Robust Model Predictive Control of Linear Time Invariant System with Disturbances

Abstract

In recent few years authors investigate methods for establishing and solving stability of model predictive control (MPC) with linear time-invariant systems. In this paper a modification of a predictive control scheme to ensure robust feasibility and stability for a case of discrete time linear systems with disturbances is submitted. A new cost function, which penalizes a deviation of state trajectories from a robust invariant set and deviation from some ideal control law is presented. The cost allows formulation of robustly stable MPC that can be solved via min-max optimization using a single linear program. The theory of robust invariant sets is investigated and applied. Further an algorithm for finding robust invariant and feasible sets, that are used in the new linear cost function, is presented. These suggested robust invariant sets have suitable properties and are less computational intensive. A min-max formulation of MPC than involves minimization of the maximum of disturbance influence, which range over a given polytope of possible disturbance realizations, is presented. In general, the solution of this problem is computationally demanding with large number of unknown parameters. The appropriate use of the performance criteria with respect to the receding horizon strategy, low complexity robust invariant sets and feedback policy with consideration of only the extreme realizations of disturbance signals can decrease this number of parameters. An efficient solution of the robust min-max optimization defined for mixed 1/∞ norm is given.