Design a lag-lead compensator for the system of Problem 8.2-2. Note that \(G_{p}(\mathrm{~s})=1 /\left(s^{2}+sight)\), \(T=0.2 \mathrm{~s}\), and \(H=1\).

(a) Use the lag compensator of Problem 8.4-2 and increase its gain by a factor of 2.

(b) Cascade a lead compensator to achieve a phase margin of 60 degrees for the lag-lead combination.

(c) Plot the step responses for this system for the lag compensator of Problem 8.4-2, the lead compensator of 8.6-2, and the lag-lead design of this problem.

Problem 8.2-2

Consider the system of Fig. P8.2-2 with \(T=0.2\).

\[
G(z)=\frac{z-1}{z} z\left[\frac{1}{s^{2}(s+1)}ight]=\frac{0.01873 z+0.01752}{(z-1)(z-0.8187)}
\]

Fig. P8.2-2

Problem 8.4-2

To satisfy the steady-state constraints for the system of Problem 8.2-3, the de gain of the digital controller must be equal 4.

Problem 8.6-2

Repeat Problem 8.4-2 using a phase-lead controller. In part (c), the overshoot is approximately 26 percent.

Transcribed Image Text:

D(z) T 1-87 S m(1) C(s) 1 s(x+1)