Minimax Adaptive Control for a Finite Set of Linear Systems

TL;DR

This paper introduces a minimax adaptive control method for finite sets of linear systems, ensuring robustness and optimality through a new game-theoretic approach and semi-definite programming.

Contribution

It presents a novel zero-sum dynamic game formulation for adaptive control with bounded l2-gain, including explicit bounds and a controller that transitions to H-infinity control.

Findings

Achieves bounded l2-gain from disturbances to errors.

Provides explicit upper bounds via semi-definite programming.

Controller converges to standard H-infinity control after parameter estimation.

Abstract

An adaptive controller with bounded l2-gain from disturbances to errors is derived for linear time-invariant systems with uncertain parameters restricted to a finite set. The gain bound refers to the closed loop system, including the non-linear learning procedure. As a result, robustness to unmodelled dynamics (possibly nonlinear and infinite-dimensional) follows from the small gain theorem. The approach is based on a new zero-sum dynamic game formulation, which optimizes the trade-off between exploration and exploitation. An explicit upper bound on the optimal value function is stated in terms of semi-definite programming and a corresponding simple formula for an adaptive controller achieving the upper bound is given. Once the uncertain parameters have been sufficiently estimated, the controller behaves like standard H-infinity optimal control.