New Model Reduction Approaches For Spatio-temporal Systems And Its Applications In The Modelling Of Aluminum Alloy Rolling Processes

Many advanced manufacturing processes, such as rolling processes, semiconductor manufacturing and high accuracy positioning of flexiable arm belong to complex spatio-temporal systems that their states, controls, output and process parameters may vary temporally and spatially. The mechanistic dynamical modeling of spatio-temporal systems typically leads to various partial differential equations (PDEs). These PDEs are infinite-dimensionl in nature, and it is diffcult for prediction, control and optimization of the spatio-temporal systems because of their unavailable analytical solution, the nonlinearities, uncertainties and energy field coupling.In practical engineering applications, the model reduction of infinite-dimensional systems to finite-dimensional systems is needed because of finite actuators/sensors and limited computing powers. Conventional time/space discretization approaches, such as finite difference method (FDM) and finite element method (FEM), often lead to high-order models, which are not suitable for simulations and control design of spatio-temporal systems. Modeling by spectral method can obtain much lower dimensional models than conventional methods. However, it is only appropriate to modeling a kind of PDE system typically involves spatial differential operators with eigenspectra that can be partitioned into a slow and a fast complements and the dimension of the reduced model is not the lowest for a given accuracy. Thus, lower-dimensional approximate modeling with less computation cost to reveal the dynamics of DPSs is a very difficult and challenging problem in engineering and science.Some studies have been carried out for new model reduction approaches of spatio-temporal systems and its application in aluminum alloy rolling processes. A smaller class of new spatial basis functions is obtained by the transform from spectral basis functions. Based on new basis functions, lower-dimensional ODE systems with satisfied accuracy are developed to approximate the dynamics of spatio-temporal systems.The studies of this note mainly contain the following four parts: With the known DPSs, new spatial orthogonal basis functions are obtained by linear combinations from spectral basis functions, where the combinations matrix are developed by balanced truncation for linear terms. That the model error based on new basis function is small than spectral basis functions with the same order is proved theoretically. Based on the obtained new basis functions, a hybrid intelligent model is developed to approximate the spatio-temporal systems. Simulations for a typical spatio-temporal system are used to demonstrate the effectiveness of the proposed approach.For unknown DPSs, the EEFs and the corresponding temporal coefficients are obtained by POD from spatio-temporal measured output, thus a linear ODE system is used to approximate the temporal dynamics for nonlinear systems. Improved EEFs are obtained from spatial basis transform and transformation matrix is obtained by balanced truncation for the linear ODE system. That the model error based on improved EEFs is small than the same order EEFs is also proved theoretically After time/space separation based on the improved EEFs, traditional intelligent models are used to identify the dynamics of the unknown spatio-temporal processes.For the known DPSs, new spatial orthogonal basis functions are obtained by basis function transforms from spectral basis functions, and transformation matrix are developed by optimization method. A simple error functions related to transform matrix are derived strictly for optimization. The algorithm based on PSO is proposed for optimization of an orthogonal transformation matrix. Lower-dimensional optimal spatial basis functions are obtained from spectral basis function by transform. Using the optimal basis functions for expansions and nonlinear Galerkin method, lower-dimensional ODE systems can be derived.For four-high mill of aluminum alloy hot rolling, a thermal mechanically coupled model is obtained for the deformation of working rolls. The new model reduction approaches based on balanced truncation are used for low dimensional modeling of the thermal deformation and elastic deformation. Based on actual production data, low dimensional intelligent models of the transverse thickness distributions of aluminum alloy hot rolled strip by synthsis of thermal deformation and elastic deformation of work rolls. The predictions show that it has a good agreement with the requirements of the engineering applications.