EC-404 Control System - Dec 2024 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
   aDefine transfer function. Derive the expression for closed loop transfer function.
   How to answer

   • Define transfer function as ratio of Laplace of output to input with zero initial conditions
   • Draw general closed-loop block with forward path G(s) and feedback H(s)
   • Write equations C(s)=G(s)E(s), E(s)=R(s)-H(s)C(s)
   • Derive C(s)/R(s)=G(s)/(1+G(s)H(s)), unity feedback case
2. 7 Marks
   bFind the transfer function of given circuit.

   

   How to answer

   • Convert circuit to s-domain: $R_1 \parallel 1/SC_1$, $SL_1 \parallel R_2$
   • Find equivalent series and shunt impedances
   • Apply voltage divider $V_o(s)/V_1(s)=Z_2/(Z_1+Z_2)$
   • Simplify to rational function in $s$
3. 7 Marks
   aFind the transfer function for the block diagram shown as below.

   

   How to answer

   • Reduce inner loops G1 with H1 and G2 with H2 to G1/(1+G1H1) and G2/(1+G2H2)
   • Combine cascaded blocks and resolve outer feedback H3
   • Apply block reduction / Mason's rule step by step
   • Write final C(s)/R(s) expression
4. 7 Marks
   bDetermine the transfer function of signal flow graph as shown in figure by using Mason's gain formula.

   

   How to answer

   • Identify nodes 1 to 6, forward path gains and loops
   • List forward paths Pk and individual loops L1, L2, etc.
   • Compute graph determinant Delta and path cofactors Delta_k
   • Apply Mason's formula T = sum(Pk*Delta_k)/Delta
5. 7 Marks
   aExplain about various test signals used in control system.
   How to answer

   • Define step, ramp, parabolic, impulse and sinusoidal signals with Laplace forms
   • Give mathematical and graphical representation of each
   • Explain use for transient and steady-state analysis
   • Mention which signal tests which error constant
6. 7 Marks
   bThe closed loop transfer function of a unity feedback control system is given by $$\frac{C(s)}{R(s)} = \frac{20}{s^2 + 16s + 25}$$ Determine:

   1. i)Damping ratio
   2. ii)Natural undamped resonance frequency
   3. iii)Percentage peak overshoot
   How to answer

   • Compare denominator with $s^2+2\zeta\omega_n s+\omega_n^2$ to get $\omega_n$ and $\zeta$
   • Compute $\omega_n = 5$, $\zeta = 1.6$ indicating overdamped
   • Compute % peak overshoot formula $e^{-\pi\zeta/\sqrt{1-\zeta^2}}$ (zero for >1)
   • Interpret results for second-order unity feedback
7. 7 Marks
   aWhat is root locus? Describe steps involved in sketching root locus.
   How to answer

   • Define root locus as locus of closed-loop poles with varying gain K
   • List construction rules: poles, zeros, asymptotes, breakaway, angles
   • Explain steps for sketching in sequence
   • Mention use for stability and design
8. 7 Marks
   bApply Routh Hurwitz stability criteria to determine the stability for given characteristic equation. $$s^5 + s^4 + 2s^3 + 2s^2 + 3s + 5 = 0$$
   How to answer

   • Form Routh array for $s^5+s^4+2s^3+2s^2+3s+5=0$
   • Handle special cases: zero row / auxiliary equation if needed
   • Check first-column sign changes to count RHP poles
   • Conclude stability / instability
9. 7 Marks
   aWrite a short note on polar plot.
   How to answer

   • Define polar plot as G(jw) magnitude vs phase for w 0 to infinity
   • Explain procedure to plot magnitude and angle
   • Mention features: start/end points, crossings
   • State use for stability and frequency response
10. 7 Marks
    bState Nyquist stability criterion and describe procedure of assessment of relative stability using Nyquist plot.
    How to answer

    • State Nyquist criterion Z = N+P with encirclement of -1+j0
    • Describe Nyquist path and mapping procedure
    • Explain gain margin, phase margin for relative stability
    • Relate encirclements to stability assessment
11. 7 Marks
    aDiscuss types of compensation techniques and design of phase-lag compensators.
    How to answer

    • List series, parallel, feedback and lag, lead, lag-lead compensation
    • Explain phase-lag characteristics and transfer function
    • Give design steps using Bode plot for lag compensator
    • Mention effects on steady-state error and stability
12. 7 Marks
    bWrite a short note on Bode Plot.
    How to answer

    • Define Bode plot as log magnitude and phase vs log frequency
    • Explain procedure: corner frequencies, slopes, asymptotes
    • Mention determination of GM, PM and stability
    • Give example sketch for typical factor
13. 14 Marks
    Derive the relationship between '$\alpha$' and '$\phi m$' for a lead compensator.
    How to answer

    • Write lead compensator $G_c(s)=(1+Ts)/(1+\alpha Ts)$ with $\alpha<1$
    • Derive max phase $\phi_m$ at $\omega_m$ by differentiating phase
    • Show $\sin\phi_m=(1-\alpha)/(1+\alpha)$ and $\omega_m$
    • Relate $\alpha$ to required phase lead
14. 7 Marks
    aConvert the following transfer function into state-space form: $$\frac{V(s)}{U(s)} = \frac{k(c_2s + c_1)}{s^3 + a_3s^2 + a_2s + a_1}$$
    How to answer

    • Write transfer function in controllable/observable canonical form
    • Define state vector and derive A matrix from denominator coefficients
    • Form B matrix from numerator coefficients $k,c1,c2$
    • Write output matrix C and D
15. 7 Marks
    bObtain the solution of state equation and write the properties of state transition matrix.
    How to answer

    • Write state equation solution $x(t)=e^{At}x(0)+\int e^{A(t-\tau)}Bu d\tau$
    • Derive state transition matrix $\phi(t)=e^{At}=L^{-1}[(sI-A)^{-1}]$
    • List properties: $\phi(0)=I$, $\phi^{-1}=\phi(-t)$, semigroup
    • Mention significance for stability and response