摘要
许多制造过程具有时间和空间耦合的特征,属于时空耦合系统,典型的例子有轧制过程、半导体制造过程、柔性操作手的高精度定位等。这些过程在数学上由偏微分方程描述,在控制领域称为分布参数系统。由于这类过程的时空耦合特点,所以本质上是无穷维的,且存在非线性、不确定性和多场耦合,很难获得其解析解,因此对于这类时空耦合过程的系统进行快速仿真分析和控制器设计是非常困难的。
工程上,一般对无穷维非线性时空耦合系统进行降维处理,即利用有穷维系统来近似原系统。传统时空离散方法如差分法和有限元法均只能得到阶数非常高的近似模型,不适用于时空耦合系统的快速仿真和控制器设计。谱方法能得到比传统时空离散方法维数低得多的近似模型,然而只适用于一类能进行快慢分离的系统,且在通常情况下,得到的近似模型的维数不是最低的。因此,针对一大类非线性时空耦合过程,通过有效的计算方法,建立既保证建模精度,维数又最低,适用于系统快速仿真和控制器设计的低维近似模型,不但是工程上亟待解决的难题,也是一个科学上的挑战。
针对这一难题,本文主要研究非线性时空耦合系统降维新方法及其在铝合金板带轧制过程建模中的应用。通过由谱方法导出的空间正交基函数进行线性变换得到个数较少的新基函数。基于获得的新基函数,直接投影或者结合智能技术获得原系统的一个非常低维的近似。提出的降维新方法能得到比基于传统降维方法更低维的动态模型,而且能保持较好的精度。
本文主要研究工作包括以下方面：
针对模型已知的非线性时空耦合系统,对谱方法导出的空间正交基函数进行线性组合得到一组新的正交基函数,空间基函数组合矩阵由时空耦合系统线性部分的平衡截断方法得到,严格地证明了基于新空间基函数的建模误差比基于同阶谱方法建模的误差小。基于获得的新空间基函数,利用混合智能建模方法来近似时空耦合系统,对典型时空耦合系统的降维结果说明了理论结果的正确性,而且表明基于新基函数的低维模型的精度优于更高维基于传统谱方法的降维模型的精度。
针对模型未知的非线性时空耦合系统,利用正交分解技术获得该系统的实验特征函数和对应的时间系数后,建立原非线性系统的一个线性近似。对上述线性系统利用平衡截断方法得到基函数转换矩阵,从而将实验特征函数转换得到一组个数较少的改进实验特征函数,严格地证明了基于改进实验特征函数的建模误差比基于同阶实验特征函数的建模误差小。基于新的实验特征函数进行时空分离后,利用传统智能方法对系统的动态进行低维建模,仿真结果说明了理论结果的正确性,且表明基于改进实验特征函数的低维模型能达到基于初始实验特征函数的更高维模型的精度。
针对模型已知的时空耦合系统,通过对谱方法导出的空间正交基函数进行转换得到一组新的个数较少的空问正交基函数,其中空间基函数转换矩阵由最优化方法获得。经过严格地推导得到简明且易于优化计算的的误差函数,提出了求解优化算法计算正交的转换矩阵。得到最优的转换矩阵后,利用空间基函数转换可以得到最优空间基函数。基于获得的最优空间基函数,利用时空分离结合非线性Galerkin方法,可以获得到在一定精度要求下的最低维近似动态模型。
针对铝合金热轧四辊轧机,建立了工作辊的热力耦合变形模型,然后采用平衡截断降维方法分别对轧辊的热变形和弹性变形进行了低维建模。基于实际生产数据并综合轧辊的热变形和弹性变形,建立了板带横向厚度分布的低维智能模型。对铝合金板带的横向厚度分布的预测表明建立的低维近似模型与实际测量数据吻合较好,能满足工程应用的要求。
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.
引文
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