12/6/04
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Advanced Embedded Systems Design
Lecture 14 Implementation of a PID controller
BAE 5030

003
Fall 2004
Instructor: Marvin Stone
Biosystems and Agricultural Engineering
Oklahoma State University
12/6/04
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Goals for Class Today
•
Questions over reading / homework
(
CAN Implementation
)
•
Zigbee and 802.14.5
–
(Kyle)
•
PID implementation
(Stone)
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Elements of a feedback control system
•
Review elements and
variables
Gc
G
2
G
3
Error
Manipulated
Variable
D
Controlled
Variable
out
+
+
G
1
out

+
out
Load
in
Setpoint
set
out
out_measured
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)
1
(
)
1
(
2
3
2
2
3
1
G
G
G
G
G
G
G
G
G
c
c
set
c
in
out
)
(
_
measured
out
set
c
G
D
2
1
DG
G
in
out
3
_
G
out
measured
out
Output (
out
) is readily calculated as a function of:
Load (
in
) and
Setpoint (
set
)
Manipulation is a simple function of the controller TF and error.
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Digital form of a classic feedback controlled system
•
If sampling rate is fast and holds are employed, this
system approaches the analog system
Gc
G
2
G
3
Error
Manipulated
Variable
D* D
Controlled
Variable
out
+
+
G
1
out

+
out
Load
in
Setpoint
set
out
out_measured
out_measured
*
Computer based controller
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One of the conventional models used to express a PID controller is:
dt
de
edt
e
K
M
d
t
t
i
c
0
1
rate
Derivitive
rate
Reset
signal
Error
e
gain
Controller
K
on
Manipulati
M
d
i
c
Time Domain PID Controller Equation
Where:
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Derivitive Form of a PID Controller
A convenient way to implement this equation in a controller is as the
derivative of manipulation known as the velocity form of the equation
as shown below:
2
2
dt
e
d
e
dt
de
K
dt
dM
d
i
c
In a practical system this equation will work well and does not require
any steady

state references, but eliminating the
i
and
d
term
completely results in:
dt
de
K
dt
dM
c
de
K
dM
c
or,
This equation has no positional reference and error accumulation is a
problem. Use velocity form only for PI or PID modes.
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Conversion of the DE to a Difference Equation
To begin the conversion of the PID equation to a difference
equation, the equation is multiplied by
dt
.
dt
de
d
edt
de
K
dM
d
i
c
Note that since
M
is a differential and
e
ss
is zero, this equation
conveniently applies to the absolute variables as well as the
deviation variables.
For small
D
t
, the equation can be approximated as:
D
D
D
D
D
D
t
t
K
m
d
i
c
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Representation with Discrete Time Variables
Each of the differences (
D
⤠捡渠扥b灲敳獥搠慳摩獣牥瑥⁶慬略a潦o
敡捨映瑨攠v慲楡扬敳(
m
and
) at the times 0, 1, and 2 as shown
below:
t
0
t
2
t
1
M
1
2
0
M
0
M
1
M
2
The equation can be simplified with the assumption that
D
t
is
constant:
D
D
D
D
D
D
t
t
K
m
d
i
c
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Discrete form of PID controller
Replacement of the differences (
D
⤠)楴h瑨攠e楳捲i瑥
v慲楡扬敳e⠠
m and
) results in:
D
D
D
D
1
2
2
1
2
1
2
)
(
t
t
K
m
m
d
i
c
D
D
0
1
1
2
2
1
2
1
2
)
(
t
t
K
m
m
d
i
c
D
D
0
1
2
2
1
2
1
2
2
)
(
t
t
K
m
m
d
i
c
Note that
D
楳i慳au浥搠瑯b攠愠捯n獴慮s⸠⁉.
D
t
varies, the equations
should be derived with that in mind.
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Discrete form of PID controller
This equation can be solved for the current manipulation,
m
2
, in terms
of values known at time
t
2
:
m
1
,
e
2
,
e
1
, and
e
0
.
The other parameters in the equation are constants.
D
D
D
D
0
1
2
1
2
2
1
1
t
t
t
t
K
m
m
d
d
d
i
c
Where
C
1
,
C
2
, and
C
3
are constants and the current manipulation is
expressed in terms of known values, the current and past errors.
0
3
1
2
2
1
1
2
c
c
c
K
m
m
c
or,
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Translation of PID Equation into Algorithm
This equation may be translated directly into a computer language, for
example:
m2 = m1 + k*(C1*e2
–
C2*e1 + C3*e0);
Within a computer program, the current error
is calculated from the current measurement of
the controlled variable and the setpoint, for example:
e2 = T_setpoint
–
T_measured;
The current manipulation
m2
is then computed using the previous
controller equation, and finally, at the end of the time step, each of the
variables is shifted forward for the next calculation.
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Translation of PID Equation into Algorithm
For example in C, the code might look like:
measure_and_manipulate() //Call once per delta T
begin
T_measured = measure_T(); //Get the measured temperature
e2 = T_setpoint
–
T_measured //Calculate the current error
m2 = m1 + k*(C1*e2
–
C2*e1 + C3*e0); //Calculate the manipulation
set_manupilation(m2); //Output the manipulation
e0 = e1; //Shift the error and manipulation
e1 = e2; //forward one time step
m1= m2;
end;
Note that the time step
D
t
is controlled by the time required to execute
the loop. C1,C2 and C3 are all functions of
D
t
. The equation will
probably be executed as floats! (Or very special care must be taken
with scaling.
12/6/04
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Assignment
•
Complete CAN message demo
•
Turn in course portfolio by 5:00 PM Wednesday Dec.
8th
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