Assessment and Future Directions of Nonlinear Model Predictive ...

118 F.A.C.C. Fontes, L. Magni, and É. GyurkovicsThe MPC algorithm performs according to a receding horizon strategy, asfollows.1. Measure the current state of the plant x ∗ (t i ).2. Compute the open-loop optimal control ū :[t i ,t i + T c ] → IR n solution toproblem P(t i ,x ∗ (t i ),T c ,T p ).3. Apply to the plant the control u ∗ (t) :=ū(t; t i ,x ∗ (t i )) in the interval [t i ,t i +δ)(the remaining control ū(t),t≥ t i + δ is discarded).4. Repeat the procedure from (1.) for the next sampling instant t i+1 (the indexi is incremented by one unit).The resultant control law u ∗ is a “sampling-feedback” control since duringeach sampling interval, the control u ∗ is dependent on the state x ∗ (t i ). Moreprecisely the resulting trajectory is given byx ∗ (t 0 )=x t0 , ẋ ∗ (t) =f(t, x ∗ (t),u ∗ (t)) t ≥ t 0 ,whereu ∗ (t) =k(t, x ∗ (⌊t⌋ π )) := ū(t; ⌊t⌋ π ,x ∗ (⌊t⌋ π )) t ≥ t 0 .and the function t ↦→ ⌊t⌋ π gives the last sampling instant before t, thatis⌊t⌋ π := max{t i ∈ π : t i ≤ t}.iSimilar sampled-data frameworks using continuous-time models and samplingthe state of the plant at discrete instants of time were adopted in [2, 6, 7, 8, 13]and are becoming the accepted framework for continuous-time MPC. It canbe shown that with this framework it is possible to address —and guaranteestability, and robustness, of the resultant closed-loop system — for a very largeclass of systems, possibly nonlinear, time-varying and nonholonomic.3 Nonholonomic Systems and Discontinuous FeedbackThere are many physical systems with interest in practice which can only be modelledappropriately as nonholonomic systems. Some examples are the wheeledvehicles, robot manipulators, and many other mechanical systems.A difficulty encountered in controlling this kind of systems is that any linearizationaround the origin is uncontrollable and therefore any linear controlmethods are useless to tackle them. But, perhaps the main challenging characteristicof the nonholonomic systems is that it is not possible to stabilize it if justtime-invariant continuous feedbacks are allowed [1]. However, if we allow discontinuousfeedbacks, it might not be clear what is the solution of the dynamicdifferential equation. (See [4, 8] for a further discussion of this issue).A solution concept that has been proved successful in dealing with stabilizationby discontinuous feedbacks for a general class of controllable systemsis the concept of “sampling-feedback” solution proposed in [5]. It can be seenthat sampled-data MPC framework described can be combined naturally with