The open-loop transfer function of a system is G(s)=\frac{500 K}{(s+3)(s+7)(s+12)} \quad \text { and } \quad H(s)=1 (i)State the rules for sketching a Bode Plot (ii) Explain the terms: gain margin and phase margin. Find the frequency domain transfer function for the open loop system. Based on Bode plot in Figure Q2(c) below, estimate graphically the phase andgain margins.

) Find the range of gain k, to yield stability for the closed loop system with aunity feedback. -) A plant has following transfer function. G(s)=\frac{1}{\left(s^{3}+2 s^{2}+5 s\right)} use the Ziegler-Nichols tuning rules (second method) to design a PID controller,and clearly state the gain values. [The Ziegler-Nichols tuning rules (second method) recommends the following settings: K, = 0.6K., T, = 0.5T. andT, = 0.125T, where Kp is the proportional gain, T; is the integral time constant(reset time), Ta is the derivative time constant (rate time), K is the critical gain and T is the corresponding period of oscillation].