Adjoint-based parameter and state estimation in 1-D magnetohydrodynamic (MHD) flow system

Previous Article
Frequency $H_{2}/H_{∞}$ optimizing control for isolated microgrid based on IPSO algorithm
JIMO Home
This Issue
Next Article
The modified inertial relaxed CQ algorithm for solving the split feasibility problems

October 2018, 14(4): 1579-1594. doi: 10.3934/jimo.2018022

Adjoint-based parameter and state estimation in 1-D magnetohydrodynamic (MHD) flow system

Zhigang Ren ^1, , Shan Guo ^2, , Zhipeng Li ^3, and Zongze Wu ^1,,

1.	School of Automation, Guangdong University of Technology, Guangzhou, Guangdong, China
2.	School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China
3.	School of Control Science and Engineering, Shandong University, Jinan, Shandong, China

* Corresponding author: Zongze Wu

Received April 2017 Revised October 2017 Published January 2018

Fund Project: This work is supported by the National Natural Science Foundation of China grants (61703114,61673126,61473253) and the Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University, China (ICT170288, ICT170301)

In this paper, an adjoint-based optimization method is employed to estimate the unknown coefficients and states arising in an one-dimensional (1-D) magnetohydrodynamic (MHD) flow, whose dynamics can be modeled by a coupled partial differential equations (PDEs) under some suitable assumptions. In this model, the coefficients of the Reynolds number and initial conditions as well as state variables are supposed to be unknown and need to be estimated. We first employ the Lagrange multiplier method to connect the dynamics of the 1-D MHD system and the cost functional defined as the least square errors between the measurements in the experiment and the numerical simulation values. Then, we use the adjoint-based method to the augmented Lagrangian cost functional to get an adjoint coupled PDEs system, and the exact gradients of the defined cost functional with respect to these unknown parameters and initial states are further derived. The existed gradient-based optimization technique such as sequential quadratic programming (SQP) is employed for minimizing the cost functional in the optimization process. Finally, we illustrate the numerical examples to verify the effectiveness of our adjoint-based estimation approach.

Keywords: Magnetohydrodynamic, parameter estimation, state estimation, adjoint method, inverse problem, data assimilation.

Mathematics Subject Classification: Primary: 49J20, 65K10; Secondary: 65L09.

Citation: Zhigang Ren, Shan Guo, Zhipeng Li, Zongze Wu. Adjoint-based parameter and state estimation in 1-D magnetohydrodynamic (MHD) flow system. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1579-1594. doi: 10.3934/jimo.2018022

References:

[1]	J. Baker and P. D. Christofides, Drag reduction in transitional linearized channel flow using distributed control, International Journal of Control, 75 (2002), 1213-1218. doi: 10.1080/00207170210163631.
[2]	Q. Chai, R. Loxton, K. L. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486. doi: 10.3934/jimo.2013.9.471.
[3]	Q. Chai, R. Loxton, K. L. Teo and C. Yang, Time-delay estimation for nonlinear systems with piecewise-constant input, Applied Mathematics and Computation, 219 (2013), 9543-9560. doi: 10.1016/j.amc.2013.03.015.
[4]	S. D. Conte and D. K. Kahaner, Numerical Analysis, John Wiley and Sons Ltd., 2003.
[5]	P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001.
[6]	K. Debbagh, P. Cathalifaud and C. Airiau, Optimal and robust control of small disturbances in a channel flow with a normal magnetic field, Physics of Fluids, 19 (2007), 014103. doi: 10.1063/1.2429038.
[7]	Y. Ding and S. S. Wang, Identification of manning's roughness coefficients in channel network using adjoint analysis, International Journal of Computational Fluid Dynamics, 19 (2005), 3-13. doi: 10.1080/10618560410001710496.
[8]	J. P. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics: with Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, 2004. doi: 10.1017/CBO9780511616945.
[9]	L. D. Landau, J. Bell, M. Kearsley, L. Pitaevskii, E. Lifshitz and J. Sykes, Electrodynamics of Continuous Media, Elsevier, 2013.
[10]	Q. Lin, R. Loxton, C. Xu and K. L. Teo, Parameter estimation for nonlinear time-delay systems with noisy output measurements, Automatica, 60 (2015), 48-56. doi: 10.1016/j.automatica.2015.06.028.
[11]	Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.
[12]	C. Liu, R. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 163 (2014), 957-988. doi: 10.1007/s10957-014-0533-7.
[13]	U. Müller and L. Bühler, Magnetofluiddynamics in Channels and Containers, Springer Science & Business Media, 2013.
[14]	I. Munteanu, Boundary feedback stabilization of periodic fluid flows in a magnetohydrodynamic channel, IEEE Transactions on Automatic Control, 58 (2013), 2119-2115. doi: 10.1109/TAC.2013.2244312.
[15]	I. M. Navon, X. Zou, J. Derber and J. Sela, Variational data assimilation with an adiabatic version of the nmc spectral model, Monthly Weather Review, 120 (1992), 1433-1446. doi: 10.1175/1520-0493(1992)120<1433:VDAWAA>2.0.CO;2.
[16]	V. T. Nguyen, D. Georges and G. Besançon, State and parameter estimation in 1-D hyperbolic PDEs based on an adjoint method, Automatica, 67 (2016), 185-191. doi: 10.1016/j.automatica.2016.01.031.
[17]	S. Qiana and H. H. Bau, Magneto-hydrodynamics based microfluidics, Mechanics Research Communications, 36 (2009), 10-21. doi: 10.1016/j.mechrescom.2008.06.013.
[18]	S. Qian and H. H. Bau, Magneto-hydrodynamic stirrer for stationary and moving fluids, Sensors and Actuators B: Chemical, 106 (2005), 859-870. doi: 10.1016/j.snb.2004.07.011.
[19]	Z. Ren, C. Xu, Q. Lin and R. Loxton, A gradient-based kernel optimization approach for parabolic distributed parameter control systems, Pacific Journal of Optimization, 12 (2016), 263-287.
[20]	E. Schuster, L. X. Luo and M. Krstić, MHD channel flow control in 2D: Mixing enhancement by boundary feedback, Automatica, 44 (2008), 2498-2507. doi: 10.1016/j.automatica.2008.02.018.
[21]	K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991.
[22]	V. Tsyba and A. Y. Chebotarev, Optimal control asymptotics of a magnetohydrodynamic flow, Computational Mathematics and Mathematical Physics, 49 (2009), 466-473.
[23]	R. Vazquez, E. Schuster and M. Krstic, A closed-form full-state feedback controller for stabilization of 3D magnetohydrodynamic channel flow, Journal of Dynamic Systems, Measurement, and Control, 131 (2009), 041001.
[24]	C. Xu, E. Schuster, R. Vazquez and M. Krstic, Stabilization of linearized 2D magnetohydrodynamic channel flow by backstepping boundary control, Systems & Control Letters, 57 (2008), 805-812. doi: 10.1016/j.sysconle.2008.03.008.

