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PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL
Ivo BUKOVSKÝ1
Jiří BÍLA1
Noriasu HOMMA2
Ricardo RODRIGUEZ1
1Czech Technical University in Prague
2Tohoku University, Japan
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PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL
• We consider sample-by-sample adaptation of discrete-time models and controllers by gradient descent
2( )( 1) ( ) ;
... adaptable parameter of a model or controler
kk ki i
i
thi
Qw w i
w
w i
weight update system
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• Stability monitoring and maintenance of weight update system of adaptively tuned models and controllers significantly contributes to a stable and convergent control loop
PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL
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PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL
• In the paper, we introduce derivation of stability condition for gradient-descent tuned models and controllers
• The approach is valid for models and controllers that are nonlinear (incl. linear), but they are linear in parameters– Not suitable for conventional neural networks (MLP,
RBF)– Suitable for Higher-Order Neural Units (HONU, also
known as polynomial neural networks) (not limited to)
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PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL
Further in this presentation
• Fundamental gradient descent schemes for adaptive identification and control
• Static or dynamic Higher Order Neural Units (HONU)
• Stability conditions for static and dynamic HONU and its maintenance at every adaptation step
• Demonstration of achievements with ONU( NOx prediction – EME I, lung motion prediction, nonlinear control loop of a laboratory system)
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Plant
Adaptive model-linear
- neural network,
+-
2( )( 1) ( ) ;
kk ki i
i
ew w i
w
( )ku( )krealy
Fundamental gradient descent schemes for adaptive identification and control
Plant Identification by Gradient Descent
( )ke
( )ky
... neural weights
(adaptable parameter)
... control variableu
w
weight update system
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(Základní schemata adaptivní identifikace a řízení gradientovými metodami)
Automatické ladění adaptivního stavového regulátoru
Regulovanásoustava
Adaptivní regulátor-lineární
- polynomiální-- klasická neuronová síť
2( )( 1) ( ) ;
kk ki i
i
ew w i
w
( )kv ( )krealy
Referenční model (požadované chování regulované soustavy)
+-
Žádaná hodnota
+-
( )kdesiredy
... adaptovatelný parametr
(váhy u neuronových sítí)
... žádaná hodnota
w
v
Systém adaptovaných
vah
( )ke
Žádaný průběh chování
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Fundamental gradient descent schemes for adaptive identification and control (continue)
Tuning of Adaptive Controller in a Feedback Control Loop with Gradient Descent
Plantadaptive controller
- linear PID - neural network,
2( )( 1) ( ) ;
kk ki i
i
ew w i
w
( )kv
( )krealy
Model of desired behavior
+-
( )kdesiredy
... neural weight ,
(adaptable parameter)
... desired value
w
v
+-
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2( )( 1) ( ) ;
kk ki i
i
ew w i
w
Plant
Adaptive model-linear
- neural network,
+-
( )ku( )krealy
Fundamental gradient descent schemes for adaptive identification and control (continue)
Updating Control Inputs Directly by Gradient Descent
( )ke
( )ky
2( )( 1) ( ) c
kck ki
eu u
w
+-
( )kdesiredy
eC(k)
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The question is:
• How do we assure stability of nonlinear adaptive control loop?• The ways is to assure stability and convergence of adaptive
components in a control loop (plant model + controller)• What nonlinear model to use?
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• MLP or RBF networks as models and controllers– Not linear in parameters– Guaranteeing stability is complicated (not
suitable for undergraduate level, difficult for PhD students from non-heavy-math schools)
– Guaranteeing stability is complicated and theoretically heavy for practicioners (thus not attractive for practice)
Static & Dynamic Higher-Order Neural Units
How do we assure stability of the nonlinear adaptive control loop? What model to choose?
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Static & Dynamic Higher-Order Neural Units
How do we assure stability of the nonlinear adaptive control loop? What model to choose?
2( )( 1) ( ) ;
kk ki i
i
ew w i
w
Weight-update system:
Example of 2nd-order HONU: 1
( ...)
( ...)
