E. C. Biscaia Jr ., A. R. Secchi, L. S. Santos

E. C. Biscaia Jr., A. R. Secchi, L. S. Santos

Programa de Engenharia Química (PEQ) – COPPE – UFRJ

Rio de Janeiro - Brazil

Dynamic Optimisation Using Wavelets Bases

Aims of the Contribution

]t,p),t(u),t(y),t(x),t(x[Jmin f

)]p,t),t(u),t(y),t(x[f)t(x

0t,p),t(u),t(y),t(xg

0,),(),(),( ttutytxh

ff fe x t y t u t[ ( ), ( ), ( ), ] 0

00 x)t(x

Improve numerical methods for solving dynamic optimisation problems:

s.t

Sequential Method


)]p,t),t(u),t(y),t(x[fdt)t(dx



0]),(),(),([ fff tutytxe

00 x)t(x

)t(u)t(u),t(usn10

fn ttttts 210

Control variables are discretized and dynamic model is solved numerically at each iteration of the NLP

Sequential Method






00 x)t(x

)t(u)t(u),t(usn10

fn ttttts 210

discretization in time domain ns stages

Control variables are discretized and dynamic model is solved numerically at each iteration of the NLP

Sequential Method






00 x)t(x

)t(u)t(u),t(usn10

fns ttttt 210

1t

2t

1snt

sntcontrol profile

(parameterization)

Sequential Method






00 x)t(x

)t(u)t(u),t(usn10

fns ttttt 210

1t

2t

1snt

snt

decision variables

Sequential Method






00 x)t(x

)t(u)t(u),t(usn10

fns ttttt 210

1t

2t

1snt

snt

decision variables

NLP solver

Calculates optimal

control profile

Sequential Method






00 x)t(x

)t(u)t(u),t(usn10

fns ttttt 210

1t

2t

1snt

snt

decision variables

NLP solver

Calculates optimal

control profile

Successive Refinement

Initial profile

NLP solver

Refinement

NLP solver

Wavelets Sequential Method






00 x)t(x

NLP solver







00 x)t(x

NLP solver

Wavelets

Improving Adaptation of discrete points at each

iteration







00 x)t(x )t(u)t(u),t(usn10

fns ttttt 210

NLP solver

Wavelets


iteration







00 x)t(x )t(u)t(u),t(usn10

fns ttttt 210

NLP solver

Wavelets

new mesh

1t

2t

1snt

snt

3t


iteration

Wavelets AnalysisConsidering a function , it can be transformed into wavelet domain as:

)(tu

)t(),t(ud m,nm,n details


)(tu


k n

Tn m n m

n m

u t d t D t2 2 1

, ,

1 0

( ) ( )

control variable

Inner product


)(tu

where is the maximum level resolution. k

],,,,[ 12,2,,0,212,2,,0,212,1,,1,10,1 nkknn


k n

Tn m n m

n m

u t d t D t2 2 1

, ,

1 0

( ) ( )

control variable

],,,,[ 12,2,,0,212,2,,0,212,1,,1,10,1 nkknn dddddddD

Vector of wavelets details

Inner product

Resolution Position

kns 2

Wavelets Analysis

Haar wavelet has been considered:

10,0121,1210,1

toutt

tt

Wavelets Analysis


10,0121,1210,1

toutt

tt

mtnnmn 22 2/, )12(,,0 nm

Wavelets Analysis


10,0121,1210,1

toutt

tt

jnkmkjmn ,,0, ,,

mtnnmn 22 2/, )12(,,0 nm

Orthogonal basis

)t(u)t(u),t(usn10

fns ttttt 210

Wavelets Analysis

)t(u)t(u),t(usn10

fns ttttt 210

Control profile

NLP solver

Wavelets Analysis

NLP solver

)t(u)t(u),t(usn10

fns ttttt 210

Control profile

NLP solver

How Wavelets Work

NLP solver

Wavelets

Wavelets

Iteration 1

Iteration 2

Wavelets Thresholding Analysis

],,,,[ 12,2,,0,212,2,,0,212,1,,1,10,1 nkknn dddddddD



],,,,[ 12,2,,0,212,2,,0,212,1,,1,10,1 nkknn dddddddD

]dd,,dd,dd,d[D nkknn ,,,,,,,,,,,,,th

12202122021211101

Thresholding: some details are eliminated.



