ECE Question bank for IV sem And VI Sem And VIII Sem : January 2014

CLASS: II YEAR / IV SEMESTER ECE

SUBJECT CODE AND NAME: EC 2255 – CONTROL SYSTEMS

UNIT I - CONTROL SYSTEM MODELING

PART – A

1. What is control system?

A system consists of a number of components connected together to perform a specific function. In a system when the output quantity is controlled by varying the input quantity then the system is called control system.

2. What are the two major types of control systems? May 2010

The two major types of control system are open loop and closed loop systems.

3. Name any two dynamic models used to represent control systems.

May 2013

4. Define open loop and closed loop systems.

· The control system in which the output quantity has no effect upon the input quantity are called open loop control system. This means that the output is not feedback to the input for correction.

· The control system in which the output has an effect upon the input quantity so as to maintain the desired output values are called closed loop control system.

6. What are the advantages of closed loop control system?

· The closed loop systems are accurate.

· The closed loop systems are accurate even in the presence of non linearity’s.

· Closed loop systems are less affected by noise and parameter variation.

· The ratio of the output to input variations is very low.

7. Give some examples for open and closed loop systems. Dec 2009

· Temperature control system

· Traffic control system

· Numerical control system.

8. What are the components of feedback control system?

The components of feedback control system are

· Plant

· Feedback path elements

· Error detector and controller.

9. What is feedback? What type of feedback is employed in control system? May 2011

The feedback is a property of the system by which it permits the output to be compared with input so that appropriate controlling action can be decided. Negative is employed in control system.

10. What are the advantages of feedback control? May / June 2009

· Rejection of disturbance signal.

· Accuracy in tracking steady state value.

· Low sensitivity to parameter variations.

11. Why negative feedback is preferred in control systems?

The negative feedback results in better stability in steady state and rejects any disturbance signals. It also has low sensitivity to parameter variations. Hence negative feedback is preferred in closed loop systems.

12. What is the effect of positive feedback on stability?

The positive feedback increases the error signal and drives the output to instability. But sometimes the positive feedback is used in minor loops in control systems to amplify certain internal signals or parameters.

13. Distinguish between open loop and closed loop systems. May/June 2009, 2010, 2011

S.No	Open loop system	Closed loop system
1.	Inaccurate and unreliable	Accurate and reliable
2.	Simple and economical	Complex and costlier
3.	The changes in output due to external disturbance are not corrected	The changes in output due to external disturbances are corrected automatically
4.	They are generally stable	Great efforts are needed to design a stable System

14. What is servomechanism?

The servomechanism is a feedbeck control system in which the output is mechanical position , or time derivatives of position for eg, velocity and acceleration.

15. Write the torque balance equation of an ideal rotational mass element.

Let torque ‘T’ be applied to an ideal mass with moment of inertia ‘J’. The mass will offer an opposing torque T_i which is proportional to angular acceleration.

16. Name any two types of electrical analogous for mechanical system.

The two types of analogies for the mechanical system are force-voltage and force-current analogy.

17. Write the electrical analogous elements in force – voltage analogy for the elements of mechanical translational system.

Force, f	Voltage, e
Velocity, v	Current, i
Displacement, x	Charge, q
Mass, M	Inductance, L
Frictional coefficient, B	Resistance, R
Stiffness, K	Inverse of capacitance, 1/C
Newton’s second law	Kirchoff’s voltage law,

18. Write the electrical analogous elements in force – current analogy for the elements of mechanical translational system.

Force, f	Current, i
Velocity, v	Voltage, e
Displacement, x	Flux, Φ
Mass, M	capacitance, C
Frictional coefficient, B	Conductance, G
Stiffness, K	Inverse of inductance, 1/C
Newton’s second law	Kirchoff’s current law,

19. Write the electrical analogous elements in torque – voltage analogy for the elements of mechanical rotational system.

Torque, T	Voltage, e
Angular velocity, ω	Current, i
Moment of inertia, J	Inductance, L
Angular displacement,	Charge, q
Stiffness of spring, K	Inverse of capacitance, 1/C
Frictional coefficient, B	Resistance, R
Newton’s second law	Kirchoff’s voltage law,

20. Write the electrical analogous elements in force – current analogy for the elements of mechanical rotational system.

Torque, T	Current, i
Angular velocity, ω	Voltage, e
Angular displacement,	Flux, Φ
Moment of inertia, J	Capacitance, C
Frictional coefficient, B	Conductance, G
Stiffness of spring, K	Inverse of inductance, 1/C
Newton’s second law	Kirchoff’s current law,

21. Define transfer function. Dec 2006, 2009, May 2008, 2009, 2010, 2011

The transfer function of a system is defined as the ratio of the Laplace transform of output to Laplace transform of input with zero initial conditions.

22. Name two types of electrical analogous for mechanical system. Dec 2009

The two types of analogies for the mechanical system are Force voltage and force current analogy.

23. What is a block diagram?

A block diagram of a system is a pictorial representation of the functions performed by each component of the system and shows the flow of signals. The basic elements of block diagram are block, branch point and summing point.

24. What are the components of the block diagram? Nov 2011

The basic elements are Block, branch and summing point.

25. State block diagram simplification rule for removing feedback loop May,2009

Proof

C=(R-CH)G

C=RG-CGH

C+CGH=RG

C(1+HG)=RH

(C/R)=(G/1+GH)

26. What is the rule for moving the summing point ahead of a block?

27. What is the transmittance?

The transmittance is the gain acquired by the signal when it travels from one node to another node in signal flow graph.

28. What is the sink and source?

Source is he input node in the signal flow graph and it has only outgoing branches.

Sink is an output node in the signal flow graph and it has only incoming branches.

29. What is signal flow graph? Dec 2009

A signal flow graph is a diagram that represents a set of simultaneous algebraic equations. By taking Laplace transform the time domain differential equations governing a control system can be transferred to a set of algebraic equations in s-domain.

30. List two advantages of signal flow graph.

Using Mason’s gain formula the overall gain of the system can be computed easily.
This method is simpler than the block diagram reduction techniques.

31. Write Masons Gain formula. Dec 2009, May 2006, 2009, 2011,2013

Masons Gain formula states that the overall gain of the system is

T = 1/ ∆Σk Pk ∆k

k- no. of forward paths in the signal flow graph.

Pk- Forward path gain of k th forward path

∆ = 1-[sum of individual loop gains] + [sum of gain products of all possible

combinations of two non touching loops]-[sum of gain products of

All possible combinations of three non touching loops] +…

∆k - ∆ for that part of the graph which is not touching k th forward path.

32. Define non-touching loop.

The loops are said to be non touching if they do not have common nodes.

33. What are the advantages of closed loop control system? May / June 2012 Nov/ Dec 2012

· The closed loop systems are accurate.

· The closed loop systems are accurate even in the presence of non-linearities.

· The closed loop systems are less affected by noise.

· The sensitivity of the systems may be made small to make the system more stable.

34. What are the properties of signal flow graphs? Apr/May 2010 May/June 2006, 2012

Signal flow graph is applicable to linear systems.
It consists of nodes and branches.

A node is a point representing a variable or signal.
A branch indicates functional dependence of one signal on the other.

The algebraic equations must be in the form of cause and effect relationship.

35. List the basic elements used for modeling a mechanical rotational system. Apr 2010

o Mass with moment of inertia J

o Dash-pot with rotational frictional coefficient B

o Torsional spring with stiffness K.

36. Write down the transfer function of the system whose block diagram is shown below. Nov/ Dec 2012

May/June 2010, 2011

Apr/May 2008, 2011

May/June 2012

May/June 2010 Nov/ Dec 2011

Apr/ May 2010

June 2010 / EEE

May 2008, 2011

11. (i) Derive the transfer function for Armature controlled DC motor. (8) Nov/ Dec 2011

(ii) List the properties of signal flow graph? (8)Nov/ Dec 2011

(iii) Derive the transfer function for Field controlled DC motor. (8)Apr/ May 2010

12. (i) Write the rules for block diagram reduction techniques.

(8) Nov/ Dec 2011

(ii) What are the components of feedback control system? Explain in details. (8)

13. Construct a signal flow graph for armature controlled DC motor. (16) May / June 2009

14. Describe the force-voltage and force-current analogy with example. (16) May 2010

15. Draw the equivalent mechanical system of the system shown in the figure. Write the set of equilibrium equations for it and obtain electrical analogous circuits using (i) F-V analogy (ii) F-I analogy. (16) May / June 2009

16. (i) Reduce the block diagram shown in figure and obtain its closed loop transfer function

. (8) May / June 2009

(ii) Find

by using Mason’s gain formula for the signal flow graph shown in figure. (8) May / June 2009

17. Write the equations of motion are S-domain for the system shown in figure. Determine the transfer function of the system. (16) May / June 2009

18. Obtain the transfer function of the mechanical system. (12) May / June 2009

19. Derive the transfer function of the given system by

i) Block diagram reduction technique (8)

ii) Signal flow graph. (8)

May / June 2009

20. (i) Determine the transfer function for the system having the block diagram as shown in figure. (8) Nov/ Dec 2007 Apr/ May 2010

(ii) Determine the transfer function of the network in figure. (8)Apr/ May 2010

21. Draw the force voltage and force current analogy circuits for the mechanical translational system given below in figure. (16) Apr/ May 2010

22. Obtain the transfer function

for the network shown in figure using signal flow graph technique. (16) Nov/ Dec 2007 Apr/ May 2010

23. (i) Using block diagram reduction technique find the closed loop transfer function

of the system whose block diagram is shown below. (8) May / June 2012

(ii) Construct the signal flow graph for the following set of simultaneous equations.

X₂ = A₂₁X₁ + A₂₃X₃ X₃ = A₃₁X₁ + A₃₂X₂ + A₃₃X₃

X₄ = A₄₂X₂ + A₄₃X₃

And obtain the overall transfer function using Mason’s gain formula. (8)

24. In the system shown in figure below R, L and C are electrical parameters while K, M, and B are mechanical parameters. Find the transfer function

for the system where E₁(t) is input voltage while x(t) is the output displacement. (16) Nov/ Dec 2012

25. A block diagram shown below. Construct the equivalent signal flow graph and obtain

Using Mason’s formula. (8) Nov/ Dec 2012

UNIT II - TIME RESPONSE ANALYSIS

PART – A

1. What is time response?

The time response is the output of the closed loop system as a function of time. It is denoted by C(t). It is given by inverse Laplace of the product of input and transfer function of the system.

2. What are the time domain specifications? May 2009

· Delay time

· Rise time

· Peak time

· Maximum overshoot

· Settling time

3. What is transient and steady state response? Nov/Dec 2012

The transient response is the response of the system when the input changes from one state to another. The response of the system t time infinity is called steady-state response.

4. For what purpose standard test signals are used?

While analyzing the systems it is highly impossible to each one of it as an study the response. Hence the analysis points of view, those signals, which are most commonly used as reference inputs, are called as standard test signals.

5. Name the test signals used in time response analysis. May 2011

The commonly used test input signals in control system are impulse, step, ramp, acceleration and sinusoidal signals.

