EX-405 Control System - Nov 2023 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
   aWhat is a mason gain formula? Explain each component of the formula and mention its advantages over block diagram reduction techniques.
   How to answer

   • State Mason's gain formula T = (1/Delta) sum Pk Delta_k
   • Define forward path Pk, loop gains, Delta = 1 - sum L1 + sum L2 - ...
   • Define Delta_k as cofactor removing touching loops
   • Steps to apply to SFG
   • Advantages over block reduction: no stepwise reduction, systematic, complex MIMO, computer friendly
2. 7 Marks
   bObtain the overall transfer function C/R from the signal flow graph shown.

   

   How to answer

   • Draw given SFG with nodes and gains G1-G8, H2, H3
   • Identify forward paths and individual loops
   • Compute Delta and Delta_k
   • Apply Mason's formula C/R = sum Pk Delta_k / Delta
   • Simplify to overall transfer function
3. 7 Marks
   aDetermine the transfer function for the block diagram shown in Figure.

   

   How to answer

   • Redraw block diagram with G1, G2, G3, H1, H2
   • Shift / combine summing points and pickoff points
   • Reduce inner loops with feedback
   • Combine cascade blocks
   • Obtain C(s)/R(s) expression
4. 7 Marks
   bExplain time response of first order system to unit step and unit ramp input. Find the steady state error response for both.
   How to answer

   • Model first order system T(s)=1/(Ts+1)
   • Unit step response c(t)=1-exp(-t/T), error ess
   • Unit ramp response c(t)=t-T+T exp(-t/T), steady error = T
   • Plot/explain transient and steady state
   • Compare steady state errors for both inputs
5. 7 Marks
   aWhat are the standard test signals used in time domain analysis, explain each one in detail?
   How to answer

   • Define impulse, step, ramp, parabolic signals with Laplace
   • Give mathematical and graphical representation
   • Explain use in time domain analysis
   • Explain transient vs steady state testing
   • Give example of order suitability
6. 7 Marks
   bDiscuss the derivative and integral control in detail.
   How to answer

   • Define PD/PI actions: derivative on error rate, integral on accumulation
   • Give transfer functions Kd*s and Ki/s
   • Explain effect on transient, steady error, stability
   • Explain practical implementation issues
   • Compare applications
7. 7 Marks
   aComment on the stability of the system whose characteristic equation is given by $$s^5 + 2s^4 + 3s^3 + 6s^2 + 2s + 1 = 0.$$
   How to answer

   • Form Routh array for s^5+2s^4+3s^3+6s^2+2s+1=0
   • Check first column sign changes and zero rows
   • Identify auxiliary equation if needed
   • Conclude number of RHP poles
   • Comment on absolute/relative stability
8. 7 Marks
   bSketch the root locus for a system having transfer function as $$G(s)=\frac{k(s+1)}{(s+3)}$$
   How to answer

   • Identify pole at s=-3 and zero at s=-1, gain k
   • Plot real-axis locus, breakaway, angles
   • Sketch root locus branches vs k
   • Mark stability range
   • Conclude locus shape
9. 7 Marks
   aConsider the following polar plot shown in Figure below. If now a pole at origin and a pole at $s = -1/T_2$ are added, sketch the polar plot.

   

   How to answer

   • Describe given GH-plane polar plot (omega=0 to infinity)
   • Add 90 deg per pole at origin and lag of 1/(1+T2 s)
   • Rotate start/end by -180 deg total
   • Sketch modified polar plot shape
   • Note intersections with axes
10. 7 Marks
    bSketch the bode plot for the following transfer function and determine phase margin and gain margin. $$G(s)=\frac{75(1+0.2s)}{[s(s^2+16s+100)]}$$
    How to answer

    • Rewrite G(s)=75(1+0.2s)/[s(s^2+16s+100)] in Bode form
    • Identify corner frequencies 5 rad/s and 10 rad/s
    • Draw asymptotic magnitude and phase plots
    • Find gain crossover, phase crossover
    • Read gain margin and phase margin
11. 7 Marks
    aDraw the polar plot for open loop transfer function for unity feedback system.
    How to answer

    • State open-loop TF assumed for unity feedback (type/order)
    • Compute magnitude and phase vs omega
    • Tabulate and plot Im vs Re polar curve
    • Mark omega=0 and omega=infinity points
    • Comment on stability indication
12. 7 Marks
    bState and explain Nyquist stability criteria.
    How to answer

    • State Nyquist criterion Z = N + P
    • Explain encirclement of -1+j0 point
    • Explain mapping of Nyquist contour
    • Steps to check stability
    • Give example interpretation
13. 7 Marks
    aWhat is lag-lead compensator? Under what conditions it is employed? Explain.
    How to answer

    • Define lag-lead compensator TF with both lag and lead sections
    • Give pole-zero arrangement and Bode characteristics
    • Explain when both transient and steady state improvement needed
    • Give design conditions/applications
    • Compare with lag-only/lead-only
14. 7 Marks
    bObtain the solution of state equation and list the properties of state transition matrix.
    How to answer

    • Write state equation x_dot=Ax+Bu, y=Cx+Du
    • Solve using Laplace: x(t)=phi(t)x0+integral phi...
    • Define STM phi(t)=exp(At)=L^-1[(sI-A)^-1]
    • List properties: phi(0)=I, phi^-1=phi(-t), phi(t1+t2)=...
    • Mention significance
15. 14 Marks
    Write a short notes on any two:

    1. i)Eigen values and Eigen vectors
    2. ii)Servomotors
    3. iii)Gain and Phase Margin
    4. iv)Relative stability
    How to answer

    • Choose any two topics from four options
    • Give definition, formula/diagram for each
    • Explain key features and applications
    • Give example where relevant
    • Conclude with significance in control