The Reduced Rank of Ensemble Kalman Filter to Estimate the Temperature of Non Isothermal Continue Stirred Tank Reactor

Kalman filter is an algorithm to estimate the state variable of dynamical stochastic system. The square root ensemble Kalman filter is an modification of Kalman filter. The square root ensemble Kalman filter is proposed to keep the computational stability and reduce the computational time. In this paper we study the efficiency of the reduced rank ensemble Kalman filter. We apply this algorithm to the non isothermal continue stirred tank reactor problem. We decompose the covariance of the ensemble estimation by using the singular value decomposition (the SVD), and then we reduced the rank of the diagonal matrix of those singular values. We make a simulation by using Matlab program. We took some the number of ensemble such as 100, 200 and 500. We compared the computational time and the accuracy between the square root ensemble Kalman filter and the ensemble Kalman filter. The reduced rank ensemble Kalman filter can’t be applied in this problem because the dimension of state variable is too less.


Introduction
Kalman filter is an algorithm to estimate the state variable of the stochastic dynamical linear system.This algorithm combines the mathematical model with the measurement data (Lewis,[6]).Kalman filter has been applied in various problems such as the estimation of the water level in the river, wave of ocean and the tide.There are many modification of Kalman filter algorithm, such as the ensemble Kalman Filter (Burger et al. [3]), the reduced rank square root covariance filter (Verlaan [8]), the square root ensemble Kalman filter (Evensen,[4]) and the variance reduced ensemble Kalman filter (Heemink,[5]).These modifications have been done to avoid the convergence of algorithm, to reduce the computation time, to decrease the error of estimation and other.In the Ensemble Kalman filter, it is generated an ensemble value as initial estimation of state variable and an ensemble measurement data based on real measurement data (Burger et al., [3].So in the ensemble Kalman filter, we need more computational time than the Kalman filter.The square root ensemble Kalman filter is proposed to keep the computational stability and the variance reduced ensemble Kalman filter is proposed to keep the computational stability and reduce the computational time of square root ensemble Kalman filter (Heemink,[5]).In this paper we study the efficiency of the reduced rank ensemble Kalman filter.We apply this algorithm in non linear dynamic stochastic system such as the non isothermal continue stirred tank reactor.

Methods
The mathematical model of non isothermal continue stirred tank reactor is (Qu,[7]) In this paper we estimate the concentration of reactance, , temperature of tank reactor, and the temperature of cooling jacket of reactor, if we can measure the concentration of reactance, , and temperature of tank reactor, or we just can measure temperature of tank reactor, .Because the non isothermal continues stirred tank reactor is a non linear dynamic system, then we estimate those three parameters by using the square root Ensemble Kalman filter.
The ensemble Kalman filter and the square root ensemble Kalman filter are modified of Kalman filter.Kalman filter is an algorithm to estimate the state variable of the stochastic dynamical linear system.This algorithm combines the mathematical model with the measurement data (Lewis,[6]).Kalman filter has been applied in various problems such as the estimation of the water level in the river, wave of ocean and the tide.There are many modification of Kalman filter algorithm, such as the ensemble Kalman Filter (Evensen, [4]), the reduced rank square root covariance filter (Verlaan, [8]), the square root ensemble Kalman filter (Evensen, [4]) and the variance reduced ensemble Kalman filter (Heemink,[5]).These modifications have been done to avoid the convergence of algorithm, to reduce the computation time, to decrease the error of estimation and other.In the Ensemble Kalman filter, it is generated an ensemble value as initial estimation of state variable and an ensemble measurement data based on real measurement data (Evensen [4]).So in the ensemble Kalman filter, we need more computational time than the Kalman filter.The square root ensemble Kalman filter is proposed to keep the computational stability and the variance reduced ensemble Kalman filter is proposed to keep the computational stability and reduce the computational time of square root ensemble Kalman filter (Heemink,[5]).
At the previous research, we had applied the ensemble Kalman filter in the non isothermal continue stirred tank (Baehaqi,[2]).We compared the accuracy and efficiency between the Ensemble Kalman Filter with the Unscented Kalman filter.The ensemble Kalman filter need less computational time than the unscented Kalman filter, but the unscented Kalman filter more accuracy than the ensemble Kalman filter.We also applied the reduced rank ensemble Kalman filter in the diffusion problem (Albab,[1]).In that research, we reduced 30 rank of covariance from 100.We compare the computational time and efficiency between the ensemble Kalman filter, the square root ensemble Kalman filter and the reduced rank ensemble Kalman filter.From the simulation we obtained that the square root ensemble Kalman filter need more computational time than others, the reduced rank ensemble Kalman filter has more accurate than others but need more computational time than the ensemble Kalman filter.So in this paper, we combined those research, we estimate the variable state of non isothermal continue stirred tank by using the reduced rank ensemble Kalman filter.
The square root ensemble Kalman filter is a modification of ensemble Kalman filter algorithm.In this algorithm we write the covariance error of ensemble Kalman filter as square root matrix.The square root ensemble Kalman filter is usually more stable than the ensemble Kalman filter.Suppose, it is given stochastic dynamic system and measurement equation Let, we want estimate the state variable based on the measurement data by using the square root ensemble Kalman filter, then we use the algorithm of square root ensemble Kalman filter as follows:

