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where the vertices are computed as (cf. (27)) K =(R+B⊤PB )−1B⊤PA , K =B+E, (37) ζ,j j ζ,j j ζ,j ζ,j j ζ,j w,j j j where Bj+ denotes the Moore-Penrose pseudo-inverse of Bj, and Pj is the solution to (28) at the system vertices (Aζ,j,Bζ,j) given the weights Qj and Rj, such that Kζ,1 =Kζ(ρ1,min,ρ2,min,...,ρL,min), Kζ,2 =Kζ(ρ1,max,ρ2,min,...,ρL,min), . Kζ,N−1 = Kζ(ρ1,max,ρ2,max,...,ρL−1,max,ρL,min) Kζ,N =Kζ(ρ1,max,ρ2,max,...,ρL,max) (38) The final control law, which implements the feedback gains via convex combination, is then given from (25)–(38) as N u(k) := uζ(k) = u∗ζ(k) − ξj(ρ(k))Kζ,j(ζj(k) − ζj∗(k)) + Kw,jw(k). (39) j=1 The desired tracking values in the control law (39) can be obtained using an approximated frozen model, where ζj∗ = (x∗,0) and u∗ζ = u∗, with the pair (x∗(k),u∗(k)) computed explicitly online, at each iteration given a desired conversion per pass Xs, as follows. Set x∗1(k + 1) = 95 x∗1(k) = x1(k) (since the dynamics are relatively slow) and x∗2(k) = x2(k) and substitute Xs and x∗1(k + 1) into (17) and(19) to obtain x∗2(k + 1) = ((1 + x∗1)/(1 − Xs) − 1)2 during charging or x∗2(k + 1) = ((1 − Xs)/(Xs + 1/x∗1))2 when discharging. The corresponding ideal reference concentrations (c2,c3,c4,c5)∗(k) can then be computed from x∗1(k), x∗2(k) using the procedure of Section 2.3. Given (c2, c3, c4, c5)∗(k), the reference varying parameters (ρ1, ρ2, ρ3, ρ4, ρ5)∗(k) can be 100 computed via (10),(22). Finally, the reference control input u∗(k) = (x∗2(k+1)−(1−τρ∗2(k))x∗2(k)− τρ∗4(k)I(k))/(τρ∗3(k)) can be found via solution to (12). We note here, that this convex combination of vertices approach offers reduced computational complexity by sacrificing some performance (both in terms of conservativeness of the bounding convex polytope and accuracy of the reconstruction through linear combinations); however, this is 105 desired as part of the design brief and will be demonstrated in Section 4 as a justified investment, due to comparable performance. As an additional benefit, the analysis, in terms of system con- 13PDF Image | Electrolyte Flow Rate Control Vanadium Redox Flow Batteries
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