Assessment and Future Directions of Nonlinear Model Predictive ...

624 S.V. Raković and D.Q. Maynez 0 N (x)⊕R ⊆ Z f ⊕R = X f ⊆ T. Similarly, U(X 0 i (·),µ0 i (·)) ⊆ U for all i ∈ N N−1because v 0 i (x) ∈ V yields that v0 i (x) ⊕ U ν ⊆ V ⊕ U ν ⊆ U for all i ∈ N N−1 .Finally, by Proposition 2 it follows that {Ay + Bµ 0 i (y; x)+w | (y, w) ∈ X 0 i (x) ×W} ⊆X 0 i+1 (x) for all i ∈ N N−1. Clearly, an analogous observation holds forany arbitrary couple (v,z) ∈V N (x) given any arbitrary x ∈X N .We consider the following implicit robust model predictive control law κ 0 N (·)yielded by the solution of P S N (x):κ 0 N (x) v 0 0(x)+ν(x − z 0 (x)) (33)We establish some relevant properties of the proposed controller κ 0 N (·) byexploitingthe results reported in [9].Proposition 3. (i) For all x ∈R, V 0 N (x) =0, z0 (x) =0, v 0 (x) ={0, 0,...,0}and κ 0 N (x) =ν(x). (ii) Let x ∈X N and let (v 0 (x),z 0 (x)) be defined by (28),then for all x + ∈ Ax + Bκ 0 N (x) ⊕ W there exists (v(x+ ),z(x + )) ∈V N (x + ) andV 0 N (x + ) ≤ V 0 N (x) − l(z 0 (x),v 0 0(x)). (34)The main stability result follows from Theorem 1 in [9] (definition of robustexponential stability of a set can be found in [9, 14]):Theorem 2. The set R is robustly exponentially stable for controlled uncertainsystem x + = Ax + Bκ 0 N (x)+w, w ∈ W. The region of attraction is X N .The proposed controller κ 0 N (·) results in a set sequence {X0 0 (x(i))}, where:X 0 0(x(i)) = z 0 (x(i)) ⊕R, i ∈ N (35)and z 0 (x(i)) → 0 exponentially as i →∞. The actual trajectory x(·) {x(i)},where x(i) is the solution of x + = Ax+Bκ 0 N (x)+w at time i ∈ N, correspondingto a particular realization of an infinite admissible disturbance sequence w(·) {w i }, satisfies x(i) ∈ X0 0 (x(i)), ∀i ∈ N. Proposition 3 implies that X0 0 (x(i)) ⊆X N , ∀i ∈ N and Theorem 2 implies that X0 0 (x(i)) →R(where R⊆T) asi →∞exponentially in the Hausdorff metric.4.2 Illustrative ExampleOur illustrative example is a double integrator:[ ] [ ]1 1 1x + = x + u + w (36)0 1 1with w ∈ W { w ∈ R 2 : |w| ∞ ≤ 0.2 }, x ∈ X {x ∈ R 2 ||x| ∞ ≤ 20, x 1 ≤1.85, x 2 ≤ 2}, u ∈ U {u ||u| ≤2} and T { x ∈ R 2 : |x| ∞ ≤ 3 }∩X ,where x i is the i th coordinate of a vector x. The cost function is defined by (26)with Q = 100I, R = 100; the terminal cost V f (x) is the value function (1/2)x ′ P f xfor the optimal unconstrained problem for the nominal system. The horizon isN = 8. The tube cross-section R is constructed by using methods of [15, 16].