show all references

References:

[1]	J. Baker and P. D. Christofides, Drag reduction in transitional linearized channel flow using distributed control, International Journal of Control, 75 (2002), 1213-1218. doi: 10.1080/00207170210163631.
[2]	Q. Chai, R. Loxton, K. L. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems, Journal of Industrial and Management Optimization, 9 (2013), 471-486. doi: 10.3934/jimo.2013.9.471.
[3]	Q. Chai, R. Loxton, K. L. Teo and C. Yang, Time-delay estimation for nonlinear systems with piecewise-constant input, Applied Mathematics and Computation, 219 (2013), 9543-9560. doi: 10.1016/j.amc.2013.03.015.
[4]	S. D. Conte and D. K. Kahaner, Numerical Analysis, John Wiley and Sons Ltd., 2003.
[5]	P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001.
[6]	K. Debbagh, P. Cathalifaud and C. Airiau, Optimal and robust control of small disturbances in a channel flow with a normal magnetic field, Physics of Fluids, 19 (2007), 014103. doi: 10.1063/1.2429038.
[7]	Y. Ding and S. S. Wang, Identification of manning's roughness coefficients in channel network using adjoint analysis, International Journal of Computational Fluid Dynamics, 19 (2005), 3-13. doi: 10.1080/10618560410001710496.
[8]	J. P. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics: with Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, 2004. doi: 10.1017/CBO9780511616945.
[9]	L. D. Landau, J. Bell, M. Kearsley, L. Pitaevskii, E. Lifshitz and J. Sykes, Electrodynamics of Continuous Media, Elsevier, 2013.
[10]	Q. Lin, R. Loxton, C. Xu and K. L. Teo, Parameter estimation for nonlinear time-delay systems with noisy output measurements, Automatica, 60 (2015), 48-56. doi: 10.1016/j.automatica.2015.06.028.
[11]	Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.
[12]	C. Liu, R. Loxton and K. L. Teo, Switching time and parameter optimization in nonlinear switched systems with multiple time-delays, Journal of Optimization Theory and Applications, 163 (2014), 957-988. doi: 10.1007/s10957-014-0533-7.
[13]	U. Müller and L. Bühler, Magnetofluiddynamics in Channels and Containers, Springer Science & Business Media, 2013.
[14]	I. Munteanu, Boundary feedback stabilization of periodic fluid flows in a magnetohydrodynamic channel, IEEE Transactions on Automatic Control, 58 (2013), 2119-2115. doi: 10.1109/TAC.2013.2244312.
[15]	I. M. Navon, X. Zou, J. Derber and J. Sela, Variational data assimilation with an adiabatic version of the nmc spectral model, Monthly Weather Review, 120 (1992), 1433-1446. doi: 10.1175/1520-0493(1992)120<1433:VDAWAA>2.0.CO;2.
[16]	V. T. Nguyen, D. Georges and G. Besançon, State and parameter estimation in 1-D hyperbolic PDEs based on an adjoint method, Automatica, 67 (2016), 185-191. doi: 10.1016/j.automatica.2016.01.031.
[17]	S. Qiana and H. H. Bau, Magneto-hydrodynamics based microfluidics, Mechanics Research Communications, 36 (2009), 10-21. doi: 10.1016/j.mechrescom.2008.06.013.
[18]	S. Qian and H. H. Bau, Magneto-hydrodynamic stirrer for stationary and moving fluids, Sensors and Actuators B: Chemical, 106 (2005), 859-870. doi: 10.1016/j.snb.2004.07.011.
[19]	Z. Ren, C. Xu, Q. Lin and R. Loxton, A gradient-based kernel optimization approach for parabolic distributed parameter control systems, Pacific Journal of Optimization, 12 (2016), 263-287.
[20]	E. Schuster, L. X. Luo and M. Krstić, MHD channel flow control in 2D: Mixing enhancement by boundary feedback, Automatica, 44 (2008), 2498-2507. doi: 10.1016/j.automatica.2008.02.018.
[21]	K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991.
[22]	V. Tsyba and A. Y. Chebotarev, Optimal control asymptotics of a magnetohydrodynamic flow, Computational Mathematics and Mathematical Physics, 49 (2009), 466-473.
[23]	R. Vazquez, E. Schuster and M. Krstic, A closed-form full-state feedback controller for stabilization of 3D magnetohydrodynamic channel flow, Journal of Dynamic Systems, Measurement, and Control, 131 (2009), 041001.
[24]	C. Xu, E. Schuster, R. Vazquez and M. Krstic, Stabilization of linearized 2D magnetohydrodynamic channel flow by backstepping boundary control, Systems & Control Letters, 57 (2008), 805-812. doi: 10.1016/j.sysconle.2008.03.008.

