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Chapter 4. Explicit/Multi-Parametric MPC Control of PSA Systems tioned operational constraints in an optimization framework is formulated in Eq. (5.20). Here, y is the control variable, the hydrogen purity, yr is the set point purity, and u is the manipulative variable, the adsorption time. Furthermore, y has been scaled as y = (yoriginal − 0.9999)×105, which also makes yr as zero. The same scaling rule is employed while performing the system identification step to maintain consistency. The model mismatch error, e(k) (= yplant - y) is taken into consideration to ensure offset free tracking. N and M represents the prediction and control horizon, fixed to values 4 and 2, respectively. The sampling interval is taken as one complete PSA cycle, where k is the particular sampling instant. min u Z = s.t. (y(k) − y (k)) (y(k) − y (k)) + k=1 k=0 ∆u(k) R∆u(k) x(k + 1) = y(k) = N−1 M−1 ∑rTr∑T Ax(k) + bu(k) cx(k) + e(k) ≤ 115 In order to obtain the explicit control laws for the MPC problem in Eq. (4.26) specified above, the POP toolbox [122, 117, 118] is employed. In the first step which is performed offline, the MPC problem is re-formulated as a multi- parametric quadratic programming problem [119, 120, 121]. Table 4.7 shows the number of parameters obtained for each of the reduced SS model derived in sec- tion 4.4. For example, the 8th order SS model requires 11 parameters, including 8 parameters x1 , x2 , . . ., x8 for each state ranging from -100 to 100, one for u ranging from 55 to 115, one for y ranging from -100 to 10, and the remaining one for yr ranging from -0.1 to 0.1. In the next step, the multi-parametric program- 55 ≤ u(k) y(k) ≤ 1 (4.26) 89PDF Image | Operation and Control of Pressure Swing Adsorption Systems
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