Control System Question Bank Wbut

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1. Introduction 1. What is Control System? Differentiate between Continuous Control System and Discrete-Data Control System with example. 2. Explain open loop & closed loop control system. State their merits & demerits. Distinguish between closed-loop and open-loop system. 3. What do you mean by feedback control system? What are the basic components of closed-loop control system? Discuss the function of these components. 4. What do you mean by LTI system.

2. Mathematical Modeling 1. 2. 3. 4. 5.

What do you mean by “Mathematical Model” of a system? Define transfer function of a system. What do you mean by “type” and “order” of a system? Write Force-Voltage and Force-Current analogy. Write down the differential equations governing the behavior of mechanical system shown in the following fig. Also draw an analogous electrical circuit. K J1

J2 Θ output

T Applied torque

6. Determine the differential equation describing the complete dynamics of the mechanical system. Also develop the electrical analog circuit based on force – voltage analogy.

K2

B1 M2

K1

B2

X1

M1 X2

2 7. Draw the electrical analogy, by force-current (f-i) analogy, of the mechanical system shown in figure.

K1 M2

x2

K2

M1 x1

f 8. Write the differential equations of the mechanical system shown in figure 1. Also obtain an analogous electrical circuit. X1

F(t)

X2

F12 K

M2

M1

F1

9. Find out the transfer function

VI

K

F2

Vo ( s ) for the electrical network shown in Figure-1 VI ( s)

Vo

3 V ( s) 10. Derive the transfer function, o , for the electrical network shown in below. VI ( s)

R1

C1

VI

Vo

R2

11. The unit step response of system is given by 5 5 C(t) = + 5t − e − 2t 2 2 Find the transfer function of the system.

3. Block diagram Reduction 1. What do you mean by “Block Diagram” of a system? 2. Using block diagram reduction technique, find the overall transfer function of the systems whose block diagram is given below i. G4

C(s) R(s)

+ -

G1

+ -

G2 H2

H1 ii.

+

G3

4

4. Signal Flow Graph 1. 2. 3. 4.

What is Signal Flow Graph? State & explain Mason’s gain formula. Write down the merits of Mason’s gain formula. Draw a signal flow graph for the systems shown below & hence obtain the transfer function using Mason’s gain formula.

i. G4 R(s) +

+

-

-

+ G1

G2

G3 +

H1

ii.

iii. Find C(s)/U(s)

C(s)

5 iv. G4

R(s) +

+

-

-

C(s)

+

G1

G2

G3

+ -

H3

-

H2

H1

v.

5. Time response and Steady-state error 1. What do mean by steady state and transient state? 2. What are the standard tests signals used for analyzing a control system. 3. Derive the expression of unit step response of a typical 1st order system. Plot the response. What do you mean by time constant? 4. Derive the expression of unit step response of an under-damped 2nd order system. Plot the response. 5. Draw the typical response curve of a second order system for a unit step input. Define delay time, rise time, peak time, peak overshoot, settling time & steady state error indicating them on the response curve. Derive the expressions of these quantities. 6. What are the effects of adding a pole & adding a zero to a (i) first order (ii) second order system. 7. Give the definition of the error constants KP, KV, Ka.

6 8. A unity feedback system is characterized by an open loop transfer function K G(s) = s ( s + 10) Determine the gain K so that the system will have a damping ratio of 0.5. For this value of K determine settling time, peak overshoot, peak time & steady state error for a unit step input. 9. The maximum overshoot for a unity feedback control system having its forward path transfer function as G( s) =

K s (Ts + 1)

is to be reduced from 60% to 20%. The system input is a unit step function. Determine the factor by which K should be reduced to achieve aforesaid reduction. [Ref. B. S. Manke] 10. Consider the position control system shown in fig. R(s)

+ -

+ -

12

1 s ( s + 1)

C(s)

αs

Determine the settling time and maximum overshoot in the response to a step input for ∝=0. Repeat if ∝=2. Determine the value of ∝ so that the damping ratio of the closed-loop transfer function may be increased to 0.6. What are the values of the settling time, maximum overshoot and the steady state error to a unit step input?

i. ii. iii.

11. For the system shown below determine the value of a & b so that the system has damping ratio of 0.7 & ω n =4 rad/sec.

R(s) +

+

-

-

a s+2

1 s

C(s)

b

Obtain the rise time, peak time, settling time & steady state error of the system for a unit step input.

