New Methods for Design of Full- and Reduced-Order Observers and Observer-Based Controllers for Systems with Slow and Fast Modes

摘要

This dissertation addresses the design of observer and observer-based controllers for singularly perturbed linear systems. To that end, we present an algorithm for the recursive solution of the singularly perturbed algebraic Sylvester equation. Due to the presence of a small singular perturbation parameter that indicates separation of the system variables into slow and fast, the corresponding algebraic Sylvester equation is numerically ill-conditioned. The observer driven controller design of singularly perturbed linear systems with the observer design done using the algebraic Sylvester equation is extremely ill-conditioned since the observer has to be much faster than the feedback system. The proposed method for the recursive reduced-order solution of the algebraic Sylvester equations removes ill-conditioning and iteratively obtains the solution in terms of four reduced-order numerically well-conditioned algebraic Sylvester equations corresponding to slow and fast variables. The convergence rate of the proposed algorithm is O(epsilon), where is a small positive singular perturbation parameter.;The new design technique for full-order Luenberger observers for systems with slow and fast modes is presented. The existing methods are able to design independent slow and fast observers with O(epsilon) accuracy only, where epsilon is a small positive singular perturbation parameter. In this dissertation, the design of independent slow and fast reduced-order observers was performed with the exact accuracy. The results obtained are extended to the design of corresponding observer driven controllers. The design allows complete time-scale separation for both the observer and controller through the complete and exact decomposition into slow and fast time scale problems. This method reduces both off-line and on-line computations. The effectiveness of the new methods is demonstrated through both theoretical and simulation results.;The results obtained for the full-order observer of singularly perturbed linear systems are extended to design of reduced-order observers (using both the Sylvester equation and Luenberger observer formulations) and the design of corresponding controllers for singularly perturbed systems. In such design additional computational advantages are achieved due to the use of the reduced order observers. Several cases of reduced-order observer designs are considered depending on the measured state space variables: only all slow variables are measured, only all fast variables are measured, some combinations of the slow and fast variables are measured.