Estimation of River Current Using Kalman Filter Finite Element Method. Abstract

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1 Kawahara Lab (February, ) Chuo Univ. JAPAN Estimation of River Current Using Kalman Filter Finite Element Method Shin-ichiro SEKIGUCHI Department of Civil Engineering, Chuo University, Kasuga --7, Bunkyo-ku, Tokyo -8, JAPAN d@educ.kc.chuo-u.ac.jp Abstract This paper presents an estimation technique using the Kalman filter finite element method ( KF- FEM ). Estimation value that is not observed is estimated by KF-FEM, which is the combination method of the Kalman filter and the finite element method. The Kalman filter is employed for the solution of time series analysis and the finite element method is space analysis. In this paper the shallow water equation is applied as the state equation. The flow in Arakawa river in Japan has been computed as an actual model. Keywords : Kalman Filter, Finite Element Method, Shallow Water Equation, INTRODUCTION Arakawa River, Boundary Condition The Kalman filter is a linear system, which is applied to the state space and orthogonal projection methods invented by R.E.Kalman and R.S.Bucy. The linear system is asymptotically stable with controllability and observability. There are a lot of physical quantities which change irregularly on the earth. It is impossible to forecast the value completely in the future of these irregular changes. On the other hand, obtaining the estimation value in the future according to a past observation value of the time series becomes possible by applying a filtering theory. Its most immediate applications are the control of complex dynamic systems such as continuous manufacturing processes, aircraft, ships, and spacecraft. In order to control a dynamic system, the Kalman filter theory must be introduced. For these applications, it is not always possible to measure every variable control. The Kalman filter provides a means for inferring the missing information from indirect measurements. The Kalman filter is used for predicting the likely future courses of dynamic systems that people are not likely to control, such as the river flow at the time of flood, the trajectories of celestial bodies, and the prices of traded commodities. As the numerical example, this paper presents the estimation of the river current. The Arakawa river is analyzed to obtain the performance of the KF-FEM. The actual data obtained on September th in at two measured points are used. Velocity at one point is estimated based at observation data on another point. The estimation value compared with the observation data.

2 THE KALMAN FILTER. State Space Model The Kalman filter is based on a set of two systems. The system equation can be expressed state of the phenomena. The observation equation is dependent on the observation data and measuring points. System equation is as follows ; and observation equation is x k+ = F k x k + G k w k () y k = H k x k + v k () where x k is state vector at time k, F k is state transition matrix which represents the finite element equation, G k is driving matrix and w k is a system noise, and v k is observation vector at time k, H k is observation matrix and v k is an observation noise, respectively. System noise w k is assumed : E{w k } = () and observation noise v k is cov{w k,w j } = E{w k,w T j } = Q k δ kj () E{v k } = () with cov{v k,v j } = E{v k,v T j } = R k δ kj () where δ kj is the Kronecker s delta function. δ kj = cov{w k,v j } = (7) { k = j otherwise. Assumption The optimal estimate ˆx k is the average of x k giving the observation data Y k, The covariance P k is written as follows ; ˆx k = E{x k Y k } (8) P k = cov{x k Y k } = E{(x k ˆx k )(x k ˆx k ) T } (9)

3 where P k is called estimated error covariance. The estimate x k is the average of x k giving the observation data Y k, The covariance Γ k is written as follows ; x k = E{x k Y k } () Γ k = cov{x k Y k } where Γ k is called predicted error covariance. The whole process can be defined by those equations. = E{(x k x k )(x k x k )T } (). Formulation The Bayes rule is shown as follows ; P(x k Y k ) = P(y k x k )P(x k Y k ) P(y k Y k ) () where Optimal estimated value ˆx k, Kalman-gain K k, estimated error covariance P k and predicted error covariance Γ k+ are derived in the following forms. ˆx k = x k + K k (y k H k x k) () K k = Γ k Hk T (R k + H k Γ k Hk T ) () P k = (I K k H k )Γ k () Γ k+ = F k P k Fk T + G kq k G T k () where Q k is system error covariance and R k is observation error covariance. FINITE ELEMENT METHOD. Basic Equation In this paper, basic equation is employed for the liner shallow water equation as follows ; u i + g(η + h + z),i Al(u i,j + u j,i ) = (7) η + {(η + h)u i },i = (8) where u i is water velocity of X and Y direction, η is water elevation, h is water depth, g is the gravitational acceleration, Al is the coefficient of kinematic eddy viscosity, and z is bed elevation, respectively.