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12. For the system shown in figure below, determine the values of k and T so that the maximum overshoot in the unit step response is 0.254 and peak time is 35. Obtain rise time and settling time. R(s)

+

C(s)

k s ( sT + 1)

-

13. Find the error coefficients of the following system.

14. Find the static error constants & the steady state error of a unity feedback linear 10 control system whose forward path transfer function is G ( s ) = , when s ( s + 1) subjected to a polynomial input r(t)=a0+a1t+a2t2. 15. A control system with a PD controller is shown in the figure, Find the values of KP & Kd so that the ramp error constant is 1000 & the damping ratio is 0.5

+ -

Kp + Kd s

1000 s(s+10)

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Stability and Routh-Hurwitz Criterion 1. What do you mean by absolute stability and relative stability. 2. State BIBO stability criterion. Show that for a bounded input –bounded output α

stable system ∫ g (τ )dτ is finite, where g(t) is the impulse response of the system. 0

3. State & explain Routh’s stability criterion. What are the limitations of Routh Hurwitz criteria? 4. Using Routh-Hurwitz criterion, determine the stability of the closed-loop system that has the following characteristic equation. i. s 4 + 6s 3 + 21s 2 + 36s + 20 = 0 ii. s 5 + 6s 4 + 3s 3 + 2s 2 + s + 1 = 0 iii. s 5 + s 4 + 2s 3 + 2s 2 + 3s + 5 = 0 iv. s 6 + 2s 5 + 8s 4 + 12s3 + 20s 2 + 16s + 16 = 0 5. The open loop transfer function of a unity feedback control system is given by G( s) =

K ( s + 2)( s + 4)( s 2 + 6s + 25)

By applying Routh’s Criterion determine i. The range of K for which the closed loop system will be stable. ii. The value of K which will cause sustained oscillation in the closed loop system. What are the corresponding oscillation frequencies. 6. Determine the range of values of K (K>0) such that the characteristic equation s3 + 3(K+1)s2 + (7K+5)s + (4K+7) =0 has roots more negative than s = -1.

7. Root-locus 1. Sketch the root locus diagram as K is varied from zero to infinity for the system whose open loop transfer function is given by G ( s) H ( s) =

K s ( s + 1)( s + 3)

Determine (i) the value of K for ξ=0.5 (ii) the value of K for marginal stability (iii) the value of K at s= -4. (iv) gain margin and phase margin [Ref. B. S. Manke]

9 2. Sketch the root locus diagram as K is varied from zero to infinity for the system whose open loop transfer function is given by G ( s) H ( s) =

K s ( s + 2)( s + 4)

Determine the value of K to have 40% overshoot for unit step input. [Ref. B. S. Manke] 3. Applying the rules of construction, draw the root-loci of the following systems whose open-loop transfer function is i. ii. iii.

K s ( s + 6)( s 2 + 4 s + 13)

K s( s + 4)( s 2 + 4s + 13)

(

K s2 + 4

)

s ( s + 2)

[Ref. B. S. Manke]

8. Frequency Response 1. Define different frequency domain specifications for a second order system. 2. The closed loop transfer function of a feedback control system is given by C ( s) 1 = R( s ) (1 + 0.01s )(1 + 0.06s + 0.1s 2 ) Determine the Resonant peak, Resonant frequency & bandwidth of the system. 3. Without Bode plot, from definition determine the value of K for a unity feedback control system having open loop transfer function G ( s) H ( s) =

K s ( s + 2)( s + 4)

such that (a) gain margin=20 dB (b) phase margin=60°. [Ref. B. S. Manke]

9. Bode Plot 1. What are the advantages of dB plotting? 2. Draw the Bode Plot of the following systems having open loop transfer function as follows. Find out the Gain margin & phase margin from the plot and comment on stability of the system. i. G ( s) H ( s ) = ii. G ( s) H ( s) =

2 ( s + 0.25 )

s ( s + 1)( s + 0.5) 2

48 ( s + 10 ) s ( s + 20)( s 2 + 2.4s + 16)

[Ref. B. S. Manke]

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3. The open-loop transfer function of a control system is K jω ( j 0.1ω + 1)( jω + 1)

G ( jω ) H ( jω ) =

Determine the value of K so that (a) gain margin=15 dB (b) phase margin=60° [Ref. B. S. Manke] 4. For the following Bode plots, find the transfer function. i. Bode Diagram 30

-20 dB/dec 24.1 20

-40 dB/dec

Magnitude (dB)

10

0

8 -20 dB/dec

-10

-12.5 -20

-40 dB/dec

-20.5 -30

-40 0 10

1

2

10

10

Frequency (rad/sec)

[Ref. B. S. Manke] ii. db 32

-6db/octave 6db/octave

25 12db/octave

0.1

0.5

1

5

log ω

10. Nyquist Plot 1. 2. 3. 4.

State and explain Nyquist stability criterion. Define Nyquist contour. Explain how it helps to determine stability. State the “Principle of argument” & its extension to Nyquist criterion. What do you mean by minimum and nonminimum phase system?

