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where Xc and Xd denote the conversion per pass during charging and discharging respectively. Substitution of x1, x2 (6) in (18) and noting their positive, non-zero bounds yields x2 =SOC+Xc−Xc·SOC x1 SOC − Xc · SOC = 1 − SOC − Xd + Xd · SOC . 1−SOC +Xd ·SOC (19) Denote the conversion per pass as X. Following substitution of (17) in (19) and suitable rearrange- ment, we obtain X = 1 − 1+√x1 , X= c 1+√x2 during charging (20) 1− x2 Xd = √x1 , during discharging The objective can then be formulated as a tracking control problem, whereby the flow rate or input, u, is manipulated such that the tracking error, e(k) = Xs − X(k) is driven to zero for a desired conversion factor Xs. We then define a “performance” output, yp = Xc or yp = Xd during charging or discharging respectively, as 1+ x2 yp=ρ5 0x, (21) C (ρ) ρ5,d = x1(1 + √x2), (22) In order to achieve asymptotic tracking of a reference, r = Xs, we can add an integral action state, σ, to (12), satisfying σ(k + 1) = σ + τ(r − yp). (23) Hence, we define an augmented state, ζ := x satisfying σ where, √ √ x2− x1 1−x2 x1 ρ5,c = x1(1 + √x2), with ρ5,c, ρ5,d corresponding to a respective conversion rate, Xc, Xd in (20). ζ(k+1)= A(ρ) 0 ζ+ B(ρ) u+ E(ρ) w+ 0 r, (24) −τC(ρ) 1 0 0 τ Aζ Bζ 80 where the 0 matrices are of appropriate dimension. Using the augmented LPV state space model of (24) we can now design a state feedback controller to achieve offset-free tracking. 9PDF Image | Electrolyte Flow Rate Control Vanadium Redox Flow Batteries
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