Robust and optimal stabilization of uncertain linear systems using LQR methods

Robust and Optimal stabilization of a class of a Linear Time Invariant (LTI) systems is discussed which exhibit linear time varying (LTV) behavior due to the presence of parametric variations and uncertainties. Linear quadratic methods offer global optimal control solutions for LTI systems. Such methods offer optimal solutions only locally for systems which become LTV due to parametric uncertainties. We propose that such an LTV system can be divided into two or more LTI systems in terms of the operating conditions ranging from nominal to most uncertain. In our proposed approach, two linear quadratic regulators would each be separately designed for nominal operating conditions and the uncertain conditions in a system. It is shown that the switchings between the two regulators depend upon the size of the uncertainty. A machine learning algorithm such as the support vector machine has been used to design a switching surface as a function of only the parametric uncertainties of the system. Extended high gain observers are used to estimate the parametric uncertainties needed for switching between the two regulators. Simulation results are included to demonstrate the performance of the proposed approach.