Assessment and Future Directions of Nonlinear Model Predictive ...

620 S.V. Raković and D.Q. Mayneassumption that the couple (A, B) is controllable. With system (17) we associatethe corresponding nominal system:z + = Az + Bv (18)where z ∈ R n is the current state, v ∈ R m is the current control action and z +is the successor state of the nominal system. Let φ(i; x, π, w) denote the solutionat time i of (17) when the control policy is π {µ 0 (·),µ 1 (·),...,µ N−1 (·)}, thedisturbance sequence is w and the initial state is x at time 0. If the initial stateof nominal model is z at time 0 then ¯φ(k; z,v) denotes the solution to (18) attime instant k, given the control sequence v {v 0 ,v 1 ...v N−1 }.Definition 1. [1] A set Ω ⊂ R n is a robust positively invariant (RPI) setfor system x + = f(x, w) and constraint set (X, W) if Ω ⊆ X and f(x, w) ∈Ω, ∀w ∈ W, ∀x ∈ Ω.AsetΩ ⊂ R n is a positively invariant (PI) set for system x + = f(x) andconstraint set X if Ω ⊆ X and f(x) ∈ Ω, ∀x ∈ Ω.A set Ω ⊂ R n is a robust control invariant (RCI) set for system x + =f(x, u, w) and constraint set (X, U, W) if Ω ⊆ X and for every x ∈ Ω thereexists a u ∈ U such that f(x, u, w) ∈ Ω, ∀w ∈ W.AsetΩ ⊂ R n is a control invariant (CI) set for system x + = f(x, u) andconstraint set (X, U) if Ω ⊆ X and for every x ∈ Ω there exists a u ∈ U suchthat f(x, u) ∈ Ω.If the set Ω is a RCI set for system x + = f(x, u, w) and constraint set (X, U, W),then there exists a control law ν : Ω → U such that the set Ω is a RPI setfor system x + = f(x, ν(x),w) and constraint set (X ν , W) withX ν {x ∈X | ν(x) ∈ U}. Thecontrollawν(·) is any control law satisfying:ν(x) ∈U(x), U(x) {u ∈ U | f(x, u, w) ∈ Ω, ∀w ∈ W}, x∈ Ω (19)An interesting observation [7, 14] is recalled next:Proposition 2. Let Ω be a RPI set for system x + = Ax + Bν(x) +w andconstraint set (X ν , W), whereX ν {x ∈ X | ν(x) ∈ U}. Letalsox ∈ z ⊕Ω andu = v+ν(x−z). Then for all v ∈ R m , x + ∈ z + ⊕Ω where x + Ax+Bu+w, w ∈W and z + Az + Bv.Proposition 2 allows us to exploit a simple parameterization of the tube-policypair (X,π) as follows. The state tube X = {X 0 ,X 1 ,...,X N } is parametrized by{z i } and R as follows:X i z i ⊕R (20)where z i is the tube cross–section center at time i and R is a set representingthe tube cross–section. Thecontrollawsµ i (·) defining the control policy π ={µ 0 (·),µ 1 (·),...,µ N−1 (·)} are parametrized by {z i } and {v i } as follows:µ i (y) v i + ν(y − z i ), y ∈ X i , (21)for all i ∈ N N−1 ,wherev i is the feedforward component of the control law andν(·) is feedback component of the control law µ i (·). With appropriate constraints