IV.5.2: Bode Stability Criterion

Consider a simple first order plus dead time process to be controlled by a proportional controller:

(a) Closed Loop	(b) Opened Loop
Fig. IV.14: Example of first order system for studying Bode stability criterion

The open-loop transfer function for this system is given by

(IV.71)

The Bode plot of the above open loop transfer function is given by the following figure.

Fig. IV.15: Bode plot

We are interested to know the frequency where the phase shift is . Numerically it can be solved by the equation

(IV.72)

The frequency is and the value of at is observed to be which can also be found numerically by,

(IV.73)

The above exercise indicates that in order to obtain at this frequency one needs to set the value of as,

(IV.74)

At this juncture, one needs to perform a thought experiment in order to understand the Bode stability criterion. Let us set the value of controller gain, and let us “open up” the feedback loop as indicated in the figure before. Suppose, we vary the setpoint as a sinusoidal function . As the loop is open, the error will be equal to the setpoint and thereby yield an output,