EC-404 Control System - Jun 2022 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aExplain the classification of control systems.
   How to answer

   • Define open-loop vs closed-loop systems
   • Classify by feedback, linearity, time-variance, continuity, etc.
   • Give examples of each type
2. 7 Marks
   bDefine transfer function and determine the transfer function of RLC series circuit if the voltage across the capacitor is a output variable and input is voltage source $v(s)$.
   How to answer

   • Define transfer function as ratio of Laplace output to input with zero initial conditions
   • Write RLC series differential equation in s-domain
   • Derive Vc(s)/V(s) = (1/sC)/(R+sL+1/sC)
3. 7 Marks
   aDiscuss basis for framing the rules of block diagram reduction technique? What are drawbacks of the block diagram reduction technique?
   How to answer

   • State need for block diagram reduction rules
   • List shifting of summing point and take-off point rules
   • Discuss drawbacks like loss of physical insight, tedious for complex systems
4. 7 Marks
   bFind the transfer function for the block diagram shown as below.

   

   How to answer

   • Identify forward blocks G1, G2, G3 and feedbacks H1, H2
   • Apply series, parallel and feedback reduction steps
   • Obtain overall C(s)/R(s) expression
5. 4 Marks
   aExplain about various test signals used in control systems?
   How to answer

   • Define purpose of test signals
   • Describe step, ramp, parabolic and impulse signals with Laplace forms
   • Explain use in transient and steady-state analysis
6. 10 Marks
   bThe closed loop transfer function of a unity feedback control system is given by $C(s)/R(s) = 20/(s^2+16s+25)$ Determine:

   1. i)Damping ratio
   2. ii)Natural undammed resonance frequency
   3. iii)Percentage peak overshoot
   4. iv)Expression for error response
   How to answer

   • Compare denominator with $s^2+2\zeta\omega_n s+\omega_n^2$ to find $\zeta, \omega_n$
   • Calculate peak overshoot $M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}}$
   • Write error response expression for second-order system
7. 5 Marks
   aWhat are the necessary conditions to have all the roots of characteristics equation in the left half of s-plane?
   How to answer

   • State Routh-Hurwitz necessary conditions
   • All coefficients positive and non-zero
   • All powers of s present in characteristic equation
8. 9 Marks
   bFor the unity feedback system the open loop T.F. $$G(s)=\frac{K}{s(1+0.6s)(1+0.4s)}$$ Determine

   1. a)Range of values of K
   2. b)Marginal K
   3. c)Frequency of sustained oscillations
   How to answer

   • Form characteristic equation $1+G(s)=0$
   • Build Routh array and apply $K>0$ conditions
   • Find marginal K for zero row and frequency from auxiliary equation
9. 7 Marks
   aWrite short notes on various frequency domain specifications.
   How to answer

   • Define resonant peak, resonant frequency, bandwidth, gain and phase margins
   • Give formulas in terms of damping ratio
   • Explain significance in frequency response
10. 7 Marks
    bDerive expression for resonant peak and resonant frequency and hence establish correlation between time and frequency response.
    How to answer

    • Start from second-order closed-loop magnitude $M(j\omega)$
    • Differentiate to find resonant frequency $\omega_r = \omega_n\sqrt{1-2\zeta^2}$
    • Derive $M_r = 1/(2\zeta\sqrt{1-\zeta^2})$ and correlate with time response
11. 14 Marks
    The open loop transfer function of a system is $$G(s)=\frac{K}{s(1+s)(1+0.1s)}$$ Determine the value of K such that

    1. i)Gain Margin = 10dB
    2. ii)Phase Margin = 50 degree
    How to answer

    • Plot Bode / Nyquist to find phase cross-over and gain cross-over frequencies
    • Apply GM = -20log|G| and PM = 180+angle conditions
    • Solve for K for GM=10dB and PM=50 degree
12. 7 Marks
    aWhat is lag-lead compensator? Under what conditions it is employed?
    How to answer

    • Define lag-lead compensator with pole-zero combination
    • Explain need to improve both transient and steady-state
    • List conditions for employment
13. 7 Marks
    bDerive the relationship between '$\alpha$' and '$\Phi_{m}$' for a lead compensator.
    How to answer

    • Write lead compensator transfer function with $\alpha<1$
    • Find frequency of maximum phase lead $\omega_m$
    • Derive $\sin\Phi_m = (1-\alpha)/(1+\alpha)$ relation
14. 8 Marks
    aObtain the solution of state equation and list the properties of state transition matrix.
    How to answer

    • Write state equation solution $x(t)=e^{At}x(0)+\int e^{A(t-\tau)}Bu d\tau$
    • Define state transition matrix $\phi(t)=e^{At}$
    • List properties like $\phi(0)=I$, $\phi^{-1}(t)=\phi(-t)$
15. 6 Marks
    bTest the given system for Observability using Kalman's test. $$A=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -6 & -11 & -6 \end{bmatrix}, B=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$$
    How to answer

    • State Kalman's observability condition with $Q_o = [C^T A^TC^T ...]$
    • Substitute given A, B matrices (C missing)
    • Check rank for full observability