Profile control

Experimental measurements of DH in a-Si H using SIMS were first performed by Carlson and Magee (1978). A sample is grown that contains a thin layer in which a small amount (=1-3 at. %) of the bonded hydrogen is replaced with deuterium. When annealed at elevated temperatures, the deuterium diffuses into the top and bottom layers and the deuterium profile is measured using SIMS. The diffusion coefficient is obtained by subtracting the control profile from the annealed profile and fitting the concentration values to the expression, valid for diffusion from a semiinfinite source into a semi-infinite half-plane (Crank, 1956),... 

Here the problem is given as an initial value problem, although the concepts can easily be generalized to boundary value problems and even partial differential equations. Note also that both continuous variables, x (parameters), and functions of time, U(t) (control profiles), are included as decision variables. Constraints can also be enforced over the entire time domain and at final time. 

On the other hand, the optimal control problem with a discretized control profile can be treated as a nonlinear program. The earliest studies come under the heading of control vector parameterization (Rosenbrock and Storey, 1966), with a representation of U t) as a polynomial or piecewise constant function. Here the mode is solved repeatedly in an inner loop while parameters representing V t) are updated on the outside. While hill climbing algorithms were used initially, recent efficient and sophisticated optimization methods require techniques for accurate gradient calculation from the DAE model. 

The embedded model approach represented by problem (17) has been very successful in solving large process problems. Sargent and Sullivan (1979) optimized feed changeover policies for a sequence of distillation columns that included seven control profiles and 50 differential equations. More recently, Mujtaba and Macchietto (1988) used the SPEEDUP implementation of this method for optimal control of plate-to-plate batch distillation columns. 

Using piecewise constant control profiles and orthogonal collocation on finite elements, this approach was further developed by Renfro (Renfro, 1986 Renfro et al, 1987) to deal with much larger problems. More recent simultaneous applications that involve SQP, orthogonal collocation, and piecewise constant control profiles have been presented by Patwardhan et al (1988) for online control, and by Eaton and Rawlings (1988) for optimization of batch crystallizers. These studies have shown that simultaneous approaches can be applied successfully to small-scale applications with complex constraints. 

However, all of these studies determine only approximate or parameterized optimal control profiles. Also, they do not consider the effect of approximation error in discretizing the ODEs to algebraic equations. In this section we therefore explore the potential of simultaneous methods for larger and more complex process optimization problems with ODE models. Given the characteristics of the simultaneous approach, it becomes important to consider the following topics ... 

In summary, formulation of (27) with appropriate placement of finite elements works well for parameter optimization problems. In the next subsection, however, we consider additional difficulties when control profiles are introduced. Stated briefly, the reason for these difficulties lies in the nature of the discretized variational conditions of (16). As shown in Logsdon and Biegler (1989), optimality conditions for parameter optimization problems take the form of two point boundary value problems. For optimal control... 

However the accurate treatment of state variable inequality constraints presents a few problems. Parameter optimization problems obtained by discretizing the control profile generally allow inequality constraints to be active only at a finite set of points, simply because a finite set of decisions cannot influence an infinite number of values (i.e., keeping the state fixed at every point in a finite time period). 

However, if we are interested in an accurate representation of the exact optimal control profile, the problem becomes more complicated. First, from (16) we recognize the possibility that the optimal solution can cause time-dependent inequality constraints to become active for a finite period of time. 

For these time periods, the ODEs and active algebraic constraints influence the state and control variables. For these active sets, we therefore need to be able to analyze and implicitly solve the DAE system. To represent the control profiles at the same level of approximation as for the state profiles, approximation and stability properties for DAE (rather than ODE) solvers must be considered. Moreover, the variational conditions for problem (16), with different active constraint sets over time, lead to a multizone set of DAE systems. Consequently, the analogous Kuhn-Tucker conditions from (27) must have stability and approximation properties capable of handling ail of these DAE systems. 

Finally, finite elements are added as decision variables in (27) not just to ensure accurate approximation (of the state and control profiles), but also to provide optimal points of discontinuity for the control profile. This dual purpose led Cuthrell and Biegler (1987) to distinguish some elements as finite-and super-elements. These roles can be combined, however, if one considers the NLP formulation of the optimal control problem given below ... 

