(IV.75)

Now, suppose two events occur simultaneously

•  The setpoint perturbation is stopped

•  The feedback loop is reconnected

Then, the error signal will remain as . In other words, the response of the system will continue to oscillate with constant amplitude even when the setpoint signal is withdrawn.

Alternatively, if we choose the value of controller gain less than , (say ) then

(IV.76)

If we repeat the above thought experiment, the output signal will take the form

(IV.77)

Upon closing the loop and withdrawing the setpoint perturbation, the new value for the error for the next cycle will be that will eventually yield an output response of and so on. It is evident that the amplitude of the error signal would diminish at every cycle and eventually lead to zero.

In case we choose the value of controller gain greater than , (say ) then

(IV.78)

The same thought experiment would lead to ever increasing error signal because the amplitude ratio is greater than 1.

Hence the above thought experiment indicates that we have been able to find a combination of frequency and controller gain such that the of the process becomes 1 and phase shift becomes simultaneously at that combination . The output response shows a sustained oscillation with a time period at this combination. Any frequency, , will lead to oscillation with increasing amplitude and eventually will lead to instability. Hence, the frequency is termed as the crossover frequency , the gain value is termed as ultimate gain and is called the ultimate period of oscillation of the closed loop system.

The conclusion drawn from the above thought experiment is the Bode Stability Criterion and can be stated as follows -A feedback control system is unstable if the amplitude ratio of the corresponding open loop transfer function is greater than one at the crossover frequency. The value of controller gain is the decisive factor in order to ensure its stability.

It is further understood from eq. (IV.72) that large dead time leads to smaller value crossover frequency. In other words, even a low frequency signal will be able to destabilize such process.