Control optimization criterion

To define the cost function that will drive the control optimization, we briefly outine the mapping from a constraint graph model of hardware behavior to a control implementation. The details are presented in Chapters 6 and 8 we summarize in this section the major results as background for defining the control optimization criterion. 

In relative scheduling, the activation of an operation is specified with respect to the completion of a set of anchors with data-dependent execution delays. In particular, the start time T v) for a vertex v e is defined in terms of the anchor offsets 12 = ra(v) a -4(v), where o a(v) represents the time at which V can begin execution after the completion of a, and A v) represents the anchors in the fan-in of v. 

6(a) denotes the (data-dependent) execution delay of the anchor a. A minimum relative schedule 12min is one with minimum offsets, which guarantees the minimum overall latency (T(vn) — T(uo)) for all profiles of execution delays 6(a), /ae A.  

For a particular anchor a, represents its maximal offset and is given by the equation  

Given a relative schedule, the control implementation approach of Chapter 8 generates a control circuit to activate operations according to the schedule. The control circuit can be modeled as consisting of two components an offset control for each anchor A and a synchronization control for each vertex d G 1, as described below  

---

The optimal control of a process can be defined as a control sequence in time, which when applied to the process over a specified control interval, will cause it to operate in some optimal manner. The criterion for optimality is defined in terms of an objective function and constraints and the process is characterised by a dynamic model. The optimality criterion in batch distillation may have a number of forms, maximising a profit function, maximising the amount of product, minimising the batch time, etc. subject to any constraints on the system. The most common constraints in batch distillation are on the amount and on the purity of the product at the end of the process or at some intermediate point in time. The most common control variable of the process is the reflux ratio for a conventional column and reboil ratio for an inverted column and both for an MVC column. 

Recently there has been great interest in discrete-time optimal control based on a one-step ahead optimization criterion, also known as minimum variance control. A number of different approaches for minimum variance control has been developed in the last decade. MacGregor (51) and Palmor and Shinnar (52) have provided overviews of these minimum variance controller design techniques. 

Reactors are considered in Sections 16.11.6.1 throngh 16.11.6.33 vessels are discussed in Sections 16.11.6.34 and 16.11.6.35. For most process eqnipment, equipment is sized based on the optimization criterion of cost. For reactors, the optimization criterion may be selectivity, yield, flexibility, ability to control, or cost. Section 16.11.6.1 lists the general rules of thumb. Section 16.11.6.2 gives examples of the reactor type based on the type of reaction. Sections 16.11.6.3 onward give rules of thumb organized by reactor type. 

Another form of conflict resolution is compromise. Poon [1983a, 1987] proposed that an optimal controller might counterbalance the metabolic needs versus energetic needs of the body, and the resulting compromise would determine the ventilatory response. The tug-of-war between the two conflicting control objectives may be represented by a compound optimization criterion which reflects the balance between the chemical and mechanical costs of breathing ... 

A majority of the modules requires certain input parameters, which have to be defined in a user-written configuration file, and is read by the main python module main.py. The configuration file specifies all class objects, modules, and submodules that are desired for optimization process. It also contains important preferences concerning the system (e.g., inpul/output paths, number of computer cores, batch system), the optimization (e.g., algorithm, step length control, stopping criterion, initial parameters, constraints), and the optimization problem (e.g., objective functions, the loss function s target values). When molecular simulations are performed, all desired properties and parameters of the thermodynamic system have to be defined (e.g., ensemble, temperatures, pressures, physical properties to be fitted, number of molecules, box size, number of MD/MC steps, time step). Hence, the file is divided into three blocks. If more than one substance is considered in the optimization, one block for each substance has to be indicated. 

The chapter is organized as follows. We describe the criterion that will drive the control optimization in Section 9.1. We then present the concept of synchronization redundancy in Section 9.2, and show how they can be used to reduce the control cost Finally, we present in Section 9.3 a technique called control synchronization that introduces redundancies by lengthening and serializing the constraint graph hardware model. 

Thickener control philosophies are usually based on the idea that the Iindertlow density obtained is the most important performance criterion. The o ertlow clarity is also a consideration, but this is generally not as critical. Additional factors which must be considered are optimization of tlocciilant usage and protection of the raking mechanism. 

Control philosophies for clarifiers are based on the idea that the overflow is the most important performance criterion. Underflow density or suspended sohds content is a consideration, as is optimal use of flocculation and pH control reagents. Automated controls are of three basic types (I) control loops that optimize coagulant, flocculant, and pH control reagent additions (2) those that regulate underflow removal and (3) rake drive controls. Equahzation of the feed is provided in some installations, but the clarifier feed is usually not a controlled variable with respect to the clarifier operation. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

The need for formal logics in optimization and the need for unattended optimization is probably the largest in chromatography, especially in liquid chromatography. Figure 13 shows a schematic representation of a chromatographic system with the controllable and uncontrollable, or fixed factors The output signal is a sequence of clock-shaped peaks, which represent the separated compounds. A first problem encountered in optimization is to decide which parameter or criterion will be optimized. In spectrometry, the criterion is more or less obvious e.g. sensitivity. In chromato-... 

Studies in optimization-X Questing control with an economic criterion (with R.N. Schindler). Chem. Eng. ScL 22,345-352 (1967). 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

---