6. Define step signal.

It is the sudden application of the input at a specified time. Mathematically it can be expressed as

R(t) = A for t > 0

R(t) = 0 for t< 0.

If A= 1,then it is called unit step function denoted by u(t)

7. Define ramp signal.

It is the constant rate of change in input. I.e. gradual application of the input. Magnitude of ramp input is nothing but its slope.

Mathematically it can be expressed as

R(t) = At for t > 0

R(t) = 0 for t< 0.

If A= 1,then it is called unit ramp function.

8. Define parabolic signal.

It is a signal, in which the instantaneous value varies as square of time from an initial value of zero at t =0.

Mathematically it can be expressed as

R(t) = A t²/2 for t > 0

R(t) = 0 for t< 0.

If A= 1,then it is called unit parabolic function.

9. Define step signal, Ramp signal and parabolic signal and impulse signal.

The step signal is a signal whose value changes from 0 to A and remains constant at A for t > 0.

A ramp signal is a signal whose value increases linearly with time from an initial value of zero at t = 0.

It is a signal in which the instantaneous value varies as square of the time from an initial value of zero at t = 0.

A signal which is available for very short duration is called impulse signal. Ideal impulse signal is a unit impulse signal which is defined as a signal having zero values at all time except at t = 0. At t = 0 the magnitude becomes infinite.

10. Define peak time and peak overshoot.

11. Define peak time. May / June 2009

12. What is meant by peak overshoot? Nov 2010

Peak time: it is the time taken for the response to raise from 0 to 100%, the very first time (or) it is the time taken for the response to reach peak overshoot, M_p

Peak overshoot: It is defined as the ratio of the maximum peak value to final value, where maximum peak value is measured from final value.

13. Define damping ratio.

The damping ratio is defined as the ratio of actual damping to critical damping.

14. Define damping.

Every system has the tendency to oppose the oscillatory behavior of the system, which is called damping.

15. An increase in damping ratio increase rise time.

16. How is system classified depending on the value of damping?

Depending on the value of damping, the system can be classified into the following four cases

Case 1 : Undamped system, _ = 0

Case 2 : Underdamped system, 0 < _ < 1

Case 3 : Critically damped system, _ = 1

Case 4 : Over damped system, _ > 1.

17. Define sensitivity of the control system.

An effect in the system performance due to parameter variations can be studied mathematically defining the term sensitivity of a control system. The change in a particular variable, due t parameter can be expressed in terms of sensitivity.

18. Define natural frequency.

The frequency of oscillations under damping ratio =0 condition is called natural frequency.

19. What will happen to the stability of the system, if closed loop poles moves in the left half way from imaginary axis?

As closed loop poles moves in the left half way from imaginary axis in the s-plane, transients die out more quickly, making system more stable.

20. What happens to damping ratio and rise time if bandwidth is increased?

A large bandwidth corresponds to a small rise time, or fast response. So bandwidth varies inversely proportional to the speed of response. So as bandwidth is increased, the damping ratio and rise time both reduces.

21. A system is critically damped. How will the system behave, if the gain of the system is increased?

A system is critically damped means, gain is at its marginal value and system closed loop poles are on the imaginary axis. If gain is increased beyond this marginal value, the closed loop poles on the imaginary axis gets shifted in the right half of the s-plane making the system unstable in nature.

22. What is damped frequency of oscillation?

In under damped system the response is damped oscillatory. The frequency of Damped oscillation is given

23. Sketch the response of a second order under damped system.

24. List the time domain specifications. May/ June 2009 Dec 2006, 2011

The time domain specifications are

(i) Delay time (ii) Rise time (iii) Peak time (iv) Maximum overshoot (v) Settling time.

25. Define rise time, delay time, peak time May/ June 2009, 2010, 2012, Nov 2012

Rise time is the time taken for response to raise from 0 to 100% for the

very first time.

Delay time is the time taken for response to reach 50% of the final value, for the very first time.

Peak time is the time taken for the response to reach the peak value for the very first time (or) It is the time taken for the response to reach peak overshoot, Mp .

26. What is meant by stable system?

To get the desired output, system must pass through transient period. Transient response must vanish after some time to get the final value closer to the desired value. Such systems in which transient response dies out after some time is called stable systems.

27. What is steady state error? May/ June 2006

The steady state error is the value of error signal e(t), when t tends to infinity . The steady state error is a measure of system accuracy. These errors arise from the nature of inputs, type of system and from non-linearity of system components.

28. What is meant by type number of the system? What is its significance?

The type number is given by number of poles of loop transfer function at the origin. The type number of the system decides the steady state error.

29. How are control systems classified in accordance with the number of integrations in the open loop transfer function?

Control systems are classified in accordance with the number of integrations in the open-loop transfer function as

Type - 0 system.

Type – 1 system.

Type - 2 system.

30. What are static error constants? May 2009, 2011

Positional error constant

Velocity error constant

Acceleration error constant

These constants are associated with steady state error in a particular type of system and for the standard input

31. List the advantages of generalized error constants. Apr 2012, Nov/Dec 2012

1. Generalized error series gives error signal as a function of time.

2. Using generalized error constants the steady state error can be determined for any

type of input but static error constants are used to determine m state error when the

input is anyone of the standard input.

32. List the disadvantages of static error coefficients.

The disadvantages of static error coefficients are:

Method does not provide variation of error with respect to time, which will be otherwise very useful from design point of view.
Method cannot give the error if inputs are other than standard inputs.
Most of the times, method gives mathematical answer of the precise value of the error.
The method is applicable only to stable system.

33. What are generalized error constants? May/ June 2012

They are the coefficients of generalized series. The generalized error series is given by e(t) = C₀r(t) + C₁dr(t)/dt + ( C₂/ 2! ) dr²(t)/dt² + ………….. + (C_n / n!) drⁿ(t)/dtⁿ…

The coefficients C₀, C₁, C₂,…,C_n are called generalized error coefficients or dynamic error coefficients.

34. Determine error coefficients for the system having

35. What are zero and poles? Apr / May 2010

The zero of a function, F(s) is the value at which the function, F(s) becomes zero, where F(s) is a function of complex variable s. The pole of a function, F(s) is the value at which the function, F(s) becomes infinite, where F(s) is a function of complex variable s.

36. Why the zeros on the real axis near the origin are generally avoided in design?

The closer the zero at the origin, the more pronounced is the peaking phenomenon. Hence the zeros on the real axis near the origin are generally avoided in design.

37.What is meant by order of a system? Nov/Dec 2006

The order of the system is given by the order of the differential equation governing the system. If the system is governed by n^th order differential equation then the system is called n^th order system.

The order of the system is given by the order of the differential equation, governing the system. It is also given by the maximum power of S in the denominator polynomial of the transfer function. The maximum power of S also gives number of poles of the system and so the order of the system is also given by number of poles of the transfer function.

38. What is positional error coefficient?

Steady state error of the system for a step input is 1/(1+ Kp). where Kp is the positional error coefficient. The positional error coefficient is given by

39. What is velocity error coefficient?

Steady state error of the system for a ramp input is 1/( Kv). where Kv is the velocity error coefficient. The velocity error coefficient is given by

40. What is acceleration error coefficient?

Steady state error of the system for a step input is 1/(Ka). where Ka is the acceleration error coefficient. The acceleration error coefficient is given by

41. When will the concept of Kp, Kv, Ka applicable?

The concept of Kp, Kv, Ka is applicable only if the system is represented in its simple form, and only when the system is stable.

42. What is called a proportional plus integral controller?

In an integral error compensation scheme, the output response depends in some manner upon the integral of the actuating which produces an output signal of two terms, one proportional to the actuating signal and the other proportional to tits integral. Such a controller is called proportional plus integral controller.

43. What is called a PID controller?

To increase the damping factor of the dominant poles of a PI controlled system, it is combined with a derivative error scheme. Such a controller is called a PID controller.

44. What is the advantage of PD controller?

The advantage of PD controller is that as the damping increases due to compensation, with in remaining fixed, the system settling time reduces.

45. What is the effect of PD controller on the system performance?

The effect of PD controller is to increase the damping ratio of the system and so peak overshoot is reduced.

46. What is the effect of PI controller on the system performance?

The PI controller is increases the order of the system by one, which results in reducing the steady state error. But the system becomes less stable than the original system.

47. Why derivative controller is not used alone in control systems?

The derivative controller produces a control action based on the rate of change of error signal, and it does not produce corrective measures for any constant error. Hence derivative controller is not used alone in the control system.

48. Draw the functional block diagram of PID controller.

49. State the desired feature of PID controller. Apr / May 2010

50. Why derivative controller is not used in control systems? May/ June 2012

The derivative controller produces a control action based on rate of change of error signal and it does not produce corrective measures for any constant error. Hence derivative controller is not used in control systems.

51. The bock diagram shown in fig represents a heating oven. The set point 1000^◦C. What is the steady-state temperature? Apr/May 2010

52. What is the integral time square error of the second order system with step input having damping coefficient g and undamped natural frequency ω_n? Nov 2007, EEE

53. Draw the response of a first order system for step input.

54. Obtain the response of a first order system for subjected to parabolic input. May / June 2009

55. State the use of dynamic error series.

56. With reference to time response of a control system, define Rise time. Nov 2011

57. With reference to time response of a control system, define Peak time. Nov 2012

58. The damping ratio and natural frequency of oscillation of a second order system is 0.5 and 0.8 rad/sec respectively. Calculate resonant peak and resonant frequency.

59. Determine the error coefficients for the system having

Apr/May 2010

60. Determine the damping ratio and natural frequency of oscillation of

May 2013

62. What are steady state and transient responses of a control system? Nov 2012

61. Give the steady state errors to a various standard inputs for type -2 system. May 2013

PART – B

1. Derive the expressions and draw the response of first order system for unit step input. (8) June 2010 / EEE

2. Draw the response of second order system for critically damped case and when input is unit step. (8) June – 2010 / EEE

3. Derive the expressions for second order system for under damped case and when the input is unit step.

(8) May 2011/ EEE / ECE

4. Derive the expressions / Derive the time response for second order system for undamped case and when the input is unit step. (8) Apr / May 2010

5. Define and derive the following terms of a second order system subjected to step input.

1. Rise time

2. Peak time

3. Settling time

4. Overshoot (16) May / June 2009

6. A unity feedback system has

. Determine type of the system, all error coefficients and error for ramp input with magnitude 4.

(8) May / June 2009

7. A second order system is given by,

. Find its rise time, peak time, peak overshoot and settling time if subjected to unit step input. Also calculate expression for it’s output response. (8) May / June 2009

8. The forward path transfer function of an unity feedback control system is given by

. Obtain an expression for unit step response of the system. (8) Apr/May 2010

9. Consider a second order model

. Find the response y(t) to input of unit step function. (16) May / June 2013

10. The unit impulse response of a unit feedback control system is given by

. Find the open loop transfer function. (16) May / June 2013

11. A system has an open loop transfer function

with unity feedback when K and T are positive constants. Determine the factor by which K should be multiplied to reduce the overshoot from 85% to 35%.

