A Comparative Study between Wavelet-Adaptive Multiple Shooting and

A Comparative Study between Wavelet-Adaptive Multiple Shooting and Single Shooting Implemented in a Matlab-EMSO Environment Lizandro S. Santos, Argimiro R. Secchi, Evaristo C. Biscaia Jr. Chemical Engineering Program PEQ/COPPE Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil. Outline Outline Introduction Dynamic Optimization with Wavelets Algorithm Structure Results Conclusions Outline Goals Implementation of Wavelet-Adaptive Algorithm[1] to solve optimal control problems using both single shooting and multiple shooting strategies; Using software EMSO as modeling environment for dynamic optimization formulation; Comparing CPU performance of wavelet-adaptive single and multiple shooting strategies. [1] Santos et al., 2014, Computer Aided and Chemical Engineering, v. 33, p. 247 - 252. Goals Mathematic Formulation Dynamic Optimization General Definition tf min min t0 , x t0 , t f , x t f , p L t , x t , u t , p dt u t ,t f ,p u t ,t f ,p t0 Meyer term Lagrange term subject to h t , x t , x t , u t , p 0 equality g t , x t , x t , u t , p 0 inequality

0 e tf ,x tf u min u t u max p min p p max x0 x0 t0 , t0 t t f Mathematic Formulation terminal u t R nu x t R nx p R np Dynamic DynamicOptimization OptimizationMethods Methods Bellman (HJB) Indirect Methods Pontryagin (EL) Dynamic Optimization (Sequential) Single Shooting Direct Methods Multiple Shooting Simultaneous Dynamic Optimization Methods Sequential Method (Single Shooting) Control vector parameterization (e.g., piecewise constant approximation) 1, , ns ns Time domain discretization

Number of stages u t i se i 1 t i , i 1, , ns Stage 2 Stage 1 Stage 3 2 u 3 1 t0 0 1 time 2 tF Sequential Method Single Shooting Sensitivity Equations for i 1, ns i Non-Linear Programming h x t , x t , u t dt u ex.: Interior Point, SQP i 1 end * gu x t ,u t u t ,x t ,

Sequential Integration 2 for i 1, ns u i 3 3 h x t , x t , u t dt 1 t0 0 1 time 2 tF f Sequential Method i 1 end Multiple Shooting Sensitivity and Constraints for i 1, ns i Non-Linear Programming h x t , x t , u t dt u ex.: Interior Point, SQP i 1 end * gu x t , u t

u t , x t , xo,i x i 1 , i 2, ns parallel Parallel Integration 2 xo ,2 xo ,1 t0 i 3 3 h x t , x t , u t dt xo ,3 1 0 for i 1, ns 1 time 2 tF f Multiple Shooting i 1 end Characteristics Single Shooting Multiple Shooting (ii) Results in a smaller NLP if compared (i) A larger NLP is formed, more with multiple shooting. sensitivity information is required, and the number

of SQP iterations may not be reduced. (ii) Sequential approach may be (ii) Multiple shooting can solve unstable unable to deal with unstable systems systems. (frequently happens in reactive systems and control problems). (iii) More parallelizable. Characteristics Wavelet Analysis for Image and Signal Processing Wavelet Decomposition Original Image Processed Image Resolution levels Noisy signal Wavelet Adaptation Wavelet-Adaptive How Wavelets Dynamic Work Optimization t0 t1 t2 tns t f Iteration 2 u( t 0 ),u( t1 ) u( t ns ) Wavelets NLP solver Iteration 1 Wavelets NLP solver Control profile Wavelet Adaptation Wavelets Thresholding Analysis Considering a function u (t ) , it can be transformed into wavelet domain as (discrete wavelet transform): dn,m u(t ), n,m (t ) DWT(u) control variable k n 2 2 1

