dynamic neural network control (dnnc): a non-conventional neural network model
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Dynamic Neural Network Control (DNNC): A Non-Conventional Neural Network Model Masoud Nikravesh EECS Department, CS Division BISC Program University of California Berkeley, California Abstract: - PowerPoint PPT PresentationTRANSCRIPT
Dynamic Neural Network Control (DNNC): A Non-Conventional Neural Network Model
Masoud Nikravesh
EECS Department, CS Division
BISC Program
University of California
Berkeley, California
Abstract:
In this study, Dynamic Neural Network Control methodology for model identification
and control of nonlinear processes is presented. The methodology uses several
techniques: Dynamic Neural Network Control (DNNC) network structure,
neuro-statistical (neural network & non-parametric statistical technique such as
ACE; Alternative Conditional Expectation) techniques, model-based control
strategy, and stability analysis techniques such as Liapunov theory. In this
study, the DNNC model is used because it is much easier to update and adapt the
network on-line. In addition, this technique in conjunction with
Levenberge-Marquardt algorithm can be used as a more robust technique for network
training and optimization purposes. The ACE technique is used for scaling the
networks input-output data and can be used to find the input structure of the
network. The result from Liapunov theory is used to find the optimal neural
network structure. In addition, a special neural network structure is used to
insure the stability of the network for long-term prediction. In this model, the
current information from the input layer is presented into a pseudo hidden layer.
This model minimizes not only the conventional error in the output layer but also
minimizes the filtered value of the output. This technique is a tradeoff between
the accuracy of the actual and filtered prediction, which will result in the
stability of the long-term prediction of the network model. Even though, it is
clear that DNNC will perform better than PID control, it is useful to compare PID
with DNNC to illustrate the extreme range of the non linearity of the processes
were used in this study. The integration of the DNNC and the
shortest-prediction-horizon nonlinear model-predictive control is a great
candidate for control of highly nonlinear processes including biochemical
reactors.
References:
1. M. Nikravesh, A. E. Farell, T. G. Stanford, Control of Nonisothermal CSTR with
time varying parameters via dynamic neural nework control (DNNC), Chemical
Engineering Journal, vol. 76, 2000, pp. 1-16.
2. M. Nikravesh, Artificial neural networks for nonlinear control of industrial
processes, " Nonlinear Model Based Process Control", Book edited by Ridvan Berber
and Costas
Karavaris, NATO Advanced Science Institute Series, Vol 353, 1998 Kluwer Academic
Publishers, pp. 831-870
3. S. Valluri, M. Soroush, and M. Nikravesh, Shortest-prediction-horizon nonlinear
model-predictive control, Chemical Engineering science, Vol 53, No2, pp. 273-292,
1998.
Dynamic Neural Network Control (DNNC):
A Non-Conventional Neural Network Model
Masoud Nikravesh
EECS Department, CS Division
BISC Program
University of California
Berkeley, California
Dynamic Neural Network Control (DNNC)
1. Introduction
2. Theory
3. Applications and Results
4. Conclusions
5. Future Works
u(k)
y(k+1)
u(k-1)
u(k-M+1)
u(k-N+1)
ym (k) B1B2
W2
W1(N+1)
W11
W12
W1M
W1N
1
1
1
BM
BN
1
Q P
P
yu
d
d
e’
ysp
y
+
+
+
+
-
-
w
IMC
Q1 P
P
yu
d
d
e’
ysp
y
+
+
+
+
-
-Q2
+
-
w1 w
w2
Modified IMC, Zheng
et al. (1994)
To address integral windup.
Q’1 P
h(x(t-)
yu
d
d
e’
ysp
y
+
+
+
+
-
-Q’2
+
-
w’1 w
w’2
Non-linear model state-feedback control structure
x=f(x)+g(x) ux
P
On-Line Adaptation
FilterFilter
SetpointSetpoint
TrajectoryTrajectory
PController
Model
PressurePressure
PressurePressure
q
FilterFilter
SetpointSetpoint
TrajectoryTrajectory
Controller
Model
CA
CA
qc
CA
y k j a y k jii
j( ) ( )*
u(k - i + j) + + d(k + j)
1
y k j y k j a u k j io ii j
N*( ) ( ) ( )
1
y k j a u k N jo N( ) ( ) 1
d k k km( ) ( ) ( ) = y - y
Model Predictive Control
Y
Time i
ai
k = discrete time
y(k) = model output u(k) = change in the input (manipulated variable) defined as u(k)- u(k-1) d(k) = unmodelled disturbance effects on the output ai = unit step response coefficients N = number of time intervals needed to describe the process dynamics (Note: )
ym(k) = current feedback measurement y* (k+j) = predicted output at k+j due to input moves up to k.
