H₂/H_∞ formulation of LQR controls based on LMI for continuous-time uncertain systems

Abstract

The classical theory of linear matrix inequality (LMI)-based linear quadratic regulator (LQR) control that is well established in literature does not provide, in the controller project phase, the occurrence of exogenous inputs acting on the plants. Therefore, this study proposes sufficient conditions for the robust synthesis of a mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ control of the LQR problem. A new mixed polytopic and norm-bounded uncertainties representation for linear time-invariant uncertain systems allows to express different uncertainty models in one single LMI by using the S-procedure. As an additional objective, we propose the study of the robustness of controllers concerned with the sensibility of their coefficients, in such a way that the variations in controller gain matrix are described by norm-bounded uncertainties. A practical implementation in Quanser${}^{\circledR }$ active suspension evaluates the control projects through the rejection level of vibrations caused by track irregularities.

Keywords:

• Active suspension
• controllers robustness
• LMI-based LQR
• mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ control

Acknowledgements

This work was supported by Brazilian National Council for Scientific and Technological Development (CNPq)  under research fellowships 301227/2017-9, 309872/2018-9, and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

Disclosure statement

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

Additional information

Notes on contributors

Rodrigo da Ponte Caun

Rodrigo da Ponte Caun received the B.Sc. degree from School of Engineering, São Paulo State University (UNESP), Ilha Solteira, Brazil, in 2006, the M.Sc. degree from School of Electrical and Computer Engineering, University of Campinas (UNICAMP), Campinas, Brazil, in 2008, and the D.Sc. degree from School of Engineering, São Paulo State University (UNESP), Ilha Solteira, Brazil, in 2019, all in Electrical Engineering. In 2011, he joined the Department of Electronic Engineering, Federal University of Technology - Parana (UTFPR), Toledo, Brazil. He was vice Coordinator (2012-2012) and Coordinator (2013-2013) of the undergraduate degree in Electronic Engineering. Additionally, he was Director of Business and Community Affairs (2014-2016). In 2016, he joined the Department of Electrical Engineering, Federal University of Technology - Parana (UTFPR), Apucarana, Brazil, where he is currently an Adjunct Professor. His interests include robust and quadratic linear control based on linear matrix inequalities.

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 FEIS-UNESP in 1989.

Marcelo Carvalho Minhoto Teixeira

Marcelo Carvalho Minhoto Teixeira holds a degree from the Escola de Engenharia de Lins (EEL), Brazil, completed in 1979. He has a Master’s degree from Universidade Federal do Rio de Janeiro (UFRJ–COPPE), Brazil, completed in 1982. He concluded his Doctorate at the Pontifı́cia Universidade Católica do Rio de Janeiro (PUC/RJ),Brazil, in 1989, in Electrical Engineering. In 1982, he joined the Department of Electrical Engineering, FEIS, UNESP - School of Engineering, São Paulo State University, Ilha Solteira, Brazil, where he currently holds a Professorship. In 1996 and 1997, he was Visiting Professor at the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, United States. He was Coordinator (2005–2007) and vice Coordinator (2000–2005 and 2007–2010) of the Postgraduate Program in Electrical Engineering at FEIS-UNESP. He has was a member (2009–2012) of the Brazilian Evaluation Committee of Postgraduate Courses, IV Engineering CAPES. Additionally, he was the Coordinator (1991–1993) and vice Coordinator (1989–1991 and from 1993–1995) of the Undergraduate Program in Electrical Engineering at FEIS-UNESP. His interests include control theory and applications, education in control, neural networks, variable-structure systems, linear matrix inequality, fuzzy systems, and switched systems.