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Licensed Unlicensed Requires Authentication Published by De Gruyter May 17, 2019

A State Estimation Method Based on Integration of Linear and Extended Kalman Filters

M. Farsi and M. Dehghan Manshadi


The Linear Kalman Filter (LKF) and Extended Kalman Filter (EKF) as correction-prediction observers are two common model-based filters, which are widely used to estimate the unmeasurable states and parameters in the chemical processes. The main disadvantages of LKF and EKF are low accuracy and reliability of estimation and high computational time, respectively. The main object of this work is the modification of conventional Kalman filter to cover the disadvantages of LKF and EKF. The proposed method is planned based on the systematic updating the Jacobian matrix of estimator applied on the nonlinear systems. In this regard, a linearity index is defined based on the characteristics of outputs to determine switching time between LKF and EKF to avoid unnecessary updating the Jacobian matrix. The performance of the proposed method is compared with the LKF and EKF considering four chemical benchmarks as the nonlinear state space models. Altought EKF and proposed filter present similar precision to estimate unmeasurable states, the proposed method has a lower computational time.



Outlet cross-sectional area of tanks


Cross section area of ith reactor


Linearized matrix of measurements function

CAmol lit1

Concentration of A in the reactor

CBmol lit1

Concentration of B in the reactor

CPJ kg1K1

Specific Heat capacity of mixture

CpjJ kg1K1

Specific Heat capacity of coolant


Disturbance in the system

EJ mol1

Activation energy of reaction




Flowrate of jacket


The acceleration due to gravity


Pump constants


Kalman gain matrix

Kilit mol1s1

Reaction pre-exponential factor


Recurrence relationship of fourth-order Runge-Kutta




Adaptation gain matrix


Covariance matrix of model noise


Covariance matrix of measurement noise

RJ mol1K1

Universal gas constant


Slope of the state variations

t (sec)





Input of the system


Heat transfer coefficient


Percent of maximum speed in pump i




States of the system


Measured states of the system (outputs)

Greek letter

The threshold for second derivative


Process-noise vector


Measurement-noise vector

ΔHJ mol1

Heat of reaction





Artificial Neural Networks


Extended Kalman Filter


Linear Kalman Filter


Neuro Extended Kalman Filter


Neuro High Gain Observer


Unscented Kalman Filters


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Received: 2018-09-03
Revised: 2019-04-29
Accepted: 2019-04-30
Published Online: 2019-05-17

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