Output mini-max control for polynomial systems: analysis and applications

Abstract

This paper presents a solution to a robust optimal regulation problem for a nonlinear polynomial system affected by parametric and matched uncertainties, which is based only on partial state information. The parameters describing the dynamics of the nonlinear polynomial plant depend on a vector of unknown parameters, which belongs to a finite parametric set, and the application of a certain control input is associated with the worst or least favourable value of the unknown parameter. A high-order sliding mode state reconstructor is designed for the nonlinear plant in such a way that the previously designed control can be applied for a system with incomplete information. Additionally, the matched uncertainty is also compensated by means of the same output-based regulator. The obtained algorithm is applied to control an uncertain nonlinear inductor circuit of the third order and a mechanical pendulum of the third order, successfully verifying the effectiveness of the developed approach.

Keywords:

• mini-max control
• nonlinear systems
• sliding mode observers
• electrical circuit

Acknowledgements

M. Jiménez-Lizárraga was supported by project PAICyT CE851-11 and by the National Council of Science and Technology of Mexico (CONACYT) under project 169734. M. Basin was supported by the National Council of Science and Technology of Mexico (CONACYT) under project 129081.

Additional information

Notes on contributors

Manuel Jiménez-Lizárraga

Manuel Jiménez-Lizárraga received the BS degree in electrical engineering from Instituto Tecnologico de Culiacán, México, the MS and Ph.D. degrees on Automatic Control from CINVESTAV-IPN México, in 1996, 2000 and 2006 respectively. He is currently with the Faculty of Physical and Mathematical Sciences of Autonomous University of Nuevo León, México. His research interests include uncertain differential games, robust and optimal control and sliding mode observers.

Michael V. Basin

Michael V. Basin received his Ph.D. degree in Physical and Mathematical Sciences with major in Automatic Control and System Analysis from the Moscow Aviation Institute in 1992. His work experience includes Senior Scientist position in the Institute of Control Science (Russian Academy of Sciences) in 1992–1996, Visiting Professor position in the University of Nevada at Reno in 1996–1997 and Full-Time Tenured Professor position in the Autonomous University of Nuevo Leon, Mexico, from 1998. Dr. Basin published a monograph, more than 100 research papers in international referred journals and more than 130 papers in Proceedings of the leading IEEE and IFAC conferences and symposiums. He is the author of the ‘New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems’, published by Springer. His works are cited more than 950 times. He is the Editor-in-Chief of Journal of The Franklin Institute, an Associate Editor of Automatica, International Journal of Systems Science, IET Control Theory and Applications, and other journals. Dr. Basin serves as a member of IEEE Control System Society Technical Committee on Intelligent Control, Program Committee member of IEEE Conference on Decision and Control 2008, IEEE Conferences on Control Applications 2009, 2012, IEEE Workshops on Variable Structure Systems 2010, 2012, 2014. Dr. Basin was awarded a title of Highly Cited Researcher by Thomson Reuters (International Science Institute), the publisher of Science Citation Index, in 2009; he is a Senior Member of the IEEE Control Systems Society and a regular member of the Mexican Academy of Sciences. His research interests include optimal filtering and control problems, stochastic systems, time-delay systems, identification, sliding mode control and variable structure systems.

Victoria Celeste Rodríguez Carreón

Victoria Celeste Rodríguez Carreón received her degree in mathematics in 2008, and at present is pursuing a Ph.D. in industrial physics engineering at the Faculty of Physics and Mathematics Science from the Autonomous University of Nuevo Leon, Mexico. Her research interests include differential games, stochastic systems and control problems.

Pablo Cesar Rodríguez Ramírez

Pablo Cesar Rodríguez Ramírez received his degree in mathematics at the Faculty of Physics and Mathematics Science from the Autonomous University of Nuevo Leon, Mexico, in 2008, is currently pursuing a Ph.D. in industrial physics engineering in the same unit. His research interests include optimal filtering and control problems, stochastic systems, identification and sliding mode control, under the tutelage of Dr. Michael V. Basin.