KALMAN AND PARTICLE FILTERS

Transcription

1 KALMAN AND PARTICLE FILTERS H. R. B. Orlande, M. J. Colaço, G. S. Duliravich, F. L. V. Vianna, W. B. da Silva, H. M. da Fonseca and O. Fudym Department of Mechanical Engineering Escola Politécnica/COPPE Federal University of Rio de Janeiro, UFRJ Rio de Janeiro, RJ, Brazil 1

2 SUMMARY State estimation problems Kalman filter Particle filter Examples: Lumped system and 1D Heat conduction Applications Conclusions

3 STATE ESTIMATION PROBLEM State Evolution Model: x = f (x -1, u -1, v -1 ) Observation Model: z h ( x, n ) x R n x = state variables to be estimated u R n p = input variable v R n v z R n z n R n n = state noise = measurements = measurement noise Subscript = 1, 2,, denotes an instant t in a dynamic problem

4 STATE ESTIMATION PROBLEM Definition: The state estimation problem aims at obtaining information about x based on the state evolution model and on the measurements given by the observation model.

5 BAYESIAN FRAMEWORK The solution of the inverse problem within the Bayesian framewor is recast in the form of statistical inference from the posterior probability density, which is the model for the conditional probability distribution of the unnown parameters given the measurements. The measurement model incorporating the related uncertainties is called the lielihood, that is, the conditional probability of the measurements given the unnown parameters. The model for the unnowns that reflects all the uncertainty of the parameters without the information conveyed by the measurements, is called the prior model.

6 BAYESIAN FRAMEWORK The formal mechanism to combine the new information (measurements) with the previously available information (prior) is nown as the Bayes theorem: posterior ( x) ( x z) ( x) ( z x) () z where posterior (x) is the posterior probability density, (x) is the prior density, (z x) is the lielihood function and (z) is the marginal probability density of the measurements, which plays the role of a normalizing constant.

7 STATE ESTIMATION PROBLEM State Evolution Model: x = f (x -1, u -1, v -1 ) Observation Model: z h ( x, n ) The evolution-observation model is based on the following assumptions : (i) The sequence x for = 1, 2,, is a Marovian process, that is, ( x x, x,, x ) ( x x ) (ii) The sequence z x the history of for = 1, 2,, is a Marovian process with respect to, that is, ( z x, x,, x ) ( z x ) 0 1 (iii) The sequence x depends on the past observations only through its own history, that is, ( x x, z ) ( x x ) 1 1: 1 1

8 STATE ESTIMATION PROBLEM State Evolution Model: x = f (x -1, u -1, v -1 ) Observation Model: z h ( x, n ) Different problems can be considered: 1. The prediction problem, concerned with the determination of ( x z 1: 1) ; 2. The filtering problem, concerned with the determination of ( x z 1: ) ; 3. The fixed-lag smoothing problem, concerned the determination of ( x z 1: p ), where p 1 is the fixed lag; 4. The whole-domain smoothing problem, concerned with the determination of z1: K { zi, i 1,, K} ( x z ), where is the complete sequence of 1: K measurements.

9 FILTERING PROBLEM By assuming that ( x z ) ( x ) posterior probability density ( x z ) is available, the is then obtained with Bayesian filters in two steps: prediction and update 1:

10 (x 0 ) (x 1 x 0 ) Prediction (x 1 ) Update (z 1 x 1 ) (x 1 z 1 ) (x 2 x 1 ) Prediction (x 2 z 1 ) Update (z 2 x 2 ) (x 2 z 1:2 )

11 THE KALMAN FILTER Evolution and observation models are linear. Noises in such models are additive and Gaussian, with nown means and covariances. Optimal solution if these hypotheses hold. State Evolution Model: x = F x 1 + G u 1 + s 1 +v 1 Observation Model: z H x n F and H are nown matrices for the linear evolutions of the state x and of the observation z, respectively. G is matrix that determines how the control u affects the state x. Vector s is assumed to be a nown input. Noises v and n have zero means and covariance matrices Q and R, respectively.

