1. Chapter 4. Time Response
I may not have gone where I intended to go, but I think I have ended up where I needed to be.
Pusan National University
Intelligent Robot Laboratory

2. Table of Contents Introduction Poles, Zeros and System Response
First-order System
Second-order System: Introduction
General Second-order System
Underdamped Second-order System
System Response with Additional Poles
System Response with Zeros
Effects of Nonlinearities upon Time Response
Laplace Transform Solution of State Equations
Time Domain Solution of State Equations

3. Introduction Control System Definition
Analysis of system transient response
Step response of the first and second-order systems
Poles and zeros of a transfer function

4. Poles, Zeros and System Response
Output response of a system
Forced response (steady-state response, particular solution)
Natural response (homogeneous solution)
Forced response + natural response
Poles of a transfer function
S that causes the transfer function to become infinite.
Zeros of a transfer function
S that causes the transfer function to become zero.

5. Poles, Zeros and System Response
Poles and zeros of a first-order system : An example
Figure 4.1
a. System showing input and output;
b. pole-zero plot of the system

6. Poles, Zeros and System Response
Poles and zeros of a first-order system: An example
Figure 4.1 c. Evolution of system response
A pole of the input function -> forced response
A pole of the transfer function -> natural response
A pole on the real axis -> exponential response
Zeros and poles -> amplitudes.

7. Poles, Zeros and System Response
Example 4.1
Given the system of Figure 4.2, write the output, , in general terms.
Specify the forced and natural parts of the solution
Figure 4.2 System for Example 4.1

8. First-Order System A first-order system without zeros
(b)
Figure 4.3 a. First-order system b. pole plot

9. First-order System Time constant, Rise time, Tr Settling time, Ts
Time to decay to 37 % of its initial value
or to rise to 63 % of its final value.
: the exponential frequency
Rise time, Tr
Time for the waveform to go from
0.1 to 0.9 of its final value
Settling time, Ts
Time for the response to reach, and stay within, 2 % of its final value
Figure 4.4
First-order system response to a unit step

10. First-Order System First-order transfer function via testing
No overshoot
Nonzero initial slope
Figure 4.5
Laboratory results of a system step response test

11. Second-order System: Summary
Figure 4.6 Second order system, pole plots, and step responses

12. Second-order System: Overdamped
Over damped response
Two system real poles
An exponential natural response
whose exponential frequency is equal to the pole location

13. Second-order System: Underdamped
Under damped response
Two system complex poles
Real part: exponential decay of the sinusoid's amplitude
Imaginary part: the frequency of the sinusoidal oscillation

14. Second-order System: Envelope
Under damped response
Second-order step response components generated by complex poles
Sinusoidal Freq.: damped frequency of oscillation,
Figure 4.7 Second-order step response components generated by complex poles

15. Second-order System: Example 4.2
By inspection, write the form of the step response of the system in Figure 4.8
Factoring the denominator of the transfer function in Figure 4.8,
We find the poles to be
Real part, -5, is the exponential frequency for the damping.
Imaginary part, 13.23, is the radian frequency for the sinusoidal oscillations
is a constant plus an exponentially damped sinusoid.
Figure 4.8 System for Example 4.2

16. Second-order System: Undamped
Un-damped response
Two system imaginary poles
A sinusoidal natural response whose frequency is equal to the location of the imaginary poles.
Not decay

17. Second-order S.: Critically damped
Critically damped response
Two system multiple real poles
An exponential and an exponential multiplied by time.

18. Second-order System: Summary
Figure 4.9 Step response for second-order system damping cases
Over damped responses Poles: Natural response:
Under damped responses Poles: Natural response:
Un-damped responses Poles: Natural response:
Critically damped responses Poles: Natural response:

19. The general Second-order System
Natural frequency,
Frequency of oscillation of the system without damping.
Damping ratio,
General second-order system
Without damping,

20. The general Second-order System
By definition, the natural frequency, is the oscillation of this system.
Poles of this system are on the axis at
Assuming an under damped system,
the complex poles have a real part, , equal to –a/2
From which
Second-order transfer function finally looks like this:

21. The general Second-order System
Example 4.3
Given the transfer function of find and
General second-order transfer function
, from which
Also,
Substituting the value of

22. The general Second-order System
Figure Second-order response as a function of damping ratio

23. The general Second-order System
Example 4.4
For each of the system shown in Figure 4.11, find the value of
Report the kind of response expected
Figure System for Example 4.4