In the absence of any additional information, it is assumed that
dkjdk () ()
N ifor a a Ni
y ky ky k
y k N
y K P
u k u k u k u k Nu k u k u k u k Nu k u k u k u k N
u k N u k N u
( )( )( )
( )
( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) (
123
1 2 1 01 1 2 02 1 3 0
1 2 k N u k
u k P N u k
aaa
a
ad k
N
P
3 0
0 0 0 1
111
1
1
1
2
3
1
) ( )
( ) ( )( )
(13)
y d = U a1
y(k + j) = y(k + j) - y (k + j)o
The Backpropagation Neural Network
z = x w + x w + + x w + = 1 ,..., P.
(i)1 2 N1 2
( ) ( ) ( )i iN
i
for i
=
x xx
x x
ww
1 2
1
2 N
1
2zz
z
xx x
xwP
N
N
P P P N
(1)
( )
( )
(1) (1) (1)
( ) ( ) ( )
( ) ( ) ( )
2 22
2 2
1
11
1
y F z1 ( ).z = X w = X w
1
w = | T
wT
X = | 1 1X
Comparing DMC with the neural network, the following analogy may
be drawn,
y z
U X
w ad
1 1
.
The DNNC Process Model
y k y k a u k d k( ) ( ) ( ) ( ) 1 1 11(k)
e (k) ( ) ( )k a u k 1 1
y k k u k k u k k u kk u k N k u k N
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )
1 1 21
a + a + a + ...+ a + a + d(k + 1)
1 2 3
N N
d k y k y km( ) ( ) ( ) 1 1 1
d k k k( ) ( ) ( ) 1 y - (k) y(k)m
d k k( ) ( ) 1 (k) - (k) y + (k) d(k)m
d k k( ) ( ) 1 y (k) + (k) d(k).m
The DNNC Process Model
y k k u k k u k k u kk u k N
k k
uy
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( )
1 1 21
a + a + a + ...+ a (k) u(k - N + 2) + a + y (k) + d(k)
= a(k) (k) + (k) d(k)
1 2 3
N-1 N
m
T
y k g uy( ) ( 1 (k),a(k))
State-Space Representation of DNNC
y k w w uy B B( ) ( ) 1 2 1 1 2 T
uy = [ u(k) u(k -1) u(k - N + 2) u(k - N + 1) y (k)]
w = [ w1 w1 ... w1 w1 ] m
T
1 1 2 N N+1T
( )ze ee e
z z
z z
y k w w uy B B
uy
( ) ( )
1 2 1 1 2
= [u(k) u(k -1) u (k - N + 2) u(k - N + 1) y(k)]
w = [ w1 (w1 - w1 ) ... (w1 - w1 ) w1 ]
T
T
1 1 2 1 N N-1 N+1T
x k f x k g x k u k
y k h z x k u k
y k h x k u k
( ) ( ( )) ( ( ), ( )),
( ) ( ), ( ) ,
( ) ( ( ), ( ))
1
1
1
x k x k x k x k x k
u k u k N y k
N NT
T
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
1 2 1
2 1
= u(k -1)
x k x k x k x k x k
k u k N y k
k k k u k
N NT
T
T
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ), ( ))
1 1 1 1 1
1 2 1
1 2 1
= u(k) u
= u(k) x x h(x1 N-2
State-Space Representation of DNNC
State-Space Representation of DNNC
f x k x k x k x k
g x k u k u k h x k u k
h x k u k y k w w uy B B
NT
T
( ( )) ( ) ( ) ( )
( ( ), ( )) ( ) ( ( ), ( ))
( ( ), ( )) ( ) ( )
0 0
0 0 0 0
1
1 2 2
2 1 1 2 T
Stability of the DNNC Process Model
J
w w y k
Y
N N N
0 0 0 0 0 01 0 0 0 0 00 1 0 0 0 0
0 0 0 0 0 00 0 0 1 0 0
1 1 11 2 3 2 1 2 12 ( )
y k w uy B
w w y k
w w w
j
j j
j j j
1
1 1
1 1
1
1 1
2 12
1 1
( ) ( )
( )
,...,
,
,
T
N -1
= w21 1 121 y k w N( ) .
