1. Youngjune, Han young@ssu.ac.kr
Chapter 4 Time Response
Youngjune, Han

2. Chapter Objectives
How to find the time response from the transfer function
How to use poles and zeros to determine the response of a control system
How to describe quantitatively the transient response of first- and second- order system
How to approximate higher-order systems as first or second order
How to view the effects of nonlinearities on the system time response
How to find the time response from the state-space representation

3. Introduction
In chapter 2, how transfer function can represent linear, time-invariant system
In chapter 3, systems were represented directly in the time domain via the state and output equation
Chapter 4 is devoted to the analysis of system transient response

4. Poles, zeros, and System response
Output response of a system = the forced response + natural response
Solving a differential equation
Taking the inverse Laplace transform
Laborious and time-consuming
Using the attribute qualitative to describe the method
Using poles and zeros, and their relationship to the time response of a system

5. Poles, zeros, and System response
Poles of a Transfer Function
Values of the Laplace transform variable, s, that cause the transfer function to become infinite
Any roots of the denominator of the transfer function that are common to roots of the numerator
Zeros of a Transfer Function
Values of the Laplace transform variable, s, that cause the transfer function to become zero
Any roots of the numerator of the transfer function that are common to roots of denominator

6. Poles, zeros, and System response
Pole and zeros of a First-Order System: An Example
Poles and Zeros are plotted on the complex s-place
Using an X for the poles
Using a O for the zeros
Finding the unit step response of the system

7. Poles, zeros, and System response
A pole of the input function  generating the form of the forced response
A pole of the transfer function  generating the form of the natural response
A pole on the real axis  generating an exponential response of the form e-at (where –a is the pole location on the real axis)
The zeros and poles  generating the amplitudes for both the forced and natural response

8. First-Order System First-order system without zeros
 Laplace transform of step response
Pole at –a generates response Ke-at

9. First-Order System Significance of parameter a, exponential frequency
When t=1/a, e-at|t=1/a = e-1 =0.37
c(t)|t=1/a =1-e-at|t=1/a = = 0.63

10. First-Order System Three transient response performance specifications
Time Constant, 1/a
the time for e-at to decay to 37% of its initial vaue
the time it takes for the step response to rise to 63% of the system response
Rise Time, Tr
the time for the waveform to go from 0.1 to 0.9 of its final value
Settling Time, Ts
the Time for the response to reach, and stay with, 2% of its final value

11. First-Order System
First-order Transfer Functions via Testing

12. Second-Order System
Introduction

13. Second-Order System Overdamped Response, 4.7(a) 
A pole at the original that comes from the unit step input: generating the constant forced response
Two system poles on real axis: generating an exponential natural response whose exponential frequency is equal to the pole location 

14. Second-Order System Underdamped Response4.7(c) 
A pole at the original that comes from the unit step input: generating the constant forced response
Two complex poles: generating an exponential natural response
An exponentially decaying amplitude, time constant: the reciprocal of the real part of the system pole
A sinusoidal waveform, frequency: the imaginary part of the system pole 

15. Second-Order System
Underdamped Response, 4.7(c)

16. Second-Order System Undamped Response4.7(c) 
A pole at the original that comes from the unit step input: generating the constant forced response
Two complex poles at imaginary axis: generating an exponential natural response
zero decaying amplitude
a sinusoidal waveform, frequency: the imaginary part of the system pole 

17. Second-Order System Critically damped Response4.7(c) 
A pole at the original that comes from the unit step input: generating the constant forced response
Two multiple real poles at real axis: generating an exponential natural response
An exponential and an exponential multiplied by time
Exponential frequency is equal to the location of the real poles 

18. The General Second-Order System
Two physically meaningful specification: natural frequency and damping ratio
Natural Frequency, ωn
The frequency of oscillation of the system without damping
For exampling, RLC circuit with the resistance shorted
Damping ratio,ζ
The ratio of the exponential decay frequency of the envelope to the natural frequency

19. The General Second-Order System
Consider the general system,
Without damping,
 the natural frequency 
Assuming an underdamped system,
General second-order transfer function
A real part, σ, equal to –a/2

20. The General Second-Order System
Second-order response as a function of damping ratio

21. Underdamped Second-Order System
A detailed description of the underdamped response is necessary for both analysis and design.
Finding the step response for the general second-order system
Expanding by partial fraction

22. Underdamped Second-Order System
Taking the inverse Laplace transform

23. Underdamped Second-Order System
Other parameters associated with the underdamped response:
Peak time : the time required to reach the first, or maximum, peak
Percent overshoot (%OS) : The amount that the waveform overshoots the steady-state, or final, value at peak time, expressed as a percentage of the steady-state value
Settling time : the time required for the transient’s damped oscillations to reach and stay within ±2% of the steady-state value
Rise time : the time required for the waveform to go 0.1 of the final value to 0.9 of the final value

