Assessment and Future Directions of Nonlinear Model Predictive ...

354 A. Alessandri, M. Baglietto, and G. BattistelliSumming up, the following receding-horizon estimation procedure has to beapplied at any time instant t = N,N +1,....Procedure 11. Given the observations vector y t t−N , compute the optimal estimate ˆπ◦ t,t thatminimizes the distance measure (6), i.e.,ˆπ ◦ t,t=arg minˆπ t,t∈P td(y t t−N , ˆπ t,t) .2. Set ˆγ ◦ t,t = r α,ω (ˆπ ◦ t,t) .3. Given the optimal estimate ˆγ ◦ t,t , the observations vector y t−ωt−N+α ,andtheprediction ¯x t−N+α , compute the optimal estimateˆx ◦ t−N+α,t =arg min J (ˆx t−N+α,t , ¯x t−N+α ,y t−ωˆxt−N+α , ˆγ ◦ )t,tt−N +α,tthat minimizes cost (9) under the constraints (8).4. Given the optimal estimates ˆx ◦ t−N+α,t and ˆλ ◦ t−N+α,t , compute the prediction¯x t−N+α+1 as¯x t−N+α+1 = A(ˆλ ◦ t−N+α,t) ˆx ◦ t−N+α,t .The procedure is initialized at time t = N with an a-priori prediction ¯x α .It is important to note that the form of the set P t plays a central role inthe possibility of computing the minimum in step 1 in a reasonable time. Infact, if the cardinality of the set P t grows very rapidly with the size N ofthe observations window or with the number L of possible discrete states, sucha computation may become too time-demanding (this happens, for example,when the system can switch arbitrarily at every time step). Such issues can beavoided if the a-priori knowledge on the evolution of the discrete state leads toa considerable reduction of the number of admissible switching patterns. Thisis the case, for example, when the size N + 1 of the observations window issmaller than the minimum admissible number of steps between one switch andthe following one. In fact, under such an assumption, the cardinality of the setP t is L[(L − 1)N +1] (see[2]).As to step 3, since cost (9) depends quadratically on the estimate ˆx t−N+α,t ,by applying the first order optimality condition a closed-form expression can bederived for the optimal estimate ˆx ◦ t−N+α,t . More specifically, along the lines of[1], where non-switching linear systems were considered, the following propositioncan be easily proved.Proposition 1. Suppose that µ>0 or that rank { F (ˆγ ◦ t,t) } = n . Then cost (9)has a unique minimum point given byˆx ◦ t−N+α,t = [ µI + F (ˆγ ◦ t,t) ⊤ F (ˆγ ◦ t,t) ] −1 [µ ¯xt−N+α + F (ˆγ ◦ t,t) ⊤ y t−ωt−N+α].