1. Chapter 5 Frequency-Domain Analysis
NUAA-Control System Engineering
Chapter 5 Frequency-Domain Analysis

2. Content in Chapter 5
5-1 Frequency Response (or Frequency Characteristics)
5-2 Nyquist plot and Nyquist stability criterion
5-3 Bode plot and Bode stability criterion

3. 5-1 Frequency Response

4. A Perspective on the Frequency-Response Design Method
The design of feedback control systems in industry is probably accomplished using frequency-response methods more than any other.
Advantages of frequency-response design:
-It provides good designs in the face of uncertainty in the plant model
-Experimental information can be used for design purposes.
For systems with poorly known or changing high-frequency resonances, we can temper their feedback compensation to alleviate the effects
Raw measurements of the output amplitude and phase of a plant undergoing a sinusoidal input excitation are sufficient to design a suitable feedback control.
-No intermediate processing of the data (such as finding poles and zeros) is required to arrive at the system model. 4

5. Frequency response
The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal.
G(s)
H(s)
For a LTI system, when the input to it is a sinusoid signal, the resulting output , as well as signals throughout the system, is sinusoidal in the steady-state;
The output differs from the input waveform only in amplitude and phase.

6. The closed-loop transfer function of the LTI system:
For frequency-domain analysis, we replace s by jω:
The frequency-domain transfer function M(jω) may be expressed in terms of its magnitude and phase:
magnitude
phase

7. The magnitude of M(jω) is
Gain characteristic
The phase of M(jω) is
Phase characteristic
Gain-phase characteristics of an ideal low-pass filter

8. Example. Frequency response of a Capacitor
Consider the capacitor described by the equation
where v is the input and i is the output. Determine the sinusoidal steady-state response of the capacitor.
Solution.
The transfer function of the capacitor is So
Computing the magnitude and phase, we find that

9. Phase characteristic:
Gain characteristic:
Phase characteristic:
Output:
For a unit-amplitude sinusoidal input v, the output i will be a sinusoid with magnitude Cω, and the phase of the output will lead the input by 90°.
Note that for this example the magnitude is proportional to the input frequency while the phase is independent of frequency.

10. Frequency-Domain Specifications
Cutoff
rate
Resonant peak
Resonant frequency
Bandwidth
Typical gain-phase characteristic of a control system

11. Frequency response of a prototype second-order system
Closed-loop transfer function:
Its frequency-domain transfer function:
Define

12. The magnitude of M(ju) is
The phase of M(ju) is
Resonant peak
The resonant frequency of M(ju) is
With , we have
Since frequency is a real quantity, it requires So

13. According to the definition of Bandwidth
With , we have

14. For a prototype second-order system ( )
Resonant peak
Resonant frequency
Bandwidth

15. Correlation between pole locations, unit-step response and the magnitude of the frequency response

16. Example. The specifications on a second-order unity-feedback control system with the closed-loop transfer function
are that the maximum overshoot must not exceed 10 percent, and the rise time be less than 0.1 sec. Find the corresponding limiting values of Mr and BW analytically.
Solution.
Maximum overshoot:
Rise time:

17. Based on time-domain analysis, we obtain and
Frequency-domain specifications:
Resonant peak
Bandwidth

18. Effects of adding a zero to the OL TF
Open-loop TF：
Closed-loop TF：
Adding a zero at
Open-loop TF：
The additional zero changes both numerator and denominator.
Closed-loop TF：

19. As analyzing the prototype second-order system, using similar but more complicate calculation, we obtain
Bandwidth
where
For fixed ωn and ζ, we analyze the effect of

20. The general effect of adding a zero the open-loop transfer function is to increase the bandwidth of the closed-loop system.

21. Effects of adding a pole to the OL TF
Open-loop TF：
Closed-loop TF：
Adding a pole at
Open-loop TF：
Closed-loop TF：

22. The effect of adding a pole the open-loop transfer function is to make the closed-loop system less stable, while decreasing the bandwidth.

23. 5-2 Nyquist Plot and Nyquist Criterion

24. Nyquist Criterion What is Nyquist criterion used for? G(s) H(s)
Nyquist criterion is a semigraphical method that determines the stability of a closed-loop system;
Nyquist criterion allows us to determine the stability of a closed-loop system from the frequency-response of the loop function G(jw)H(j(w)

25. Review about stability
Closed-loop TF:
Characteristic equation (CE):
Stability conditions:
Open-loop stability: poles of the loop TF G(s)H(s) are all in the left-half s-plane.
Closed-loop stability: poles of the closed-loop TF or roots of the CE are all in the left-half s-plane.

26. Definition of Encircled and Enclosed
Encircled: A point or region in a complex function plane is said to be encircled by a closed path if it is found inside the path.
Enclosed: A point or region in a complex function plane is said to be encircled by a closed path if it is encircled in the countclockwise(CCW) direction.
Point A is encircled in the closed path;
Point A is also enclosed in the closed path;

27. Number of Encirclements and Enclosures
Point A is encircled once;
Point B is encircled twice.
Point C is enclosed once;
Point D is enclosed twice.

28. Mapping from the complex s-plane to the Δ(s) -plane
Exercise 1: Consider a function Δ(s) =s-1, please map a circle with a radius 1 centered at 1 from s-plane to the Δ(s)-plane .
s-plane
Δ( s)-plane
Mapping

29. Principle the Argument
Let be a single-valued function that has a finite number of poles in the s-plane.
Suppose that an arbitrary closed path is chosen in the s-plane so that the path does not go through any one of the poles or zeros of ;
The corresponding locus mapped in the plane will encircle the origin as many times as the difference between the number of zeros and poles (P) of that are encircled by the s-plane locus .
In equation form:
N - number of encirclements of the origin by the plane locus
Z - number of zeros of encircled by the s-plane locus
P - number of poles of encircled by the s-plane locus

30. Nyquist Path
A curve composed of the imaginary axis and an arc of infinite radius such that the curve completely encloses the right half of the s-plane .
s-plane
Nyquist path is in the CCW direction
Since in mathematics, CCW is traditionally defined to be the positive sense.
Note Nyquist path does not pass through any poles or zeros of Δ(s); if Δ(s) has any pole or zero in the right-half plane, it will be encircled by

31. Nyquist Criterion and Nyquist Diagram
Δ( s)-plane
s-plane
Nyquist Path
G( s)H(s)-plane
Nyquist Diagram:
Plot the loop function to determine the closed-loop stability
Critical point:
(-1+j0)

32. Nyquist Criterion and G(s)H(s) Plot
s-plane
G( s)H(s)-plane
Nyquist Path
G(s)H(s) Plot
The Nyquist Path is shown in the left figure. This path is mapped through the loop tranfer function G(s)H(S) to the G(s)H(s) plot in the right figure. The Nyquist Creterion follows:

33. Nyquist Criterion and Nyquist Plot
s-plane
G( s)H(s)-plane
Nyquist Path
Nyquist Plot
N - number of encirclements of (-1,j0) by the G(s)H(s) plot
Z - number of zeros of that are inside the right-half plane
P - number of poles of that are inside the right-half plane
The condition of closed-loop stability according to the Nyquist Creterion is:

34. has the same poles as , so P can be obtained by counting the number of poles of in the right-half plane.

35. An example Consider the system with the loop function
Matlab program for Nyquist plot
(G(s)H(s) plot)
>>num=5;
>>den=[ ];
>>nyquist(num,den);
Question 2: what if
Question 1: is the closed-loop system stable?
N=0, P=0,
N=-P, stable

36. 1. With root locus technique:
>>num=1;
>>den=[ ];
>>rlocus(num,den);
For K* varies from 0 to ∞, we draw the RL
When K*=8 (K=1.6), the RL cross the jw-axis, the closed-loop system is marginally stable.
For K*>8 (K>1.6), the closed-loop system has two roots in the RHP and is unstable.

37. >>K=1;
>>num=5*K;
>>den=[ ];
>>nyquist(num,den);
2. With Nyquist plot and Nyquist criterion:
K=1
No pole of G(s)H(s) in RHP, so P=0;
Nyquist plot does not encircle (-1,j0), so N=0
Thus N=-P
The closed-loop system is stable

38. >>K=1.6;
>>num=5*K;
>>den=[ ];
>>nyquist(num,den);
2. With Nyquist plot and Nyquist criterion:
K=1.6
No pole of G(s)H(s) in RHP, so P=0;
The Nyquist plot just go through
(-1,j0)
The closed-loop system is marginally stable

39. >>K=4;
>>num=5*K;
>>den=[ ];
>>nyquist(num,den);
2. With Nyquist plot and Nyquist criterion:
K=4
No pole of G(s)H(s) in RHP, so P=0;
Nyquist plot encircles (-1,j0) twice, so N=2
Thus Z=N+P=2
The closed-loop system has two poles in RHP and is unstable

40. Nyquist Criterion for Systems with Minimum-Phase Transfer Functions
What is called a minimum-phase transfer function?
A minimum-phase transfer function does not have poles or zeros in the right-half s-plane or on the jw-axis, except at s=0.
Consider the transfer functions
Both transfer functions have the same magnitude for all frequencies
But the phases of the two transfer functions are drastically different.

