PROPORTIONAL-INTEGRAL (PI) CONTROL and response of PI controller (For unit step change) (Transfer function of PI controller)

 PROPORTIONAL-INTEGRAL (PI) CONTROL and response of PI controller (For unit step change) (Transfer function of PI controller)

This mode of control is described by the relationship

In this case, we have added to the proportional action term, K_c another term that is proportional to the integral of the error. , The values of K_c and 𝜏_I may be varied by two knobs in the controller. 

To visualize the response of this controller, consider the response to a unit step change in error, as shown in Fig.1. This unit-step response is most directly obtained by inserting Є = 1 into eq. (1), which yields

Notice that p changes suddenly by an amount K_c and then changes linearly with time at a rate Kc/𝜏_I.

The response of the PI controller is as shown in the below fig.

From the above response it is clear that for a unit step change in error, the output signal takes step change of amount K_c and after that linearly increases with ‘t’ with a slope of 𝜏_I.

Transfer function of PI controller

Introducing deviation variables in eq.(1) P = p -p_s

Transfer function will become

 

Some manufacturers prefer to use the term reset rate, which is defined as the reciprocal of 71. The integral adjustment knob on a controller may be marked in terms of integral time or reset rate. The calibration of the proportional and integral knobs is often checked by observing the jump and slope of the step response shown in Fig.