Homework 5: Frequency Response Cascade Compensator Design¶
Problems¶
The uncompensated loop-transfer function of a system is $\(G(s)H(s)=\frac{2}{s(s+2)}.\)\( Assuming unity-gain feedback, design a cascade lead compensator to achieve a velocity constant \)K_v$ of 20 and a phase margin of 45°.
Design a lag compensator for the system of Question 1 to achieve the same design constraints. Compare the relative merits of the two approaches.
The open-loop transfer function of a position control system is $\(Go(s)=\frac{25}{s\left(1+\frac{1}{4}s\right)\left(1+\frac{1}{16}s\right)}.\)$ The system’s gain and phase margin are to exceed 1.5 and 15° respectively. Determine whether these specification are satisfied and if not design a lead compensator to meet the specifications and also to maintain the open-loop gain at 25.
Using frequency response methods, design a compensator to achieve a step-response with a rise time \(t_r \le 0.4\) s, a peak overshoot \(\%OS \le 20\%\) and a step error constant \(K_p = 20\) for the system with plant transfer function $\(\frac{3}{(s+1)(s+3)}.\)\( Estimate the closed-loop bandwidth of the compensated system and the resonant peak \)M_{\mathrm{max}}$·
A type 2 servomechanism has transfer function $\(G(s)H(s) = \frac{0.25}{s^2(1+0.25s)},\;H(s)=1.\)\( Show the effect on stability of adding the cascade lead network: \)\(D(s)=\frac{1}{16}\left(\frac{1+4s}{1+0.25s}\right)\)\( and a pre-amplifier with gain \)K_p = 16$.
Repeat the design of Question 4 using the w-transform method to determine the parameters of a suitable digital compensator. Assume that the sampling frequency \(\omega_s = 10\omega_{\mathrm{Bw}}\). Write down the difference equation of the resulting digital compensator.