12 THE KALMAN FILTER Prediction: x F x = F x G u 1 + s 1 +v 1 P F P F Q T 1 Update: T 1 K P H H P H R T x x K ( z H x ) P I - K H P

13 THE PARTICLE FILTER Monte-Carlo techniques are the most general and robust for non-linear and/or non-gaussian distributions. The ey idea is to represent the required posterior density function by a set of random samples (particles) with associated weights, and to compute the estimates based on these samples and weights. Introduced in the 50 s, but no much used until recently because of limited computational resources. Particles degenerated very fast in early implementations, i.e., most of the particles would have negligible weight. The resampling step has a fundamental role in the advancement of the particle filter.

14 Sampling Importance Resampling (SIR) Algorithm (Ristic, B., Arulampalam, S., Gordon, N., 2004, Beyond the Kalman Filter, Artech House, Boston)

15 Sampling Importance Resampling (SIR) Algorithm (Ristic, B., Arulampalam, S., Gordon, N., 2004, Beyond the Kalman Filter, Artech House, Boston)

16 Sampling Importance Resampling (SIR) Algorithm (Ristic, B., Arulampalam, S., Gordon, N., 2004, Beyond the Kalman Filter, Artech House, Boston) Although the resampling step reduces the effects of degeneracy, it introduces other practical problems: Limitation in the parallelization. Particles that have high weights are statistically selected many times: Loss of diversity, nown as sample impoverishment, specially if the evolution model errors are small.

17 Sampling Importance Resampling (SIR) Algorithm (Ristic, B., Arulampalam, S., Gordon, N., 2004, Beyond the Kalman Filter, Artech House, Boston) Step 1 For i 1,, N draw new particles x i density x x i 1 from the prior and then use the lielihood density i i to calculate the correspondent weights w z x Calculate the total weight Step 2 T w N i1 i. w and then normalize i 1 i the particle weights, that is, for i 1,, N let w Tw w Step 3 Resample the particles as follows : Construct the cumulative sum of weights (CSW) by i computing ci ci 1 w for i 1,, N, with c0 0. Let i 1 and draw a starting point u 1 from the uniform distribution U 0, For j 1,, N 1 N 1 Move along the CSW by maing u u N j While uj ci mae ii 1. Assign sample Assign sample x j j w x i 1 N j 1 1 Weights are easily evaluated and importance density easily sampled. Sampling of the importance density is independent of the measurements at that time. The filter can be sensitive to outliers. Resampling is applied every iteration, which can result in fast loss of diversity of the particles.

18 EXAMPLE: Lumped System where d ( t ) ( ) m () t mq t for t > 0 dt h 0 for t = 0 ( t) T( t) T 0 0 T T h m cl

19 EXAMPLE: Lumped System Two illustrative cases are examined: (i) Heat Flux q(t) = q 0 constant and deterministically nown; (ii) Heat Flux q(t) = q 0 f(t) with unnown time variation. Plate is made of aluminum ( = 2707 gm -3, c = 896 Jg -1 K -1 ), with thicness L = 0.03 m, q 0 = 8000 Wm -2, =20 o C, h = 50 Wm -2 K -1 and T 0 = 50 o C. Measurements of the transient temperature of the slab are assumed available. These measurements contain additive, uncorrelated, Gaussian errors, with zero mean and a constant standard deviation s z. The errors in the state evolution model are also supposed to be additive, uncorrelated, Gaussian, with zero mean and a constant standard deviation s.

20 (i) Heat Flux q(t) = q 0 constant and deterministically nown The analytical solution for this problem is given by: mt q0 mt ( t) 0e (1 e ) (A.3) h The only state variable in this case is the temperature ( t ) since the applied heat flux q 0 is constant and deterministically nown, as the other parameters appearing in the formulation. By using a forward finite-differences approximation for the time derivative in equation (A.1.a), we obtain: mq 0 (1 mt) 1 t (A.4) h Therefore, the state and observation models given by equations (3.a,b) are obtained with: x [ ] F [(1 mt)] q 0 s m t H 2 [1] Q s h [ ] R 2 z [ s ] (A5.a-f) x = F x 1 + G u 1 + s 1 +v 1 z H x n

