Showing papers by "Wpmh Maurice Heemels published in 1999"

Linear complementarity systems

Abstract: We introduce a new class of dynamical systems called "linear complementarity systems." The time evolution of these systems consists of a series of continuous phases separated by "events" which cause a change in dynamics and possibly a jump in the state vector. The occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming. The framework we describe is suitable for certain situations in which both differential equations and inequalities play a role; for instance, in mechanics, electrical networks, piecewise linear systems, and dynamic optimization. We present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions.

Linear complementarity systems : a study in hybrid dynamics

Abstract: Technological innovation pushes towards the consideration of dynamical systems of a mixed continuous and discrete nature, which are called “hybrid systems” Hybrid systems arise, for instance, from the combination of an analog continuous-time process and a digital time-asynchronous controller Many consumer products (cars, micro-wave units, washing machines and so on) are controlled by digital embedded software, rendering the overall process a system with mixed dynamics Also many physical systems display hybrid behavior: the description of multi body dynamics depends crucially on the presence or absence of a contact, models of friction phenomena distinguish between slip and stick phases and electrical circuits contain switching elements like diodes that can be blocking (open circuit) or conducting (short circuit) From these examples it is obvious that a too general study of hybrid systems will lack decisive power: it will not result in detailed information on individual elements in the studied class Therefore, one has to consider a subclass of hybrid systems carrying a clear additional structure allowing analysis of its behavior (eg well-posedness, simulation methods, stability) and facilitating systematic controller synthesis However, the chosen subclass must also contain many interesting examples from an application point of view The class of (linear) complementarity systems satisfies both requirements and is the subject of the thesis Complementarity systems are described by differential equations, inequalities and logic expressions and form dynamical extensions of the linear complementarity problem (LCP) of mathematical programming The study of the complementarity class is motivated by a broad range of physically interesting systems that can be reformulated in terms of the complementarity formalism Examples include mechanical systems subject to unilateral constraints, Coulomb friction or one-sided springs; electrical networks with diodes; control systems with saturation or deadzones; piecewise linear and variable structure systems; relay systems; hydraulic processes with one-way valves; and sets of equations resulting from optimal control problems with state or control constraints Moreover, in Chapter 6 it is shown that the class of “projected dynamical systems” also fits into the complementarity framework To obtain a well-founded theory, it is essential to define a physically relevant solution concept and answer the classical questions of existence and uniqueness of solutions Because of the “jump-phenomena” in the system variables and the multimodal behavior, formulating a solution concept for linear complementarity systems (LCS) is non-trivial The solution trajectories are defined by combining a hybrid point of view and a distributional framework After the formal introduction of the solution concept, connections are established with the existing literature on mechanical systems and electrical circuits It is shown that the proposed solution concept is not an artificial one, but that it is in accordance with well-known rules specified for subclasses of complementarity systems

Projected dynamical systems in a complementarity formalism

Abstract: 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.

Well-posedness of linear complementarity systems

Abstract: In hybrid systems theory one often assumes (i) existence of solutions (ii) uniqueness of solutions and (iii) non-Zenoness (i.e. at most a finite number of events in a finite time interval). Sufficient conditions for these properties are rarely given. We present the state-of-the-art of the well-posedness results for the linear complementarity class of hybrid systems. Results on global existence of solutions and exclusion of accumulations of event times are given. Moreover, we present several examples of hybrid systems showing the interaction between the solution concept, non-Zeno assumptions and well-posedness.

The nature of solutions to linear passive complementarity systems

Abstract: Linear passive systems with complementarity conditions (as an application, one may consider linear passive networks with ideal diodes) are studied. For these systems contained in the linear complementarity class of hybrid systems, existence and uniqueness of solutions are established. Moreover, the nature of the solutions is characterized. In particular, it is shown that derivatives of Dirac impulses cannot occur and Dirac impulses and jumps in the state variable can only occur at t=0. These facts reduce the 'complexity' of the solution in a sense. Finally, we give an explicit characterization of the set of initial states from which no Dirac impulses or discontinuities in the state variable occur. This set of 'regular states' turns out to be invariant under the dynamics.

Applications of complementarity systems

Abstract: The class of complementarity systems has been analyzed in considerable detail as a special subclass of hybrid systems. The goal of this paper is to motivate the ongoing research by showing that many physically relevant systems fit in the framework of complementarity systems.

Hybrid systems : Modelling embedded controllers

Abstract: This paper indicates several possibilities to model embedded controllers as hybrid systems. It is shown that hybrid systems, the combinations of time-continuous and discrete-event systems, allow many powerful mathematical descriptions. Instances of hybrid systems include embedded controllers, supervisory systems and asynchronous processes. Although simulation programs are currently becoming more powerful for dealing with hybrid systems, analysis and synthesis are still open questions.