Numerical Algorithms of the Discrete Coupled Algebraic Riccati Equation Arising in Optimal Control Systems

*e 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. *e 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. *ey are all monotonous and convergent. Numerical examples demonstrate the convergence effect of the presented algorithms.


Introduction and Preliminaries
e discrete coupled Riccati equation is usually encountered in optimal control and filter design problems in control theory [1][2][3][4][5][6][7][8][9], particularly in the jump-linear quadratic optimal control problem [10]. Consider the following jumplinear system: with initial state x(0) � x 0 , r(0) � r 0 , where x k ∈ R n is the plant state, u k ∈ R m is the control vector, and y k ∈ R q is the process output. Here, k is the time index, r k is the form process taking values in the finite set S � 1, 2, . . . , s { }, and r k is a finite-state discrete-time Markov chain with transition probabilities.
Pr r k+1 � j r k � i � e ij , 1 ≤ i, j ≤ s, e ii > 0. (2) 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 A i ∈ R n×n is a constant matrix, B i ∈ R n×m , Q i ∈ R n×n is a symmetric positive definite matrix, i ∈ S, F i � P i + j≠i e ij P j is the coupled term, e ij are real nonnegative constants defined as e ij ≡ (e ij /e ii ) with the properties e ij ∈ [0, 1], e ii > 0, and j∈S e ij � 1, and P i ∈ R n×n denotes the symmetric positive definite solution of the DCARE. Applying Woodbury matrix equality

DCARE (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][12][13][14][15], trace and eigenvalue bounds [16][17][18][19][20][21][22][23], and the existence and uniqueness [24][25][26]. Besides these results, numerical solutions of Riccati equations are very important and have been studied by many scholars [27][28][29][30][31][32][33][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][25][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. R denotes the real number field. R n×m denotes the set of n × m real matrices. For X � (x ij ) ∈ R n×n , let X T and X − 1 denote the transpose and inverse of the matrix X, respectively. e inequality X > ( ≥ )0 means X is a symmetric positive or semidefinite matrix, and the inequality X > ( ≥ )Y means X − Y is a symmetric positive (semi-) definite matrix. e identity matrix with appropriate dimensions is represented by I. Lemma 1 (see [36]). If A, B ∈ R n×n are symmetric positive definite matrices, then Lemma 2 (see [22]). Let matrices A, X, R, Y ∈ R n×n with X, Y > 0, R ≥ 0, and X > ( ≥ )Y. en, with strict inequality if A is nonsingular, and X > Y.

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: Step 2: compute From Algorithm 1, we get an increasing and bounded iterative sequence, which is convergent to the positive definite solution of DCARE (5).

Theorem 1.
Let P i ( * ) be the positive definite solution of DCARE (5) and Q i > 0. e iterative sequences P i (k) and F i (k) are generated by the iterative (8) with 0 ≤ ω ≤ 1, and then Proof. Since P i ( * ) is positive definite solution of DCARE (5), then Since ω − 1 ≤ 0 and (12), we get 2 Mathematical Problems in Engineering that is, Suppose that According to (16) and Lemma 2, we get From (15), (17), (18), we get Mathematical Problems in Engineering us, the proof of induction is completed. Because P i (k) and F i (k) are monotonically increasing and they are bounded, then lim k⟶∞ P i (k) and lim k⟶∞ F i (k) exist. As k ⟶ ∞, Algorithm 1 gives us, lim k⟶∞ P i (k) � P i ( * ).
We can choose (21) as starting value and get the following algorithm.

Algorithm 2
Step 1: 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 P i ( * ) be the positive definite solution of DCARE (5) and Q i > 0. e iterative sequences P i (k) and F i (k) are generated by iterative (23) with 0 ≤ ω ≤ 1, and then Proof. According to (21) and (23), we have Since ω − 1 ≤ 0, with (25) and (26) we get that is, Suppose that According to (30) and Lemma 2, we get From (29), (31), (32), we get us, the proof of induction is completed. Because P i (k) and F i (k) are monotonically decreasing and they are bounded, then lim k⟶∞ P i (k) and lim k⟶∞ F i (k) exist. In a similar way as the proof of (20), as k ⟶ ∞, Algorithm 2 gives lim k⟶∞ P i (k) � P i ( * ).

Remark 1. In Algorithm 2, if B i B T
i is singular, we can choose a suitable G so that (G − 1 + B i B T i ) − 1 is nonsingular, as in eorem 2.

Remark 2.
In Algorithm 1, the sequence P i (k) in (8) with the initial value P i (0) � Q i converges monotonically to a positive definite solution of DCARE (5), and so does the sequence P i (k) 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 e ij � 0(j ≠ i), 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 ω � 1, 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 0 ≤ ω ≤ 1, but we can run the two algorithms with ω > 1 in practical computation. And, we have faster convergence speed if appropriate parameters are selected. We will illustrate it in the following examples.

Numerical Examples
In this section, the following numerical examples are presented to show the effectiveness of our results.

Mathematical Problems in Engineering
Example 1 (see [26]). Consider DCARE (5) Since there are two equations in the DCARE, the superiority of the ω in Algorithm 1 is not obvious. So, we choose ω � 1 here. After 9 steps of iteration of (8), we obtain the solution P i of DCARE (5).
and the residual ‖A T i [P i (9) + j≠i e ij P j (9) However, it needs 47 steps of iteration for the algorithm in [26] Because 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 ω. Although we only prove the convergence of Algorithm 1 with 0 ≤ ω ≤ 1, from Table 1, we find the convergence rapid is the fastest when ω is 1.8. After 31 steps of iteration of (8) with ω � 1.8, we obtain the solution P i of DCARE (5 Example 3 (see [26] and the residual ‖A T i [P i (4)+ j≠i e ij P j (4)] − 1 + B i B T i } − 1 A i + Q i − P i (4)‖ is 9.070366679964081e − 011.
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
e authors declare that they have no conflicts of interest.