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I u EOCV ρ,x VRB Dynamics: c ̇ = f(c,Q,I) Q=u EOCV = g(c) Signal Processing Model: x = h(EOCV ) ρ = l(x,cideal) Controller Model: x(k + 1) = A(ρ)x(k) + B(ρ)u(k) + E(ρ)w(k) yp(k) = C(ρ)x(k) σ(k + 1) = σ + τ(r − yp) w=I Output: u = κ(x,w,ρ) I Figure 2: VRB Control System Block Diagram 3.2. Scheduled State Feedback Design A simplified block diagram for the overall control scheme is shown in Figure 2. We employ the augmented state-feedback tracking controller with disturbance accommodation: u(k) = u∗(k) − Kx(k)(x(k) − x∗(k)) − Kσ(k)σ − Kw(k)w(k), (25) where x∗ is the desired state value and u∗ is the corresponding control input. The feedback gain, Kw(k) = Kw(ρ(k)) = B+(ρ(k))E(ρ(k)) in (25), is a standard implementation of disturbance accommodation, where B+ denotes the Moore-Penrose pseudo-inverse of B. The feedback gains, Kx(k) = Kx(ρ(k)),Kσ(k) = Kσ(ρ(k)), are designed to achieve certain performance characteristics and are defined as K K :=Kζ, (26) xσ which is an LQR for the augmented system (24), provided (Aζ(ρ),Bζ(ρ)) is stabilisable (i.e., controllable, see Section 3.3) for all ρ(k). By updating measurements of ρ = ρ(k), and hence (Aζ(ρ),Bζ(ρ)), at each time step k, the feedback gain, Kζ = Kζ(ρ(k)) in (26), can be computed as ⊤ −1⊤ Kζ =(R+BζPBζ) BζPAζ, (27) 10PDF Image | Electrolyte Flow Rate Control Vanadium Redox Flow Batteries
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