5.8.  THE ITAE PERFORMANCE CRITERION FOR OPTIMIZING THE TRANSIENT RESPONSE

The ITAE performance index, as defined by Eq. (5.46), is considered further in this section. As discussed in Section 5.6, this performance index does not penalize large initial errors that are unavoidable. However, it does penalize long-duration transients. We now consider the form that the system transfer function should take, for various order systems, in order to achieve zero stedy-state step and ramp error systems and minimize the ITAE.

In the case of the zero steady-state step error system, Eq. (5.52) shows that the form of the system transfer function is given by

The procedure used to produce a table of standard system transfer functions of the form C(s)/R(s) was to vary each coefficient in Eq. (5.56) separately until the integral of time multiplied by the absolute value of error became a minimum. Then the successive coefficients were varied in sequence to minimize the ITAE value.

If this criterion is applied to the second-order system described by Eq. (5.38), the optimum damping ratio is found to be 0.7. A listing of system transfer functions, C(s)/R(s), has been prepared by Graham and Lathrop [2], They show the optimum form of the denominator, for systems whose transfer functions are of the form given by Eq (5.56), which will minimize the integral of Eq. (5.46). For example, the optimum form for a second-order system is given by

where ζ = 1.4/2 = 0.7. The optimum form for a third-order system is given by

Table 5.5, which has been obtained from Reference 2, shows the optimum denominator transfer function for systems through the eighth order which will minimize the integral of Eq. (5.46). These standard forms provide a quick and simple method for synthesizing an optimum dynamic response.

In the case of the zero steady-state ramp error systems, the system transfer function was shown in Eq. (5.54) to be given by

Figure 5.9   Control system containing two pure integrations.

The objective here also is to obtain a set of standard forms for the denominators of the system transfer functions given by Eq. (5.59). Table 5.6, obtained from Reference 2 was obtained in a similar way to Table 5.5.

The ITAE criterion is applied to several problems in the problem section of this chapter (see Problems 5.38–5.42, and illustrative problems I5.9 and I5.10.

The ITAE criterion is a straightforward method for optimizing the transient response of a system when the transfer function is known. Generally, it produces smaller overshoots and oscillations than the other criteria presented. It should be emphasized, however, that the ITAE solution is very sensitive and may not be useful for certain systems where most of the system poles and zeros and gains may be specified initially. For the latter case, designers do not have the flexibility for selecting as many of the parameters as they might wish.