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

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

**SOLUTION:**

Therefore,

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, *G _{p}*(

**(a)** Determine the gain crossover frequency, *ω _{c}*, 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, *G _{c}*(

is to be added in series with the process’ transfer function, *G _{p}*(

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