Discrete-time ℋ₂ and ℋ_∞ poly-quadratic filter designs for polytopic LPV systems

Abstract

This paper addresses novel discrete-time ${\mathcal{H}}_2$ and ${\mathcal{H}}_{\mathrm{\infty }}$ filter designs for polytopic linear parameter-varying (LPV) systems. A less conservative approach is employed to obtain the synthesis conditions, such that the polytopic LPV system state-space representation may have all its matrices subject to time-varying parameters and be solved by an efficient procedure. Parameter-dependent Lyapunov functions are used to derive both ${\mathcal{H}}_2$ and ${\mathcal{H}}_{\mathrm{\infty }}$ discrete-time filter design conditions resulting in a poly-quadratic approach in terms of linear matrix inequalities. A multi-objective ${\mathcal{H}}_2/{\mathcal{H}}_{\mathrm{\infty }}$ solution is also provided. The proposed conditions have the advantage of recovering the established poly-quadratic concept as a particular case, which was an unresolved problem until the present paper. Moreover, using the addressed approach, it is possible to obtain improved quadratic ${\mathcal{H}}_2$ and ${\mathcal{H}}_{\mathrm{\infty }}$ filters. Numerical examples illustrate the proposed design methods applicability as an observer to estimate LPV system states and its performance in comparison with other approaches.

Keywords:

• Discrete-time LPV systems
• filter design
• ${\mathcal{H}}_2$ performance
• ${\mathcal{H}}_{\mathrm{\infty }}$ performance
• linear matrix inequalities

Disclosure statement

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

Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analysed in this study.

Additional information

Funding

Matheus S. de Oliveira and Kayol S. Mayer are supported in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil(CAPES) – Finance Code 001.

Notes on contributors

Matheus S. de Oliveira

Matheus S. de Oliveira received the B.Eng. degree in Control and Automation in 2015 and the M.S degree in Electrical Engineering in 2018 from the Pontifical Catholic University of Rio Grande do Sul - PUCRS. He is currently working toward a Ph.D. degree in Electronic Engineering at the Instituto Tecnológico de Aeronáutica - ITA. His main research interest is robust control, including uncertain systems, LPV systems, performance criteria and LMIs.

Kayol S. Mayer

Kayol S. Mayer received the B.Sc. degree in Automation and Control Engineering from the Pontifical Catholic University of Rio Grande do Sul, in 2016, and the M.Sc. degree in Electrical Engineering, also from PUCRS, in 2018. Currently, he is a Ph.D. Candidate in Electrical Engineering at the University of Campinas - Unicamp. His current research interests include wireless and optical communications, adaptive digital signal processing, and machine learning.

Renan L. Pereira

Renan L. Lima Pereira received the B.Sc. degree in electrical engineering from Federal University of Maranhão in 2009, and the M.Sc and Ph.D. degrees in electronic engineering from Instituto Tecnológico de Aeronáutica in 2011 and 2014, respectively. Since 2018, he is professor at the Instituto Tecnológico de Aeronáutica. His research interests include robust control, gain-scheduled control and optimisation.