This self-contained book gives a detailed treatment of optimal control theory that enables readers to formulate and solve optimal control problems. With a strong emphasis on problem solving, it provides all the necessary mathematical analyses and derivations of important results, including multiplier theorems and Pontryagin's principle. The text presents various examples and basic concepts of optimal control and describes important numerical methods and computational algorithms for solving a wide range of optimal control problems, including periodic processes.
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This chapter introduces optimal control with the help of several examples taken from chemical engineering applications. The examples elucidate the use of control functions to achieve what is desired in those applications. The mathematical underpinnings illustrate the formulation of optimal control problems. The examples help build up the notion of objective functionals to be optimized using control functions.
1.1 Definition
An optimal controlis a function that optimizes the performance of a system changing with time, space, or any other independent variable. That function is a relation between a selected system input or property and an independent variable. The appellation “control” signifies the use of a function to control the state of the system and obtain some desired performance. As a subject, optimal control is the embodiment of principles that characterize optimal controls, and help determine them in what we call optimal control problems.
Consider a well-mixed batch reactor, shown in Figure 1.1, with chemical species A and B reacting to form a product C. The reactivities are dependent on the reactor temperature, T, which can be changed with time, t. At any time, however, the temperature is the same or uniform throughout the reactor because of perfect mixing. Such a system is described by the mass balances of the involved species or the equations of change. They are differential equations, which have time as the independent variable in the present case.
Figure 1.1 A batch reactor operating with temperature as a function of time over a certain time duration
An optimal control problem for the batch reactor is to find the temperature versus time function, the application of which maximizes the product concentration at the final time tf. That function is the optimal control
among all possible control functions, such as those shown in Figure 1.2.
Figure 1.2 Optimal control
and other possible control functions Ta(t)–Td(t)
Let us formulate the above problem for the elementary reaction
where a, b, and c are the stoichiometric coefficients of the species A, B, and C. Denoting their respective concentrations by x, y, and z at any time t, the batch reaction process may be described by the following equations of change:
(1.1)
(1.2)
(1.3)
with the initial conditions
In the above equations, k0 is the Arrhenius constant, E is...
