Direct Model Reference Adaptive Control in Infinite-Dimensional Hilbert Space

Though great advances have been reported in adaptive control of single-input/single-output (SISO) systems and some multi-input/multi-output (MIMO) systems, some precise a priori structural information of the plant (at least the order) is needed for most of the methods proposed. This is unsatisfactory in some applications because of unmodeled dynamics and structure and noisy operating environment. In fact, in many high performance control system designs, as for example the control of large space structures, the distributed nature of the plant must be taken into account. These distributed parameter systems are frequently modeled by partial differential equations. Therefore, they must be analyzed in the appropriate infinite-dimensional state space. A particular approach based on model reference adaptive control (MRAC) with command generator tracker (CGT) concepts, adopts a set of assumptions that are not dependent on the system dimension. The method has been applied successfully to some finite-dimensional systems and shows promise for the infinite-dimensional state space generalizations as well. In this paper, the scheme is modified in order to make the transition of this theory from finite dimensions to the infinite-dimensional Hilbert Space, mathematically rigorous. Four main technical difficulties for such a transition are discussed: coercivity of the solution in the Lyapunov equation, application of the Invariance Principle in infinite dimensions, the strict positive realness condition, and the existence and the uniqueness of solutions. We investigate some of the ramifications and the remedies of these issues. Robustness with respect to state and output perturbations and parameter variations is also discussed to demonstrate the practicality of this controller operating under realistic conditions.