State-space Modeling And Advanced Control For Fluid Flow And Heat Transfer Of Radiating Participating Magnetohydrodynamics

Radiation magnetohydrodynamics(Radiation-Magnetohydrodynamics,Radiation-MHD or R-MHD)model describes the flow of electrically conducting fluids in the presence of magnetic fields.An important application of R-MHD is the modeling of plasma physics,ranging from plasma confinement for thermonuclear fusion to astrophysical plasma dynamics.R-MHD is also used to model the crystal growth dynamic,for instance,the oxide crystal growth process.The R-MHD model consists of a nonself-adjoint,nonlinear system of partial differential equations(PDEs)that couple the Navier-Stokes equations,the energy equation and the radiative transfer equation.This thesis is divided into four parts:(1)We construct continue-time and discrete-time state-space representations for 2D R-MHD cavity flow in the case of strong magnetic field(as characterized by Hartmann number equal to 100).(2)We design and verify a temperature boundary feedback controller for 2D R-MHD cavity flow.(3)We design and verify a preview controller for continuous-time state-space models.(4)We design and verify a robust D stability controller for uncertain discrete-time state-space models.To obtain the state-space representation for 2D R-MHD cavity flow,applying reasonable assumptions,the mathematical model for fluid flow and heat transfer of viscous incompressible R-MHD is developed in Cartesian coordinates considering complicated thermal radiative transfer of the magnetic fluid and boundary walls.The processes,which are called distributed parameter systems,are greatly impacted by time and space.The spatial discretization of the model is performed using collocation spectral method(CSM),which provides the state-space form of the R-MHD model.We will obtain the continuous-time and discrete-time state-space models when the terms with time derivation are not and are discretized,respectively.Finally,transforming the 2D system into 1D system with tensor product,we obtain the state-space representation of the processes for the following research.For the study of temperature boundary feedback control,we use the L2-norm of first-order spatial derivatives of velocity and temperature field as a measure of mixing.The discrete-time state-space representation of the flow and heat transfer processes of R-MHD cavity flow is selected as the subject.A feedback control law that maximizes the measure related to mixing and minimizes the control and sensing efforts are designed for 2D cavity flow.Firstly,choosing the energy function as the combination of the kinetic and internal energy of the flow,we can give an upper bound on a performance cost index.The performance cost index is the combination of the energy function and the mixing measure function.The temperature feedback control law is obtained to minimize the value of performance cost index based on minimum principle.Secondly,applying the control law as the temperature boundary conditions,we simulate temperature field and flow patterns with the CSM solver for R-MHD equations.We analyze the effects of optical thickness on flow patterns and temperature field,and give the effectiveness of mixing enhancement by temperature boundary feedback control.For the study of preview control,choosing the continuous-time state-space representation of the flow and heat transfer processes of R-MHD cavity flow as the subject,an optimal preview controller is designed considering the future information.The future information is provided by radiating participating MHD evolution with time.The controller can improve the inherent hysteresis and the transient response of radiating participating MHD process,and reduce the control energy.Velocity,temperature and radiation intensity are defined as state,and the pressure,temperature boundary conditions and magnetic field intensity are defined as control input.Firstly,based on preview control theory,an augmented error system is constructed by combining the tracking error and the state.A sufficient condition is obtained,based on optimal control theory,to minimize the performance cost index.The performance cost index consists of the tracking error and the time derivative of control.Secondly,the state-space equation and the sufficient condition are treated by singular value decomposition.Thus,the problem of designing an optimal preview controller is transformed into the resolution problem of differential Riccati equation and ordinary differential equations.The resolution problem of differential Riccati equation can be obtained by solving algebraic Riccati equation and differential Lyapunov equation.Finally,we analysis how the preview length affect the control and why the initial segments appear shocks.For the study of robust D stability control,choosing the uncertain discrete-time state-space representation of the flow and heat transfer processes of R-MHD cavity flow as the subject,a necessary and sufficient D-stability condition and a sufficient D-stabilization condition are proposed by using the state augmentation and Schur component techniques.Firstly,we consider the uncertainties exist in all coefficient matrices with limited energy.The robustness analysis problem can be divided into two cases:right-singular case and left-singular case based on the form of difference matrix E.A necessary and sufficient robust stability condition is proposed for the first and second cases in certain area by Schur component technique,respectively.Secondly,by using a state augmentation and Schur component technique,the sufficient D stabilization conditions are obtained for both cases.Both the robust D-stability and robust D-stabilization conditions are proposed in terms of linear matrix inequalities(LMIs).The innovations include the following aspects:(1)The state-space representation for fluid flow and heat transfer of viscous incompressible R-MHD is developed by making full use of the inherent advantages of CSM.It has high-order accuracy and exponential convergence in theory.The state-space model provides a research basis for the future research with different real intent.(2)A temperature boundary feedback control law is firstly proposed for mixing enhancement in 2D R-MHD cavity flow.(3)In order to improve the hysteresis of R-MHD process,the tracking performance,the transient response and reduce the controlling consumption,a preview control law is obtained considering future information.(4)Choosing the uncertain discrete-time state-space representation of the flow and heat transfer processes of R-MHD cavity flow as the subject,a necessary and sufficient D-stability condition and a sufficient D-stabilization condition are proposed by using the strict LMI method.