Design of a mixed ℋ₂/ℋ_∞ gain scheduling state derivative feedback controller for linear parameter-varying systems

Abstract

In this paper, the mixed ${\mathcal{H}}_2/{\mathcal{H}}_{\mathrm{\infty }}$ guaranteed cost for continuous-time linear parameter-varying (LPV) systems is proposed. In addition, the gain scheduling (GS) strategy and the state derivative feedback (SDF) were considered. The time-varying parameter is assumed to be available for online measurement, and the conditions are based on linear matrix inequalities (LMIs). To improve the behaviour of the system, the $\mathcal{D}$-stability is used. Examples show that the proposed conditions achieved good results in reducing the disturbance effect in the system. Furthermore, with the proposed methods, it is possible to guarantee the asymptotic stability and a mixed ${\mathcal{H}}_2/{\mathcal{H}}_{\mathrm{\infty }}$ guaranteed cost minimisation.

Keywords:

• ${\mathcal{H}}_2/{\mathcal{H}}_{\mathrm{\infty }}$ guaranteed cost
• gain scheduling (GS)
• state derivative feedback (SDF)
• linear matrix inequalities (LMIs)
• linear parameter-varying (LPV) systems
• $\mathcal{D}$-stability

Disclosure statement

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

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article (and/or) its supplementary materials.

Additional information

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and by the Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brasil (CNPq) - Research Fellowships 301227/2021-8 and 309872/2018-9.

Notes on contributors

Marco Antonio Leite Beteto

Marco Antonio Leite Beteto was born in Gália-SP, Brazil, in 1994. He received the B.Sc. degree from Federal University of Technology - Paraná (UTFPR), Cornélio Procópio, Brazil, in 2017, in Control and Automation Engineering, the M.Sc. degree and the D.Sc. degree from School of Engineering, São Paulo State University (UNESP), Ilha Solteira-SP, Brazil, in 2019, in Electrical Engineering. His interests include control theory and applications, linear matrix inequality-based designs, optimal control (LQR, H2, Hoo), robust control, gain scheduling control, state derivative feedback control and fuzzy systems.

Edvaldo Assunção

Edvaldo Assunção was born in Andradina-SP, Brazil, in 1965. He received the B.Sc. degree from School of Engineering, São Paulo State University (UNESP), Ilha Solteira-SP, Brazil in 1989, the M.Sc. degree from Instituto Tecnológico de Aeronáutica, Brazil, in 1991, and the D.Sc. degree from Universidade Estadual de Campinas, Brazil, in 2000, all in Electrical Engineering. In 1992, he joined the Department of Electrical Engineering, UNESP, where he is currently an Associate Professor. His interests include control theory and applications, linear matrix inequality-based designs, robust control, gain scheduling control and fuzzy systems. Dr. Assunção received the Instituto de Engenharia de São Paulo Award from FEISUNESP in 1989.

Marcelo Carvalho Minhoto Teixeira

Marcelo Carvalho Minhoto Teixeira was born in Campo Grande, Brazil, in 1957. He received the B.Sc. degree in electrical engineering from the Escola de Engenharia de Lins, Lins,Brazil, in 1979, the M.Sc. degree in electrical engineering from the Instituto Alberto Luiz Coimbra de Pós-graduação e Pesquisa de Engenharia,Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil, in 1982, and the D.Sc. degree in electrical engineering from Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, in 1989. In 1982, he joined the Department of Electrical Engineering, Univ Estadual Paulista (UNESP), Ilha Solteira (FEIS), Brazil, where he is currently a Professor. In 1996 and 1997, he was a Visiting Scholar at the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA. He was a Coordinator, from 2005 to 2007 and a Vice Coordinator, from 2000 to 2005 and from 2007 to 2010 of the Postgraduate Program in electrical engineering at FEIS-UNESP. He was a member of the Brazilian Evaluation Committee of Postgraduate Courses IV Engineering CAPES, from 2009 to 2013. Additionally, he was the Coordinator, from 1991 to 1993 and a Vice Coordinator, from 1989 to 1991 and from 1993 to 1995 of the Undergraduate Program in electrical engineering at FEIS-UNESP. His interests include control theory and applications, neural networks, variable-structure systems, linear-matrix inequality based designs, fuzzy systems, nonlinear systems, adaptive systems, and switched systems. He was an Associate Editor for the IEEE TRANSACTIONS ON FUZZY SYSTEMS, from 2016 to 2020.