Adaptive parameter estimation of large-scale systems by reduced-order modeling

The operating environment of most large-scale or distributed parameter systems is rather poorly known. For example, the modal parameters of many proposed large flexible aerospace structures cannot be calculated with sufficient accuracy to be used in on-line controllers to meet the stringent requirements of accurate pointing and fidelity of shape. Consequently, some means of on-line adaptive parameter estimation may be necessary. Convergence of currently known adaptive parameter estimation algorithms, such as the Luders-Narendra adaptive observer, is based on exact knowledge of the total system dimension. However, this dimension may be either very large or, in the case of distributed parameter systems, infinite. This necessitates an adaptive algorithm based on a reduced-order model, i.e., a low order approximation of the actual system. Such model reduction can severely alter the convergence properties of the adaptive algorithm. Since almost all practical engineering systems can be only approximately modeled, the convergence analysis of reduced-order adaptive algorithms seems essential for the successful application of these algorithms. In this paper we analyze the convergence of the Luders-Narendra observer when based on a reduced-order model of a large-scale system. The state and parameter estimation errors are shown to be ultimately bounded by factors involving the reduced-order data and the spillover coefficients. The results of computer simulation of the observer used in a reduced-order context are presented, as well.