Initial Estimation
Based on initial estimation , where has normal Gaussian distribution with mean ̅ and covariance , we generated the ensemble of initial estimation ̂ with mean ̅ and covariance ( ̂ ( ̅ ))

Prediction Step
We predict the state ̂ based on the previous state ̂ and mathematical model of system (non isothermal continuous stirred tank reactor).Substitute the ensemble of estimation ̂ into eq.( 2) Mean of ensemble estimation: ̅ ( ̂ ) Error of ensemble ̃ ̂ ̅

Correction Step
We use measurement data to make correction on the prediction state ̂ .Generated the ensemble measurement data from eq. (3): Define ̃ ; and Matrix is a square root of covariance matrix . .

The ensemble correction estimation
We decomposed the matrix by using the singular value decomposition: [ ] ( ) Define matrix , then decompose matrix , [ ] ( ).The errors of ensemble estimations are ̃ ̃ ( ) The ensembles of estimations are Estimation state variable is mean of the ensemble estimation Finally, we substitute Eq. ( 5) into Eq.( 4) in the prediction step and continue with the correction step such that we get the state estimation ̂ , in time step .

Result and Discussion
Here, we make a simulation using Matlab program.
We took some the number of ensemble such as 100, 200 and 500.We compare the computational time and the accuracy between the reduced rank ensemble Kalman filter, the square root ensemble Kalman filter and the ensemble Kalman filter.
Before we applied the reduced rank ensemble Kalman filter, we rewrite Eq. ( 1) as a discrete time non linear dynamic stochastic system Define [ ] and the eq.( 5) has a same form as eq.( 2).To make a correlation between the measurement data and the state variable which we will estimate, we define the measurement equation as stated in Eq. ( 3 Because the model, Eq.( 5) is a non linear system, we estimate state variable [ ] by using the ensemble Kalman filter, the square root ensemble Kalman filter and the reduced rank ensemble Kalman filter.
In the reduced rank ensemble Kalman filter, the rank reducing is applied in the square root inverse diagonal matrix D in Eq. ( 4) has same dimension with the dimension of state variable [ ] , that is three.Therefore, we can't apply the reduced rank ensemble Kalman filter in this non isothermal stirred tank problem.
Here we take the ensemble estimation are 100, 200 and 500.Figure 1a, 1b. Figure 2a, 2b.represented the estimation result by taking ensemble 100, matrix measurement [ ], that means, based on the data of concentration, and the tank temperature T , we estimate the concentration, the temperature of tank and the temperature of cooling jacket.Figure 1a, 1b.represent the Ensemble Kalman Filter (EnKF) and Figure 2a, 2b.represent the Square root Ensemble Kalman Filter (SQRT EnKF).The absolute error of is 0.0969, T is 0.0002 and is 0.002 for the square root ensemble Kalman filter, otherwise is 0.0414, T is 0.0001 and is 0.0008 for the ensemble Kalman filter.The computational time is 0.5781 sec and 0.56 sec for the square root ensemble Kalman filter and the ensemble Kalman filter respectively (Table 1).The others result of simulation is represented in Table 2.  Table 1 shows the SQRT EnKF need more computational time than the EnKF and the SQRT EnKF is not more accurate than the EnKF.From our simulation, we found that the data of temperature cooling jacket can't use to estimate the concentration and the temperature.In general cases, the number of ensemble influence the accuracy of estimation result, that means more the number of ensemble give more accurate the estimation, but in this problem the number of ensemble is not influence the accuracy of estimation.In SQRT EnKF, for [ ], the variable T has same error

Conclusion
From the simulation result and the analysis we conclude that: The reduced rank ensemble Kalman filter can't be applied in the non isothermal stirred tank problem, because the dimension of state variable is too less.The ensemble Kalman filter needs less computational time than the square root ensemble Kalman filter.The square root ensemble Kalman filter is not more accurate than the ensemble Kalman filter.The number of ensemble in this case is not influence the accuracy of estimation result.The type of measurement matrix or the type of variable that can be measured is not influence the accuracy of the estimation result

Table 1 .
The absolute error and the computational time

Table 2 .
We know that the variable has the largest error than others The measurement matrix

Table 2 .
The error estimation with different measurement matrix H absolute error.For further research, we can extend the application of this algorithm to more complicated problem, such as the stirred tank problem with more dimensions or other non linear stochastic dynamical system to know the influence the number of ensemble and the measurement matrix respect to the accuracy of estimation result.