Note that potential control profile discontinuities are allowed at each element location with error restrictions directly enforced for each element. For a sufficient number of elements (which can be determined by the algorithm in the previous section), the element can be as large as allowed by an active error constraint, or it can act as a degree of freedom for the control profile discontinuity, with its corresponding error constraint inactive. Otherwise, (35) is based on the implicit Runge-Kutta (IRK) or collocation... 

With four-point collocation using gaussian roots (the A-stable case), we fail to find an optimal solution. Because of this, we include an additional constraint on the control profile error (third time derivative of the control profile less than a tolerance, see Russell and Christiansen, 1978). The resulting... 

Finally, the catalyst mixing problem can be converted from an index three problem to an index zero problem by parameterizing the control profile using variable length piecewise constant functions. (This approach is acceptable because of the known form of the optimal control profile.) The solution using this approach also matches the analytical solution within numerical tolerances. 

While the reduced SQP algorithm is often suitable for parameter optimization problems, it can become inefficient for optimal control problems with many degrees of freedom (the control variables). Logsdon et al. (1990) noted this property in determining optimal reflux policies for batch distillation columns. Here, the reduced SQP method was quite successful in dealing with DAOP problems with state and control profile constraints. However, the degrees of freedom (for control variables) increase linearly with the number of elements. Consequently, if many elements are required, the effectiveness of the reduced SQP algorithm is reduced. This is due to three effects ... 

Optimal control problems have more interesting features in that control profiles are literally infinite-dimensional and attention must be paid to approximating them accurately. Here the optimality conditions can be represented implicitly by high-index DAE systems, and consequently a stable and accurate discretization is required. To demonstrate these features, the classical catalyst mixing problem of Jackson (1968) was solved with the simultaneous approach. In addition to theoretical properties of the discretization, the structure of the optimal control problem was also exploited through a chainruling strategy. 

Cuthrell, J. E., and Biegler, L. T., Simultaneous optimization and solution methods for batch reactor control profiles, Comp, and Chem. Eng. 13(1/2), 49-62 (1989). 

Figure 3.1 Concentration-distance profiles during diffusion-controlled reduction of O to R at a planar electrode. D0 = DR. (A) Initial conditions prior to potential step. Cq = 1, CR = 0 mM. (B) Profiles for O (dashed line) and R (solid line) at 1,4, and 10 ms after potential step to Es. R is soluble in solution. (C) Profiles for O and R at 1,4, and 10 ms after potential step to Es. R is soluble in the electrode. (D) Potential stepped from Es to Ef at t = 10 ms so that oxidation of R to O is now diffusion-controlled. Profiles are for 11, 15, and 20 ms after step to Es. R is soluble in the electrode.

Solute concentration fields are shown in Figure 17 for the flows in Figure 16. The diffusion-controlled profile for unidirectional solidification is un-... 

In the sequential strategy, a control (manipulated) variable profile is discretized over a time interval. The discretized control profile can be represented as a piecewise constant, a piecewise linear, or a piecewise polynomial function. The parameters in such functions and the length of time subinterval become decision variables in optimization problem. This strategy is also referred to a control vector parameterization (CVP). 

The stock is pumped through a manifold into the headbox of the paper machine, where the stock flow is decelerated and distributed over the width of the machine. Various baffles and step diffusors are used to avoid vortex flow and stagnation zones. The furnish leaves the headbox through the slice, a narrow gap with controlled profile, and impacts on one or two endless screens, the so-called papermakers wire. Water is removed from the fiber mat by the action of foils and vacuum. 

Logsdon, J.S., Efficient determination of optimal control profiles for differential and algebraic systems, PhD Thesis, (Carnegie Mellon University, USA, 1990). 

Evaluation of gradients is one of the major tasks in NLP based optimisation techniques. Rosen and Luus (1991) reviewed a number of methods for the evaluation of gradients for dynamic optimisation (optimal control) problems which uses piecewise constant optimal control profile. Some of these methods are discussed here. 

See the work of Vassiliadis (1993) for gradient evaluation methods for linear and exponentially varying control profile. See Rosen and Luus (1991) for gradient evaluations with the time invariant parameters (v) optimised and for guidelines for selecting the appropriate gradient evaluation method. 

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