Abstract

Lyapunov equations are key mathematical objects in systems theory, analysis and design of control systems, and in many applications, including balanced realization algorithms, procedures for reduced order models, Newton methods for algebraic Riccati equations, or stabilization algorithms. A new iterative accuracy-enhancing solver for both standard and generalized continuous- and discrete-time Lyapunov equations is proposed and investigated in this paper. The underlying algorithm and some technical details are summarized. At each iteration, the computed solution of a reduced Lyapunov equation serves as a correction term to refine the current solution of the initial equation. The best available algorithms for solving Lyapunov equations with dense matrices, employing the real Schur(-triangular) form of the coefficient matrices, are used. The reduction to Schur(-triangular) form has to be done only once, before starting the iterative process. The algorithm converges in very few iterations. The results obtained by solving series of numerically difficult examples derived from the SLICOT benchmark collections for Lyapunov equations are compared to the solutions returned by the MATLAB and SLICOT solvers. The new solver can be more accurate than these state-of-the-art solvers and requires little additional computational effort.

Highlights

  • Lyapunov equations are key mathematical objects in systems theory, analysis and design of systems, and in many applications

  • The results obtained by solving series of numerically difficult examples derived from the SLICOT benchmark collections for Lyapunov equations are compared to the solutions returned by the MATLAB and SLICOT solvers

  • The best available algorithms for solving Lyapunov equations with dense coefficient matrices, based on the orthogonal reduction to the real Schur(-triangular) form are used in the implementation

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Summary

Introduction

Lyapunov equations are key mathematical objects in systems theory, analysis and design of (control) systems, and in many applications. Solving these equations is an essential step in balanced realization algorithms [1,2], in procedures for reduced order models for systems or controllers [3,4,5,6,7], in Newton methods for algebraic Riccati equations (AREs) [8,9,10,11,12,13,14], or in stabilization algorithms [12,15,16]. A necessary and sufficient condition for asymptotic stability of system (3) is that for any symmetric positive definite matrix Y, denoted Y > 0, there is a unique

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