Stability and controllability of planar bimodal linear complementarity systems

The object of study of this paper is the class of hybrid systems consisting of so-called linear complementarity (LC) systems, that received a lot of attention recently and has strong connections to piecewise affine (PWA) systems. In addition to PWA systems, some of the linear or affine submodels of the LC systems can ‘live’ at lower-dimensional subspaces and re-initializations of the state variable at mode changes is possible. For LC systems we study the stability and controllability problem. Although these problems received for various classes of hybrid systems ample attention, necessary and sufficient conditions, which are explicit and easily verifiable, are hardly found in the literature. For LC systems with two modes and a state dimension of two such conditions are presented

On the controllability of bimodal piecewise linear systems

This paper studies controllability of bimodal systems that consist of two linear dynamics on each side of a given hyperplane. We show that the controllability properties of these systems can be inferred from those of linear systems for which the inputs are constrained in a certain way. Inspired by the earlier work on constrained controllability of linear systems, we derive necessary and sufficient conditions for a bimodal piecewise linear system to be controllable.Natl Sci Fdn; Univ Penn, Sch Engn & Appl Sci

On the Zeno behavior of linear complementarity systems

In this paper, the so-called Zeno phenomenon is addressed for linear complementarity systems which are interconnections of linear systems and complementarity conditions. We present some sufficient conditions for absence of Zeno behavior. It is also shown that the zero state, which is the most obvious candidate for being a Zeno state, cannot be a Zeno state in certain cases.

Existence and uniqueness of solutions for a class of piecewise linear dynamical systems

We consider the class of dynamical systems that arises when inputs and outputs of a linear system are connected pairwise by means of piecewise linear algebraic relations. It is not assumed that these relations define inputs in terms of outputs or vice versa; in particular, the relations need not be Lipschitzian. We obtain conditions for existence and uniqueness of solutions of such dynamical systems in the class of piecewise Bohl functions.

A time-stepping method for relay systems

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.

The nature of solutions to linear passive complementarity systems

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.

Stability and controllability of planar bimodal linear complementarity systems

The object of study of this paper is the class of hybrid systems consisting of so-called linear complementarity (LC) systems, that received a lot of attention recently and has strong connections to piecewise affine (PWA) systems. In addition to PWA systems, some of the linear or affine submodels of the LC systems can ‘live’ at lower-dimensional subspaces and re-initializations of the state variable at mode changes is possible. For LC systems we study the stability and controllability problem. Although these problems received for various classes of hybrid systems ample attention, necessary and sufficient conditions, which are explicit and easily verifiable, are hardly found in the literature. For LC systems with two modes and a state dimension of two such conditions are presented

On the dissipativity of uncontrollable systems

This paper deals with dissipativity of uncontrollable linear time-invariant systems with quadratic supply rates and storage functions. A definition of dissipativity appropriate for this class of systems is given. We present a necessary and sufficient condition for dissipativeness in the single input / single output case.