Application of the ITAE Performance Criterion

to Achieve Robust PID Control

The integral of time multiplied by the absolute value of error (ITAE) criterion should be used in the design of the PID controller for the inverted pendulum system. Refer to your textbook for a description of the criterion.

Representing the inverted pendulum system with G(s), which you must determine, and using the PID controller

determine the closed-loop transfer function, T₁(s), without prefiltering [G_p(s) = 1]. The closed-loop transfer function is given by

.

The resulting equation will be fourth order. The optimum transfer equation based on the ITAE criterion for the fourth order system is given by

.

(Modern Control Systems, Dorf)

You must select in order to meet the settling time requirement. T_s is given by

and z will be near 0.8. Therefore, can be determined.

Equating the denominators of T₁(s) and T(s) makes it possible to solve for K_D, K_P, and K_I. Once those coefficients are known, the prefilter, G_p(s), can be solved for from

to achieve the desired ITAE response.

Disclaimer: This is an initial solution method. In order to ensure you make no mistakes, you should verify everything included herein before proceeding with this method. z is very difficult to solve for so you should probably assume that the value given is good enough. However, it is not impossible to solve for so if you feel confident, be my guest.