],,,,[ 12,2,,0,212,2,,0,212,1,,1,10,1 nkknn dddddddD

]dd,,dd,dd,d[D nkknn ,,,,,,,,,,,,,th

12202122021211101

)t(DtuTThTh New thresholded

control profile

Thresholding: some details are eliminated.


Thresholding strategiesThresholding:

(i)decomposition of the data ; (ii)comparing detail coefficients with a given threshold value and shrinking coefficients close to zero, eliminating data noise effects (DONOHO and JOHNTONE, 1995):



Visushrink (DONOHO, 1992):

n m n md u tt, ,( ), ( )

, , ,ˆ , n m n m n md Thr d d

,,

,

0, if,

1, if

n mn m

n m

dThr d

d

)ln(2ˆ sn

11, : 0,1, , 2 1

ˆ0.6745

nn mmedian d m

standard deviation of a white noise

details coefficients



Visushrink (DONOHO, 1992):

Fixed user specified (SCHLEGEL and MARQUARDT,2004 and BINDER, 2000):

n m n md u tt, ,( ), ( )

, , ,ˆ , n m n m n md Thr d d

,,

,

0, if,

1, if

n mn m

n m

dThr d

d

)ln(2ˆ sn

e

11, : 0,1, , 2 1

ˆ0.6745

nn mmedian d m

standard deviation of a white noise

details coefficients

How Wavelets Work

)t(u)t(u),t(usn10

fns ttttt 210

Control profile

NLP solver

NLP solver

Wavelets

Wavelets

Sequential AlgorithmIncorporate the

Visushrink threshold procedure and

compare with other fixed threshold

parameters;

Observe if the CPU is affected by changes of

threshold rule.

Improve, at each iteration, the estimate

of control profile.

Algorithm and Parameters

1. Integrator: Runge Kutta fourth order (ode45 Matlab);

2. Optimisation: Interior Point (Matlab) was used as NLP solver;

3. Wavelets: Routines of Matlab 7.6;

4. Stop Criteria

1

i iobj obj

iobj

F F

F

Flowsheet of Wavelet Refinement Algorithm

0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

Initial control Profile

u

t

Example: Semi-batch Isothermal Reactor (SRINIVASAN et al. ,2003)

Constant by parts interpolation


0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010 Optimal control profile 8 stages

u

t



0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010Optimal control profile

16 stages

u

t



0 50 100 150 200 2500.0000

0.0005

0.0010

0.0015

0.0020

0.0025

n = 4

n (Resolution)

n = 1n = 2

n = 3

Adaptive Visushrink Threshold16 stages

wavele

t deta

ils

t

Visushrink Threshold Fixed Threshold

0 50 100 150 200 2500.0000

0.0005

0.0010

0.0015

0.0020

0.0025Fixed threshold

16 stagesw

avele

t deta

ils

t



0 50 100 150 200 2500.0000

0.0005

0.0010

0.0015

0.0020

0.0025Thresholded details

16 stages

wavele

t deta

ils

t



0 50 100 150 200 2500.0000

0.0005

0.0010

0.0015

0.0020

0.0025Thresholded details

16 stagesw

avele

t deta

ils

t


0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

Optimal control profile 20 stagesu

t

0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010


t




0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010


t

0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010


t


Locations of discontinuity points ~ large details coefficients



0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

Optimal control profile 16 stages

u

t

0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

Optimal control profile 16 stages

u

t


0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010


t

0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010


t


Case Studies

Semi-batch Isothermal Reactor (SRINIVASAN et al., 2003))

A B C B D2

Bu F tf C t( )max ( )

0 25 50 75 100 125 150 175 200 225 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

u [L.m

in-1

]

t [min]

