1. Hanani binti Abdul Wahab 24 September 2008
BDA3073
Hanani binti Abdul Wahab
24 September 2008

2. Review
Root locus provides a powerful tool for selecting control system parameters that yield desirable transient performance.
The root locus rules of construction show the effects of poles, zeros, and gain changes on the root loci (the closed loop poles).
We can design control systems from the root loci by selecting the best closed loop poles and finding the related parameter.

3. Try This!
Sketch a root locus for a control system with unity feedback

4. rlocus command

5. Sisotool

6. Command rlocfind Command Help rlocfind
Find root locus gains for a given set of roots.
[k,poles]=rlocfind(sys) is used for interactive gain selection from the root locus plot of the SISO system sys generated by rlocus. rlocfind puts up a crosshair cursor in the graphics window which is used to select a pole location on an existing root locus. The root locus gain associated with this point is returned in K and all the system poles for this gain are returned in poles.

7. Command rlocfind Command Help rlocfind (cont’d)
[k,poles]=rlocfind(sys,p) takes a vector p of desired root locations and computes a root locus gain for each of these locations (i.e., a gain for which one of the closed-loop roots is near the desired location). The j-th entry of the vector K gives the computed gain for the location p(j), and the j-th column of the matrix poles lists the resulting closed-loop poles.

8. Frequency Response Methods
Bode Plot

9. Outline What is frequency response Rules for Constructing Bode Plot
Frequency Response Plots Bode and Nyquist (Polar)
Rules for Constructing Bode Plot

10. What is frequency response
An important alternative approach to system analysis and design is the frequency response method.
The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal. We will investigate the steady-state response of the system to the sinusoidal input as the frequency varies.

11. Advantages of the frequency response method
The sinusoidal input signal for various ranges of frequency and amplitude is readily available.
It is the most reliable and uncomplicated method for the experimental analysis of a system.
Control of system bandwidth.
The TF describing the sinusoidal steady-state behaviour of the system is easily obtained by replacing s with jω in the system TF.

12. Frequency response plots
Polar plot
The TF G(s) can be described in the frequency domain by
The above equation is used for the polar plot representation of the frequency response in the polar plane.
polar plane
Alternatively, the TF G(jω) can be represented by

13. Frequency response plots (cont’d)
Example – polar plot
Consider a simple RC circuit. The TF of the system is
The sinusoidal steady-state TF is
The polar plot is obtained from
polar plot
To draw the polar plot, R(ω) and X(ω) at typical frequencies, e.g., ω=0, ∞, are to be determined.

14. Frequency response plots (cont’d)
Limitations of polar plots:
The addition of poles and zeros requires the recalculation of the frequency response.
The effect of individual poles and zeros is not indicated.
A more widely used graphical tool to plot frequency response is the Bode diagram.
Bode plot
The TF in the frequency domain can be written as
For a Bode diagram, we normally use
Magnitude versus ω and phase versus ω are plotted separately.

15. Bode diagram Advantages of Bode plots:
Multiplication of magnitudes can be converted into addition by virtue of the definition of logarithmic gain.
Straight-line asymptotes are simple to be used for sketching an approximate log-magnitude curve.
The use of a logarithmic scale for the frequency is a more judicious choice than a linear scale of frequency as this expands the low frequency range, which is more important in practical systems.
An interval of two frequencies with a ratio equal to 10 is called a decade.
The slope of the asymptotic line in the figure is -20dB/decade.

16. Rules for Constructing Bode Plot
To draw Bode Plot there are four steps:
1.Rewrite the transfer function in proper form.
2.Separate the transfer function into its constituent parts.
3.Draw the Bode diagram for each part.
4.Draw the overall Bode diagram by adding up the results from part 3.

17. 1.Rewrite the transfer function in proper form.
Make both the lowest order term in the numerator and denominator unity.
Example 1:

18. 2.Separate the transfer function into its constituent parts.
The next step is to split up the function into its constituent parts.  There are seven types of parts:
A constant
Poles at the origin
Zeros at the origin
Real Poles
Real Zeros
Complex conjugate poles
Complex conjugate zeros

19. 2.Separate the transfer function into its constituent parts.
Example 2:
This function has a constant of 2, 
a zero at s=-10, and  poles at s=-3 and s=-50.

20. 3.Draw the Bode diagram for each part.

