Assessment and Future Directions of Nonlinear Model Predictive ...

242 L. Magni and R. Scattoliniu = κ MPC (x) (8)In order to guarantee the stability of the origin of the closed-loop system (3),(8), many different choices of the stabilizing control law κ f (·), of the terminal setX f and of the terminal cost function V f have been proposed in the literature, see[37], [40], [2], [6], [28], [32], [7], [16], [14], [20]. Irrespective of the specific algorithmapplied, a general result can be stated under the following assumptions whichwill always be considered in the sequel.Assumption 3.1. l(x, u) is Lipschitz with Lipschitz constant L l and is such thatα l (|x|) ≤ l(x, u) ≤ β l (|(x, u)|) where α l and β l are K functions.Assumption 3.2. Let κ f (·), V f (·), X f be such that1. X f ⊆ X, X f closed, 0 ∈ X f2. κ f (x) ∈ U, ∀x ∈ X f3. κ f (x) is Lipschitz in X f with Lipschitz constant L κf4. f(x, κ f (x)) ∈ X f , ∀x ∈ X f5. α Vf (|x|) ≤ V f (x) ≤ β Vf (|x|) ,α Vf and β Vf K functions6. V f (f(x, κ f (x))) − V f (x) ≤−l(x, κ f (x)), ∀x ∈ X f7. V f is Lipschitz in X f with Lipschitz constant L VfTheorem 1. Let X MPC (N) be the set of the states such that a feasible solutionfor the FHOCP exists. Given an auxiliary control law κ f , a terminalset X f , a terminal penalty V f and a cost l(·, ·) satisfying Assumptions3.1, 3.2, the origin is an asymptotically stable equilibrium point for the closedloopsystem formed by (3) and (8) with output admissible set X MPC (N) andV (¯x, N) :=J(¯x, u o t,t+N−1 ,N) is an associated Lyapunov function. Moreover ifα l (|x|) =α l |x| p ,β Vf (|x|) =β Vf |x| p ,p>0, then the origin is an exponentiallystable equilibrium point in X MPC (N).Proof of Theorem 1. First note thatV (x, N) :=J(x, u o t,t+N−1 ,N) ≥ l(x, κMPC (x)) ≥ α l (|x|) (9)Moreover, letting u o t,t+N−1 be the solution of the FHOCP with horizon N attime t, in view of Assumption 3.2ũ t,t+N =[u o t,t+N−1,κ f (x(t + N))]is an admissible control sequence for the FHOCP with horizon N +1withJ(x, ũ t,t+N ,N +1)=V (x, N) − V f (x(t + N)) + V f (x(t + N +1))+l(x(t + N),κ f (x(t + N))) ≤ V (x, N)so thatV (x, N +1)≤ V (x, N), ∀x ∈ X MPC (N) (10)with V (x, 0) = V f (x), ∀x ∈ X f .Then