- Research
- Open access
- Published:
Perturbation analysis of the stochastic algebraic Riccati equation
Journal of Inequalities and Applications volume 2013, Article number: 580 (2013)
Abstract
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.
1 Introduction
In this paper we consider a general class of continuous-time stochastic algebraic Riccati equations
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 .
In essence, SARE (1a)-(1b) is a rational Riccati-type matrix equation associated with the operator
where the affine linear operators , , , and are defined by
We say that X is the maximal solution (or the greatest solution) of SARE (1a)-(1b) if it satisfies (1a)-(1b) and for any satisfying and (1b), i.e., X is the maximal solution of with the constraint (1b). Furthermore, it is easily seen that SARE (1a)-(1b) also contains the continuous-time algebraic Riccati equation (CARE)
with , , and , and the discrete-time algebraic Riccati equation (DARE)
with and , as special cases.
Matrix equations of the type (1a)-(1b) are encountered in the indefinite linear quadratic (LQ) control problem [1], 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 [[3], Definition 5.2].)
Definition 1.1 Let be a solution to SARE (1a)-(1b), and , where . The matrix X is called a stabilizing solution for â„› if the spectrum of the associated operator with respect to X defined by
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 [3]. See also [[1], 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 [11]. 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 [6], matrix sign function method [19], disk function method [20], 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 [3] and the interior-point algorithm presented in [1], 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) [23].
Due to the effect of roundoff errors or the measurement errors of experimental data, small perturbations are often incorporated in the coefficient matrices of SARE (1a)-(1b), and hence we obtain the perturbed SARE
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.
To facilitate our discussion, we use to denote the Frobenius norm and to denote the operator norm induced by the Frobenius norm. For and , the Kronecker product of A and B is defined by , and the operator is denoted by . It is known that
where , , , and is the Kronecker permutation matrix which maps into for a rectangle matrix A, i.e.,
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
Assume that is the unique stabilizing solution to SARE (1a)-(1b) and is a symmetric solution of perturbed SARE (5a)-(5b), that is,
where the two operator and are given by
and two affine linear operators , are defined by
for all . Let
The purpose of this section is to derive a perturbation equation of ΔX from SAREs (1a)-(1b) and (5a)-(5b). For the sake of perturbation analysis, we adopt the following notations:
and
Moreover, let
and by the definition of Ψ, we define
Note that and . Substituting (11) into (8), we observe that
Thus far, we have not specified the relation between and . Such a tedious task can be turned into a breeze by repeatedly applying the matrix identities [29]
To begin with, assume that ΔR and ΔD are sufficiently small so that is invertible. We see that the product
It follows that
since . Next, from (11) we can see that
Applying (15), we obtain the linear equation
where
It follows from (16) that
Equipped with this fact, we now are going to derive a perturbation equation in terms of ΔX by using ΔA, ΔB, ΔC, ΔD, ΔS, ΔR, δS, and δQ. It should be noted that
with
and
with
It then is natural to express the left-hand side of (17) by ΔΦ and ΔΨ such that
with
Observe further that
Upon substituting (10) into and , we have
so that the structure of E in (17) can be partitioned into linear equations
that is, .
Lemma 2.1 Let X be the stabilizing solution of SARE (1a)-(1b) and be a symmetric solution of perturbed SARE (5a)-(5b). If , then ΔX satisfies the equation
where
where , the matrices ΔA, ΔB, and so on are given by (9)-(12).
Note that and are not dependent on ΔX, is a linear function of ΔX, and is a function of ΔX with degree at most 2. Assume that the linear operator of (4) is invertible. It is easy to see that the perturbed equation (20) is true if and only if
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
Let be a continuous mapping defined by
We see that any fixed point of the mapping f is a solution to the perturbed equation (22). Our approach in this section is to present an upper bound for the existence of some fixed points ΔX. It starts with the discussion that the mapping f given by (23) satisfies
Define linear operators , , and by
and the scalars ω, μ, ν, τ, η by
From (21a) we then have
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 [[30], Theorem 3.9].
Lemma 3.1 Let A and B be two matrices in . Then and .
