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Dynamic Phase Boundary Estimation using Electrical Impedance Tomography By Umer Zeeshan Ijaz, Control Engineering Lab, Department of Electronic Engineering, Cheju National University, Cheju 690-756, Korea Thesis Defense (Supervised by Professor Kyung Youn Kim) Dated: 13.11.2007

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2 CONTENTS Introduction Electrical Impedance Tomography Boundary Representation –Fourier Coefficients –Front Points Extended Kalman Filter Kinematic Models Interacting Multiple Model Scheme Unscented Kalman Filter Gauss-Newton Unscented Kalman Filter

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3 INTRODUCTION Chemical engineers frequently encounter the flow of a mixture of two fluids in Liquid-gas or liquid-vapor mixtures condensers and evaporators gas-liquid reactors combustion systems transport of some solid materials slurry of the solid particles in a liquid, and pumping the mixture through a pipe Liquid-liquid mixtures in emulsions as well as liquid-liquid extraction. Types of Flows

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4 Electrical Impedance Tomography (EIT) is a imaging modality in which the internal resistivity distribution is reconstructed based on the measured voltages on the surface object. COMPUTER Reconstruction Algorithm Reconstruction Algorithm Interface with Instrument Interface with Instrument V I Concept of electrical impedance tomography WHAT IS EIT?

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5 The forward problem calculates the voltages on the electrodes by using the injected current and assumed resistivity distribution. The inverse problem reconstructs the resistivity distribution by using the voltage measurements on the electrodes. Forward vs. inverse problem for EIT FORWARD SOLVER VS INVERSE SOLVER Inverse Solver Forward Solver An iterative inverse solver

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6 Governing Equation derived from Maxwell Equation Boundary Conditions: Complete Electrode Model MATHEMATICAL MODEL: FORWARD SOLVER Between electrodes, no current crosses the boundary if the impedance outside the imaged volume is much greater than that inside There is an existence of a thin, high-impedance layer beneath electrodes delivering current. This layer may be modelled as the limit of a thin layer of thickness d and impedance z/d as d goes to zero. (use ohms law) Beneath electrodes, neither potential nor the current crossing the boundary is known. Net current crossing the boundary beneath an electrode is equal to the current being delivered to it by tomograph electronics Constraints: For the solution to be unique

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7 FEM DISCRETIZATION OF FORWARD PROBLEM Electrode area Basis function N-Nodes, L-Electrodes, m-elements P-Patterns 1 2 3 4 N 1 L 2 Potential inside: Potential on electrodes:

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8 CURRENT INJECTION PROTOCOL Current frame Trigonometric (L-1 Curent Patterns) Opposite (L/2 Current Patterns) Adjacent (L Current Patterns)

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9 Truncated Fourier Coefficients Approach(Close Boundary) BOUNDARY INTERFACE REPRESENTATION 1/2

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10 BOUNDARY INTERFACE REPRESENTATION 2/2 Front Points Approach (Open Boundary) σ=σ 0 σ=σ 1 A0A0 A1A1 C

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11 Inverse Solver Forward Solver Inverse Solver Forward Solver BOUNDARY INTERFACE FORWARD SOLVER 1/2 Changes required Analytical Jacobians Boundary to Resistivity Profile Mapping (Forward Solver)

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12 s1s1 s2s2 AlAl ArAr σlσl σrσr SlSl SrSr C l (s) NiNi NjNj BOUNDARY INTERFACE FORWARD SOLVER 2/2 (a) description of interface with front points /fourier coefficients b) mesh elements above the interface/inside the target are assigned one conductivity value ; (c) mesh elements below the interface/outside the target are assigned second conductivity value ; (d) mesh elements lying on the interface are assigned area average conductivity values assigned using equation ; and (e) final conductivity values at the end of assignment. (a) (b)(c) (d) (e)

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13 State Space Model Random-walk model Nonlinear measurement equation Linearizing the measurement equation about the predicted mean in the previous step Regularization INVERSE SOLVER

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14 Measurement Update Time Update Jacobian Forward Solver Predefined EXTENDED KALMAN FILTER (Front Points)

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15 Figure 2.14. Reconstructed images for scenario 2 with 3% white Gaussian noise. The solid line represents the true interface and the dotted line represents the estimated interface. EXTENDED KALMAN FILTER (Front Points) Results 3% Noise 10-Front Points, Contrast Ratio of 1:100, Moving every 4 Current Patterns (First two modes of cosine and sine with additional cosine in image reconstruction)