u t detail coefficients Inner product dn,m n,m (t ) DT (t ) iDWT(d) n1 m0 Resolution Position Vector of wavelets details k where k is the maximum level resolution. ns 2 D [d1,0 , d1,1, , d1,2 n 1 , d 2,0, , d 2,2 n 1 , , d 2 k ,0, , d 2 k ,2 n 1 ] [ 1,0 , 1,1,, 1,2 n 1 , 2,0,, 2,2 n 1 , , 2 k ,0,, 2 k ,2 n 1 ] Wavelet Adaptation Wavelets Thresholding Analysis Haar[1] wavelet has been considered: 0 t 1 2 1, t 1, 1 2 t 1 0, t 0 ou t 1 n,m 2n / 2 2n t m n,m , j ,k 0, m 0, , (2n 1) m k , n j Orthogonal basis Daubechies, I., 1992, Ten Lectures on Wavelets, Philadelphia, Society for Industrial and Applied Mathematics. [1] Wavelet Adaptation Wavelets Thresholding Analysis dn,m u(t ), n,m (t ) details D [d1,0 , d1,1,, d1,2 n 1 , d 2,0,, d 2,2 n 1 , , d 2 k ,0,, d 2 k ,2 n 1 ]

Thresholding: some details are eliminated. Dth [ d1 ,0 ,d1 ,1 , ,d1 ,2n 1 ,d2 ,0 , ,d2 ,2 n 1 , ,d2k ,0 , ,d2 k ,2 n 1 ] Th u ThT t D ( t ) Wavelet Adaptation New thresholded control profile Wavelets Thresholding Thresholding Strategies Analysis soft threshold:[2] d n,m u(t ), n,m (t ) d n, m Thr S d n,m , 0 d n, m detail coefficients ThrS dn,m , if d n, m if d n,m , if d n, m Choice of threshold: Visushrink[3] Sureshrink[4] dn,m Thr S d n,m , Cross Validation[5] Fixed Threshold[6]

Wieland B., 2009, PhD Thesis, Universitt Ulm, Deutschland. [3] Donoho, D.L. and Johnstone, I.M., 1994, Biometrika, v. 81, pp. 425-455. [4] Donoho, D.L., 1995, IEEE Transactions on Information Theory, v. 41, n. 3, pp. 613-627. [5] Jansen, M., 2000, PhD Thesis. [6] Schlegel, M, et al., 2005, Computer and Chemical Engineering, v. 29, pp. 1731-1751. [2] Wavelet Adaptation Control Vector Parameterization Shrink Threshold (CVPS) ncvps argmin n Wavelets Daubechies s.t. min max n max card d n card d n, card d n compression parameterization d Thr d, ncvps iDWT d 1.4 1.2 1 0.8 n 0.6

m ax 0.4 card dn card dn, card dn 0.2 0 1 2 3 4 5 6 Threshold Wavelet Adaptation 7 8 9 10 11 Algorithm Structure (MATLAB) Initial Conditions t0 , x 0 Adaptive Algorithm u t Based on wavelets analysis Santos et al. (2011, 2014) Optimal Conditions x,u,p,J

EMSO NLP Enviroment for including DAE model Algorithm Structure Case Studies Studies Case 1 Batch Reactor Problem (CIZNIA et al,. 2006) 2 Continuous State Constraint Problem (JACOBSON & LELE, 1969) 3 Temperature Control (TOMLAB MANUAL, 2010) 4 Linear Bang Bang Control (LUSS, 2002) 5 Pressure-constrained Batch Reactor (HUANG, 2002) 6 Singular Control 5 (LUSS, 2002) 7 Time Free Bang Bang Control (LIANG et al. 2003) 8 Drug Displacement Problem (LUSS, 2002) 9 Lee & Ramirez Bioreactor Q = 0 (BALSA-CANTO et al., 2001) 10 Lee & Ramirez Bioreactor Q = 2.5 (BALSA-CANTO et al, 2001) 11 Non-linear CSTR (BALSA-CANTO et al., 2001) 12 CSTR Reactor (SRINAVASAN et al., 2003a) 13 Batch Fermentator Penicilin (BANGA et al., 2005) 14 The mixed catalyst problem (BELL e SARGENT, 2000) 15 Benoit Chachuat Example - (CHACHUAT, 2006) 16 Singular Control 4 (LUSS, 2002) 17 Singular Control 2 (LUSS, 2002) 18 Quadruple Integral (TOMLAB MANUAL, 2010) 19 Singular Control 3 (LUSS, 2002) 20 Park and Ramirez Bioreactor (BALSA-CANTO et al., 2001) control state 1 1 1 1 1

