## Control System Engineering - May 2016

### 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.

### Do as directed (short questions)

**1(a)** List out the features of negative feedback in a closed loop system.(1 marks)
**1(b)** What is steady-state error?(1 marks)
**1(c)** Define pole, zero and order of a control system.(1 marks)
**1(d)** What are the advantages of state space analysis?(1 marks)
**1(e)** Draw the block diagram/signal flow graph representation of the system described by the state model $ \begin{bmatrix}
x_1\\\\
x_2\\\\
x_3
\end{bmatrix}=\begin{bmatrix}
a_1 & a_2 & 0\\\\
1 & 0 & 1\\\\
0 & 1 & 0
\end{bmatrix}\begin{bmatrix}
x_1\\\\
x_2\\\\
x_3
\end{bmatrix}+\begin{bmatrix}
1\\\\
0\\\\
0
\end{bmatrix}u\ \text{and}\ y=x_3. $(1 marks)
**1(f)** The closed loop transfer function of a second order system is given by $ \dfrac{200}{s^2+20s+200} $. Determine the damping ratio and natural frequency of oscillation.(1 marks)
**1(g)** What is polar plot? Draw the polar plot of G(s) = 1/(1 + sT).(1 marks)
**1(h)** With graphical representation, explain How the roots of characteristic equation are related to stability?(1 marks)
**1(i)** What is Nyquist stability criterion?(1 marks)
**1(j)** How will you find the gain K at a point on root locus?(1 marks)
**1(k)** Sketch the frequency response (bode) plot of G(s) = 1/(1 + sT).(1 marks)
**1(l)** What is the effect on system performance, when a proportional
controller is introduced in a system?(1 marks)
**1(m)** State the transfer function of lead compensator and draw its pole-zero plot.(1 marks)
**1(n)** What is PD-Controller and what are its effect on system performance?(1 marks)
**2(a)** For the given mechanical translation system as shown in Fig. 1. Write down differential equations, represents in Force-Voltage analogy.

**2(b)**Obtain the state space representation of armature controlled DC motor with load. Consider armature current i

_{a}, the angular displacement of shaft θ , and the speed dθ/dt as state variables, and θ as output variable.(4 marks)

### Solve any one question from Q.2(c) & Q.2(d)

**2(c)** A linear feedback control system has the block diagram shown in Fig. 2. Using block diagram reduction rules, obtain overall transfer function C(s)/R(s) .

**2(d)**For the signal flow graph shown in Fig. 3, using Masson's gain formula determine the overall transmission C/R. (7 marks)

### Solve any three question from Q.3(a), Q.3(b), Q.3(c) & Q.3(d), Q.3(e), Q.3(f)

**3(a)** Define thermal resistance and thermal capacitance. Also derive the transfer function of Thermometer placed in water bath as a Thermal system.(3 marks)
**3(b)** Find the position, velocity and acceleration error constants, for a unity feedback control system has the open loop transfer function G(s) = 10(s+2)/s^{2}(s+1).(4 marks)
**3(c)** Obtain the state model and give block diagram representation for a
system whose closed-loop transfer function is given as, $$\dfrac{Y(s)}{U(s)}=\dfrac{10(s+4)}{s(s+1)(s+3)}$$(7 marks)
**3(d)** Using suitable diagram derive the transfer function of liquid level
system with interaction.(3 marks)
**3(e)** A unity feedback system has a open loop transfer function of $ G(s)=\dfrac{20(s+5)}{s(s+0.1)(s+3)} $. Determine the steady-state error for parabolic input.(4 marks)
**3(f)** The open-loop transfer function of a unity feedback system is given by G(s) = K/s (sT+1). where K and T are positive constant. By what factor should the amplifier gain K be reduced, so that the peak overshoot of unit step response of the system is reduced from 75% to 25%.(7 marks)

### Solve any three question from Q.4(a), Q.4(b), Q.4(c) & Q.4(d), Q.4(e), Q.4(f)

**4(a)** State advantages and limitations of Routh's stability criterion.(3 marks)
**4(b)** Using R-H criterion determine the relation between K and T so that
unity feedback control system whose open-loop transfer function given is stable $ G(s)=\dfrac{K}{s[s(s+10)+T]}. $(4 marks)
**4(c)** Investigate the stability of a closed-loop system whose open-loop transfer function is $ G(s)H(s)=\dfrac{5}{s(1+5s)} $ using Nyquist stability criterion.(7 marks)
**4(d)** What do you understand by absolute stability and relative stability? Which method indicates what type of stability?(3 marks)
**4(e)** Explain, How the gain and phase margin are obtained from Nyquist
Plots?(4 marks)
**4(f)** Draw the Bode plot for a system having $ G(s)H(s)=\dfrac{100}{s(s+1)(s+2)}. $ Find out Gain margin, Phase margin, Gain crossover frequency and phase cross over frequency.(7 marks)

### Solve any three question from Q.5(a), Q.5(b), Q.5(c) & Q.5(d), Q.5(e), Q.5(f)

**5(a)** What is breakaway and breakin point? How to determine them?(3 marks)
**5(b)** Determine the relation between the phase margin and damping ratio for an underdamped second-order system.(4 marks)
**5(c)** Sketch the root locus of the system whose open-loop transfer function is $ G(s)=\dfrac{K}{s(s+2)(s+4)}. $ Find the values of K so that the damping ratio of the closed-loop system is 0.5.(7 marks)
**5(d)** How will you obtain the transfer function from Bode magnitude plot?(3 marks)
**5(e)** What is lag compensator? With respect to the electrical equivalent phase-lag compensator state the transfer function, draw its pole-zero plot, and the bode plot of lag compensator.(4 marks)
**5(f)** Design suitable lead compensator for a system with unity feedback and having open-loop transfer function $$G(s)=\dfrac{K}{s(s+8)}.$$

to meet the following specifications:

(i) Percentage peak overshoot = 9.5%

(ii) Natural frequency of oscillation, ω_{n} = 12 rad / sec

(iii) Velocity error constant, K_{v} ≥ 10.(7 marks)