24. The general Second-order System
Using the values of a and b from each of the system of Figure 4.11,
we find for system , which is thus over damped.
Since for system , which is thus critically damped
And for system , which is thus under-damped, since

25. Underdamped Second-order System
The transform of the response, , is the transform of the input times the transfer function
where it is assumed that (the under damped case)

26. Underdamped Second-order System
Taking the inverse Laplace transform
Figure Second-order under damped response for damping ratio values

27. Underdamped Second-order System
Figure Second-order under damped response specifications

28. Underdamped Second-order System
Second-order under damped response specifications
Rise time, : the time required for the waveform to go from 0.1 of the final value to 0.9 of the final value
Peak time, : the time required to reach the first or maximum, peak
Percent overshoot, : the amount that the waveform overshoots the steady-state, or final, value at the peak time, expressed as a percentage of the steady-stat value
Settling time, : the time required for the transient’s damped oscillations to reach and stay within of the steady-state value

29. Underdamped Second-order System
Evaluation of
Assuming zero initial conditions
Completing squares in the denominator
Therefore
Settling the derivative equal to zero yields

30. Underdamped Second-order System
Evaluation of
From Figure 4.13 the percent overshoot is given by
The term is found by evaluating at the peak time,
Figure 4.13,

31. Underdamped Second-order System
For the unit step,
Figure percent overshoot versus damping ratio

32. Underdamped Second-order System
Evaluation of
Stays within of the steady-state value,
Assuming that at the settling time
Damping ratio varies from 0 to 0.9 => the numerator varies from 3.91 to 4.74

33. Underdamped Second-order System
Evaluation of
Stays within of the steady-state value,
Figure Normalized rise time versus damping ratio for a second-order underdamped response

34. Underdamped Second-order System
Example 4.5
Given the transfer function
Find
is 0.475
is 2.838

35. Underdamped Second-order System
is second
Using the table
The normalized rise time is approximately 2.3 seconds
Dividing by yields second

36. Underdamped Second-order System
The pole plot for a general, under damped second-order system
The radial distance from the origin to the pole is the natural frequency,
and the
is the imaginary part of the part and is called
the damped frequency of oscillation
is the magnitude of the real part of the pole and is the exponential
damping frequency
Figure 4.16
Pole plot for an under damped second-order system

37. Underdamped Second-order System
Note :
Figure 4.17
Lines of constant peak time, , settling time, , and percent overshoot,

38. Underdamped Second-order System
Step responses of second-order under damped system as poles move
Figure 4.18 With constant real part (constant )
Figure 4.19 With constant imaginary part (constant )

39. Underdamped Second-order System
Step responses of second-order under damped system as pole move
Figure 4.20 With constant real part (constant )

40. Underdamped Second-order System
Example 4.6
Given the pole plot shown in Figure 4.21, find
The damping ratio is given by
The natural frequency, , is the radial
distance from the origin to the pole, or
The peak time is
The percent overshoot is
The approximate settling time is
Figure 4.21 Pole plot for Example 4.6

41. Underdamped Second-order System
Example 4.7
Given the system shown in Figure 4.22, find J and D to yield 20%
overshoot and a settling time of 2 seconds for a step input of torque
The transfer function for the system is
From the transfer function,
Figure 4.22 Rotational mechanical system for Example 4.7

42. Underdamped Second-order System
From the problem statement,
Or Hence,
, 20% overshoot implies
From the problem statement, K=5 N-m/rad, D=1.04 N-m/rad, J=0.26kg-m^2

43. System Response with Additional Poles
Dominant pole
Under certain conditions, a system with more than two pole or with zeros can be approximated as a second-order system that has just two complex dominant pole
Assuming that the complex poles are at
the real pole is at
In the time domain,

44. System Response with Additional Poles
Case , and is not much larger than
Case , and is much larger than

45. System Response with Additional Poles
Case ,
Figure 4.23 Component responses of a three-pole system : component responses

46. System Response with Additional Poles
Assume a step response, , of a three-pole system
where we assume that the non-dominant pole is located at -c on the real axis
The steady-state response approaches unity
As the non-dominant pole approaches , or ,
Thus, for this example, D, the residue of the non-dominant pole and its response,
becomes zero as the non-dominant pole approaches infinity.