DNNC Controller
u k
k Bw w
w
C C
( )
( )(
,
1 2
21
1 1
- B ) uy1
T
wwNT
1C
1,11,21,N-1 = [ w w ... w 11 ,] uyC T = [ u(k -1) u(k - 2) ... u(k - N + 2) u(k - N +1) y(k)]
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )k k y k d k y k
d k y k y k
set
m
1 1
1 0 5( ) .z ln 1- z1+ z
Stability of the DNNC Process Model
x k
ww
x kx kx k
x kx kx k
R
k
N NN
N
N
Nset
( )
( )( )( )
( )( )
( ) ( )
1
11
1 0 0 0 0 00 1 0 0 0 0
0 0 0 0 0 00 0 0 1 0 00 0 0 0 0 0
00
00
1 2 3 2 11
1
1
2
3
2
1
=
+
jj
set
ww
R
k Bw
B
w
1
1
1
2
1
1
2 1
1
,
( )
Stability of the DNNC Process Model
J = U
1 2 3 2 11
1
11
1 0 0 0 0 00 1 0 0 0 0
0 0 0 0 0 00 0 0 1 0 00 0 0 0 0 0
N NNw
w
NNNNN
1
12
23
210
Extension of the DNNC Model to the MIMO Case in IMC Framework
y k w W uy k B B
uy
uy j
( ) ( ( ) )
( )
1 2 1 1 2 F
= [ uy (k) uy (k) ... u (k) ]
= [ u (k) u (k) ... u (k - N + 1) y (k) ]
(1) (2) (j) T
(j) (j) (j)i m
(j) T
y(k + 1) = [ y (k + 1) y (k + 1) . . . y (k + 1) ] (1) (2) (j) T
W
ww w
w w w
j
j j j j
1 =
w w
1 1 1 2 1
2 1 2 2
1 2
, , ,
, ,
, , ,
w i j, [ w w . . . w w ]i,1 i,2 i,N i,NT
j j 1
Neuro-Statistical Model
W J J J + I e T -1 T2
W JT J J + e T T -1 T T
T V
1
V = ij1
2 1me ei k
k m
m
j k
V = I2
W W k
Distribution ofMin
Distribution ofMax
Distribution ofMean
Mean+Std(Mean)Mean-Std(Mean)
Max-Std(Max)Min+Std(Min)
Distribution ofMin
Distribution ofMax
Distribution ofMean
Mean+2*Std(Mean)Mean-2*Std(Mean)
Min+2*Std(Min)Max-2*Std(Max)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0Actual
Neu
ral N
etw
ork
Pred
ictio
n
Actual Mean Mean+Std Mean-Std Upper Lower
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20Data Points
Neu
ral N
etw
ork
Pred
ictio
n
Actual Mean Mean+Std Mean-Std Upper Lower
0 5 10 15 20 0 0.5 1
1.5 2
2.5 3
3.5 4
4.5
Actual
Upper
Lower
Mean
MPV
F(k)F(k-1)F(k-2)F(k-3)
F(k-4)
P(k)P(k-1)P(k-2)
P(k-4)P(k-3)
P(k+1)
u(k)y(k+1)
u(k-1)
u(k-M+1)
u(k-N+1)
ym (k)
nonlinear transfer function
lineartransfer function
B1 B2
W2
W1(N+1)
W11
W12
W1M
W1N
1 1
Pc(k+1)
F(k)F(k-1)F(k-2)F(k-3)
F(k-4)
P(k)P(k-1)P(k-2)
P(k-4)P(k-3)
P(k+1)
Z(-1)Z(-2)
Z(-5)Z(-4)
Z(-3)
u(k)
y(k+1)
u(k-1)
u(k-M+1)
u(k-N+1)
ym (k) B1B2
W2
W1(N+1)
W11
W12
W1M
W1N
1
1
1
BM
BN
1
720
740
760
780
800
820
0 250 500 750Sampling Time
Wel
lhea
d Pr
essu
re, p
si
Actual data
Neural NetworkPrediction
F(k)F(k-1)F(k-2)F(k-3)
F(k-4)P(k)P(k-1)P(k-2)
P(k-4)P(k-3)
P(k+1)
720
740
760
0 250 500 750Sampling Time
Wel
lhea
d Pr
essu
re, p
si
Solid Line: ActualThin Line: Network Prediction
Pc(k+1)
F(k)F(k-1)F(k-2)F(k-3)
F(k-4)
P(k)P(k-1)P(k-2)
P(k-4)P(k-3)
P(k+1)
Z(-1)Z(-2)
Z(-5)Z(-4)
Z(-3)
u(k)
y(k+1)
u(k-1)
u(k-M+1)
u(k-N+1)
ym (k) B1B2
W2
W1(N+1)
W11
W12
W1M
W1N
1
1
1
BM
BN
1
Typical multi-layer DNNC process model.
