Assessment and Future Directions of Nonlinear Model Predictive ...

88 P. Mhaskar, N.H. El-Farra, and P.D. Christofideswhere x(t) ∈ IR n denotes the vector of continuous-time state variables, u σ (t) =[u 1 σ (t) ···um σ (t)]T ∈U σ ⊂ IR m denotes the vector of constrained manipulatedinputs taking values in a nonempty compact convex set U σ := {u σ ∈ IR m :‖u σ ‖≤u maxσ }, where‖·‖is the Euclidian norm, u maxσ > 0 is the magnitude ofthe constraints, σ :[0, ∞) →Kis the switching signal which is assumed to bea piecewise continuous (from the right) function of time, i.e., σ(t k ) = lim σ(t)t→t + kfor all k, implying that only a finite number of switches is allowed on any finiteinterval of time. p is the number of modes of the switched system, σ(t), whichtakes different values in the finite index set K, represents a discrete state thatindexes the vector field f(·), the matrix g(·), and the control input u(·), whichaltogether determine ẋ.Consider the nonlinear switched system of Eq.21, with a prescribed switchingsequence (including the switching times) defined by T k,in = {t k in1,t k in2,...} andT k,out = {t k out,t 1 k out,...}. Also, assume that for each mode of the switched2system, a Lyapunov–based predictive controller of the form of Eqs.16-20 has beendesigned and an estimate of the stability region generated. The control problemis formulated as the one of designing a Lyapunov-based predictive controller thatguides the closed–loop system trajectory in a way that the schedule described bythe switching times is followed and stability of the closed–loop system is achieved.The main idea (formalized in Theorem 2 below) is to design a Lyapunov–basedpredictive controller for each constituent mode in which the switched systemoperates, and incorporate constraints in the predictive controller design whichupon satisfaction ensure that the prescribed transitions between the modes occurin a way that guarantees stability of the switched closed–loop system.Theorem 2. Consider the constrained nonlinear system of Eq.10, the controlLyapunov functions V k , k =1, ··· ,p, and the stability region estimates Ω k , k =1, ··· ,p under continuous implementation of the bounded controller of Eqs.5-6with fixed ρ k > 0, k =1, ··· ,p.Let0