Chattering-Free Simulation for Hybrid Dynamical Systems Semantics and Prototype Implementation

Wherever continuous and discrete dynamics interact, hybrid systems arise. This is especially the case in many technological systems in which logic decision-making and embedded control actions are combined with continuous physical processes. Also for many mechanical, biological, electrical and economical systems the use of hybrid models is essential to adequately describe their behaviour. To capture the evolution of these systems, mathematical models are needed that combine in one way or another the dynamics of the continuous parts of the system with the dynamics of the logic and discrete parts. These mathematical models come in all kinds of variations, but basically consist of some form of differential or difference equations on the one hand and automata or other discrete-event models on the other hand. The collection of analysis and synthesis techniques based on these models forms the research area of hybrid systems theory, which plays an important role in the multi-disciplinary design of many technological systems that surround us. This paper presents an overview from the perspective of the control community on modelling, analysis and control design for hybrid dynamical systems and surveys the major research lines in this appealing and lively research area.

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Incremental Graphical Asymptotic Stability for Hybrid Dynamical Systems

Feedback Stabilization of Controlled Dynamical Systems - Lecture Notes in Control and Information Sciences ◽

10.1007/978-3-319-51298-3_9 ◽

2017 ◽

pp. 231-262

Author(s):

Yuchun Li ◽

Ricardo G. Sanfelice

Keyword(s):

Dynamical Systems ◽

Asymptotic Stability ◽

Hybrid Dynamical Systems

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Theory of hybrid dynamical systems and its applications to biological and medical systems

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2010.0237 ◽

2010 ◽

Vol 368 (1930) ◽

pp. 4893-4914 ◽

Cited By ~ 65

Author(s):

Kazuyuki Aihara ◽

Hideyuki Suzuki

Keyword(s):

Nonlinear Dynamics ◽

Dynamical Systems ◽

Mathematical Modelling ◽

Systems Theory ◽

Control Systems ◽

Hybrid Systems ◽

Hybrid Dynamical Systems ◽

Medical Systems ◽

Theme Issue ◽

And Control

In this introductory article, we survey the contents of this Theme Issue. This Theme Issue deals with a fertile region of hybrid dynamical systems that are characterized by the coexistence of continuous and discrete dynamics. It is now well known that there exist many hybrid dynamical systems with discontinuities such as impact, switching, friction and sliding. The first aim of this Issue is to discuss recent developments in understanding nonlinear dynamics of hybrid dynamical systems in the two main theoretical fields of dynamical systems theory and control systems theory. A combined study of the hybrid systems dynamics in the two theoretical fields might contribute to a more comprehensive understanding of hybrid dynamical systems. In addition, mathematical modelling by hybrid dynamical systems is particularly important for understanding the nonlinear dynamics of biological and medical systems as they have many discontinuities such as threshold-triggered firing in neurons, on–off switching of gene expression by a transcription factor, division in cells and certain types of chronotherapy for prostate cancer. Hence, the second aim is to discuss recent applications of hybrid dynamical systems in biology and medicine. Thus, this Issue is not only general to serve as a survey of recent progress in hybrid systems theory but also specific to introduce interesting and stimulating applications of hybrid systems in biology and medicine. As the introduction to the topics in this Theme Issue, we provide a brief history of nonlinear dynamics and mathematical modelling, different mathematical models of hybrid dynamical systems, the relationship between dynamical systems theory and control systems theory, examples of complex behaviour in a simple neuron model and its variants, applications of hybrid dynamical systems in biology and medicine as a road map of articles in this Theme Issue and future directions of hybrid systems modelling.

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