(16) May / June 2009, Nov 2010

12. A system has an open loop transfer function

with unity feedback when K and T are positive constants. By what factor should the amplifier gain be reduced so that the peak overshoot of unit step response of the system is reduced

from 75% to 25%. (8) May / June 2012

13. A unity feedback system has

. Determine the steady state errors for unit step, unit ramp and unit acceleration input. Also determine the damping ratio and natural frequency of the dominant roots. (16) May / June 2009

14. Discuss construction and working principle of stepper motor.

(16) Apr / May 2010

15. Derive the expression to find steady state error of a closed loop control system. (6)

16. The closed loop transfer function of a second order system is given by

. Determine the damping ratio, natural frequency of oscillations, rise time, settling time and peak overshoot. (10) Nov/ Dec 2011

17. Determine error coefficients for a system whose open loop transfer function is

. Also compute steady state error if the input to the system is

(8) Apr / May 2010

18. Find the static error coefficients for a system whose G(S) H(S) =

and also find the steady state error for r(t)=1+ t +

(8) May / June 2009

19. Certain measurements were conducted on servo mechanisms which show the system response as

when subjected to a unit step input.

1. Find the expression for closed loop transfer function.

2. Obtain the undamped natural frequency and damping ratio.

20. For a servomechanisms with open loop transfer function

. What type of input signal gives constant steady state error and calculate its value. (8)

21. The unity feedback system is characterized by an open loop transfer function

. Determine the gain K, so that the system will have a damping ratio of 0.5. For this value of K, determine the settling time, peak overshoot and time to peak overshoot for unit step input. (8) Apr / may 2011, Nov 2012

22. A unity feedback system has the forward transfer function

. The input

is applied to the system. Determine the minimum value of K₁, if the steady error is to be less than 0.1. (8) Apr / May 2011, Nov 2012

23. A certain negative feedback system has the following forward path transfer function

. The input

is applied to the system. Determine the minimum value of K, if the steady error is to be less than 1. (8) Apr / May 2012

24. Explain P, PI, PID, I, PD controllers. (4X4) (16) Apr/May 2011 Nov/Dec 2012

25. Explain in detail the system response with PI, PD and PID controllers. (16) Nov 2010

26. Discuss the effect of derivative control on the performance of a second order system. (8) Apr / May 2012

27. Figure shows PD controller used for a system.

Determine the value of Td so that system will be critically damped. Calculate its settling time. (8) Apr / May 2012

28. Consider a unity feedback system with open loop transfer function,

. Design a PID controller to satisfy the following specifications:

The steady state error for unit ramp unit should be less than 0.08
Damping ratio = 0.8 and
Natural frequency of oscillation = 2.5 rad/sec

State the expressions for the transfer function of the PID controller and for the open loop transfer function of the compensated system. (16) Nov / Dec 2011

29. An unity feedbeck control system has

. By using derivative control the damping ratio is to be made to 0.8. Determine the value of

and compare the rise time, peak time and maximum overshoot of the system. (i) without derivative control , (ii) With derivative control. The input to the system is unit step. (16) Apr/May 2010

30. Describe the different transient response characteristics of a control system. (8) Apr/May 2010

31. Determine K to limit the error of a system for input

Having

(8) Apr/May 2010

32. The overall transfer function of the control system is given by

. Find

as well as steady state error, if the input is

(16) May / June 2009

33. The system shown in fig has the following specifications

. Find the value of

to meet the specification of system. (16) Apr/May 2010

34. (i) For a servomechanisms with open loop transfer function(S)=

.What type of input signal gives constant steady state error and calculate its value. (8)

(ii) Find the static error coefficients for a system whose G(S) H(S) =

and also find the steady state error for r(t)=1+ t +

(8) May / June 2009

35. (i) Obtain the response of unity feedback system whose open loop transfer function is G(S) =

and When the input is unit step. (8)

(ii) A unity feedback system has an amplifier with gain KA=10 and gain ratio G(S) =

in the feed forward Path .A derivative feedback, H(S) =S K_o is introduced as a minor loop around G(S).Determine the derivative feed back constant ,K_o, so that the System damping factor is 0.6. (8)

36. Explain P, PI, PID, I, PD controllers. (4X4) Apr/May 2010 Nov/Dec 2012

37. (i) A unity feedback system has G(S) =

. Determine type of the system, all the error coefficients and error for ramp input with magnitude 4.

(ii) A second order system is given by

.Find its rise time, peak time, peak overshoot and settling time if subjected to unit step input. Also calculate expression for its output response. (10) May/ June 2009

38. A unity feedback system has G(S) =

. Determine the steady state errors for unit step, unit ramp and unit acceleration input. Also determine the damping ratio and natural frequency of the dominant roots. (16) May/ June 2009

39. Determine error coefficients for a system whose open loop transfer function G(S)H(S)=

. Also compute steady state error if the input to the system is a₀+a₁t+a₂t². (8) Apr / May 2010

40. Find the steady state error system whose G(S) H(S) =

and also find the steady state error if the input is r(t) =1+ t +t² May / June 2009

41. Obtain the steady state error for unit step, ramp input and parabolic input in terms of the transfer function. (16) May / June 2006, 2010 Nov/ Dec 2006, 2007, 2011

42. Determine the time response specifications and expression for output for unit step input to a system having equation as follows

+16y = 9x. (16) Nov/ Dec 2007

43. (i) Discuss the effect on the performance of a second order control system of the proportional derivative control. (8) May / June 2009, 2012

(ii) Figure shows PD controller used for the system. Determine the value of T_dso that system will be critically damped. Calculate it’s settling time. (8) May / June 2009, 2012

44. With suitable block diagrams and equations, explain the following types of controllers

employed in control systems:

1. Proportional controller

2. Proportional plus integral controller

3. PID controller

4. Integral controller (16) Nov / Dec 2012

45. The unity feedback system is characterized by an open loop transfer function

. 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 and time to peak overshoot for a unit step input. (16) Nov / Dec 2012

46. A unity feedback system has the forward transfer function

. the input

is applied to the system. Determine the minimum value of K₁, if the steady state error is to be less than 0.1. (16) Nov / Dec 2012

UNIT III - FREQUENCY RESPONSE ANALYSIS

PART –A

1. What is meant by frequency response of system?

The magnitude and phase relationship between the sinusoidal input and the steady state output of a system is termed as the frequency response. In linear time invariant systems, the frequency response is independent of the amplitude and phase of the input signal.

2. What are the advantages of frequency response analysis?

The absolute and relative stability of the closed loop system can be estimated from the knowledge of the open loop frequency response.
The practical testing of system can be easily carried with available sinusoidal signal generators and precise measurement equipments.
The transfer function of complicated functions can be determined experimentally by frequency response tests.
The design and parameter adjustment can be carried more easily.
The corrective measure for noise disturbance and parameter variation can be easily carried.
It can be extended to nonlinear systems.
The apparatus required for obtaining frequency response is simple and inexpensive and easy to use.

3. Give the limitations of frequency response analysis.

The methods considered somewhat “old” and outdated in view of extensive methods developed for digital computer simulation and modeling.

Obtaining frequency response practically is fairly time consuming.

For an existing system, obtaining frequency response is possible only if the time constants are up to few minutes.

4. List the frequency domain methods to find the stability of the system.

The commonly used frequency domain methods to sketch the frequency response of the systems are

Bode plot

Polar plot

Nyquist plot

Nichol’s chart

5. Define an octave.

The range of frequencies ω₂= 2ω₁ is called an octave.

6. What are the frequency domain specifications? Nov/Dec 2006

The frequency domain specifications indicates the performance of the system in frequency domain, and they are

· Resonant peak, M_r

· Cut-off rate,

· Resonant frequency, ω_r

· Gain margin, K_g

· Bandwidth, ω_b

· Phase margin, γ

7. Write short notes on the correlation between the time and frequency response.

There exists a correlation between time and frequency response of first or second order systems. The frequency domain specifications can be expressed in terms of the time domain; there is a corresponding resonant peak in frequency domain. For higher order systems there is no explicit correlation between time and frequency response. But if there is a pair of dominant complex conjugate poles, then the system can be approximated to second order system and the correlation between time and frequency response can be eliminated.

8. What are minimum phase systems?

The minimum phase systems are systems with minimum phase transfer functions. In minimum phase transfer functions, all poles and zero will lie on the left half of S-plane.

9. What is all pass system?

All pass systems are systems will all pass transfer functions. In some systems, the property of unit magnitude at all frequencies applies to all transfer functions with this property are called all-pass systems. Antisymmetric pole-zero patterns for every pole in the left half of S-plane, there is a zero in the mirror image position with respect to imaginary axis.

10. Define non-minimum phase transfer function?

A transfer function which has one or more zeros in the right half of the S-plane is known as non-minimum phase transfer function.

11. Define minimum phase transfer function.

A transfer function which has least (minimum) phase angle range for a given magnitude curve is called a minimum phase transfer function.

12. What is bode plot?

The bode plot is a frequency response plot of the transfer function of a system. It consists of two plots-magnitude plot and phase plot. The magnitude plot is a graph between magnitude of a system transfer function in db and the frequency. The phase plot is a graph between the phase or argument of a system transfer function in degrees and the frequency. Usually, both the plots are plotted on a common x-axis in which the frequencies are expressed in logarithmic scale.

13. Define a decade in bode plot.

20 log G(jω) = -20 log ωT.

= -20 log ω - 20logT.

The plot of above equation is straight line with a slope –20dB per unit change in log ωA unit change in 20 log ω means Log(ω₂/ ω₁) = 1 Or ω₂= 10 ω₁ this range of frequency is called a decade.

14. How static error coefficients can be determined in bode plots?

The steady state error of a closed loop system depends on the system type and gain. The static error coefficents cn be determined by these two characteristics viz. type and gain. For any given log magnitude curve, the system type and gain can be determined

Positional error coefficient is determined by type 0 system

Velocity error coefficient is determined by type 1 system

Accelerational error coefficient is determined by type 2 system.

15. Give the factors of G(jw) used in the construction of bode plots.

The factors of G(jw) used in the construction of bode plot are:

Constant gain K

Poles at the origin

Poles on the real axis

Zero on the real axis

Complex conjugate poles, zeros.

16. What is approximate bode plot?

In approx. bode plot the magnitude plot of first and second order factors are approximated by two straight lines, which are asymptotes to exact plot. One straight line is at 0db, for the frequency. For the frequency range 0 to Wc and the other straight line is drawn with a slope of 20db/dec for frequency range Wc of 10^(infinity). Here Wc is corner frequency.

17. What is sensitivity?

All physical elements have properties that change with environment and age. A good control system should be very sensitive to these parameters variations while being able to follow the command responsively. This is called sensitivity.

18. What are the advantages of bode plot? May / June 2006, 2009

Transfer function of system can be obtained from bode plot.

Data for constructing complicated polar and nyquist plots can be easily obtained from bode plot.

It indicates how system should be compensated to get desired response.

Relative stability of system can be studied by calculating G.M. and P.M. from bode plot.

19. Define cut-off rate.

The slope of the log-magnitude curve near the cut-off frequency is called cut-Off rate.

