Joe Zhou, Caleb Rush
May 5, 2006
Chapter 3: Control Design
1
Introduction
The project design continues with the final design of the temperature broad being drawn up
in eagle. Work also continues on finishing drawing the design for the motor board up in
eagle. Meanwhile we have estimated new values for variables in the transfer function of the
temperature board and put those in a simulink simulation. We will not actually know these
values till the hardware is built.
2
Body
2.1
Controller Redesign
1) Control Gain KI and Kp and Kd calculation:
Figure 1.1: Result from SISO tool
K
e − sτ d which is identical
τ bs + 1
to the one of the previous lab, and the plant has parameters: K=1.24, τ b = 195.65 ,
We expect the transfer function of the TCL board is G =
τ d = 3.5 . Because we assumed there is no delay, the new transfer function of the TCL
board, which also known as the plant G, becomes G =
K
. Furthermore, the settling
τ bs + 1
time of the system must be greater than τ b . We chose it to be 200 seconds. Of course,
the settling time of the actual hardware could be smaller once we confirm the actual
value of τ b . Use sisotool in MATLAB to obtain the values of KI and Kp and Kd with
settling time of 200 seconds.
According to figure 1.1, the transfer function of the close loop system is
T = 3.42 *
(1 + 77 s + 39 s 2 )
s
Because Equation 1.1 is in the form of
188.76.
(Equation 1.1)
, so Kd =3.42, Kp=263.34, Ki =
2) Simulations:
The parameters of the plant K and τ b will change in the new design because we replace the
power resistor with a Peltier cooler. The expected K value of the new design will be double
of the previous one since the power of the cooler is twice as much as the power resistor; the
new τ b value is expected to be half of the previous one since the temperature response is
much faster due to the higher power input. However, since the same LM 35 sensors will still
be used in the new design, the time delay of the sensor τ d should remain the same. By
modifying the plant model with new K and τ b values and tuning the gains, we discovered
that when Kd =0.5, Kp=3.9448, Ki = 0.0245, the system had the best performance.
Figure 2.1: New design schematics in Simulink
Figure 2.2: New design performance in simulation
Figure 2.2 suggests that the settling time of the system is around 85 second. As you may see,
the desire temperature difference is set to be 0.25. It takes about 85 seconds to reach 0.23
which indicates the signal first time enters the 2% steady state error envelope. Moreover, the
system has about 4% overshoot. As we zoom in, the peak of the signal is at 0.2539 which is
about 4% overshoot.
3) Sensitivity to parameter changes and performance prediction:
* All values of Kp, Kd, and Ki in table 2.1-2.3 must multiplied by a 0.005 backoff gain to
obtain the actual Kp, Kd, and Ki values of the controller.
Kp
788.96
394.48
197.24
Increase 50%
No change
Decrease 50%
Settling Time (sec)
85
290
415
Table 2.1: Changes in Kp and its corresponding settling time*
y = 0.0002x 2 - 0.7478x + 555
R2 = 1
Kp vs Ts
450
400
350
300
Ts
250
200
150
100
50
0
0
200
400
600
800
Kp
Figure 2.1: Kp vs Settling Time
1000
According to figure and table 2.1,
decreasing Kp will lead to increasing
settling time. Based on the obtained
equation from figure 2.1, which represents
the relationship between Kp and the
system settling time, calculate the
derivative of the equation will give the
sensitivity of Kp to the system settling
time. Results are shown in table 2.6.
Kd
Settling Time (sec)
Increase 50%
No change
Decrease 50%
200
100
50
290
290
300
Table 2.2: Changes in Kd and its corresponding settling time*
y = 0.0013x 2 - 0.4x + 316.67
R2 = 1
Kd vs Ts
According to figure and table 2.2, by
decreasing Kd the settling time slightly
increases. Based on the obtained equation
from figure 2.1, which represents the
relationship between Kd and the system
settling time, calculate the derivative of
the equation will give the sensitivity of Kd
to the system settling time. Results are
shown in table 2.6.
