Comments on ‘Less conservative conditions for robust LQR-state-derivative controller design: an LMI approach’ and new sufficient LMI conditions for invertibility of a convex combination of matrices

Abstract

This note is concerned with conditions on a set of non-singular matrices $A_i\in {\mathbb{R}}^{n\times n}$, $i=1,2,\dots ,r$, so that any convex combination of these matrices is also non-singular. The first part of the note points out that Theorem 2.3 in a previous paper [Beteto et al. (2021). Less conservative conditions for robust LQR-state derivative controller design: An LMI approach. International Journal of Systems Science] provides only necessary conditions, which are not sufficient in the general case. In the second part, some stability results based on Linear Matrix Inequalities (LMIs) for a class of fractional order systems are used to establish new sufficient conditions. Numerical examples are presented for illustration. The results suggest that the new LMI conditions may be less conservative compared to a test proposed in the literature on P-matrices, and also to a positive-definiteness test based on matrix cross-products.

Keywords:

• Linear quadratic regulator (LQR)
• linear matrix inequalities (LMIs)
• state derivative feedback (SDF)
• robust control
• full rank conditions of convex combinations of matrices

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

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. Furthermore, R. K. H. Galvão, M. C. M. Teixeira and E. Assunção acknowledge the support of Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil) under grants 303393/2018-1, 309872/2018-9 and 301227/2017-9 (Research Fellowships).

Notes on contributors

Roberto Kawakami Harrop Galvão

Roberto Kawakami Harrop Galvão received the B.Sc. degree in Electronic Engineering (with Summa cum Laude honours) and the M.Sc. and Ph.D. degrees in Systems and Control from Instituto Tecnológico de Aeronáutica (ITA), Brazil, in 1995, 1997 and 1999, respectively. Since 1998 he has been with the faculty of the Electronic Engineering Department of ITA, where he is currently a Professor of Systems and Control. In 2001 he spent a sabbatical leave at the Cybernetics Department of the University of Reading, UK. Dr. Galvão is a recipient of the Montenegro Award for Teaching Excellence at ITA. He has published more than 130 papers in peer-reviewed journals and has supervised or co-supervised more than 50 MSc dissertations and 20 theses. His research interests include model predictive control and robust control.

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, São Paulo State University (UNESP), School of Engineering, Ilha Solteira, SP, 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.

Tomasz Szulc

Tomasz Szulc received the M.Sc. and Ph.D. degrees from the Institute of Mathematics of Adam Mickiewicz University (Poznan, Poland) in 1975 and 1982, respectively. Since 1975 he has been working at the Institute of Mathematics. In 1977, the Institute became part of the Faculty of Mathematics and Physics, from which the Faculty of Mathematics and Computer Science of the Adam Mickiewicz University emerged in 1993, where he works currently. In 1992 he received a postdoctoral degree (habilitation) in the Faculty of Mathematics and Physics of the Adam Mickiewicz University. The areas of his scientific interests are the applications of linear algebra, matrix theory and numerical linear algebra. He collaborated and still collaborates with many world-renowned specialists in these fields. He is the author or co-author of nearly 50 publications.

Edvaldo Assunção

Edvaldo Assunção was born in Andradina-SP, Brazil, in 1965. He received the B.Sc. degree in Electrical Engineering from School of Engineering, São Paulo State University (UNESP), Ilha Solteira-SP, Brazil in 1989, the M.Sc. degree in Systems and Control from Instituto Tecnológico de Aeronáutica (ITA), Brazil, in 1991, and the D.Sc. degree in Electrical Engineering from Universidade Estadual de Campinas (UNICAMP), Brazil, in 2000. 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 in 1989.

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 from School of Engineering, São Paulo State University (UNESP), Ilha Solteira-SP, Brazil, in 2019, in Electrical Engineering. He is currently a PhD student in Electrical Engineering Program at FEIS-UNESP. 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.