Figure 1. Estimation of initial condition $u(x,0)$ (scenario 1).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Estimation of initial condition $B(x,0)$ (scenario 1).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. Iteration process of cost function value (scenario 1).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. Estimation of state $u(x,t)$ (scenario 1).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. Estimation of state $B(x,t)$ (scenario 1).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. Estimation of initial condition $u(x,0)$ (scenario 2).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7. Estimation of initial condition $B(x,0)$ (scenario 2).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8. Iteration process of cost function value (scenario 2).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9. Estimation of state $u(x,t)$ (scenario 2).

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10. Estimation of state $B(x,t)$ (scenario 2).

Figure Options

Download full-size image

Download as PowerPoint slide

Table .

Algorithm 1. Gradient-based optimization procedure for solving Problem P$_0$.
Step 1: Choose the initial guess $\nu$, $\nu_m$ and $u_{0}^{in}(x)$, $B_{0}^{in}(x)$.
Step 2: Solve the 1-D MHD model (1) forward in time corresponding to initial guess values from $t=0$ to $t=T$, and solve the adjoint coupled PDE systems (7) backward in time corresponding to initial guess values from $t=T$ to $t=0$ to obtain $u(x, t)$, $B(x, t)$, $\lambda_1(x, t)$, $\lambda_2(x, t)$.
Step 3: Compute cost functional and its gradients according to (4) and (5)-(6).
Step 4: Use the gradients information obtained in $\mathbf{Step 3}$ to perform an optimality test. If $\nu$, $\nu_m$ and $u_{0}^{in}(x), B_0^{in}(x)$ are optimal, then exit; Otherwise, go to $\mathbf{Step 5}$.
Step 5: Use the gradient information obtained in $\mathbf{Step 3}$ to calculate a searching direction.
Step 6: Perform a line search to determine the optimal step length.
Step 7: Compute the new points $\nu$, $\nu_m$ and $u_{0}^{in}(x)$, $B_{0}^{in}(x)$ and return to $\mathbf{Step 2}$.

