Chapter 2 Linear Control System .ppt

Lcs course elec

1. Control Systems Department of Electrical Engineering

2. Continuous Time System
Response
Control Systems Department of Electrical Engineering

3. Control Systems Department of Electrical Engineering
Response of 1
st
Order Systems
In a 1
st
order system, the input r(t) and output y(t) are related by
a differential equation of the form
The corresponding transfer function is
The zero state and zero input response can be calculated as

4. Control Systems Department of Electrical Engineering
Response of 1
st
Order Systems
For a step input and zero initial conditions
Y(s) contains only zero state component given as:
Time solution is given as:

5. Control Systems Department of Electrical Engineering
Response of 1
st
Order Systems
1
st
order system step response with zero initial conditions

6. Control Systems Department of Electrical Engineering
Response of 1
st
Order Systems
For non zero initial conditions
The corresponding time response is

7. Control Systems Department of Electrical Engineering
Response of 1
st
Order Systems
1
st
order system step response with non-zero initial conditions

8. Control Systems Department of Electrical Engineering
Response of 1
st
Order Systems
Time constant of stable 1
st
order system is
It is the rate of decay of
The transfer function of the 1
st
order system can be written
in the form

9. Control Systems Department of Electrical Engineering
Drill Problems
Find and sketch the response of the system with the
following transfer function, input and initial conditions.

10. Control Systems Department of Electrical Engineering
Drill Problems
Find and sketch the response of the system with the
following transfer function, input and initial conditions.

11. Control Systems Department of Electrical Engineering
Drill Problems
Find and sketch the response of the system with the
following transfer function, input and initial conditions.

12. Control Systems Department of Electrical Engineering
Drill Problems
Find and time constant of the following system

13. Control Systems Department of Electrical Engineering
Drill Problems
Find and time constant of the following system

14. Control Systems Department of Electrical Engineering
Drill Problems
Find the time constant of the following system

15. Control Systems Department of Electrical Engineering
Response of 2
nd
Order Systems
In a 2
nd
order system, the input r(t) an d output y(t) are related
by a differential equation of the form
The corresponding transfer function is
The zero state and zero input response can be calculated as

16. Control Systems Department of Electrical Engineering
Response of 2
nd
Order Systems
Characteristic polynomial of the second order system is
Roots of the characteristic polynomial are s
1and s
2given as:

17. Control Systems Department of Electrical Engineering
Overdamped Response
If the Characteristic polynomial roots are Real and Distinct, the unit
step response of the system is termed as Over Damped response.

18. Control Systems Department of Electrical Engineering
Overdamped Response
If the Characteristic polynomial roots are real and distinct, the unit
step response of the system is termed as Over Damped response.
What happens if the initial conditions are not zero?

19. Control Systems Department of Electrical Engineering
Critically Damped Response
If the Characteristic polynomial roots are Real & Repeated, the unit
step response of the system is termed as Critically Damped response.

20. Control Systems Department of Electrical Engineering
Critically Damped Response
If the Characteristic polynomial roots are real & repeated, the unit step
response of the system is termed as Critically Damped response.

21. Control Systems Department of Electrical Engineering
Under Damped Response
If the Characteristic polynomial roots are Complex, the unit step
response of the system is termed as Under Damped response.

22. Control Systems Department of Electrical Engineering
Under Damped Response
If the Characteristic polynomial roots are Complex, the unit step
response of the system is termed as Under Damped response.

23. Control Systems Department of Electrical Engineering
Under Damped Response1,2
18sj    
8
( ) 1 cos 8 sin 8
8
( ) 1 1.06 cos 8 19.47
t
to
c t e t t
c t e t



  


  

24. Control Systems Department of Electrical Engineering
Undamped Response
If the Characteristic polynomial roots are Imaginary, the unit step
response of the system is termed as Undamped response.1,2
3sj
Undamped response:
Poles: two imaginary poles at s
1, s
2
Natural response:sustained oscillations at 3 rad/sec

25. Control Systems Department of Electrical Engineering
Underdamped Natural Frequency & Damping Ratio
•Underdamped 2
nd
order systems have a natural response that is
described by a radian frequency of oscillation ωand an exponential
constant σ.
•We can find ωandσfrom characteristic polynomial

26. Control Systems Department of Electrical Engineering
Underdamped Natural Frequency & Damping Ratio
The standard 2
nd
order system transfer function with natural
frequency and damping ratio is of the form
Thetwoquantitiesζandω
ncanbeusedtodescribethe
characteristicsofthesecond-ordertransientresponsejustas
timeconstantsdescribethefirst-ordersystemresponse.
NaturalFrequencyisthefrequencyofoscillationofthe2
nd
ordersystemwithoutdamping.
DampingRatiodetermineshowfasttheoscillationsdecayto
steadystatevalue.

27. Control Systems Department of Electrical Engineering
Underdamped Natural Frequency & Damping Ratio
•With underdamped 2
nd
order system response, a more useful form is
described by the underdamped natural frequency ω
nand the damping
ratio ξ.
•The quantities ωand σare related to underdamped natural frequency
ω
nand the damping ratio ξ.

28. Control Systems Department of Electrical Engineering
Underdamped Natural Frequency & Damping Ratio
•For ξ between 0 and 1, the characteristic roots lie on a
circle of radius ω
nabout the origin in the left half of the
s-plane.
•For ξ = 0 the roots are on the
imaginary axis and for ξ = 1
both roots are on the negative
real axis and repeated.

29. Control Systems Department of Electrical Engineering
Underdamped Natural Frequency & Damping Ratio
•For ξ between 0 and 1, the characteristic roots lie on a
circle of radius ω
nabout the origin in the left half of the
s-plane.
•For ξ = 0 the roots are on the
imaginary axis and for ξ = 1
both roots are on the negative
real axis and repeated.
•The underdamped natural
frequency ω
n is the radian
frequency at which the
oscillations would occur if the
damping ratio ξ = 0
•The damping ratio is related to
damping angle i.e.

