Assessment and Future Directions of Nonlinear Model Predictive ...

Robustness and Robust Design of MPC 243V (x, N +1)≤ V (x, N) ≤ V f (x) ≤ β Vf (|x|) , ∀x ∈ X f (11)FinallyV (x, N) =l(x, κ MPC (x)) + J(f(x, κ MPC (x)),u o t+1,t+N−1 ,N − 1)≥ l(x, κ MPC (x)) + V (f(x, κ MPC (x)),N)≥ α l (|x|)+V (f(x, κ MPC (x)),N), ∀x ∈ X MPC (N) (12)Then, in view (9), (11) and (12) V (x, N) is a Lyapunov function and in view ofLemma 1 the asymptotic stability in X MPC (N) and the exponential stability inX f are proven. In order to prove exponential stability in X MPC (N), let B ρ bethe largest ball such that B ρ ∈ X f and ¯V be a constant such that V (x, N) ≤ ¯Vfor all x ∈ X MPC (N). Now define( ) ¯Vᾱ 2 =maxρ p ,β V f,then it is easy to see [27] thatV (x, N) ≤ ᾱ 2 |x| p , ∀x ∈ X MPC (N) (13)4 Robustness Problem and Uncertainty DescriptionLet the uncertain system be described byx(k +1)=f(x(k),u(k)) + g(x(k),u(k),w(k)), k ≥ t, x(t) =¯x (14)or equivalentlyx(k +1)= ˜f(x(k),u(k),w(k)), k ≥ t, x(t) =¯x (15)In (14), f(x, u) is the nominal part of the system, w ∈M W for some compactsubset W⊆R p is the disturbance and g(·, ·, ·) is the uncertain term assumed tobe Lipschitz with respect to all its arguments with Lipschitz constant L g .The perturbation term g(·, ·, ·) allows one to describe modeling errors, aging,or uncertainties and disturbances typical of any realistic problem. Usually, onlypartial information on g(·, ·, ·) is available, such as an upper bound on its absolutevalue |g(·, ·, ·)| .For the robustness analysis the concept of Input to State Stability (ISS)isapowerful tool.Definition 5 (Input-to-state stability). The systemx(k +1)=f(x(k),w(k)),k ≥ t, x(t) =¯x (16)with w ∈M W is said to be ISS in Ξ if there exists a KL function β, andaKfunction γ such that|x(k)| ≤β(|¯x| ,k)+γ (‖w‖) , ∀k ≥ t, ∀¯x ∈ Ξ