9813284 Drakunov The objective of this research project is to develop robust and computationally feasible control schemes which can successfully deal with Distributed Parameter Systems(DPS). The main difficulties to control DPS for example, a diffusion equation modeling the evolution of the temperature field in arc welding, or a Timoshenko equation modeling the vibrations of a flexible manipulator, are complexity and strong uncertainty. Although one can assume for simulation purposes that the geometry and properties of the heated (or cooled) materials are known, or that the manipulator payload may be ignored, in practice they vary in a wide range. An accepted model for a DPS is obtained by (1) selecting a suitable set of boundary conditions that describe a closely related problem; (2) solving the associated eigenvalue/eigenfuction problem; and (3) invoking the assumed-mode method. The result is an infinite system of uncoupled, ordinary differential equations for the so called system modes. In practice, one truncates the model and fulfills the control design using a finite number of equations. To validate such an approach in general, the effects of unmodeled dynamics, that is, control and observation spillover, need to be examined. On the other hand, there exists the sliding mode control methodology, which is well developed for finite dimensional systems but relatively very little has appeared in the literature for infinite dimensional systems. The appeal of this approach is its robutsness to matched disturbances and parameter variations. The design idea is based on the following procedure: in order to solve a control problem such as the stabilization of the weld width or heat penetration, the objective is formulated as a certain function of the system states which defines a desired manifold. A control is found such that the set in the state space where this relation is true forms a sliding manifold, that is, an integral manifold reachable in finite tim e. Once in the sliding mode, the dynamic behavior is prescribed by the choice of switching surfaces, and the system is inherently insensitive to that class of parameter variations and external disturbances which is implicit in the system input channels -- the so called matched uncertainty. The inherent insensitivity of a system with sliding modes to parameter variations and disturbances eliminates the need for exact modeling - a very desirable property that has quite remarkable theoretical and practical implications. In this research project, the generalization of this method is developed with mathematical rigor to further study and synthesize efficient control and sensing algorithms for DPS. ***
