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Finally, because all of the control variables appear linearly, problem (3.1) is a singular optimal control problem. In singular control problems, the optimal control profiles cannot be determined directly from the stationarity condition of the Hamiltonian. The Euler-Lagrange equations obtained after applying the maximum principle to (3.1) are high index in nature and ill-conditioned. This doesn’t affect optimal controls that lie at their bounds where solution is “bang-bang”. However, a bang-bang solution may not always be guaranteed with complex nonlinear state equations of PSA system as the Hamiltonian derivative w.r.t. controls can be zero for some time interval, leading to a singular control profile. Repeated time differentiations of the Hamiltonian derivative can recover the control, but identifying the beginning and the end of a singular arc is often difficult. Applying orthogonal collocation to singular problems can also reflect this behavior with an ill-conditioned reduced Hessian and solutions characterized by oscillations that do not abate with increasing mesh refinement [102]. To address singular problems, several approaches have been suggested which propose regularizations to improve eigenvalues of the reduced Hessian, and to guarantee a unique solution [97, 180, 101]. In particular, regularizations have been performed through coarse discretization of the control profile [161]. Applying a coarse control discretization hampers rapid decay of the eigenvalues of reduced Hessian, thus allowing a reasonable solution to the singular problem. Hence, we adopt this relatively simple regularization heuristic to ameliorate the singular nature of the control problem (3.1). Such an approach, coupled with moving finite element strategy, improves the bang-bang nature of the optimal solution, locates singular arcs and minimizes their length with fewer finite elements. 3.4 Solution Strategy Chapter 3. PSA Superstructure 46PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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