Bu F t( )Optimal Control Profile: 128 stages

0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010 16 stages

u

t [min]

Control profile evolution

0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010 20 stages

u

t [min]


0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.001030 stages

u

t [min]


0 50 100 150 200 250

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010 34 stages

u

t [min]


Results: Semi-batch Isothermal Reactor

Uniform mesh with 128 stages

Fixed Threshold 10-4



Visushirink

Iter.Total CPU

nsTotal CPU

nsTotal CPU

nsTotal CPU

nsTotal CPU

ns

1 8 8 8 82 16 16 16 163 22 22 24 164 26 30 32 205 28 36 36 306 32 42 40 347 36 46 428 42 58 549 461011

1 128 0.76 46 0.85 58 0.91 54 0.22 34

Reference CPU time: Uniform mesh

Bioreactor problem (CANTO et al. , 2001))

Sf

u F tJ M t

( )max ( )

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

u [L

.h-1

]

t [h]

Su F t( )Optimal Control Profile: 128 stages

M: monomerS: substrate


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0 16 stages

u [L.h

-1]

t [h]


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0 20 stages

u [L.h

-1]

t [h]


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0 26 stages

u [L.h

-1]

t [h]


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0 34 stages

u [L.h

-1]

t [h]


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0 46 stages

u [L.h

-1]

t [h]


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0 64 stages

u [L.h

-1]

t [h]

Results: Bioreactor problem)





Visushirink

Iter.Total CPU

nsTotal CPU

nsTotal CPU

nsTotal CPU

nsTotal CPU

ns

1 8 8 8 82 16 16 16 163 26 24 32 204 30 32 38 265 38 42 46 346 44 48 58 467 48 56 66 648 56 74 729 70 84

1 128 0.82 70 1.32 74 1.58 84 0.37 64


Mixture of Catalysts (BELL and SARGENT, 2000)

fu t

C t( )

max[ ( )]

catu t

cat cat1

( )1 2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

u

t

Optimal Control Profile: 64 stages

CBAcatcat 21


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 16 stages

u

t


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 20 stages

u

t


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 26 stages

u

t

Results: Bioreactor problem




Visushirink

Iter.Total CPU

nsTotal CPU

nsTotal CPU

nsTotal CPU

ns

1 8 8 82 16 16 163 20 20 204 22 22 265 28 30

1 64 0.43 28 0.56 30 0.14 26


Conclusions

1. Two threshold procedures were analyzed here: Fixed and Visuhrink (level dependent threshold). According to the results, Visushrink strategy is able to denoise the details in a more efficient way than Fixed strategy, that is more conservative. We have shown that the choice of a threshold procedure can improve the wavelet adaptation algorithm;

Conclusions


2. Other examples from literature ( BINDER et al., 2000; SRINIVASAN et al. , 2003; SCHLEGEL, 2004) was solved and have been presented similar results: a considerable improvement of CPU time when Visushrink is used.

Conclusions


2. Other examples from literature ( BINDER et al., 2000; SRINIVASAN et al. , 2003; SCHLEGEL, 2004) was solved and have been presented similar results: a considerable improvement of CPU time when Visushrink is used.

3. Other wavelets thresholding strategies (as Sureshrink and Minimaxi) has been investigated, however in some cased these strategies have undesirable results, with worse performance than Fixed strategy.

Future Works

1. This algorithm will be used to solve more complex problems with several control variables in order to improve the sequential adaptation of each control profile. Our expectation is to observe the intensification of threshold influence for these problems.

Future Works

1. This algorithm will be used to solve more complex problems with several control variables in order to improve the sequential adaptation of each control profile. Our expectation is to observe the intensification of threshold influence for these problems.

2. As observed here, wavelets are able to detect discontinuity points and therefore the location of different control arcs. A more sophisticated interpolation of control profile will be implemented in these regions with aims to reduce he number of stages and consequently decision variables.

Thank You

E. C. Biscaia Jr ., A. R. Secchi, L. S. Santos

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