Abstract
The discrete coupled algebraic Riccati equation (DCARE) has wide applications in robust control, optimal control, and so on. In this paper, we present two iterative algorithms for solving the DCARE. The two iterative algorithms contain both the iterative solution in the last iterative step and the iterative solution in the current iterative step. And, for different initial value, the iterative sequences are increasing and bounded in one algorithm and decreasing and bounded in another. They are all monotonous and convergent. Numerical examples demonstrate the convergence effect of the presented algorithms.
1. Introduction and Preliminaries
The discrete coupled Riccati equation is usually encountered in optimal control and filter design problems in control theory [1–9], particularly in the jump-linear quadratic optimal control problem [10]. Consider the following jump-linear system:with initial state , , where is the plant state, is the control vector, and is the process output. Here, is the time index, is the form process taking values in the finite set , and is a finite-state discrete-time Markov chain with transition probabilities.
Minimizing the cost criterion of system (1) reduces to solving coupled algebraic Riccati-like equations. After some transformation, the coupled algebraic Riccati-like equations turn the following discrete coupled algebraic Riccati equation (DCARE)where is a constant matrix, , is a symmetric positive definite matrix, , is the coupled term, are real nonnegative constants defined as with the properties , , and , and denotes the symmetric positive definite solution of the DCARE. Applying Woodbury matrix equalityDCARE (3) turns to
Because of the importance of Riccati equations in control theory and control engineering, a lot of research studies about Riccati equations have been devoted to this field, such as solution bounds [11–15], trace and eigenvalue bounds [16–23], and the existence and uniqueness [24–26]. Besides these results, numerical solutions of Riccati equations are very important and have been studied by many scholars [27–34] because the numerical solutions of the Riccati equations are necessary in some practical engineering, such as finding the optimal state feedback controller in the optimal control system. Especially, for the DCARE, fixed point iterative algorithms are given in [24–26]. Stein iterations are presented in [35] which are based on the properties of a Stein equation. Among these results, we find less work has been done to discuss the numerical solution of the DCARE. Considering the importance and necessity of the numerical solutions of the DCARE, we propose two algorithms to discuss the numerical solution of the DCARE.
In this paper, we first propose an iterative algorithm with a parameter for solving the DCARE and prove its monotonically convergence. Second, we give an upper solution bound of the DCARE, by which another iterative algorithm is presented, and the proof of its monotonically convergence is given. For different initial values, the iterative sequences are increasing and bounded in one algorithm and decreasing and bounded in another. Last, numerical examples are given to illustrate the convergence effect of the two algorithms.
We first introduce some symbol conventions. denotes the real number field. denotes the set of real matrices. For , let and denote the transpose and inverse of the matrix , respectively. The inequality means X is a symmetric positive or semidefinite matrix, and the inequality means is a symmetric positive (semi-) definite matrix. The identity matrix with appropriate dimensions is represented by .
Lemma 1 (see [36]). If are symmetric positive definite matrices, then
Lemma 2 (see [22]). Let matrices with , , and . Then,with strict inequality if is nonsingular, and .
2. Main Results
In [25, 26], the authors have derived several solution bounds by which iterative algorithms have been proposed, but there are many restrictions in these algorithms. In this part, we first present an iterative algorithm for DCARE (5) which do not have any restrictions.
Algorithm 1. Step 1: set , , , . Step 2: computeFrom Algorithm 1, we get an increasing and bounded iterative sequence, which is convergent to the positive definite solution of DCARE (5).
Theorem 1. Let be the positive definite solution of DCARE (5) and . The iterative sequences and are generated by the iterative (8) with , and then
Proof. Since is positive definite solution of DCARE (5), thenso . Therefore,Since and (12), we getthat is, Suppose thatAccording to (16) and Lemma 2, we getFrom (15), (17), (18), we getThus, the proof of induction is completed. Because and are monotonically increasing and they are bounded, then and exist. As , Algorithm 1 givesThus, .
Theorem 2. Let be the positive definite solution of DCARE (5), then
Proof. If , then by Lemma 1, we getWhen , ; therefore, (22) changes to (21) in this case.
We can choose (21) as starting value and get the following algorithm.
Algorithm 2. Step 1: set , , , . Step 2: compute From Algorithm 2, we get a decreasing and bounded iterative sequences, which is convergent to the positive definite solution of DCARE (5).
Theorem 3. Let be the positive definite solution of DCARE (5) and . The iterative sequences and are generated by iterative (23) with , and then
Proof. According to (21) and (23), we haveSince , with (25) and (26) we getthat is, Suppose that According to (30) and Lemma 2, we getFrom (29), (31), (32), we getThus, the proof of induction is completed. Because and are monotonically decreasing and they are bounded, then and exist. In a similar way as the proof of (20), as , Algorithm 2 gives .
Remark 1. In Algorithm 2, if is singular, we can choose a suitable so that is nonsingular, as in Theorem 2.
Remark 2. In Algorithm 1, the sequence in (8) with the initial value converges monotonically to a positive definite solution of DCARE (5), and so does the sequence in (23) with the initial value . But the two positive definite solutions may be different. Whether the positive definite solution of DCARE (5) is unique or not, a problem needs to be discussed further.
Remark 3. When , DCARE (5) changes to the discrete algebraic Riccati equation. And iterative sequences (8) and (23), respectively, in Algorithm 1 and Algorithm 2, become the iterative (17) and iterative (28) in [22], which means that the algorithms of the DCARE in this paper are generalizations of the discrete algebraic Riccati equation. Moreover, when , the iterative (8) and (23) are extensions on the discrete coupled algebraic Riccati equation of the work of [22].
Remark 4. In this paper, we only prove Algorithms 1 and 2 are convergent under the condition , but we can run the two algorithms with in practical computation. And, we have faster convergence speed if appropriate parameters are selected. We will illustrate it in the following examples.
3. Numerical Examples
In this section, the following numerical examples are presented to show the effectiveness of our results.
Example 1 (see [26]). Consider DCARE (5) withSince there are two equations in the DCARE, the superiority of the in Algorithm 1 is not obvious. So, we choose here. After 9 steps of iteration of (8), we obtain the solution of DCARE (5).and the residual is .
However, it needs 47 steps of iteration for the algorithm in [26] to get the iteration solution of DCARE (5).
Example 2. Consider DCARE (5) withBecause the restrictions of the algorithms in [25, 26] are not met for this case, the algorithms in [25, 26] cannot work.
For Algorithm 1, the steps of iteration and the residual are presented in Table 1 with different parameter . Although we only prove the convergence of Algorithm 1 with , from Table 1, we find the convergence rapid is the fastest when is 1.8. After 31 steps of iteration of (8) with , we obtain the solution of DCARE (5).and the residual is .
Example 3 (see [26]). Consider the DCARE (5) with and are the same as Example 2.
For Algorithm 2, since is singular, by choosing , then Algorithm 2 can work now. After 4 steps of iteration of (23) with , we obtain the solution of DCARE (5):and the residual is .
However, it needs 18 steps of iteration for the algorithm in [26] to get the iteration solution of DCARE (5).
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
The work was supported in part by the National Natural Science Foundation for Youths of China (11801164), National Natural Science Foundation of China (11971413), the Key Project of National Natural Science Foundation of China (91430213), the General Project of Hunan Provincial Natural Science Foundation of China (2015JJ2134), and the General Project of Hunan Provincial Education Department of China (15C1320).