Table Options

Download as excel

Algorithm 1. Gradient-based optimization procedure for solving Problem P$_0$.
Step 1: Choose the initial guess $\nu$, $\nu_m$ and $u_{0}^{in}(x)$, $B_{0}^{in}(x)$.
Step 2: Solve the 1-D MHD model (1) forward in time corresponding to initial guess values from $t=0$ to $t=T$, and solve the adjoint coupled PDE systems (7) backward in time corresponding to initial guess values from $t=T$ to $t=0$ to obtain $u(x, t)$, $B(x, t)$, $\lambda_1(x, t)$, $\lambda_2(x, t)$.
Step 3: Compute cost functional and its gradients according to (4) and (5)-(6).
Step 4: Use the gradients information obtained in $\mathbf{Step 3}$ to perform an optimality test. If $\nu$, $\nu_m$ and $u_{0}^{in}(x), B_0^{in}(x)$ are optimal, then exit; Otherwise, go to $\mathbf{Step 5}$.
Step 5: Use the gradient information obtained in $\mathbf{Step 3}$ to calculate a searching direction.
Step 6: Perform a line search to determine the optimal step length.
Step 7: Compute the new points $\nu$, $\nu_m$ and $u_{0}^{in}(x)$, $B_{0}^{in}(x)$ and return to $\mathbf{Step 2}$.

[1]	Simon Hubmer, Andreas Neubauer, Ronny Ramlau, Henning U. Voss. On the parameter estimation problem of magnetic resonance advection imaging. Inverse Problems & Imaging, 2018, 12 (1) : 175-204. doi: 10.3934/ipi.2018007
[2]	Dominique Chapelle, Philippe Moireau, Patrick Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 65-84. doi: 10.3934/dcds.2009.23.65
[3]	Ferenc Hartung. Parameter estimation by quasilinearization in differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1611-1631. doi: 10.3934/dcdsb.2013.18.1611
[4]	Alessandro Corbetta, Adrian Muntean, Kiamars Vafayi. Parameter estimation of social forces in pedestrian dynamics models via a probabilistic method. Mathematical Biosciences & Engineering, 2015, 12 (2) : 337-356. doi: 10.3934/mbe.2015.12.337
[5]	Azmy S. Ackleh, Jeremy J. Thibodeaux. Parameter estimation in a structured erythropoiesis model. Mathematical Biosciences & Engineering, 2008, 5 (4) : 601-616. doi: 10.3934/mbe.2008.5.601
[6]	Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (3) : 553-576. doi: 10.3934/mbe.2012.9.553
[7]	Robert Azencott, Yutheeka Gadhyan. Accurate parameter estimation for coupled stochastic dynamics. Conference Publications, 2009, 2009 (Special) : 44-53. doi: 10.3934/proc.2009.2009.44
[8]	Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1
[9]	Stephane Chretien, Paul Clarkson. A fast algorithm for the semi-definite relaxation of the state estimation problem in power grids. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2018161
[10]	Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75
[11]	Azmy S. Ackleh, H.T. Banks, Keng Deng, Shuhua Hu. Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations. Mathematical Biosciences & Engineering, 2005, 2 (2) : 289-315. doi: 10.3934/mbe.2005.2.289
[12]	Krzysztof Fujarewicz, Krzysztof Łakomiec. Parameter estimation of systems with delays via structural sensitivity analysis. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2521-2533. doi: 10.3934/dcdsb.2014.19.2521
[13]	Andrea Arnold, Daniela Calvetti, Erkki Somersalo. Vectorized and parallel particle filter SMC parameter estimation for stiff ODEs. Conference Publications, 2015, 2015 (special) : 75-84. doi: 10.3934/proc.2015.0075
[14]	Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[15]	Claude Stolz. On estimation of internal state by an optimal control approach for elastoplastic material. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 599-611. doi: 10.3934/dcdss.2016014
[16]	Peng Cui, Hongguo Zhao, Jun-e Feng. State estimation for discrete linear systems with observation time-delayed noise. Journal of Industrial & Management Optimization, 2011, 7 (1) : 79-85. doi: 10.3934/jimo.2011.7.79
[17]	Elena K. Kostousova. State estimation for linear impulsive differential systems through polyhedral techniques. Conference Publications, 2009, 2009 (Special) : 466-475. doi: 10.3934/proc.2009.2009.466
[18]	W. Y. Tan, L.-J. Zhang, C.W. Chen. Stochastic modeling of carcinogenesis: State space models and estimation of parameters. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 297-322. doi: 10.3934/dcdsb.2004.4.297
[19]	Ariel Cintrón-Arias, Carlos Castillo-Chávez, Luís M. A. Bettencourt, Alun L. Lloyd, H. T. Banks. The estimation of the effective reproductive number from disease outbreak data. Mathematical Biosciences & Engineering, 2009, 6 (2) : 261-282. doi: 10.3934/mbe.2009.6.261
[20]	Johnathan M. Bardsley. An efficient computational method for total variation-penalized Poisson likelihood estimation. Inverse Problems & Imaging, 2008, 2 (2) : 167-185. doi: 10.3934/ipi.2008.2.167

2018 Impact Factor: 1.025

Tools

Metrics

Tools

Article outline

Figures and Tables

2018 Impact Factor: 1.025

Tools

Figures and Tables

2018 Impact Factor: 1.025

American Institute of Mathematical Sciences