# Control Systems Jan 2007

Note : Answer any FIVE full questions.

1 a. Explain the difference between open loop and closed loop control systems, with suitable examples.

b.  For the mechanical system shown in the figl(b), draw the force-voltage analogous electrical system and determine the displacements as function of time at A & B; also draw approximately these displacements. The applied force is a unit impulse. F(t) – £ (t) N. Mi = 4 kgms Di = 5 N / m / sec /ftVHitrn u/VTtn li = 100 cms h~ 25 cms M(s)=NgD” M2 – 24 kgms D2 ~ 60 N / m / sec K2 = 1-25 cms/N (spring compliance)

2 a. For a negative feedback control system, starting from fundamentals, show that the closed loop transfer function M(s) is given by tzy>4+JvO’ where G^= */D, ; h^=N/d,

b. The performance equations of a controlled system are given by the following set of C(s) linear algebraic equations. Draw the block diagram and determine —- by reducing

the block diagram in steps.

Ei(s) – R(s) – H3(s) C(s)

E2(s) – E,(s)-H,(s)E4(s)

E3(s) = Gi(s)E2(s)-H2(s)C(s)

E4(s) – G2(s) E3(s)

C(s) = G3(s) E4(s)

3 a. For the circuit shown in the Fig.3(a), write the performance equations considering the voltage and current variables as indicated, draw the corresponding signal flow graph I (s)/ 3V using Mason’s Gain formula. mo and determine

Rx =lOOkn ; R2=50kn ; R, = 40AQ ; =10juF’; C2=5^iF

b. Briefly explain the following with examples :

i) Part of signal flow graph not touching a forward path.

ii)   Mixed Node.

4 a. What are impulse and step signals? How are they defined mathematically? What are their Lap lace Transformations?

b.  Starting from fundamentals, derive an expression for the step response of a typical under damped second order closed loop control system. Show the typical variation of the response and mark the settling time on a 5% tolerance basis.

5 a. What are static error co-efficients? Derive expressions for the same.

b. A negative feedback control system has gfr)= (,K A and »(*)=—k- +S + 1) s + 4

Determine the range of k for the absolute stability of the system ; also determine the frequency of sustained self oscillations for the limiting value of k.

6 a.  State the rules for the construction of Root Loci of the characteristic equation of a feedback control system.

b. For a negative feedback control system, G(s) = k// 2 A and H(s) = y~—\ w Zs(s2+4s + \3)   w (s + 4) Obtain the root locus for the root of the characteristic equation and plot the same using a scale of 1 unit of Real s – 2 cm and 1 unit of Imaginary s = 2 cm.

7 a. For a closed ioop control system,, H (s) ~ 1. Determine the Resonant

^(^ + 8) Peak and Resonant Frequency.K. 1

b. A negative feedback control system has G(s) – ^^and H(s) = – Obtain the complete Nyquist plot (G H Locus ) and discuss the stability of the system (with respect to the variable parameter k).

8 a.     Show that for a unity feedback control system with G(j)  G(jwc) = r , where ‘ Wc* is the phase cross over frequency. ab(a + b)

b.  Given G(V) = 2)^ + 50^ + 200) ^or a ur^t^ feedback control system, draw the Bode Plots and hence determine the Phase Margin and Gain Margin.