(Solved): (a) Use the Nyquist Stability Criterion to determine the Nyquist Locus and sk ...

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(a) Use the Nyquist Stability Criterion to determine the Nyquist Locus and sketch a clearly labelled Nyquist Diagram for the system described by the open loop transfer function, \( K(s) G(s) H(s) \) : \[ K(s) G(s) H(s)=\frac{k}{(s+2)(s+4)(s+6)} \] where \( K(s)=k \). Based on analysis of the Nyquist Diagram: (i) Comment on the stability of the closed loop system. (ii) Determine the range of \( k \) to ensure a stable closed loop response. (iii) Given that \( k \) is set to 48 : a. Calculate the Gain Margin. b. Calculate the DC Gain (i.e. the gain as \( \omega \) tends to \( 0^{+} \)) and then deduce the Phase Margin. [20 marks] (b) To improve steady state performance, the proportional controller, \( K(s)=k \), is replaced with a controller where; \[ K(s)=\frac{48(s+6)}{s} \] Use the Nyquist Stability Criterion to determine the Nyquist Locus and sketch a clearly labelled Nyquist Diagram for the new system. (i) Calculate the Gain and Phase Margin. (ii) Comment on system stability if there was also a small pure time delay included within the open loop transfer function.