Adaptive Robust Optimal Control of Constrained Continuous-Time Linear Systems: A Functional Constraint Generation Approach

We study the adaptive robust optimal control (AROC) problem for linear systems as an extension of finite-dimensional adaptive robust optimization problems. Given a continuous-time linear system under uncertain disturbance inputs with robust state and control constraints, the AROC problem finds the optimal control law adaptive to disturbance trajectories, which achieves the lowest worst-case cost. Then, the functional constraint generation (FCG) algorithm is designed, which extends the well-known constraint generation approach to the infinite-dimensional problem. The FCG algorithm consists of: first, a master problem that finds the optimal control solution under a collection of disturbance trajectories selected from the uncertainty set, and second, a subproblem that finds the worst-case disturbance trajectory for a given control solution and adds it to the master problem. Considering each iteration in the FCG algorithm as an operator updating control solutions, we prove that this operator has a unique fixed point as the optimal solution of the AROC problem. Further, by the monotone convergence theory of operators, we prove that the FCG algorithm converges to the optimal solution of the AROC problem. This result establishes the consistency between the convergence properties of the constraint generation approach for the infinite-dimensional optimization problem and finite-dimensional counterpart.