EX-405 Control System - Jun 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
   aObtain the expression for $y(t)$ which is satisfying the differential equations $\frac{d^2y(t)}{dt^2}+6\frac{dy(t)}{dt}+8y(t)=16e^{-t}$ Neglect initial conditions.
   How to answer

   • Take Laplace transform with zero initial conditions.
   • Obtain Y(s) = 16/((s+1)(s+2)(s+4)).
   • Partial fraction expansion.
   • Inverse Laplace to get y(t) for t>=0.
2. 7 Marks
   bFor the mechanical system shown, obtain the F-V analogous electrical network.

   

   How to answer

   • Write mechanical equilibrium equations for M1 and M2.
   • Map force to voltage, velocity to current for F-V analogy.
   • Map mass to inductance, damper to resistance, spring to capacitance.
   • Draw series-parallel electrical network with loops for x1, x2.
3. 7 Marks
   aUsing Mason's gain formula to find out C/R, of the signal flow graph shown.

   

   How to answer

   • Identify forward paths and individual loops.
   • Find non-touching loops and determinant Delta.
   • Find path co-factors Delta_k.
   • Apply Mason's formula C/R = sum P_k Delta_k / Delta.
4. 7 Marks
   bDerive the expressions for peak time and rise time in terms of $\xi$ and $w_n$ for a second order control system.
   How to answer

   • Write standard second-order step response with damping ratio.
   • Differentiate and set to zero for peak time.
   • Obtain Tp = pi / (wn*sqrt(1-xi^2)).
   • Derive rise time Tr expression in terms of xi and wn.
5. 7 Marks
   aFor a control system in fig., find the values of $K_1$ and $K_2$ that $M_p = 25\%$ and $T_p = 4$ sec. Assume unit step input.

   

   How to answer

   • Find closed-loop transfer function C(s)/R(s).
   • Compare with standard second-order form to get xi, wn.
   • Use Mp and Tp formulas to solve for xi, wn.
   • Solve for K1 and K2.
6. 7 Marks
   bDetermine static error coefficients of a unity feedback system with $G(s)=\frac{K}{s^2+(s+20)(s+30)}$.
   How to answer

   • Identify type and order from G(s).
   • Compute position coefficient Kp = lim G(s).
   • Compute velocity coefficient Kv = lim sG(s).
   • Compute acceleration coefficient Ka = lim s^2 G(s).
7. 7 Marks
   a$s^6+4s^5+3s^4-16s^2-64s-48=0$. Find the number of roots of this equation with positive real part, zero real part and negative real part.
   How to answer

   • Form Routh array for 6th order polynomial.
   • Handle missing s^3 term and zero row if any.
   • Count sign changes in first column.
   • Infer numbers with positive, zero and negative real parts.
8. 7 Marks
   bThe open loop transfer function of a control system is given by $G(s)=\frac{K}{s(s+2)(s^2+6s+25)}$ sketch the complete root locus as K is varies from 0 to infinity.
   How to answer

   • Find poles, zeros and number of branches.
   • Draw real-axis sections, asymptotes and centroid.
   • Find breakaway points and jw crossings.
   • Sketch complete locus for K 0 to infinity.
9. 7 Marks
   aWhat are the effects adding open loop poles and zero on the nature of the root locus and on system?
   How to answer

   • State effect of adding pole on locus shift and stability.
   • State effect of adding zero on locus shift and stability.
   • Explain with angle condition and examples.
   • Summarize relative stability implications.
10. 7 Marks
    bFor a particular unity feedback system $G(s)=\frac{242(s+5)}{s(s+1)(s^2+5s+121)}$ Sketch the Bode Plot. Find $\omega_{gc}$ and $\omega_{pc}$, G.M., P.M. Comment on stability.
    How to answer

    • Write G(jw) and corner frequencies.
    • Draw asymptotic magnitude and phase plots.
    • Find gain crossover, phase crossover, GM, PM.
    • Comment on closed-loop stability.
11. 7 Marks
    aExplain the concept of gain margin and phase margin. Explain how these values help in studying relative stability.
    How to answer

    • Define gain margin and phase margin.
    • Show them on Bode/Nyquist plots.
    • Relate positive margins to relative stability.
    • Explain use for design specifications.
12. 7 Marks
    bFor a certain control system $G(s)H(s)=\frac{K}{s(s+2)(s+10)}$ Sketch the Nyquist plot and hence calculate the range of values of K for stability.
    How to answer

    • Map Nyquist contour avoiding pole at origin.
    • Plot G(jw)H(jw) and mirror for negative frequencies.
    • Apply Nyquist stability criterion N = Z-P.
    • Find K range for no encirclement of -1+j0.
13. 7 Marks
    aFind the eigen values, eigen matrix and model matrix for, $A=\begin{bmatrix}0 & 0 & 1\\2 & 0 & 0\\8 & 2 & -5\end{bmatrix}$.
    How to answer

    • Form characteristic equation |lambda I - A|=0.
    • Solve cubic for eigenvalues.
    • Find eigenvectors for each eigenvalue.
    • Form eigen matrix and modal matrix.
14. 7 Marks
    bWrite down properties of state transition matrix.
    How to answer

    • Define state transition matrix phi(t)=exp(At).
    • List properties: phi(0)=I, phi inverse, composition.
    • Include differentiation and similarity properties.
    • Brief justification of each property.
15. 14 Marks
    Write a short notes on (any two)

    1. i)Tacho-Generators
    2. ii)Polar plot
    3. iii)AC servomotors
    4. iv)Proportional, derivative and integral control
    How to answer

    • Select any two topics from list.
    • Give principle, construction/working and diagram.
    • Mention transfer function or applications.
    • Conclude with advantages/uses in control.