Outline
• Advantages/disadvantages.
• Design procedures.
• Examples.
Frequency Response
Controller Design
M. Sami Fadali
Professor of Electrical Engineering
UNR
1
2
Design Procedure
Frequency Response Design
1. Select
Advantage: Familiar design procedure.
and obtain the transfer function
.
Disadvantages:
2. Bilinearly transform
1. Indirect design: controller distortion.
into
using
2. Requires experience.
3. Familiar criteria (PM, GM) have different values
from their analog counterparts for the same
performance.
MATLAB
>> gd=c2d(g,T)
>> d2c(gd,'tustin')
3
4
Design Procedure (Cont.)
Deign Specifications
3. Draw the Bode plot of
, and use
analog frequency response methods to
design a controller
that satisfies the
frequency domain specifications
• PM, GM , BW
• Use frequency response design
procedures for analog systems (e.g. lag,
lead, lag-lead).
4. Transform the controller back into the zplane using
= gain crossover (0 dB magnitude)
frequency
• BW increases with
5. Verify that the performance obtained is
satisfactory.
5
6
Solution
Example 6.15
• T = 0.1 s, obtain z-transfer function
Z
Consider the cruise control system of
Example 3.2, where the analog process is
• Bilinear transformation
Transform the corresponding
to the
-plane by considering both
and
. Evaluate the role of the sampling
period by analyzing the corresponding Bode
plots.
.
• T = 0.01 s, obtain z-transfer function
• Bilinear transformation
7
.
8
Bode Plots
,
&
Discussion
For 1
2
• Pole in -plane in the same position as the pole
in the s-plane.
• Have TF zero while
does not (different
frequency response from the analog system
• Greater influence of the zero on the system
dynamics when the sampling period is smaller.
• Distortion in the low frequency range is negligible.
• Gain as goes to zero is unity as is the DC gain
of the analog system.
Bode Diagram
Magnitude (dB)
0
G1(w)
-20
G(s)
-40
G 2(w)
-60
-80
0
Phase (deg)
-45
-90
-135
-180
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
Frequency (rad/sec)
9
10
Example 6.16
Solution
• For 10% overshoot,
DC motor speed control system: (type 0)
analog plant has the transfer function
Design a digital controller by using
frequency response methods to obtain: (i)
zero steady-state error due to a unit step,
(ii) an overshoot less than 10%, (iii) a
settling time of about 1
we calculate
• Choose
Z
.
.
11
.
12
Bilinear Transformation
Bode Plot
Bode Diagram
Gm = 25.7 dB (at 32.2 rad/sec) , Pm = 61.3 deg (at 4.86 rad/sec)
100
, pole-zero cancellation (simple design)
Magnitude (dB)
50
C(w)G(w)
0
G(w)
-50
-100
-150
0
Phase (deg)
-45
-90
-135
-180
-225
-270
-2
10
13
Step Response
0
10
1
10
2
10
3
10
4
10
5
10
Frequency (rad/sec)
14
Example 6.17
DC motor speed control system: (type 0)
analog plant has the transfer function
Step Response
System: Gcl
Peak amplitude: 1.07
Overshoot (%): 7.17
At time (sec): 0.56
1.2
-1
10
System: Gcl
Settling Time (sec): 0.818
Amplitude
1
0.8
Design a digital controller by using
frequency response methods to obtain: (i)
zero steady-state error due to a unit step,
(ii) an overshoot less than 10%, (iii) a
settling time of about 1
s
Use
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
15
16
Solution
Design
• TF of analog system, ADC and DAC
• Cancel two poles with zeros.
• Add poles at zero and
Z
.
.
.
• Poles almost in the same locations as poles of
.
to make the gain
• Consider RHP zero at
crossover frequency about 5 rad/s.
17
Bode plots of
&
18
Closed-loop Step Response
Step Response
Bode Diagram
Gm = 9.7 dB (at 18.2 rad/sec) , Pm = 59.9 deg (at 3.99 rad/sec)
System: Gcl
Peak amplitude: 1.08
Overshoot (%): 8.09
At time (sec): 0.4
40
Magnitude (dB)
20
1.2
C(w)G(w)
0
System: Gcl
Settling Time (sec): 0.718
1
G(w)
-20
Amplitude
-40
-60
0
0.8
0.6
Phase (deg)
-45
0.4
-90
-135
0.2
-180
-225
0
-1
10
0
10
1
10
2
10
3
10
4
10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Frequency (rad/sec)
19
20