Model predictive control: for want of a local control Lyapunov function, all is not lost

We present stability results for unconstrained discrete-time nonlinear systems controlled using finite-horizon model predictive control (MPC) algorithms that do not require the terminal cost to be a local control Lyapunov function. The two key assumptions we make are that the value function is bounded by a K_∞ function of a state measure related to the distance of the state to the target set and that this measure is detectable from the stage cost. We show that these assumptions are sufficient to guarantee closed-loop asymptotic stability that is semiglobal and practical in the horizon length and robust to small perturbations. If the assumptions hold with linear (or locally linear) K_∞ functions, then the stability will be global (or semiglobal) for long enough horizon lengths. In the global case, we give an explicit formula for a sufficiently long horizon length. We relate the upper bound assumption to exponential and asymptotic controllability. Using terminal and stage costs that are controllable to zero with respect to a state measure, we can guarantee the required upper bound, but we also require that the state measure be detectable from the stage cost to ensure stability. While such costs and state measures may not be easy to construct in general, we explore a class of systems, called homogeneous systems, for which it is straightforward to choose them. In fact, we show for homogeneous systems that the associated K_∞ functions are linear, thereby guaranteeing global asymptotic stability. We discuss two examples found elsewhere in the MPC literature, including the discrete-time nonholonomic integrator, to demonstrate our methods. For these systems, we give a new result: They can be globally asymptotically stabilized by a finite-horizon MPC algorithm that has guaranteed robustness. We also show that stable linear systems with control constraints can be globally exponentially stabilized using finite-horizon MPC without requiring the terminal cost to be a global control Lyapunov function.