Abstract
This paper addresses the state estimation for a class of nonlinear time-varying stochastic systems with both uncertain dynamics and unknown measurement bias. A novel extended state based Kalman filter (ESKF) algorithm is developed to estimate the original state, the uncertain dynamics and the measurement bias. It is shown that the estimation error of the proposed algorithm is bounded in the mean square sense. Also, the estimation of the measurement bias asymptotically converges to its true value, such that the influence of measurement bias is eliminated. Furthermore, the asymptotic optimality of the estimation result is proved while the uncertain dynamics approaches to a constant vector. Finally, a simulation study for harmonic oscillator system model is provided to illustrate the effectiveness of proposed method.
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References
Luenberger, D. G. (1966). Observers for multivariable systems. IEEE Transactions on Automatic Control, 11(2), 190–197.
Kalman, R. E. (1959). A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82(1), 35–45.
Han, J. (1995). A class of extended state observers for uncertain systems. Control and Decision, 10(1), 85–88.
Gao, Z. (2003). Scaling and bandwidth-parameterization based controller tuning. In Proceedings of the American Control Conference (pp. 4989–4996). Denver, CO, USA.
Xue, W., Huang, Y., & Gao, Z. (2016). On ADRC for non-minimum phase systems: canonical form selection and stability conditions. Control Theory and Technology, 14(3), 199–208.
Chen, S., Bai, W., Hu, Y., Huang, Y., & Gao, Z. (2020). On the conceptualization of total disturbance and its profound implications. Science China Information Sciences, 63(2), 129201.
Huang, Y., & Han, J. (2000). Analysis and design for the second order nonlinear continuous extended states observer. Science Bulletin, 45(21), 1938–1944.
Zhao, Z., & Guo, B. (2018). A novel extended state observer for output tracking of MIMO systems with mismatched uncertainty. IEEE Transactions on Automatic Control, 63(1), 211–218.
Zheng, Q., Gaol, L.Q., & Gao, Z. (2007). On stability analysis of active disturbance rejection control for nonlinear time-varying plants with unknown dynamics. In Proceedings of the 46th IEEE Conference on Decision and Control (pp. 3501–3506). New Orleans, LA, USA.
Yang, X., & Huang, Y. (2009). Capabilities of extended state observer for estimating uncertainties. In Proceedings of the American Control Conference (pp. 3700–3705). St Louis, MO, USA.
Sira-Ramírez, H., Linares-Flores, J., García-Rodríguez, C., & Contreras-Ordaz, M. A. (2014). On the control of the permanent magnet synchronous motor: an active disturbance rejection control approach. IEEE Transactions on Control Systems Technology, 22(5), 2056–2063.
Huang, Y., & Xue, W. (2014). Active disturbance rejection control: methodology and theoretical analysis. ISA Transactions, 53(4), 963–976.
Zhu, E., Pang, J., Sun, N., Gao, H., Sun, Q., & Chen, Z. (2014). Airship horizontal trajectory tracking control based on active disturbance rejection control (ADRC). Nonlinear Dynamics, 75(4), 725–734.
Qiu, D., Sun, M., Wang, Z., Wang, Y., & Chen, Z. (2014). Practical wind-disturbance rejection for large deep space observatory antenna. IEEE Transactions on Control Systems Technology, 22(5), 1983–1990.
Zheng, Q., & Gao, Z. (2018). Active disturbance rejection control: some recent experimental and industrial case studies. Control Theory and Technology, 16(4), 301–313.
Chen, J., Ren, B., & Zhong, Q.-C. (2016). UDE-based trajectory tracking control of piezoelectric stages. IEEE Transactions on Industrial Electronics, 63(10), 6450–6459.
Sun, L., Li, D., Zhong, Q., & Lee, K. Y. (2016). Control of a class of industrial processes with time delay based on a modified uncertainty and disturbance estimator. IEEE Transactions on Industrial Electronics, 63(11), 7018–7028.
Soffker, D., Yu, T., & Mller, P. C. (1995). State estimation of dynamical systems with nonlinearities by using proportional-integral observer. International Journal of Systems Science, 26(9), 1571–1582.