1 1 1 2 2 4 1 1 1 1 1 1 1 1 1 2 3 2 3 3 3 2 2 8 9 8 3 4 3 3 5 3 4 4 5 path const. end const. control arcs control arcs/control 0 1 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0

0 0 1 0 0 0 2 2 0 0 0 2 0 0 2 0 0 4 0 0 1 1 1 2 2 2 2 2 4 5 10 3 3 3 3 3 3 4 4 5 1.0 1.0 1.0 2.0 2.0 2.0 2.0 2.0 2.0 2.5 2.5 3.0 3.0 3.0 3.0 3.0

3.0 4.0 4.0 5.0 J 0.0695 0.1728 203.8656 41.3488 11.7270 0.7540 5.7571 30.1400 21.8169 221.2858 6.1515 0.4317 88.0364 0.0480 4.5859 0.1194 0.4446 1.0000 1.2523 32.6860 Results: overview Thresholdings with Single Shooting SS CPU 1 Batch Reactor Problem (CIZNIA et al,. 2006) 2 Continuous State Constraint Problem (JACOBSON & LELE, 1969) 3 Temperature Control (TOMLAB MANUAL, 2010) 4 Linear Bang Bang Control (LUSS, 2002) 5 Pressure-constrained Batch Reactor (HUANG, 2002) 6 Singular Control 5 (LUSS, 2002) 7 Time Free Bang Bang Control (LIANG et al. 2003)

8 Drug Displacement Problem (LUSS, 2002) 9 Lee & Ramirez Bioreactor Q = 0 (BALSA-CANTO et al., 2001) 10 Lee & Ramirez Bioreactor Q = 2.5 (BALSA-CANTO et al, 2001) 11 Non-linear CSTR (BALSA-CANTO et al., 2001) 12 CSTR Reactor (SRINAVASAN et al., 2003a) 13 Batch Fermentator Penicilin (BANGA et al., 2005) 14 The mixed catalyst problem (BELL e SARGENT, 2000) 15 Benoit Chachuat Example - (CHACHUAT, 2006) 16 Singular Control 4 (LUSS, 2002) 17 Singular Control 2 (LUSS, 2002) 18 Quadruple Integral (TOMLAB MANUAL, 2010) 19 Singular Control 3 (LUSS, 2002) 20 Park and Ramirez Bioreactor (BALSA-CANTO et al., 2001) 0.12 0.16 1.64 1.68 0.25 0.27

0.98 0.94 0.54 0.39 0.61 0.10 1.24 0.10 0.29 0.60 0.15 0.12 0.20 0.86 VS ns 24 53 93 82 35 47 53 55 81 120 518 37 39 32 64 64 51 34 62 84 CPU 0.06 0.03 0.28 0.3 0.18 0.18 0.41 0.22 0.59 0.36 0.5 0.15 0.94 0.15 0.32 0.49 0.26 0.2 0.28

1.23 CVPS ns 53 25 28 99 27 35 31 27 87 123 454 39 33 40 51 18 46 32 63 75 CPU 1.9 0.69 0.69 0.28 0.25 0.21 0.35 0.53 0.23 0.55 0.28 0.07 0.95 0.16 0.29 0.86 0.22 0.16 0.5 0.93 Fixed ns 52 97 52 57 39 40 29 38 65

