This paper establishes equivalences among five classes of hybrid systems: mixed logical dynamical (MLD) systems, linear complementarity (LC) systems, extended linear complementarity (ELC) systems, piecewise affine (PWA) systems, and maxmin-plus-scaling (MMPS) systems. Some of the equivalences are established under (rather mild) additional assumptions. These results are of paramount importance for transferring theoretical properties and tools from one class to another, with the consequence that for the study of a particular hybrid system that belongs to any of these classes, one can choose the most convenient hybrid modeling framework.

We establish equivalences among five classes of hybrid systems, that we have encountered in previous research: mixed logical dynamical systems, linear complementarity systems, extended linear complementarity systems, piecewise affine systems, and max-min-plus-scaling systems. These results are of paramount importance for transferring properties and tools from one class to another. 1

The following five classes of hybrid systems were recently proved to be equivalent: linear complementarity (LC) systems, extended linear complementarity (ELC) systems, mixed logical dynamical (MLD) systems, piecewise affine (PWA) systems, and max-minplus-scaling (MMPS) systems. Some of the equivalences were obtained under additional assumptions, such as boundedness of system variables. In this paper, for linear or hybrid plants in closed-loop with a model predictive control (MPC) controller based on a linear model and fulfilling linear constraints on input and state variables, we provide a simple and direct proof that the closed-loop system (cl-MPC) is a subclass of any of the former five classes of hybrid systems. This result opens the use of tools developed for hybrid systems (such as stability, robust stability, and safety analysis tools) to study closed-loop properties of MPC.

Projected dynamical systems have been introduced by Dupuis and Nagurney as dynamic extensions of variational inequalities. In the systems and control literature, complementarity systems have been studied as input/output dynamical systems whose inputs and outputs are connected through complementarity conditions. We show here that, under mild conditions, projected dynamical systems can be written as complementarity systems. Keywords: variational inequalities, complementarity, discontinuous dynamical systems, systems theory, optimization. 1 Introduction In this paper, we connect two classes of discontinuous dynamical systems. One is the class of projected dynamical systems introduced by Dupuis and Nagurney [4] and further developed by Nagurney and Zhang [14]. These systems are described by differential equations of the form x(t) = \Pi K (x(t); \GammaF (x(t))); (1) Dept. of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, an...

The aim of this course is to introduce some fundamental concepts from the area of hybrid systems, that is dynamical systems that involve the interaction of continuous (real valued) states and discrete (finite valued) states. Applications where these types of dynamics play a prominent role will be highlighted. We will introduce general methods for investigating properties such as existence of solutions, reachability and decidability of hybrid systems. The methods will be demonstrated on the motivating applications. Students who successfully complete the course should be able to appreciate the diversity of phenomena that arise in hybrid systems and how discrete “discrete ” entities and concepts such as automata, decidability and bisimulation can coexist with continuous entities and

This paper describes a comprehensive and systematic framework for building mixed continuous/discrete, i.e., hybrid physical system models. Hybrid models are a natural representation for embedded systems (physical systems with digital controllers) and for complex physical systems whose behavior is simplified by introducing discrete transitions to replace fast nonlinear dynamics. In this paper we focus on two classes of abstraction mechanisms, viz., time scale and parameter abstractions, discuss their impact on building hybrid models, and then derive the transition semantics required to ensure that the derived models are consistent with physical system principles. The transition semantics are incorporated into a formal model representation language, which is used to derive a computational architecture for hybrid systems based on hybrid automata. This architecture forms the basis for a variety of hybrid simulation, analysis, and verification algorithms. A complex example of a colliding rod system demonstrates the application of our modeling framework. The divergence of time and behavior analysis principles are applied to ensure that physical principles are not violated in the definition of the discrete transition model. The overall goal is to use this framework as a basis for developing systematic compositional modeling and analysis schemes for hybrid modeling of

Abstract — In this paper we focus on observability for hybrid systems in the mixed-logic dynamical form. We show that the maximal set of observable states, that is usually non convex and disconnected, can be represented as the union of finitely many polytopic regions. The argument, that is based on multi-parametric programming theory, is constructive and provides an algorithm for the computation of the regions. I.

An extension of the linear complementarity problem (LCP) of mathematical programming is the so-called rational complementarity problem (RCP). This problem occurs if complementarity conditions are imposed on input and output variables of linear dynamical input /state/output systems. The resulting dynamical systems are called linear complementarity systems. Since the RCP is crucial both in issues concerning existence and uniqueness of solutions to complementarity systems and in time simulation of complementarity systems, it is worthwhile to consider existence and uniqueness questions of solutions to the RCP. In this paper necessary and sufficient conditions are presented guaranteeing existence and uniqueness of solutions to the RCP in terms of corresponding LCPs. Using these results and proving that the corresponding LCPs have certain properties, we can show uniqueness and existence of solutions to linear mechanical systems with unilateral constraints, electrical networks with diodes, an...

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