21 (ii) Heat Flux q(t) = q 0 f(t) with unnown time variation The analytical solution for this problem is given by: t mt mq0 mt ( t) e 0 e f ( t) dt h t0 (A.6) In this case, the state variables are given by the temperature ( ) and the function that t gives the time variation of the applied heat flux, that is, f ( t) f.as in the case examined above, the applied heat flux q 0 is constant and deterministically nown, as the other parameters appearing in the formulation. By using a forward finite-differences approximation for the time derivative in equation (A.1.a), we obtain the equation for the evolution of the state variable ( ) : t mq (1 m t) t f h (A.7) A random wal model is used for the state variable f ( t ) f, which is given in the form: f f 1 1 (A.8) where -1 is Gaussian with zero mean and constant standard deviation s rw.

22 (ii) Heat Flux q(t) = q 0 f(t) with unnown time variation Therefore, the state and observation models given by equations (3.a,b) are obtained with: x f F mq (1 ) h mt t s 0 0 (A.9.a-c) 2 H s 0 2 [1 0] Q R 2 s z 0 s rw [ ] (A.9.d-f) x = F x 1 + G u 1 + s 1 +v 1 z H x n

23 EXAMPLE: 1D Heat conduction Linear Heat Conduction Problem 2 1 T T t x 2 in 0 < x < L, for t > 0 T 0 at x=0, for t > 0 T T * T T * at x=l, for t > 0 for t=0, in 0 < x < L Explicit finite-differences: 1 T FT S Τ T T 1 N F (1 2 r) r r (1 2 r) r r (1 2 r) r r (1 2 r) S 0 0 rt *

24 EXAMPLE: 1D Heat conduction Concrete with thermal diffusivity = 4.9x10-7 m 2 /s Standard-deviation for the measurement errors = 2 o C Final time = 250 seconds Measurements available in the region every 1 second L = 0.1 m N = 50 internal nodes

25 T, C T, C Exact Temperature x, m Time, s Measured Temperature x, m Time, s

26 T(x=0.01 m), C Kalman Filter (Standard-deviation for the evolution model errors of 1 o C) Exact Measured Predicted 99% Confidence Interval Time, s

27 T(x=0.01 m), C Kalman Filter (Standard-deviation for the evolution model errors of 0.5 o C) Exact Measured Predicted 99% Confidence Interval Time, s

28 T(x=0.01 m), C Kalman Filter (Standard-deviation for the evolution model errors of 5 o C) Exact Measured Predicted 99% Confidence Interval Time, s

29 T(x=0.01 m), C Particle Filter (Standard-deviation for the evolution model errors of 1 o C) Particle Filter with Re-sampling - Number of Particles = 50 Exact Measured Predicted 99% Confidence Interval CPU Time Kalman filter: 0.5 s Particle filter: 6.9 s Time, s

30 T(x=0.01 m), C Particle Filter (evolution model errors with uniform distribution in [-1,1] o C) Particle Filter with Re-sampling - Number of Particles = 50 Exact Measured Predicted 99% Confidence Interval Time, s

31 3 Capacidade Volumetric Térm Heat ica Capacity Volum (Jm etrica -3o C (J/m. C) -1 ) Thermal Condutividade Conductivity térm ica (Wm (W -1o /m. C) C -1 ) EXAMPLE: 1D Heat conduction Nonlinear Heat Conduction Problem T T C( T ) K ( T ) t x x in 0 < x < L, for t > 0 T 0 x at x=0, for t > 0 T ( T ) q * x at x=l, for t > 0 T T * for t=0, in 0 < x < L Graphite 6x10 6 5x C T A A e T A 1 2 4x10 6 3x10 6 Thermal Condutividade conductivity térm ica Heat Capacidade capacity térm ica T B B e T B 1 2 2x x Temperature peratura ( o ( C)

32 EXAMPLE: 1D Heat conduction T* = 20 o C q* = 10 5 W/m 2 L = 0.01 m 50 finite-volumes Final time = 90 s Time step = 1 s Errors in the state evolution and observation models were supposed to be additive, Gaussian, uncorrelated, with zero mean and constant standard-deviations Standard-deviation for the state evolution model = 5 o C Standard-deviation for the observation model = 10 o C