302
300
298
Ts
296
294
292
290
288
286
0
50
100
150
200
250
Kd
Figure 2.2: Kd vs Settling Time
Ki
9.7866
4.8933
2.4467
Increase 50%
No change
Decrease 50%
Settling Time (sec)
240
290
170
Table 2.3: Changes in Ki and its corresponding settling time*
Ki vs Ts
y = -8.0745x 2 + 108.31x - 46.676
R2 = 1
350
According to figure and table 2.3, by
decreasing Ki the settling time increases
then decreases. Based on the obtained
equation from figure 2.1, which represents
the relationship between Ki and the
system settling time, calculate the
derivative of the equation will give the
sensitivity of Ki to the system settling
time. Results are shown in table 2.6.
300
250
Ts
200
150
100
50
0
0
2
4
6
8
10
12
Ki
Figure 2.3: Ki vs Settling Time
Increase 50%
No change
Decrease 50%
K
2.48
1.24
0.62
Settling Time (sec)
% Overshoot
31
85
170
0.8
4
1.24
Table 2.4: Changes in K, its corresponding settling time and % overshoot
K vs Ts y = 50.295x2 - 230.65x + 293.67
K vs % Over Shoot
5
160
4.5
140
4
120
3.5
% Over Shoot
Ts (sec)
R2 = 1
180
100
80
60
y = -3.7808x 2 + 11.484x - 4.4267
R2 = 1
3
2.5
2
1.5
40
1
20
0.5
0
0
0.5
1
1.5
2
2.5
0
3
0
K (DC gain)
0.5
1
1.5
2
2.5
3
K (DC gain)
Figure 2.4: Change in K vs
corresponding Settling Time
Figure 2.5: Change in K vs
corresponding % overshoot
According to table 2.4, figure 2.4 and 2.5, by decreasing K, the settling time will increase, and
the % overshoot of the system will increase then decrease. Based on the obtained equation
from figure 2.4 and 2.5, which represents the relationship between K and the system settling
time, and the relationship between K and the system % overshoot respectively, calculate the
two derivatives of the equation will give the sensitivity of K to the system settling time and
% overshoot respectively. Results are shown in table 2.7 and 2.8.
Increase 50%
No change
Decrease 50%
Tao b
391.3
195.65
97.83
Settling Time (sec)
% Overshoot
125
85
33
9.4
4
0
Table 2.5: Changes in τ b , its corresponding settling time and % overshoot
Tao B vs Ts y = -0.0011x 2 + 0.8587x - 40.342
Tao B vs % Over Shoot
y = -5E-05x2 + 0.0542x - 4.8673
R2 = 1
R2 = 1
140
10
9
120
8
7
% Over Shoot
Ts (sec)
100
80
60
6
5
4
3
40
2
20
1
0
0
0
100
200
300
Tao B (Time Constant)
Figure 2.6: Change in τ b vs
corresponding Settling Time
400
500
0
100
200
300
Tao B (Time Constant)
Figure 2.7: Changes in τ b vs
corresponding % overshoot
400
500
According to table 2.5, figure 2.6 and 2.7, by decreasing τ b , the settling time will decrease,
and the % overshoot of the system will decrease as well. Based on the obtained equation
from figure 2.6 and 2.7, which represents the relationship between τ b and the system
settling time, and the relationship between τ b and the system % overshoot respectively,
calculate the two derivatives of the equation will give the sensitivity of τ b to the system
settling time and % overshoot respectively. Results are shown in table 2.7 and 2.8.
Tao d
Increase 50%
No change
Decrease 50%
Setting Time (sec)
7
3.5
1.75
% Over Shoot
85
85
87
0.52
4
1.44
Table 2.5: Changes in τ d , its corresponding settling time and % overshoot
y = 0.2177x2 - 2.2857x + 90.333
R2 = 1
Tao d vs Ts
Tao d vs Ts
y = -0.468x 2 + 3.92x - 3.9867
R2 = 1
4.5
87.5
4
87
3.5
3
Ts (sec)
Ts (sec)
86.5
86
85.5
2.5
2
1.5
85
1
0.5
84.5
0
84
0
1
2
3
4
5
6
7
Tao d (Time Delay)
Figure 2.8: Change in τ d vs
corresponding Settling Time
8
0
1
2
3
4
5
6
Tao d (Time Delay)
Figure 2.9: Changes in τ d vs
corresponding % overshoot
According to table 2.5, figure 2.8 and 2.9, by decreasing τ d , the settling time will slightly
increase, and the % overshoot of the system will increase then decrease. Based on the
obtained equation from figure 2.8 and 2.9, which represents the relationship between τ d and
the system settling time, and the relationship between τ d and the system % overshoot
respectively, calculate the two derivatives of the equation will give the sensitivity of τ d to the
system settling time and % overshoot respectively. Results are shown in table 2.7 and 2.8.