It immediately follows that the matrices δQ and δS, defined by (10), satisfy
Assume that the scalar satisfies
Then is bounded by
From (21b) we see that
and also from (18) and (19) we have
It follows that
where the positive scalar δ is defined by
Also, from (28) and Lemma 3.1 we know that and . This implies that is nonsingular,
Similarly, we have
Assume that
It then follows from Lemma 3.1 and (21c) that
and from (9), (11) and (31) that
Upon substituting (34), (35) and (38) into (37), we see that
Finally, by (26), (29) and (32), we arrive at the statement
where
Consider the quadratic equation
It is true that if
then the positive scalar denoted by
is a solution to (41). Let be a compact subset of given by
It can be seen that in (39)
It then follows from the Brouwer fixed-point theorem (see [31]) that the continuous mapping f has a fixed point , that is, condition (22) automatically holds.
Observe also that if , then
This implies that assumption (36) is true, if assumption (42a)-(42b) is true.
4 Stability analysis
We have shown that the mapping f given by (23) has a Hermitian fixed point . This further implies that perturbed SARE (1a)-(1b) has a Hermitian solution . In this section, we want to discuss the stability of the solution , i.e., show that the solution is the unique maximal solution to SARE (1a)-(1b). Let Ï’ and Î be two operators defined by
with the notations Φ and Ψ given in Definition 1.1. It follows that the operator defined by (4) can also be written as
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 [3].
Theorem 4.1 The linear operator given by (44) is stable, i.e., , if and only if and for all .
When small perturbations are taken into consideration, the perturbed operator of can be expressed by
where and for all . Define the quantity
for and
where the set . It should be noted that if , and , then
where the value is defined by [27]
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 [23]
Suppose that the linear operator given by (44) is stable, and let
Then
We now apply Theorem 4.2 to (46) and obtain that
Hence, if a perturbation matrix satisfies
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 [23]
Suppose that the linear operator is stable, and let the scalars and ψ be defined as in (48). If the perturbation matrices satisfy
then is c-stable and the perturbed linear operator defined by (45) is also stable, i.e., .
Upon substituting for X in and of (11), we shall have
Also, corresponding to , the perturbed and of Ψ and Φ, respectively, can be expressed in terms of the formulae
with
Let . Since , it follows from (38), (50) and (51) that
Thus and are bounded by the inequalities
Here, the above upper bounds are obtained by simplifying those given by (30) and (31). Let
where and are defined by (38) and Θ is defined to be the right-hand side of (49), that is,
We then have
It follows that if the condition
or, equivalently,
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)
Let X be the stabilizing solution of (1a)-(1b). Let ω, , , δ, , , , α, ε, f, γ, , , , , Θ be the scalars defined by (25), (27), (33), (35), (38), (40), (53) and (54), respectively. Define
If the perturbed quantities of the coefficients of (5a)-(5b) are sufficiently small, for example, , such that
then perturbed SARE (5a)-(5b) has the unique stabilizing solution , and
5 Condition number of the SARE
In the study of a computational problem, a fundamental issue is to determine the condition number of a problem to be the ratio of the relative change in the solution to the relative change in the argument. Applying the theory of condition number given by Rice [32], we define the condition number of the stabilizing solution X of SARE (1a)-(1b) by
where the set of perturbed matrices is defined by
with
and , , , , , , , κ are positive parameters. Then (56) gives the absolute condition number if
and gives the relative condition number if
It follows from (22) and (24a)-(24d) that
where the linear operators and are defined by
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 [34] to make this presentation more self-contained.
Theorem 5.1 Let be a linear operator and
for all and . Then the optimal solution to the problem
exists for some and . Furthermore, if the linear operator is a positive operator with respect to any , that is, for all , we have
Then there exists an optimal solution to problem (60) such that is symmetric.
Proof Since â„’ is a linear operator on a finite dimensional space, it is clear that the optimal solution of (60) exists. Assume that solves this optimization problem. Let and be the matrix representation of the operator â„’ such that
By (59) and (61), we have
Note that
It follows that if , by (62), we see that is another optimal solution for (60). This proves the first part of the theorem.
For the second part, if there exists a symmetric optimal solution, then it completes the proof. Otherwise, from the first part, we know that there exists an optimal solution with , and to (60). Let . We have the following matrix decomposition:
where is a zero matrix with size , Q is an orthogonal matrix, and for . Let
be a real symmetric matrix. Since , it is true that
Using the fact that , we see that and
If , then , which implies that is a symmetric optimal solution to (60) (see [[25], Lemma A.1]). This completes the proof. □
With the existence theory established above, it is interesting to note that the condition number defined by (56) can be written as
Note that the second equality in (63) is only an application of linearity of the norm. (For the proof, see Lemma A.1.) Observe further that the inverse operator of (4) satisfies
since for all . It follows that
Also, it is known that the inverse operator is positive [3, Corollary 3.8]. It follows that is also a positive operator. Now, applying Theorem 5.1 to the operator in (63), we obtain the equality
where the extended set is defined by
On the other hand, observe that the matrix representation of the operation in (4) can be written in terms of . Corresponding to (24a)-(24d) and (58a)-(58b), we let
and
It follows that
Based on the above discussion, we have the following result.