30. Control Systems Department of Electrical Engineering
Underdamped Natural Frequency & Damping Ratio
Consider the 2
nd
order system with the transfer function

31. Control Systems Department of Electrical Engineering
Underdamped Natural Frequency & Damping Ratio
The standard 2
nd
order system transfer function with natural
frequency and damping ratio is of the form

32. Control Systems Department of Electrical Engineering
Underdamped Natural Frequency & Damping Ratio
•Set of normalized step response curves for underdamped 2
nd
order
systems

33. Control Systems Department of Electrical Engineering
Find and sketch the response of the system with following
Transfer function, input and initial conditions.
Drill Problems

34. Control Systems Department of Electrical Engineering
Find and sketch the response of the system with following
Transfer function, input and initial conditions.
Drill Problems

35. Control Systems Department of Electrical Engineering
Determine which of the systems are overdamped, critically
damped and under damped.
Drill Problems

36. Control Systems Department of Electrical Engineering
Determine which of the systems are overdamped, critically
damped and under damped.
Drill Problems

37. Control Systems Department of Electrical Engineering
Determine which of the systems are overdamped, critically
damped and under damped.
Drill Problems

38. Control Systems Department of Electrical Engineering
For the 2
nd
order system with the following transfer function,
determine the under damped natural frequency, the damping
ratio, and the oscillation frequency.
Drill Problems

39. Control Systems Department of Electrical Engineering
For the 2
nd
order system with the following transfer function,
determine the under damped natural frequency, the damping
ratio, and the oscillation frequency.
Drill Problems

40. Control Systems Department of Electrical Engineering
For the 2
nd
order system with the following transfer function,
determine the under damped natural frequency, the damping
ratio, and the oscillation frequency.
Drill Problems

41. Control Systems Department of Electrical Engineering
Find the constant k for which the system with Transfer
Function T(s) has the given 2
nd
order response property
Drill Problems

42. Control Systems Department of Electrical Engineering
Find the constant k for which the system with Transfer
Function T(s) has the given 2
nd
order response property
Drill Problems

43. Control Systems Department of Electrical Engineering
Find the constant k for which the system with Transfer
Function T(s) has the given 2
nd
order response property
Drill Problems

44. Control Systems Department of Electrical Engineering
The quality of the performance of a stable system is
characterized by the rise time, peak time, overshoot and
settling time.
Rise Time, Overshoot & Settling Time
Rise Time: It is the interval of time required for the step
response of the system to go from 10% to 90% of its final
value.

45. Control Systems Department of Electrical Engineering
Step Response & Damping Ratio

46. Control Systems Department of Electrical Engineering
Rise Time & Damping Ratio
The rise time increases as damping ratio increases.

47. Control Systems Department of Electrical Engineering
Rise Time, Overshoot & Settling Time
Peak Time: It is the time required for the step response of
the system to reach the first peak. 2
1
p
d
n
T





48. Control Systems Department of Electrical Engineering
Rise Time Vs damping Ratio

49. Control Systems Department of Electrical Engineering
Rise Time, Overshoot & Settling Time
Overshoot: It is the percentage difference between the
maximum and steady state values of the step response.  
2
/1
% 100OS e
 


50. Control Systems Department of Electrical Engineering
Rise Time, Overshoot & Settling Time  
2
/1
% 100OS e
 


51. Control Systems Department of Electrical Engineering
Rise Time, Overshoot & Settling Time
Settling Time: It is the minimum time required before the
system response remains within 5% of the final value.

52. Control Systems Department of Electrical Engineering
Rise Time, Overshoot & Settling Time

53. Control Systems Department of Electrical Engineering
Example
Given the pole plot find ξ, ω
n, T
p, %OS and T
sfrom pole location.
Φ

54. Control Systems Department of Electrical Engineering
Alternate Expression for Step Response
For an under-damped second order system the transfer
function is of the form
The step response has the Laplace transform given as:
For unit step input or command A = 1 and so

55. Control Systems Department of Electrical Engineering
The time response is
Alternate Expression for Step Response

56. Control Systems Department of Electrical Engineering
Drill Problems
Using the curves given in the text, find approximately the
percent overshoot, rise time and settling time of the
following systems when driven by the step input.

57. Control Systems Department of Electrical Engineering
Drill Problems
Using the curves given in the text, find approximately the
percent overshoot, rise time and settling time of the
following systems when driven by the step input.

58. Control Systems Department of Electrical Engineering
Drill Problems
Using the curves given in the text, find approximately the
percent overshoot, rise time and settling time of the
following systems when driven by the step input.

59. Control Systems Department of Electrical Engineering
Drill Problems
Using the curves given in the text, find approximately the
percent overshoot, rise time and settling time of the
following systems when driven by the step input.

60. Control Systems Department of Electrical Engineering
Higher Order System Response
The zero-state response to a unit impulse input of third and
higher order systems consists of sum of term, one term for
each characteristic root.

61. Control Systems Department of Electrical Engineering
Higher Order System Response
Consider the system transfer function

62. Control Systems Department of Electrical Engineering
The Concept of Dominant Poles
•Thepolesnear to the jωaxis are called thedominant poles.
•Thepoleswhich have very small real parts have small
damping ratio.
•3
rd
order system
•Complex poles are dominant
•Can be approximated with 2nd
order system

63. Control Systems Department of Electrical Engineering
The Concept of Dominant Poles
•3
rd
order system
•Real pole is dominant
•Can be approximated with 1
st
order system
•3
rd
order system
•All three poles are dominant