ψ 12 (t) F + ( F + F ) Ψ 12 (t) f ( 1 + F + F ) Ψ(t) f + Ψ12 (t) F r ( 1 + F + F ) Ψ(t) + Ψ12 (t) F := ω 1 (t, ). (1) Similarly, H(t) r ( 1 + F + F ) Ψ(t) + Ψ22 (t) F := ω 2 (t, ). (11) Since J () = I n and ω 1 (t,)= the quantity τ := sup { t T : ω 1 (t, ) J 1 (t) < 1 } is correctly defined. Hence, as a corollary of (9) (11) we can formulate the following perturbation result: Theorem 1. For t [, τ) the spectral norm of the perturbation P (t) satisfies the estimate P (t) (ω 2 (t, )+ω 1 (t, ) P(t) ) J 1 (t) f(t, ) := 1 ω 1 (t, ) J 1. (12) (t) To represent the norm Ψ(t) of the perturbation Ψ(t) involved in the expressions for ω i by the known quantities A, S, Q, F consider the perturbed system (2) Ψ (t) + Ψ (t) = (M + M) ( Ψ(t) + Ψ(t) ), [ A M := Q ] S A T. We have Ψ(t) = t e M(t s) M e (M+ M)s ds

5 M. Konstantinov, V. Angelova / Sensitivity analysis of differential matrix Riccati equation 299 Table 1 Power series Log norm Jordan (1) Jordan (2) Schur c 1 1 cond(y ) cond(y ) 1 β M µ(m) α(m) α(m)+d m α(m) γ 1 ν p m l and t Ψ(t) M e M(t s) e (M+ M)s ds t M g(t s) ( ) Ψ(s) + g(s) ds, (13) where M := M and g(t) is an upper bound for e Mt, i.e., e Mt g(t). There are several bounds g for the norm of the matrix exponential based on the power series for e Mt, the logarithmic norm of M, the Jordan form and the Schur decompositions of M, see [8]: p 1 g(t) = c e βt (γt) k /k!, k= where the values of the constants c, β, γ and p are given in table 1. Here µ(m) is the maximum eigenvalue of the matrix (M + M H )/2, J = Y 1 MY is the Jordan canonical form of M and m 1 is the dimension of the maximum block in J (the matrix Y is chosen so that the condition number cond(y ) = Y Y 1 is minimized), α(m) is the spectral abscissa of M, i.e., the maximum real part of the eigenvalues of M, d m := cos(π/(m + 1)), and T = U H MU = Λ + N is the Schur decomposition of M, where U is unitary, Λ is diagonal and N is strictly upper triangular matrix (the matrix U is chosen so that the norm ν = N of the matrix N is minimized), and l = min{s: N s = } is the index of nilpotency of N. For each estimate g we have Ψ(t) u(t), where u is the solution to the majorant integral equation corresponding to (13): t u(t) = M g(t s) ( g(s) + u(s) ) ds. 4. Numerical example Consider the third order matrix differential Riccati equation of type (1) with matrices A = VA V, F = VF V, S = VS V, Q = VQ V, where V is the

6 3 M. Konstantinov, V. Angelova / Sensitivity analysis of differential matrix Riccati equation Table 2 t δ P δ P1 δ P2 δ P Table 3 k δ P1 δ P2 δ P3 of same order as δ P of same order as δ P of same order as δ P 1 exceeds δ P by 1 order of same order as δ P of same order as δ P 2 exceeds δ P by 3 orders exceeds δ P by 1 order exceeds δ P by 1 order 3 exceeds δ P by 5 orders exceeds δ P by 2 orders exceeds δ P by 2 orders elementary reflection V = I 3 2vv T /3, v := [1,1,1] T and the matrices A, F, S, Q are chosen as A = diag ( 1 1 k, 2, 3 1 k), F = diag(,,), S =diag ( 1 k,1,1 k), Q =diag ( 3 1 k,5,7 1 k) for some positive integer k. The sensitivity of this equation increases with increasing k. The perturbations in the data are taken as: A = V A V, F = V F V, S = V S V, Q = V Q V,where A =diag(3 1 k,2,1 k )1 j,forj=1, 8,...,2, and F =, S =, Q =. The exact relative perturbation P (t) δ P := P (t) in the solution of DMRE is compared with the estimate f(t, ) δ P3 := P (t) following from the above given estimate (12), the estimate δ P1 following from the linear estimate K L of Kenney and Hewer [3] and the estimate δ P2 resulting from the non-linear bound proposed by Konstantinov and Pelova [4]. In table 2 the results

7 M. Konstantinov, V. Angelova / Sensitivity analysis of differential matrix Riccati equation 31 for k = 1, j = 6 are presented and in table 3 the results for k =, 1, 2, 3 and j = 1, 8, 6, 4, 2 are generalized. The results show that with the deterioration of the conditioning of the system (i.e., of the matrix A) the estimate δ P3 from (12) is superior to the other two methods with respect to the closeness to the estimated quantity and size of the domain of validity. Moreover the estimate proposed in this paper has the advantage that it is not related with the solution of the DMRE and hence with the problem of possible loss of symmetry and divergence of the numerical procedure. References [1] G. Golub and C. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, MD, 1983). [2] B. Kågström, Bounds and perturbation bounds for the matrix exponential, BIT 17 (1977) [3] C. Kenney and G. Hewer, The sensitivity of the algebraic and differential Riccati equation, SIAM J. Control Optim. 28 (199) [4] M. Konstantinov and G. Pelova, Sensitivity of the solution to differential matrix Riccati equations, IEEE Trans. Autom. Control 36 (1991) [5] M. Konstantinov, P. Petkov, D. Gu and I. Postlethwaite, Perturbation techniques for linear control problems, Report 95-7, Department of Engineering, Leicester University (February 1995). [6] M. Konstantinov, I. Popchev and V. Angelova, A new approach to the sensitivity analysis of differential matrix Riccati equation, in: National Conference Automatica 94, Sofia, Bulgaria (1994). [7] P. Lancaster, Theory of Matrices (Academic Press, New York, 1969). [8] P. Petkov, N. Christov and M. Konstantinov, Computational Methods for Linear Control Systems (Prentice-Hall, Hemel Hempstead, 1991). [9] C. Van Loan, The sensitivity of the matrix exponential, SIAM J. Numer. Anal. 14 (1977)