An iterative algorithm for periodic sylvester matrix equations

The problem of solving periodic Sylvester matrix equations is discussed in this paper. A new kind of iterative algorithm is proposed for constructing the least square solution for the equations. The basic idea is to develop the solution matrices in the least square sense. Two numerical examples are presented to illustrate the convergence and performance of the iterative method.

1. Introduction. The common periodic Sylvester matrix equations are the matrix equations given by where the matrices A j , B j , C j ∈ R n×n are given coefficient matrices and the unknown matrices X j ∈ R n×n are to be determined. All of the mentioned matrices follow a T cycle, i.e., A j+T = A j , B j+T = B j ,C j+T = C j and X j+T = X j .
Compared with linear time-invariant systems, linear time-varying systems are more theoretically challenging since it is generally impossible to know the state transition matrices. Hence, linear time-varying systems, including linear periodic systems as special cases, have received considerable attention during the past several decades [16,26,27]. However, the analysis of time varying systems, especially periodic systems, often involves the solution of periodic matrix equations. Periodic matrix equations play fundamental roles in the study of (discrete-time) linear periodic systems. [22] proposes an effective algorithm for the periodic discretetime Riccati equation arising from a linear periodic time-varying system (A k , B k ),

LINGLING LV, ZHE ZHANG, LEI ZHANG AND WEISHU WANG
which is particularly efficient when A k is time-invariant and B k is periodic, and has special property associated with the problem of spacecraft attitude control using magnetic torques. [4] discuss the solvability of the Sylvester-like matrix equation AX + f (X)B = C through an auxiliary standard (or generalized) Sylvester matrix equation. [10] proposes a gradient based iterative method to find the solutions of the general Sylvester discrete-time periodic matrix equations, which is proven that the proposed iterative method can obtain the solutions of the periodic matrix equations for any initial matrices.
For this reason, many researchers have carried out their works for finding analytical and numerical solutions of many kinds of matrix equations. [28] is concerned with iterative methods for solving a class of coupled matrix equations including the well-known coupled Markovian jump Lyapunov matrix equations as special cases. [19] develops a gradient based and a least squares based iterative algorithms for solving matrix equation AXB + CX T D = F . [8] applies a hierarchical identification principle to study solving the Sylvester and Lyapunov matrix equations. [23] presented a family of iterative algorithms for the matrix equation AX = F and the coupled Sylvester matrix equations utilizing the properties of the eigenvalues related to the symmetric positive definite matrices. By constructing an objective function and using the gradient search, a gradient-based iteration is established in [9] for solving the coupled matrix equations A i XB i = F i , i = 1, 2, ..., p. By using the hierarchical identification principle and introducing the convergence factor and the iterative matrix, a family of inversion-free iterative algorithms is proposed in [24] for solving nonlinear matrix equations X + A T X −1 A = I. M. Dehghan and M. Hajarian proposed some iterative algorithms based on the conjugate gradient (CG) method for solving the system of generalized Sylvester matrix equations( [5]), coupled Sylvester matrix equations( [6]) and the second-order Sylvester matrix equation EV F 2 − AV F 2 − CV = BW ( [7]), which are applications of CG in the area of solving time-invariant matrix equations. There are still many papers that are available for reference(one can see [25,15,1,2,3,17,18,20,21]).
Periodic Sylvester matrix equation, one of the most straightforward extension of time-variant matrix equation, has vital applications for analysis and design of linear periodic systems. Taking advantage of the CG method, M.Hajarian proposed an iterative algorithm for solving the problem in [11]. It can be proved that M.Hajarian's algorithm is valid. However, the number of iteration of its operation process is relatively large on account of that the computational method of iteration step is not so ideal. Recently, we have proposed a new iterative method for coupled matrix equations in [14], where time-invariant unknown matrices are restricted by several time-invariant matrix equations at the same time. Based on the idea presented in [14] and [12], we could build a new method which can be used to solve the matrix equation in time-variant form. And in this paper, we develop a new iterative algorithm for periodic Sylvester matrix equation (1), which has better convergence property and less computational burden compared with existed methods.
Here, we give descriptions of some symbols which will be encountered in the rest of this paper. tr(A) means the trace of matrix A. For the space R n×n , an inner product A, B indicates tr(B T A) for all A, B ∈ R n×n . In this sense, norm A is a Frobenius norm of matrix A. By contrast, the symbol A 2 denotes the spectral norm of matrix A. For matrices M and N , their Kronecker product is expressed as M ⊗ N . For matrix

AN ITERATIVE ALGORITHM FOR PERIODIC SYLVESTER MATRIX EQUATIONS 415
vec(X) is the column stretching operation of X, which is According to the property of Kronecker product, for matrices M , N and X with appropriate dimension, holds.
2. Main results. For this type of periodic Sylvester matrix equation, least squares method is a widely adopted technique to solve it. Its ultimate goal is seeking the matricx sequence X j for j = 0, 1, · · · , T − 1 to minimize the following index function: From this point of view, we have: Immediately, for j = 0, 1, · · · , T −1, the least squares solution ( Based on above-mentioned theoretical basis, the iteration method to solve the periodic Sylvester matrix equation via the least squares method can be expressed as the following algorithm.

