Transcript No Slide Title

Objectives in modeling
Analytical models
Process analysis: studies of reaction mechanisms, kinetics,
parameter estimation
Process design
Process optimization
Process on-line monitoring
Process control
Input - output models
Process on-line monitoring
Process control
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Fuzzy logic models
In fuzzy logic models input and output spaces are covered or approximated with discourses of fuzzy sets labeled as linguistic variables
For example, if Ai  X is an i-th fuzzy set it is defined as an ordered
pair:

 
Ai  xt ,  A  xt  xt   X , t  0, t f
where x(t) is a scalar value of an input variable at time t, and A is
called a membership function which is a measure of degree of membership of x(t) to Ai expressed as a scalar value between 0 and 1.
Typical membership functions have a form of a bellshaped or
Gaussian, triangular, square, truncated ramp and other forms
B
A
S
T
Izraziti (Booleovi) skupovi
S
T
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Neizraziti (“fuzzy”) skupovi
Grafički (Venn-ov) dijagram prikaza dvaju izrazitih (A) i neizrazitih (B)
skupova S i T. Izraziti skupovi su disjunktni, a kod neizrazitih skupova
neki elementi istovremeno pripadaju u oba skupa. Stupanj pripadnosti
neizrazitom skupu izražena je intenzitetom sive boje.
B) skup “hladna voda”
A) skup “tekuća voda”
(T)
(T)
1
1
0
0
-20
4
0
T / 0C
20
-20
0
20
T / 0C
Prikazi funkcija pripadnosti (T) fizikalne veličine
temperatura T skupovima (A) “tekuća voda” i (B) “hladna
voda”
Gaussian membership functions
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Fuzzy Logic Inference Systems
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( Mamdani Model )
Logical rules with
linguistic variables
X
AX
AY
Input
space of
physical
variables
Input
space of
linguistic
variables
Output
space of
linguistic
variables
Y
Output
space of
physical
variables
Input output relationships are modeled by fuzzy inference 7
system, FIS.
It is based on fuzzy logic reasoning which is a superset of
classical Boolean logic rules for crisp sets.
Elementary logic operations with fuzzy sets are:
fuzzy intersection or conjunction ( Boolean AND )
 Ai x  Aj x  T  Ai x,  Aj x
A typical choice of T-norm operator is a minimum function
corresponding to Boolean AND, i.e.:
Ai x AND Aj x  minAi x, Aj x
and standard choice to Boolean OR and NOT:
Ai x OR Aj x  maxAi x, Aj x
NOT Ax  1  Ax 
Process of mapping scalar between input and output sets by8
Fuzzy Inference System.
Fuzzification
x(t)
Fuzzy inference
Defuzzification
y(t)
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Sugeno (1988) Fuzzy Inference System
Logic
relations
X
Space of
input
variables
(numbers)
AX
Z
Space of
input
logic
variables
Space of
singelton
MF
(numbers)
Y
Space of
output
variables
(numbers)
Developed for process modeling and identification.
Application in adaptive neural fuzzy logic systems ANFIS
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In Sugeno FIS for fuzzy inference
polynomial Pn approximation is applied
Y = Pn ( Z ), usually a linear model is used
Y = C1 Z + Co , C1 and Co are constants
Mapping to scalar variables is obtained by averaging
y = WT Y
Example: Fuzzy logic control of flow rate
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For example, consider a fuzzy logic model of control of a
flow rate ( position of a valve piston) based on input
values of temperature T and pH
flow rate
valve position
T
pH
BIOPROCESS
FUZZY
LOGIC
MODEL
Q
FIS model Q=f(T,pH)
FUZZY INFERENCE SYSTEM
INPUT
SPACE OF
LINGUISTIC
VARIABLES
FUZZY
RULES
OUTPUT
SPACE OF
LINGUISTIC
VARIABLES
AGGREGATION
FUZZIFICATION
DEFUZZIFICATION
INPUT DATA
T(t) pH(t)
OUTPUT DATA
Q(t)
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T
upravljački
sustav
q
z
T1
P1
P1
T2
P2
P2
T3
P3
P3
T
q1
P4
P4
q2
P5
P5
q3
P6
P6
z
M
q
“fuzifikacija”
“fuzzy”
zaključivanje
“defuzifikacija”
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 LOW pH
 LOW T

GOOD T
 HIGH T
T
T

GOOD pH
 HIGH
pH
pH
pH
T
T(t)
pH
pH(t)
List of the fuzzy rules for control of valve position
IF
IF
IF
IF
IF
T is low AND pH
T is low AND
T is high AND
T is high AND
T is good AND
is low OR good
pH is low
pH is high
pH is low
pH is good
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THEN valve is half open
THEN valve is open
THEN valve is closed
THEN valve half open
THEN valve half open
Membership function of the fuzzy sets in the output space 16

CLOSED
VALVE

OPEN
VALVE
HALF CLOSED

VALVE
Aggregation of fuzzy consequents from fuzzy inference
system FIS into a single fuzzy variable output
(t)
FIS rules
Aggregation
to output
VALVE
centroid
~ x   dx y(t) = valve position
x



y (t ) 
~ x   dx


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Slika 1. Shematski prikaz biološke živčane stanica
sinapsa
tijelo
dendrit
akson
jezgra
dendrit
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Shematski prikaz matematičkog modela živčane stanica
s1
s2
dendriti
si
sn
w1
akson
w2
wi
wn
netS=Σ si
y   netS 
Schematic representation of a neurone with a sigmoid
activation function
ACTIVATION
x1
1,2
1
x2
OUTPUT
O
x3
0,8
0,6
0,4
xi
0,2
xN
0
-6
-4
-2
0
2
4
INPUT
1
f ( s) 
1  exp( s)
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Schematic diagram of a feedforward multilayer
perceptron
X1
Y1
X2
Y2
X3
Y3
X4
I
H
O
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Model equations
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N ( l 1)
N ( l 1)