33 T, C T, C Exact Temperature x, m Time, s Measured Temperature x, m Time, s 60 80

34 T(x= m), C 600 Particle Filter with Re-sampling - Number of Particles = Exact Measured 0 Predicted 99% Confidence Interval Time, s

35 T(x=0.001 m), C 600 Particle Filter with Re-sampling - Number of Particles = Exact Measured 0 Predicted 99% Confidence Interval Time, s

36 T(x=0.002 m), C 700 Particle Filter with Re-sampling - Number of Particles = Exact Measured -200 Predicted 99% Confidence Interval Time, s

37 T(x=0.004 m), C 600 Particle Filter with Re-sampling - Number of Particles = Exact Measured -100 Predicted 99% Confidence Interval Time, s

38 APPLICATIONS Estimation of position-dependent transient heat source 2 2 T T T h g( x, y, t) C 2 2 T T t x y e e at 0 < x < L, 0 < y < L, for t > 0 ( T 0 at x = 0 and x = L, for t > 0 x ( T 0 at y = 0 and y = L, for t > 0 y ( T T 0 for t = 0, at 0 < x < L and 0 < y < L (4) H.Massard, F. sepulveda, H.R.B. Orlande and O. Fudym, 2010, Kalman Filtering for thermal diffusivity and transient source term mapping from infrared images, Inverse Problems, Design and Optimization Symposium 38 João Pessoa, Paraíba, Brazil, August 23-27, 2010

39 APPLICATIONS Estimation of position-dependent transient heat source H.Massard, F. sepulveda, H.R.B. Orlande and O. Fudym, 2010, Kalman Filtering for thermal diffusivity and transient source term mapping from infrared images, Inverse Problems, Design and Optimization Symposium 39 João Pessoa, Paraíba, Brazil, August 23-27, 2010

40 APPLICATIONS Estimation of position-dependent transient heat source T = T J P v 1 1 T P = IP v 1 P J L1 t( T1 T ) t L 0 2 t( T2 T ) t 0 0 LM t( TM T ) t H.Massard, F. sepulveda, H.R.B. Orlande and O. Fudym, 2010, Kalman Filtering for thermal diffusivity and transient source term mapping from infrared images, Inverse Problems, Design and Optimization Symposium 40 João Pessoa, Paraíba, Brazil, August 23-27, 2010

41 Estimation of position-dependent transient heat source 41 H.Massard, F. sepulveda, H.R.B. Orlande and O. Fudym, 2010, Kalman Filtering for thermal diffusivity and transient source term mapping from infrared images, Inverse Problems, Design and Optimization Symposium João Pessoa, Paraíba, Brazil, August 23-27, T T = T J P v 1 P P = IP v 1 2 M T T T T M M M a H G a H G a H G P APPLICATIONS

42 APPLICATIONS Estimation of position-dependent transient heat source H.Massard, F. sepulveda, H.R.B. Orlande and O. Fudym, 2010, Kalman Filtering for thermal diffusivity and transient source term mapping from infrared images, Inverse Problems, Design and Optimization Symposium 42 João Pessoa, Paraíba, Brazil, August 23-27, 2010

43 APPLICATIONS Estimation of position-dependent transient heat source H.Massard, F. sepulveda, H.R.B. Orlande and O. Fudym, 2010, Kalman Filtering for thermal diffusivity and transient source term mapping from infrared images, Inverse Problems, Design and Optimization Symposium 43 João Pessoa, Paraíba, Brazil, August 23-27, 2010

44 APPLICATIONS Estimation of position-dependent transient heat source H.Massard, F. sepulveda, H.R.B. Orlande and O. Fudym, 2010, Kalman Filtering for thermal diffusivity and transient source term mapping from infrared images, Inverse Problems, Design and Optimization Symposium 44 João Pessoa, Paraíba, Brazil, August 23-27, 2010

