On robust optimization and the optimal control of constrained linear systems with bounded state disturbances

The first part of this paper studies a specific class of uncertain quadratic and linear programs, where the uncertainty enters the constraints in an affine manner and the uncertainty set is a poly-tope. It is shown that one can convert the resulting semi-infinite optimization problem into a standard QP or LP with a finite number of decision variables and a finite number of constraints. This transformation is achieved in a computationally tractable way by solving as many LPs as there are constraints in the optimization problem without uncertainty. It is also shown that if the uncertainty set is given by upper and lower bounds only, then one need not solve any LPs in order to do this transformation; computing the 1-norms of the rows of the matrix by which the uncertainty enters the constraints is sufficient. The second part of the paper reviews and extends some definitions and results on input-to-state stability for nonlinear discrete-time systems. The third part of the paper shows how one can translate a class of robust finite-horizon optimal control problems (RFHOCPs) into the class of robust convex optimization problems that was studied in the first part of the paper. It is assumed that the system under consideration is linear, that there is a persistent, but bounded state disturbance that assumes values in a polytope and that there are mixed affine constraints on the input and state. By using the results from the previous sections, it is shown that one can set up a receding horizon controller (RHC) that is input-tostate stable (ISS) and guarantees robust constraint satisfaction for all time. The number of decision variables and constraints in the RFHOCP that define the RHC law increases linearly with the horizon length.