20. Define resonant peak(Mr)

It is the maximum value of magnitude of the closed loop frequency response. Larger the value of the resonant peak, more is the value of the peak overshoot of system for step input. It is a measure of relative stability of the system.

21. What is phase and gain cross over frequency? Nov/Dec 2007 Apr / May 2010

The gain cross-over frequency is the frequency at which the magnitude of the open loop transfer function is unity.

The phase cross-over frequency is the frequency at which the phase of the open loop transfer function is 180°.

22. Define gain-cross over frequency. (ωgc).

The frequency at which magnitude of G(j ω)H(j ω) is unity. I.e. 1 is called gain cross over frequency.

23. Define phase-cross over frequency. (ωgc).

The frequency at which phase angle of G(jω)H(jω) is -180˚. is called phase cross over frequency.

24. Define gain margin &phase margin. May/ June 2009 Apr / May 2010

Define gain margin G.M. in bode plot.

In root locus gain K is increased, the system stability reduces and for a certain value of K, it becomes marginally stable. (Except first and second order systems). So gain margin is defined as the margin in gain allowable by which gain can be increased till system on the verge of instability.

Define phase margin.

Phase margin is similar to the gain, it is possible to introduce phase lag in the system. I.e. negative angles without affecting magnitude plot of G(jω)H(jω). The amount of additional phase lag, which can be introduced in the system till the system reaches on the verge of instability, is called phase margin P.M.

25. How the gain margin and phase margin be improved?

The easiest way to improve G.M. and P.M. is to reduce the gain. However this increases steady state error and makes the system sluggish. Better methods are available. These methods are adding compensating networks are compensators.

26. What is bandwidth? May/ June 2009

Define bandwidth.

It is defined as the range of frequencies over which the system will respond satisfactorily. It can also be defined as range of frequencies in which the magnitude response is also flat in nature. So it is defined as range of frequencies over the magnitude of closed loop response. I.e c(j)/R(j) does not drop by more than 3db. From its zero frequency value.

27. Define resonant frequency. May/ June 2009

The frequency at which the resonant peak occurs is called resonant frequency

28. What is corner frequency? Apr 2011, Nov/Dec 2012

The magnitude plot can be approximated by asymptotic straight lines. The frequencies corresponding to the meeting point of asymptotes are called corner frequency. The slope of the magnitude plot changes at every corner frequencies.

29. What is meant by the term corner frequency?

The frequency at which change of slope from 0 db to –20db occurs is called corner frequency, denoted by ω_c.

ω_c =(1/T)

hence asymptotic i.e. approximate magnitude plot for such factor is 0 db line up to ω_c =(1/T) and line of slope –20 dB / decade. when ω > ω_c i.e. above ω_c = (1/T)

30. For a stable system the gain cross over occurs earlier than phase cross over. Justify your answer.

System is said to be stable when P.M. and G.M. are positive, while system is said to be unstable when both P.M. and G.M. are negative. Now when system is on the verge of instability, i.e. marginally stable in nature, then G.M and P.M. both are zero. This is possible when gc =pc. This condition gc =pc is useful to design the marginally stable systems. For P.M. and G.M. are positive i.e. for stable system gc <pc. While for P.M. and G.M. negative i.e. for unstable system gc >pc. In some absolutely stable system G.M. may be obtained as + , while for inherently unstable system G.M. may be obtained as -,

31. What is a polar plot?

The sinusoidal transfer function G(j) = Re[G(j)] + jIm[G(j)] G(j) = G(j) G(j) = M  From the above equations it is seen that G(j)may be represented as a phasor of magnitude M and phase angle . As the input frequency  is varied from 0 to , the magnitude M and phase angle  change and hence the tip of the phasor G(j) trace a locus in the complex plane. The locus thus obtained is known as polar plot.

32. Discuss relative stability in frequency domain.

The relative stability indicates the closeness of the system to stable region. It is an indication of the strength or degree of stability of the system .

In frequency domain the relative stability of a system can be studied from Nyquist plot. The relative stability of the system is given by closeness of the polar plot to -1+j0 point, as the polar plot gets closer to -1+j0 point the system move towards instability.

33. Why polar plots are preferred over bode plots?

A major disadvantage of bode plots is that we have two separate curves showing the variation of the gain and phase shift with frequency. A method of combining these two values in a single plot is referred to as the polar polt. For this purpose only, we prefer polar plots over bode plots. Polar plots are very useful for determining the stability of a closed loop system from its open loop frequency response.

34. Give the advantages of polar plots?

The polar plot usually requires more computation than bode plot. But it has the advantage of simultaneously providing information about gain as well as phase shift.

35. Define pure delay or transport lay.

In systems like electrical, mechanical, pneumatic systems, thereis a time delay between the application of the input and its effect on the output. This is often called

“ transport lag” or pure delay”.

36. List the procedure to sketch the polar plot of a given function.

Let G(s) be the given transfer function

 Put s=j in the given transfer function G(S) to obtain G(j)

 Evaluate G(j)and  G(j)

 At  =0 evaluate G(j)and  G(j)

 At  = evaluate G(j)and  G(j)

With these values of G(j)and  G(j) obtained for different valuesof , sketch the polar plot in polar graph paper.

37. What are the M and N circles?

The magnitude, M of the closed loop transfer function section with unity feedback will be in the form of circles in complex plane for each constant value of M. The family of these circles is called M circles. Let N= tanα where α is the phase of closed loop transfer function with unity feedback. For each constant of N, a circle can be drawn in the complex plane the family of these circles are called N circles.

38. What is meant by constant N Circles. Nov / Dec 2007, EEE

39. What happens to the damping ratio and rise time if the band width is increased?

A large BW corresponds to a small rise time or fast response so BW in inversely proportional to the speed of response, so as BW is increased damping ratio and rise time both reduces.

40. If the very low frequency asymptote magnitude plot of an unity feedback system has a slope of -40db/decade, find the standard input or inputs it can follow with any steady state errors:

At very low frequency, the magnitude plot slope is -40db/decade i.e. there are two poles at the origin and hence the system is Type 2 system. Type 2 system follows parabolic input with some error but it can follows ramp type standard inputs without any steady state errors.

41. For a stable system the gain margin and phase margin should be positive. Justify answer.

The gain margin indicates the amount of the gain which can be introduced in the system till system reaches on the verge of instability. Here positive gain margin indicates that such a gain introduction is possible till system becomes unstable i.e. system is basically stable.

Similarly margin is the amount of the lag which can be introduced till system reaches on the verge of instability so positive phase margin indicates that such a introduction possible and the system is stable. The negative gain margin and the phase margin indicates that there is no chance to introduce gain or phase lag as the system is already unstable.

42. What is Nicholas chart? Nov/Dec 2006 2012

N.B Nichols transformed the constant M and N circles to log magnitude and the phase angle coordinates and the resulting chart is known as Nichols chart. The Nichols chart consists of M and N superimposed on ordinary paths.

43. What are the advantages of Nichols chart?

· It is used to find closed loop frequency response from open loop frequency response.

· The frequency domain specifications can be determined from Nichols chart.

· The gain of the system can be adjusted to satisfy the given specifications.

44. State the uses of Nicholas chart. May/ June 2012

Give the uses of Nichols chart.

The complete closed loop frequency response can be obtained. The 3db B.W. of the closed loop system can be obtained. To design the value of K for the given Nr. The frequency Wr corresponding to the Nr for the closed loop system can be obtained. Once Mr and Wr are known the various other frequency and time domain specifications can be obtained.

45. How the closed loop frequency response is determined from the open loop

frequency response?

The G(jw) locus or the Nichols plot is sketched on the standard Nichols chart. The meeting point of M contour with G(jw) locus gives the magnitude of the closed loop system and the meeting point with N circle gives the argument or phase of the closed loop system.

46. Draw the polar plot of G(S) =. May/ June 2012

47. Draw the polar plot G(s) H(s) =. Apr / May 2010

48. Determine the frequency domain specifications of a second order system when closed loop transfer function is given by . Apr / May 2010

49. Write the MATLAB command for plotting bode diagram Nov/ Dec 2011

50. Draw the polar plot of an integral term transfer function.

May/ June 2013, ECE

51. Draw the polar plot for . May/ June 2013, EEE

52. Define compensator and give its list. May/ June 2013, EEE

What are compensators?

List the necessity of the compensating network.

What is the need for compensation? Apr / May 2010

Name the commonly used electrical compensating networks. May/ June 2009

In control systems design, under certain circumstances it is necessary to introduce some kind of corrective subsystems to force the chosen plant to meet the given specifications. These subsystems are known as compensators and their job is to compensate for the deficiency in the performance of the plant.

The compensator is a physical device. It may be an electrical network, mechanical unit, pneumatic, hydraulic or combinations of various types. The commonly used electrical compensating networks are

· Lead network or Lead compensator

· Lag network or Lag compensator

· Lag-Lead network or Lag-Lead compensator.

54. What are the two methods of specifying the performance of control system?

By a set of specifications in time domain or in frequency domain such as peak overshoot, setting time, gain margin, phase margin, steady state error etc.

By optimality of certain function e.g. en internal function.

55. Give the two approaches to the control system design problem.

There are basically two approaches to the control system design problem:

We select the configuration of the overall system by introducing compensators and then choose the parameters of the compensators to meet the given plant, we find an overall system that meets the given specifications and then compute the necessary compensations.

56. Define Lead compensator.

Gc(s) = (s+Zc) / (s + Pc) = (s+1/γ)/ (s+ 1/at), where a = Zc/Pc < 1, t > 0,

a< 1 ensures that the pole is located to the left of the zero. The compensator having a transfer of the form given above is known as a lead compensator. A lead compensator speeds up the transient response and increases the margin of stability of the system. It also helps to increase the system error constant though to a limited extent.

57. Draw the block diagram of the system with lead compensation.

58. What is a lag compensator?

Gc(s) = (s+Zc) / (s + Pc) where b = Zc/Pc > 1. b>1 ensures that pole is to the right of zero, i.e. Nearer to the origin than zero. The compensator having a transfer function of the form given above is called a lag compensator. A lag compensator improves the steady state behavior of the system while nearly preserving its transient response.

59. What is a lag lead compensator?

When both the transient and steady state response require improvement lag lead compensator is required. This is basically a lag lead compensator connected in series.

60. What are the different components by which compensators are realized?

i. Electrical components

ii. Mechanical components

iii. Pneumatic components

iv. Hydraulic or other components.

61. Give the sinusoidal transfer function of the lead compensator.

The Sinusoidal transfer function of the lead compensator is given by Ge (jw) = (1+jwt)/ (1+ jwat), a < 1 Since a < 1 the network output leads the sinusoidal input under steady state and hence the name lead compensator.

62. Write the transfer function of a lead compensator network. Apr / May 2010

Transfer function of a lead compensator g_c(s) =

63. Why lag compensator is called so?

The sinusoidal transfer function of the lag network is given by

Ge(jw) = (1+jwt)/(1+jwbt), since b > 1 the steady state output has a lagging phase angle with respect to sinusoidal input and hence the name lag network.

64. What is meant by compensation?

All the control systems are designed to achieve specific objectives. The certain requirements are defined for the control system. If a system is to be redesigned so as to meet the required specifications, it is necessary to alter the system by adding an external device to it. Such a redesign of a system using an additional device is called compensation.