Sira-Ramírez, H. (2018). From flatness, GPI observers, GPI control and flat filters to observer-based ADRC. Control Theory and Technology, 16(4), 249–260.
Schrijver, E., & van Dijk, J. (2002). Disturbance observers for rigid mechanical systems: equivalence, stability, and design. Journal of Dynamic Systems, Measurement, and Control, 124(4), 539–548.
Hu, Q., Li, B., & Qi, J. (2014). Disturbance observer based finite-time attitude control for rigid spacecraft under input saturation. Aerospace Science and Technology, 39, 13–21.
Yang, J., Li, S., Sun, C., & Guo, L. (2013). Nonlinear-disturbance-observer-based robust flight control for airbreathing hypersonic vehicles. IEEE Transactions on Aerospace and Electronic Systems, 49(2), 1263–1275.
Li, S., Yang, J., Chen, W.-H., & Chen, X. (2014). Disturbance Observer-based Control. Boca Raton: CRC Press.
Chen, W., Yang, J., Guo, L., & Li, S. (2016). Disturbance-observer-based control and related methods – an overview. IEEE Transactions on Industrial Electronics, 63(2), 1083–1095.
Bai, W., Xue, W., Huang, Y., & Fang, H. (2018). On extended state based Kalman filter design for a class of nonlinear time-varying uncertain systems. Science China Information Sciences, 61(4), 042201.
Xue, W., Zhang, X., Sun, L., & Fang, H. (2020). Extended state filter based disturbance and uncertainty mitigation for nonlinear uncertain systems with application to fuel cell temperature control. IEEE Transactions on Industrial Electronics, 67(12), 10682–10692.
Zanetti, R., & Bishop, R. H. (2012). Kalman filters with uncompensated biases. Journal of Guidance Control and Dynamics, 35(1), 327–335.
Bembenek, C., Chmielewski, T.A., & Kalata, P.R. (1998). Observability conditions for biased linear time invariant systems. In Proceedings of the American Control Conference (pp. 1180–1184). Philadelphia, PA, USA.
Zhao, S., Duncan, S. R., & Howey, D. A. (2017). Observability analysis and state estimation of lithium-ion batteries in the presence of sensor biases. IEEE Transactions on Control Systems Technology, 25(1), 326–333.
Friedland, B. (1969). Treatment of bias in recursive filtering. IEEE Transactions on Automatic Control, 14(4), 359–367.
Keller, J. Y., & Darouach, M. (1997). Optimal two-stage Kalman filter in the presence of random bias. Automatica, 33(9), 1745–1748.
Zhang, L., Lv, M., Niu, Z., & Rao, W. (2014). Two-stage cubature Kalman filter for nonlinear system with random bias. In International Conference on Multisensor Fusion and Information Integration for Intelligent Systems. Beijing, China.
Zhang, X., Xue, W., Fang, H., Yang, J., & Li, S. (2020). Extended state based Kalman filter for uncertain systems with bias. IFAC-PapersOnLine.
Julier, S.J., & Uhlmann, J.K. (1997). A non-divergent estimation algorithm in the presence of unknown correlations. In Proceedings of the American Control Conference (pp. 2369–2373). Albuquerque, NM, USA.
Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory (pp. 234–235). New York: Academic Press.
Guo, L. (1992). Shi Bian Sui Ji Xi Tong (Time-varying Stochastic Systems) (pp. 36–38). Changchun: Jilin Science and Technology Press (In Chinese).
Wu, J., Elser, A., Zeng, S., & Allgöwer, F. (2016). Consensus-based distributed Kalman-Bucy filter for continuous-time systems. IFAC-PapersOnLine, 49(22), 321–326.
Acknowledgements
This work was partly supported by National Key R&D Program of China (No. 2018YFA0703800), the National Nature Science Foundation of China (Nos. 11931018, 61633003-3) and the Beijing Advanced Innovation Center for Intelligent Robots and Systems (No. 2019IRS09).
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Zhang, X., Xue, W. & Fang, HT. On extended state based Kalman filter for nonlinear time-varying uncertain systems with measurement bias. Control Theory Technol. 19, 142–152 (2021). https://doi.org/10.1007/s11768-021-00034-2
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DOI: https://doi.org/10.1007/s11768-021-00034-2