J.M. Schumacher

Complementarity modeling of hybrid systems

A complementarity framework is described for the modeling of certain classes of mixed continuous/discrete dynamical systems. The use of such a framework is well known for mechanical systems with inequality constraints, but we give a more general formulation which also applies, for instance, to switching control systems. The main theoretical results in the paper are concerned with uniqueness of smooth continuations; the solution of this problem requires the construction of a map from the continuous state to the discrete state. A crucial technical tool is the so-called linear complementarity problem from mathematical programming, and we introduce various generalizations of this problem.

On linear passive complementarity systems

We study the notion of passivity in the context of complementarity systems, which form a class of nonsmooth dynamical systems that is obtained from the coupling of a standard input/output system to complementarity conditions as used in mathematical programming. In terms of electrical circuits, the systems that we study may be viewed as passive networks with ideal diodes. Extending results from earlier work, we consider here complementarity systems with external inputs. It is shown that the assumption of passivity of the underlying input/output dynamical system plays an important role in establishing existence and uniqueness of solutions. We prove that solutions may contain delta functions but no higher-order impulses. Several characterizations are provided for the state jumps that may occur due to inconsistent initialization or to input discontinuities. Many of the results still hold when the assumption of passivity is replaced by the assumption of “passifiability by pole shifting”. The paper ends with some remarks on stability.

A Direct Approach to Compensator Design for Distributed Parameter Systems

We present a direct approach to finite-order compensator design for distributed parameter systems, i.e., one that is not based on reduced order modelling. Instead, we use a parametrization around an initial compensator which displays both controller order and closed-loop stability in a convenient way. The main result is an existence theorem which holds for a wide class of linear time-invariant systems (parabolic, delay, damped hyperbolic). The most important assumptions are: bounded inputs and outputs, finitely many unstable modes, completeness of eigenvectors. An example is included to illustrate the feasibility of our method for purposes of design.

Complementarity systems in optimization

Abstract.Complementarity systems consist of ordinary differential equations coupled to complementarity conditions. They form a class of nonsmooth dynamical systems that is of use in mechanical and electrical engineering as well as in optimization and in other fields. The paper illustrates how complementarity systems arise in mathematical programming by means of a number of examples of various nature. This is followed by a brief survey of the results that are available concerning existence, uniqueness, and generation of solutions. The emphasis in this paper is on linear complementarity systems.

Impulsive-smooth behavior in multimode systems part I: state-space and polynomial representations

Abstract A ‘switched’ or ‘multimode’ system is one that can switch between various modes of operation. We consider here switched systems in which the modes of operation are characterized as linear finite-dimensional systems, not necessarily all of the same McMillan degree. When a switch occurs from one of the modes to another of lower McMillan degree, the state space collapses and an impulse may result, followed by a smooth evolution under the new regime. This paper is concerned with the description of such impulsive-smooth behavior on a typical interval. We propose an algebraic framework, modeled on the class of impulsive-smooth distributions as defined by Hautus. Both state-space and polynomial representations are considered, and we discuss transformations between the two forms.

Finite-dimensional regulators for a class of infinite-dimensional systems

Abstract We show that the ‘direct approach’, developed by the author for stabilization of certain classes of distributed parameter systems, can be extended to cover regulation problems as well. An iterative design algorithm is presented, together with proof that the algorithm will converge after a finite number of steps. The procedure is illustrated with an example of a constant disturbance acting on a delay system.

Algebraic Necessary and Sufficient Conditions for the Controllability of Conewise Linear Systems

The problem of checking certain controllability properties of even very simple piecewise linear systems is known to be undecidable. This paper focuses on conewise linear systems, i.e., systems for which the state space is partitioned into conical regions and a linear dynamics is active on each of these regions. For this class of systems, we present algebraic necessary and sufficient conditions for controllability. We also show that the classical results of controllability of linear systems and input-constrained linear systems can be recovered from our main result. Our treatment employs tools both from geometric control theory and mathematical programming.

Asymptotic analysis of linear feedback Nash equilibria in nonzero-sum linear-quadratic differential games

In this paper, we discuss nonzero-sum linear-quadratic differential games. For this kind of games, the Nash equilibria for different kinds of information structures were first studied by Starr and Ho. Most of the literature on the topic of nonzero-sum linear-quadratic differential games is concerned with games of fixed, finite duration; i.e., games are studied over a finite time horizon tf. In this paper, we study the behavior of feedback Nash equilibria for tf→∞.In the case of memoryless perfect-state information, we study the so-called feedback Nash equilibrium. Contrary to the open-loop case, we note that the coupled Riccati equations for the feedback Nash equilibrium are inherently nonlinear. Therefore, we limit the dynamic analysis to the scalar case. For the special case that all parameters are scalar, a detailed dynamical analysis is given for the quadratic system of coupled Riccati equations. We show that the asymptotic behavior of the solutions of the Riccati equations depends strongly on the specified terminal values. Finally, we show that, although the feedback Nash equilibrium over any fixed finite horizon is generically unique, there can exist several different feedback Nash equilibria in stationary strategies for the infinite-horizon problem, even when we restrict our attention to Nash equilibria that are stable in the dynamical sense.

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.

Impulsive-smooth behavior in multimode systems: part II: minimality and equivalence

This is the second part of a two-part study of linear multimode systems. In the first part, it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant, and both first-order and polynomial representations were introduced that satisfy this requirement. Here we determine the conditions under which first-order representations are minimal. We also show how two minimal representations of the same behavior are related; this leads in particular to an appropriate state-space isomorphism theorem. The minimality conditions are given a dynamic interpretation.