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Control Theory / Théorie de Contrôle
(Kirsten Morris, Organizer)

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FRANCIS CLARKE, Université Lyon
Robustness of discontinuous feedbacks

A differential equation with discontinuous right-hand side would appear in general to be unduly sensitive to initial conditions, or to the presence of perturbation or error terms. For this reason, one might doubt that discontinuous stabilizing feedback would be effective in the presence of such terms. Yet a large class of such feedbacks constructed through ``proximal aiming'' has desirable features in this regard. We describe this precisely, as well as another approach which limits the discontinuities to a small tractable set.

GEIR E. DULLERUD, Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, Illinois  61801
Distributed control of inhomogeneous systems, with boundary conditions

This paper considers control design for distributed systems, where the controller is to adopt and preserve the distributed spatial structure of the system. Specifically systems that are inhomogeneous with respect to the spatial variable, and may have imposed boundary conditions, are considered. Operator theoretic tools for working with these systems are developed, and lead to convex conditions for analysis and synthesis with respect to the l2-induced norm. Our general motivation is to develop tools for dealing with physical systems that are inherently distributed and are to be controlled using a distributed strategy, as well as the increasing number of systems that are distributed because they are comprised of distributed interacting subsystems which have only local control available, but nonetheless require global performance objectives to be achieved.

BRUCE FRANCIS, Electrical and Computer Engineering, University of Toronto Toronto, Ontario
Applications of postmodern control theory

One can identify at least three eras in linear control design in this century: classical, modern, and postmodern. Postmodern refers to the multivariable design approach developed mostly in the 1980s starting from Zames' famous 1981 paper.

This talk will begin with a historical review from the viewpoint of a volume in preparation of 25 seminal control papers of this century. Then some case studies will be presented, including telerobotics, large flexible space structures, and web-handling systems.

LEON GLASS, McGill University
Control of complex rhythms in medicine: theory, experiment, and clinical applications

The human body in health and disease exhibits a host of complex rhythms. Most are normal, but others may indicate serious illness or even lead to sudden death. I will review efforts underway to characterize complex body rhythms, and to understand their underlying mechanisms. If the origin of the rhythms are understood (or perhaps even if they are not) it may be possible to control the rhythms to establish normal functioning or to prevent the onset of more dangerous rhythms. There is a well-established medical device industry that successfully controls abnormal cardiac and neural rhythms, but there is often minimal theoretical analysis of underlying mechanisms. However, various strategies to control cardiac and neural heart rhythms have been proposed that are based on nonlinear mathematical models of the abnormal rhythms.

KAZIFUMI ITO, North Carolina State University, North Carolina, USA
Receding horizon optimal control problem for infinite dimensional control systems

We consider the receding horizon optimal control problem with terminal cost chosen as control Liapunov function and analyze the asymptotic behavior of solutions for regulator as well as disturbance attenuation problems.

ANDREW LEWIS, Queens University
Affine connection control systems

The affine connection formalism is a powerful one in the study of control theory for a large class of mechanical systems, namely those which are Lagrangian with Lagrangian equal to kinetic minus potential energy. The tools of affine differential geometry enable an investigation of some basic questions surrounding control theory for such systems, and topics such as controllability, optimal control, and feedback equivalence will be discussed.

KIRSTEN MORRIS, Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario
Comparing finite-dimensional models for infinite-dimensional systems

A practical and important class of control problems concern systems modelled by partial differential equations. In general, it is necessary to use a numerical approximation to simulate the response of the system and to compute controllers for the system. A scheme that yields good results when used for simulation may be inappropriate for use in controller design. For instance, a scheme for which the simulation results converge quickly may have a corresponding controller sequence that converges very slowly or not at all. These issues will be discussed with respect to several common applications. Criteria appropriate for evaluation of an approximation scheme will be discussed.

HITAY OZBAY, Department of Electrical Engineering, Ohio State University Columbus, Ohio  43210, USA
Finite dimensional H-infinity controllers for a class of infinite dimensional systems

Over the last decade or so several different methods have been developed to solve the H-infinity optimal controller design problem for infinite dimensional systems. One of these methods is called the ``skew Toeplitz'' approach: it was shown that the singular values and vectors of an infinite dimensional ``Hankel+Toeplitz'' operator (which determines the optimal H-infinity controller) are computed from invertibility conditions of a skew Toeplitz operator. Under certain mild assumptions these invertibility conditions can be reduced to finitely many interpolation conditions. Thus the H-infinity controller can be derived by solving a finite set of linear equations. In this presentation we will give a simplified formula for H-infinity controllers for single input single output plants for which inner outer factorizations can be done numerically. The special structure of this controller expression will be exploited to derive finite dimensional (rational) approximations of the optimal controller. Direct approximation of the controller will be compared to indirect approximations. Numerical examples will be given from time delay systems.

DAVID RUSSELL, Virginia Polytechnic and State University, Virginia, USA
Forced waves in a supported nonlinear elastic beam

This paper is concerned with forced propagation of constant velocity waves in an elastic beam modelled by a nonlinear partial differential equation of type corresponding to the full von Karman model in plate theory. An elastic beam, supported on a flat, inelastic surface and subject to a vertical gravitational force, is subjected to a compressive force, tending to create buckling away from the surface. A heavy moving weight forces the buckled state to move at constant velocity c. Using analytical and computational studies we examine the properties of the resulting travelling wave as they depend on the various parameters present in the problem and on the propagation velocity c. These properties include the geometric form of the travelling wave and its stability aspects.

RON STERN, Department of Mathematics and Statistics, Concordia University Montreal, Quebec  H4B 1R6
Some issues in discontinuous feedback design

In problems of stabilization and optimal control, it is generally the case that effective continuous feedback controls do not exist. In recent years, the tools of nonsmooth analysis have been employed in order to produce discontinuous feedbacks. In this talk,the application of these techniques to a problem of state constrained optimal control will be described.