65. What are the two situations in which compensation is required?

There are two situations in which compensation is required:

· The system is absolutely unstable and the compensation is required to stabilize it as well as to achieve a specified performance.

· The system is stable but the compensation is required to obtain the desired performance.

66. List the types of compensation.

· Series Compensation

· Parallel compensation

· Series-parallel compensation

67. What is the importance of compensation? May/ June 2009

i. When the system is stable, compensation is provided to obtain the desired

performance.

ii. When the system is absolute unstable, then compensation is required to stabilize the system and also to meet the desired performance.

68. Mention the need for lead compensation. Apr / May 2010

The lead compensation increases the bandwidth and improves the speed of response. When the given system is stable/unstable and requires improvement in transient state response then lead compensation is employed.

69. What is lag-compensation?

The lag compensation is a design procedure in which a lag compensator is introduced in the system so as to meet the desired specifications.

70. What is lead compensation?

The lead compensation is a design procedure in which a lead compensator is introduced in the system so as to meet the desired specifications.

71. Write the advantages and disadvantages of lead compensation technique. Nov / Dec 2006

72. What is series compensation?

The series compensation is a design procedure in which a compensator is introduced in series with plant to alter the system behavior and to provide satisfactory performance.

73. What is parallel compensation?

The feedback compensation is a design procedure in which a compensator is introduced in the feedback path so as to meet the desired specifications. It is also called parallel compensation.

74. When lag /lead /lag-lead compensation employed?

· Lag compensation is employed for a stable system for improvement in steady state performance.

· Lead compensation is employed for stable/unstable system for improvement in transient-state performance.

· Lag-lead compensation is employed for stable/unstable system for improvement in both steady-state and transient state performance.

75. Obtain the maximum phase lead angle of a lead network. May / June 2009

76. Draw the circuit diagram of lead network.

77. Draw the circuit diagram of lead network and draw its ploe – zero

diagram. Apr / May 2011

78. Write the transfer function of lag network and draw its ploe – zero

diagram. May / June 2013,EEE

79. What are the observations that are made from the Bode’s plot of the lag

compensated system?

· The cross over frequency is reduced.

· The high frequency end of the lag-magnitude plot has been raised up by a dB gain of 20log (1/a).

80. What is the effect of lead compensator and lag compensator on system

bandwidth?

Lead compensator increases the system bandwidth whereas Lag compensator reduces the system bandwidth.

81. Distinguish between lead compensator and lag compensator.

Lead compensator Lag compensator

1. Increases system bandwidth Reduces system bandwidth

2. Increases speed of response slows down speed of response

82. What are the forms in which frequency domain specification are given in cascade compensation?

Phase margin Φpm or resonant peak Mr – indicative of relative stability.
Bandwidth ω0 or resonant frequency ωr - indicative of rise time and settling time.
Error constant – indicative of steady state error.

83. How is the cascade compensation carried out in frequency domain?

The frequency domain compensation may be carried out using Nyquist plots, Bode plots or Nichol’s chart. The advantages of the Bode plots are that they are easier to draw and modify.

84. Give the effects of Lead compensation.

· The lead compensator adds dominant zero and a pole. This increases the damping of the closed loop system.

· It improves the phase margin of the closed loop system.

· The steady state error does not get affected.

· It increases bandwidth of the closed loop system. More the bandwidth the faster is the response.

· The increased damping means less rise time and less settling time. Thus there is improvement in the transient response.

85. Give the effects and limitations of lag compensator.

Lag compensator allows high gain at low frequencies, thus it is basically a low pass filter. Hence it improves steady state performance.
The system becomes more sensitive to the parameter variations.
Reduced bandwidth means slower response. Thus rise time and settling time are usually longer.
The attenuation characteristics are used for compensation.
Lag compensator approximately acts as PI controller and thus tends to make they system less stable.
The attenuation due to lag compensator shifts the gain crossover frequency to a lower frequency point. Thus the bandwidth of the system gets reduced.

86. Explain how the lead compensation is done using Bode plots.

The lead compensation on Bode’s plot proceeds by adjusting the system error constant to the desired value. The phase margin of the uncompensated system is then checked, if found satisfactory, the lead compensation is designed to meet the specified phase margin.

87. What are the observations that are made from the Bode plots of the lead

compensated system?

1. The phase cross over frequency is increased.

2. The high frequency end of the log-magnitude plot has been raised up by a dB-gain of 20log (1/a).

88. Discuss cascade compensation in time domain.

Cascade compensation in time domain is conveniently carried out by the root locus technique. In this method of compensation, the original design specifications on dynamic response are converted into ε and ωn of a pair of desired complex conjugate closed loop poles based on the assumption that the system will be dominated by these complex poles and therefore its dynamic behavior can be approximated by that of a second order system.

89. What is called compensation?

To meet independent specifications, a second order system requires to be modified. This modification is termed as compensation. It should allow for high open-loop gains to meet the specified steady state accuracy and yet preserve a satisfactory dynamic performance.

90. What are the different compensation techniques?

a) Derivative error compensation.

b) Derivative output compensation.

c) Integral error compensation.

91. What is derivative output compensation?

A system is said to possess a derivative output compensation when the generation of its output depends in some way upon the rate at which the controlled variable is changing.

PART –B

1. Plot the bode diagram for the following transfer function and obtain the gain and phase cross over frequencies G(S) =

. (16) May/ June 2010

2. The open loop transfer function of a unity feedback system is G(S) =

Sketch the polar plot and determine the gain margin and phase margin. (16) Apr / May 2008 May/June 2009 Nov/Dec 2011

3. Sketch the bode plot and hence find gain cross over frequency , Phase cross over frequency ,Gain margin and phase margin G(S) =

(16) Nov/Dec 2006, 2011 Apr / May 2011

4. Sketch the bode magnitude plot G(S) =

(8) May/June 2012

5. Sketch the polar plot for the following transfer function and find gain cross over frequency , Phase cross over frequency ,Gain margin and phase margin G(S) =

. (16)

6. Construct the polar plot for the function GH(S) =

.Find gain cross over frequency, Phase cross over frequency, Gain margin and phase margin. (16)

7. Plot the bode diagram for the following transfer function and obtain the gain and phase cross over frequencies. G(S) =

. Determine the value of K for a gain cross over frequency of 20 rad /sec. (16)

8. Give G(s)=

find K for the following two cases (i) Gain margin equal to 6 db

(ii) Phase margin equal to 45˚. (16) Nov/Dec 2012

9. Draw the pole-zero diagram of a lead compensator. Propose lead compensation using electrical network. Derive the transfer function. Draw the bode plots. (16) Nov/Dec 2012

10. Sketch the polar plot for the following transfer function and find gain cross over frequency, phase cross over frequency, gain margin and phase margin.

G(S) = (16)

11. A unity feedback system has open loop transfer function G(S) =

. Using Nichol’s charts determine the closed loop frequency response and estimate all the frequency domain specifications. (16)

12. Sketch the Bode plot find K when phase margin = 30˚.

G(S) =

. (16) Apr / May 2010

13. Sketch the Bode plot find K when gain margin = 10db. G(S) =

. (16) Nov/Dec 2007

14. A unity feedback system has an open loop transfer function G(S) =

. Design a suitable phase lag compensators to achieve the following specifications Kv = 8 and Phase margin 40 deg with usual notation. (16)

15. Consider a type 1 unity feedback system with an OLTF G(S) =

.The system is to be compensated to meet the following specifications Kv > 5sec and PM>43 deg .Design suitable lag compensators. (16)

16. Draw the bode plot G(s) =

. (16) May/June 2009

17. Sketch the polar plot G(S) =

(16) May/June 2009

18. Plot the bode diagram for the following transfer function and obtain the gain and phase margin G(S) =

. (16) May/June 2009

19. Design a suitable lead compensator for a system with unity feedback and having open loop transfer function G(S) =

to meet the specifications as damping ratio = 0.5 and undamped natural frequency = 2 rad / sec. (16) May/June 2009

20. Sketch the bode plot G(S) =

(16) May/June 2009 Nov/Dec 2011

21. Discuss in detail about the design of a lag-lead compensator. Design the elements of the network and sketch the bode plot. (16) May/June 2009

22. Explain step involved in the design of a lag compensator.

(16) Nov/Dec 2007 Apr / May 2010

23. Sketch the bode plot for the following transfer function and find Gain margin and phase margin G(S)H(S) =

. (16) Apr / May 2010

24. A unity feedback system has an open loop transfer function G(S) =

Design a suitable phase lead compensators to achieve the following specifications (i) K_v = 20 sec^-1 (ii) Phase margin = +50˚ (iii)Gain margin ≥+10 db. (16) Apr / May 2010

25. A unity feedback system has an open loop transfer function

G(S) =

Design a suitable phase lead compensators to achieve the following specifications (i) K_v = 12 sec^-1 (ii) Phase margin = 40˚ . (16) May/June 2009

26. Write the short notes on correlation between the time and frequency response? (8)

27. The open loop transfer function of a unity feedback system is

G(S) =

Sketch the polar plot and determine the gain margin and phase margin. (16) Apr / May 2011

28. Sketch the Bode plot for the transfer function and determine the value of K for the gain cross over frequency of 5 rad/sec. (16) May / June 2013, EEE

29. Sketch the polar plot for the following transfer function and determine the gain and phase margin

(16) May / June 2013, EEE

30. Explain in detail the design procedure of lead compensator using Bode plot. (16) May / June 2013, ECE

31. Consider a unity feedback open loop transfer function

. Draw

the Bode plot and find the phase margin and gain cross over frequencies, phase and

gain margin and stability of the system. (16) May / June 2013, ECE

32. Draw the pole-zero diagram of a lead compensator. Propose lead compensation using

electrical network. Derive the transfer function. Draw the bode plots. (16) Nov/ Dec 2012

33. Given

, find K for the following two cases:

(i) Gain Margin equal to 6dB

(ii) Phase Margin equal to 45⁰ (16) Nov/ Dec 2012

34. Explain the use of M circles and N circles for the study of frequency response analysis of feedback system? (8) May/June 2012

UNIT IV - STABILITY ANALYSIS

PART- A

1. Define parameter variations.

The parameters of any control system cannot be constant through its entire life. There are always changes in the parameters due to environmental changes and other disturbances. These changes are called parameter variations.

2. Define sensitivity of a control system.

An effect in the system performance due to parameter variations can be studied mathematically defining the tern sensitivity of a control system. The change in particular variable due to parameter can be expressed in terms of sensitivity.

3. What is stability? May / June 2006

For a bounded input signal, if the output has constant amplitude oscillations may be stable.

4. What are the necessary conditions for stability of control systems.

May 2009, E&I

5. State Routh stability criteria. Nov/Dec 2006 ,Apr / May 2010

Routh’s criterion states that, the necessary and sufficient condition for the stability is that, all the elements in the first column of the Routh’s array be positive. If the condition is not met, the system is unstable, and the number of sign changes in the elements of the first column of Routh’s array corresponds to the number of roots of characteristic equation in the right half of the S-plane.

