Assessment and Future Directions of Nonlinear Model Predictive ...

Minimum-Distance Receding-Horizon State Estimation 353the a-priori knowledge of the evolution of the discrete state. Furthermore, let usdenote by G t the set of admissible switching patterns in the restricted interval[t − N + α, t − ω] . For the sake of simplicity, let us suppose that such a-prioriknowledge does not diminish with time, i.e., P t+1 ⊆P t for t = N,N +1,...,or,less restrictively, P t ⊆P N . In accordance with the minimum-distance criterionproposed in Section 2, at every time instant t = N,N +1,..., we shall addressthe minimization of the loss functiond(yt−N t , ˆπ t,t )= ∥ [I − P (ˆπt,t )] yt−Nt ∥ . (6)Let us now consider step ii). At any time t = N,N +1,..., the objective is tofind estimates of the continuous state vectors x t−N+α ,...,x t−ω on the basis ofthe measures collected in an observations window [t−N +α, t−ω],ofa“prediction”¯x t−N+α , and of the estimate ˆγ t,t of the switching pattern γ t obtained instep i). Let us denote by ˆx t−N+α,t ,...,ˆx t−ω,t the estimates (to be made at timet) of x t−N+α ,...,x t−ω , respectively. We assume that the prediction ¯x t−N+α isdetermined via the noise-free state equation by the estimates ˆx t−N+α−1,t−1 andˆλ t−N+α−1,t−1 ,thatis,¯x t−N+α = A(ˆλ t−N+α−1,t−1 ) ˆx t−N+α−1,t−1 , t = N +1,N +2,... . (7)The vector ¯x α denotes an a-priori prediction of x α .A notable simplification of the estimation scheme can be obtained by determiningthe estimates ˆx t−N+α+1,t ,...,ˆx t−ω,t from the first estimate ˆx t−N+α,tvia the noise-free state equation, that is,ˆx i+1,t = A(ˆλ i,t ) ˆx i,t , i = t − N + α,...,t− ω − 1. (8)By applying (8), it follows that, at time t, only the estimate ˆx t−N+α,t has to be determined,whereas the estimates ˆx t−N+α+1,t ,...,ˆx t−ω,t can be computed via (8).As we have assumed the statistics of the disturbances and of the initial continuousstate to be unknown, a natural criterion to derive the estimator consistsin resorting to a least-squares approach. Towards this end, following the lines of[1, 3], at any time instant t = N,N +1,... we shall address the minimizationof the following quadratic cost function:J (ˆx t−N+α,t , ¯x t−N+α ,y t−ωt−N+α , )ˆγ t,t = µ ‖ ˆxt−N+α,t − ¯x t−N+α ‖ 2+t−ω∑i=t−N+α∥ y i − C(ˆλ i,t ) ˆx i,t∥ ∥∥2where µ is a non-negative scalar by which we express our belief in the prediction¯x t−N+α as compared with the observation model. It is worth noting that µcould be replaced with suitable weight matrices, without involving additionalconceptual difficulties in the reasoning reported later on. Note that, by applying(8), cost (9) can be written in the equivalent formJ (ˆx t−N+α,t , ¯x t−N+α ,y t−ωt−N+α , )ˆγ t,t = µ ‖ ˆxt−N+α,t − ¯x t−N+α ‖ 2+ ∥ ∥ yt−ωt−N+α − F (ˆγ t,t ) ˆx t−N+α,t∥ ∥2. (10)(9)