4 . Boundary Condition On the boundary Γ, boundary conditions are given as follows ; u i = û i on Γ u u n = u i n i = û n on Γ n whereˆmeans the specified value on the boundary and n i denotes the direction cosines of the unit outward normal of the boundary, respectively.. Finite Element Equation The discretization in space direction is carried out applying the Galerkin method. The velocity and water elevation on a triangle element using linear interpolation function. For the weighting function the same function as the interpolation function is employed. The explicit Euler method is applied to the discretization in time direction. The finite element equation can be derived as follows ; Mu n+ Mv n+ = { M t(alh xx + AlH yy )}u n talh yx v n tgs x η n tg(h + z)s x (9) = talh xy u n + { M t(alh xx + AlH yy )}v n tgs y η n tg(h + z)s y () Mη n+ = ths x u n ths y v n + Mη n () where M is the selective lumping coefficient and written as ; M = e M + ( e)m () in which e is the lumping parameter adjusting the stability of computation. THE KALMAN FILTER FINITE ELEMENT METHOD. State Transition Matrix F k The finite element equation is applied to the state transition matrix in Kalman filter. From the finite element equations eqs. (9)-(), the state transition matrix is given as ; u v η n+ = M { M t(alh xx + AlH yy )} M talh yx M tgs x M talh xy M { M t(alh xx + AlH yy )} M tgs y M ths x M ths y M M tg(h + z) S x S y u v η n () where the coefficient matrix that is the part of the underline is represented by F k. This matrix is stationary and independent of time series. The other part is represented by Z n. This part is the section of bed elevation.

5 . Algorithm The algorithm of the Kalman filter finite element method is written as follows.. Initial value setting Γ = V ˆx = x. Calculate the Kalman-gain K k = Γ k H T k (R k + H k Γ k H T k ). Calculate the estimated error covariance P k = (I K k H k )Γ k. Calculate the predicted error covariance Γ k+ = F k P k F T k + G k Q k G T k. Convergence criterion if tr[p k+ ] - tr[p k ] ε then go to else go to. Calculate the estimate value x n = F n ˆx n + Z n 7. Calculate the optimal estimated value ˆx n = x n + K n(y n H n x n ) NUMERICAL EXAMPLE. Arakawa River Arakawa river is located in Tokyo in Japan. It is important to regulate the flood flow of the river. Sometimes good observation data can t be obtained because of bad weather, or personal error, etc. In this paper, non observed value is computed using the Kalman filter finite element method. The flow of Arakawa river can be estimated. The total length of Arakawa river is about 7km. The length of analytical area was taken to be 8km. This distance was decided because of the position of the observation points. The observation data obtained on September th in are used. The observation point is set entrance of analytical area.

6 . Finite Element Mesh As numerical examples, Arakawa river current is analyzed by the Kalman filter finite element method. The observation data are measured on velocity and water elevation at two points on the Arakawa river. The finite element mesh and position of observation points are shown in Figure-. Total number of nodes are and elements are 77, respectively, Figure : Finite Element Mesh.... Velocity (m/sec). Velocity (m/sec)... Sep (time) Sep (time) Inflow-Velocity Outflow-Velocity Water Depth (m) Water Depth (m) Sep (time) Sep (time) Inflow-Water Depth Outflow-Water Depth Figure : ( by Arakawa River Upper Reaches Works Office )

7 . Parameter In this paper, time increment t is.(s), gravitational acceleration is 9.8(m/s ), lumping parameter is.9, observation error covariance R is. and diagonal of system error covariance Q is., respectively. NUMERICAL RESULT The estimation value was computed at point No.. The comparison between the estimation value and the observation data is represented in Figure, Estimation Value Estimation Value... Velocity (m/sec). Water Depth (m)... Sep (time) Sep (time) Velocity at Point. Water Depth at Point. Figure : and Estimation Value 7 CONCLUSION The estimation value of water elevation has been obtained in good agreement with the observed data. There is a little discrepancy in velocity. It is important how to set boundary condition. The result changes fairly depending on how to set the boundary condition. Some improvement must be followed. First, the algorithm of this paper requires a lot of computational time. Second, there are few observation points relative to the computational area. Third, the finite element mesh is too large. These problems will be future work. It s necessary to improve the algorithm of the Kalman filter finite element method. This problem is very important and difficult. In this research, The method was applied to the algorithm, which is separated into two parts. It s necessary to increase the numbers of observation points. References. J.Matsumoto, T.Umetsu and M.Kawahara : Shallow Water and Sediment Transport Analysis by Implicit FEM, Journal of Applied Mechanics, (). R.Suga, K.Yonekawa and M.Kawahara : Estimation of Tidal Current using Kalman Filter Finite Element Method with AIC, Second M.I.T.Conference on Computational Fluid and Solid Mechanics,(). R.E.Kalman and R.S.Bucy : New Results in Linear Filtering and Prediction Theory, Trans. ASME,J. Basic Eng., vol.8d,no.,pp.9-8,(9) 7