Theorem 5.2 The condition number given by (56) has the explicit expression . In particular, we have the relative condition number
6 Numerical experiment
In this section we want to demonstrate the sharpness of perturbation bound (55) and its relationship with the relative condition number (64). Based on Newton’s iteration [3], a numerical example, done with coefficient matrices, is illustrated. The numerical algorithm is described in Algorithm 1. The corresponding stopping criterion is determined when the value of the Normalized Residual (NRes)
is less than or equal to a prescribed tolerance.
Example 1 Given a parameter , for some , let the matrices A, B, C, D be defined by
and the matrices S, R, H be defined by
It is easily seen that the unique stabilizing and maximal solution is
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).
Firstly, we would like to evaluate the accuracy of the perturbation bound with the fixed parameter , i.e., , and different weighted coefficients, , for . It can be seen from Table 1 that the values of the relative errors are closely bounded by our perturbation bounds of (55). In other words, (55) does provide a sharp upper bound of the relative errors of the stabilizing solution X.
Secondly, we want to investigate how ill-conditioned matrices affect the quantities of perturbation bounds. In this sense, the weighted coefficients are fixed to be 10−15, i.e., . The relationships among relative errors, perturbation bounds, and relative condition numbers are shown in Table 2. Due to the singularity of the matrix R caused by parameter r, the accuracy of the perturbation bounds is highly affected by the singularity. When the value of m increases, the perturbation bound is still tight to the relative error. Also, it can be seen that the number of accurate digits of the perturbation bounds is reduced proportionally to the increase of the quantities of the relative condition numbers. In other words, if the accurate digits of the perturbation bound are added to the digits in the relative condition numbers, this number is almost equal to 16. (While using IEEE double-precision, the machine precision is around .) This implies that the derived perturbation bound of (55) is fairly sharp.
7 Conclusion
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).
Appendix
We provide here a proof of the condition given by (63).
Lemma A.1 Let , , ℳ, , ℋ, , be the operators defined by (58a)-(58b), (24a)-(24d) and (4), and let , be defined by (57). Then the following equality holds:
Proof For any , , we see that
where . It follows that
On the other hand, for any fixed , choose any perturbation matrices
and therefore
It is true that and this gives the fact that
Hence
Comparison of (66) and (67) gives (65). □
References
Rami MA, Zhou XY: Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls. IEEE Trans. Autom. Control 2000, 45(6):1131–1143. 10.1109/9.863597
El Bouhtouri A, Hinrichsen D, Pritchard AJ: On the disturbance attenuation problem for a wide class of time invariant linear stochastic systems. Stoch. Stoch. Rep. 1999, 65(3–4):255–297.
Damm T, Hinrichsen D: Newton’s method for a rational matrix equation occurring in stochastic control. 332/334. Proceedings of the Eighth Conference of the International Linear Algebra Society 2001, 81–109.
Hinrichsen D, Pritchard AJ:Stochastic . SIAM J. Control Optim. 1998, 36: 1504–1538. 10.1137/S0363012996301336
Lancaster P, Rodman L Oxford Science Publications. In Algebraic Riccati Equations. Clarendon, New York; 1995.
Mehrmann VL Lecture Notes in Control and Information Sciences 163. In The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution. Springer, Berlin; 1991.
Zhou K, Doyle JC, Glover K: Robust and Optimal Control. Prentice Hall, Upper Saddle River; 1996.
Benner, P, Laub, AJ, Mehrmann, V: A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: continuous-time case. Technical Report SPC 95_22, Fakultät für Mathematik, TU Chemnitz-Zwickau, 09107 Chemnitz, FRG. http://www.tu-chemnitz.de/sfb393/spc95pr.html (1995).
Benner, P, Laub, AJ, Mehrmann, V: A collection of benchmark examples for the numerical solution of algebraic Riccati equations II: discrete-time case. Technical Report SPC 95_23, Fakultät für Mathematik, TU Chemnitz-Zwickau, 09107 Chemnitz, FRG. http://www.tu-chemnitz.de/sfb393/spc95pr.html (1995).
Sima V Monographs and Textbooks in Pure and Applied Mathematics 200. In Algorithms for Linear-Quadratic Optimization. Dekker, New York; 1996.