Ts
Kp
Kd
Ki
0.0004
0.0026
-16.149
Table 2.6: Sensitivity of parameters of
the controller to the settling time
According to table 2.6, Ki is the most
sensitive parameter to the system since
changing Ki will lead to the greatest
changing rate in the system settling time.
This is indicated in table 2.6. Ki has the
largest absolute value among the three
parameters in the controller. The negative
sign indicates, increasing Ki will lead to a
decreasing in settling time.
7
8
Ts
K
Tao b
Tao d
100.59
-0.0022
0.4354
Table 2.7: Sensitivity of parameters of
the plant to the settling time
K
Tao b
Tao d
% Overshoot
-7.5616
-1.00E-04
-0.936
Table 2.8: Sensitivity of parameters of
the plant to the % overshoot
According to table 2.7, K is the most
sensitive parameter to the system since
changing K will lead to the greatest
change in settling time since changing K
leads to the largest absolute value in
changing Ts. On the other hand, τ b is the
least sensitive parameter because it has the
least impact on the settling time of the
system.
According to table 2.8, K is the most
sensitive parameter to the system since
changing K will lead to the greatest
change in % overshoot. On the other
hand, τ b is the least sensitive parameter
because it has the least impact on the %
overshoot of the system.
3) Performance Prediction and Comparison:
Simulation with saturation and delay
Tao b / 2 (sec)
Zero Steady State Error
Equilibrium Temp Difference *100 (C)
% Over Shoot
Settling Time within 2% steady state error (sec)
Previous Design
195.65
0
0.25
0
206
New Design
97.83
0
0.25
4
85
Table 3.1: Performance difference between previous and new design
4) Stability Margins:
According to figure 1.1, the locations of the zeros are at -0.025+0.005j and -0.025-0.005j.
There is only one pole at the origin. Therefore, the system will always be on the left half
plane regardless the change of the gain. In other words, the system will always be stable.
2.2
Hardware Architecture
An H-Bridge was determined to be the best design for controlling the Peltier cooler on the
temperature board. The final version of the temperature board schematic is shown below.
Figure 1 – Temperature Board Schematic
The stray pin on the lower left will possibly be connected as another ground lead and 2 pins
will be added by the aluminum block for the peltier cooler. After this schematic was
completed the printed circuit board layout was completed with all wires traced, which will be
shown in appendix. At this point, the rewiring processes of the temperature control board
are completed. The design of the paralleled op amp of the motor control board is completed
as well; however, wiring processes on the printed circuit board are continued.
2.3
Software Architecture
As was done last quarter, a USB I/O card will be used to interface between the boards and
the computer where the controller will run in Matlab. Simulink will be used to assist in gain
tuning. Up to this point, there is no coding is required.
Risk and Hazard
2.4
The major risk is a possibility for burns. The aluminum block will be able to heat up to 80 or
100 degrees Celsius. Other than that, the temperature board should be totally safe.
Moreover, we must ensure the H-Bridge is wired correctly or the control circuit of the
Peltier cooler in figure 1 will get fired.
2.5
Revised Project Plan
The project plan was revised to be more realistic for the current situation. The table and
flow chart from the project plan have been updated and shown below.
Past
4/8
4/6
Future
2.3
4/15
4.2
4/19
5/19
5/26
4.1
3.1
1.1
2.1 and 2.2
1.1
5/16
5.2
3.3
5.3 and 5.4
3.2
4/7
4/12
4/18
5/20
5/13
4.3
5/22
5.1
Only Tasks Related To Temperature Control Board are in progress
Only Tasks Related To Motor Control Board are in progress
Both Tasks Related To Temperature Control Board and Motor Control Board are in progress
Critical Path
Path
3
Technical Obstacles
Works on the motor control board are continued even thought they are not mentioned in
Mile Stone 3. The rewiring processes of the motor board are much more complicated than
expected. Moreover, when it comes to ordering hardware, often time we are unable to select
the best parts for the design due to their unavailability.
4
Team Management
The group has worked well together. There have been no problems in sharing work,
communicating or helping each other.
5/30
Appendix:
Figure 2 – Temperature Board Layout