Since the new ω_g should be 10 rad/sec, the required additional phase at ω_g, to maintain the specified PM, is 45 - (180 - 198) = 63° . With safety margin 2°,

$\displaystyle \alpha_2 = \left( {\frac{{1 - \sin (65^o )}}{{1 + \sin (65^o )}}} \right) = 0.05$

And

$\displaystyle 10 = \frac{1}{{\tau_2 \sqrt \alpha_2 }}$

which gives $\tau_2=0.45$ . However, introducing this compensator will actually increase the gain crossover frequency where the phase characteristic will be different than the designed one. This can be seen from Figure 3.

$\includegraphics[width=5.0in]{m5l8fig3.eps}$

Figure 3: Frequency response of the system in Example 1 with only a lead compensator

The gain crossover frequency is increased to 23.2 rad/sec. At 10 rad/sec, the phase angle is -134° and gain is 12.6 dB. To make this as the actual gain crossover frequency, lag part should provide an attenuation of -12.6 dB at high frequencies.

At high frequencies the magnitude of the lag compensator part is $1/\alpha_1$ . Thus ,

$\displaystyle 20 \log_{10} \alpha_1 = 12.6$

which gives $\alpha_1=4.27$ . Now, $1/\tau_1$ should be placed much below the new gain crossover frequency to retain the desired PM. Let $1/\tau_1$ be 0.25. Thus

$\displaystyle \tau_1= 4$

The overall compensator is

$\displaystyle C(s) = 30 \; \frac{1+4s}{1+17.08 s} \; \frac{1+0.45 s}{1+0.0225 s}$