Laub AJ: A Schur method for solving algebraic Riccati equations. IEEE Trans. Autom. Control 1979, 24(6):913–921. 10.1109/TAC.1979.1102178
Ammar G, Benner P, Mehrmann V: A multishift algorithm for the numerical solution of algebraic Riccati equations. Electron. Trans. Numer. Anal. 1993, 1: 33–48.
Ammar G, Mehrmann V: On Hamiltonian and symplectic Hessenberg forms. Linear Algebra Appl. 1991, 149: 55–72.
Benner P, Mehrmann V, Xu H: A new method for computing the stable invariant subspace of a real Hamiltonian matrix. J. Comput. Appl. Math. 1997, 86: 17–43. 10.1016/S0377-0427(97)00146-5
Benner P, Mehrmann V, Xu H: A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils. Numer. Math. 1998, 78(3):329–358. 10.1007/s002110050315
Bunse-Gerstner A, Byers R, Mehrmann V: A chart of numerical methods for structured eigenvalue problems. SIAM J. Matrix Anal. Appl. 1992, 13: 419–453. 10.1137/0613028
Bunse-Gerstner A, Mehrmann V: A symplectic QR like algorithm for the solution of the real algebraic Riccati equation. IEEE Trans. Autom. Control 1986, 31(12):1104–1113. 10.1109/TAC.1986.1104186
Mehrmann V: A step toward a unified treatment of continuous and discrete time control problems. Linear Algebra Appl. 1996, 241–243: 749–779.
Byers R: Solving the algebraic Riccati equation with the matrix sign function. Linear Algebra Appl. 1987, 85: 267–279.
Benner, P: Contributions to the numerical solutions of algebraic Riccati equations and related eigenvalue problems. PhD thesis, Fakultät für Mathematik, TU Chemnitz-Zwickau, Chemnitz, Germany (1997)
Chu EK-W, Fan H-Y, Lin W-W: A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations. Linear Algebra Appl. 2005, 396: 55–80.
Chu EK-W, Fan H-Y, Lin W-W, Wang C-S: Structure-preserving algorithms for periodic discrete-time algebraic Riccati equations. Int. J. Control 2004, 77(8):767–788. 10.1080/00207170410001714988
Chiang C-Y, Fan H-Y: Residual bounds of the stochastic algebraic Riccati equation. Appl. Numer. Math. 2013, 63: 78–87.
Konstantinov M, Gu D-W, Mehrmann V, Petkov P Studies in Computational Mathematics 9. In Perturbation Theory for Matrix Equations. North-Holland, Amsterdam; 2003.
Sun J-G: Perturbation theory for algebraic Riccati equations. SIAM J. Matrix Anal. Appl. 1998, 19(1):39–65. 10.1137/S0895479895291303
Sun J-g: Sensitivity analysis of the discrete-time algebraic Riccati equation. 275/276. Proceedings of the Sixth Conference of the International Linear Algebra Society 1998, 595–615. Chemnitz, 1996
Sun J-g: Residual bounds of approximate solutions of the algebraic Riccati equation. Numer. Math. 1997, 76(2):249–263. 10.1007/s002110050262
Sun J-g: Residual bounds of approximate solutions of the discrete-time algebraic Riccati equation. Numer. Math. 1998, 78(3):463–478. 10.1007/s002110050321
Riedel KS: A Sherman-Morrison-Woodbury identity for rank augmenting matrices with application to centering. SIAM J. Matrix Anal. Appl. 1992, 13(2):659–662. 10.1137/0613040
Stewart GW, Sun JG Computer Science and Scientific Computing. In Matrix Perturbation Theory. Academic Press, Boston; 1990.
Ortega JM, Rheinboldt WC Classics in Applied Mathematics 30. In Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia; 2000. Reprint of the 1970 original
Rice JR: A theory of condition. SIAM J. Numer. Anal. 1966, 3: 287–310. 10.1137/0703023
Sun J-g: Condition numbers of algebraic Riccati equations in the Frobenius norm. Linear Algebra Appl. 2002, 350: 237–261. 10.1016/S0024-3795(02)00294-X
Xu S: Matrix Computation in Control Theory. Higher Education Press, Beijing; 2010. (In Chinese)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Chiang, CY., Fan, HY., Lin, M.M. et al. Perturbation analysis of the stochastic algebraic Riccati equation. J Inequal Appl 2013, 580 (2013). https://doi.org/10.1186/1029-242X-2013-580
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-580