Optimization and Systems Theory Seminar
Friday, November 22, 2002, 11.00-12.00, Room 3721, Lindstedtsvägen 25

Assistant Professor Maurice Heemels
Control Systems Group
Deptartment of Electrical Engineering
Eindhoven University of Technology
Eindhoven, The Netherlands
E-mail: m.heemels@tue.nl

Complementarity systems and other hybrid model classes - Well-posedness issues

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. 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).

Several hybrid modelling frameworks have been introduced recently ranging from the general hybrid automaton model to more dedicated descriptions like piecewise affine systems. In this presentation we consider a hybrid model class consisting of so-called linear complementarity systems, which are close to piecewise affine systems. Complementarity systems show up naturally in the physical multi-modal systems mentioned before, but also in situations where piecewise linear elements like saturations, dead zones or relays play a role or in sets of equations resulting from optimal control problems with state or control constraints like in closed-loop MPC systems.

The presentation starts by showing how the models use the complementarity to capture the non-smoothness in the applications. Complementarity is defined between pairs of variables and is given by nonnegativity conditions and a complementary zero structure of the variables like in the voltage/current-relation of an ideal diode and the constraint/Lagrange multiplier-description in optimization. The framework was used initially in continuous time, which required an in-depth study of the dynamical behavior (state jumps, Zenoness, splitting of trajectories, etc.) of these systems. As a consequence, the issue of well-posedness (existence and uniqueness of a trajectory giving an initial conditions) received a lot of attention and the main results will be discussed. In discrete-time the issue is far less complex.

To obtain discrete-time counterparts of complementarity systems, discretization / time-stepping methods have been studied for the purpose of sampled-data control and simulation. We studied the convergence of these non-smooth dynamical systems when the discretization parameter (typically the sample time) goes to zero. For the resulting discrete-time complementarity systems we considered the relationships to other hybrid model classes like piecewise affine systems as defined first by Sontag and mixed logic dynamical systems as introduced by Bemporad and Morari. As methods were found to transfer the models into other forms, the (MPC) control and verification tools as developed for e.g. MLD systems can be directly applied. Also ideas used for analyzing for instance stability for piecewise affine systems can be used now for complementarity systems. Some possibilities are presented. Finally, the lines of future work and corresponding projects are discussed that range from fundamental research to industrial application.

Calendar of seminars
Last update: November 14, 2002 by Anders Forsgren, anders.forsgren@math.kth.se.