Robust adaptive control in Hilbert Space

Abstract

Though great advances have been reported in the field of adaptive control in the past decade, some precise a priori structural information of the plant (at least the order) remains essential for most of the methods proposed. This is unsatisfactory in some applications because of the unmodeled dynamics, uncertain plant structure, and the noisy operating environment. In fact, for many high-performance control system designs, for example, control of large space structural systems, the distributed nature of the plant must be taken into account. These distributed parameter systems are frequently modelled by partial differential equations. Therefore, they must be analyzed in the appropriate infinite-dimensional state space. A particular approach, model reference adaptive control with command generator tracker concepts, adopts a set of assumptions that are not system dimension dependent. The method has been applied successfully to some finite-dimensional systems and shows promise for the infinite-dimensional state space generalization. This paper elevates the theory from the finite dimensions to the finite-dimensional Hilbert Space. The key obstacles for such a transition are noted. This then necessitates a modification of the adaptive control law. Under the modified scheme, the error signal is shown driven to a residue set asymptotically, the size of which depends on how close the nominal closed-loop plant is to positive realness. An added bonus is the robustness with respect to bounded state and output disturbances as well as model perturbation. All these properties are not true in general for the finite-dimensional control law. The example of damped beam equation is included to illustrate the techniques.