nstability Illustration
Instability is caused by a condition where for some frequency the transfer function is such that feedback to the error summer actually increases the error because of the gain and the phase shift.Now,if there is any frequency for which this condition exists,then oscillations will always start and grow at that frequency.To illustrate this,we have assumed that a small transient oscillation is introduced from external sources as a disturbance in r.Assumed that at this frequency the gain is greater than 1-say,2-and the phase shift is ,that is,a lag of .Let us study the result,frozen instant by instant.The summer algebraically subtracts the feedback from the input,giving the error signal e1 of the next instant in Figure 12.15a.In the next instant,Figure 12.15b,the transient is gone but the feedback,b2,is the original e1 amplified by 2 and phase-shifted by .This passes through the negative summing point and become e2.Figure 12.15c shown e2 amplified by 2 and phase-shifted by again becoming b3 and then e3. Thus,you can see that the error is actually growing!The control system is forcing the oscillating error to increase,instant by instant.Of course,in actually,it happens smoothly,and the output would look something like Figure 12.8,for oscillating growth.If there is any frequency where such conditions for growth exist,the system is unstable and something like random noise will eventually set the system into growing oscillation.When a process-control installation is designed,one has the objective of regulation the controlled variable without instability in the loop.Stability can be ensured by designing the controller gains so that oscillation growth is never favorable,according to certain criteria.
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