2. The Frequency-Response Design Method
Feedback Control Systems
Chapter 6
The Frequency-Response Design Method

3. Chapter 6
The Frequency-Response Design Method
Frequency Response
A linear system’s response to sinusoidal inputs is called the system’s frequency response.
Frequency response can be obtained from knowledge of the system’s pole and zero locations.
Consider a system described by
where
With zero initial conditions, the output is given by

4. Steady-state response
Chapter 6
The Frequency-Response Design Method
Frequency Response
Assuming all the poles of G(s) are distinct, a partial fraction expansion of the previous equation will result in
Transient response
Steady-state response
where

5. Frequency Response Examine But the input is Therefore Magnitude Phase
Chapter 6
The Frequency-Response Design Method
Frequency Response
Examine
Magnitude
Phase
But the input is
Therefore
Meaning?

6. Chapter 6
The Frequency-Response Design Method
Frequency Response
A stable linear time-invariant system with transfer function G(s), excited by a sinusoid with unit amplitude and frequency ω0, will, after the response has reached steady-state, exhibit a sinusoidal output with a magnitude M(ω0) and a phase Φ(ω0) at the frequency ω0.
The magnitude M is given by |G(jω)| and the phase Φ is given by G(jω), which are the magnitude and the angle of the complex quantity G(s) evaluated with s taking the values along the imaginary axis (s = jω).
The frequency response of a system consists of the frequency functions |G(jω)| and G(jω), which describe how a system will respond to a sinusoidal input of any frequency.

7. Chapter 6
The Frequency-Response Design Method
Frequency Response
Frequency response analysis is interesting not only because it will help us to understand how a system responds to a sinusoidal input, but also because evaluating G(s) with s taking on values along jω axis will prove to be very useful in determining the stability of a closed-loop system.
As we know, jω axis is the boundary between stability and instability. Therefore, evaluating G(jω) along the frequency band will provide information that allows us to determine closed-loop stability from the open-loop G(s).

8. Frequency Response Φ M = D
Chapter 6
The Frequency-Response Design Method
Frequency Response
Given the transfer function of a lead compensation
(a) Analytically determine its frequency response characteristics and discuss what you would expect from the result. Φ
M = D
At low frequency, ω→0, M→K, Φ→0.
At high frequency, ω→∞, M→K/α, Φ→0.
At intermediate frequency, Φ > 0.

9. Chapter 6
The Frequency-Response Design Method
Frequency Response
(b) Use MATLAB to plot D(jω) with K = 1, T = 1, and α = 0.1 for 0.1 ≤ ω ≤ 100 and verify the prediction from (a).
Using bode(num,den), in this case bode([1 1],[0.1 1]), MATLAB produces the frequency response of the lead compensation.

10. Chapter 6
The Frequency-Response Design Method
Frequency Response
For second order system having the transfer function
we already plotted the step response for various values of ζ.
The damping and rise time of a system can be determined from the transient-response curve.

11. Chapter 6
The Frequency-Response Design Method
Frequency Response
The corresponding frequency response of the system can be found by replacing s = jω
This G(jω) can be plotted along the frequency axis, for various values of ζ.

12. Chapter 6
The Frequency-Response Design Method
Frequency Response
The damping of a system can be determined from the peak in the magnitude of the frequency response curve.
The rise time can be estimated from the bandwidth, which is approximately equal to ωn. ?
 The transient-response curve and the frequency-response curve contain the same information.

13. Chapter 6
The Frequency-Response Design Method
Frequency Response
Bandwidth is defined as the maximum frequency at which the output of system will track an input sinusoid in a satisfactory manner.
By convention, the bandwidth is the frequency at which the output is attenuated to a factor of times the input.
The maximum value of the frequency-response magnitude is referred to as the resonant peak Mr.

14. Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
Advantages of working with frequency response in terms of Bode plots:
Dynamic compensator design can be based entirely on Bode plots.
Bode plots can be determined experimentally.
Bode plots of systems in series can be simply added, which is quite convenient.
The use of a logarithmic scale permits a much wider range of frequencies to be displayed on a single plot compared with the use of linear scales.
Bode plot of a system is made of two curves,
The logarithm of magnitude vs. the logarithm of frequency, log M vs. log ω, or also Mdb vs. log ω.
The phase versus the logarithm of frequency Φ vs. log ω.

15. Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
For root locus design method, the open-loop transfer function is written in the form
For frequency-response design method, s is replaced with jω to write the transfer function in the Bode form

16. Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
To draw the Bode plot of a transfer function, it must be rewritten in magnitude equation and phase equation, i.e.,
Then
Phase Equation
Magnitude Equation
Magnitude Equation
(log)
Magnitude Equation
(db)

17. Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
Examining all the transfer functions we have dealt with so far, all of them are the combinations of the following four terms:
Gain
Pole or zero at the origin
Simple pole or zero
Quadratic poles or zeros
Once we understand how to plot each term, it will be easy to draw the composite plot, since log M and Φ are the additive combination of the magnitude logarithms and the phases of all terms.

18. Bode Plot Techniques Gain Magnitude Phase Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
Gain
Magnitude
Phase

19. Bode Plot Techniques Pole or zero at the origin Magnitude Phase
Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
Pole or zero at the origin
Magnitude
Phase

20. Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
As example, Bode plot of a zero (jω) at origin will be as follows:

21. Bode Plot Techniques Approximation Magnitude Phase Simple pole or zero
Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
Simple pole or zero
Approximation
Magnitude
Phase
The point where ωτ = 1 or ω = 1/τ is called the break point.

22. Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
As example, Bode magnitude plot of a simple zero (jωτ+1) is given below, with τ = 10.
The break point lies at ω = 1/τ = 0.1.
Correction of Asymptote

23. Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
The corresponding Bode phase plot of a simple zero (jωτ+1) is given as:
Corrections of Asymptotes by 11°, at ω = 0.02 and ω = 0.5.
Corresponds to 1/5ωbreak and 5ωbreak

24. Bode Plot Techniques Quadratic poles or zeros
Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
Quadratic poles or zeros
Asymptotes can be used for rough sketch.
Afterwards, correction must be made according to the value of damping factor ζ.

25. Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques

26. Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques 5

27. Chapter 6
The Frequency-Response Design Method
Bode Plot: Example
Plot the Bode magnitude and phase for the system with the transfer function (rough composite acceptable)
ωb1 = 0.5, ωb2 = 10, and ωb3 = 50
Convert the function to the Bode form,
One pole at the origin
5 terms will be drawn separately and finally composited

28. Bode Plot Techniques : Rough composite ωb1 = 0.5 ωb2 = 10 ωb3 = 50
Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
ωb1 = 0.5
ωb2 = 10
ωb3 = 50 60
–20 db/dec 40
0 db/dec 20
–20 db/dec db
–40 db/dec
–20
–40
: Rough composite

29. Bode Plot Techniques : Rough composite : 3-db-corrected composite
Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
ωb1 = 0.5
ωb2 = 10
ωb3 = 50 60 40
+3 db
–3 db 20 db
–3 db
–20
–40
Final Result
: Rough composite
: 3-db-corrected composite

30. Bode Plot Techniques : Rough composite ωb1 = 0.5 ωb2 = 10 ωb3 = 50 2
Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
ωb1 = 0.5
ωb2 = 10
ωb3 = 50 2
2.5
–11°
–11°
+11° 10
+11°
0.1
–11° 50
+11°
250
: Rough composite

31. Bode Plot Techniques : Rough composite : 11°-corrected composite
Chapter 6
The Frequency-Response Design Method
Bode Plot Techniques
ωb1 = 0.5
ωb2 = 10
ωb3 = 50
: Rough composite
Final Result
: 11°-corrected composite

32. Chapter 6
The Frequency-Response Design Method
Homework 8 (1/2)
No.1. Draw (manually) the Bode plot for the following open-loop transfer functions in logarithmic and semi-logarithmic scale accordingly. Draw only the rough composite.
No.2. Next slide.

33. Chapter 6
The Frequency-Response Design Method
Homework 8 (2/2)
No.2.
Derive the transfer function of the electrical system given above.
If R1 = 10 kΩ , R2 = 5 kΩ and C = 0.1 μF, draw (manually) the Bode plot of the system. Draw only the rough composite
Deadline: Tuesday, 5 December 2017.