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 (e.g. 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.