Modern complex control systems usually require more sophisticated performance criteria than those presented so far. As seen in the previous section, error and the time at which it occurs are very important factors that usually must be considered simultaneously. A performance index is a single measure of a system’s performance that emphasizes those characteristics of the response that are deemed to be important. The notion of a performance index is very important in estimator design using linear-state-variable feedback, which is presented in Sections 8.1 through 8.6, and in optimal control theory, where the system is designed to optimize this performance index given certain constraints. This subject will be discussed fully in Chapter 11.

Refrences 1 and 2 discuss an entire class of performance indices that are various functions of error and/or time. This section reviews those that are useful in the design of practical linear control systems.

A fairly useful performance index is the *integral of the absolute magnitude of the error* (IAE) criterion, which is

By utilizing the magnitude of the error, this integral expression increases for either positive or negative error, and results in a fairly good underdamped system. For a second-order system, *S*_{1} has a minimum for a damping ratio of approximately 0.7.

Another useful performance index is the *integral of the ...*

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