Question

A unity feedback control system has the open loop TF as: \(G(s)=\frac{K(s+a+1)(s+b)}{s(s+a)(s+a+2)}\)

a) Find analytical expressions for the magnitude and phase response for \(\mathrm{G}(\mathrm{s}) .\left[K=K_{1}\right]\)

b) Make a plot of the log-magnitude and the phase, using log-frequency in rad/s as the ordinate. \(\left[K=K_{1}\right]\)

c) Sketch the Bode asymptotic magnitude and asymptotic phase plots. \(\left[K=K_{1}\right]\)

d) Compare the results from \((a),(b)\), and \((c) .\left[K=K_{1}\right]\)

e) Using the Nyquist criterion, find out if system is stable. Show your steps. \(\left[K=K_{1}\right]\)

f) Using the Nyquist criterion, find the range of \(K\) for stability.

g) Use bode plots to calculate the gain and phase margins if the value of \(K\) is:

- \(0.1 K_{1}\)

- \(10 K_{1}\)

- \(100 K_{1}\)

h) Use Matlab to check the results from (g).

i) Use Matlab to display the gain and phase margins on:

- Nyquist plot

- Bode plot