Assessment and Future Directions of Nonlinear Model Predictive ...

Minimum-Distance Receding-Horizon State Estimation 355Let us now define the following quantitiesf min△= minγ∈G Nσ [F (γ)] ,¯f △ =max ‖F (γ) − F (γ ′ )‖ ,γ,γ ′ ∈G Na △ =maxλ∈L ‖A(λ)‖ ,f △ =maxγ∈G N‖F (γ)‖ ,△ρ w = sup ‖w‖,w∈Wā△= maxλ,λ ′ ∈L ‖A(λ) − A(λ′ )‖ .h △ =maxγ∈G N‖H(γ)‖ ,△ρ v =sup ‖v‖ ,v∈VNote that, for the sake of compactness, the dependence of f min , h , f ,and ¯fon the size N of the observations window and on the scalars α and ω has beenomitted.In order to show the convergence properties of the proposed estimator, thefollowing assumptions are needed.A1. W and V are bounded sets.A2. System (3) is (α, ω)-mode observable in N +1 steps.A3. For any γ ∈G N ,wehave rank{F (γ)} = n .Clearly, Assumption A1 ensures that ρ w < +∞ and ρ v < +∞ . As to AssumptionA3, it ensures that the considered system is observable in the restrictedwindow [t − N + α, t − ω] with respect to the continuous state for any switchingpattern γ t and hence that f min > 0.We are now ready to state the following theorem.Theorem 2. Suppose that Assumptions A1, A2, and A3 are satisfied. Then the△norm of the estimation error e t−N+α = xt−N+α − ˆx ◦ t−N+α,t is bounded aboveas‖e t−N+α ‖≤ζ t−N+α , t = N,N +1,... .The sequence {ζ t } is defined recursively asζ α = d α ,ζ t = cζ t−1 + d, t= α +1,α+2,... . (11)whereµac =µ + fmin2 ,{1( )(µd =ā + f ¯f)µ + fmin2 a α ρ x + aα − 1a − 1 ρ w + µρ w(+f h √ N − α − ωρ w + √ ) }N − α − ω +1ρ v ,{1d α =µ + fmin2 µ ‖x α − ¯x α ‖ + f ¯f+f()a α ρ x + aα − 1a − 1 ρ w(h √ N − α − ωρ w + √ N − α − ω +1ρ v] } .