1
2
The Application of State Variable Feedback Control to
Fermentation Process
Pipat Titapand ’ and Ake Chaisawadi 2
King Mongkut’s Institute of Technology Thonburi
Abstract
The PID control technique uses open loop transfer function analysis and feedback
compensator design. By using this technique, it is difficult to control non-linear and time varying
systems, then a closed loop pole placement technique has been introduced in modern control
process and it is widely used in the state variable feedback control.
The application of state variable feedback control to the fermentation process is pro-
posed in this paper. A microprocessor based state variable feedback controller has been designed
and constructed for controlling a baker yeast fermentation process system fed by molasses. The
controller receives state variables from a sensor and the variables are provided in matrix form.
This paper also includes state observer method for calculate the gain of the controller.
The experimental results show that this controller can control multivariable at the same
time, ensure the accuracy of set point value, and transmit data to computer for analysis by serial
port of controller. This design emphasizes on easy data entering. The results can be shown by
LCD display or printer. This method also increases quality control for the process.
Graduate Student, Department of Electrical Engineering
Associate Professor, Department of Control System and Instrumentation Engineering
X=AX+BU (3.1)
U=-GX (3.2)
i=[~-BG]X (3.3.a)
X(s) = [SI - A + BG]-’ X(0) (3.3.b)
det[SI- A + BG] = Sk i-&S’-’ -&Sk-* +...+&_$+& =o
x = TX2 = TAT-’
(3.6.a)
(3.6.b)
6
B = TB
T = [@VI-
&I Q zo Controllability Matrix
Q=[B AB A’B . . ..A”-‘B]
w so Triangular Toeplitz Matrix
1 1 a, a, . . . . a,_,
0 1 a, . . . . ak_*
0 0 1 . . . .w =
I. .. .
0 0 . . . . 1
Transformation Matrix ?.z'h%fiM~%l 2 a$.¶~@ Companion form zo
- a , . . . .
0 . . . .
1 . . . .
- ‘k
0
1
(3.6.C)
(3.6.d)
(3.6.e)
(3.7)
(3.6)
7
(3.9)
1 g,, s,, ..... s,0 0 0 . . . . . 0
- -B G = .
s,, &.. . .
Sk 1 = . , .. .0 0 0 . . . . .
--A-BG=
-a, -s, - a2 -S, . . . ..-ak -i,1
1 0 . . . . . 0
0 1 . . . . .
0 0
(3.10)
8
9, = a, -a,
-_u = -GX = #I(;) z-G x
(3.12.a)
(xmb)
Gt =Tt ct = T’(k) (3.12.C)
G ’ =[(QFV)‘j-’ (&) (3.12.d)
2,..... =.
-22 _
7r -
442 z. . . . . . . . . . . . +A+&_ _-t _
4. . . . . . u+ :.. [y-c,i,]B.2_ -4 _ (3.13)
(3.14)
(3.15)i = FZ+Gy+HU
e1
H
e = . . . =
e2
X, 2,. . . . . . . . . . . . .x,-i2:
9
(3.16)
.e2 = Fe, + (A,, - LC,A&C, + FLC,)X, +(A22 - LC,A,, - F)X2
+(B2 - LC,B, - H)U (3.17.a)
5 = [A,, - LC,A,, + FL+-’
H = B2 - LC,B,
F = A22 - LC,A,,
(3.17.b)
(3.17.c)
(3.17.d)
dm.m73 (3.17.a) -wwiierLh
e2 = Fe, (3.18)
xr-m~aJnl5 (2.18) w¶~hil eigen uosLuw%l F +um@J~~ua~Luw'ifl L &¶~~L~~n
L LI&U-?IHU~~~ eigen YEN Lam% F ki n"i rank WIN Observability Matrix N, LY%J k-l
do N, = (A,2)T,CT A22T A,,CT . . . . . (A22T)k - l
A,2TC,T 1 (3.19)
i = Fiz+(A,, - LC,A,,)C,-‘y+ HU (3.20.a)
_k, = Ly+Z (3.20.b)
10
i = [A-BG]X
i- = LY+Z
i = FZ+GY+HU
det[SI - A + BG] = det[SI - F] = 0
(3.21)
(3.22)
(3.23)
(3.24)
Clock Pulse
ILimit raise
pulse
ILimit Lower
pulse
Set Point +-
Error Signal
I I
Calculate Controlledx(k+l)=Ax(k)+BU(k) Gain Matrix Variable
), U(k)= -Gx(k)Equation : X1-3.4
-+ (G)Equation: I
1 353.12 1 +
_
36 ~_................._. ......................... .................................................. ........ ..... .._......._......._. ............................................................................................................... ................................. i/
31 /
26
21
16
7
6.5
6
5.5
5
4.5
IS
I7
I9
21
2 23
”5 25
27
29
I5
I7
19
21
=! 23I3 253
27
29
31
33
35
37
39
41
43
45
47
I ILlB zJ6 ns”
7 .-
9 --
II .-
I3 _-
I5 .-
I7 .-
I9 .-
21 .-
; 23 .-
2.5 25-- --
27 --
29 .-
31 _-
33 ._
35 .-
37 ._
39 --
41 --
43 --
45 .-
47 .-
I IcljP3”
18
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 4 1 L - - - i~ State
Time (Hours)
?tid 14 ni~~~qu?nms?u’YoJ~~nl~~~u~~~nlu~~~~~~ul~ 2 %I,
5.7
5.6
5.2
Time ( Hours)
Ii%I 4.00-E 3.00.g 2.003 1.00$ 0.00
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