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given constant matrices Aj, Bj, Cj, Ej of compatible dimension. The number of vertices is denoted byN=2L =32andthefunctionsξj :P→Rsatisfyξj(ρ)≥0,andNj=1ξj(ρ)=1,forall ρ ∈ P. Hence, for each ρ ∈ P, (A(ρ), B(ρ), C(ρ), E(ρ)) lies in the convex hull Co{ (A1, B1, C1, E1), (A2,B2,C2,E2), ..., (AN,BN,CN,EN) }. In order to explicitly handle the dimension of ρ (30), such that the mapping P → R is possible, and hence the combinations of matrices, e.g., ξjAj (32), are possible, we need to express the functions ξj(ρ) as a normalised linear combination of the varying parameter ρ. Hence, if we consider all combinations of the maximum and minimum values for the elements of ρ(k) (see (31)), then we can define ξj(ρ) := ξj(ρ(k)) := ξj(k) as where ξN (k) = (1 − φ1(k))(1 − φ2(k)) · . . . · (1 − φL(k)), φi(k) ρi,max −ρi(k), for i=1,...,L. (34) vertices, where ξ1(k) = φ1(k)φ2(k) · . . . · φL(k), ξ2(k) = (1 − φ1(k))φ2(k) · . . . · φL(k), . ξN−1(k) = (1 − φ1(k))(1 − φ2(k)) · . . . · (1 − φL−1(k))φL(k), (33) ρi,max − ρi,min Then, using (12), (31)–(34), the state matrices can be computed as a convex combination of their A1 =A(ρ1,min,ρ2,min,...,ρL,min), A2 =A(ρ1,max,ρ2,min,...,ρL,min), . AN−1 =A(ρ1,max,ρ2,max,...,ρL−1,max,ρL,min) AN =A(ρ1,max,ρ2,max,...,ρL,max) (35) 90 A similar treatment can be performed for the matrices B(ρ), C(ρ) and E(ρ) in (12) and hence Aζ(ρ), Bζ(ρ) in (24). Provided the pairs (Aζ,j,Bζ,j) (formed analogously to (32) given (24)) are stabilisable, for j = 1, . . . , N , feedback gains are then implemented as (see, e.g., [12]) NN Kζ = ξj(ρ)Kζ,j, j=1 Kw = ξj(ρ)Kw,j, (36) j=1 12PDF Image | Electrolyte Flow Rate Control Vanadium Redox Flow Batteries
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