Design a PI controller using Bode techniques for the plant:
such that the following performance specifications are met:
System is stable.
What will be an ideal response?
The specs are first mapped into the frequency domain, specifically onto the Bode LogMagnitude (LM) plot.
A plot of the uncompensated system (plant) and the spec system is shown below.
The PI compensator adds a low frequency slope of –1 LM/decade, adds a zero, and adds gain to an uncompensated system. To make the uncompensated system match the spec we must “bend” the low frequency part of the LM plot up to have a slope of –1 at frequencies of 10 r/s and less. This can be accomplished by adding the PI transfer function:
The value for K is 20 (alog(1.3)). This is the amount of gain required to vertically shift the uncompensated system plot until it coincides with the spec plot. The resulting PI compensator transfer function is:
The final control system is shown in the following figure.
The unit step response and the unit ramp error responses are presented below.
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