6. What are the conditions for a linear time invariant system to be stable?

A linear time- invariant system is stable if the following two notions of system stability are satisfied. I. When the system is by a bounded input, the output is bounded. II. In the absence of the input, the output tends towards zero irrespective of initial conditions.

7. What do you mean by asymptotic stability?

In the absence of the input, the output tends towards zero (the equilibrium state of the systems) irrespective of initial conditions. This stability is known as asymptotic stability.

8. How the system is classified based on stability?

Based on the stability, the system can be classified as

· Absolute stable system.

· Conditionally stable system.

· Unstable system.

· Marginally stable or critically stable system.

9. Define BIBO Stability? Nov/Dec 2006 Apr / May 2010

A linear relaxed system is said to have BIBO stability if every bounded (finite) input results in a bounded (finite) output.

10. What is meant by unstable system?

A linear time invariant system is said to be unstable if

· The system produces unbounded output for a bounded input.

· In absence of the input, output nay not be returning to zero.

11. What is meant by critically or marginally stable system?

A linear time invariant system is said to be critically or marginally stable, if for a bounded input, its output oscillates with constant frequency and amplitude. Such oscillations of output are called undamped oscillations or sustained oscillations.

12. What is the necessary condition for stability?

The necessary condition for the stability is that all the co-efficient of the characteristic polynomial be positive.

13. State the requirement for BIBO stability.

The requirement for BIBO stability is that

Where m(τ) is the impulse response of the system.

14. State Hurwitz criterion.

The necessary and sufficient conditions to have all roots of the characteristic equation in left half of the s-plane is that, the sub-determinants DK, k = 1,2,…..,m obtained from Hurwitz’s determinant must be positive.

15. Define absolute stable.

Absolutely stable with respect to a parameter of the system, if it is stable for all values of this parameter.

16. What do you mean by relative stability?

Relative stability is a quantitative of how fast the transients die out in the system. If it is stable for all values of this parameter.

17. What does the positive ness of the coefficients of characteristic equation indicate?

· The positive-ness of the coefficients of characteristic equation is necessary as well as sufficient condition for stability of system of first and second order.

· The positive-ness of the coefficients of the characteristic equation ensures the negative-ness of the real parts of the complex roots for third and higher order systems.

18. State the conditions under which the coefficients can be zero or negative.

· One or more roots have positive real parts.

· A root (or roots) at origin i.e. SK = 0 and hence an = 0.

· Sl = 0 for some l, which implies the presence of roots on the jw axis.

19. How the roots of characteristic equation are related to stability?

If the roots of characteristic equation has positive real part then the impulse response of the system is not bounded (the impulse response will be finite as t tends to infinity.) hence the system will be unstable. If the roots have negative real parts then impulse response is bounded. ( the impulse response becomes zero as t tends to infinity). Hence the system will be stable.

20. What is the relation between stability and coefficient of characteristic polynomial?

If the coefficients of characteristic polynomial are negative or zero, then some of the roots lie on the right half of the S-plane. Hence the system is unstable. If the coefficients k of characteristic polynomial are positive and if no coefficient is zero, then there is a possibility of the system to be stable, provided all the roots are lying on left half of S-plane.

21. What will be the nature of impulse response when the roots of characteristic equation are lying on imaginary axis?

If the roots of characteristic equation lie on imaginary axis, then the impulse response is oscillatory.

22. What will be the nature of impulse response when the roots of characteristic equation are lying on right half of the S-plane?

When the roots are lying on the real axis, i.e on the right half of the S-plane, the response is exponentially increasing. When the roots are complex conjugate and lying on the right half of the S-plane, the response is oscillatory with exponentially increasing amplitude.

23. What is ROUTH stability criterion?

ROUTH stability criterion states that, the necessary and sufficient condtition for stability is that all of the elements in the first column of the routh’s array be positive. If this condition is not met, then the system is unstable, and the number of sign changes in the elements of the first column corresponds to the number of roots of characteristic equation in the right half of the S-plane.

24. What is auxiliary polynomial?

In the construction of the Routh array, a row of all zero indicates the existence of an even polynomial as a factor of the given characteristic equation. In an even polynomial, the coefficient of auxilary polynomial are given by the elements of the row just above the row of all zeros.

25. What is quadrantal symmetry?

The symmetry of roots with respect to both real and imaginary axis is called quadrantal symmetry.

26. Give an application of Routh Stability criterion,

The routh Stability criterion is frequently used for the determination of the condition of stability of linear feedback control systems.

27. The Routh-Hurwitz criteria gives absolute stability. Justify your answer.

Basically Routh-Hurwitz is a time domain method. It only gives the indication about the locations of the roots of the characteristic equation in the S-plane. It does not give the information about the actual locations and the types of roots. As the actual locations of the roots are unknown, it is impossible to calculate the parameters required for the prediction of the relative stability. I.e. gain margin , phase margin etc.

28. The addition of a pole will make a system more stable. Justify your answer.

This is false statement. When the pole is added to the system, it drives the root locus towards imaginary axis, they become dominant and hence relative stability of the system decrease. It makes the system more oscillatory. So addition of pole makes the system unstable and not stable.

29. What do you mean by root locus technique?

Root locus technique provides a graphical method of plotting the locus of the roots in the S-plane as a given system parameter, is varied over the complete range of values(may be from zero to infinity). The roots corresponding to a particular value of the system parameter can then be located on the locus or the value of the parameter for a desired root location can be determined from the locus.

30. In the routh array what conclusion you can make when there is a row of all zeros?

All zero row in routh array indicates the existence of an even polynomial as a factor of the given characteristic equation. The even polynomial may have roots on imaginary axis.

31. What is limitedly stable system?

For a bounded input signal, if the output has constant amplitude oscillations, then the system may be stable or unstable, under some limited constraints. Such a system is called limitedly stable system.

32. How will you find the root locus on real axis?

To find the root locus on real axis, choose a test point on the real axis. If the total number of poles and zeros on the real axis to the right of this test point is odd number, then the test point lies on the root locus. If it is even number means, then the test point does not lie on the root locus.

33. The root locus gives the bird’s eye view of the stability of the systems. Justify your answer.

The above statement is false. The root locus gives the locus of the characteristic equation as gain K is varied from zero to infinity. So from root locus we can get the exact locations of roots for any value of k. similarly for any value of K, the nature of the roots were real complex conjugates can be predicted. From this we get the idea about transient response nature of the system, including the settling time of the system. It can give range of values of K for which system remians stable. Thus root locus gives total information about the stability of the system. Hence it is said that root locus gives inside analysis of the stability and not the birds eye view of the stability.

34. Whether root locus gives the idea about the steady state error?

The root locus gives the information about the nature of the roots of the characteristic equation. So from it, root locus gives the idea about the overall stability of the system and the nature of the transient response. But it cannot give idea about the steady state errors. The steady state error depends partly on the system gain K, hence we may say that root locus gives the partly information about steady state errors.

35. What is centroid? How the centroid is calculated?

The meeting point of asymptotes with real axis is called centroid. The centroid is given by

Centroid () = sum of real parts of poles – sum of real parts of zeros

P-Z

Where P= number of poles

Z = number of zeros.

36.The roots of characteristic equation will be real and equal for a system. Justify your answer.

The above statement is false, because the underdamped system is the one having exponential decaying. It oscillates with decreasing amplitude and settles down. For the real and equal roots, the response is exponential but fast and without any oscillations. It is called critically damped. So real and equal roots does not represent underdamped system.

37. The root locus gives the idea about the transient response of the system. Justify your answer.

The root locus is the locus of the roots of the characteristic equation as K (gain) varies from 0 to infinity. The transient response is totally dependent on the nature of the roots of the characteristic equation, real -negative roots give exponentially decaying. Real- positive roots gives exponentially increasing., complex with negative real parts give decaying oscillatory transient response. The information about nature of roots is provided by root locus. Hence it gives idea about the transient response of the system.

38. A system having repeated roots of charcteristic equation on imaginary axis is stable. State true or false.

Answer: False. Because when there are repeated roots on the imaginary axis, the output contains a term like t e-at and t tends to , such term makes the system output controllable. Hence sush system is unstable.

39. What are asymptotes? Give the formula to calculate the angle of asymptotoes.

Asymptotes are straight lines, which are parallel to root locus going to infinity and meet the root locus at infinity.

Angle of asymptoes = +(2q+1) 180 where P- no. of poles, Z- no.of zeros

P-Z.

40. What are breakaway points?

Points at which multiple roots of the characteristic equation occur are called breakaway points. A breakaway point may involve two or more than two branches.

41.State the rule for obtaining breakaway point.

· Construct the characteristic equation 1+ G(s)+H(S) = 0 of the system.

· From this equation, separate the terms involving K and terms involving S. Write the value of K in terms of S. i.e. K = f(s)

· Differentiate the above equation with respect to S. equate it to zero. I.e.dK/ds = 0.

· Roots of the equation dK/ds = 0 gives us the breakaway points.

42. How do you find the crossing point of root locus in imaginary axis?

Method (i) by routh hurwitz criterion.

Method (ii) By letting S= jw. In the characteristic equation and separate the real and imaginary part. These two equations are equated to zero. Solve the two equations for ω and K. the value of ω gives the point where the root locus crosses the imaginary axis and the value of K is the gain corresponding to the crossing point.

43. If a system is having complex conjugate with positive real point, then the system is said to be unstable.

44. If a system is having complex conjugate with negate real point, then the system is said to be absolute stable.

45. Give the formula for finding angle of departure at complex pole.

46. Give the formula for finding angle of arrival at complex pole.

47. What is magnitude criterion of root locus?

48. What is angle criterion of root locus?

The angle criterion of root locus states that S =Sa will be a point on root locus if for that value of S, the argument or phase of G(S)H(S) is equal to an odd multiple of 180

49. What is dominant pole?

The dominant pole is a pair of complex conjugate pole which decides transient response of the system. In higher order systems, the dominant poles are very close to origin and all other poles of the system are widely separated and so they have less effect on transient response of the system.

50. How will you find the gain K at a point on the root locus?

The gain K at a point S= Sa on root locus is given by

K= product of length of vectors from open loop zero to the point Sa

product of length of vectors from open loop poles to the point Sa

51. Give the effect of addition of poles on the root locus.

· Root locus shift towards imaginary axis.

· System stability relatively decreases.

· System becomes more oscillatory in nature.

· Range of operating value of K for system stability decreases.

52. Give the effect of addition of zero on the root locus.

· Root locus shift to left away from imaginary axis.

· Relative stability of the system increases.

· System becomes less oscillatory in nature.

· Range of operating value of K for system stability increases.

53. State the advantages of root locus method.

· Root locus analysis helps in deciding the stability of the control systems with time delay.

· Information about settling time of the system also can be determined from the root locus.

· The absolute stability of the system can be predicted from the location of the roots in the S-plane.

54. Define gain margin in Nyquist plot.

Gain margin is the amount of gain in decibels(db) that is allowed to be increased in the loop before the closed loop system reaches stability.

