7.11.  ILLUSTRATIVE PROBLEMS AND SOLUTIONS

This section provides a set of illustrative problems and their solutions to supplement the material presented in Chapter 7.

I7.1. A closed-loop unity–feedback control system used to position a load has a forward transfer function given by

Image

The time constant of the open-loop process is 2 sec. Determine the time constant of the closed-loop control system.

SOLUTION:

Image

Therefore,

Image

So, the time constant of the closed-loop control system is 0.1818 sec.

I7.2. We wish to analyze the performance of a unity-feedback second-order control system whose forward transfer function represents a process, Gp(s), given by:

Image

(a)  Determine the gain crossover frequency, ωc, analytically without using the graphical Bode plot.

(b)  Determine the phase margin and gain margin of this control system analytically without using the graphical Bode plot.

(c)  A phase-lead network compensation, Gc(s), given by

Image

is to be added in series with the process’ transfer function, Gp(

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