140 324 30 40 27 49 42 53 35 36 86 CPU normalized by the uniform discretization with 128 stages CPU 0.62 0.44 0.33 0.48 0.44 0.62 0.65 0.81 0.89 0.71 0.72 0.38 0.98 0.72 0.33 0.91 0.49 0.36 0.48 1.22 ns 62 88 62 88 48 55 41 43 72 123 351 44 54 37 56 62 66 51 66 95

Results (Single Shooting: 1 CPU, Multiple Shooting: 2 CPUs) Illustrative example: Semi-batch Isothermal Reactor [2] max C (t f ) u 0.0010 Initial control Profile 0.0008 Piecewise-constant interpolation u 0.0006 u FB (t ) 0.0004 0.0002 0.0000 0 50 100 150 200 250 A B C 2B D t [2] Srinavasan, B. et al., 2003, Computer and Chemical Engineering, v. 27, pp. 1-26. Illustration Illustration Illustration of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor [2] Optimal control profile 8 stages 0.0010 0.0008 u 0.0006 0.0004

0.0002 0.0000 0 50 100 150 200 250 t [2] Srinavasan, B. et al., 2003, Computer and Chemical Engineering, v. 27, pp. 1-26. Illustration Illustration Illustration of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor [2] Optimal control profile 16 stages 0.0010 0.0008 0.0008 0.0006 0.0006 0.0004 0.0004 u u 0.0010 0.0002 0.0002 0.0000 0.0000 0 50

100 150 200 250 Analytical 0 25 50 75 100 125 150 175 200 225 250 t [2] t Srinavasan, B. et al., 2003, Computer and Chemical Engineering, v. 27, pp. 1-26. Illustration Example 1 [3] t 3 min 2 2 x t u t dt t 0 dx 1 x x u , dt x t 1, t 0,3 x 0 x0 0.05 x t 1 x 3 0 1 u t 1 [3] Diehl et al., (2006), Fast Direct Multiple Shooting Algorithms for Optimal Robot Control Example 1 Example 1 [1] State variable at the first iteration of SS method State variable at the first iteration

of MS method State Variable 1 for: 0th iteration State Variable 1 for: 0th iteration 0.075 20 0.07 0.065 x(t) x(t) 15 10 0.06 0.055 5 0.05 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 t time time t Uncontrollable growth of x for any x 0 0.062 [1]

Diehl et al., (2006), Fast Direct Multiple Shooting Algorithms for Optimal Robot Control Example 1 2 2.5 3 Example 1 [1] Optimal control profile Optimal state profile Dynamic control profile for: 39 stages State Variable 1 for: 4th iteration 0.02 0.05 0 0.04 u(t) u x(t) -0.02 -0.04 -0.06 0.03 0.02 -0.08 0.01 -0.1 -0.12 [1] 0 0.5 1 1.5 time t

2 2.5 3 0 0.5 1 1.5 time t Diehl et al., (2006), Fast Direct Multiple Shooting Algorithms for Optimal Robot Control Example 1 2 2.5 3 Example 1 [1] Single Shooting [1] Multiple Shooting iterations CPU(s) stages J CPU(s) Stages J 1 10 8 0.006313 10 8

0.006313 2 17 16 0.006216 17 16 0.006216 3 30 19 0.006197 34 19 0.006197 4 70 28 0.006186 92 28 0.006186 5 131 39 0.006183 234 39 0.006183

Diehl et al., (2006), Fast Direct Multiple Shooting Algorithms for Optimal Robot Control Example 1 Example 2 min x3 t f dx1 x2 dt dx2 u dt dx3 x12 , dt 1 u t 1 Example 2 t 0,5 Example 2 Evolution of Control Profiles Trajetoria de controle refinada para: 16 estagios 1 u (t) 0.5 0 Iteration 2 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t Trajetoria de controle refinada para: 23 estagios 1 u(t) 0.5 Iteration 3 0 -0.5 0 0.5 1 1.5 2 2.5 t Example 2 3 3.5 4 4.5 5 Example 2 Evolution of Control Profiles Trajetoria de controle refinada para: 23 estagios 1 u(t) 0.5 Iteration 4 0 -0.5 0 0.5 1

