EX-405 Control System - Jun 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
   aDistinguish between Open loop control system and closed loop control system.
   How to answer

   • Definition of open loop system with block diagram and examples
   • Definition of closed loop system with feedback, block diagram and examples
   • Comparison on feedback, accuracy, stability, cost, sensitivity
   • Advantages/disadvantages summary
2. 7 Marks
   bFind the transfer function using Mason's gain equation.

   

   How to answer

   • Identify forward paths, individual loops and non-touching loops from signal flow graph
   • Apply Mason's gain formula $T = \frac{1}{\Delta}\sum P_k\Delta_k$
   • Compute $\Delta$ and $\Delta_k$ for the graph
   • Simplify to get $C(s)/R(s)$
3. 7 Marks
   aDetermine the transfer function $X_1(s)/F(s)$ for the system shown below.

   

   How to answer

   • Draw free-body diagrams and write force balance equations for M1 and M2
   • Take Laplace transform with zero initial conditions
   • Eliminate X2(s) to relate X1(s) and F(s)
   • Write final $X_1(s)/F(s)$ in standard form
4. 7 Marks
   bThe block diagram of a unity feedback (negative) system is shown in figure. Determine the steady state error for unit ramp input when K=400. Also determine the value of K for which the steady state error to unit ramp will be 0.02.

   

   How to answer

   • Find open-loop transfer function and system type for ramp input
   • Compute velocity error constant $K_v = \lim_{s\to 0} sG(s)$
   • Compute $e_{ss}=1/K_v$ for K=400
   • Solve $K$ for specified $e_{ss}=0.02$
5. 7 Marks
   aExplain time response of first order system to unit step and unit ramp input. Find the steady state error response for both.
   How to answer

   • Write first-order transfer function $1/(Ts+1)$ and step/ramp responses
   • Plot/explain transient with time constant T
   • Derive steady-state errors using final value theorem
   • Compare zero error for step vs constant error T for ramp
6. 7 Marks
   bDiscuss the proportional and integral control in detail.
   How to answer

   • Principle and control law of proportional control
   • Principle and control law of integral control
   • Effect on steady-state error, stability and transient
   • Applications and limitations
7. 7 Marks
   aDetermine the RH stability of given characteristic equation, $s^4 + 8s^3 + 18s^2 + 16s + 5 = 0$.
   How to answer

   • Form Routh array for $s^4+8s^3+18s^2+16s+5=0$
   • Check first-column sign changes
   • Conclude about RH-plane poles / stability
   • State number of unstable roots if any
8. 7 Marks
   bSketch the root locus for the unity feedback system whose open loop transfer function is $G(s)H(s) = K/[s (s+4) (s^2+4s+20)]$
   How to answer

   • Locate open-loop poles/zeros and root-locus branches
   • Draw asymptotes, centroid, breakaway points and imaginary-axis crossing
   • Sketch complete root locus for $K>0$
   • Comment on stability with gain
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

   • Recall given polar plot shape in GH-plane (omega=0 to infinity)
   • Add -90 deg per pole at origin and lag of $1/(1+j\omega T_2)$
   • Rotate/shape plot for added two poles
   • Sketch resulting polar plot with start/end angles
10. 7 Marks
    bDescribe the procedure for developing the polar plot
    How to answer

    • Definition of polar plot $G(j\omega)$ magnitude vs phase
    • Steps: substitute $s=j\omega$, compute magnitude and phase vs omega
    • Tabulate/plot for $\omega=0$ to $\infty$ on polar coordinates
    • Interpretation for stability
11. 7 Marks
    aDraw the Nyquist plot for the system whose open loop transfer function is $G(s)H(s) = K/[s(s+2) (s+10)]$ Determine the range of K for which the closed loop system is stable.
    How to answer

    • Map Nyquist path and plot $G(s)H(s)=K/[s(s+2)(s+10)]$ for $\omega$
    • Find intersection with negative real axis
    • Apply Nyquist stability criterion $Z=N+P$
    • Solve range of K for Z=0 (stable closed loop)
12. 7 Marks
    bSketch the bode plot for the following transfer function and determine phase margin and gain margin. $G(s) = 75 (1+0.2s) / [s (s^2+16s+100)]$
    How to answer

    • Rewrite $G(s)=75(1+0.2s)/[s(s^2+16s+100)]$ in Bode standard form
    • Plot asymptotic magnitude and phase vs log frequency
    • Find gain crossover, phase crossover, GM and PM
    • Comment on stability from margins
13. 7 Marks
    aDraw the electrical circuit diagram that represents the Lead-Lag compensator and explain in detail.
    How to answer

    • Circuit diagram of Lead-Lag compensator using R-C network / op-amp
    • Transfer function form with lead and lag time constants
    • Working in different frequency regions
    • Design considerations and uses
14. 7 Marks
    bGiven the state equations below, write the transfer function for this system. Assume the output variable in the transfer function is $y(t)$ which is equal to the state variable $x_1$. $$\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -6 & -3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 0 \\ 2u(t) \end{bmatrix}$$
    How to answer

    • Write state model $\dot{x}=Ax+Bu$, $y=Cx$ with given A, B, C
    • Use $T(s)=C(sI-A)^{-1}B$ formula
    • Compute $(sI-A)^{-1}$ and multiply
    • Simplify to rational transfer function
15. 14 Marks
    Write a short notes on any two

    1. i)Tacho-generators
    2. ii)PID Controller
    3. iii)Stability by root loci
    4. iv)Relative stability
    How to answer

    • Select any two topics from the four options
    • Give definition, principle/block diagram and working for each
    • Mention key formulas, characteristics and applications
    • Conclude with significance in control systems