24. Underdamped Second-Order System
Other parameters associated with the underdamped response

25. Underdamped Second-Order System
Evaluation of Tp
Assuming zero initial conditions
Completing squares in the denominator
Setting the derivative equal to zero,
 Peak Time(Tp):

26. Underdamped Second-Order System
Evaluation of %OS
Given by
Cmax is found by evaluating c(t) at the peak time,
 Percent overshoot, %OS for the unit step( cfinal = 1 ):

27. Underdamped Second-Order System
Evaluation of Ts
 Settling Time (Ts):
 an approximation of Settling Time(Ts):

28. Underdamped Second-Order System
Evaluation of Tr

29. Underdamped Second-Order System
Relating peak time, percent overshoot, and settling time to the location of the poles
 the radial distance from the original to the pole
= natural frequency( ) 

30. Underdamped Second-Order System
Tp: Inversely proportional to the imaginary part of the pole
 Horizontal lines: lines of constant peak time
Ts: Inversely proportional to the real part of the pole
 Vertial lines: lines of constant settling time
Since , radial line = lines of constant
Since percent overshoot is only a function of , radial line = lines of constant percent overshoot

31. Underdamped Second-Order System
Lines of constant peak time, settling time, percent overshoot

32. Underdamped Second-Order System
Lines of constant peak time, settling time, percent overshoot

33. Example Transient Response through component design
The transfer function
From the transfer function,
From the problem statement, 

34. Example Transient Response through component design
A 20% overshoot 
From the problem statement, K= 5N-m/rad
 D=1.04 N-m-s/rad, and J=0.26kg-m2

35. System Response with additional Poles
A system with more than two poles or zeros
Can be approximated as a second-order system that has just two complex dominant poles
Consider a three-pole system with complex poles and a third poles on the real axis
Complex poles:
Real pole:
The output transform:
 In time domain,

36. System Response with additional Poles
Figure 4.23 Component responses of a three-pole system: a. pole plot; b. component responses: non-dominant pole is near dominant second-order pair (Case I), far from the pair (Case II), and at infinity (Case III)

37. System Response with additional Poles
Case 1: αr = αr1 and is not much larger than ζωn
The real pole’s transient response will not decay to insignificance at the peak time or settling time generated by the second-order pair.
 Cannot be represented as a second-order system
Case 2, 3: αr ≫ζωn
The pure exponential will die out much rapidly than the second-order underdamped step response  parameters as percent overshoot, settling time, and peak time will be generated by the second-order underdamped step response component
Can be represented as a pure second-order system
If the real pole is five times farther to the left than the dominant poles, the system can be represented by its dominant second-order pair of poles

38. System Response with zeros
Considering the general effect of a zero
Adding a zero to the transfer function
The response: the derivative of the original response and a scaled version of the original response
If a , the negative of the zero, is very large, the response is approximately aC(s).
As a become smaller, the derivative term contributes more to the response and has a greater effect.
If a is negative, placing the zero in the right half-plane, the response may begin to turn toward the negative direction  nonminimum-phase system

39. System Response with zeros
A two-pole system with a poles at (-1+j2.828)
Figure 4.25 Effect of adding a zero to a two-pole system
Figure 4.26 Step response of a nonminimum-phase system

40. 4.10 Laplace Transform Solution of State Equation

41. 4.10 Time Domain Solution of State Equation
Assuming a homogeneous state equation of the form
Assuming a series solution
Substituting X(t) into homogeneous state eq.

42. 4.10 Time Domain Solution of State Equation
Equating like coefficients yields

43. 4.10 Time Domain Solution of State Equation
From previous eq.
X(0) = bo
Therefore,
where eAt is state-transition matrix
where Φ(t)= eAt

44. 4.10 Time Domain Solution of State Equation
Let us now solve the forced, or nonhomo-geneous, problem.
Rearrange and multiply both sides by eAt

45. 4.11 Time Domain Solution of State Equation
The solution in the time domain
The state-transition matrix

46. Case Studies: Antenna Control: Open-Loop Response
Figure 4.32 Antenna azimuth position control system for angular velocity:
a. forward path; b. equivalent forward path

47. Case Studies: Antenna Control: Open-Loop Response
(a) Predicting the nature of the unit step response
(b) the damping ratio and natural frequency
 and
(c) the angular velocity response to a step input  

48. Case Studies: Antenna Control: Open-Loop Response
(d) converting the transfer function to state space
The transfer function
Taking the inverse Laplace transform
Defining the phase variable as
State equation and output equation