Assessment and Future Directions of Nonlinear Model Predictive ...

On Disturbance Attenuation of Nonlinear Moving Horizon Control 289Proof: By the first assumption, the triple (γ o ,Q o ,Y o ) renders (18) feasible forall k ≥ 0 and (19d) feasible for all k ≥ 1. Moreover, at each t k ≥ t 0 ,wecanalways find r c > 0 satisfying (19c). At time t 0 ,ifx e (t 0 ) is bounded, we can definer 0 := x e (t 0 ) T Q −1o x e (t 0 ) satisfying (19b) for some ɛ ≥ 0. The feasibility of (18)leads to (5) with the pair (x T e Q−1 o x e,γ o ), which implies that x(1) is boundedif the disturbance is bounded in the amplitude. By induction, we can concludethat there exist r c > 0andɛ ≥ 0 such that (r k ,γ o ,Q o ,Y o ) construct a feasiblesolution to (19) at each t k ≥ t 0 ,wherer k := x e (t k ) T Q −1o x e(t k ) ✷We now give the following moving horizon algorithm for the tracking problemconsidered, which is computationally tractable:Step 1. Initialization. Choose r c and (q 1 ,q 2 ).Step 2. At time t 0 .Getx e (t 0 ),u j0,max and Ω 0 .Takeɛ = 0 and solve (19) without(19d) to obtain (r 0 ,γ 0 ,Q 0 ,Y 0 ). If the problem is not feasible, increaseɛ>0. Set K 0 = Y 0 Q −10 , P 0 = Q −10 , p 0 = V 0 (x e (t 0 )) and go to Step 4.Step 3. At time t k >t 0 .Getx e (t k ),u jk,max and Ω k .Takeɛ = 0 and solve (19) toobtain (r k ,γ k ,Q k ,Y k ). If the problem is not feasible, increase ɛ>0. SetK k = Y k Q −1k, P k = Q −1kand prepare for the next time instant accordingto (8).Step 4. Compute the closed-loop control asu e (t) =K k x e (t), ∀t ∈ [t k ,t k+1 ). (21)Replace t k by t k+1 and continue with Step 3.3.2 Closed-Loop PropertiesThe above algorithm provides an (almost) optimal solution to the optimizationproblem (19) at each sampling time t k ≥ t 0 ,denotedby(r k ,γ k ,Q k ,Y k ). Theclosed-loop control is then given by (21) with K k = Y k Q −1k.Wefirstdiscussthesatisfaction of the control constraints (16). The result is obvious: if the followinginequality∣ eTj (K k x e (t)+u d (t)) ∣ ≤ uj,max , ∀t ∈ [t k ,t k+1 ),j=1, 2, ··· ,m 2 (22)is satisfied, then, the control constraints are respected. According to the algorithm,the on-line optimization procedure shapes first the state ellipsoid to meetthe control constraints. If it fails, the ellipsoid will be enlarged by some ɛ>0.After the successful optimization, the satisfaction of the control constraints canbe checked by (22). The conservatism involved in the ellipsoid evaluation of thecontrol constraints is to some extent reduced, since the control constraints arein general given in a polytopic form [CGW06]. Hence, we can state the followingresults according to the discussion in Section 2.