Assessment and Future Directions of Nonlinear Model Predictive ...

352 A. Alessandri, M. Baglietto, and G. BattistelliIn the light of Lemma 2, provided that the initial continuous state x 0 is “farenough” from the origin, if one applies the minimum-distance criterion (4), theactual switching pattern π N cannot be confused with another switching patternπ that is distinguishable from π N according to Definition 1. Note thatthe boundedness of the sets W and V ensures the finiteness of all the scalarsδ max (π, π ′ ) . Furthermore, the satisfaction of the joint observability condition ensuresthat the minimum singular value σ {[I − P (π)]F (π N )} is strictly greaterthan 0. It is important to note that such a value represents a measure of theseparation between the linear subspaces ¯S(π) and ¯S(π N ) , i.e., the greater is theangle between such subspaces the greater is the value of σ {[I − P (π)]F (π N )} .Recalling the notion of observability given in Definition 2, Lemma 2 leads ina straightforward way to the following theorem.Theorem 1. Suppose that the sets W and V are bounded and that the noisefreesystem (3) is (α, ω)-mode observable in N +1 steps. If the initial continuousstate x 0 satisfies the condition‖x 0 ‖ >ρ x△=δ max (π, π)+δ max (π ′ ,π)maxπ, π ′ ∈P Nσ {[I − P (π ′ ,)]F (π)}r α,ω (π) ≠ r α,ω (π ′ )then r α,ω (ˆπ N )=r α,ω (π N ) .Thus, provided that the initial continuous state x 0 is “far enough” from theorigin, the minimum-distance criterion (4) leads to the exact identification ofthe discrete state in the interval [α, N − ω].3 A Receding-Horizon State Estimation SchemeIn this section, the previous results are applied to the development of a recedinghorizonscheme for the estimation of both the discrete and the continuous state.In Section 2, it has been shown that under suitable assumptions, given theobservations vector yt−N t , it is possible to obtain a “reliable” estimate of thediscrete state in the restricted interval [t−N +α, t−ω] . As a consequence, at anytime instant t = N,N +1,..., the following estimation scheme can be adopted:i) estimate the switching pattern in the restricted interval [t − N + α, t − ω]on the basis of the observations vector in the extended interval [t − N,t] ; ii)estimate the continuous state in the restricted interval [t − N + α, t − ω] byminimizing a certain quadratic cost involving the estimated discrete state.△Let us first consider step i). Towards this end, let us denote as γ t = r α,ω (π t )the switching pattern in the restricted interval [t − N + α, t − ω]. Furthermore,let us denote by ˆλ t−N,t ,...,ˆλ t,t , ˆπ t,t ,and ˆγ t,t the estimates (made at timet) of λ t−N ,...,λ t , π t ,and γ t , respectively. In order to take into account thepossibility of a time-varying a-priori knowledge on the discrete state, let us considerthe set P t of all the admissible switching patterns at time t , i.e., the set ofall the switching patterns in the observations window [t − N,t] consistent with