41. A minimum-phase system (all zeros in the LHP) with a given magnitude curve will produce the smallest change in the associated phase, as shown in G1.

42. Consider the loop transfer function:
If L(s) is minimum-phase, that is, L(s) does not have any poles or zeros in the right-half plane or on the jw-axis, except at s=0
Then P=0, where P is the number of poles of Δ(s)=1+G(s)H(s), which has the same poles as L(s).
Thus, the Nyquist criterion (N=-P) for a system with L(s) being minimum-phase is simplified to

43. Nyquist criterion for systems with minimum-phase loop transfer function
For a closed-loop system with loop transfer function L(s) that is of minimum-phase type, the system is closed-loop stable , if the Nyquist plot (L(s) plot) that corresponds to the Nyquist path does not enclose (-1,j0) point. If the (-1,j0) is enclosed by the Nyquist plot, the system is unstable.
The Nyquist stability can be checked by plotting the segment of L(jw) from w= ∞ to 0.

44. Example
Consider a single-loop feedback system with the loop transfer function
Analyze the stability of the closed-loop system.
Solution.
Since L(s) is minimum-phase, we can analyze the closed-loop stability by investigating whether the Nyquist plot enclose the critical point (-1,j0) for L(jw)/K first.
w=∞:
w=0+:

45. The frequency is positive, so
the Nyquist plot does not enclose (-1,jw);
stable
the Nyquist plot goes through (-1,jw);
marginally stable
unstable
the Nyquist plot encloses (-1,jw).

46. By root locus technique
>>z=[]
>>p=[0, -2, -10];
>>k=1
>>sys=zpk(z,p,k);
>>rlocus(sys);

47. Gain Margin and Phase Margin
Relative Stability
Gain Margin and Phase Margin
For a stable system, relative stability describes how stable the system is.
In time-domain, the relative stability is measured by maximum overshoot and damping ratio.
In frequency-domain, the relative stability is measured by resonance peak and how close the Nyquist plot of L(jw) is to the (-1,j0) point.
The relative stability of the blue curve is higher than the green curve.

48. (for minimum-phase loop transfer functions)
Gain Margin (GM)
(for minimum-phase loop transfer functions)
Phase crossover
L(jw)-plane
Phase crossover frequency ωp
For a closed-loop system with L(jw) as its loop transfer function, it gain margin is defined as

49. Gain margin represents the amount of gain in decibels (dB) that can be added to the loop before the closed-loop system becomes unstable.

50. (for minimum-phase loop transfer functions)
Phase Margin (PM)
(for minimum-phase loop transfer functions)
Gain margin alone is inadequate to indicate relative stability when system parameters other the loop gain are subject to variation.
With the same gain margin, system represented by plot A is more stable than plot B.
Gain crossover frequency ωg
Phase margin:

51. Example Consider the transfer function
Draw its Nyquist plot when w varies from 0 to ∞.
Solution.
Substituting s=jw into G(s) yields:
The magnitude and phase of G(jw) at w=0 and w=∞ are computed as follows.
Thus the properties of the Nyquist plot of G(jw) at w=0 and w=∞ are ascertained.
Next we determine the intersection…

52. If the Nyquist plot of G(jw) intersects with the real axis, we have
This means that the G(jw) plot intersects only with the real axis of the G(jw)-plane at the origin.
Similarly, intersection of G(jw) with the imaginary axis:
which corresponds to the origin of the G(jw)-plane.
The conclusion is that the Nyquist plot of G(jw) does not intersect any one of the axes at any finite nonzero frequency.
At w=∞,
At w=0, 52

53. Example Consider a system with a loop transfer function as
Determine its gain margin and phase margin.
Solution.
Phase crossover frequency ωp:
Gain margin:
Gain crossover frequency ωg:
Phase margin:

54. Advantages of Nyquist plot:
-By Nyquist plot of the loop transfer function, the closed-loop stability can be easily determined with reference to the critical point (-1,j0).
-It can analyze systems with either minimum phase or nonminimum phase loop transfer function.
Disadvantages of Nyquist plot:
-By Nyquist plot only, it is not convenient to carry out controller design.

55. 5-3 Bode Plot

56. Bode Plot
The Bode plot of the function G(jw) is composed of two plots:
-- the amplitude of G(jw) in decibels (dB) versus log10w or w
-- the phase of G(jw) in degrees as a function of log10w or w.
Without loss of generality, the following transfer function is used to illustrate the construction of the Bode Plot
where K, T1, T2, τ1, ζ, ωn are real constants. It is assumed that the second-order polynomial in the denominator has complex conjugate zeros.