( )
k
k
k
y
y
u
( )kx
( )sk ny
0
r rn n
i j iji j i
x x w
20 0 0 1 0 2 i j ny x x x x x x x x x 0,0 0,1 0,2 i,j n,nw w w w w
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“axis of adapted neural weights”
LNU
HONU
convetional NN
2( )k
k
e
0
Approximation strength of neural networks can be improved by adding more neurons or even layers, GA, PSO,…
Static & Dynamic Higher-Order Neural Units (continue)
Sketch of optimization error surfacesLinear x MLP Networks x HONU
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Static & Dynamic Higher-Order Neural Units (continue)
Static MLP vs. QNU as MISO models of hot steam turbine averaged data (“steady states”, batch training by Levenberg-Marquardt)
• double hidden layer FFNN
• single hidden layer FFNN
• static QNU• measured data
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Static & Dynamic Higher-Order Neural Units (continue)Respiration time series: Training Accuracy for Predicting Exhalation Time -Instances of trained neural architectures trained from different initial conditions by L-M algorithm
2-hidden-layer static MLPs (static feedforward networks)
1-hidden-layer static MLPs (static feedforward networks)
static
QNUs
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0 50 100 150
0
20
40
60
80
100
trénovacích epoch
JRNN
JDLNU
JDQNU
trénovacích epoch
Trénování predikce Mackey-
Glass
0 50 100 150
0
20
40
60
80
100
trénovacích epoch
JDQNU
JDLNU
JRNN
Trénování predikce polohy plic0 20 40 60
0
20
40
60
80
trénovacích epoch
JRNN
JRNN
JDQNU
Trénování predikce nelineárního periodického
signálu
0 50 100 150
0
20
40
60
80
100
trénovacích epoch
JRNN
JDLNU
JDQNU
trénovacích epoch
Static & Dynamic Higher-Order Neural Units (continue)
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0,0 0 0 0,1 0 1 0,2 0 2
2, ,... i j i j n n n
y w x x w x x w x x
w x x w x
0,0
0,1
0,20 0 0 1 0 2
,
n n
n n
w
w
wy x x x x x x x x
w
rowx colW
1( ...)
( ...)
( )
k
k
k
y
y
u
( )kx
( )sk ny
0
r rn n
i j iji j i
x x w
Static & Dynamic Higher-Order Neural Units (continue)
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Stability of weight-update system
• Condition for STATIC HONU
• Condition for DYNAMICAL HONU
( ) ( ) 1k k 1 M colx rowx
( )( ) ( ) ( ) 1
kk n k kse
rowx1 M colx rowx
colW
,
HONU
1( ...)
( ...)
( )
k
k
k
y
y
u
( )kx
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0 50 100 150 200 250 300 350 400-2
-1
0
1One Epoch of GD Adaptation of Recurrent QNU to Predict MacKey-Glass Equation (training data vs. neural output)
0 50 100 150 200 250 300 350 400-0.5
0
0.5Prediciton Error during the Epoch of Adaptation
0 50 100 150 200 250 300 350 4000.98
1
1.02
1.04
k
Spectral Radius during the Epoch (stability of weight update system at each adaptation step)
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0 100 200 300 400 500 600 700 800
-2
0
2
k
GD Adaptation of Recurrent QNU to Predict MacKey-Glass Equation (training data vs. neural output)
0 100 200 300 400 500 600 700 800-10
0
10
k
Prediciton Error during Adaptation
0 100 200 300 400 500 600 700 800
1
1.5
2
k
Spectral Radius during Adaptation (stability of weight update system at each adaptation step)
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600 620 640 660 680 700 720 740 760 780 800
-2
0
2
k
GD Adaptation of Recurrent QNU to Predict MacKey-Glass Equation (training data vs. neural output)
600 620 640 660 680 700 720 740 760 780 800-10
0
10
k
Prediciton Error during Adaptation
600 620 640 660 680 700 720 740 760 780 800
1
1.05
1.1
k
Spectral Radius during Adaptation (stability of weight update system at each adaptation step)
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Achievements with QNU
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250 300 350 400
-1
-0.5
0
0.5
1
1.5
t [min]
NOx,CO prediction – EME I
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trénování testování
Obr. 1: Dobře natrénovaná síť TptRNN pro 3-minutovou predikci klouzavých 3-minutových průměrů NOx, externí měřené vstupy jsou klapky a výkon , (klouzavé průměry se počítají jako průměry předchozích, současných a následujících hodnot, při intervalu predikce 3 minuty to znamená, že externí vstupy jsou již dostupné ale model v principu predikuje 3-minutový průměr který má být za 2 minuty), včase cca 415 ignoruje výpadek měření NOx a výstup modelu dobře nahrazuje měření.
