On the inherent robustness of optimal and suboptimal MPC

The stability of nominal model predictive control (MPC) is well characterized in the literature. Lyapunov theory forms a framework that unites both linear and nonlinear control problems, and it is amenable to a wide variety of new MPC formulations (see (Rawlings and Mayne, 2009, Ch. 2) and the references it contains). In all practical implementations, however, robustness to disturbances is a requirement. In some forms of MPC, robustness is not guaranteed; in Grimm, Messina, Tuna, and Teel (2004) it was shown that while linear quadratic MPC with polyhedral constraints is robust, there exist nominally stable implementations of nonlinear MPC that are unstable for arbitrarily small perturbations. Thus, it is vital to study the robustness of nonlinear MPC. One method to deal with disturbances is a robust MPC formulation. In this formulation, the control problem is formulated such that, for all possible disturbances in a disturbance set known a priori, all constraints are satisfied. Because the robustness properties are usually guaranteed only when disturbances lie within these bounds, conservative disturbance estimates are often used. In Limón Marruedo, Álamo, and Camacho (2002), a constrainttightening method is proposed to maintain recursive feasibility. In (Rawlings and Mayne, 2009, Ch. 3), a robust controller that optimizes over a set of control policies to recursively satisfy constraints is proposed. Min-max MPC is a robust control strategy that minimizes the controller’s cost under the worst-case disturbance. However, the authors in Yu, Reble, Chen, and Allgöwer (2014) noted that min-max MPC formulations are computationally expensive to solve and often result in poor performance. A different approach is to characterize the conditions under which nominal MPC is inherently robust. The authors in Grimm et al. (2004) demonstrated that continuous