JAMES M . DOUGLAS
MORTON M. DENN
OPTIMAL DESIGN AND CONTROL Variational calculus looks much harder than i t is. This paper shows how the math derives from intuitive concepts and applies it t o simple common problems such as optimizing chemical reactor design and control system design .
BY VARIATIONAL METHODS ncreasing competition and smaller profit margins have
I always provided the chemical industry with incentives to improve process design and control techniques. Although engineers have long sought to optimize system performance, only in recent years have methods become available which are sufficiently powerful to treat the complex problems encountered in industrial systems. The mathematical advances of greatest interest to chemical engineers may roughly be divided in two categories. The iirst includes procedures for finding the maximum of a multivariable function. Wilde (37) has treated these procedures in detail. The second category deals with problems in which the system equations must be considered simultaneously with the optimization problem. In controller design, for example, response time depends on the mathematical description of the system. Problems in the second category may often be treated either by d p a m i c programming or variational methods. Information on dynamic programming is abundant and easily accessible (7, 4), but variational methods have been presented in a manner which is frequently confusing and often a t a level of mathematical sophistication which obscures the basic simplicity of the principles employed. This paper is an attempt to demonstrate the ease with which very powerful tools of optimization can be developed from intuitive concepts and to show how they may be applied to simple, common problems. An elementary proof of the maximum principle is given, and some examples of the synthesis of optimal control systems and the optimal design of chemical reactors are discussed in detail. The examples have been chosen to demonstrate the technique without introducing the complexity of large systems. The methods are equally applicable, however, to any process for which a suitable mathematical model exist The Variational Equation
Suppose that we wish to optimize a system which is continuous in some variable, such as time or length. A plug flow reactor, for example, is described by a set of differential equations in which the independent variable is reactor length or time. The time behavior of a batch reactor or continuous flow stirred tank is also described by ordinary differential equations. Our goal is choosing the control or design variables which optimize some measure of system performance.
For simplicity we shall restrict our attention to systems described by two variables, say x1 and xe, and two control or design variables, U I and u2. It may be convenient to think of the control of a continuous flow siirred tank in which a single reaction occurs. Then, X I would be reactant concentration, x2 the temperatye, U I and uz the flow rates of coolant and reactant, respectively. Such a system is described by the equations
dxi - = fl(X1, dt
x2, u1, ue)
Xl(0)
=
(14
XI0
and the performance index by
E =
Lo
5(x1, xe, u1, u d
dt
(2)
Assume that we wish to select u1 and uz in the interval 0 t 8, in order to minimize E . In terms ofareactor, we wish to select the flow rates over the entire period of operation to minimize some appropriate cumulative measure of error, such as deviation from a preset opera. ting condition. The simplest intuitive scheme for selecting the optimal functions ul(t) and uz(t) is first to select some specific values, say ril(t) and a&). Equations 1 may then be solved to give values f l ( t ) and a&), and the performance index takes on a particular value, which we may denote as &[ax,rill. We can then ask whether small changes in u1 and us will increase or decrease E . Assume that we select the new values
<