55. Define phase margin in Nyquist plot.

Phase margin may be defined as the angle in degrees through which the G(j ω)H(j ω) plot must rotate about origin in order that the gain cross over point on the locus passes through the point (-1+j0)

56. What are the observations that are made from the polar plot?

· Addition of a pole at the origin to a transfer function rotates the polar plot at zero and infinite frequencies by a further angle of -90⁰

· Addition of a non zero pole to a tranfer function results in further rotation of the polar plot through angle of -90⁰ as ω →α

57. On what theorem the Nyquist stability criterion is based on?

The Nyquist stability criterion is based on Cauchy’s residue theorem of complex variables which is referred to as the principle of argument.

58. State the principle of argument.

The principle of argument can be stated as follows:

Let G(s) be a single valued function that has finite poles in the S-plane. Suppose that an arbitrary closed path Ta is chosen in the S-plane so that the path does not go through any one of the poles or the zeros of Q(s); the corresponding locus mapped in the Q(s) plane will encircle the origin as many times as the difference between the number of the zeros and the number of poles of Q(s) that are encircled by the S-plane locus Ta.

59. Discuss Nyquist stability criterion.

Nyquist has used the mapping theorem or the principle of argument effectively to develop a criterion to study the stability of control system in the frequency domain. He has suggested to select a single valued function F(s) as 1+G(s)H(s), where G(S)H(s) is open loop transfer function of the system. F(s) = G(s)H(s)

Poles of 1+G(s)H(s) = poles of G(s)H(s) = open loops poles.

They are known to us, but the zeros of 1+G(s)H(s) are unknown to us.

For stability all the zeros of 1+G(s)H(s) must be in left half of s-plane. Instead of analyzing whether all the zeros are located in the left half of the s-plane, it is better to the presence of any one zero of 1+G(s)H(s) in the right half of the s-plane making system unstable.

So Nyquist has suggested to select a T(s) path which will encircle the entire half of the s-plane. This path is called Nyquist path and should not be changed except small modifications.

60. List the steps to solve problems by Nyquist criterion.

Step1: count how many no. of poles of G(s)H(s) are in the right half of the s-plane i.e, with the positive real part. This is the value of p.

Step2: Decide the stability criterion as N= (-P) i.e., how many times Nyquist plot should encircle -1+j0 point for absolute stability.

Step3: select Nyquist path as per the function G(s)H(s).

Step4: Analyse the sections as starting point and terminating point. Last section analysis not required.

Step5: Mathematically find out Wpc and the intersection of Nyquist plot with negative real axis by rationalizing G(jw)H(jw).

Step6: With the number of encirclements N of -1+j0 by Nyquist plot. If this matches with the criterion decided in step 2 system is stable, otherwise the system is unstable.

61. List the advantages of Nyquist plot.

o It gives same information about absolute stability as provided by rouths criterion.

o Useful for determining the stability of the closed loop system from open loop transfer function without knowing the roots of characteristic equation.

o Information regarding frequency response can be obtained.

o Very useful for analyzing conditionally stable systems.

o It also indicates relative stability giving the values of G.M. and P.M.

62. Define phase cross over frequency in nyquist plot.

The phase cross over frequency pc is the frequency at which phase cross over point or where  G(j)H(j) = 180

63. Define phase cross over point in nyquist plot.

It is the point in the G(j) plane at which the nyquist plot G(j)H(j) intersects the negative real axis.

64. What is root locus? Nov/Dec 2012

The path taken by a root of characteristic equation when open loop gain K is varied from 0 to infinity is called root locus.

65. What are root loci? May / June 2009

The path taken by the roots of the open loop transfer function when the loop gain is varied from 0 to α are called root loci.

66. What is angle of criterion of root locus?

67. What is a dominant pole?

The dominant point is a pair of complex conjugate pole which decides transient response of the system. In higher order systems the dominant poles are very close to origin and all other poles of the system are widely separated and so they have less effect on transient response of the system.

68. What is characteristics equation?

The denominator polynomial C(s)/R(s) is the characteristic equation of the system.

69. What is a centroid?

The meeting point of asymptotes with real axis is called centroid.

Centroid=sum of poles-sum of zeros /n-m

70. Define a breakaway& break in point.

At breakaway point the root locus breaks from the real axis to enter into the complex plane. At breakin point the root locus enters the real axis from complex plane

71. What are asymptotes? How will you find the angle of asymptotes? Apr / May 2010

Asymptotes are straight lines which are parallel to root locus going to infinity an meet the root locus at infinity.

Angle--+or - 180(2q+1)/n-m, q------0,1,2,----(n-m)

72. What is Nyquist stability criterion? May/ June 2009 Apr / May 2010 Nov/Dec 2006, 2012

If G(s) H(s) contour in the G(s) H(s) palne corresponding to Nyquist contour in s-plane encircles the point -1+j0 in the anti clockwise direction as many times as the number of right half s-planes poles of G(s) H(s).Then the closed loop system is stable.

73. What is angle criterion for root locus? May/ June 2009

The angle criterion states that S=S_awill be a point on root locus if for that value of S argument or phase of G)S)H(S) is equal to an odd multiple of 180®.

74. Define the root loci and root contour. May 2009, E&I

75. What are the effects of addition of open loop pole? Apr 2010

76. Sketch the response of the system with reference to the stability in case if the

1. Root lies in left half of S plane

2. On the imaginary axis

3. Right half of S plane May / June 2009

77. What are the disadvantages of Hurwitz criterion. May 2009

78. State the magnitude and the angle condition of root locus. Apr 2010

79. Using routh criterion, determine the stability of the system represented by the characteristics equation Nov / Dec 2010

80. State the rule for obtaining the breakaway point in Root locus. Apr/ may 2011

81. State any two limitations of routh stability criterion. Nov / Dec 2011

82. State the advantages of Nyquist stability criterion over that of Routh’s criterion. Nov 2012

83. State the method of obtaining the gain K at appoint on Root locus.

Nov / Dec 2007, EEE

84. At breakaway point in Root loci several branches combines. Why? Nov /Dec 2006, EEE

85. How the angle of arrival at zero is obtained in Root locus?

Apr / May 2005, EEE

PART-B

1. Examine the stability of Routh’s criterion

(8) May 2009

2. Find the range of K, so that system with tha characteristics equation,

will be stable Routh’s criterion. (8) May 2009

3. F(S)= S⁶+S⁵-2S⁴-3S³-7S²-4S-4=0. Find the number of roots falling in the RHS plane and LHS plane. (8)

4. F(S)= S⁶+2S⁵+8S⁴+12S³+20S²+16SS+16=0. Find the number of roots falling in the RHS plane and LHS plane. (6) May/June 2012

5. Draw the Nyquist plot for the system whose open loop transfer function is G(S) H(S) =

Determine the range of K for which the closed loop system is stable. (16) May/June 2009, 2012 Apr 2010

6. Construct Nyquist plot for a feedback control system whose open loop transfer function is given by G(S)H(S) =

comment on the stability of open loop and closed loop transfer function. (16)

7. Sketch the Nyquist plot for a system with the open loop transfer function G(S) H(S)=

.Determine the range of values of K for which the system is stable. (16) Nov/Dec 2012

8. Sketch the root locus for the unity feedback system whose open loop transfer function is G(S) =

. (16) Nov/Dec2006 June 2006

9. Sketch the root locus for the unity feedback system whose open loop transfer function is G(S) =

. (10) Nov/Dec 2007, 2012

10. Sketch the root locus for the unity feedback system whose open loop transfer function is G(S) =

(16)

11. Sketch the root locus for the unity feedback system whose open loop transfer function is G(S) =

. (16)

12. Sketch the root locus for the unity feedback system whose open loop transfer function is G(S) =

(10) May/June 2012

13. Sketch the Nyquist plot determine the stability of the system

G(S) H(S)=

. (16) Nov/Dec 2007 May/June 2009

14. Using Routh criterion Determine the stability of the system whose characteristics equation is S⁵+S⁴+2S³+2S²+3S+15=0. (8) May/June 2009

15. Explain in detail about Root locus method. (16) May/June 2009

16. (i) List the rules for constructing root locus. (12)

(ii) Write the procedure for constructing root locus. (4) Nov/Dec 2011

17. Sketch the Nyquist plot for a system with the open loop transfer function G(S) =

. Determine the range of values of K for which the system is stable. (16) Apr / May 2010

18. Draw the Nyquist plot for the system whose open loop transfer function is G(S) H(S) =

(16) Apr / May 2010

19. Draw the Nyquist plot for the system whose open loop transfer function is G(S) H(S) =

(16) Nov/Dec2006

20. Sketch the root locus for the unity feedback system whose open loop transfer function is G(S) =

. (10) Apr / May 2010

21. Using Routh criterion Determine the stability of the system whose

characteristics equation is

1. S⁵+2S⁴+3S³+6S²+10S+15

2. S⁵+6S⁴+15S³+30S²+44S+24 (8+8) May/June 2006

22. Using Routh criterion Determine the stability of the system whose characteristics equation is S⁵+S⁴+2S³+3S+5=0. (8) Nov/Dec 2011

23. Consider the sixth order system with the characteristic equation

. Use Routh-Hurwitz criterion to examine the stability of the system. (16) May / June 2013

24. Sketch the root locus of the system having

(16) May / June 2013

25. Determine the range of K for stability of unity feedback system whose open loop transfer function is

using Routh stability criterion. (8) Nov / Dec 2012

26. Draw the approximate root locus diagram for a closed loop system whose loop transfer function is given by

comment on the stability. (8) Nov / Dec 2012

27. Sketch the Nyquist plot for a system with open loop transfer function

and determine the range of K for which the system is stable. (16) Nov / Dec 2012

28. Sketch the root locus of the system whose open loop transfer function is

. Find the value of K so that the damping ratio is 0.5 (16) May / June 2013, EEE

29. Construct the Routh Hurwitz array and determine the stability of the system represented by the characteristic equation and comment on the location of roots.

(i)

(ii)

(16)May / June 2013, EEE

UNIT V - STATE VARIABLE ANALYSIS & DIGITAL CONTROL SYSTEMS

PART-A

1. Define ‘state’ and ‘state variables’. Nov/Dec 2012

2. What is state?

3. What is state variable?

The state is the condition of a system at any instant, t. The state of dynamic system is defined as a minimal set of variables such that the knowledge of these variables at t =to together with the knowledge of inputs t > 0 completely determine the behavior of the system for t > to.

A set of variable which describes the state of the system at any time instant are called state variables. The variables involved in determining the state of dynamic system are called state variables. Generally x1(t),x2(t),x3(t)…….xn(t) are called state variables.

4. What is state vector?

The state vector x(t) is the vector sum of all the state variables.

5. What is state space?

The space whose coordinate axes are nothing but the ‘n’ state variables with time as the implicit variable is called state space.

6. What are the advantages of state space analysis?

· It can be applied to non-linear as well as time varying systems.

· Any type of input can be considered for designing the system.

· It can be conveniently applied to multiple input multiple output systems.

· The state variables selected need not necessarily be the physical quantities of the system

· The state space analysis can be performed with initial conditions.

· Using this analysis the internal state of the system at any time instant can be predicted.

