Design constrained optimization

The majority of minimization routines are designed for unconstrained optimization, in which the control algorithm is free to select any parameters it wishes. Only a minority can handle constrained optimization. 

Closed-loop multivariable boiler control has to be planned and performed carefully because plant operators are not traditionally willing to reduce air-fuel ratios due to concerns about CO and other symptoms associated with Oz-deficient combustion. Model predictive control (MPC) is by far the most widely used technique for conducting multivariable boiler optimization and control. Forms of MPC that are inherently multivariable and that include real-time constrained optimization in the design are best suited for boiler application. 

Boston, J. F., "Algorithms for Distillation Calculations with Bounded-Variable Design Constraints and Equality-or Inequality-Constrained Optimization", Paper presented at Houston AIChE Meeting, April 1979. 

Although the value of on-line constrained optimization is expUcitly identified in this passage, the concept of moving horizon is missing. That is, perhaps, due to the fact that the author was concerned with the design of autopilots for airplane landing, a task that has a finite duration T. The algorithm described above is, essentially, the mathematical equivalent of MFC for a batch chemical process. 

The situation is quite different when inequality constraints are included in the MPC on-line optimization problem. In the sequel, we will refer to inequality constrained MPC simply as constrained MPC. For constrained MPC, no closed-form (explicit) solution can be written. Because different inequahty constraints may be active at each time, a constrained MPC controller is not linear, making the entire closed loop nonlinear. To analyze and design constrained MPC systems requires an approach that is not based on linear control theory. We will present the basic ideas in Section III. We will then present some examples that show the interesting behavior that MPC may demonstrate, and we will subsequently explain how MPC theory can conceptually simplify and practically improve MPC. 

This completes the derivation of the PLQ model which has briefly been analyzed in our previous studies [58, 59]. The discussed behavior of the Pade-designed, biologically optimized effect Ep(D) yields the sought requirements for the two as miptotes S[T (D) e ° and S f fD) at small and large values of D, respectively. At large doses, by setting y = fSDo, we can alternatively write Sf (D) Thus, when y is constrained... 

In all of these alternatives, the design team selects acceptable temperature levels and flow rates of the recirculating fluids. These are usually limited by the rates of reaction, and especially the need to avoid thermal runaway or catalyst deterioration, as well as the materials (rf construction and the temperature levels of the available cold process streams and utilities, such as cooling water. It is common to assign temperatures on the basis of these factors earily in process synthesis. However, as optimization strategies are perfected, temperature levels are varied within bounds. See Chapters 10 and 18 for discussions of the use of optimization in process synthesis and optimization of process flowsheets, as well as Example 6.3 to see how constrained optimization is applied to design an ammonia cold-shot converter. 

If we consider human error to be a process of variation akin to the genetic variability inherent in mutation, it is possible to see that some error may well be advantageous to performance and therefore adaptive in nature. We already know that some error is deleterious to performance and safety, whereas a significant proportion of error is neutral. The next logical step is to accept that, indeed, some error must have adaptive benefits. To this end, in systems that attempt to standardize and constrain performance to eliminate error, are we simply interfering with natural processes designed to optimize performance ... 

The task of finding the design point involves the solution of the following constrained optimization problem ... 

The most important and challenging problems in active and passive stmctural control systems are the formulation and solution of optimal control and nonlinear constrained optimization needed to develop appropriate closed loop feedback control algorithms and the optimal placement, which is the central focus of this book. State-of-the-art techniques for optimal design of passive and active control systems are described in detail in various chapters written by researchers aroimd the world. I welcome this new book for offering a very good overview of the current developments in the field. 

In addition to above, complexity of individual components, costs, technology risks and failure effects should also be considered. In other words the allocation must be done in such a way that other design constrains are not violate. A case-by-case optimization is often possible (Birolini 2006). 

As the limit state function is in general nonlinear it is not possible to know the design point in advance and this has to be found iteratively. The design point is thus, the solution to the constrained optimization problem ... 

The introduction of inequality constraints results in a constrained optimization problem that can be solved numerically using linear or quadratic programming techniques (Edgar et al., 2001). As an example, consider the addition of inequality constraints to the MFC design problem in the previous section. Suppose that it is desired to calculate the M-step control policy AU(k) that minimizes the quadratic objective function J in Eq. 20-54, while satisfying the constraints in Eqs. 20-59, 20-60, and 20-61. The output predictions are made using the step-response model in Eq. 20-36. This MFC... 

Figure 8 Sidechain rotamer libraries may constrain optimization problems in protein design. The stereo figure shows 140 superimposed phenylalanine residues randomly taken from high resolution structures in the protein structure database. As is clearly seen, not only is the roatation about the C -C bond discretized to three rotamers, but also the plane of the aromatic ring is constrained...

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