olj k   f  Wijl  oi(l 1) k    jl  netl  Wijl  oi(l 1) k    jl
i 1
 i 1

 T  
1
E    y  t   y  t 
2 k
E
Wi , j   
Wi , j
Methods of adaptation:
On-line back propagation of error with use of momentum term
Batch wise use of conjugate gradients ( Ribiere-Pollack, LevebergMarquard)
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NN models for process control
NNARX: Regressor vector:
 t   yt 1 yt  na ut  nk ut  nb  nk 1
T
Predictor:


y t   y t t 1,  NN  t , 
NNOE: Regressor vector:


T
 t   yt 1 yt  na ut  nk ut  nb  nk 1
Predictor:

y t   NN  t , 
Inverse neural network control
Compensation of
process noise ?
n
Y
XI
NN-1
Input information on reference
transients of output variables
PROCESS
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Inverse neural network control coupled with a
PID feedback loop
n
XI
NN-1
PROCESS
Y
PID
+
-
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Internal model control structure
n1
NN -1
xI
n2
n3
PROCESS
Y
-
NN
+
Chemostat as a single input single output SISO system
D
S
CHEMOSTAT
NN
cS
dcX
 D  cX  M 
 cX
dt
K M  cS
dcS
cS
 D  cS 0  cS   YSX   M 
 cX
dt
K M  cS
cS
dcP
 YPX   M 
 cX  D  cX
dt
K M  cS
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CHEMOSTAT SISO MODELS
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NN
cS k  1  NN cS k , cS k 1, D(k ), D(k 1)
NN-1
Dk 1  NN
1
cS k 1, cS k , cs (k 1), D(k )
Responses of concentration of substrate chemostat to a sine
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perturbation of reference concentration obtained with direct inverse
control. Reference signal is plotted as a solid curve and response is
dotted. Frequency of perturbations are A: 0,0125 min-1; B: 0,025
min-1; C: 0,2 min-1; D: 0,1 min-1
10
9
A
B
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7
6
5
4
3
2
1
0
12
0
20
40
60
80
100
120
140
120
140
160
180
200
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D
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C
10
8
7
8
6
5
6
4
4
3
2
2
1
0
0
20
40
60
80
100
120
140
160
180
200
0
0
20
40
60
80
100
160
180
200
Responses of substrate (s), dilution rate (D), product (p), and
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biomass (x) under direct inverse neural network control. Reference
signal is a series of square impulses of substrate. The chemostat
responses are dotted lines and the reference is a solid line.
s
D
4
3.9
3.8
3.7
3.6
x
p
3.5
3.4
3.3
3.2
3.1
0
20
40
60
80
100
120
140
160
180
200
Responses of substrate under direct inverse neural (
network control and internal model (
….)
….) control .
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10
8
6
4
2
0
0
20
40
60
80
100
120
140
160
180
200
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Comparison of direct inverse neural network control and
internal model neural network control with 7,5% relative
standard noise in substrate measurement
S
0
100
Time (min)
200
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NN from B. yeast production in deep jet bioreactor
(Podravka)
EtOH
3-run
2-run
1-run
15 h
15 h
15 h
Measured
NN model
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Adaptive neuro fuzzy inference system ANFIS
Integration of neural networks with fuzzy logic
modeling.
ANFIS does not require prior selection of fuzzy logic
variables
ANFIS does not require prior logic inference rules
ANFIS requires only sets of input and output
training data ( like for NN modeling )
ANFIS has Sugeno structure of fuzzy logic systems
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ANFIS provides fuzzy logic clustering of data to
artificial linguistic variables.
ANFIS provides adaptive membership functions
for definition of association of data to linguistic
variables (fuzzy variables).
ANFIS provides combinatorial generation of
logical relations for mapping between input and
output fuzzy sets.
ANFIS provides adaptation of parameters in
Sugeno mapping.
ANFIS provides back propagation method for
adaptation of model to training data.
ANFIS model of chemostat D(k)=f [ Sref,S(k),S(k-1)]
Input MF
Sref
output MF
not
or
and
S(k)
D(k)
output
S(k-1)
input
rules
Sugeno i/o mapping
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DEMODemo
PROGRAMS
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Conclusions
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Neural networks NN and Fuzzy logic inference (FIS) systems are
very practical methods for modelling and control of bioprocesses.
Advanced computer supported instrumentation for physical,
chemical and biological variables provide large data banks applicable
for training NN and FIS models.
NN and FIS are best suited for on-line monitoring, soft identification
and nonlinear multivariable adaptive control.
Unlike analytical models, NN and FIS can be developed without “a
priori” fundamental knowledge of a process.
Analytical models are “very expensive” to develop, but they are the
most valuable engineering tool.
NN and FIS can integrate knowledge in a very general form.
Information from on-line instruments, image analysis and
human experience can be easily incorporated.
Analytical models are excellent for extrapolation in the entire
process space, while NN and FIS are the best at interpolation in
the training set and need to be tested for extrapolation outside
training.
Integration of NN and FIS into Adaptive Neural Fuzzy Inference
Systems ANFIS leads to models which combine the best
properties of NN and FIS.
ANFIS are highly adaptive like NN, they are transparent for
logical rules like FIS, automatically generate linguistic variables
and logical rules, and are trained to extensive process data.
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Model verification of NN, FIS and ANFIS is
the most important step before their
application in laboratory and industrial
practice.