45 APPLICATIONS Estimation of position-dependent transient heat source H.Massard, F. sepulveda, H.R.B. Orlande and O. Fudym, 2010, Kalman Filtering for thermal diffusivity and transient source term mapping from infrared images, Inverse Problems, Design and Optimization Symposium 45 João Pessoa, Paraíba, Brazil, August 23-27, 2010

46 Line Heat Sin APPLICATIONS Estimation of the location of the solidification front and the intensity of a line heat sin Solid Liquid T (r,t) T Interface 0 S(t) Silva, W. B. Orlande, H. R. B.; Colaço, M. J., Fudym, O., 2011, Application Of Bayesian Filters To A One- Dimensional Solidification Problem, 21st Brazilian Congress of Mechanical Engineering, Natal, RN, Brazil. 46

47 APPLICATIONS The mathematical formulation for the solid phase is given as,, 1 Ts r t 1 Ts r t r in 0 r S( t) and t 0 r r r s t (1.a) while the liquid phase is described as, T r, t 1 Tl r t 1 l r in S( t) r and t 0 (1.b) r r r l t T r, t T in r and t 0 (1.c) l i T r, t T in t 0 and r 0 (1.d) l i At the interface between liquid and solid phases, the following conditions must be satisfied T r, t T r, t T in r S( t) and t 0 (1.e) s l m Ts r, t Tl r, t St () s l L in r S( t) and t 0 r r t (1.f) 47

48 APPLICATIONS An analytical solution of this problem can be obtained for this physical problem and it is given by [8]: 2 Q s r 2 Ts r, t Tm Ei Ei 0 r S( t) 4s 4st 2 Ti T m r Tl r, t Ti E ( ) 2 i S t r 4 s st E i l (2.a) (2.b) where the eigenvalues λ and the solidification front S (t) are given by 2 s Q 2 l Ti T e m e l 2 L 4 s E i l 2 s s (3.a) S( t) 2 st (3.b) In the above equations T i is the uniform initial temperature, T m is the melting temperature of the material, L is the s and l are the thermal conductivities of the solid and latent heat of solidification of the material, is the density, liquid phases, respectively, s and l are the thermal diffusivities of the solid and liquid phases, respectively, and 48 T s and T l are temperatures of the solid and liquid phases, respectively.

49 APPLICATIONS Estimation of the location of the solidification front and the intensity of a line heat sin The physical problem defined by Eqs. (1.a-f) was solved analytically, where we used the following data, 2 2 corresponding to solidifying water: Ti 25 C, Tm 0 C, m s , m l , w s 2.22 s s mc, w l 0.61, g, J 3 L 80. The line heat sin was supposed to have a constant value equals to W Q = 50. mc m g m In this wor, the measurements (for the observation model) were obtained at r=0.01 m. The simulated noisy measurements were uncorrelated, additive, Gaussian, with zero mean and constant standard deviation equal to 5% of the maximum temperature. Figures 3.a,b show the transient measurements obtained after applying such constant line heat sin, with and without errors, respectively. Silva, W. B. Orlande, H. R. B.; Colaço, M. J., Fudym, O., 2011, Application Of Bayesian Filters To A One- Dimensional Solidification Problem, 21st Brazilian Congress of Mechanical Engineering, Natal, RN, Brazil. 49

50 APPLICATIONS Estimation of the location of the solidification front and the intensity of a line heat sin Silva, W. B. Orlande, H. R. B.; Colaço, M. J., Fudym, O., 2011, Application Of Bayesian Filters To A One- Dimensional Solidification Problem, 21st Brazilian Congress of Mechanical Engineering, Natal, RN, Brazil.

