2_Dynamic Learning Rate (ηD) for Recurrent_diapositivas_XP

8/4/2019 2_Dynamic Learning Rate (D) for Recurrent_diapositivas_XP

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Dynamic Learning Rate (D) for Recurrent High Order

Neural Observer (RHONO): Anaerobic Process Application

K. J. Gurubel1, E. N. Snchez1 and S. Carlos-Hernndez2

2011 International Joint Conference on Neural Networks

San Jose, California July 31 - August 5, 2011

1Cinvestav Unidad Guadalajara

2Cinvestav Unidad Saltillo

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In order to increase the performance of the neuronalobserver, a dynamic learning rate (D) for the EKFtraining algorithm is proposed which depends on

operations condition for a system under control, in orderto improve the learning of the neuronal network inpresence of disturbances. Considering that the pH is on-line measured, this variable is used to determine the

proposed D.

2

OBJECTIVE


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INTRODUCTION

Anaerobic digestion is a biological process in which organicmatter (substrate) is degraded by anaerobic bacteria(biomass), in absence of oxygen. Such degradationproduces biogas, consisting primarily of methane (CH4)

and carbon dioxide (CO2), and stable organic residues.

Anaerobic process is a complex and sequential processwhich occurs in four basic stages: Hydrolysis, Acidogenesis,Acetogenesis y Methanogenesis. Each stage has a specific

dynamics; methanogenesis which is the slowest oneimposes the dynamics of the process and is considered asthe critical stage. Then, special attention is focused onmethanogenesis. This process is developed in a continuousstirred tank reactor (CSTR) with biomass filter.

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INTRODUCTION

Different biogas sensors have been developed in order tomeasure CH4; however, substrate and biomass measuresare more restrictive.

A nonlinear discrete-time neural observer for unknownnonlinear systems in presence of external disturbancesand parameter uncertainties is proposed in a previouswork for estimate unmeasurable variables.

The neural observer is based on a discrete-time recurrenthigh-order neural network trained with an extendedKalman-filter based algorithm. The objective is toestimate biomass concentration, substrate degradationand inorganic carbon in the anaerobic process.

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MATHEMATICAL MODEL

The biological phenomena are modeled by ordinarydifferential equations, which represent the dynamicalpart of the process

5

ZZDdt

dZ

ICICD

XRRXRXRRdt

dIC

SSDXRXRdt

dS

Xkdt

dX

SSDXRdtdS

Xkdt

dX

inin

inin

inin

d

inin

d

,

,

,

,

,

22311152232

221142232

2222

111161

1111


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MATHEMATICAL MODEL

where X1, corresponding to hydrolytic, acidogenic andacetogenic bacteria and X2, corresponding tomethanogenic bacteria. On the other hand, the organicload is classified in S1, the components equivalentglucose, which model complex molecules and S2 , the

components equivalent acetic acid. 1 is the growth rate(Haldane type) of X1 (h1), 2 the growth rate (Haldanetype) of X2 (h1), kd1 the death rate of X1 (mol L1), kd2the death rate of X2 (mol L1), Din the dilution rate(h1), S1in the fast degradable substrate input (mol L1),S2in the slow degradable substrate input (mol L1), ICinorganic carbon (mol L1), Z the total of cations(mol L1), ICin the inorganic carbon input (mol L1), Zinthe input cations (mol L-1), is a coefficient consideringlaw of partial pressure for the dissolved CO2 andR1, . . .,R6 are the yield coefficients.

6

1X2

X

1S

2S

1

2

1X

2X

1h

1h

1X

1dk

1Lmol 2dk2X

1Lmol

1h

inD

inS1

inS

2

1Lmol

1Lmol

1Lmol

1Lmol

1Lmol

1Lmol

inIC

IC

inZZ

61,..., RR


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The discrete-time RHONO estimates the variables of themethanogenesis stage: biomass ( X2), substrate ( S2) andinorganic carbon ( IC). The observability property of thisanaerobic digestion process was analyzed in a previous work.

DISCRETE-TIME RHONO

7

Observer scheme

2X

2S

IC


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DISCRETE TIME RHONO

Observer structure

8

,

1

,

1

,

1

3

2

35

2

34

233

2

3231

2222

24

2322

222212

122

14

1322

122112

kegkICkICSwkDkICSw

kXSwkICSwkICSwkIC

kegkSkSSw

kICSwkSSwkSSwkS

kegkDkXSw

kICSwkXSwkXSwkX

inin

in

in


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DISCRETE TIME RHONO

where wij is the respective on-line adapted weight vector;X2, S2 and IC are the estimated states; S(.) is the sigmoidfunction defined as S(x)= tanh(x); Din is the inputdilution rate and e is the output error.

