Assessment and Future Directions of Nonlinear Model Predictive ...

Optimal Online Control of Dynamical Systems Under Uncertainty 329feedback along a realized path (τ,y ∗ τ(·)), τ ∈ T h . Effort on treating admissiblebut not realized positions (τ,y τ (·)) is not spent.In the sequel we concentrate on a linear optimal control problem with parametricuncertainty. The feedback will be constructed on the base of optimalopen-loop control strategies. Note that closed-loop formulation as in [5, 6] forthe problem under consideration (on finite-time interval T ) results again in constructionof the functional u 0 (τ,y τ (·)) for all y τ (·) ∈ Y τ , τ ∈ T h .3 Optimal Online Control with Open-Loop StrategiesConsider a linear time-varying control system and a linear sensor:ẋ(t) =A(t)x(t)+B(t)u(t)+w(t), y(t) =C(t)x(t)+ξ(t) (4)with piecewise continuous matrix functions A(t) ∈ R n×n ,B(t) ∈ R n×r ,C(t) ∈R q×n , t ∈ T .Let the disturbance w(t), t ∈ T , and the initial state x(t ∗ )havetheformw(t) =M(t)v, t ∈ T ;x(t ∗ )=x 0 + Gz,where M(t) ∈ R p×l , t ∈ T , is a piecewise continuous matrix function, x 0 ∈ R n ,G ∈ R n×k ; v ∈ R l and z ∈ R k are unknown bounded parameters:v ∈ V = {v ∈ R l : w ∗ ≤ v ≤ w ∗ }; z ∈ Z = {z ∈ R k : d ∗ ≤ z ≤ d ∗ }.Drawing analogy with stochastic uncertainty, we call the sets Z, V aprioridistributions of the initial state and the disturbance parameters. A set Γ = Z×Vis called the aprioridistribution of the parameters γ =(z,v).Let U = {u ∈ R r : u ∗ ≤ u ≤ u ∗ }, Ξ = {ξ ∈ R q : ξ ∗ ≤ ξ ≤ ξ ∗ }, X ∗ = {x ∈R n : g ∗i ≤ h ′ i x ≤ g∗ i ,i= 1,m}, whereh i ∈ R n , g ∗i