Assessment and Future Directions of Nonlinear Model Predictive ...

618 S.V. Raković and D.Q. Maynewhere x ∈ R n is the current state, u ∈ R n is the current control input and x +is the successor state and f : R n × R m × R p → R n ; the bounded disturbancew is known only to that extent that it belongs to the compact set W ⊂ R p thatcontains the origin in its interior. The system is subject to the following set ofhard constraints:(x, u, w) ∈ X × U × W (2)where X and U are compact (closed and bounded) sets respectively, each containingthe origin in its interior. Additionally it is required that the state trajectoriesavoid a predefined open set O, generally specified as the union of a finitenumber of open sets, introducing an additional state constraintx/∈ O, O ⋃O j , (3)j∈N qThe hard state constraints (2) and (3) can be converted into a single non–convexstate constraint:x ∈ X O X \ O (4)Let W ¯W N denote the class of admissible disturbance sequences w {w(i) |i ∈ N N−1 }.Letφ(i; x, π, w) denote the solution at time i of (2) when the controlpolicy is π, the disturbance sequence is w and the initial state is x at time 0; apolicy π is a sequence of control laws, i.e. π {µ 0 (·),µ 1 (·),...,µ N−1 (·)} whereµ i (·) is the control law (mapping state to control) at time i.Given a set X ⊂ R n and a control law µ : X → U where U ⊂ R m we define:X + F(X, µ, W), F(X, µ, W) {f(x, µ(x),w) | (x, w) ∈ X × W} (5)U(X, µ) {µ(x) | x ∈ X} (6)Robust model predictive control is defined, as usual, by specifying a finite-horizonrobust optimal control problem that is solved on-line. In this paper, the robustoptimal control problem is the determination of an appropriate tube, defined as asequence X {X 0 ,X 1 ,...,X N } of sets of states, and an associated control policyπ = {µ 0 (·),µ 1 (·),...,µ N−1 (·)} that minimize an appropriately chosen costfunction and satisfy the following set of constraints, for a given initial conditionx ∈ X O , that generalize corresponding constraints in [6]:x ∈ X 0 , (7)X i ⊆ X O , ∀i ∈ N N−1 (8)X N ⊆ X f ⊆ T ⊆ X O , (9)U(X i ,µ i ) ⊆ U, ∀i ∈ N N−1 (10)F(X i ,µ i , W) ⊆ X i+1 , ∀i ∈ N N−1 (11)where F(X i ,µ i , W) andU(X i ,µ i ) are defined, respectively, by (5) and (6), T ⊆X O and X f ⊆ T are a target set and its appropriate subset. It is assumed thatX O ≠ ∅ and moreover that the set T is assumed to be compact (i.e. closed andbounded) and convex set containing the origin in its interior. The relevance ofthe constraints (7)– (11) is shown by the following result [6, 11]: