IV.5 Stability of feedback control system

Frequency response analysis is an useful tool for designing feedback controllers. It enables the designer to study the stability characteristics of a closed loop system using Bode or Nyquist plots and also to select the appropriate design values of controller parameters.

IV.5.1 Bode Diagram of closed loop process

IV.5.1.1: Bode diagram of PID controler

It is also worth to study the Bode diagram of PID controller in the following figure.

Fig. IV.13(a): Bode Diagram of PID controller

The PID controllers have the following characteristics:

And

It is left to the reader to exercise how to arrive at above eqs. Note that the PID controllers ideally have three asymptotes.

•  At with a slope

•  At with a slope

•  At with a zero slope.

IV.5.1.2: Bode diagram of Processes in series

Let us have a control loop with two components, viz ., one PI controller and one first order plus dead time model.

The open loop transfer function is

The above open loop transfer function is a combination of four individual processes in series, viz., pure gain, pure dead time, first order system and PI controller. Two time constants are observed in the series that would yield the location of corner frequencies viz ., or

Fig. IV.13(b): Bode plot of processes in series given in example

The above figure shows the Bode plot of the open loop process indicated in this example. AR and phase shift for all individual transfer functions as well as the overall transfer function have been indicated along with the location of corner frequencies.