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Lung Tumor Motion Prediction
0 500 1000 1500 2000 2500 3000 3500-2
-1
0
1
2
Late
ral a
xis
[mm
]
0 500 1000 1500 2000 2500 3000 3500-10
-5
0
5
Ceph
aloc
auda
l axi
s [m
m]
0 500 1000 1500 2000 2500 3000 3500-2
-1
0
1
2
k
Ante
ropo
ster
ior A
xis
[mm
]
y1
y2
y3
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20 40 60 80 100 120
t [sec]
-8
-6
-4
-2
0
2
4
6testing MAE= 0.853120295578 [mm], RMSE= 1.14143756682, treatment time = 86[sec], computing time= 83.385[sec]
20 40 60 80 100 120t [sec]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0absolute value of prediction error
Lung Tumor Motion Prediction by static QNU
sampling 15 Hz, epochs=100, Ntrain=360, 492 neural weights
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Lung Tumor Motion Prediction by static QNU
10^0 10^1 10^2
epochs
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040 Averaged normalized SSE of Retrainings
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Nonlinear Control Loop of a Laboratory System
[ ] Ladislav Smetana: Nonlinear Neuro-Controller for Automatic Control,Laboratory System, Master’s Thesis, Czech Tech. Univ. in Prague, 2008.
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Nonlinear Control Loop of a Laboratory System
PID Control and Nonlinearity of the Plant
0 100 200 300 400 500 600-30
-25
-20
-15
-10
-5
0
t [s]
y [c
m]
Prubeh PID regulace v zavislosti na hloubce ponoru batyskafu, serizeno na hloubku 20 cm
5 cm
10 cm
15 cm20 cm
25 cm
0 100 200 300 400 500 600-40
-35
-30
-25
-20
-15
-10
-5
0
t [s]
y [c
m]
Prubeh PID regulace v zavislosti na hloubce ponoru batyskafu, serizeno na hloubku 10 cm
5 cm
10 cm
15 cm20 cm
25 cm
Tunned PID controller for 10 cm
30
Tunned PID controller for 20 cm
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Nonlinear Control Loop of a Laboratory System
0 100 200 300 400 500 600-30
-25
-20
-15
-10
-5
0
t [s]
y [c
m]
Prubeh PID regulace v zavislosti na hloubce ponoru batyskafu, serizeno na hloubku 20 cm
5 cm
10 cm
15 cm20 cm
25 cm
0 100 200 300 400 500 600-40
-35
-30
-25
-20
-15
-10
-5
0
t [s]
y [c
m]
Prubeh PID regulace v zavislosti na hloubce ponoru batyskafu, serizeno na hloubku 10 cm
5 cm
10 cm
15 cm20 cm
25 cm
310 20 40 60 80 100 120 140
-25
-20
-15
-10
-5
0
t [s]
y [c
m]
Prubeh regulace neuro-regulatoru zavislosti na hloubce ponoru batyskafu
5 cm
10 cm
15 cm20 cm
25 cm
QNU as Adaptive Controller (simplest gradient descent)
Linear PID
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Nonlinear Control Loop of a Laboratory System
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False Neighbor Analysis is a single-scale analysis
x yyf )(x
( )
( ) ( )
i
j i
x
x x
( )
( ) ( )
i
j i
y
y y
To train neural networks , input (state) vector must be estimated to minimize uncertainty in training data
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Děkuji za pozornost
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y=f(x)x input data y output data
False Neighbors
1 2 IF AND
THEN and are False Neighbors
=> How much is correct Rx and Ry? - we do not know
=> Let's characterize false neighbors over whole intervals
of Rx and Ry, an
x yR y y R 1 2
1 2
x x
x x
d not just for their single setup
False Neighbor Analysis is a single-scale analysis
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Slope of FN in Log-Log plot
FN = 4.2239*log2(id) - 4.5879
-2
0
2
4
6
8
1 1.5 2 2.5 3
log2(id)
log2(FN) Linear (log2(FN))
q(k ) c r (k )H
( )
( )
log
log log
k
k
q
c H r
MULTI-SCALE ANALYSIS approach (MSA)
number of false neighbours on a main diagonal
0
50
100
150
1 2 3 4 5 6
id...index of a diagonal cell
FN
• To characterize a system over the range of setups
• Power law
•What is the fundamental idea?