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16 Bubble moving with constant velocityBubble moving with constant acceleration Bubble expanding with constant velocity Bubble expanding with constant acceleration KINEMATIC MODEL (Fourier Coefficients)

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17 Distinguishability can be defined as a measurement ability to differentiate between homogeneous and inhomogeneous conductivities inside the domain. Power distinguishability is defined as the measured power change between the homogeneous and inhomogeneous cases, divided by the power applied in homogeneous case. OPTIMAL CURRENT PATTERN (Front Points) 1. Trigonometric method with first 2 modes of cosine and sine (4 injections; 5 EKF states with repeated use of the first cosine) 2. Opposite method with e1-e9 and e5-e13 pairs (2 injections; 5 states with repeated use of e1-e9, e5-e13, e1-e9) 3. Cross method with e3-e7, e5-e13 pairs (2 injections; 5 states with repeated use of e3-e7, e5-e13, e3-e7) 4. Opposite method with e3-e11, e7-e15 pairs (2 injections; 5 states with repeated use of e3-e11, e7-e15, e3-e11) 5. Opposite method with e3-e11, e7-e15, e5-e13 pairs (3 injections; 5 states with repeated use of e3-e11, e7-e15). 1 2 3 4 5 1% Noise

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18 a) Bubble moving with constant velocity b) Bubble expanding with constant velocity c) Bubble moving with constant acceleration d) Bubble expanding with constant acceleration Solid Line: True Boundary Dotted Line: Estimated Boundary Solid Line: Kinematic Model Dotted Line: Random-Walk Model KINEMATIC MODEL RESULTS (Fourier Coefficients) 6-Fourier Coefficients, Contrast Ratio of 1:10 6, Moving every current Pattern

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19 T.U M.U EKF1 T.U M.U EKF2 T.U M.U EKF3 IMM SCHEME (Fourier Coefficients) 1/3 *As error decreases, modelling probability increases

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20 IMM SCHEME (Fourier Coefficients) 2/3 EKF2EKF3 EKF1 Transition Probability Predefined Mixing of estimates and error covariances

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21 IMM SCHEME (Fourier Coefficients) 3/3 Interacting/Mixing of the Estimates Filter 1Filter M Linearization State Estimation Combination Model Probability Update One-cycle flow diagram of the inverse solver with the IMM scheme. EKF1 EKF2 EKF3

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22 EKF1EKF2EKF3IMM IMM SCHEME RESULTS (Fourier Coefficients) 6-Fourier Coefficients, Contrast Ratio of 1:10 6, Moving after 8 current patterns

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23 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++ ++ + + ++ ++ + + ++ ++ + + ++ ++ + + ++ ++ + + ++ + + + + ++ ++ + + + ++ + + + ++ + + + + + + + +++ ++ + + + ++ + + + ++ + + + ++ + + + + ++ + + ++ + + + + + + + +++ + +++ + + + +++ + ++ ++ + ++ ++ ++ + + + + + ++ ++ +++ + + + + ++ ++ +++ + + + + + ++ ++ + +++ Actual (Sampling) true mean mean covariance Linearized (EKF)UT UT mean UT covariance sigma points transformed sigma points EKF mean EKF covariance UNSCENTED TRANSFORM An example of unscented transform for mean and covariance propagation: a) actual; (b) first-order linearization (EKF); and (c) unscented transform (a)(b)(c)

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24 UNSCENTED KALMAN FILTER (1/4) Generate 2n+1 sigma points where n is the size of augmented vector Each point is the augmented vector Run the state equation Calculate predicted mean and covariance State Space Model:

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25 Run the measurement equation and find the mean Time update complete Create covariance matrices The sigma points should move towards the mean and at the same time, the sigma points on x domain should move towards the mean UNSCENTED KALMAN FILTER (2/4)

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26 Calculate the gain and update the estimates and error covariance matrices UNSCENTED KALMAN FILTER (3/4) Actual measurement True value Define weights where Composite scaling parameter Spread of sigma points, usually1e- 3 Usually zero Usually 2 for Gaussian distribution Measurement update complete

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27 + - State Equation Nonlinear Measurement Equation [FEM Forward Solver] Weighted MeanWeighted Covariance Kalman Gain k=k+1 k UNSCENTED KALMAN FILTER (4/4) Block diagram of unscented Kalman filter for phase boundary estimation