1.5 2 2.5 3 3.5 4 4.5 5 t Trajetoria de controle: 49 estagios 1 u(t) 0.5 Iteration 5 0 -0.5 0 0.5 1 1.5 2 2.5 t Example 2 3 3.5 4 4.5 5 Example 2 Single Shooting Multiple Shooting

iterations CPU(s) stages J CPU(s) stages J 1 7 8 0.277441 7 8 0.277441 2 33 16 0.269938 36 16 0.269938 3 106 27 0.268688 149 27 0.268688 4

256 43 0.268481 358 43 0.268481 5 266 53 0.268481 511 53 0.268481 Example 2 Example 3 [3] J max xc t f V t f x a k1xa xb t 0 250 u xa V x b k1xa xb 2k2 xb 2 u xb,in xb V V u 1 xc xa,oVo xaV V

xd 1 xa xb,in xb V xa,o xb,in xb,o Vo 2V 0 u 0, 001 xb t f 0, 025 xd t f 0,15 [2] Srinavasan, B. et al., 2003, Computer and Chemical Engineering, v. 27, pp. 1-26. Example 3 Example 3 -4 Dynamic control profile for: 8 stages x 10 Control variable 10 5 0 0 50 150 200 Time Wavelets aproximation and details coefficients -4

Wavelets coefficients Control variable x 10 Dynamic control profile for: 14 stages -4 15 x 10 55 00 1 2 3 4 5 6 7 8 9 Points 50 -3 x 10 100 150 200 250 Time Wavelets aproximation and details coefficients 2 1.5 1 250 Legend Ap

Dt.1 Dt.2 Dt.3 stages discrete points 10 10 0 lets coefficients 100 Example 3 stages discrete points Legend Ap Dt.1 Dt.2 Dt.3 Example 3 -4 Dynamic control profile for: 24 stages Control variable x 10 10 5 0 0 50 Control variable Wavelets coefficients -3 x 10-4 x 10 3 100 150 200 Time Wavelets aproximation and details coefficients Dynamic control profile for: 36 stages

10 2 5 1 stages discrete points 0 0 0 0 5 50 avelets coefficients -3 4 x 10 10 100 15 20 200 Points 150 Time Wavelets aproximation and details coefficients 3 stages discrete points 2 1 Example 3 250 Legend Ap Dt.1 Dt.2 Dt.3 Dt.4 25 250 Legend Ap Dt.1

Dt.2 Dt.3 Dt.4 Example 3 Single Shooting Multiple Shooting iterations CPU(s) stages J CPU(s) stages J 1 8 4 0.4287 11 4 0.4287 2 42 8 0.4308 43 8 0.4308 3 236 14 0.4316

155 14 0.4316 4 1016 24 0.4316 443 24 0.4316 5 1433 36 0.4317 1073 36 0.4317 Example 3 Example 4 [4] J max x1 t f t 0 84 x1 k1x1 k2 x2 k3 h x 2 u k4 x2 x 3 x2 x2 t 50 x3 t 2100 0 u [4] Banga, J.R., et al. 2005, Journal of Biotechnology, v. 117, pp. 407-419. Example 4

Example 4 [4] Dynamic control profile for: 16 stages Control variable 20 15 10 5 0 Control variable Wavelets coefficients 0 30 30 20 20 10 avelets coefficients 20 30 40 50 60 70 Time Wavelets aproximation and details coefficients Dynamic control profile for: 24 stages stages discrete points 100 -10 0 0 0 [4] 10 2 10 4 20 6 30

