Optimality of robust disturbance-feedback strategies

Summary In this paper, robust disturbance-feedback strategies for finite time-horizon problems are studied. Linear discrete-time systems subject to linear control, state constraints, and quadratic objective functions are considered. In addition, persistent disturbances, which enter the system additively and are contained in a polytopic set, act on the system. The synthesis of robust strategies leads in the case of the traditional robust state-feedback and open-loop min-max strategies to, respectively, nonconvex problems or conservatism. However, robust disturbance-feedback problems can easily be reformulated as convex problems and solved by tractable linear matrix inequalities. Hence this approach bypasses the nonconvexity issue while maintaining the advantages of feedback strategies. As a key result, it is shown that both sources of conservatism attributed to this approach, namely, the relaxation method and the affine parametrization, can be removed at the expense of an increase in computational effort.