Assessment and Future Directions of Nonlinear Model Predictive ...

330 R. Gabasov, F.M. Kirillova, and N.M. DmitrukThe first condition gives admissibility of the control strategy while the secondone ensures its optimality.It can be shown that a control u(t|τ,y ∗ τ (·)), t ∈ [τ,t∗ ], is admissible if togetherwith u ∗ (t), t ∈ [t ∗ ,τ[, it steers the nominal systemat the instant t ∗ on the terminal setẋ 0 (t) =A(t)x 0 (t)+B(t)u(t), x 0 (t ∗ )=x 0X ∗ 0 (τ) ={x ∈ R n : g ∗i − ˆβ i (τ) ≤ h ′ ix ≤ g ∗ i − ˆα i (τ), i = 1,m}.Here the tightened terminal constraints are determined by the estimatesˆα i (τ) =maxh ′ ix, x ∈ ˆX ∗ (τ); ˆβi (τ) =minh ′ ix, x ∈ ˆX ∗ (τ); i = 1,m, (5)of the a posteriori distribution ˆX ∗ (τ) = ˆX ∗ (τ,y ∗ τ (·)) of terminal states x ∗(t ∗ )ofthe uncertain systemẋ ∗ (t) =A(t)x ∗ (t)+w(t), x ∗ (t ∗ )=Gz,(z,v) ∈ ˆΓ (τ).The (current) optimal open-loop control u 0 (t|τ,y ∗ τ (·)), t ∈ [τ,t∗ ], therefore, isa solution to the determined optimal control problemc ′ x(t ∗ ) → max, ẋ(t) =A(t)x(t)+B(t)u(t), x(τ) =x 0 (τ), (6)x(t ∗ ) ∈ X0 ∗(τ), u(t) ∈ U, t ∈ [τ,t∗ ].Extremal problems (5) are called the optimal observation problems accompanyingthe optimal control problem under uncertainty. Problem (6) is called theaccompanying optimal control problem.The optimal open-loop control is fed into the system until the next measurementis processed: u ∗ (t) =u 0 (t|τ,y ∗ τ(·)), t ∈ [τ + s(τ),τ + h + s(τ + h)[. Heres(τ)