Dynamic Optimization and Control: A Variational Approach

Dynamic Optimization and Control: A Variational Approach

Dynamic Optimization and Control: A Variational Approach

Dynamic Optimization and Control: A Variational Approach

Excerpt

The term "control system" generally brings to mind a servomechanism or a simple temperature regulator. Such a device is characterized by the fact that it incorporates logic which detects when the output differs from the desired value and feeds back to the input a signal proportional to this difference, which in turn causes the output to change in the direction tending to decrease this difference. Thus-eventually-the output is made to approximate the desired value to an arbitrary tolerance. Such elementary application of the feedback principle is an adequate guide to the design of the simple control systems mentioned above, but in the more complex control problems such as arise in chemical process control, inventory control, and scheduling of manufacturing, as well as in stabilization and guidance of vehicles, a deeper theory is needed.

In the more general control situations the problem is still one of adjusting the inputs (adjustment based on the measurement of system outputs and the knowledge of the environment interacting with the system) so as to optimize a performance criterion. Now, however-in contrast to the servomechanism problem-it is no longer obvious when the optimum has been reached and what changes in the several inputs will result in an improvement. Thus there arises the need for developing a theory for the design of a controller to carry out optimization under such circumstances.

The second complication arises from the greater importance of system dynamics. Negative feedback can be counted on to null the error-eventually-possibly after a long time. But time is precious, and today's control system is required to optimize not just the steady state behavior-for steady state may never be reached since, generally, the desired outputs are stochastic time functions-but rather a time average or a time integral of the expected value of the performance criterion.

It is these two aspects of the design of complex control systems that give rise to the problem of dynamic optimizations which is the central theme of this monograph. The problem which is treated can be abstracted as follows: Given a description of the performance criterion, the dynamics of the element being controlled and its environment (the description may be only probabilistic); to find the control law expressing the control inputs as a function of system . . .

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