1(a) Explain the effect of addition of pole and zero to the system

1(b) Explain any five rules of Root Locus Plot in detail.

1(c) Define Gain margin and Phase margin. Explain how these margins are used for stability analysis.

1(d) Explain the Mason’s gain formula with reference to Signal Flow Graph Technique

1(e) Explain needs of compensation in control system also explain different types of Compensation with suitable example.

2(a) Using block reduction technique, obtain the transfer function

2(b) Construct SFG for the following set of equation.

i) Y2 = G1Y1 – G2Y4

ii) Y3 = G3Y2 + G4Y3

iii) Y4 = G5Y1 + G6Y3, Where Y4 is the output.

Obtain the overall transfer function by using Mason’s gain formula.

3(a) Explain Controllability and Observability with the necessary condition for stability and Check Controllability and Observability for the system $$ x = \begin{bmatrix} 0 & 6 & -5\\\ 1 & 0 & 2\\\ 3&2&4 \end{bmatrix} \space x + \begin{bmatrix} 0 \\\ 1 \\\ 2 \end{bmatrix} \space u $$

$$ y = \begin{bmatrix} 1 & 3 & 0 \end{bmatrix} \space x$$

3(b) Explain PID Controller and Model Predictive control system in detail? Also list its advantages.

4(a) Construct the Routh array and determine the stability of the system whose characteristics equation is $s^6 + 2s^5 + 8s^4 + 12s^3 + 20s^2 + 16s + 16 = 0$

4(b) Sketch the root locus for a unity feedback control system and forward transfer function is G(s) = $\frac{k(s+3)}{s(s+2)(s+1)(s+4)}$

Find the frequency and gain K for which the root locus crosses the imaginary axis. For what range of k is the system stable?

5(a) Construct the Bode Plot for the open loop transfer function. G(s) = $\frac{288(s+4)}{5(s+1)(s^2 +48s + 144)}$ H(s) = 1. Comment on Stability.

5(b) State and Prove properties of State Transition matrix. Obtain the state model for the system with transfer function : $\frac{Y(s)}{V(s)} = \frac{3s+4}{s^2 +5s +6}$

6(a) Sketch the Nyquist plot for a given open loop transfer function $\frac{Y(s)}{V(s)} = \frac{1}{(s+1)(s+2)}$

And comment on the stability of the system.

6(b) A unity feedback system has G(s) = $\frac{20(s+1)}{(s^2)(s+2)(s+4)}$

Find :

i. All static error co-efficient (Kp, Kv, Ka).

ii. Steady State Error of ramp i/p with magnitude 4.