Assessment and Future Directions of Nonlinear Model Predictive ...

Minimum-Distance Receding-Horizon State Estimation 351Definition 2. System (3) is said to be (α, ω)-mode observable in N +1 steps if,for every couple π, π ′ ∈P N such that r α,ω (π) ≠ r α,ω (π ′ ) , π is distinguishablefrom π ′ (or, equivalently, π and π ′ are jointly observable).According to Definition 2, if system (3) is (α, ω)-mode observable, then differentswitching patterns in the interval [α, N − ω] generate different observationsvector in the interval [0,N] , provided that the initial continuous state is not null.As a consequence, the switching pattern r α,ω (π N ) can be determined uniquelyfrom the observations vector y0N . In fact, there could be more than one switchingpattern π such that y0N ∈ ¯S(π) ; however, they all correspond to the sameswitching pattern in the restricted interval [α, N − ω].With these observability results in mind, let us now focus on the noisy system(1). Clearly, if the noise vectors are not identically null, in general the noisyobservations vector y0N does not belong to the linear subspace ¯S(π N ).However,if the noise vectors are “small,” it is reasonable to think that y0N is “close” (insome sense) to such a set. This simple intuition leads us to adopt a minimumdistancecriterion for the estimation of the switching pattern. Towards this end,given a generic switching pattern π ∈P N , let us denote as d(y0 N ,π) the distancebetween the observations vector y0 N and the linear subspace ¯S(π). Clearly,d(y0 N ,π) can be obtained asd(y0 N ,π)=∥ ∥ [I − P (π)] y0N ∥where P (π) is the matrix of the orthogonal projection on ¯S(π), Then we shallconsider as an estimate of π N the switching pattern ˆπ N such thatˆπ N =arg min d(y0 N ,π) . (4)π∈P NIt is important to note that such a criterion can be always applied regardless ofthe form of the sets W and V to which the system and measurement noisesbelong. Moreover, an exact knowledge of the form of such sets is not required.Of course, it would be interesting to know whether, under suitable assumptions,the estimate ˆπ N coincides with the true switching pattern π N at least inthe restricted interval [α, N − ω] . With this respect, by defining the quantitiesδ max (π, π ′ ) =△ sup ‖ [I − P (π)] [H(π ′ )¯w + ¯v] ‖ , π,π ′ ∈P N ,¯w∈W N ; ¯v∈V N +1the following lemma can be stated.Lemma 2. Suppose that the sets W and V are bounded and consider a switchingpattern π ∈P N (with π ≠ π N ) such that π and π N are jointly observable.If the initial continuous state x 0 satisfies the condition‖x 0 ‖ > δ max(π N ,π N )+δ max (π, π N ), (5)σ {[I − P (π)]F (π N )}then we haved(y0 N ,π) >d(yN 0 ,π N ) .