S.
M. ROBERTS1 and .
I. LYVERS
TRW Computers Co., Beverly Hills, Calif.
The Gradient Method in Process Control A flexible, computable, efficient technique for handling constrained or unconstrained optima a process the shape, and orientation of both the response surface and the constraints may vary. The gradient method is a power-
controlling
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Inposition,
ful tool for ferreting out optimum conditions. The gradient method, which is readily mechanized for on-line control, can handle linear and nonlinear objective functions with linear and nonlinear constraints. In the gradient method there are three problems: • to find the direction of the gradient vector at the base point • to ascertain how far to move along the gradient vector to locate the next
point
• to determine when to terminate the calculation
Optimization Models models of chemical often formulated using three general types of equations. The first expresses the object of optimization. Commonly the objective is to maximize profit or to minimize costs. The second consists of relationships that describe the metamorphosis of raw feed material into products. Included in this category are the heat balance, material balance, and reaction rate equations. The third type expresses the constraints. These equations limit the process in some way, perhaps by upper and/or lower bounds on certain variables or conditions. A distinction between the second and third types of relationships is that the second type of relationship is true for all A material balance or heat times. balance, for example, has general validity, whereas the constraints are valid only over certain limited ranges. For example, a feed rate limitation or maximum heat transfer rate limitation may be changed by the installation of a new feed pump or larger heat transfer surface. There are several classes of variables: the manipulated, the disturbance, and the intermediate. The manipulated variables are those which the operator can set (by, say, turning valves). The disturbance variables are those over
Optimization
processes
are
1 Present address, Bonner and Moore Engineering Associates, Adams Petroleum Center, 6910 Fannin, Houston 25, Tex.
which the operator has no control, for Beexample, ambient temperature. tween these is a third class designated as intermediate variables, which may be partially controlled. Included may be such variables as catalyst activity. Computer Control and Gradient Methods. Gradient methods can be applied to computer control of chemical processes. All of the situations depicted below can be handled well by gradient methods. The use of the gradient method for computer control involves searching the mathematical model for the optimum response. The experimentation and the step-by-step movements up the response surface are done mathematically. Once the optimum is found mathematically then the process itself can be moved to the optimum conditions. If the mathematical search for the optimum takes a long time, the process conditions which triggered off the optimization routine may very well have changed by the time the optimum is found by the computer. For this case, it will pay for the computer controller to start guiding the process toward the optimum before the optimum is actually found by using intermediate results which give higher response than the current plant operating conditions. By using the mathematical model to find the optimum, the computer never sends the process outside of its constraints. After the optimum is found mathematically, the process is physically sent to the optimum point in a fashion that is always within the permissible region. The speed and the manner of moving to the optimum point must be determined by the process dynamics.
constraints as a function of the manipulatable variables. Using the response surface-constraints plot for reference (Figure 1) a classification system for steady-state optimization models is shown below as a matrix of 16 possible combinations. The conditions describing the response surface in iib column and the constraints in the jtb row identify the category i, j. Invariant response surface shape means that the relative position of any point on the response surface does not
change
with respect to the principal axes of that surface. This still allows for change of its relative position and orientation with Relarespect to the coordinate axes. tive position refers to the location of the center of the response surface relative to the axes of the plot. Orientation refers to the angles between the coordinate axes of the plot and the principal axes of the Similar statements response surface. apply for the constraints. In terms of the classification system, one can think of how to cope with this variability. A study of the process and the constraints and their relative variability will be helpful in determining whether computer control is desirable. The general approach of Box (/) has been to determine the response surface as evaluated over a relatively long period of time. By continually evaluating and re-evaluating the response surface, Box is able to follow the long-term trends in
Classification of Steady State Optimization Problems Mathematically, relationships of the
second type may be substituted into the objective function to express algebra-
ically (or graphically) the objective function as a function of the manipulatable variables. When plotted, this gives a response surface of the objective contours vs. these variables. On this same plot may be drawn the various
Figure 1. Response surface and straints plots VOL. 53, NO.
11
·
NOVEMBER 1961
con-
877
quality, and catalyst degradation, among
Response Surface and Constraints Classification Table
others.
Response Surface
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Orientation b\ Position
| .2c\
1g
.·=
ó£
Shape
®\\
Invariant Invariant Invariant
The constraints may vary owing to disturbance variables such as ambient conditions. Typical of this is the compressor capacity constraint which is a function of cooling water temperature. On the other hand, the constraints may be invariant. For example, the constraint on an over-head condenser may be a cooling duty. The quantity of water flowing over the coils from time to time may be so large that the required heat is removed despite variations in the cooling water temperature.
Varies Varies Varies
