Assessment and Future Directions of Nonlinear Model Predictive ...

12 E.F. Camacho and C. Bordonsthe origin starting from a state in the terminal region and therefore this finitehorizon cost function approximates the infinite-horizon one.These formulations and others with guaranteed stability were summarized inthe survey paper by Mayne et al. [27]. In this reference, the authors presentgeneral sufficient conditions to design a stabilizing constrained mpc and demonstratethat all the aforementioned formulations are particular cases of them.The key ingredients of the stabilizing mpc are a terminal set and a terminalcost. The terminal state denotes the state of the system predicted at the end ofthe prediction horizon. This terminal state is forced to reach a terminal set thatcontains the steady state. This state has an associated cost denoted as terminalcost, which is added to the cost function.It is assumed that the system is locally stabilizable by a control law u = h(x).This control law must satisfy the following conditions:• There is a region Ω such that for all x(t) ∈ Ω, thenh(x(t)) ∈ U (set ofadmissible control actions) and the state of the closed loop system at thenext sample time x(t +1)∈ Ω.• For all x(t) ∈ Ω, there exists a Lyapunov function V (x) such thatV (x(t)) − V (x(t +1))≥ x(t) T Rx(t)+h(x(t)) T Sh(x(t))If these conditions are verified, then considering Ω as terminal set and V (x)as terminal cost, the mpc controller (with equal values of prediction and controlhorizons) asymptotically stabilizes all initial states which are feasible. Therefore,if the initial state is such that the optimization problem has a solution, then thesystem is steered to the steady state asymptotically and satisfies the constraintsalong its evolution.The condition imposed on Ω ensures constraint fulfillment. Effectively, considerthat x(t) is a feasible state and u ∗ (t) the optimal solution; then a feasiblesolution can be obtained for x(t+1). This is the composition of the remaining tailof u ∗ (t) finished with the control action derived from the local control law h(x).Therefore, since no uncertainty is assumed, x(t+j|t+1) = x(t+j|t) for all j ≥ 1.Then the predicted evolution satisfies the constraints and x(t + P |t +1)∈ Ω,being P the prediction horizon. Thus, applying h(x(t + P |t + 1)), the systemremains in the terminal set Ω. Consequently,ifx(t) is feasible, then x(t +1)is feasible too. Since all feasible states are in X, then the system fulfills theconstraints.The second condition ensures that the optimal cost is a Lyapunov function.Hence, it is necessary for the asymptotic convergence of the system to the steadystate. Furthermore, the terminal cost is an upper bound of the optimal cost ofthe terminal state, in a similar way to the quasi-infinite horizon formulation ofmpc.The conditions previously presented are based on a state space representationof the system and full state information available at each sample time. However,most of the time the only available information is the measurement of the systemoutput. In this case the controller can be reformulated using the outputs