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10
Describing the System and Evaluating /ts Performance
Soc.1.1
The Performance Measure
In order to evaluate the performance of a system quantitatively, the
designer selects a performance measure. An optimal control is defined as one
that minimizes(or maximizes)the performance measure. In certain cases
the problem statement may clearly indicate what to select for a performance
measure,whereas in other problems the selection is a subjective matter,
For example, the statement,“Transfer the system from point A to point B
as quickly as possible,”clearly indicates that elapsed time is the performance
measure to be minimized. On the other hand, the statement,“Maintain the
position and velocity of the system near zero with a small expenditure of
control energy,”does not instantly suggest a unique performance measure.
In such problems the designer may be required to try several performance
measures before selecting one which yields what he considers to be optimal
performance. We shall discuss the selection of a performance measure in
more detail in Chapter 2.
Example 1.1-3.Let us return to the automobile problem begun in
Example 1.1-1.The state equations and physical constraints have been
defined; now we turn to the selection of a performance measure.Suppose
the objective is to make the car reach point e as quickly as possible;
then the performance measure J is given by
J=ts—to
(1.1-10)
In all that follows it will be assumed that the performance of a system is
evaluated by a measure of the form
,
中
J=h(x(ty),tj)+
g(x(t),u(t), t) dt,
(1.1-11)
t?^
where to and t, are the initial and final time; h and g are scalar functions.
ty may be specifed or“free,”depending on the problem statement.
Starting from the initial state x(to)= x。and applying a control signal
u(), for t∈[to,t?],causes a system to follow some state trajectory;the
performance measure assigns a unique real number to each trajectory of the
system.
With the background material we have accumulated it is now possible
to present an explicit statement of“the optimal control problem.”
The Optimal Control Problem
The theory developed in the subsequent chapters is aimed at solving
the following problem.