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

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## Abstract

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.

## Keywords

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

*e, e*_{c}inputs of fuzzy inference unit

*f*number of measurement points

*C*pollutant concentration

*D*_{x},*D*_{y}diffusion coefficients

*EC*concentration monitoring error

*ES*_{o}pollutant source monitoring error

*L*_{x},*L*_{y}length and width

*S*_{o}pollutant source strength

**B**_{b}sensitive matrix of WRLSA

**H**measurement matrix

**I**unit matrix

**K**gain matrix of Kalman filter

**K**_{b}gain matrix of WRLSA

**M**sensitive matrix of WRLSA

**P**covariance of state estimation errors

**P**_{b}error covariance of source estimation

**Q**process noise covariance

**R**measurement noise covariance

**S**residual variance

**Z**output vector

- \( \overline{\boldsymbol{Z}} \)
sequence of measurement residual

## Greek Symbols

*γ*weighting factor

*σ*_{q}standard deviation of process noises

*σ*_{r}standard deviation of measurement noises

*τ*time

**Ф**state transition matrix

**Ψ**input matrix

## Superscripts

- ^
monitored result in equations

^{T}transpose of a vector or a matrix

*k*the

*k*th time step

## Subscripts

- exa
exact result

- mon
monitored result

## References

- Alifanov, O. M., Artyukhin, E. A., Loginov, S. N., & Malozemov, V. V. (1981). Solution of inverse problems of heat conduction by the method of dynamic filtration.
*Journal of Engineering Physics and Thermophysics, 41*, 1260–1264.Google Scholar - Allen, C., Young, G., & Haupt, S. (2007). Improving pollutant source characterization by better estimating wind direction with a genetic algorithm.
*Atmospheric Environment, 41*, 2283–2289.CrossRefGoogle Scholar - Arystanbekova, N. K. (2004). Application of Gaussian plume models for air pollution simulation at instantaneous emissions.
*Mathematics and Computers in Simulation, 67*, 451–458.CrossRefGoogle Scholar - Chen, T., & Hsu, S. (2007). Input estimation method in the use of electronic device temperature prediction and heat flux inverse estimation.
*Numerical Heat Transfer, 52*(9), 795–815.Google Scholar - Chen, C., Liu, K., & Kehtarnavaz, N. (2016). Real-time human action recognition based on depth motion maps.
*Journal of Real-Time Image Processing, 12*, 155–163.CrossRefGoogle Scholar - Gao, S., Liu, Y., Wang, J., Deng, W., & Oh, H. (2016). The joint adaptive Kalman filter (JAKF) for vehicle motion state estimation.
*Sensors, 16*, 1103.CrossRefGoogle Scholar - Haas Laursen, D. E., Hartley, D. E., & Prinn, R. G. (1996). Optimizing an inverse method to deduce time-varying emissions of trace gases.
*Journal of Geophysical Research-Atmospheres, 101*, 22823–22831.CrossRefGoogle Scholar - Hou, L., Qian, X., Du, B., & Yuan, M. (2017). Optimization of the gas leak monitoring points distributed underground.
*Journal of Loss Prevention in the Process Industries, 47(*95–103.CrossRefGoogle Scholar - Khlaifi, A., Ionescu, A., & Candau, Y. (2009). Pollution source identification using a coupled diffusion model with a genetic algorithm.
*Mathematics and Computers in Simulation, 79*, 3500–3510.CrossRefGoogle Scholar - Liu, X., & Zhai, Z. (2007). Inverse modeling methods for indoor airborne pollutant tracking: literature review and fundamentals.
*Indoor Air, 17*, 419–438.CrossRefGoogle Scholar - Liu, W., Zhang, T., Xue, Y., Zhai, Z. J., Wang, J., Wei, Y., & Chen, Q. (2015). State-of-the-art methods for inverse design of an enclosed environment.
*Building and Environment, 91*, 91–100.CrossRefGoogle Scholar - Lu, J., Kashaev, N., & Huber, N. (2016). Optimization of crenellation patterns for fatigue crack retardation via genetic algorithm and the reduction in computational cost.
*Engineering Failure Analysis, 63(*21–30.CrossRefGoogle Scholar - Lushi, E., & Stockie, J. M. (2010). An inverse Gaussian plume approach for estimating atmospheric pollutant emissions from multiple point sources.
*Atmospheric Environment, 44*, 1097–1107.CrossRefGoogle Scholar - Ma, D., & Zhang, Z. (2016). Contaminant dispersion prediction and source estimation with integrated Gaussian-machine learning network model for point source emission in atmosphere.
