Control Vector Parameterisation

A control vector parameterisation approach [66,67] implemented with the gPROMS process modeling tool was employed to solve this dynamic optimization process [68]. The optimum values of the switching time, fj, and of the final time, tf, were determined. The optimal operating conditions were foimd for different numbers of operating cycles for either TTB or PHL as the main product. The optimum number of cycles, and hence the effective column length, is thereby determined. 

Each dynamic optimisation problem is implemented in gPROMS (Process Systems Enterprise Ltd., 2001) which incorporates a control vector parameterisation approach. The work pressure profile is considered to be a linear function of time over the entire time horizon. 

Non-linear programming technique (NLP) is used to solve the problems resulting from syntheses optimisation. This NLP approach involves transforming the general optimal control problem, which is of infinite dimension (the control variables are time-dependant), into a finite dimensional NLP problem by the means of control vector parameterisation. According to this parameterisation technique, the control variables are restricted to a predefined form of temporal variation which is often referred to as a basis function Lagrange polynoms (piecewise constant, piecewise linear) or exponential based function. A successive quadratic programming method is then applied to solve the resultant NLP. 

Optimal control of a batch distillation column consists in the determination of the suitable reflux policy with respect to a particular objective function (e.g. profit) and set of constraints. In the purpose of the present work, the optimisation problem is defined with an operating time objective function and purity constraints set on the recovery ratio (90%) and on the propylene glycol final purity (80% molar). Different basis fimctions have been adopted for the control vector parameterisation of the problem piecewise constant and linear, hyperbolic tangent function. Optimal reflux profiles are determined with the final conditions of the previous optimal reactions as initial conditions. The optimal profiles of the resultant distillations are presented on figure 2. 

Having design parameters fixed in the outer problem and with a specific choice of D° (discussed in section 7.2) the inner loop optimisation can be partitioned into M independent sequences (one for each mixture) of NTm dynamic optimisation problems. This will result to a total of ND = 2 NTm problems. In each (one for each task) problem the control vector m for each task is optimised. This can be clearly explained with reference to Figure 7.3 which shows separation of M (=2) mixtures (mixture 1 = ternary and mixture 2 = binary) and number of tasks involved in each separation duty (3 tasks for mixture 1 and 2 tasks for mixture 2). Therefore, there are 5 (= ND) independent inner loop optimal control problems. In each task a parameterisation of the time varying control vector into a number of control intervals (typically 1-4) is used, so that a finite number of parameters is obtained to represent the control functions. Mujtaba and Macchietto (1996) used a piecewise constant approximation to the reflux ratio profile, yielding two optimisation parameters (a control level and interval length) for each control interval. For any task i in operation m the inner loop optimisation problem (problem Pl-i) can be stated as ... 

Mathematically, the coupling between the mechanical and control subsystem mainly occurs in the system equations s, where the response quantities y (e. g. displacement vector) and (e. g. control forces) are determined depending on (the actual values of) the design variables w and v. These system equations often are the state-space representation as discussed in previous sections, where in the case of adaptronic structures the equations of motion and vibration are involved. So, all the remarks on modal representation, condensation, etc. apply, including proper parameterisation in the design variables. 

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