Assessment and Future Directions of Nonlinear Model Predictive ...

Sampled-Data MPC for Nonlinear Time-Varying Systems 117at time t 0 ,agivenfunctionf :IR× IR n × IR m → IR n ,andasetU ⊂ IR m ofpossible control values.We assume this system to be asymptotically controllable on X 0 and that forall t ≥ 0 f(t, 0, 0) = 0. We further assume that the function f is continuous andlocally Lipschitz with respect to the second argument.The construction of the feedback law is accomplished by using a sampleddataMPC strategy. Consider a sequence of sampling instants π := {t i } i≥0 witha constant inter-sampling time δ>0 such that t i+1 = t i +δ for all i ≥ 0. Consideralso the control horizon and predictive horizon, T c and T p ,withT p ≥ T c >δ,andan auxiliary control law k aux :IR× IR n → IR m . The feedback control is obtainedby repeatedly solving online open-loop optimal control problems P(t i ,x ti ,T c ,T p )at each sampling instant t i ∈ π, every time using the current measure of the stateof the plant x ti .P(t, x t ,T c ,T p ): Minimizesubject to:∫t+T ptL(s, x(s),u(s))ds + W (t + T p ,x(t + T p )), (2)ẋ(s) =f(s, x(s),u(s)) a.e. s∈ [t, t + T p ], (3)x(t) =x t ,x(s) ∈ X for all s ∈ [t, t + T p ],u(s) ∈ U a.e. s∈ [t, t + T c ],u(s) =k aux (s, x(s)) a.e. s∈ [t + T c ,t+ T p ],x(t + T p ) ∈ S. (4)Note that in the interval [t + T c ,t+ T p ] the control value is selected from a singletonand therefore the optimization decisions are all carried out in the interval[t, t + T c ] with the expected benefits in the computational time.The notation adopted here is as follows. The variable t represents real timewhile we reserve s to denote the time variable used in the prediction model. Thevector x t denotes the actual state of the plant measured at time t. The process(x, u) is a pair trajectory/control obtained from the model of the system. Thetrajectory is sometimes denoted as s ↦→ x(s; t, x t ,u) when we want to makeexplicit the dependence on the initial time, initial state, and control function.The pair (¯x, ū) denotes our optimal solution to an open-loop optimal controlproblem. The process (x ∗ ,u ∗ ) is the closed-loop trajectory and control resultingfrom the MPC strategy. We call design parameters the variables present in theopen-loop optimal control problem that are not from the system model (i.e.variables we are able to choose); these comprise the control horizon T c ,theprediction horizon T p , the running cost and terminal costs functions L and W ,the auxiliary control law k aux , and the terminal constraint set S ⊂ IR n .