Discrete orthogonal polynomials reduced models based on shift-transformation and discrete Walsh functions

Abstract

This paper investigates model order reduction (MOR) methods of discrete-time systems via discrete orthogonal polynomials in the time domain. First, this system is expanded under discrete orthogonal polynomials, and the expansion coefficients are computed from a linear equation. Then the reduced-order system is produced by using the orthogonal projection matrix that is defined in terms of the expansion coefficients, which can match the first several expansion coefficients of the original output, and preserve the asymptotic stability and bounded-input/bounded-output (BIBO) stability. We also study the output error between the original system and its reduced-order system. Besides, the MOR method using discrete Walsh functions is proposed for discrete-time systems with inhomogeneous initial conditions. Finally, three numerical examples are given to illustrate the feasibility of the proposed methods.

Keywords:

• Discrete-time systems
• model order reduction
• discrete orthogonal polynomials
• inhomogeneous initial conditions

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data supporting the findings of this study are available within the articles (Chahlaoui & Van Dooren, 2002; Jiang & Wang, 2019; Penzl, 2006; Zhou et al., 1994). The other data that support the findings of this study are available from the authors upon reasonable request.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11871393], International Science and Technology Cooperation Program of Shaanxi Key Research & Development Plan [grant number 2019KWZ-08], Shaanxi Natural Science Basic Research Plan [grant number 2020JQ-004], Aviation Science Foundation Project [grant number ASFC-20200014070002] and Graduate Innovation Project of Xinjiang Uyghur Autonomous Region [grant number XJ2021G022].

Notes on contributors

Zhao-Hong Wang

Zhao-Hong Wang is currently pursuing PhD at the College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, China. His research interests are theoretical studies of control systems, function approximation theory, and model order reduction.

Yao-Lin Jiang

Yao-Lin Jiang is a full professor at the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, China. He has published five books and about 310 articles in academic journals. His research interests include theoretical studies of numerical solutions of partial differential equations, model order reduction, waveform relaxation, parallel in time, optimal control problems, matrices and tensors, differential manifolds and Lie group methods for engineering problems, circuit simulation, and parallel processing.

Kang-Li Xu

Kang-Li Xu is an assistant professor at the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, China. She has published about 30 papers in academic journals. Her research interests include theoretical studies of control systems, model order reduction, differential manifolds, and Riemannian optimization.