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28 solid line : true, dotted line : EKF, dashed line : UKF Phantom Plastic Target UKF RESULTS (Fourier Coefficients) 32 Electrodes, 6-Fourier Coefficients, Contrast Ratio of 1:10 6, Moving after 6 current patterns

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29 2% Noise 3% Noise Rippled surface UKF RESULTS (Front Points) 10-Front Points, Contrast Ratio of 1:100, moving every current pattern

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30 State Equation Gauss-Newton Measurement Update GAUSS-NEWTON UNSCENTED KALMAN FILTER Offline Online State Space Model

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31 2% Noise 1% Noise 3% Noise GNUKF RESULTS (Front Points)

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32 -Analytical Jacobian used -successful till 16 Frontpoints -Contrast ratio of 1:10000 -3% Relative Noise -Current patterns reduced to 4 / target remains static with 16 electrodes configuration based on distinguishability analysis for EKF -Extended Kalman Filter and Unscented Kalman filter (recent) formulation for online monitoring -Gauss Newton Unscented Kalman filter formulation for improvement over unscented Kalman filter -With Unscented Kalman Filter and Gauss Newton Unscented Kalman Filter, image reconstruction using 1 current pattern is also possible. Front points (open boundary) Fourier coefficients (close boundary) -Analytical Jacobian used -6 coefficients to represent an elliptic object, can go for more, however, higher coefficients are quite sensitive -Contrast ratio of 1:1000000 -3% Relative Noise -Current patterns reduced to 6 / target remains static in experiments with 32 electrodes configuration. -Extended Kalman Filter and Unscented Kalman filter (recent) formulation for online monitoring -Interacting Multiple Model Scheme for time-varying process noises -Kinematic models (velocity, acceleration) done for movement of air bubbles, void fractions RESEARCH MILESTONES

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33 Any Questions?

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34 APPENDIX: Derivation of Jacobian 1/10 In some cases, the voltages are measured only at some selected electrodes, not every electrode. Also, the selected electrodes may be different at each current pattern. The measured voltages at the measurement electrodes can be obtained as where, is the number of the measurement electrodes and is the measurement matrix. The element is set to 1 if the -th electrode is measured at the -th current pattern and otherwise set to zero. Furthermore, can be extracted directly from by introducing the extended mapping matrix and where. Therefore, we have where the extended measurement matrix is defined as If the pseudo-resistance matrix defined as or is given we can calculate the Jacobian matrix. The pseudo-resistance matrix can be easily obtained during the solution of the system equation or where and

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35 APPENDIX: Derivation of Jacobian 2/10 Jacobian: Front Points Approach

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36 APPENDIX: Derivation of Jacobian 3/10 Since we are considering the stratified flow of two immiscible liquids therefore, the matrix B will be

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37 Assuming that the interface is represented by a set of linear piecewise interpolation functions:, unit pulse defined for Any small perturbation of results in small perturbation in and in where APPENDIX: Derivation of Jacobian 4/10

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38 APPENDIX: Derivation of Jacobian 5/10 Considering the interface for mesh crossing elements where For a small perturbation in only and will change The function can be expanded about the interface Finally, we have

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39 Five types of interface-crossing elements in case of an arbitrarily small perturbation of in. There are five types of interface-crossing elements when is perturbed by an arbitrarily small perturbation of. Assume that there are only two intersections of the interface and the mesh faces and the intersections and where. Recalling that the integration for each type will be evaluated as are denoted as is constant in a certain mesh, TYPE 1: TYPE 2: TYPE 3: TYPE 4: TYPE 5: APPENDIX: Derivation of Jacobian 6/10

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40 Fourier Coefficients Approach The derivative of the stiffness matrix with respect to the coefficient is APPENDIX: Derivation of Jacobian 7/10

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41 In order to obtain the Jacobian, now, let us consider the evaluation of the expression We define a new coordinate system where is the positively oriented coordinate along the closes curve, and is the coordinate outward normal from the region The perturbed boundary will be Therefore, APPENDIX: Derivation of Jacobian 8/10

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42 The Jacobian for the transformation of the coordinate will be The function can be expanded about the boundary We have APPENDIX: Derivation of Jacobian 9/10

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43 In this, is evaluated at the boundary. When differentiating with respect to, that is perturbing, we have and.On the other hand when differentiating with respect to, we have and. Finally, the derivative of the matrix with respect to the coefficients becomes where denotes the set of elements crossing If, and constant in each element, we have APPENDIX: Derivation of Jacobian 10/10

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