Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

Wang, Li

doi:https://doi.org/10.1155/2020/8627492

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Volume 2020 | Article ID 8627492 | https://doi.org/10.1155/2020/8627492

Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

Li Wang¹

Guest Editor: Ramon Costa Castelló

Received23 May 2020

Accepted03 Jul 2020

Published04 Aug 2020

Abstract

The continuous coupled algebraic Riccati equation (CCARE) has wide applications in control theory and linear systems. In this paper, by a constructed positive semidefinite matrix, matrix inequalities, and matrix eigenvalue inequalities, we propose a new two-parameter-type upper solution bound of the CCARE. Next, we present an iterative algorithm for finding the tighter upper solution bound of CCARE, prove its boundedness, and analyse its monotonicity and convergence. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.

1. Introduction

Consider the following optimal control of jump linear system described bywhere is the plant state and is the control vector. , , and is a finite state Markov jump process on , where . The quadratic performance index of system (1) is

We have with and when . The optimal state feedback controller to minimize the quadratic performance index (2) is where is the symmetric positive semidefinite solution of the continuous coupled algebraic Riccati equation (CCARE):where are real constants with the properties , and . When , the CCARE (3) changes to the following common form:

CCARE (4) is usually encountered in robust and optimal control [1–8], filter design [9], time-delay systems controller design [10], stability analysis [11–16], etc. And in these fields, it often suffices to estimate the tighter solution bounds of the algebraic Riccati equation rather than get the exact solution. For example, in [17, 18], the authors have proposed solution bounds of the continuous algebraic Riccati equation, given their applications of the new bounds in redundant control input systems, and obtained several sufficient conditions to decrease the controller gain of the new systems after the control input extension. In recent years, bounds’ estimation of solution for the CCARE has become an attractive topic, and many research approaches have been devoted to this topic. In [19–21], upper matrix bounds for the solution of the CCARE have been presented and iterative algorithms have been proposed to derive tighter upper matrix bounds. And there are many other works for studying the solutions of the CCARE, such as matrix bounds and properties [22–27], matrix eigenvalue bounds [28–30], numerical solution [31–33], and the explicit solution [34, 35].

In this paper, we construct a positive semidefinite matrix, propose a new two-parameter-type upper solution bound of the CCARE by matrix inequalities and matrix eigenvalue inequalities. Examples illuminate that the upper bound improves some recent related results. Then, according to the derived solution bound, we give an iterative algorithm, which can guarantee tighter upper solution bound of the CCARE, and prove its boundedness. Subsequently, we analysis its monotonicity and convergence.

Through this paper, let denote the set of real (complex) matrices. For , the notation is used to denote that A is a symmetric positive definite (semidefinite) matrix. Inequality means matrix is positive (semi) definite. For , let be the nonincreasing order eigenvalues of . We assume , and are the transpose, the inverse, and the spectral norm of , respectively. is called a -matrix if all its off-diagonal elements are nonpositive. It is obvious that any -matrix can be written as with is nonnegative. A -matrix is called an -matrix if . For two vectors and , is equivalent to . Symbol represents the matrix measure of and is defined as .

Lemma 1. (see [36]). For any given positive semidefinite matrix and ,if and only if the matrix is negative semidefinite.

Lemma 2. (see [37]). Let satisfy that , and let be symmetric. Then,if and only if the matrix .

Lemma 3. (see [38]). For any symmetric matrix , the following inequality holds:

Lemma 4. (see [38]). Suppose are symmetric matrices and ; then,

Lemma 5. (see [39]). If is an -matrix, then is nonnegative.

Lemma 6. (see [40]). Let and , be nonnegative; then, .

Lemma 7. (see [41]). Let both be symmetric with . Then,for all .

Lemma 8. (see [20]). Let be a given sequence of positive semidefinite matrices. Assume that there exists a positive semidefinite such that for all ,and then converges to a unique positive semidefinite .

2. Upper Solution Bounds for the CCARE

In this section, we will propose new upper matrix bounds of the solution for the CCARE (4), which improve the recent results.

Theorem 1. Let be the positive semidefinite solution of the CCARE (4). If there exist some positive definite matrices , real numbers , , and such thatand suppose is an -matrix; then, for ,where ,

Proof. where is any sufficiently small positive constant. When , from (14), we obtainAnd let . According to CARE (4), we obtainLetThen, by (16), we havewhere . According to condition (11), we get ; then,Considering , using Lemma 1 to (19) yieldsSubstituting (20) into (18) with and , we get the right-hand side of (18) is negative semidefinite. Moreover, since , according to Lemma 2, we conclude , that is,Using Lemma 3 to (21), we obtainApplying Lemma 4 to (22) yieldsTherefore, from (23), we obtain i.e.,where and . As the coefficient matrix of (24) is an -matrix, is nonnegative in terms of Lemma 5. Consequently, (24) is equivalent toby Lemma 6, where . Substituting (25) into (22) completes the proof.

