Selection of performance index

The decision on the type of performance index to be selected depends upon the nature of the control problem. Consider the design of an autopilot for a racing yacth. 

Conventionally, the autopilot is designed for course-keeping, that is to minimise the error ipeit) between that desired course ipdit) and the actual course tp (t) in the presence of disturbances (wind, waves and current). Since is fixed for most of the time, this is in essence a regulator problem. 

Using classical design techniques, the autopilot will be tuned to return the vessel on the desired course within the minimum transient period. With an optimal control strategy, a wider view is taken. The objective is to win the race, which means completing it in the shortest possible time. This in turn requires  

From equation (9.2) a general performance index could be written 

in the racing yacht example, the following state and control variables are defined X = ye(l), X2 = tpeiO, 3 = then the performance index could be expressed 

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