Assessment and Future Directions of Nonlinear Model Predictive ...

Robustness and Robust Design of MPC 2473. f(x, κ f (x)) ∈ Φ f , ∀x ∈ Φ f4. V f (f(x, κ f (x))) − V f (x) ≤−l(x, κ f (x)), ∀x ∈ Φ f5. α Vf (|x|) ≤ V f (x) ≤ β Vf (|x|), α Vf ,β Vf are K functions6. V f (·) is Lipschitz in Φ f with a Lipschitz constant L Vf7. X f := {x ∈ R n : V f (x) ≤ α v } is such that for all x ∈ Φ f ,f(x, κ f (x)) ∈ X f ,α v positive constantThen, the final theorem can be stated.Theorem 3. [24]Let X MPC (N) be the set of states of the system where thereexists a solution of the NRFHOCP. Then the closed loop system (14), (18) isISS in X MPC (N) if Assumption 6.1 is satisfied withγ ≤α − α vL Vf L N−1fThe above robust synthesis method ensures the feasibility of the solution througha wise choice of the constrains (ii) and(iii) intheNRFHOPC formulation.However, the solution can be extremely conservative or may not even exist, sothat less stringent approaches are advisable.7 Robust MPC Design with Min-Max ApproachesThe design of MPC algorithms with robust stability has been first placed in anH ∞ setting in [44] for linear unconstrained systems. Since then, many papershave considered the linear constrained and unconstrained case, see for example[41]. For nonlinear continuous time systems, H ∞ -MPC control algorithms havebeen proposed in [3], [30], [8], while discrete-time systems have been studied in[29], [15], [17], [33], [36]. In [29] the basic approach consists in solving a minmaxproblem where an H ∞ -type cost function is maximized with respect to theadmissible disturbance sequence, i.e. the ”nature”, and minimized with respectto future controls over the prediction horizon. The optimization can be solvedeither in open-loop or in closed-loop. The merits and drawbacks of these solutionsare discussed in the sequel.7.1 Open-Loop Min-Max MPCAssume again that the perturbed system is given byx(k +1)= ˜f(x(k),u(k),w(k)), k ≥ t, x(t) =¯x (19)where now ˜f(·, ·, ·) is a known Lipschitz function with Lipschitz constant L ˜fand˜f(0, 0, 0) = 0. The state and control variables must satisfy the constraints (4),while the disturbance w is assumed to fulfill the following hypothesis.Assumption 7.1. The disturbance w is contained in a compact set W and thereexists a K function γ(·) such that |w| ≤γ(|(x, u)|).