Abstract
It is well-known that the stability of a first-order autonomous system can be determined by testing the symmetric positive definite solutions of associated Lyapunov matrix equations. However, the research on the constrained solutions of the Lyapunov matrix equations is quite few. In this paper, we present three iterative algorithms for symmetric positive semidefinite solutions of the Lyapunov matrix equations. The first and second iterative algorithms are based on the relaxed proximal point algorithm (RPPA) and the Peaceman–Rachford splitting method (PRSM), respectively, and their global convergence can be ensured by corresponding results in the literature. The third iterative algorithm is based on the famous alternating direction method of multipliers (ADMM), and its convergence is subsequently discussed in detail. Finally, numerical simulation results illustrate the effectiveness of the proposed iterative algorithms.
Highlights
Linear matrix equations are often encountered in various engineering disciplines, such as control theory [1, 2], system theory [3, 4] and stability analysis [5, 6]
Motivated by the relaxed proximal point algorithm (RPPA) in [16], we shall present an iterative algorithm for the constrained Lyapunov matrix equations and discuss its convergence
We evaluate the performance of the three iterative algorithms designed in this paper for solving the Lyapunov matrix equations, and report some preliminary numerical results. e codes of our algorithms were written entirely in Matlab R2014a, and were executed on a inkPad notebook with Pentium(R) Dual-Core CPU T4400@ 2.2 GHz, 2 GB of memory
Summary
Linear matrix equations are often encountered in various engineering disciplines, such as control theory [1, 2], system theory [3, 4] and stability analysis [5, 6]. Niu et al [9] and Xie and Ma [10] extended the method in [8] by designing some accelerated gradient-based iterative algorithms for problem (1) without X∗ ∈ Sm+ , which often have more efficient numerical performance by choosing suitable parameters. Based on the hierarchical identification principle and a new fixed point equation, Sun et al [7] designed two least-squares iterative algorithms for problem (1) without X∗ ∈ Sm+. Ke and Ma [12] extended the alternating direction method of multipliers (ADMM), which is a famous numerical method in separable optimization programming, to solve problem (1) with the nonnegative constraint X∗ ≥ 0, which is obviously motivated by the design principle proposed by Xu et al [13].
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