7. How the modal matrix is determined? May / June 2012

The modal matrix M can be formed from eigenvectors. Let m₁, m₂, m₃….. m_n be the eigenvectors of a nth order system. Now the modal matrix M is obtained by arranging aii the eigenvectors column wise. ie M = [m₁ m₂ m₃ …… m_n]

8. Mention the need for state variables. Nov / Dec 2010

9. Write the properties of state transition matrix? Apr / May 2010

· Ф(0) = e^{A x 0}= 1 (Unit matrix)

· Ф(t) = (e^-At)^-1 = [Ф(-t)]^-1

_·Ф(t₁+t₂) = e^A(t₁^+t₂⁾ = e^At₁ e^At₂ = Ф(t₁) Ф(t₂) = Ф(t₂) Ф(t₁)

10. What are phase variables?

The phase variables are defined as the state variables which are obtained from one of the system variables and its derivatives.

11. Name the methods of state space representation for phase variables. Apr / May 2011

· Bush form or companion form

· By using mason’s gain formula

· By using laplace transform

12. Determine the controllability of the system described by the state equation. Apr / May 2010

13. Define controllability and observability.

A system is said to be completely state controllable if it is possible to transfer the system state from any initial state X(t_o) at any other desired state X(t) in specified finite time by a control vector U(t).

A general nth order multi-input linear time invariant system X = AX +Bu. Is completely controllable if and only if the rank of the composite matrix Qc = [ B:AB : A²B:…….. :A^n-1B] is n

A system is said to be completely observable if every state X(t) can be completely identified by measurements of the output Y(t) over a finite time interval.

A general nth order multi-input multiple output linear time invariant system X = AX +Bu. Y= CX is completely observable if rank of the composite matrix Qc = [ C^T:A^TC^T:…….. :(AT)^n-1B] is n

14. List the methods used to test the stability of discrete time system.

· Jury’s stability test.

Bilinear transformation.
Root locus technique.

15. What is the effect of pole zero cancellation in transfer function?

If cancellation of pole zero occurs in the transfer function of a system, then the system will be either not state controllable or unobservable depending on how the state variables are defined ( or chosen)

16. What is sampled data control system? Nov/Dec 2012

In a control system, if the signal in any part of the system is discrete then the entire system is said to be sampled data system.

17. What is the condition to be satisfied for a sampled data system to be stable?

The poles of the pulse transfer function H(z) must lie inside z-plane unit circle.

18. What is the characteristic equation of a sampled data system?

The denominator polynomial of a closed loop pulse transfer function H(z) is known as the characteristic equation.

19. When a control system can be called as sampled data control system?

Any control system can be called as sampled data control system, when ever,

A digital system (computer/ microprocessor/microcontroller) becomes part of control system.

Control components are on the time sharing mode.

Control signals are discrete or digital signals.

20. Distinguish between sampled data systems and continuous-time systems.

Control system components of sampled data control system are able to handle discrete (digital) signals. On the other hand, continuous time system components can handle analog signals. Similarly output signals of sampled data system components are discrete (digital) signals.

21. What is digital controller?

A digital device used to generate control signal for which error signal is given as input.

22. List the advantages and disadvantages of sampled data control system.

Advantages	Disadvantages
Accuracy increased while compared with analog system	Digital conversion and reconstruction may introduce noise.
Speed increased and flexibility improved.	Improper selection of sampling period may leads to instability.
Digital transducers have better resolution.
Highly immune to noise.
Linearizable becomes simple.
Free from transmission noise.
Complex algorithms can easily be implemented.

23. Distinguish between analog and digital controllers.

Digital controller	Analog controller
Complex control algorithms can also be simply implemented.	Complex circuits required, some time it may not permit to implement.
Less cost than an analog controller	Costlier.
Fast acting.	Slow.
Control algorithms can be modified simply by changing software.	New circuit is required if algoritms modified.
Time sharing of controller by many system is possible.	Separate controllers are I need for multiple system environment.

24. What is discrete signal sequence f(k)?

A discrete signal sequence or discrete time signal f(k) is function of independent variable k is an integer.

25. What is impulse response?

The output (response) of a system when the input is impulse signal is known as impulse response.

26. What is weighting sequence?

The impulse response of a linear discrete time system is called weighting sequence.

27. What is pulse transfer function?

It is the mathematical model of discrete time system. It is the impulse response of the system represented in the z-domain. It is also defined as the ratio of z-transform of output signal to the z-transform of input signal of the system.

28. What is pulse transfer function?

It is the mathematical model of discrete time system. It is the impulse response of the system represented in the z-domain. It is also defined as the ratio of z- transform of output signal to the z-transform of input signal of the system.

Pulse transfer function H(z) = C(z)/R(z).

Where C(z) is z-transform of output signal.

R(z) is z-transform of input signal.

29. State sampling theorem.

A continuous time signal can be completely represented in its samples and recovered back if the sampling frequency Fs≥2Fmax where Fs is the sampling frequency and Fmax is the maximum frequency present in the signal.

30. What are sampling and sampler?

Sampling of a signal is a process by which analog signals are sampled at predetermined intervals to convert into discrete time signals. The device used to perform sampling is called sampler.

31. What is periodic sampling?

Sampling of a signal at uniform equal intervals is called periodic sampling. The uniform equal interval T is called period.

32. What is meant by quantization? May / June 2012

The process of approximating a discrete time continuous valued signal into a discrete valued signal is called quantization. If the sampled analog value lies in between two digital adjacent values then the sampled analog value will be represented by a digital value which is nearer to the analog value than the other. This process of approximation is called quantization.

33. What is coding?

Representation of sampled data by n bit binary number is called coding.

34. What is hold circuit?

What are hold circuits and explain it.

A device used to convert digital signal into analog signal.

The function of the hold circuit is to reconstruct the signal which is applied as input to the sampler. The simplest holding device holds the signal between two consecutive instants at its preceded value. Till next sampling instant is reached.

35. What is zero-order hold?

It is a hold circuit. The output of the hold circuit is analog signal whose magnitude equal to latest sampled value till next sample occurs.

36. What is first order hold?

The output of the first order hold is constructed from latest two samples (current and previous samples). The slope of the output signal is determined by this current and previous sample.

37. What are the problems encountered in a practical hold circuits?

Hold mode may drop occurs nonlinear variation during sampling aperture, error in the periodicity of sampling.

38. What is acquisition time?

Time taken by an analog to digital converter to sample the signal, to quantize it and to code it is known as acquisition time.

39. Define aperture time.

It is the duration of sampling of analog signal.

40. What is settling time?

It is the time taken by a digital to analog converter to convert the given digital signal into analog signal magnitude and be remain with in the tolerance is called settling time.

41. What is hold mode droop?

There is no droop in an ideal hold circuit. The change in signal magnitude during hold mode of a hold circuit is called hold mode droop.

42. What are the problems that may occur in a practical hold circuit?

· Hold mode droop may occur.

· Nonlinear variation during sampling aperture.

· Error in the periodicity of sampling.

43. How the high frequency noise in the output hold circuits can be filtered?

The control system components act as low pass filter. Hence the high frequency signals are automatically filtered.

44. What is alias in sampling process? Nov/ Dec 2011

PART-B

1. Obtain the state space representation of

(a) Armature controlled DC motor. (8)

(b) Field controlled dc motor. (8) May/June 2012

2. Explain in brief about the different types of system realization methods.

3. Explain in detail the state space representation for continuous time systems. (8) Nov / Dec 2010

4. Explain in detail the state space representation for discrete time systems. (8) Nov / Dec 2010

5. Obtain the state space model of the system with transfer function

in phase variable form. (16)

6. The state model of the system is given by

Check for controllability. (8) May/June 2010

7. A system characterized by the following state equation

; T>0,

Y = [1 0]

Find (i) Transfer function of the system

(ii) State transition matrix. (16)

8. A control system is described by the differential equation

= u(t) where y(t) is the observed output and u(t) is the input describe the system in the state variable form i.e ., X = AX +BU, Y =CX + DU

Check for controllability and observability. (16)

9. Consider the following transfer functions. Obtain the state space representation of these systems using controllable canonical form.

(8)

10. (a) State cayley- Hamilton theorem.

(b) State the properties of state transistion matrix.

X=AX+BU, Y =CX+DU. (16)

11. A system is represented by the state equation

, where

. Determine the transfer function of the system. (16) May / June 2013

12. A system is characterized by the transfer function

. Identify the first state as the output. Determine whether or not the system is completely controllable and observable. (16) May / June 2013

13. Obtain the state space representation for the electrical network shown in figure below (8) Apr / May 2010

14. The state space representation of a system is given below:

u y=

Obtain the transfer function. (16) Nov/Dec 2012

15. The state space representation of a system is given below:

Obtain the transfer function. (16) Nov / Dec 2012, Apr 2011

16. The state model of the system is given by

u y=

Check for controllability and observability. (8) Nov/Dec 2012

17. The state model matrices of a system are given below:

A =

B =

and C =

Evaluate the observability of the system using Gilbert’s test. (10) May/June 2012

18. Determine the state controllability and observability of the system described by

(16) Nov / Dec 2010

19. Find the controllability of the system described by the following equation:

(6) May / June 2012

20. Write the state equations for the system shown below in which X₁, X₂, and X₃ constitute the state vector.

Determine whether the system is completely controllable and observable. (16) Apr 2011

21. Find the state variable equation for a mechanical system (spring – mass – damper system) shown below.

(8) Nov / Dec 2011

22. A LTI system is characterized by the state equation

where ‘U’ is a unit step function. Compute the solution of this equation by assuming initial condition

. Use inverse Laplace transform technique. (8) Nov / Dec 2011

23. Using cascade method decompose the transfer function

and obtain the state model. (8) Apr / May 2010

24. (i) Determine the transfer matrix from the data given below:

A =

B =

C = [1 1] D = 0. (8)

(ii) The transfer function of a control system is given by

Check for controllability. (8) Apr / May 2010

25. Obtain the state space representation of armature controlled DC motor with load shown below.

Choose the armature current i_a the angular displacement of shaft Ө and the speed

as state variables and as Ө output variables. (16) May/June 2012

26. Obtain the z- domain transfer function of the system shown below.

(8) Nov/Dec 2012

27. Write the state equations in phase variable form for a system with the differential equation.

+19

+13y=13

+26u (16) Nov/Dec 2011

28. Check if all the roots of the following characteristics equations lie within the unit circle

(i) Z³-0.2z²-0.25z+0.05=0

(ii) Z⁴-1.7z³+1.04z²-0.268z+0.024=0. (16) Nov/Dec 2011

29. For the following transfer functions obtain the state space representation of this systems in controllable canonical form

(i) T (s) =

(ii) T (s) =

(16) May/June 2010

30. Derive the expression for the sampling theorem and draw and explain sample and hold circuit. (8)

31. Explain briefly about sampled data control system. (8)

32. A sampled data control system is shown in the figure below

Find the open loop pulse transfer function, if the controller gain is unity with sampling period time 0.5 seconds. (16) Nov /Dec 2011

ECE Question bank for IV sem And VI Sem And VIII Sem

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Wednesday, 29 January 2014