Extended Hamiltonian algorithm for the solution of discrete algebraic Lyapunov equations

Abstract

In this paper, we use a second-order learning algorithm for solving the numerical solution of the discrete algebraic Lyapunov equation. Specifically, Extended Hamiltonian algorithm based on the manifold of positive definite symmetric matrices is provided. Furthermore, this algorithm is compared with the Euclidean gradient algorithm, the Riemannian gradient algorithm and the two traditional iteration methods. Simulation examples show that the convergence speed of the Extended Hamiltonian algorithm is the fastest one among these algorithms.

Introduction

There has been a rapid development in the applications of the discrete algebraic Lyapunov equation (DALE), such as the signal processing, the system stability analysis and the optimal design of robust controllers [1], [2]. And, in the past decades, there has been an increasing interest in the solution problems of the DALE [3], [4], [5], [6]. The matrix bounds problems for the solution of the DALE were discussed in [7], [8], [9]. And, the eigenvalue estimation for the solution to the DALE was investigated in [10], [11]. The semi-global stabilization problem of the DALE for the discrete time linear periodic system was considered in [12]. Some effective iterative methods, such as the Smith accelerative iteration (SAI) [13] and the traditional FPIM [14] were given for the numerical solution of the discrete algebraic Lyapunov equation. Recently, the Euclidean gradient algorithm (EGA) and the Riemannian gradient algorithm (RGA) for the numerical solution of the discrete algebraic Lyapunov equation with a cost function of the Riemannian distance on the curved Riemannian manifold were proposed by Duan and Sun [15].

However, it is well known that the EGA and RGA are first-order learning algorithms, hence the convergence speeds of the EGA and RGA are very slow. The inclusion of a momentum term has been found to increase the rate of convergence dramatically [16], [17]. Based on this, Fiori developed the Extended Hamiltonian algorithm (EHA) on manifold, which is a second-order learning algorithm [18]. The second-order learning algorithm converges faster than the first-order learning for optimization problem if certain conditions are satisfied.

In this paper, we will apply the EHA to calculate the numerical solution of the DALE. Furthermore, we give two simulations to show the efficiency of our algorithm.

The rest of the paper is organized as follows. Section 2 introduces some fundamental knowledge on manifold, which will be used throughout the paper. Section 3 presents Extended Hamiltonian algorithm on manifold of positive definite symmetric matrices and Section 4 illustrates the convergence speeds of the EHA compared with the EGA and the RGA using two numerical examples in 7-dimensional and 10-dimensional discrete linear systems. Section 5 concludes the paper and suggests further research topics along the line of the present research.