8 10 12 14 40 60 70 Points 50 Time Wavelets aproximation and details coefficients 80 Legend Ap Dt.1 Dt.2 Dt.3 Dt.4 16 80 60 40 stages discrete points 20 Banga, J.R., et al. 2005, Journal of Biotechnology, v. 117, pp. 407-419. 0 Example 4 Legend Ap Dt.1 Dt.2 Dt.3 Dt.4 Example 4 [4] Dynamic control profile for: 38 stages Control variable 30 20 10 0 [4] Wavelets coefficients Wavelets Controlcoefficients variable

0 10 20 30 40 50 60 70 Time Dynamic control profile for: 60 coefficients stages Wavelets aproximation and details 80 30 15 Legend Ap Dt.1 Dt.2 Dt.3 Dt.4 Dt.5 70 80 35 40 30 80 60 20 40 10 20 stages discrete points 0 0 -20 00 10 5

20 10 40 50 60 20 25 30 Time Points Wavelets aproximation and details coefficients 60 40 stages discrete points 20 Banga, J.R., et al. 2005, Journal of Biotechnology, v. 117, pp. 407-419. 0 -20 Example 4 Legend Ap Dt.1 Dt.2 Dt.3 Dt.4 Dt.5 Example 4 [4] Single Shooting [4] Multiple Shooting iterations CPU(s) stages J CPU(s) stages J 1

89 8 17.6223 56 8 17.6223 2 211 16 17.6321 144 16 17.6321 3 1128 24 17.6383 854 24 17.6383 4 1820 38 17.6388 1543 38 17.6388 5 3753 60

17.6389 3219 60 17.6389 Banga, J.R., et al. 2005, Journal of Biotechnology, v. 117, pp. 407-419. Example 4 Conclusions The wavelet-adaptive algorithm generated a non-uniform control parameterization for both multiple shooting and single shooting strategies allowing a faster convergence to the optimal solution; The combination of wavelet-adaptive strategy with parallel integration makes multiple shooting attractive and a good alternative for single shooting method; It has been observed that single shooting is faster than multiple shooting for small and smooth NLP problems; however, for large-dimension or non-smooth problems multiple shooting might be faster than single shooting approach. Conclusions A Comparative Study between Wavelet-Adaptive Multiple Shooting Optimal Wavelet-Threshold Selection to Solve and Single Shooting Implemented in a Matlab-EMSO Dynamic Optimization ProblemsEnvironment Thank you for your attention! [email protected] Visushrink Threshold (VS) Minimizes the probability that any noise sample will exceed the threshold. Given a sequence of variables, the expected value of the maximum increase with the length of the signal: VS 2 ln(ns ) median

d n 1,m : m 0,1, , 2n 1 1 0.6745 Standard deviation estimate for normal distribution of small details Inverse of cumulative normal distribution Sureshrink Threshold (SS) Sureshrink is an adaptive thresholding strategy. It is based on Steins Unbiased Risk Estimator[7] (SURE), a method for estimating the optimal threshold at each wavelet level: ns SURE d = ns 2 # i : di min d , i i 1 where 1, if x is true # x 0, if x is false SS min SURE [7] Stein, C M., 1981, The Annals of Statistics, v. 9, pp. 1135-1151 2 Cross Validation Threshold (CV) In short, the concept of CV is given by the construction of several estimates, never using the whole dataset. Using these estimates one predicts what the expelled data could have been and compares the prediction with the actual values of the expelled data.[8] ui 0.5 ui 1 ui 1 , i 2, ,ns 1 1 CV ns ui u ti di DWT vi

vi ,ui 1 ,u i ,ui 1 , ,uns ns 1 where 2 di di i 2 CV min CV Wieland B., 2009, Speech signal noise reduction with wavelets, PhD Thesis, Universitt Ulm, Deutschland [8] Fixed Threshold Use a user specified fixed threshold rule.[9] The threshold does not change with the wavelet level. The threshold value must, at least, be set between max (d) and min (d). min d [9] fixed max d Schlegel, M, et al., 2005, Computer and Chemical Engineering, v. 29, pp. 1731-1751.

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