Perturbation analysis of the stochastic algebraic Riccati equation
© Chiang et al.; licensee Springer. 2013
Received: 16 September 2013
Accepted: 14 November 2013
Published: 11 December 2013
In this paper we study a general class of stochastic algebraic Riccati equations (SARE) arising from the indefinite linear quadratic control and stochastic problems. Using the Brouwer fixed point theorem, we provide sufficient conditions for the existence of a stabilizing solution of the perturbed SARE. We obtain a theoretical perturbation bound for measuring accurately the relative error in the exact solution of the SARE. Moreover, we slightly modify the condition theory developed by Rice and provide explicit expressions of the condition number with respect to the stabilizing solution of the SARE. A numerical example is applied to illustrate the sharpness of the perturbation bound and its correspondence with the condition number.
MSC: Primary 15A24; 65F35; secondary 47H10, 47H14.
where , , , , , respectively. Moreover, and are symmetric matrices. Here we denote (respectively, ) if M is symmetric positive definite (respectively, positive semidefinite). The unknown is a symmetric solution to SARE (1a)-(1b). Let be the set of all symmetric real matrices. For any , we write if .
with and , as special cases.
Matrix equations of the type (1a)-(1b) are encountered in the indefinite linear quadratic (LQ) control problem , and the disturbance attenuation problem, which is in deterministic case the control theory, for linear stochastic systems with both state- and input-dependent white noise. For example, see [2–4]. For simplicity, we only consider one-dimensional Wiener process of white noise in this paper; it is straightforward but tedious to extend all perturbation results presented in this paper for multi-dimensional cases. In the aforementioned applications of linear stochastic systems, a symmetric solution X, called a stabilizing solution, to SARE (1a)-(1b) ought to be determined for the design of optimal controllers. This stabilizing solution plays a very important role in many applications of linear system control theory. The definition of a stabilizing solution to SARE (1a)-(1b) is given as follows. (See also [, Definition 5.2].)
is contained in the open left half plane, i.e., .
Note that if in (1a)-(1b), then it is easily seen from Definition 1.1 that the matrix is a stabilizing solution to SARE (1a)-(1b) or, equivalently, CARE (2) if and only if . Therefore, Definition 1.1 is a natural generalization of the definition of a stabilizing solution to CARE (2) in classical linear control theory. Moreover, a necessary and sufficient condition for the existence of the stabilizing solution to a more general SARE is derived in Theorem 7.2 of . See also [, Theorem 10]. In this case, it is also shown that if SARE (1a)-(1b) has a stabilizing solution , then it is necessarily a maximal solution and thus unique [1, 3].
The standard CARE (2) and DARE (3) are widely studied and play very important roles in both classical LQ and control problems for deterministic linear systems [5–7]. In the past four decades, an extensive amount of numerical methods were studied and developed for solving the CARE and DARE (see [8–10] and the references therein). There are two major methodologies among these numerical methods or algorithms. One is the so-called Schur method or invariant subspace method, which was first proposed by Laub . According to this methodology, the unique and non-negative definite stabilizing solution of the CARE (or DARE) can be obtained by computing the stable invariant subspace (or deflating subspace) of the associated Hamiltonian matrix (or symplectic matrix pencil). Some variants of the invariant subspace method, which preserve the structure of the Hamiltonian matrix (or symplectic matrix pencil) by special orthogonal transformations in the whole computational process, are considered by Mehrmann and his coauthors [12–18]. The other methodology comes from the iterative method, for example, it is referred to as Newton’s method , matrix sign function method , disk function method , and structured doubling algorithms [21, 22] and references therein. So far there has been no sources in applying the invariant subspace methods for solving SARE (1a)-(1b), since the structures of associated Hamiltonian matrix or symplectic matrix pencil are not available. Only the iterative methods, e.g., Newton’s method  and the interior-point algorithm presented in , can be applied to computing the numerical solutions of SARE (1a)-(1b). Recently, normwise residual bounds were proposed for assessing the accuracy of a computed solution to SARE (1a)-(1b) .
where , , , , , and are perturbed coefficient matrices of compatible sizes. The main question is under what conditions perturbed SARE (5a)-(5b) still has a stabilizing solution . Moreover, how sensitive is the stabilizing solution of original SARE (1a)-(1b) with respect to small changes in the coefficient matrices? This is related to the conditioning of SARE (1a)-(1b). Therefore, we will try to answer these questions for SARE (1a)-(1b) in this paper. For CARE (2) and DARE (3), the normwise non-local and local perturbation bounds have been widely studied in the literature. See, e.g., [24–26]. Also, computable residual bounds were derived for measuring the accuracy of a computed solution to CARE (2) and DARE (3), respectively [27, 28]. To our best knowledge, these issues have not been taken into account for constrained SARE (1a)-(1b) in the literature.
where the matrix has 1 as its entry and 0’s elsewhere.
This paper is organized as follows. In Section 2, a perturbation equation is derived from SAREs (1a)-(1b) and (5a)-(5b) without dropping any higher-order terms. By using Brouwer fixed point theorem, we obtain a perturbation bound for the stabilizing solution of SARE (5a)-(5b) in Section 3. In order to guarantee the existence of the stabilizing solution of perturbed SARE (5a)-(5b), some stability analysis of the operator is established in Section 4. A theoretical formula of the normwise condition number of the stabilizing solution to SARE (1a)-(1b) is derived in Section 5. Finally, in Section 6, a numerical example is given to illustrate the sharpness and tightness of our perturbation bounds, and Section 7 concludes the paper.
