Showing papers by "Wpmh Maurice Heemels published in 2000"

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.

On the equivalence of classes of hybrid systems : mixed logical dynamical and complementarity systems

Abstract: We establish the equivalence of five classes of hybrid dynamical systems: mixed logical dynamical systems, linear complementarity systems, extended linear complementarity systems, piecewise affine systems and max-min-plus-scaling systems.

A time-stepping method for relay systems

Abstract: In this paper we will analyze a time-stepping method for the numerical simulation of dynamical systems containing Coulomb friction or relay characteristics. Time-stepping techniques replace the original dynamical system by a sequence of algebraic problems, that have to be solved for each time-step. For relay systems the one-step problem can be reformulated as a linear complementarity problem for which a wide range of solution algorithms already exists. As the event times at which the relay switches are "overstepped," the consistency of the method in the sense of the convergence of a sequence of approximations to an actual solution of the relay system can be put into question. However, in this paper we show that the proposed method is consistent even in the case that the event times accumulate (Zeno behavior). By an example we will illustrate how the method deals with Zeno trajectories.

Well-posedness of a class of linear networks with ideal diodes

Abstract: We consider electrical networks containing linear elements, independent voltage/current sources and ideal diodes. As a test of model validity, we have shown the well-posedness (in the sense of existence and uniquness of solutions) of such network models under a condition on the zero structure at infinity of the underlying linear system. It is also shown that this condition is implied by passivity. As an additional result, the set of initials states for which the corresponding solution trajectory is impulse-free is explicitly characterized.

From Lipschitzian to non-Lipschitzian characteristics: continuity of behaviors

Abstract: Linear complementarity systems are used to model discontinuous dynamical systems such as networks with ideal diodes and mechanical systems with unilateral constraints. In these systems mode changes are modeled by a relation between nonnegative, complementarity variables. We consider approximating systems obtained by replacing this non-Lipschitzian relation with a Lipschitzian function and investigate the convergence of the solutions of the approximating system to those of the ideal system as the Lipschitzian characteristic approaches to the (non-Lipschitzian) complementarity relation. It is shown that this kind of convergence holds for linear passive complementarity systems for which solutions are known to exist and to be unique. Moreover, this result is extended to systems that can be made passive by pole shifting.

A time-stepping method for relay systems

Abstract: In this paper we will analyze a time-stepping method for the numerical simulation of dynamical systems containing Coulomb friction or relay characteristics. Time-stepping techniques replace the original dynamical system by a sequence of algebraic problems, that have to be solved for each time-step. For relay systems the one-step problem can be reformulated as a linear complementarity problem for which a wide range of solution algorithms already exists. As the event times at which the relay switches are \"overstepped,\" the consistency of the method in the sense of the convergence of a sequence of approximations to an actual solution of the relay system can be put into question. However, in this paper we show that the proposed method is consistent even in the case that the event times accumulate (Zeno behavior). By an example we will illustrate how the method deals with Zeno trajectories.

Dynamical analysis of linear passive networks with ideal diodes

Abstract: The use of time-stepping methods for the simulation of the transient behaviour of circuit models with ideal diodes is by now well established. Little study has been made so far of the question in what sense the approximated time functions converge to the true solution of the network model. To answer this, one of course needs to establish first what should be understood by the transient "true solution" of such a network of a mixed discrete and continuous nature. In this paper we develop a solution concept and we prove that under mild conditions solutions exist, are unique, and are continuous except for a (possible) jump at the initial time. In a companion paper we discuss the convergence of a time-stepping algorithm.