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22 ,

SX

IC )(S)tanh()( xxS inD

ijw

e


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THE EKF TRAINING ALGORITHM

In this work, we use an EKF-based training algorithmdescribed by

where is the output estimation error and

is the weight estimation error covariance matrix, isthe weight (state) vector, is the respective number ofneural network weights

10

i i i iK k P k H k M k

1T

i i i i i iP k P k K k H k P k Q k

1

, 1, ,T

i i i iiM k R k H k P k H k i n

pe k i iL LiP k

iL

iw

iL

kykykekekKkwkw iii

,1


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THE EKF TRAINING ALGORITHM

is the plant output, is the neural networkoutput, , is the number of states, is the Kalmangain matrix, , is the NN weight estimation noisecovariance matrix, , is the error noise covariance,

and is a matrix, given as follows:

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T

ij

ij

y kH

w k

py

py

iL p

iK

i iL L

iQ

p p

iR

iL piH

where and .

n

1, ,i n 1, , ij L


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STATE ESTIMATION WITH

The process model and the observer are implementedusing Matlab/SimulinkTM. Nominal values of weredetermined in a previous work for each one of the

estimated variables as: 1 for the estimation of X2 , 2 forthe estimation of S2 and 3 for the estimation of IC.

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2

X

2S IC


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In order to test the observer sensitivity to input changes,a disturbance on the input substrate (270% S2in increase)is incepted at t = 200 h

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0 500 1000 15000.05

0.1

0.15

0.2

0.25

0.3

Time (h)

S2in(mol/L)

inS2

Disturbance in S2ininS2


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0 500 1000 15000.005

0.01

0.015

Time (h)

X2

(UA)

X2

Estimated

X2

System

0 500 1000 1500

0.01

0.02

0.03

Time (h)

S2

(mol/L)

S2 EstimatedS

2System

0 500 1000 1500

0.1

0.2

0.3

Time (h)

IC

(mol/L)

IC2

Estimated

IC2

System

State estimation


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States estimation are obtained by the RHONO, with anerror when the step on the input substrate is incepted,which is eliminated in the steady state. This error could

be due to the observer structure; it is possible that theNN is not able to learn all the nonlinear dynamicsrelated to the variables.

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DYNAMIC LEARNING RATE

pH is a sensible and determining variable in the processfor the adequate growth of bacteria and then for thewastes transformation; therefore it is used to determineD as proportional ( ) to the on-line measurement

substrate pH

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6 6.5 7 7.5 80.8

0.9

1

1.1

1.2

1.3

pH

D

pHD


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DYNAMIC LEARNING RATE

In order to implement this time varying learning rate,the EKF training algorithm is modified as follows:

The other components of algorithm remain unchanged.

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,1 kekKkwkwiiDiii


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SIMULATION RESULTS

With the purpose to analyze the effects ofD, three testsare realized with different initial values (D0), andconsidering the same disturbance on S2in. First, D0 equal

to the nominal values (), second, D0 100 % larger thannominal and third, D0 50 % smaller than nominal .These initial values are selected heuristic ally.

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inS2


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TEST 1: D0 =

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0 500 1000 1500

0.95

1

Time (h)

1

0 500 1000 1500

0.48

0.5

Time (h)

2

0 500 1000 1500

9.5

10

Time (h)

3

D3

D2

D1

Dynamic trajectories D


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20

0 500 1000 15000.005

0.01

0.015

Time (h)

X2

(UA)

X2

Estimated

X2

System

0 500 1000 1500

0.01

0.02

0.03

Time (h)

S2

(m

ol/L)

S2

Estimated

S2

System

0 500 1000 1500

0.1

0.2

0.3

Time (h)

IC

(mol/L)

IC2

Estimated

IC2

System

State estimation

TEST 1: D0 =


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and S2 are estimated with a smaller error when thestep on the input substrate is incepted. As is easy to seethe error vanishes quickly. This error could be inducedby the abrupt change in the system conditions. Since the

pH measure is directly related to IC , this variable isestimated with a negligible error during all thesimulation. In general, these estimation errors could bedue to the observer structure, which is a simple one. Theproposed

D

with initial nominal values produces adiminution in the transient error as compared withresults calculated with .