2 Perturbation equation
that is, .
where , the matrices ΔA, ΔB, and so on are given by (9)-(12).
Thus far, we have not specified the condition for the existence of the solution ΔX in (22). In the subsequent discussion, we shall limit our attention to identifying the condition of the existence of a fixed point of (23), that is, to determine an upper bound on the size of ΔX.
3 Perturbation bounds
We now move into more specific details pertaining to the discussion of the fixed point of the continuous mapping f. Before doing so, we need to describe an important property of the norm of the product of two matrices and repeatedly employ it in the following discussion. For the proof, the reader is referred to [, Theorem 3.9].
Lemma 3.1 Let A and B be two matrices in . Then and .
It then follows from the Brouwer fixed-point theorem (see ) that the continuous mapping f has a fixed point , that is, condition (22) automatically holds.
This implies that assumption (36) is true, if assumption (42a)-(42b) is true.
4 Stability analysis
We then have the following important result addressing the condition for a linear operator to be stable. To see a few necessary and sufficient conditions on the stability, we refer to the results and proofs given in .
Theorem 4.1 The linear operator given by (44) is stable, i.e., , if and only if and for all .
Here, () denote the eigenvalues of .
The connection between and the maximum of the scalar function on can be established in the following form.
Theorem 4.2 
then (47) implies that the matrix must be c-stable.
We now turn to a key stability test of the operator , the striking tool of our stability analysis.
Theorem 4.3 
then is c-stable and the perturbed linear operator defined by (45) is also stable, i.e., .
holds, then corresponding to Theorem 4.3, the perturbed linear operator with respect to is stable. In other words, the matrix must be the unique stabilizing (and maximal) solution to perturbed SARE (5a)-(5b).
We now have all the materials needed for the existence of a stabilizing solution of (5a)-(5b).
Theorem 4.4 (Perturbation bound)
5 Condition number of the SARE
In order to derive the explicit expression for the condition number of the stabilizing solution X of (1a)-(1b), we require a theorem concerning the form of the optimal solution. This theorem can be regarded as a theoretical extension of the results discussed in [25, 33]. Most strategies have been established earlier by using much heavier machinery. Since this theorem is most relevant to our stability analysis, we briefly outline a direct proof with ideas from  to make this presentation more self-contained.
Then there exists an optimal solution to problem (60) such that is symmetric.
It follows that if , by (62), we see that is another optimal solution for (60). This proves the first part of the theorem.
If , then , which implies that is a symmetric optimal solution to (60) (see [, Lemma A.1]). This completes the proof. □
Based on the above discussion, we have the following result.
6 Numerical experiment
Let the perturbed coefficient matrices ΔA, ΔB, ΔC, ΔD, ΔS, ΔR and ΔH be generated using the MATLAB command randn with the weighted coefficient . That is, the matrices ΔA, ΔB, ΔC, ΔD, ΔS, ΔR and ΔH are generated in forms of , respectively. Since ΔR and ΔH are required to be symmetric, we need to fine-tune the perturbed matrices ΔR and ΔH by redefining ΔR and ΔH as and , respectively. Now, let , which are coefficient matrices of SARE (5a)-(5b).
Relative errors and perturbation bounds
6.28 × 10−5
5.74 × 10−4
6.34 × 10−6
5.22 × 10−5
1.14 × 10−6
8.42 × 10−6
1.61 × 10−7
7.42 × 10−7
1.05 × 10−8
5.58 × 10−8
Relative errors, perturbation bounds and relative condition numbers
5.94 × 10−15
1.64 × 10−14
5.84 × 101
5.28 × 10−14
9.47 × 10−14
5.56 × 102
4.75 × 10−13
7.85 × 10−13
5.56 × 103
4.58 × 10−12
7.40 × 10−12
5.56 × 104
4.85 × 10−11
7.26 × 10−11
5.56 × 105
4.56 × 10−10
7.22 × 10−10
5.57 × 106
4.69 × 10−9
8.61 × 10−9
5.57 × 107
While doing numerical computation, it is important in practice to have an accurate method for estimating the relative error and the condition number of the given problems. In this paper, we focus on providing a tight perturbation bound of the stabilizing solution to SARE (1a)-(1b) under small changes in the coefficient matrices. Also, some sufficient conditions are presented for the existence of the stabilizing solution to the perturbed SARE. The corresponding condition number of the stabilizing solution is provided in this work. We highlight and compare the practical performance of the derived perturbation bound and condition number through a numerical example. Numerical results show that our perturbation bound is very sensitive to the condition number of the stabilizing solution. As a consequence, they provide good measurement tools for the sensitivity analysis of SARE (1a)-(1b).
We provide here a proof of the condition given by (63).
Comparison of (66) and (67) gives (65). □
The authors wish to thank the editor and two anonymous referees for many interesting and valuable suggestions on the manuscript. This research work is partially supported by the National Science Council and the National Center for Theoretical Sciences in Taiwan. The first author was supported by the National Science Council of Taiwan under Grant NSC 102-2115-M-150-002. The second author was supported by the National Science Council of Taiwan under Grant NSC 102-2115-M-003-009. The third author was supported by the National Science Council of Taiwan under Grant NSC 101-2115-M-194-007-MY3.
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