Optimum control

The general problem of determining optimal process control can be summarized as follows:

Given a process defined by its model, find among the admissible commands the one that allows both :

• verify given initial and final conditions;

• to satisfy various constraints;

• to optimize a chosen criterion.

This definition calls for a few comments:

• Any search for control, and a fortiori for optimal control, requires the manipulation of mathematical expressions, in particular those characterizing the evolution of the process, i.e. its model. The choice of model is therefore of prime importance. If it is too simple, it will not characterize the process sufficiently well, and if it is unnecessarily sophisticated, it will lead to very complex calculations;

• control is generally subject to various constraints linked to its implementation (limited acceleration, limited ramp-up speed, limited flow rate, excluded discontinuities, limited reservoir capacity, etc.), itself linked to the equipment available for implementation;

• process variables may be subject to various constraints related to saturation, safety, construction, comfort, cost... ;

• the initial and final states of the process may also be subject to various constraints linked to the initial conditions and the objective to be achieved. For example, a helicopter taking off from one ship and landing on another, both of which are in motion;

• the criterion to be optimized must correspond to the expression of a carefully studied choice. It may be linked to the values of the state and the control taken at given times, linked to an integral of a function of these variables over a fixed or non-fixed time interval, or both;

• the existence of a control satisfying a given objective implies that the process is controllable, an assumption that will be made implicitly and systematically.

In the following study, we shall confine ourselves to the case of continuous-state systems corresponding to physical processes. So as not to make the text too long, only the fundamental results most commonly used in practice are presented.