## Control System Engineering - Dec 2015

### Electronics and Comm. Engg. (Semester 4)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1(a)** Write short notes on open loop control systems and closed loop control systems. Discuss their advantages and disadvantages.(7 marks)
**1(b)** Obtain system transfer function C(s)/R(s) using block diagram reduction
technique for the system shown in figure 1.(7 marks)
**2(a)** Derive Correlation Between Transfer Functions and State-Space Equations.(4 marks)

### Solved any one question from Q.2(b) & Q.2(c)

**2(b)** Explain Mason's gain formula.(3 marks)
**2(c)** Determine the state space model of the system shown in figure 2.(7 marks)

### Solved any one question from Q.3 & Q.4

**2(d)** Define transfer function.
Obtain the transfer
of the system defined by $$\begin{bmatrix}
\dot{x_1}\\
\dot{x_2}\\
\dot{x_3}
\end{bmatrix}=\begin{bmatrix}
-1 & 1 & 0\\
0 & -1 & 1\\
0 & 0 & -2
\end{bmatrix}\begin{bmatrix}
x_1\\
x_2\\
x_3
\end{bmatrix}+\begin{bmatrix}
0\\
0\\
1
\end{bmatrix}u\ \ \ \ \ y=\begin{bmatrix}
1 & 0 & 0
\end{bmatrix}\begin{bmatrix}
x_1\\
x_2\\
x_3
\end{bmatrix}$$(7 marks)
**3(a)** Define steady state error and derive the expressions for error constants K _{p } , K _{ v } and K_{ a } corresponding to step, ramp and parabolic input respectively.(7 marks)
**3(b)** Obtain the values of delay time t _{ d } , rise time t _{ r } , peak time t _{ p } , settling time t _{ s }
and peak overshoot M p for the given open loop transfer function of a unity feedback control system G(s)=16/s (s+6)(7 marks)
**4(a)** Derive the expressions
Of Rise time, Peak time and Peak overshoot for the system having close loop transfer function $$T(s)=\dfrac{C(s)}{R(s)}=\dfrac{{\omega _{n}}^{2}}{s62+2\xi \omega _ns+{\omega _{n}}^{2}}.$$(7 marks)

### Solved any one question from Q.5 & Q.6

**4(b)** The open loop transfer function of a unity feedback system is given by $$G(s)=\dfrac{k}{s(1+Ts)}$$ where k and T are constants. By what factor should the amplifier gain be reduced so that the peak overshoot of the system is reduced from 60% to 15% ?(7 marks)
**5(a)** Using Routh's criterion check the stability of a system whose characteristic equation is given by s ^{ 6 } +2s ^{ 5 } +8s ^{ 4 } +12s ^{ 3 } +20s ^{ 2 } +16s+16=0(7 marks)
**5(b)** What is Root locus? Sketch the Root locus plot for the unity feedback system having $$G(s)=\dfrac{K}{s(s+1)(s+3)(s+4)}$$(7 marks)
**6(a)** Determine range of k for system stability, for the given characteristic equation of Feedback control system s ^{ 4 } +2s ^{ 3 } +(4+k)s ^{ 2 } +9s+25=0(7 marks)

### Solved any one question from Q.7 & Q.8

**6(b)** Sketch the Root locus plot for the unity feedback system having an open loop transfer function $$G(s)=\dfrac{K}{s(s+3)(s^2+2s+2)}$$.(7 marks)
**7(a)** State and explain compensator? Explain Phase-Lead compensator in detail.(7 marks)
**7(b)** The feed forward transfer function of a close loop system is G(s)=1/s(s+1) and feedback transfer function is H(s) =1/(s+2).

(i) Draw the polar plot of G(s)H(s).

(ii) Find ? corresponding to $$\angle G(j\omega )H(j\omega )=180^{\circ}.$$

(iii) Find $$| G(j\omega )H(j\omega )|$$ corresponding to frequency obtain in (ii).(7 marks)
**8(a)** Draw the Nyquist plot for G(s)=1/s(s-1) and comment on system stability.(7 marks)
**8(b)** Determine gain margin and phase margin using bode plot for the system having transfer function $$G(s)H(s)=\dfrac{1}{s(1+s)(1+0.1s)}$$ and comment on stability.(7 marks)