Corollary 1. We show thatwhere ,is an -matrix.

Proof. (1)Since Substituting (20) into (28) and combining , we get the right-hand side of (28) is negative semidefinite. Since , according to Lemma 2, we conclude , that is, Using Lemma 4 to (29) yields that is, , i.e., where . As the coefficient matrix of (31) is an -matrix, is nonnegative in terms of Lemma 5. Consequently, (31) is equivalent to by Lemma 6, where . Substituting (32) into (29), we get .(2)(i)First, let us compare with : Considering , using Lemma 1 to (19) yields Substituting (34) into (33) with , we get the right-hand side of (33) is negative semidefinite. Thus,(ii)Next, let us compare with . For (35), assume ; then, (35) changes toBy Lemma 7, it is easily to see , then . As are -matrix, we obtainOn the contrary, for of (35), assume ; then, (35) changes toApplying Lemma 7 to (38), we conclude . Thus, we obtainwith (37). Subsequently, we obtainLastly, combining (35) with (40), we derive .

Remark 1. Due to the different proof methods, it is very hard to compare the upper bounds of Theorem 1 with the parallel results theoretically, and we will present examples to illustrate that our upper bounds are tighter than the recent results for some cases.

3. Iterative Algorithm

According to Section 2, we give an iterative algorithm as following. The algorithm is based on upper bounds (12) from Theorem 1. And we will discuss the following conclusions in the same condition as Theorem 1.

Algorithm 1. Let , , and is from Theorem 1. For ,where .
Next, we will discuss the boundedness of Algorithm 1 and analyse its monotonicity and convergence.

Theorem 2. Let be the iteration in Algorithm 1. If the CCARE (4) has positive semidefinite solution , then

Proof. We proof the result by induction.(i)From (25), we get . Furthermore, where . According to Theorem 1, we get .(ii)Suppose for all , then . In a similar way to the proof of (18), we havewhere . Considering , using Lemma 1 to (19) yieldSubstituting (45) into (44) with and , we get the right-hand side of (44) is negative semidefinite. Moreover, since , according to Lemma 2, we get . This completes the induction.

Theorem 3. Let be the iteration in Algorithm 1. If , where and are from Theorem 1, then the sequence is monotone decreasing, i.e.,

Proof. We proof the result by induction.(i).(ii)Suppose for all , thenTherefore, the proof is complete.
According to Lemma 8, combining Theorems 2 and 3, we get the following conclusion.

Theorem 4. Let be the iteration in Algorithm 1. If , where and are from Theorem 1, then the sequence is monotone decreasing, and converges to the unique positive semidefinite matrices .

Now, we present examples to illustrate the effectiveness of the main results, and the stopping criterion in Algorithm 1 is set as

Example 1. (see [42]). Consider the CCARE (4) withChoose ; , and by Theorem 1, we obtainBy Corollary 1, we obtainBy Theorem 3.2 in [21], taking , , we obtainBy Theorem 2.2 in [42], taking , , we obtainObviously, we see and , that is, our bounds in Theorem 1 are tighter than [21, 42].
Sincethen Algorithm 1 is expected to be strictly monotone decreasing and converges to tighter upper bounds. By computation, the upper solution bounds of CCARE (4) areafter 6 iterations, which are tighter than and , respectively.

Example 2. Consider the CCARE (4) withChoosing , , and , by Theorem 1, we obtainBy Corollary 1, we obtainBy Theorem 3.1 in [21], taking , , we obtainBy Theorem 2.2 in [42], taking , , , we obtainObviously, we see and , that is, our bounds in Theorem 1 are tighter than [21, 42].
Sincethen Algorithm 1 is expected to be strictly monotone decreasing and converges to tighter upper bounds. By computation, the upper solution bounds of CCARE (4) areafter 7 iterations, which are tighter than , , and , respectively.

4. Conclusions

In this paper, a new two-parameter-type upper solution bound of the CCARE has been proposed. Next, an iterative algorithm for finding the tighter upper solution bound of CCARE has been presented, and its boundedness, monotonicity, and convergence have been proved. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.

Data Availability

All data generated or analyzed during this study are included in this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the National Natural Science Foundation for Youths of China (11801164), National Natural Science Foundation of China (11971413), Key Project of National Natural Science Foundation of China (91430213), General Project of Hunan Provincial Natural Science Foundation of China (2015JJ2134), and General Project of Hunan Provincial Education Department of China (15C1320).

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Copyright © 2020 Li Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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