![Page 37: PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL Ivo BUKOVSKÝ 1 Jiří BÍLA 1 Noriasu HOMMA 2 Ricardo RODRIGUEZ 1 1 Czech Technical University in Prague](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d355503460f94a0c5f2/html5/thumbnails/37.jpg)
MULTI-SCALE ANALYSIS approach (MSA)• What is the fundamental idea?
q(k ) c r (k )H
q … quantityH … characterizing exponentr(k) … discretely growing radius
r(k)=2,4,8
•To characterize a system over the range of intervals•The power-law concept
![Page 38: PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL Ivo BUKOVSKÝ 1 Jiří BÍLA 1 Noriasu HOMMA 2 Ricardo RODRIGUEZ 1 1 Czech Technical University in Prague](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d355503460f94a0c5f2/html5/thumbnails/38.jpg)
MULTI-SCALE ANALYSIS approach (MSA)• What is the fundamental idea?
q(k ) c r (k )H
q … quantityH … characterizing exponentr(k) … discretely growing radius
r(k)=2,4,8
•To characterize a system over the range of intervals
•The power-law concept
![Page 39: PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL Ivo BUKOVSKÝ 1 Jiří BÍLA 1 Noriasu HOMMA 2 Ricardo RODRIGUEZ 1 1 Czech Technical University in Prague](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d355503460f94a0c5f2/html5/thumbnails/39.jpg)
MULTI-SCALE ANALYSIS approach (MSA)• What is the fundamental idea?
k r(k) q A q B
1 2 4 22 4 13 113 8 44 44
r(k)=2,4,8
log2(qB) = 2.2297*log2(r) - 1.1531
log2(qA) = 1.7297*log2(r) + 0.2605
0.9
1.9
2.9
3.9
4.9
1 1.5 2 2.5 3log2(r(k))
q(k ) c r (k )H
![Page 40: PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL Ivo BUKOVSKÝ 1 Jiří BÍLA 1 Noriasu HOMMA 2 Ricardo RODRIGUEZ 1 1 Czech Technical University in Prague](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d355503460f94a0c5f2/html5/thumbnails/40.jpg)
MULTI-SCALE ANALYSIS approach (MSA) (cont.)
• How can MSA help to create better neural network models?
j =1 j =2 j =3 j =4 j =5
i=1
max FN (highest chance that y1≠y2
when x1=x2 )
i=2 FN (2,2)
i=3 FN (3,3)
i=4 FN (4,4,)
i=5
min FN (lowest chance that y1≠y2
when x1=x2 )
FN (i ,j ) … count of False Neighbors for Rx (i ) and Ry ( j )
Rx(i
)
Ry ( j )
Smallest Rx - maximum of different states of a system
Largest Rx - minimum of different states of a system
Smallest Ry - maximum of recognized different outputs
Largest Ry - minimum of recognized different outputs
FN decrease
FN decreases
ffecf F
N d
ecre
ase
ffecf F
N d
ecre
ase
( )f yxj
i
False Neighbors Matrix:
Multiscale False Neighbor Approach
![Page 41: PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL Ivo BUKOVSKÝ 1 Jiří BÍLA 1 Noriasu HOMMA 2 Ricardo RODRIGUEZ 1 1 Czech Technical University in Prague](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649d355503460f94a0c5f2/html5/thumbnails/41.jpg)
MULTI-SCALE ANALYSIS approach (MSA) (cont.)
• What are other potentrials for MSA for signal processing?
• MSA based signal processing
• Variance Fractal Dimension Trajectory (VFDT)
• Mutual Information
– Multiscale approach to calculate mutual information itself
– Mutual information of VFDT processed signals
• Everywhere, where a common analysis is subject to a
single-parameter setup and changing the setup disqualifies
the analysis results.