*Journal of Hazardous Materials, 311(*237–245.CrossRefGoogle Scholar - Ma, D., Tan, W., Zhang, Z., & Hu, J. (2017). Parameter identification for continuous point emission source based on Tikhonov regularization method coupled with particle swarm optimization algorithm.
*Journal of Hazardous Materials, 325(*239–250.CrossRefGoogle Scholar - Mach, T., Reichel, L., Van Barel, M., & Vandebril, R. (2016). Adaptive cross approximation for ill-posed problems.
*Journal of Computational and Applied Mathematics, 303(*206–217.CrossRefGoogle Scholar - Pan, H., Lü, Z., Lin, W., Li, J., & Chen, L. (2017). State of charge estimation of lithium-ion batteries using a grey extended kalman filter and a novel open-circuit voltage model.
*Energy, 138*, 764–775.Google Scholar - Pazos, F., & Bhaya, A. (2015). Adaptive choice of the Tikhonov regularization parameter to solve ill-posed linear algebraic equations via Liapunov Optimizing Control.
*Journal of Computational and Applied Mathematics, 279(*123–132.CrossRefGoogle Scholar - Richardson, R. R., & Howey, D. A. (2015). Sensorless battery internal temperature estimation using a Kalman filter with impedance measurement.
*IEEE Transactions on Sustainable Energy, 6*, 1190–1199.CrossRefGoogle Scholar - Ristic, B., Gunatilaka, A., & Wang, Y. (2017). Rao–Blackwell dimension reduction applied to hazardous source parameter estimation.
*Signal Processing, 132(*177–182.CrossRefGoogle Scholar - Saidi, M. H., Sajadi, B., & Molaeimanesh, G. R. (2011). The effect of source motion on contaminant distribution in the cleanrooms.
*Energy and Buildings, 43*, 966–970.CrossRefGoogle Scholar - Shankar Rao, K. (2007). Source estimation methods for atmospheric dispersion.
*Atmospheric Environment, 41*, 6964–6973.CrossRefGoogle Scholar - Shih, Y., Chiu, C., & Wang, O. (2007). Dynamic airflow simulation within an isolation room.
*Building and Environment, 42*, 3194–3209.CrossRefGoogle Scholar - Sportisse, B. (2007). A review of current issues in air pollution modeling and simulation.
*Computational Geosciences, 11*, 159–181.CrossRefGoogle Scholar - Thomson, L. C., Hirst, B., Gibson, G., Gillespie, S., Jonathan, P., Skeldon, K. D., & Padgett, M. J. (2007). An improved algorithm for locating a gas source using inverse methods.
*Atmospheric Environment, 41*, 1128–1134.CrossRefGoogle Scholar - Tran, T. H., Pham, D. T., Hoang, V. L., & Nguyen, H. P. (2014). Water pollution estimation based on the 2D transport–diffusion model and the Singular Evolutive Interpolated Kalman filter.
*Comptes Rendus Mécanique, 342*, 106–124.CrossRefGoogle Scholar - Wang, Y., Zhang, R., Zhang, Z., & Wang, F. (2017a). Leakage risk quantitative calculation model and its application for anaerobic reactor.
*Journal of the Taiwan Institute of Chemical Engineers, 77(*152–160.CrossRefGoogle Scholar - Wang, X., Wang, G., Chen, H., & Zhang, L. (2017b). Real-time temperature field reconstruction of boiler drum based on fuzzy adaptive Kalman filter and order reduction.
*International Journal of Thermal Sciences, 113(*145–153.CrossRefGoogle Scholar - Wang, X., Zhang, D., Zhang, L., & Jiang, C. (2018). Real-time thermal states monitoring of absorber tube for parabolic trough solar collector with non-uniform solar flux.
*International Journal of Energy Research, 42*, 707–719.CrossRefGoogle Scholar - Wei, Y., Zhou, H., Zhang, T. T., & Wang, S. (2017). Inverse identification of multiple temporal sources releasing the same tracer gaseous pollutant.
*Building and Environment, 118(*184–195.CrossRefGoogle Scholar - Yang, X., Yang, Z., Yin, X., & Li, J. (2008). Chaos gray-coded genetic algorithm and its application for pollution source identifications in convection–diffusion equation.
*Communications in Nonlinear Science and Numerical Simulation, 13*, 1676–1688.CrossRefGoogle Scholar - Yang, F., Fu, C., & Li, X. (2017). The method of simplified Tikhonov regularization for a time-fractional inverse diffusion problem.
*Mathematics and Computers in Simulation, 144*, 219–234.Google Scholar - Zhang, T., & You, X. (2014). A simulation-based inverse design of preset aircraft cabin environment.
*Building and Environment, 82(*20–26.CrossRefGoogle Scholar