21. 3.Draw the Bode diagram for each part.
Cont’d

22. Draw Bode Plot Procedure
0 db
Slope=
-20n db/dec
Magnitude
Phase
-90n° wb
0 db
Slope=
+20n db/dec
Magnitude
Phase
+90n°

23. Draw Bode Plot Procedure
Phase 0°
-90n°
0.1wb
10wb wb
0 db
Slope=
-20n db/dec
Magnitude
Slope=
+20n db/dec wb
0 db
Magnitude
Phase 0°
+90n°
0.1wb
10wb

24. Draw Bode Plot Procedure
Phase 0°
-180n°
0.1wb
10wb wb
0 db
Slope=
-40n db/dec
Magnitude
Slope=
+40n db/dec wb
0 db
Magnitude
Phase 0°
+180n°
0.1wb
10wb

25. Bode Plot Procedure The following example illustrates this procedure:
1.Rewrite the transfer function in proper form.
2.Separate the transfer function into its constituent parts.

26. Bode Plot Procedure
3.Draw the Bode diagram for each part.

27. Bode Plot Procedure
4.Draw the overall Bode diagram by adding up the results from part 3.

28. Check your AnsWer!
Step 2: Separate the transfer function into its constituent parts.
The transfer function has 2 components:
A constant of 3.3
A pole at s= -30

29. Bode Plot Procedure
Create the Phase Plot

30. Bode Plot Procedure
The Phase plot compared to the computer generated plot then becomes:

31. Draw the Bode Diagram for the transfer function:
Try This!
Draw the Bode Diagram for the transfer function:
                         
REMEMBER the four steps:
1.Rewrite the transfer function in proper form.
2.Separate the transfer function into its constituent parts.
3.Draw the Bode diagram for each part.
4.Draw the overall Bode diagram by adding up the results from part 3.

32. Check your AnsWer!
Step 1: Rewrite the transfer function in proper form.  
Make both the lowest order term in the numerator and denominator unity.  The numerator is an order 0 polynomial, the denominator is order 1.

33. Check your AnsWer! Step 3: Draw the Bode diagram for each part.
The constant is the cyan line (A quantity of 3.3 is equal to 10.4 dB).  The phase is constant at 0 degrees.
The pole at 30 rad/sec is the blue line.  It is 0 dB up to the break frequency, then drops off with a slope of -20 dB/dec.  The phase is 0 degrees up to 1/10 the break frequency (3 rad/sec) then drops linearly down to -90 degrees at 10 times the break frequency (300 rad/sec).
Step 4:  Draw the overall Bode diagram by adding up the results from step 3.
The overall asymptotic plot is the translucent pink line, the exact response is the black line.

34. Bode and Nyquist Plot Command Help logspace
Logarithmically spaced vector.
logspace(d1,d2) generates a row vector of 50 logarithmically equally spaced points between decades and .
logspace(d1,d2,N) generates N points.

35. Bode Plot Command Help bode Bode frequency response of LTI models.
bode(sys) draws the Bode plot of the LTI model SYS (created with either tf, zpk or ss). The frequency range and number of points are chosen automatically.
bode(sys,{wmin,wmax}) draws the Bode plot for frequencies between wmin and wmax (in radians/second).

36. Bode Plot Command Help bode (cont’d)
bode(sys,w) uses the user-supplied vector w of frequencies, in radian/second, at which the Bode response is to be evaluated. See logspace to generate logarithmically spaced frequency vectors.
[mag,phase]=bode(sys,w)and[mag,phase,w]=bode(sys) return the response magnitudes and phases in degrees (along with the frequency vector w if unspecified). No plot is drawn on the screen. If sys has NY outputs and NU inputs, MAG and PHASE are arrays of size [NY NU LENGTH(W)].

37. Definition of Gain Margin and Phase Margin
The loop gain Transfer function L(s)
The gain margin is defined as the multiplicative amount that the magnitude of L(s) can be increased before the closed loop system goes unstable
Phase margin is defined as the amount of additional phase lag that can be associated with L(s) before the closed-loop system goes unstable

38. Gain and Phase margin Command Help margin
Gain and phase margins and crossover frequencies.
[Gm,Pm,Wcg,Wcp] = MARGIN(sys) computes the gain margin Gm, the phase margin Pm, and the associated frequencies Wcg and Wcp, for the SISO open-loop model sys (continuous or discrete). The gain margin Gm is defined as 1/G where G is the gain at the -180 phase crossing. The phase margin Pm is in degrees.

39. Gain and Phase margin Command Help margin (cont’d)
The gain margin in dB is derived by
Gm_dB = 20*log10(Gm)
The loop gain at Wcg can increase or decrease by this many dBs before losing stability, and Gm_dB<0 (Gm<1) means that stability is most sensitive to loop gain reduction. If there are several crossover points, MARGIN returns the smallest margins (gain margin nearest to 0 dB and phase margin nearest to 0 degrees).

40. Summary Root locus analysis Frequency response plots Nyquist (nyquist)
Bode (bode)
Gain Margin
Phase Margin