The Generalized Modified Hermitian and Skew-Hermitian Splitting Method for the Generalized Lyapunov Equation
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
The Generalized Modified Hermitian and Skew-Hermitian Splitting Method for the Generalized Lyapunov Equation Juan Zhang* and Huihui Kang Abstract: In this paper, we propose the generalized modified Hermitian and skew-Hermitian splitting (GMHSS) approach for computing the generalized Lyapunov equation. The GMHSS iteration is convergent to the unique solution of the generalized Lyapunov equation. Moreover, we discuss the convergence analysis of the GMHSS algorithm. Further, the inexact version of the GMHSS (IGMHSS) method is formulated to improve the GMHSS method. Finally, some numerical experiments are carried out to demonstrate the effectiveness and competitiveness of the derived methods. Keywords: Bilinear systems, controllability, convergence, GMHSS method, IGMHSS method, Hermitian and skew-Hermitian splitting, the generalized Lyapunov equation.
1.
INTRODUCTION
In this paper, we are concerned with the generalized Lyapunov equation as follows: m
AX + XA> + ∑ N j XN > j +C = 0,
(1)
j=1
where A, N j ∈ Cn×n and C = C> ∈ Cn×n (C is a symmetric matrix) are given matrices, m n, X ∈ Cn×n is the unknown matrix. Especially, when N j = 0 ( j = 1, 2, · · · , m), the generalized Lyapunov equation (1) degenerates to the standard Lyapunov equation. Linear matrix equations arise in different applications. The linear equation (1) studied in this paper has wide applications in the controllability and model simplification of bilinear systems [1, 2], stability analysis of linear stochastic systems [3, 4] and special linear stochastic differential equations [5]. If the generalized Lyapunov equation (1) has a solution, it is symmetric [5]. Next, we explore the origin of the generalized Lyapunov equation. It stems from a subclass of nonlinear control systems appearing in the dynamics of certain boundary controls [4, 6]. The so-called bilinear control system has been studied for many years, described by the following state-space
m x(t) ˙ = Ax(t) + ∑ N j x(t)u j (t) + Bu(t), j=1 Σ: ˆ y(t) = Cx(t), x(0) = x0 ,
(2)
where t is the time variable, x(t) ∈ Cn , u(t) ∈ Cm and y(t) ∈ Cn are the stable, input and output vectors, respectively. u j (t) is the j-th component of u(t). B ∈ Cn×m , Cˆ ∈ Cn×n and A is defined by (1). It is well-known that the reachability and observability of the system (2) exist [1, 7]. Denote P1 = eAt1 B, Pi (t1 , · · · ,ti ) = eAti [N1 Pi−1 , · · · , Nm Pi−1 ], i = 2, 3, · · · , then the reachability corresponding to (2) defined by ∞
P=∑
Z
∞
···
Z
i=1 0
0
∞
Pi Pi> dt1 · · · dti
is the solution of the generalized Lyapunov equation (1). Similarly, the observability of (2) is the solution of the dual equation for (1) m
A>Y +YA + ∑ Ni>Y Ni + Cˆ >Cˆ = 0, i=1
where Y ∈ Cn×n is the unknown matrix. Since the Lyapunov equation is a linear equation, we can turn it into a classical linear system through Kronecker product. Set
Manuscript received January 18, 2020; revised March 23, 2020; accepted April 15, 2020. Recommended by Associate Edi
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