47. System Response with Additional Poles
Example 4.8
Find the step response of each of the transfer functions shown in
through and compare them.
The step response, , for the transfer function, , can be found
by multiplying the transfer function by 1/s, a step input.
Figure 4.24 Step responses of system

48. System Response with Zeros
Effect of adding a zero to a two-pole system
The closer the zero is to the dominant poles, the greater its effect
on the transient response
As the zero moves away from the dominant poles, the response approaches
that of the two-pole system
Figure 4.25 Effect of adding a zero to a two-pole system

49. System Response with Zeros
The residue of each pole will be affected the same by the zero
The relative amplitudes remain appreciably the same
If the zero is far from the poles, then a is large compared to b and c

50. System Response with Zeros
The zero does not change the relative amplitudes of the components of the response
If we add a zero to the transfer function, yielding
Notice that the response begins to turn toward the negative direction even though the final value is positive
 non-minimum-phase system
Figure 4.26 Step response of a non-minimum-phase system

51. System Response with Zeros
Example 4.9
a. Find the transfer function, , for the operational amplifier circuit shown in Figure 4.27
The current, , through , is the same
Figure 4.27 Non-minimum-phase electric circuit

52. System Response with Zeros
But
Substituting into ,
Using voltage division,
Substituting and into
Let A approach infinity

53. System Response with Zeros
b. Let , and find the step response of the filter
Letting
Expanding into partial fractions,

54. System Response with Zeros
Or the response with a zero is
Also, from
Or the response without a zero is
Figure 4.28 Step response of the non-minimum-phase network of Figure and normalized step response of an equivalent network without the zero

55. System Response with Zeros
Example 4.10
, determine whether there is cancellation between the zero and the pole closest to the zero
And for any function for which pole-zero cancellation is valid, find the approximate response
The partial-fraction expansion of is
The residue of the pole at -3.5, which is closest to the zero at -4, is equal to 1 and is not negligible compared to the other residues
Thus, a second-order step response approximation cannot be made for

56. System Response with Zeros
The partial-fraction expansion for is
The residue of the pole at -4.01, which is closest to the zero at -4, is equal to 0.033, about two order of magnitude below any of the other residues
The response is approximately

57. Effects of Nonlinearities upon Time Response
Chapter 2 and look at the load angular velocity, ,
where
Figure 4.29 a. Effect of amplifier saturation on load angular velocity response; b. Simulink block diagram

58. Effects of Nonlinearities upon Time Response
Figure 4.30 a. Effect of deadzone on load angular displacement; b. Simulink block diagram

59. Effects of Nonlinearities upon Time Response
Figure 4.31 a. Effect of backlash on load angular displacement response; b. Simulink block diagram

60. Laplace Transform Solution of State Equations
Consider the state equation and the output equation
Taking the Laplace transform of both sides of the state equation yields
Combining all of the terms
The final solution for is
Taking the Laplace transform of the output equation yields

61. Laplace Transform Solution of State Equations
Eigen values and transfer function poles
If a system is represented in state-space
We can find the poles from

62. Laplace Transform Solution of State Equations
Example 4.11
Given the system represented in state space by
Do following:
a. Solve the preceding state equation and obtain the output for the given exponential input
b. Find the eigen values and the system poles

63. Laplace Transform Solution of State Equations
a. We will solve the problem by finding the component parts of ,
followed by substitution into

64. Laplace Transform Solution of State Equations
Since (the Laplace transform for ) is can be calculated
And using and from , respectively, we get

65. Laplace Transform Solution of State Equations
The output equation is found from or
where the pole at -1 canceled a zero at -1. Taking the inverse Laplace transform,
b. The denominator of , which is , is also the denominator of the system’s transfer function
Thus, furnishes both the poles of the system and
the eigenvalues -2,-3, and -4

66. Time Domain Solution of State Equations
The solution in the time domain is given directly by
: the state-transition matrix
: zero input response, : zero state response
Thus, for the unforced system
is the Laplace transform of the state-transition matrix

67. Time Domain Solution of State Equations
Example 4.12
For the state equation and initial state vector shown in ,
where is a unit step, find the state-transition matrix and then solve for
Since the state equation is in the from
Find the eigenvalues using Hence,
from which
Since each term of the state-transition matrix is the sum of response generated by the poles

68. Time Domain Solution of State Equations
Since each term of the state-transition matrix is the sum of response generated by the poles
And since
then

69. Time Domain Solution of State Equations
The constants are solved by taking two simultaneous equations four time
Also,
Hence, the first term

70. Time Domain Solution of State Equations
The last term
The final result is found

71. Time Domain Solution of State Equations
Example 4.13
Find the state-transition matrix of Example 4.12, using
We use the fact that is the inverse Laplace transform of
Then, first find as
From which
Finally, taking the inverse Laplace transform of each term, we obtain

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Homework: divide by 9
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