21

2X

2S

IC

TEST 1: D0 =


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22

0 500 1000 15001.8

2

2.2

Time (h)

D1

0 500 1000 15000.9

1

1.1

Time (h)

D2

0 500 1000 150018

19

20

21

Time (h)

D3

D1

D2

D3

TEST 2: D0 = 2 (100% larger)



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TEST 2: D0 = 2 (100% larger)

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0 500 1000 15000.005

0.01

0.015

Time (h)

X2

(UA)

X2

Estimated

X2

System

0 500 1000 1500

0.01

0.02

0.03

Time (h)

S2

(mol/L)

S2

Estimated

S2

System

0 500 1000 1500

0.1

0.2

0.3

Time (h)

IC

(mo

l/L)

IC2

Estimated

IC2 System

state estimation


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TEST 2: D0 = 2 (100% larger)

0 500 1000 15000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time (h)

W1

Eucliden norm of W1


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25

TEST 2: D0 = 2 (100% larger)

0 500 1000 15000

0.05

0.1

0.15

0.2

0.25

Time (h)

W2

Eucliden norm of W2


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TEST 2: D0 = 2 (100% larger)

0 500 1000 15000

1

2

3

4

5

6

7

8

Time (h)

W3

Eucliden norm of W3


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TEST 2: D0 = 2 (100% larger)

X2 and S2 are better estimated compared with resultscalculated with the initial nominal ones, when the stepon the input substrate is incepted. IC is again estimated

with a negligible error. A considerable reduction of thetransient error is obtained, improving the observerperformance.

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2X

2S

IC


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TEST 3: D0 = 0.5 (50% smaller)

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0 500 1000 15000.46

0.48

0.5

0.52

Time (h)

D1

0 500 1000 15000.23

0.24

0.25

0.26

Time (h)

D2

0 500 1000 1500

4.64.8

5

5.2

Time (h)

D3

D1

D2

D3



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0 500 1000 15000.005

0.01

0.015

Time (h)

X2

(UA)

X2

Estimated

X2

System

0 500 1000 15000

0.01

0.02

0.03

Time (h)

S2

(mol/L)

S2

Estimated

S2

System

0 500 1000 1500

0.1

0.2

0.3

Time (h)

IC

(mol/L)

IC2

Estimated

IC2

System

state estimation


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X2 and S2 estimated present an error when the step onthe input substrate is incepted. From the simulationresults of the three tests, it is clear that the best solution

is to select D larger than the nominal values.

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2X

2S


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ROBUSTNESS OF THE RHONO

On the other hand, observer tolerance to change on thesystem parameters is tested; such variation is incepted asa disturbance on the bacteria concentration. A 30%positive variation in the maximum rate growth of the

metanogenic bacteria (2max), a 15% negative variation inthe maximum rate growth of the hydrolytic, acidogenicand acetogenic bacteria (1max) and the samedisturbance on S2in are considered.

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m ax2

m ax1

inS2


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32

0 500 1000 15001.9

2

2.1

Time (h)

D1

0 500 1000 15000.95

1

1.05

Time (h)

D2

0 500 1000 150019

20

21

Time (h)

D3

D1

D2

D3

D trajectories with disturbance on thebacteria concentration


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0 500 1000 15000.005

0.01

0.015

Time (h)

X2

(UA)

X2

Estimated

X2

System

0 500 1000 1500

0

0.01

0.02

0.03

Time (h)

S2

(mol/L)

S2

Estimated

S2

System

0 500 1000 15000

0.1

0.2

0.3

Time (h)

IC(mol/L)

IC2

Estimated

IC2

System

state estimation


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D trajectories and the performance of the proposedRHONO are illustrated in the previously figuresrespectively, where it is clear that X2 , S2 and IC are wellestimated. Thus, the robustness of the proposed

RHONO to parameters variations is verified.

34

2X 2S IC


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CONCLUSIONS

A dynamic learning rate, named as D, is proposed forRHONO, trained with an EKF-based algorithm, applied toanaerobic process.

From simulation results, it is possible to see that D withinitial values larger than nominal ones contributed to the

best performance of the observer with a smaller transienterror compared with the other tests.

Simulation results also illustrate the observer robustness inpresence of disturbances on the system parameters.

Thus, it can be noticed that D is an adequate proposal toimprove the performance of the observer and to estimatethe states of the anaerobic process.

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FUTURE WORK This research will be pursued in order to evaluate the

application of the proposed D for control of theanaerobic process.

In order to improve the RHONO performance, theobserver scheme may be complemented by adding moreterms to the proposed structure.

2_Dynamic Learning Rate (ηD) for Recurrent_diapositivas_XP

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