On the time-stepping methods for linear passive networks with ideal diodes

Abstract: Simulation of switching networks is a problem that has been studied extensively in circuit theory [1, 2, 5, 11, 12, 15, 18, 25]. Roughly speaking, there are two main approaches, namely event-tracking (see e.g. [1,15]) and time-stepping methods (see [2,11,12,18] for electrical networks and [14,16,17,22,24] for unilaterally constrained mechanical systems with friction phenomena). Having a hybrid systems point of view (see for instance [21]), event-tracking methods are based on the idea of solving corresponding DAEs of the current circuit topology (called `mode' in the hybrid systems terminology), monitoring possible changes of circuit topology (mode transition), and (if necessary) determining the exact time (event time) instant of the change of topology and the next topology. Time-stepping methods di er from this scheme by regarding the whole system as a collection of di erential equations with constraints and trying to approximate the solutions of these di erential equations with constraints. As a consequnce of this point of view, there is no need to locate exact event times. However, the convergence of the approximations in a suitable sense has to be guaranteed. Since the methods seem to work well in practice, the question of convergence is usually neglected in the literature. It is the objective of this paper to provide a rigorous basis for the use of time-stepping methods in the simulation of circuits with state events. In [7] (see also [3]) the meaning of a transient true solution to the dynamical network model with ideal diodes has already been established. Using techniques borrowed from the theory of linear complementarity systems (LCS) [8,9,13,19,20], existence and uniqueness of solutions have been proven under mild conditions. Moreover, several regularity properties have been shown from which this paper will bene t. The particular time-stepping method that we will study here is based on the well-known backward Euler scheme and has been described, for instance, in [2, 11, 12] for electrical networks. Similar methods have been used in a mechanical context in [14, 16, 17, 22, 24]. The advantage of the method is that it is straightforward to implement and many algorithms (e.g. Lemke's algorithm [4], Katzenelson's algorithm [10] and others [12]) are available to solve the one-step problems consisting of linear complementarity problems (LCPs). In [11] the use of a time-stepping method based on the backward Euler scheme (or higher order linear multistep integration methods [6] like the trapezoidal rule) has already been proposed for the class of linear complementarity systems, i.e., linear time-invariant dynamical systems coupled with ideal diode characteristics (complementarity conditions). By an example, it will be shown that the method is not suited for the general class of linear complementarity systems. This example indicates also, that although the method has proven itself in practice, one should not indiscriminately apply it to general discontinuous dynamical systems. Dept. of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, and Dept. of Econometrics, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands. On leave from Dept. of Electrical Eng., Istanbul Technical University, 80626 Maslak Istanbul, Turkey. E-mail: K.Camlibel@tue.nl yDept. of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, and Dept. of Economics, Tilburg University, E-mail: w.p.m.h.heemels@tue.nl Dept. of Econometrics, Tilburg University, P.O. Box 90153 5000 LE Tilburg, The Netherlands, E-mail: jms@kub.nl Convergence problems of time-stepping methods for mechanical systems subject to unilateral constraints or friction have been studied by Stewart [22,23]. He shows that for a broad class of nonlinear constrained mechanical systems there always exists a subsequence of approximating time functions that converge to a real solution of the mechanical model. However, the convergence of the complete sequence has not been shown in [22,23]. The conditions used in [22,23] do not cover electrical networks containing ideal diodes, which form the subject of this paper. Speci cally, we will show that for the class of discontinuous dynamical systems consisting of linear electrical passive circuits with ideal diodes the backward Euler time-stepping method is consistent. To be speci c, we prove that the whole sequence (and not only a subsequence) of the approximating time functions converge to the real transient solution of the network model, when the step size decreases to zero. Although the results are written down here for networks containing ideal diodes (internally controlled switches) only, externally controlled switches can easily be included without destroying the convergence proof. The results presented here form a justi cation of the backward Euler time-stepping scheme in the eld of switched electrical networks. Such a justi cation seems required considering the problems that might occur due to changing con gurations of the network, the possibility of Dirac impulses and the discontinuities of the system's variables.

On dynamics, complementarity and passivity: Electrical networks with ideal diodes

Abstract: In this paper we study linear passive electrical circuits mixed with ideal diodes and voltage/current sources within the framework of linear complementarity systems. Linear complementarity systems form a subclass of hybrid dynamical systems and as such questions about existence and uniqueness of solution trajectories are non-trivial and will be investigated here. The nature of the behaviour is analyzed and characterizations of the inconsistent states of the network are presented. Also explicit jump rules from these inconsistent states are given in various forms. Finally, these results lead to a generalization of the notion of passivity to linear complementarity systems.

Linear complementarity systems: modelling power and well-posedness

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 timeasynchronous 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 behaviour: 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).

Continuity of smooth approximations of complementarity systems

Abstract: In a series of recent papers [4, 6, 8, 9] discontinuous dynamical systems such as networks with ideal diodes, mechanical systems with inelastic stops and feedback systems with relays have been modeled by complementarity systems. In these systems mode changes are described by a relation between nonnegative, complementary variables as depicted in Fig. 1(a). Here we consider systems obtained by approximating this relation with a Lipschitzian characteristic as shown in Fig. 1(b) or Fig. 1(c) and investigate the convergence of the solutions of the approximating system to those of the ideal system as the Lipschitzian characteristic approaches to (non-Lipschitzian) complementarity relation. Our main result stated as Theorem 2.3 below shows that this kind of continuity of the behavior of the approximating systems holds for linear passive complementarity systems for which existence and uniqueness of solutions have been established in [2, 5]. Moreover, we give necessary and suÆcient conditions for a system to be made passive by shifting its poles. It is shown that the same continuity result holds also for systems passi able by pole shifting. Continuity of linear dynamical systems have been studied before in [3, 11]. The treatment in the present paper is close to the framework of [11] in the sense that we also understand continuity as the convergence of the trajectories of the approximating systems to the trajectories of the limit system. Replacing discontinuous characteristics by smooth ones is a common practice in the simulation of discontinuous dynamical systems, see [1, 7] for instance. The results on the continuity of smooth approximations derived in the present paper provides con dence in computations based on the smoothed versions of linear passive and passi able complementarity systems and allows to draw conclusions about the behavior of the smooth approximations by studying the behavior of the idealized system.