EX-405 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
   aA unity feedback-system has an open loop transfer function $G(s)H(s)$ Design a phase lag compensator to achieve the followings specifications. Velocity error constants $K_v = 5$. Phase margin $= 45^{\circ}$.
   How to answer

   • State given specs: Kv = 5, Phase margin = 45 deg for unity feedback system
   • Steps for phase lag compensator design: determine Kc from Kv, find frequency for required phase margin, select zero-pole
   • Give transfer function form Gc(s) = Kc*beta*(s+1/T)/(s+1/(beta*T)) with design values
2. 7 Marks
   bDerive the steady state error for type one system with unit step, ramp and parabolic inputs.
   How to answer

   • Define steady state error for type-1 system in terms of error constants
   • Derive error for unit step input
   • Derive error for ramp input and parabolic input
3. 7 Marks
   aUsing Routh-Hurwitz criterion determine the relation between K and T so that unity feedback control system whose open loop transfer function given below is stable. (ii) Determine the modified relation between K and T if all the roots of characteristic equation as determined in (i) are to lie to the left of the line $S = -1$ in S-plane.
   How to answer

   • Form closed-loop characteristic equation 1+G(s)H(s)=0
   • Construct Routh array in terms of K and T
   • Apply Routh-Hurwitz conditions for stability to get K-T relation
   • Substitute s = z-1 for S = -1 shift and repeat Routh test for modified relation
4. 7 Marks
   bWhat are the important features of open loop control system.
   How to answer

   • Define open loop control system
   • List features: no feedback, simplicity, cost, accuracy, stability, sensitivity to disturbance
   • Give example implication
5. 7 Marks
   aConsider a type 1 unit feedback system with an OLTF is specified that $K_v = 12 \text{ sec}^{-1}$ and $\Phi_{pm} = 40^{\circ}$. Design lead compensator to meet the specifications.
   How to answer

   • State specs Kv = 12 sec-1, Phi_pm = 40 deg for type-1 unit feedback
   • Find uncompensated gain and phase margin, required phase lead
   • Design lead compensator: alpha, Tm, zero-pole and gain
6. 7 Marks
   bObtain the steady state erroress of type-0 type-1 and type-2 systems for standard inputs.
   How to answer

   • Define type-0, type-1, type-2 and Kp, Kv, Ka
   • Tabulate steady state errors for step, ramp, parabolic inputs
   • Interpret zero, finite and infinite errors
7. 7 Marks
   aDefine the following terms:

   1. i)Settling time and relative stability
   2. ii)phase margin and gain margin.
   How to answer

   • Define settling time (2%/5% criterion) and relative stability
   • Define phase margin and gain margin from Bode/Nyquist
   • State significance for stability
8. 7 Marks
   bDistinguish between Open loop control system and closed loop control system.
   How to answer

   • Define open loop system with block diagram concept
   • Define closed loop system with feedback concept
   • Compare on accuracy, stability, cost, complexity, sensitivity
9. 7 Marks
   aName any two compensation techniques employed in the design of control systems.
   How to answer

   • Define compensation in control systems
   • Name two techniques e.g. series/lead-lag and feedback compensation
   • State one-line purpose of each
10. 7 Marks
    bDerive the response of second order system with unit step response.
    How to answer

    • Write second-order closed-loop transfer function with damping ratio
    • Take Laplace inverse for unit step for under/critical/overdamped cases
    • Give final time response expression
11. 7 Marks
    aDerive the expression for steady state error of the closed loop system in terms of generalized error coefficients.
    How to answer

    • Define generalized error coefficients C0, C1, ...
    • Expand error transfer function E(s)/R(s) in series
    • Obtain ess = C0*r + C1*dr/dt + ... expression
12. 7 Marks
    bDefine and derive the breakaway point on the root locus.
    How to answer

    • Define breakaway point on root locus
    • State condition dK/ds = 0 from characteristic equation
    • Derive steps to compute breakaway points
13. 7 Marks
    aTest the stability of the unity feedback system G when $K = 10$ using Nyquist criterion and then find the range of K for stability.
    How to answer

    • State Nyquist stability criterion Z = P - N
    • Plot/analyze Nyquist for given G with K = 10 for encirclement
    • Find range of K for stability from intersection/gain margin
14. 7 Marks
    bDerive an expression for peak time $t_p$ of an underdamped second order systems subjected to step input.
    How to answer

    • Write underdamped second-order step response with damped frequency
    • Differentiate and set derivative to zero for first peak
    • Obtain tp = pi / (wn*sqrt(1-zeta^2))
15. 7 Marks
    aWrite short notes on:

    1. i)Need of compensator
    2. ii)Asymptotes
    How to answer

    • Explain need of compensator: improve specs, stability, margins
    • Explain asymptotes in root locus: centroid, angles, meaning
    • Give formulas for asymptote angles/centroid
16. 7 Marks
    bDerive the expression of transfer function of lead compensator. What are the effects of phase lead compensation?
    How to answer

    • Draw lead compensator circuit and write transfer function form
    • Derive Gc(s) = Kc*alpha*(1+T*s)/(1+alpha*T*s) with alpha < 1
    • State effects: increased bandwidth, improved margins, faster response