57. Substituting s=jw into G(s) yields
The magnitude of G(jw) in dB is obtained by multiplying the logarithm (base 10) of |G(jw)| by 20; we have
The phase of G(jw) is

58. In general, the function G(jw) may be of higher order and
have many more factored terms. However, the above two equations indicate that additional terms in G(jw) would simply produce more similar terms in the magnitude and phase expressions, so the basic method of construction of the Bode plot would be the same.
In general, G(jw) can contain just four simple types of factors:
1. Constant factor: K
2. Poles or zeros at the origin of order p: (jw)±p
3. Poles or zeros at s =-1/T of order q: (1+jwT )±q
4. Complex poles and zeros of order r:
(1 + j2ζω/ωn-ω2/ω2n)

59. 1. Real constant K

60. 2. Poles or zeros at the origin,
For a given p, it is a straight line with the slope:
Thus a unit change in corresponds to a change of ±20 dB in the magnitude.
So these lines pass through the 0dB axis at ω =1.

62. 3. (a) Simple zero 1+jwT
Consider the function
where T is a positive real constant.
The magnitude of G(jw) in dB is
At very low frequencies,
The two lines intersect at:
( is neglected when compared with 1.)
At very high frequencies,
(corner frequency)
This represents a straight line with a slope of 20dB

63. The steps of making of sketch of
Step 1: Locate the corner frequency w=1/T on the frequency axis;
Step 2: Draw the 20dB/decade line and the horizontal line at 0 dB with the two lines intersecting at w=1/T.
Step 3: Sketch a smooth curve by locating the 3-dB point at the corner frequency and the 1-dB points at 1 octave above and below the corner frequency.

64. The phase of G(jw)=1+jwT is
At very low frequencies,
At very high frequencies,
Since the phase of G(jw) varies from 0°to 90°, we can draw a line from 0°at 1 decade below the corner frequency to 90°at 1 decade above the corner frequency.

65. 3. (b) Simple pole, 1/(1+jwT)
Consider the function
The magnitude of G(jw) in dB is
At very low frequencies,
The two lines intersect at:
At very high frequencies,
(corner frequency)
This represents a straight line with a slope of -20dB
The phase of G(jw):
For w varies from 0 to ∞, varies from 0°to -90°.

67. 4. Complex poles and zeros
Consider the second-order transfer function
We are interested only in the case when ζ ≤ 1, since otherwise G(s) would have two unequal real poles, and the Bode plot can be obtained by considering G(s) as the product of two transfer functions with simple poles.
By letting s=jw, G(s) becomes

68. The magnitude of G(jw) in dB is
At very low frequencies,
The two lines intersect at:
At very high frequencies,
(corner frequency)
This equation represents a straight line with a slope of 40 dB decade in the Bode plot coordinates.

69. The actual magnitude curve of G(jw) in this case may differ strikingly from the asymptotic curve.
The reason for this is that the amplitude and phase curves of the second-order G(jw) depend not only on the corner frequency wn, but also on the damping ratio ζ, which does not enter the asymptotic curve.

70. The phase of G(jw) is given by

71. Example Consider the following transfer function Sketch its Bode Plot.
Solution.
Letting s=jw, we have
Reformulating it into the form for Bode Plot
where
So G(jw) has corner frequencies at w=10,2 and 5 rad/sec.

72. 1. Bode plot of K=10

73. 2. Bode Plot of the component with pole at origin : jw
magnitude curve: a straight line with slope of 20 dB/decade, passing through the w=1 rad/sec point on the 0-dB axis.
The pole at s= 0 gives a magnitude curve that is a straight line with slope of 20 dB/decade, passing through the w=1 rad/sec point on the 0-dB axis.

74. 3. Bode plot of simple zero component 1+j0.1w
Corner frequency: w=1/0.1=10 rad/sec

75. 4. Bode plot of simple pole componet 1/(1+j0.5w)
Corner frequency: w=1/0.5=2 rad/sec

76. 5. Bode plot of simple pole component 1/(1+j0.2w)
Corner frequency: w=1/0.2=5 rad/sec

77. |G(jw)|dB is obtained by adding the component curves together, point by point.
Bode Plot:
Gain crossover point: |G(jw)|dB cross the 0-dB axis
Phase crossover point: where the phase curve cross the -180°axis.

78. Nyquist Plot (Polar Plot) :
The gain-crossover point is where ,
The phase crossover point is where