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

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

 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.

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:
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##### References:
  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  S. D. Conte and D. K. Kahaner, Numerical Analysis, John Wiley and Sons Ltd., 2003. Google Scholar  P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001. Google Scholar  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. Google Scholar  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.  Google Scholar  J. P. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics: with Applications to Laboratory and Astrophysical Plasmas, Cambridge University Press, 2004. doi: 10.1017/CBO9780511616945. Google Scholar  L. D. Landau, J. Bell, M. Kearsley, L. Pitaevskii, E. Lifshitz and J. Sykes, Electrodynamics of Continuous Media, Elsevier, 2013. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  U. Müller and L. Bühler, Magnetofluiddynamics in Channels and Containers, Springer Science & Business Media, 2013. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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.  Google Scholar  K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991. Google Scholar  V. Tsyba and A. Y. Chebotarev, Optimal control asymptotics of a magnetohydrodynamic flow, Computational Mathematics and Mathematical Physics, 49 (2009), 466-473. Google Scholar  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. Google Scholar  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.  Google Scholar
 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 $\mathbf{Step 2}$.
 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 $\mathbf{Step 2}$.
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