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State Estimation Based on Unscented Kalman Filter for Burgers’ Equation

Authors: Takashi Shimizu, Tomoaki Hashimoto

Abstract:

Controlling the flow of fluids is a challenging problem that arises in many fields. Burgers’ equation is a fundamental equation for several flow phenomena such as traffic, shock waves, and turbulence. The optimal feedback control method, so-called model predictive control, has been proposed for Burgers’ equation. However, the model predictive control method is inapplicable to systems whose all state variables are not exactly known. In practical point of view, it is unusual that all the state variables of systems are exactly known, because the state variables of systems are measured through output sensors and limited parts of them can be only available. In fact, it is usual that flow velocities of fluid systems cannot be measured for all spatial domains. Hence, any practical feedback controller for fluid systems must incorporate some type of state estimator. To apply the model predictive control to the fluid systems described by Burgers’ equation, it is needed to establish a state estimation method for Burgers’ equation with limited measurable state variables. To this purpose, we apply unscented Kalman filter for estimating the state variables of fluid systems described by Burgers’ equation. The objective of this study is to establish a state estimation method based on unscented Kalman filter for Burgers’ equation. The effectiveness of the proposed method is verified by numerical simulations.

Keywords: State estimation, fluid systems, observer systems, unscented Kalman filter.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.2021955

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References:


[1] T. Hashimoto, Y. Yoshioka, T. Ohtsuka, Receding Horizon Control with Numerical Solution for Thermal Fluid Systems, Proceedings of SICE Annual Conference, pp. 1298-1303, 2012.
[2] T. Hashimoto, Y. Takiguchi and T. Ohtsuka, Receding Horizon Control for High-Dimensional Burgersf Equations with Boundary Control Inputs, Transactions of the Japan Society for Aeronautical and Space Sciences, Vol. 56, No.3, pp. 137-144, 2013.
[3] R. Satoh, T. Hashimoto and T. Ohtsuka, Receding Horizon Control for Mass Transport Phenomena in Thermal Fluid Systems, Proceedings of Australian Control Conference, pp. 273-278, 2014.
[4] T. Hashimoto, Receding Horizon Control for a Class of Discrete-time Nonlinear Implicit Systems, Proceedings of IEEE Conference on Decision and Control, pp. 5089-5094, 2014.
[5] T. Hashimoto, Optimal Feedback Control Method Using Magnetic Force for Crystal Growth Dynamics, International Journal of Science and Engineering Investigations, Vol. 4, Issue 45, pp. 1-6, 2015.
[6] T. Hashimoto, Y. Yoshioka, T. Ohtsuka, Receding Horizon Control with Numerical Solution for Spatiotemporal Dynamic Systems, Proceedings of IEEE Conference on Decision and Control, pp. 2920-2925, 2012.
[7] T. Hashimoto, Y. Takiguchi and T. Ohtsuka, Output Feedback Receding Horizon Control for Spatiotemporal Dynamic Systems, Proceedings of Asian Control Conference, 2013.
[8] T. Hashimoto, Y. Yoshioka and T. Ohtsuka, Receding Horizon Control for Hot Strip Mill Cooling Systems, IEEE/ASME Transactions on Mechatronics, Vol. 18, No. 3, pp. 998-1005, 2013.
[9] T. Hashimoto, Y. Yoshioka and T. Ohtsuka, Receding Horizon Control With Numerical Solution for Nonlinear Parabolic Partial Differential Equations, IEEE Transactions on Automatic Control, Vol. 58, No. 3, pp. 725-730, 2013.
[10] T. Hashimoto, R. Satoh and T. Ohtsuka, Receding Horizon Control for Spatiotemporal Dynamic Systems, Mechanical Engineering Journal, Vol. 3, No. 2, 15-00345, 2016.
[11] T. Hashimoto, I. Yoshimoto, T. Ohtsuka, Probabilistic Constrained Model Predictive Control for Schr¨odinger Equation with Finite Approximation, Proceedings of SICE Annual Conference, pp. 1613-1618, 2012.
[12] T. Hashimoto, Stability of Stochastic Model Predictive Control for Schr¨odinger Equation with Finite Approximation, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, Vol. 11, No. 1, pp. 12-17, 2017.
[13] T. Hashimoto, Probabilistic Constrained Model Predictive Control for Linear Discrete-time Systems with Additive Stochastic Disturbances, Proceedings of IEEE Conference on Decision and Control, pp. 6434-6439, 2013.
[14] T. Hashimoto, Computational Simulations on Stability of Model Predictive Control for Linear Discrete-time Stochastic Systems, International Journal of Computer, Electrical, Automation, Control and Information Engineering, Vol. 9, No. 8, pp. 1385-1390, 2015.
[15] T. Hashimoto, Stochastic Model Predictive Control for Linear Discrete-time Systems with Random Dither Quantization, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, Vol. 11, No. 2, pp. 130-134, 2017.
[16] T. Hashimoto, Conservativeness of Probabilistic Constrained Optimal Control Method for Unknown Probability Distribution, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, Vol. 9, No. 9, pp. 11-15, 2015.
[17] T. Hashimoto, A Method for Solving Optimal Control Problems subject to Probabilistic Affine State Constraints for Linear Discrete-time Uncertain Systems, International Journal of Mechanical and Production Engineering, Vol. 3, Issue 12, pp. 6-10, 2015.
[18] T. Hashimoto, Solutions to Probabilistic Constrained Optimal Control Problems Using Concentration Inequalities, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, Vol. 10, No. 10, pp. 441-446, 2016.
[19] H. W. Sorenson, Ed., Kalman Filtering: Theory and Application, Piscataway, NJ: IEEE, 1985.
[20] S. Julier, J. Uhlmann and H. F. Durrant-Whyte, A New Method for the Nonlinear Transformation of Means and Covariances in Filters and Estimators, IEEE Transactions on Automatic Control, Vol. 45, 2000, pp. 477-482.
[21] T. Hashimoto, T. Amemiya and H. A. Fujii, Stabilization of Linear Uncertain Delay Systems with Antisymmetric Stepwise Configurations, Journal of Dynamical and Control Systems, Vol. 14, No. 1, pp. 1-31, 2008.
[22] T. Hashimoto, T. Amemiya and H. A. Fujii, Output Feedback Stabilization of Linear Time-varying Uncertain Delay Systems, Mathematical Problems in Engineering, Vol. 2009, Article ID. 457468, 2009.
[23] T. Hashimoto and T. Amemiya, Stabilization of Linear Time-varying Uncertain Delay Systems with Double Triangular Configuration, WSEAS Transactions on Systems and Control, Vol. 4, No.9, pp.465-475, 2009.
[24] T. Hashimoto, Stabilization of Abstract Delay Systems on Banach Lattices using Nonnegative Semigroups, Proceedings of the 50th IEEE Conference on Decision and Control, pp. 1872-1877, 2011.
[25] T. Hashimoto, A Variable Transformation Method for Stabilizing Abstract Delay Systems on Banach Lattices, Journal of Mathematics Research, Vol. 4, No. 2, pp.2-9, 2012.
[26] T. Hashimoto, An Optimization Algorithm for Designing a Stabilizing Controller for Linear Time-varying Uncertain Systems with State Delays, Computational Mathematics and Modeling, Vol.24, No.1, pp.90-102, 2013.