Real-time Monitoring of Pollutant Diffusion States and Source Using Fuzzy Adaptive Kalman Filter

  • Xudong WangEmail author
  • Daqian Zhang
  • Liying Chen


An inverse analysis method for the real-time monitoring of pollutant diffusion is developed based on fuzzy adaptive Kalman filter (FAKF) coupled with weighted recursive least squares algorithm (WRLSA). In the monitoring process, the discrete diffusion states equation is established first. Then, the FAKF is adopted to realize the precise monitoring of the pollution diffusion states while the WRLSA is used to monitor the pollutant source in real time. Finally, the simulations are presented to validate the effectiveness of the technique, which shows that this technique has wide applications in situations with several different kinds of sources and measurement noises. Besides, the results demonstrate the strong robustness of this method to have great monitoring performance.


Diffusion state Fuzzy adaptive Kalman filter Inverse analysis Pollutant source Real-time monitoring 


e, ec

inputs of fuzzy inference unit


number of measurement points


pollutant concentration


diffusion coefficients


concentration monitoring error


pollutant source monitoring error


length and width


pollutant source strength


sensitive matrix of WRLSA


measurement matrix


unit matrix


gain matrix of Kalman filter


gain matrix of WRLSA


sensitive matrix of WRLSA


covariance of state estimation errors


error covariance of source estimation


process noise covariance


measurement noise covariance


residual variance


output vector

\( \overline{\boldsymbol{Z}} \)

sequence of measurement residual

Greek Symbols


weighting factor


standard deviation of process noises


standard deviation of measurement noises




state transition matrix


input matrix



monitored result in equations


transpose of a vector or a matrix


the kth time step



exact result


monitored result


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and EnvironmentSoutheast UniversityNanjingPeople’s Republic of China
  2. 2.Key Laboratory of Low-Grade Energy Utilization Technologies and Systems of Ministry of Education,School of Power EngineeringChongqing UniversityChongqingPeople’s Republic of China
  3. 3.Institute of Innovation and EntrepreneurshipChengdu Technological UniversityChengduPeople’s Republic of China

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