Assessment and Future Directions of Nonlinear Model Predictive ...

106 L. Grüne, D. Nešić, and J. Pannekusing a direct approach and full discretization that will give us one optimizationproblem per optimal control problem which can be solved using an SQP method.Therefore in Section 2 the problem, the necessary assumptions and our controlscheme will be presented. In Section 3 we review the theoretical backgroundresults about stability and inverse optimality from [14]. Having done this thenumericalimplementationwillbepresentedanddiscussedinSection4anditsperformance will be demonstrated by solving an example in Section 5. Finallyconclusions will be given in Section 6.2 Problem FormulationThe set of real numbers is denoted as R. A function γ : R ≥0 → R ≥0 is calledclass G if it is continuous, zero at zero and non-decreasing. It is of class K if itis continuous, zero at zero and strictly increasing. It is of class K ∞ if it is alsounbounded. It is of class L if it is strictly positive and it is decreasing to zero asits argument tends to infinity. A function β : R ≥0 × R ≥0 → R ≥0 is of class KL iffor every fixed t ≥ 0 the function β(·,t)isofclassK and for each fixed s>0thefunction β(s, ·) isofclassL. Given vectors ξ,x ∈ R n we often use the notation(ξ,x) :=(ξ T ,x T ) T and denote the norm by |·|.We consider a nonlinear feedback controlled plant modelẋ(t) =f(x(t),u(x(t))) (1)with vector field f : R n × U → R n and state x(t) ∈ R n ,whereu : R n → U ⊂ R mdenotes a known continuous–time static state feedback which (globally) asymptoticallystabilizes the system. We want to implement the closed loop systemusing a digital computer with sampling and zero order hold at the samplingtime instants t k = k · T , k ∈ N, T ∈ R >0 . Then for a feedback law u T (x) thesampled-data closed loop system becomesẋ(t) =f(x(t),u T (x(t k ))), t ∈ [t k ,t k+1 ). (2)Our goal is now to design u T (x) such that the corresponding sampled–datasolution of (2) reproduces the continuous–time solution x(t) of (1) as close aspossible. The solution of the system (1) at time t emanating from the initialstate x(0) = x 0 will be denoted by x(t, x 0 ). Also we will assume f(x, u(x)) tobe locally Lipschitz in x, hence a unique solution of the continuous–time closedloop system to exist for any x(0) = x 0 in a given compact set Γ ⊂ R n containingthe origin.Remark 1. The simplest approach to this problem is the emulation design inwhich one simply sets u T (x) :=u(x). This method can be used for this purposebut one can only prove practical stability of the sampled–data closed loop systemif the sampling time T is sufficiently small, see [8].In order to determine the desired sampled–data feedback u T we first search for apiecewise constant control function v whose corresponding solution approximates