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EX-405 · Control System/Unsolved PYQ Paper

EX-405 Control System - Jun 2025 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
    aFor a given rotational system, obtain the electrical analogous systems based on inverse and direct analogous methods.
    Rotational mechanical system with input torque T(t), shaft with stiffness K1 and displacement theta1(t), inertia J1 with displacement theta2(t) and friction B1 to ground, coupled via stiffness K' and damper B' to shaft with displacement theta3(t), inertia J2 with friction B2 to ground, connected via stiffness K2 to fixed wall.
    How to answer
    • Write differential equations for rotational system with inertias J1, J2, stiffness K1, K2, K', damping B1, B2, B'
    • Force-voltage (direct) analogy: torque->voltage, angular velocity->current, J->L, B->R, K->1/C
    • Force-current (inverse) analogy: torque->current, angular velocity->voltage, J->C, B->1/R, K->1/L
    • Draw both analogous electrical networks with loops/nodes corresponding to theta1, theta2, theta3
  2. 7 Marks
    bUsing Mason's gain formula, find the gain of the following system in figure below.
    Signal flow graph from R(s) to C(s) with unity branches at ends. Intermediate nodes with branches G4 (top), G1 (bottom) from first node, G5 with self-loop -H1 on top, G2 with self-loop -H2 at bottom, cross branches G4 and G7, and merging branches G6 (top) and G3 (bottom) to output node.
    How to answer
    • Identify forward paths Pk and individual loops from signal flow graph with gains G1-G7, H1, H2
    • Find non-touching loops and compute graph determinant Delta = 1 - sum L1 + sum L2 ...
    • Compute path cofactors Delta_k for each forward path
    • Apply Mason's formula T = (1/Delta) sum Pk Delta_k to get C(s)/R(s)
  3. 7 Marks
    aDerive transfer function of A.C. servomotor stating the assumption made.
    How to answer
    • State assumptions: linear magnetic circuit, negligible saturation, two-phase balanced operation, constant rotor resistance
    • Write torque-speed characteristics and motor equations for control and reference phases
    • Linearize around operating point to get transfer function Theta_m(s)/E(s)
    • Give final form Km/(s(Tm s+1)) and define motor gain and time constant
  4. 7 Marks
    bDerive the relationship maximum overshoot and damping ratio of a second order system. Draw the graph or overshoot versus damping ratio.
    How to answer
    • Write step response of standard second-order system with damping ratio zeta
    • Differentiate to find peak time and peak value c(tp)
    • Obtain Mp = exp(-pi*zeta/sqrt(1-zeta^2)) and % overshoot formula
    • Sketch Mp vs zeta decaying curve from 1 to 0
  5. 7 Marks
    aThe Fig. shows PD controller used for the system. Determine the value of $T_d$ so that system will be critically damped. Calculate its settling time.
    Block diagram: R(s) into summing junction minus feedback C(s), forward branch splits to 1 and sTd blocks summed with + sign, then into block 4/(s(s+1.6)) to output C(s) with unity feedback to first summer.
    How to answer
    • Reduce PD block (1+sTd) in forward path with plant 4/(s(s+1.6)) to closed-loop characteristic equation
    • Apply critical damping condition zeta=1 to solve for Td
    • Compute natural frequency wn and settling time ts = 4/(zeta*wn)
    • State numerical values of Td and ts
  6. 7 Marks
    bDerive the expressions for static error coefficients? How these coefficients are useful in determining steady state error? State the limitations of static error coefficient method.
    How to answer
    • Define position, velocity, acceleration error constants Kp, Kv, Ka as limits of G(s)
    • Derive ess for step, ramp, parabolic inputs in terms of Kp, Kv, Ka
    • Explain use for type 0,1,2 systems to predict finite/zero/infinite error
    • State limitations: only for unity feedback, stable systems, standard inputs
  7. 7 Marks
    aFor a unity feedback system, $$G(s) = \frac{K}{s(1+0.4s)(1+0.25s)},$$ find range of values of K, marginal value of K and frequency of sustained oscillations.
    How to answer
    • Form closed-loop characteristic equation 0.1 s^3 + ... + K = 0
    • Apply Routh-Hurwitz criterion to get stability condition on K
    • Set row of zeros for marginal K and solve auxiliary equation for oscillation frequency
    • Report range 0<K<Kmar, Kmar and omega_n
  8. 7 Marks
    bSketch the complete root locus with approximate indication of breakaway points. Comment on the stability.
    How to answer
    • Determine poles, zeros, asymptotes, centroid and real-axis segments
    • Calculate breakaway points from dK/ds=0 and angle of departure if complex poles
    • Sketch root locus branches approaching asymptotes/infinity
    • Comment on stability range of K from crossing of imaginary axis
  9. 7 Marks
    aExplain the procedure of obtaining the value of K for specified damping ratio $\xi$ from the root locus.
    How to answer
    • Draw constant zeta line at angle theta = cos^-1(zeta) from negative real axis
    • Intersection of this line with root locus gives closed-loop poles for required zeta
    • Apply magnitude condition |K G(s)H(s)|=1 at that point to find K
    • Verify with calculation steps
  10. 7 Marks
    bA unity feedback control system has $$G(s) = \frac{80}{s(s+2)(s+20)}$$ Drawn the Bode Plot. Determine G.M. P.M. $\omega_{gc}$ and $\omega_{pc}$ Comment on the stability.
    How to answer
    • Rewrite G(s) in time-constant form and identify corner frequencies 2 and 20 rad/s
    • Draw magnitude (dB) and phase asymptotes and actual Bode plot
    • Find gain crossover wgc, phase crossover wpc, GM and PM from plot
    • Comment stable if GM>0 dB and PM>0 degrees
  11. 7 Marks
    aFor a feedback control system $$G(s)H(s) = \frac{40}{(s+4)(s^2+2s+2)}$$ Find gain margin and stability from Nyquist plot.
    How to answer
    • Map Nyquist contour and plot G(jw)H(jw) for w from 0 to infinity
    • Find phase crossover where plot crosses negative real axis
    • Compute gain margin GM = 1/|G(jwpc)H(jwpc)|
    • Apply Nyquist stability criterion Z = P - N to comment on stability
  12. 7 Marks
    bWhat is Polar plot? Explain polar plots for Type 0, 1 and 2 systems.
    How to answer
    • Define polar plot as G(jw) magnitude vs phase for w 0 to infinity
    • Sketch Type 0 starting on positive real axis ending at origin
    • Sketch Type 1 starting at -90 deg infinity ending at origin
    • Sketch Type 2 starting at -180 deg infinity ending at origin
  13. 7 Marks
    aA unit step input is applied to a unity feedback control system whose open loop transfer function is given by. $$G(s) = \frac{K}{s(sT+1)}$$ Determine the values of K and T given that maximum overshoot as 26 % and resonant frequency $\omega_r = 8 rad / s$. Calculate the resonance peak $M_r$.
    How to answer
    • Relate % overshoot 26% to damping ratio zeta via Mp formula
    • Relate resonant frequency wr=8 rad/s and bandwidth to wn and zeta
    • Solve two equations for K and T using standard form K/T/(s^2+s/T+K/T)
    • Compute resonant peak Mr = 1/(2 zeta sqrt(1-zeta^2))
  14. 7 Marks
    bDefine a different steps of Lag-lead compensation network.
    How to answer
    • State need for lag-lead compensation and its transfer function form
    • List steps: determine required phase lead, place zero-pole, design lag section
    • Mention selection of beta, time constants and verification via Bode/root locus
    • Conclude with implementation check
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