Minimax Adaptive Control for a Finite Set of Linear Systems

An adaptive controller is derived for linear time-invariant systems with uncertain parameters restricted to a finite set, such that the closed loop system including the non-linear learning procedure is stable and satisfies a pre-specified l2-gain bound from disturbance to error. As a result, robustness to unmodelled (linear and non-linear) dynamics follows from the small gain theorem. The approach is based on a dynamic zero-sum game formulation with quadratic cost. Explicit upper and lower bounds on the optimal value function are stated and a simple formula for an adaptive controller achieving the upper bound is given. The controller uses semi-definite programming for optimal trade-off between exploration and exploitation. Once the uncertain parameters have been sufficiently estimated, the controller behaves like standard H-infinity state feedback.