51 Auxiliary Sampling Importance Resampling (ASIR) Algorithm (Ristic, B., Arulampalam, S., Gordon, N., 2004, Beyond the Kalman Filter, Artech House, Boston) Step 1 For i=1,...,n draw new particles x i from the prior density (x x i -1) and then calculate some characterization of x, given x i -1, as for example the mean i =E[x x i -1]. Then use the lielihood density to calculate the correspondent weights w i =(z i )w i -1 Step 2 Calculate the total weight t= i w i and then normalize the particle weights, that is, for i=1,...,n let w i = t -1 w i Step 3 Resample the particles as follows : Construct the cumulative sum of weights (CSW) by computing c i =c i-1 +w i for i=1,...,n, with c 0 =0 Let i=1and draw a starting point u 1 from the uniform distribution U[0,N -1 ] For j=1,...,n Move along the CSW by maing u j =u 1 +N -1 (j-1) While u j >c i mae i=i+1 Assign sample x j =x i Assign sample w j =N -1 Assign parent i j =i Step 4 j For j=1,...,n draw particles x from the prior density (x x ij -1), using the parent i j, and then use the lielihood density to calculate the correspondent weights w j =(z x j ) / (z ij ) Step 5 Calculate the total weight t= j w j and then normalize the particle weights, that is, for j=1,...,n let w j = t -1 w j The advantage of ASIR over SIR is that it naturally generates points from the sample at -1, which, conditioned on the current measurement, are most liely to be close to the true state. The resampling is based on some point estimate i that characterize (x x i -1), which can be the mean i =E[(x x i -1)] or simply a sample of (x x i -1). If the state evolution model noise is small, (x x i -1) is generally well characterized by i, so that the weights w i are more even and the ASIR algorithm is less sensitive to outliers than the SIR algorithm. On the other hand, if the state evolution model noise is large, the single point estimate i in the state space may not characterize well (x x i -1) and the ASIR algorithm may not be as effective as the SIR algorithm.

52 APPLICATIONS Estimation of the location of the solidification front and the intensity of a line heat sin Bayesian filter Number of Particles (NP) Time RMS error for the solidification front (m) RMS error for the line heat sin intensity (W/m) SIR min. 9x SIR min. 2x SIR min. 1x ASIR min. 7.9x Silva, W. B. Orlande, H. R. B.; Colaço, M. J., Fudym, O., 2011, Application Of Bayesian Filters To A One- Dimensional Solidification Problem, 21st Brazilian Congress of Mechanical Engineering, Natal, RN, Brazil.

53 LOCATION OF THE SOLIDIFICATION FRONTIER (m) LINE HEAT SINK (W/m) APPLICATIONS Estimation of the location of the solidification front and the intensity of a line heat sin ESTIMATED FRONTIER ASIR FILTER NP=100 REAL FRONTIER ESTIMATED FLUX ASIR FILTER NP=100 REAL FLUX TIME (seconds) TIME (seconds) Silva, W. B. Orlande, H. R. B.; Colaço, M. J., Fudym, O., 2011, Application Of Bayesian Filters To A One- Dimensional Solidification Problem, 21st Brazilian Congress of Mechanical Engineering, Natal, RN, Brazil.

54 APPLICATIONS Estimation of Unnown Heat Flux in Natural Convection PHYSICAL PROBLEM The physical problem under picture in this paper involves the transient laminar natural convection of a fluid inside a two-dimensional square cavity. The fluid is initially at rest and at the uniform temperature T c. At time zero, the bottom and top surfaces are subjected to time-dependent heat fluxes q 1 (t) and q 2 (t), respectively. The left and right surfaces are subjected to constant temperatures T c and T h, respectively. The fluid properties are assumed constant, except for the density in the buoyancy term, where we consider Boussinesq s approximation valid. Colaço, M. J., Orlande, H. R. B.; Silva, W. B., Duliravich, G., 2011, Application Of A Bayesian Filter To Estimate Unnown Heat Fluxes In A Natural Convection Problem, ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE 2011, August 29-31, Washington, DC.

55 APPLICATIONS Estimation of Unnown Heat Flux in Natural Convection Colaço, M. J., Orlande, H. R. B.; Silva, W. B., Duliravich, G., 2011, Application Of A Bayesian Filter To Estimate Unnown Heat Fluxes In A Natural Convection Problem, ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE 2011, August 29-31, Washington, DC.

56 APPLICATIONS Estimation of Unnown Heat Flux in Natural Convection Exact Estimated Colaço, M. J., Orlande, H. R. B.; Silva, W. B., Duliravich, G., 2011, Application Of A Bayesian Filter To Estimate Unnown Heat Fluxes In A Natural Convection Problem, ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE 2011, August 29-31, Washington, DC.

