Robustly Stabilizing Model Predictive Control for Nonlinear Systems with Constraints

Simone L. Oliveira and Manfred Morari, Chemical Engineering, Caltech

In this talk a technique for the design of a robustly stabilizing model predictive controller (MPC) for continuous time-varying nonlinear constrained systems which is easily implementable and computationally inexpensive will be discussed. The model used in the calculations is a family of local first order approximations of the dynamics of the plant which makes it possible for the optimization problem to be posed as a convex problem. The technique consists of a model predictive controller which is implemented in a moving horizon fashion, i.e.,a sequence of optimal control moves for the model is calculated but only the first element of the sequence is actually implemented. Then the states of the plant are either measured (state feedback case) or estimated (output feedback case) and the optimal control calculations are repeated. This procedure is performed until one sampling interval before the end of the horizon of the first optimization problem. Then the next optimal control moves calculated are implemented for the period of a whole prediction horizon and a new sequence of optimal control problems initiates. This novel ``mixed'' control strategy is shown to produce less conservative results than a previous version of the algorithm where the measurements were taken only at the end of horizons. Stability of the closed-loop system is guaranteed through the use of a ``stability constraint'' in the formulation of the optimal control problem which puts the controller in the class of finite horizon model predictive controllers with terminal constraints. The stability constraint does not introduce non-convexities in the optimization problem as it will be seen later. Moreover, a somewhat conservative version of this constraint allows the optimization problem to be posed as a simple quadratic programming (QP) problem. In the state feedback case, the resulting control algorithm is asymptotically stable under certain assumptions on the dynamics of the plant and it is able to handle bounded time-varying disturbances, time-varying parameter deviations from their nominal values and measurement bias on the state variables. In the output feedback case, satisfaction of certain conditions on the nonlinear structure of the plant, parameters, disturbances, measurement bias and state estimation errors makes it possible for the combination of robust asymptotically stable nonlinear controller and observer to result in robust asymptotic stability of the closed-loop system. Since the nonlinear structure of the dynamics of the plant does not need to be exactly known (as long as it satisfies certain assumptions), the proposed control algorithm is robustly stabilizing with respect to both parametric and structural mismatches between the plant and the family of linear models used for computation of the optimal control law. Even in the output feedback case, modeling errors and unknown time-varying parameters are allowed, i.e., both the controller and the observer are robust.