Question

# A unity feedback control system has the open loop TF as

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

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