Alternative Conditional Expectation
X Y
(Y)(X)
No. Epochs: 200No. Hidden Nodes: 10
No. Epochs: 5No. Hidden Nodes: 1
NN Prediction
-0.9-0.8-0.7-0.6-0.5
-1 -0.5 0 0.5 1x
z
-0.2-0.1
00.10.2
-1 -0.5 0 0.5 1x
Phi(x
)
-0.9-0.8-0.7-0.6-0.5
-0.2 -0.1 0 0.1 0.2Phi(x)
z
-0.9-0.8-0.7-0.6-0.5
-1 -0.5 0 0.5 1y
z
-1-0.5
00.5
1
-1 -0.5 0 0.5 1y
Phi(y
)
-0.9-0.8-0.7-0.6-0.5
-1 -0.5 0 0.5 1Phi(y)
z
-0.8 -0.75 -0.7 -0.65 -0.6 -0.55-0.9
-0.85
-0.8
-0.75
-0.7
-0.65
-0.6
-0.55
Actual
Input-Output Data,Prvious Information
Input Layer
Input Data for Current Prediction
Hidden Layer Output Layer
Prediction, Output
AC
E
Phai
-0.8 -0.75 -0.7 -0.65 -0.6 -0.55-0.9
-0.85
-0.8
-0.75
-0.7
-0.65
-0.6
-0.55
Actual
Input-Output Data,Prvious Information
Input Layer
Input Data for Current Prediction
Hidden Layer Output Layer
Prediction, OutputA
CE
Phai
u(k)
y(k+1)
u(k-1)
u(k-M+1)
u(k-N+1)
ym (k) B1B2
W2
W1(N+1)
W11
W12
W1M
W1N
1
1
1
BM
BN
1
Typical multi-layer DNNC process model.
FilterFilter
SetpointSetpoint
TrajectoryTrajectory
Controller
Model
CA
CA
qc
CA
dCdt
qV C C
ERT tA
Af A o A = k C c( ) exp( ) ( )
dTdt
qV
T T H ERT
t
h Aq
t T T
fo A
p
c pc
pc
c pccf
= k C C
C
C V q
C
c
h
( ) ( ) exp( ) ( )
exp ( ) ( )
1
h(t)
:
Fouling coefficientc(t)
:
Deactivation coefficientCA
:
effluent concentration, the controlled variableqc
:
coolant flow rate, the manipulated variableq
:
feed flow rate, disturbanceCAf
:
feed concentration Tf
:
feed temperatures Tcf
:
coolant inlet temperature
h = (t) h = ( 1 - t ) h d h h
k c= k exp -E
RT (t)o
c (t)= exp ( - t ).
Process Model
H-Injector
110
120
130
140
0 500 1000 1500 2000 2500 3000 3500Sampling Time
Inje
ctio
n R
ate,
BC
W/d
ay
ActualNetwork Prediction
LR-Injector
25
35
45
55
65
0 500 1000 1500 2000 2500 3000 3500Sampling Time
Inje
ctio
n R
ate,
BC
W/d
ay ActualNetwork Prediction
•The DNNC strategy differs from previous neural network controllers because the network structure is very simple, having limited nodes in the input and hidden layers.
•As a result of its simplicity, the DNNC design and implementation are easier than other control strategies such as conventional and hybrid neural networks.
• In addition to offering a better initialization of network weights, the inverse of the process is exact and does not involve approximation error.
•DNNC’s ability to model nonlinear process behavior does not appear to suffer as a result of its simplicity.
• For the nonlinear, time varying case, the performance of DNNC was compared to the PID control strategy.
•In comparison with PID control, DNNC showed significant improvement with faster response time toward the setpoint for the servo problem.
•The DNNC strategy is also able to reject unmodeled disturbances more effectively.
•DNNC showed excellent performance in controlling the complex processes in the region where the PID controller failed.
•It has been shown that the DNNC controller strategy is robust enough to perform well over a wide range of operating conditions.
Conclusions
Q P
P
yu
d
d
e’
ysp
y
+
+
+
+
-
-
w
IMC
Q1 P
P
yu
d
d
e’
ysp
y
+
+
+
+
-
-Q2
+
-
w1 w
w2
Modified IMC, Zheng
et al. (1994)
To address integral windup.
Q’1 P
h(x(t-)
yu
d
d
e’
ysp
y
+
+
+
+
-
-Q’2
+
-
w’1 w
w’2
Non-linear model state-feedback control structure
x=f(x)+g(x) ux
P
Future Works
•The integration of the DNNC and the shortest-prediction-horizon nonlinear model-predictive
•No assumption regarding the uncertainty in input and output
• Use of fuzzy logic techniques.
u(k)
y(k+1)
u(k-1)
u(k-M+1)
u(k-N+1)
ym (k) B1B2
W2
W1(N+1)
W11
W12
W1M
W1N
1
1
1
BM
BN
1