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Comparative Performance Evaluation of an Accuracy-Enhancing Lyapunov Solver

National Institute for Research & Development in Informatics, 011455 Bucharest, Romania
This paper is an extended version of our paper published in 2018 22nd International Conference on System Theory, Control and Computing (ICSTCC), Sinaia, Romania, 10–12 October 2018.
Information 2019, 10(6), 215;
Received: 13 May 2019 / Revised: 15 June 2019 / Accepted: 16 June 2019 / Published: 19 June 2019
(This article belongs to the Special Issue ICSTCC 2018: Advances in Control and Computers)
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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. View Full-Text
Keywords: linear multivariable systems; Lyapunov equation; numerical algorithms; software; stability linear multivariable systems; Lyapunov equation; numerical algorithms; software; stability

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Sima, V. Comparative Performance Evaluation of an Accuracy-Enhancing Lyapunov Solver. Information 2019, 10, 215.

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