EX-405 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
   aDiscuss the significance of feedback in control systems and how it improves system performance.
   How to answer

   • Define feedback and closed-loop structure
   • Effects: error reduction, disturbance rejection, sensitivity reduction
   • Stability, bandwidth and performance trade-offs
   • Examples of improvement in accuracy and transient response
2. 7 Marks
   bExplain the working of servo motors in detail. Discuss the advantages and disadvantages of servo motors.
   How to answer

   • Construction and working principle of servo motor with error detector
   • Transfer function / control operation diagram
   • Advantages: high accuracy, fast response, closed-loop control
   • Disadvantages: cost, complexity, stability and maintenance issues
3. 7 Marks
   aDetermine the value of K such that the roots of the characteristic equation given below lie to the left of line $s = 0$. $$s^3 + 7s^2 + s + K - 9 = 0$$
   How to answer

   • Form Routh-Hurwitz array for $s^3+7s^2+s+K-9=0$
   • Apply positivity conditions on first column
   • Obtain inequality for K for all roots with negative real parts
   • State allowable range of K
4. 7 Marks
   bWrite short notes on :

   1. i)Stepper motors
   2. ii)Nyquist plot
   How to answer

   • Stepper motors: principle, types, stepping operation and applications
   • Nyquist plot: mapping, encirclement of -1+j0, stability criterion
   • Key formulas / diagrams for each
   • Comparison of uses and limitations
5. 7 Marks
   aDiscuss the advantages and disadvantages of Proportional, Integral and Derivative (PID) controllers.
   How to answer

   • P, I, D control laws and transfer functions
   • Advantages: steady-state error, speed, damping
   • Disadvantages: offset, windup, noise amplification, tuning complexity
   • Summary table of effects on rise time, overshoot, steady-state error
6. 7 Marks
   bDescribe the concept of stability in control system and explain why it is important.
   How to answer

   • Definition of stable, marginally stable, unstable systems
   • Conditions: pole locations, Routh-Hurwitz, bounded input bounded output
   • Importance for safety, performance and design
7. 7 Marks
   aDescribe the procedure for constructing a bode plot. How do Bode plots represent the frequency response of a system?
   How to answer

   • Steps: transfer function in Bode form, corner frequencies, asymptotes
   • Magnitude plot: 20 dB/decade slopes and phase plot construction
   • Gain margin, phase margin from plots
   • Interpretation as frequency response
8. 7 Marks
   bExplain the concept of steady state error. The open loop transfer function of a control system is given by $$G(s)H(s) = \frac{2(s^2 + 3s + 20)}{s(s+2)(s^2 + 4s + 10)}$$ Determine the static error coefficients and steady state error to the following inputs:

   1. i)5
   2. ii)4t
   3. iii)$\frac{4t^2}{2}$
   How to answer

   • Define $K_p, K_v, K_a$ and steady-state error formulas
   • Identify system type 1 from $G(s)H(s)$ and compute coefficients
   • Compute $e_{ss}$ for step $5$, ramp $4t$ and parabolic $4t^2/2$
   • Comment on effect of type and gain
9. 7 Marks
   aExplain Eigen values and Eigen vectors. Compute the Eigen values and Eigen vectors for the system model with matrix A is given by $A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}$.
   How to answer

   • Define eigenvalues $|A-\lambda I|=0$ and eigenvectors $(A-\lambda I)x=0$
   • Solve characteristic $\lambda^2+3\lambda+2=0$ to get $\lambda=-1,-2$
   • Solve for eigenvectors corresponding to each eigenvalue
   • Physical significance in state-space stability/modes
10. 7 Marks
    bDerive the response of the second order system with unit step response.
    How to answer

    • Standard second-order closed-loop $C(s)/R(s)=\omega_n^2/(s^2+2\zeta\omega_n s+\omega_n^2)$
    • Partial fraction with unit step $R(s)=1/s$ and inverse Laplace
    • Expressions for underdamped, critically damped, overdamped cases
    • Plot characteristics: overshoot, settling time, damping ratio
11. 7 Marks
    aExplain the concept of Nyquist and relative stability with suitable examples.
    How to answer

    • Nyquist criterion $Z=N+P$ and mapping of Nyquist contour
    • Relative stability via gain margin and phase margin
    • Illustrative examples of stable/unstable encirclements
    • Interpretation for design
12. 7 Marks
    bThe open-loop transfer function of a unity feedback system is given by $G(s) = \frac{K}{s(1+0.2s)}$. Design a suitable compensator such that the system will have $K_v = 10$ and phase margin P.M. $= 50^\circ$.
    How to answer

    • Determine K from $K_v=\lim_{s\to 0}sG(s)=10$
    • Plot Bode magnitude/phase of uncompensated system and find PM
    • Calculate required phase lead and design lead compensator parameters
    • Verify compensated $K_v=10$ and P.M. $=50^\circ$
13. 7 Marks
    aDetermine the overall transfer function of the given system using Mason's gain formula.

    

    How to answer

    • Identify forward paths, individual loops and non-touching loops from graph
    • Compute $\Delta$ and $\Delta_k$ using Mason's rule
    • Form $C(s)/R(s)=\sum P_k\Delta_k/\Delta$
    • Simplify to overall transfer function
14. 7 Marks
    bExplain the concept of poles and zeros in control system analysis and design.
    How to answer

    • Define poles as roots of denominator and zeros as roots of numerator
    • Effect on stability, transient response and frequency response
    • Pole-zero plot interpretation and cancellation concept
    • Role in controller design and compensation
15. 7 Marks
    aDiscuss the compensation techniques employed in the design of control system.
    How to answer

    • Need for compensation: improve margins and steady-state accuracy
    • Lead, lag and lead-lag networks: characteristics and design steps
    • Series vs feedback compensation comparison
    • Selection guidelines with Bode/root-locus
16. 7 Marks
    bWhat is the state transition matrix? Explain the significance of the state transition matrix in predicting future state of a dynamic system.
    How to answer

    • Definition $\phi(t)=e^{At}=L^{-1}[(sI-A)^{-1}]$ and properties
    • Solution $x(t)=\phi(t)x(0)+\int$ convolution for forced response
    • Computation methods: Laplace, series, diagonalization
    • Significance in predicting free and forced future states