Study On Some Issues Of Model Reduction For Nonlinear Partial Differential Equation Dynamical Systems

ABSTRACT:Advanced Technologies such as semiconductor manufacturing, nanotechnology, biotechnology, material engineering and chemical engineering, have the typical characteristics that their inputs, outputs and even parameters can vary both temporally and spatially. Their time-space coupled nature is often described using partial differential equations (PDE). Because the spatial viariable of the nonlinear PDE system are distributed in the space, there will generate an infinite-dimensional approximated system by space/time separation using the spatial basis functions expansions. However, because of finite number of actuators and sensors for practical sensing and control, such infinite-dimensional systems need to be approximated by finite-dimensional systems for the system analysis, optimization, and control design. Thus, it is a very important and extensive problem for the model reduction of nonlinear PDE dynamical systems.This thesis will focus on the theories and methods for further model reduction of the nonlinear PDE systems.The main content includes the following aspects:1)The model reduction methods for autonomous PDE systems based on optimizationInspired by the principal component analysis, balanced truncation method for model reduction of the ODE systems, the model reduction methods for autonomous PDE systems based on optimization are proposed. Firstly, a high-dimensional ODE system is derived by the space/time separation and weighted residual method from the nonlinear PDE system. Then, a low-dimensional ODE system is obtained by the transformation from the high-dimensional ODE systems, while the transformation matrix is derived from the minization for the energy error of the two ODE dynamical systems. The obtained low-dimensional system can simplify the design of the controller for the applications of the nonlinear PDE systems in engineering.2) The model reduction methods for PDE systems using an improved error functions and optimizationNew spatial basis functions are obtained by basis functions transformation from general basis functions, while the spatial basis functions transformation matrix is derived by the optimization for an improved error function. The calculation algorithm is proposed for optimization of an orthogonal transformation matrix. Lower-dimensional optimal spatial basis functions are obtained subsequently from basis functions transformation. Using the basis functions expansions based on optimal basis functions for expansions and nonlinear Galerkin method, lower-dimensional ODE systems can be derived finally. This method can improve the performance of the modeling for the nonlinear PDE systems, its smaller predicted error and the better distribution of the error along the time will simplify the design of the controller for the applications of the nonlinear PDE systems3) The model reduction methods using the optimal combination of empirical eigenfunctions for PDE dynamical systemsTo compensate the dynamical information of the neglectful modes, the model reduction method for PDE systems based on the optimal combination of empirical eigenfunctions is proposed. The new global discrete basis functions are linear combinations of the pre-selected initial empirical functions, while the combination matrix is also obtained by the optimization method. After time/space separation using the new discrete basis functions, the lower-dimensional ODE system can be derived. This model reduction method can use the existed optimization method, and modeling performance based on the obtained new spatial basis functions has higher precisions than that based on the empirical eigenfunctions.