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Chapter 4. Explicit/Multi-Parametric MPC Control of PSA Systems represents the ith row the matrix while Y is null matrix. H GT1 ...GTq −γG −V 111 M0= . .. . . (4.10) (4.11) −γq Gq −Vq [ ]T N0 = Y, γ1S1, ..., γpSp The parameter space, or the set of x values for which the explicit relationship between [zm(x), γ(x)] and the parameter x, around the point [zm(x(0)), γ(x(0))], remains optimal is termed as a critical region. This special property is attributed to the fact that the parameter x, appears only in the constraints of the multi- parametric quadratic programming problem, Eq. 4.6, leading to a scenario, where for certain values of x, the current set of active constraints and lagrange positive condition remains valid. The mathematical definition of a critical region (CR) is given in Eq. 4.12. Rˇˇˇˇ CR = {Gzm(x) ≤ W + Sx(t), γ(x) ≥ 0} (4.12) Following this approach, the critical regions and the corresponding explicit control laws for the whole parametric space range defined by the user are obtained in the offline step [121, 120], and stored for the online usage. Online step In the closed loop environment, the plant measurements are taken as usual and passed to the mp-MPC controller, which in turn first evaluates the critical region the particular state measurement set belong to and then retrieve the cor- responding control law to be passed to the process system. Note that the online 65PDF Image | Operation and Control of Pressure Swing Adsorption Systems
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