A generalised proportional-derivative type scheme with multiple saturating structure for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with bounded inputs

Abstract

A generalised proportional-derivative type scheme for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with input constraints is developed. Its generalised design permits the choice among multiple saturating structures. Suitable functions are involved for the appropriate shaping of every error correction term, and complementarily of their addition, to achieve the concerned type of convergence (in addition to the input saturation avoidance). Such a shaping is carried out through control parameters that act as exponential weights on the gained configuration and generalised velocity errors and on their joint action. Compared to previous finite-time approaches, such exponential weights are required to satisfy generalised comparative conditions, avoiding a fixed equivalence relation among any of them. This permits to give rise to a wider spectrum of finite-time convergent closed-loop trajectories and gives an optional type of (unconventional) exponential convergence (in addition to the conventional one). The resulting extended conditions on the exponential weights and the generalised saturating structure enlarge the controller potential for performance improvement or adjustment. The generalised design is supported through a more general analysis based on more general strict Lyapunov functions, stating more solid analytical bases for possible design extensions on the accomplishment of other control objectives. The study is further supported through experimental tests on a 3-degree-of-freedom robot manipulator.

Keywords:

• Continuous control
• uniform finite-time tracking
• Euler–Lagrange systems
• bounded inputs
• strict Lyapunov functions

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement

Data available within the article or its supplementary materials.

Additional information

Notes on contributors

Arturo Zavala-Río

Arturo Zavala-Río received his BS degree in Electronic Systems Engineering and MS degree in Control Engineering from the Instituto Tecnologico y de Estudios Superiores de Monterrey, Mexico, in 1989 and 1992, respectively, and his DEA and PhD degrees in Automatic Control from the Institut National Politechnique de Grenoble, France, in 1994 and 1997, respectively. He held professor-researcher positions at Universidad Autonoma de Queretaro (1999–2000), Mexico, and Universidad Autonoma de San Luis Potosi (2001), Mexico. He has been visiting researcher at Mechanical Engineering Laboratory (1998), Japan, and Universite de Technologie de Compiegne (2001–2002; 2013–2014), France. Since 2002, he is a full-time researcher at the Instituto Potosino de Investigacion Cientifica y Tecnologica, Mexico. He has been member of the National Researcher System, Conacyt, Mexico, since 1999. His research topics focus on the modelling, analysis, and control of nonlinear systems, with particular emphasis on the control of Euler-Lagrange systems.

Griselda I. Zamora-Gómez

Griselda I. Zamora-Gómez recieved the Master and PhD degrees from Instituto Potosino deInvestigación Científica y Tecnológica A.C. (IPICyT), San Luis Potosi, Mexico, in 2015 and 2020 respectively, both of them in Control and Dynamical Systems. She is currently a Professor in the Engineering Department at Cuauhtémoc University, San Luis Potosi, Mexico. Her research interests involve dynamical and control system analysis, control design and programming for robotic systems.

Tonametl Sanchez

Tonametl Sanchez received his PhD in Electrical Engineering from the National and Autonomous University of Mexico (UNAM) in 2016 under the supervision of Prof. Jaime A. Moreno. He was a postdoctoral researcher at Institute of Engineering of UNAM, Mexico, from 2016 to 2017, and at Inria Lille - Nord Europe, France, from 2017 to 2019. Since 2019 he is associate researcher at IPICYT, Mexico. His research interests include Lyapunov function design methods, homogeneous control systems, discrete-time nonlinear control systems, and sliding mode control.

Fernando Reyes-Cortes

Fernando Reyes Cortes was born in Puebla, Mexico in 1962, and received his BSc and MSc degrees in electronics engineering in 1984 and 1989 from Benemérita Universidad Autónoma de Puebla (BUAP) and INAOE, respectively. He received the PhD in electronics at CICESE Center Research, Mexico in 1997. Since 1980, Dr. Reyes has been working at BUAP. His research interests are in the fields on control of robot manipulators with special emphasis on practical applications. Dr. Reyes has published six books and many scientific papers in national and international conferences and journals. Research interests are control of robot manipulators (regulation and tracking), visual servoing, analysis and control of dynamical systems, parameter identification, design and built of robots' prototypes and mechatronics systems.