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

2010 ◽  
Vol 368 (1930) ◽  
pp. 4893-4914 ◽  
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.

scholarly journals Stability analysis and controller synthesis for hybrid dynamical systems

2010 ◽  
Vol 368 (1930) ◽  
pp. 4937-4960 ◽  
Author(s):  
W. P. M. H. Heemels ◽  
B. De Schutter ◽  
J. Lunze ◽  
M. Lazar
Keyword(s):  
Dynamical Systems ◽  
Mathematical Models ◽  
Hybrid Systems ◽  
Discrete Event ◽  
Research Area ◽  
Hybrid Dynamical Systems ◽  
Technological Systems ◽  
Analysis And Synthesis ◽  
Control Community ◽  
The One

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.

Direct Linearization of Continuous and Hybrid Dynamical Systems

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Julie J. Parish ◽  
John E. Hurtado ◽  
Andrew J. Sinclair
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Equations ◽  
Equilibrium Configuration ◽  
Equations Of Motion ◽  
Hybrid Dynamical Systems ◽  
Linearized Equations ◽  
Direct Linearization ◽  
Taylor Series Expansions ◽  
Nonlinear Equations Of Motion ◽  
And Control

Nonlinear equations of motion are often linearized, especially for stability analysis and control design applications. Traditionally, the full nonlinear equations are formed and then linearized about the desired equilibrium configuration using methods such as Taylor series expansions. However, it has been shown that the quadratic form of the Lagrangian function can be used to directly linearize the equations of motion for discrete dynamical systems. This procedure is extended to directly generate linearized equations of motion for both continuous and hybrid dynamical systems. The results presented require only velocity-level kinematics to form the Lagrangian and find equilibrium configuration(s) for the system. A set of selected partial derivatives of the Lagrangian are then computed and used to directly construct the linearized equations of motion about the equilibrium configuration of interest, without first generating the entire nonlinear equations of motion. Given an equilibrium configuration of interest, the directly constructed linearized equations of motion allow one to bypass first forming the full nonlinear governing equations for the system. Examples are presented to illustrate the method for both continuous and hybrid systems.

Hybrid Dynamical Systems: Robust Stability and Control

2006 ◽  
Author(s):  
Cai Chaohong ◽  
Goebel Rafal ◽  
G. Sanfelice Ricardo ◽  
R. Teel Andrew
Keyword(s):  
Dynamical Systems ◽  
Robust Stability ◽  
Hybrid Dynamical Systems ◽  
Stability And Control ◽  
And Control

scholarly journals ORTHOGONAL HYBRID-FUZZY CONTROLLERS

2017 ◽  
Vol 16 (1) ◽  
pp. 25
Author(s):  
Saša Nikolić ◽  
Dragan Antić ◽  
Staniša Perić ◽  
Nikola Danković ◽  
Miodrag Spasić ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Control Systems ◽  
Control Method ◽  
Main Idea ◽  
Servo Drive ◽  
Orthogonal Functions ◽  
Fuzzy Controllers ◽  
Takagi Sugeno ◽  
Servo Drive System

The main idea of this paper is to present a possibility of application of hybrid-fuzzy controllers in control systems theory. In this paper, we have described a new method оf using orthogonal functions in control of dynamical systems. These functions generate genarilzed quasi-orthogonal filter, which are used in the concluding phase of the fuzzy controllers. Proposed hybrid-fuzzy controllers of Takagi-Sugeno type has been applied to a DC servo drive system and performed experiments have verified efficiency and improvements of a new control method.

Chaos Theory and Complexity Theory

2013 ◽  
Author(s):  
Keith Warren
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Systems Theory ◽  
Complexity Theory ◽  
Chaos Theory ◽  
Mathematical Framework ◽  
Nonlinear Interactions ◽  
Simple Systems ◽  
Over Time

Chaos theory and complexity theory, collectively known as nonlinear dynamics or dynamical systems theory, provide a mathematical framework for thinking about change over time. Chaos theory seeks an understanding of simple systems that may change in a sudden, unexpected, or irregular way. Complexity theory focuses on complex systems involving numerous interacting parts, which often give rise to unexpected order. The framework that encompasses both theories is one of nonlinear interactions between variables that give rise to outcomes that are not easily predictable. This entry provides a nonmathematical introduction, discussion of current research, and references for further reading.

scholarly journals Walking droplets through the lens of dynamical systems

2020 ◽  
Vol 34 (34) ◽  
pp. 2030009
Author(s):  
Aminur Rahman ◽  
Denis Blackmore
Keyword(s):  
Nonlinear Dynamics ◽  
Quantum Mechanics ◽  
Dynamical Systems ◽  
Systems Theory ◽  
Systems Analysis ◽  
Research Field ◽  
Physical Models ◽  
The Past ◽  
Dynamical Systems Analysis ◽  
History Of

Over the past decade the study of fluidic droplets bouncing and skipping (or “walking”) on a vibrating fluid bath has gone from an interesting experiment to a vibrant research field. The field exhibits challenging fluids problems, potential connections with quantum mechanics, and complex nonlinear dynamics. We detail advancements in the field of walking droplets through the lens of Dynamical Systems Theory, and outline questions that can be answered using dynamical systems analysis. The paper begins by discussing the history of the fluidic experiments and their resemblance to quantum experiments. With this physics backdrop, we paint a portrait of the complex nonlinear dynamics present in physical models of various walking droplet systems. Naturally, these investigations lead to even more questions, and some unsolved problems that are bound to benefit from rigorous Dynamical Systems Analysis are outlined.

Impact oscillators

1994 ◽  
Vol 347 (1683) ◽  
pp. 347-351 ◽  
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Systems Theory ◽  
Multiple Solutions ◽  
Important Class ◽  
Full Understanding ◽  
Impact Oscillators ◽  
Practical Engineering ◽  
New Phenomena ◽  
And Control

Impact oscillators are an important class of non-smooth dynamical systems whose behaviour not only exhibits typical characteristics of smooth nonlinear systems, such as generic bifurcations, multiple solutions and chaos, but also displays new phenomena. Additionally, many physical situations of practical engineering relevance have limits incorporated in their dynamic response and therefore impacts cannot be avoided. Engineers must evaluate the dangers associated with these impacts to eliminate excessive loads, reduce wear, avoid fatigue and in some cases increase passenger comfort. A full understanding of the underlying mathematics increases our ability to account for and control impacts. A brief personal review is given of the literature relating to recent advances in dynamical systems theory pertaining to impacting systems and future areas of research are highlighted.

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