11 5. What is Nichol’s chart? 6. Draw the Nyquist Plot for the following systems having open loop transfer functions as follows. ( s + 2) i. G ( s ) H ( s ) = ( s + 1)( s − 1) ( s + 0.25)

ii. G ( s) H ( s ) =

[Ref. B. S. Manke]

s ( s + 1)( s + 0.5) 2

7. Draw the Nyquist Plot for the following systems having open loop transfer functions as follows and determine the stability condition. i. G ( s) H ( s) = ii. G ( s) H ( s) =

Ks 3 ( s + 1)( s + 2) K s (T + 1) 2

[Ref. B. S. Manke]

11. State variable Method 1. 2. 3. 4.

What do you mean by state and state variable. Discuss the advantages of State Variable Method. What do you mean by controllability and observability. Obtain the state equation in matrix form for the network shown in the figure.

I1(t)

I2(t)

5. Develop a state space model for a system whose dynamics is represented by the following equation d 3 y (t ) dt

3

+3

d 2 y (t ) dt

3

+5

dy (t ) dt 3

+ 7 y (t ) = 11u (t )

6. Obtain state variable models of the systems whose transfer function s are given by i.

Y (s)

U (s)

ii. iii.

s 2 + 2s + 1

=

Y (s)

U (s) Y (s)

U (s)

s + 7 s 2 + 14 s + 8 3

= =

2s + 1 s + 2s + 3 2

1 ( s + 2 )( s + 3)( s + 4 )

12 7. A control system is described by the differential equation d 3 y (t ) = u (t ) dt 3 Where y(t) is the observed output & u(t) is the input. Describe the system in the state variable form. Calculate the state transition matrix eAt of the system. 8. First order dynamic system is represented by the differential equation

5 x (t ) + x(t ) = u (t ) y(t) = x(t) Find the transfer function of the system. 7. A system is characterized by the following state space equation ⎡• ⎤ ⎢ x1 ⎥ = ⎛ −3 1 ⎞ ⎡ x1 ⎤ + ⎡ 0 ⎤ u ⎢ • ⎥ ⎜ −2 0 ⎟ ⎢ x ⎥ ⎢1 ⎥ ⎠⎣ 2⎦ ⎣ ⎦ ⎢⎣ x2 ⎥⎦ ⎝ ⎡ x1 ⎤ y = [1 0] ⎢ ⎥ ⎣ x2 ⎦

(a) (b) (c)

t>0

Find the transfer function of the system. Compute state transition matrix. Solve the state equation for a unit step input under zero initial condition. [Ref. B. S. Manke]

8. Check for controllability and observability of a system having following coefficient matrices. 1 0⎞ ⎛ 0 ⎜ ⎟ A=⎜ 0 0 1⎟ ⎜ −6 −11 −6 ⎟ ⎝ ⎠

⎛1 ⎞ ⎜ ⎟ B = ⎜ 0 ⎟ C = (10 5 1) ⎜1 ⎟ ⎝ ⎠

[Ref. B. S. Manke]

12. PID Controller 1. What are the effects of integral control & derivative control? Why derivative control is never used alone. 2. Draw circuit for OP-AMP based realization of P, PI and PID controller. Derive expressions for Input-Output relationship of such controllers in terms of circuit parameters.

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13. Compensation Technique 1. Draw circuit for lead, lag, lead-lag compensator. Write there advantages and disadvantage. 2. Consider the system shown in the following figure.

R(s)

+

k s ( s + 1)(0.5s + 1)

C(s)

Compensate the system so that static velocity error coefficient Kv is 5/sec. The phase margin is at least 40 degree. And the gain margin is at least 10db. 3. A lead compensator required for a position control system to provide a phase lead of 350 & a gain of 6.5 dB at ω = 2.8 rad/sec. What will be the transfer function of the compensator?

14. Digital Control 1. 2. 3. 4.

State the advantages of digital control system over analog control system. Explain with a neat schematic diagram, how a digital control system works. What is sampled data system? State & explain Shanon’s sampling theorem. Express the output c(t) in the form of ZOH sampled data system as given below in the figure T ZOH C(t) e(t)

5. Determine the Z-transform of the following function i. f (k ) = u (k ) ii. f (k ) = e − ak iii. f (k ) = sin(kωt ) iv. f (k ) = b k cos(kωt ) v. f (k ) = e− ak cos(kωt )