Algorithm 1. (An iterative algorithm for equation (1))
1. Set allowed error ε, choose initial matrices X j (0) ∈ R n×n for j = 0, 1, · · · , T − 1, calculate The proposed algorithm needs about O(T n 3 ) operations at the most in each iteration step. And in the rest of this section, the convergence of the presented algorithm would be discussed.
Before further discussion, some fundamental lemmas should be put forward.
The following equation comes to true: Proof. According to step 3 of Algorithm 1, we can get Summing the above equations together gives Considering the definition of α(k) in Algorithm 1, the following equation comes to true: Then, we accomplish the proof.
Proof. Obviously, equation (3) is true for k = 0. According to the expressions of P j (k + 1) in Algorithm 1, we get Summing the above equations together from j = 0 to j = T − 1 gives According to Lemma 1, we have Thus, the proof of lemma 2 has been accomplished.
Lemma 3. For the sequences {R j (k)}, {P j (k)}, j = 0,1, · · · , T − 1, in Algorithm 1, the following relation comes to true: Proof. First of all, let where E is a n order unit matrix.

LINGLING LV, ZHE ZHANG, LEI ZHANG AND WEISHU WANG
Via some mathematical derivation, we have That means the relation holds. According to Lemma 2, for k ≥ 0, one has Then, This indicates that {J(k)} is a descent sequence, which means, for all k ≥ 0, there holds

So we have
In conclusion, in view of relations (5) and (7), we get The proof is thus finished.

LINGLING LV, ZHE ZHANG, LEI ZHANG AND WEISHU WANG
Proof. In the light of Lemma 1, there holds: Let Then relation (8) can be represented as Let us suppose Accordingly, there must exist a constant δ > 0 such that for all k ≥ 0. In view of relations (10) and (11), we have In other words, the following relation comes to true: At the same time, Lemma 3 shows that Obviously, this is a contradiction. Thus, the theorem proof has been achieved. 3. Numerical examples. In the following, two examples are given to illustrate the effectiveness and practical application of the proposed approach.
Example 1. Consider the following 3-periodic Sylvester matrix equation:

AN ITERATIVE ALGORITHM FOR PERIODIC SYLVESTER MATRIX EQUATIONS 421
Here, the coefficient matrices are respectively given by By applying the iterative algorithm given in Algorithm 1 with X 0 (0) = X 1 (0) = X 2 (0) = 10 −6 1(2), the iterative solution X j , j = 0, 1, 2, to this equation is given by: Define the relative iteration error In the meantime, we solved this equation by the algorithm raised in [11] with the same initial value and compared the solution with the result we got utilizing the method in this paper. The numerical results are shown in Fig. 1. Obviously, δ(k) decreases quickly and converges to zero along with k increases. By the comparison, it is easy to see that the proposed algorithm has much faster convergence speed than the reference item. , the multirate sampled-data system is modeled by a 2-periodic system in the form of Obviously, the open-loop system is unstable. In order to stabilize the system, find 2-periodic control law u(t) = K(t)x(t) such that the poles of the periodic close-loop system are assigned at Γ = {0.5, 0.6, 0.7, −0.6, −0.7}. Based on the previous works, periodic feedback K(t) can be characterized as: where G(t) ∈ R 2×5 be a given 2-periodic parameter matrix, and F (t) ∈ R 5×5 be a given 2-periodic matrix such that the eigenvalue set of F (1)F (0) is Γ. In the meantime, periodic pair (F (t), G(t)) should be completely observable. Specially, let AN ITERATIVE ALGORITHM FOR PERIODIC SYLVESTER MATRIX EQUATIONS 423 G(t) = 1 0 1 0 1 0 1 0 1 0 , t = 0, 1 What can be verified is that the poles assignment is valid.

4.
Conclusions. The problem of solving periodic Sylvester matrix equation is addressed in this paper. A numerical iterative algorithm based on conjugate gradient method is proposed. strict mathematical proof shows the matrix sequence generated by the given algorithm can converge to the exact solution in finite steps. At last, both numerical example and practical application are employed to demonstrate the validity of the proposed algorithm.