57 APPLICATIONS Estimation of Unnown Heat Flux in Natural Convection Exact Estimated Colaço, M. J., Orlande, H. R. B.; Silva, W. B., Duliravich, G., 2011, Application Of A Bayesian Filter To Estimate Unnown Heat Fluxes In A Natural Convection Problem, ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE 2011, August 29-31, Washington, DC.

58 APPLICATIONS Estimation and Control of the Temperature Field in Oil Pipelines The physical problem examined in this wor is based on a critical operational condition involving a pipeline shutdown situation, where the produced fluid is assumed stagnant. An optimal control approach was used to drive the predicted temperatures above a reference level. Heat flux on boundary surface Q (2) (1) Measurement Point (1) R=0 (2) R=1 X-tree Flowline Vianna, F., Orlande, H. R. B., Duliravich, G., 2010, Optimal Heating Control To Prevent Solid Deposits In Pipelines, V European Conference On Computational Fluid Dynamics - ECCOMAS CFD 2010

59 APPLICATIONS The dimensionless mathematical formulation for this onedimensional unsteady heat diffusion problem is given by 2 R, R, 1 R, 2 R R R 0 R 1, 0 R, Bi R R, Q R 1, 0 R, R 1, 0 R, is the dimensionless temperature distribution into the medium. The fluid was considered as homogeneous, isotropic and with constant thermophysical properties.

60 APPLICATIONS Control strategy is in accordance with the optimum control theory for linear problems. The aim of the associated optimal control problem is to find the control inputs u (heat flux on boundary surface) that minimizes the difference between the fluid temperature field and a desired profile. Where u* and x* refer to the steady values of the control input and state variables

61 APPLICATIONS In terms of the linear quadratic regulator problem, the optimal values of the control input are obtained by minimizing the following quadratic cost functional: where the weighting matrices Q and R are symmetric positive definite. The solution to the optimal control problem is the state feedbac control law: where the discrete-time state feedbac gain K is of the form

62 APPLICATIONS Matrix S is the steady state solution to the discrete-time Riccati equation: Thus, the control input can be calculated from the control law as However, when state variables are not directly available for control, an observer (KALMAN FILTER OR PARTICLE FILTER ) was built to estimate the state variables from the input and output variables of the system.

63 APPLICATIONS Simulated measurements with standard deviation of 3 o C in the observation model error

64 Evolution model with standard deviation of 0.01 o C KALMAN FILTER

65 Temperature Predicted with the Kalman Filter and the Exact Temperature Evolution model with standard deviation of 0.01 o C = 0.69

66 Temperature Predicted with the Kalman Filter and the Exact Temperature Evolution model with standard deviation of 0.01 o C = 1.1

67 Evolution model with standard deviation of 1 O C KALMAN FILTER

68 Temperature Predicted with the Kalman Filter and the Exact Temperature Evolution model with standard deviation of 1 o C = 0.61

69 Temperature Predicted with the Kalman Filter and the Exact Temperature Evolution model with standard deviation of 1 o C = 0.8

70 Number of Particles: 200 Evolution model with standard deviation of 1 o C PARTICLE FILTER

71 Temperature Predicted with the Particle Filter and the Exact Temperature Evolution model with standard deviation of 1 o C = 0.69

72 Temperature Predicted with the Particle Filter and the Exact Temperature Evolution model with standard deviation of 1 o C = 1.2

73 Two-dimensional case with Particle Filter Observer (3) (1) (2) Simulated measurements with standard deviation of 3 o C in the observation model error Standard deviation of the evolution model error of 1 o C. Standard deviation of the evolution model error of 3 o C.

74 CONCLUSIONS Kalman filter provides optimal solutions for linear-gaussian evolution-observation models. Particle filter is the most general and robust technique for nonlinear models and/or non-gaussian distributions. ASIR algorithm is faster and requires less particles than the SIR algorithm.

75 ACKNOWLEDGEMENTS Professors Denis Maillet, Philippe Le Masson and Yann Favennec. Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ). METTI sponsors.