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Chapter 4. Explicit/Multi-Parametric MPC Control of PSA Systems Eq. 4.6: V (x(t)) = min 1zm′ zm 2 Hzm s.t. Gzm ≤ W+Sx(t) (4.6) Here, zm represents the new set of optimization variables, while S is defined in the Eq. 4.7: S = E + GH−1F′ (4.7) In the new formulation, xt which is now acting as a parameter of the optimiza- tion problem (Eq. 4.6), appears only in the constraint equations as compared to being involved both in the objective function and constraints in the original MPC problem formulation (Eq. 4.3). Next, sensitivity analysis is performed on the Karush-Kuhn Tucker (KKT) conditions of the multi-parametric quadratic optimization problem, shown in Eq. 4.8, in order to obtain the optimal zm as an affine function of parameter xt, expressed in Eq. 4.9. Hzm + G′γ = 0 γi(Gizm − Wi − Six(t)) = 0, i = 1,...q γ ≥ 0 zm(x) = −(M0)−1N0(x − x0) + zm(x(0)) (4.8) (4.9) Here, M0 and N0 are constant matrices, as shown in Eq. 4.10 and 4.11, i γ (x) γ (x(0)) 64PDF Image | Operation and Control of Pressure Swing Adsorption Systems
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