基于Takagi-Sugeno模型的非线性系统数据驱动最优控制方法适定性的研究

We are concerned with the well-posedness of nonlinear systems optimal control, which is to be solved based on Takagi-Sugeno(T-S) fuzzy model and data-driven method. If the problem is ill-posed, we are interested in how to make it to be well-posed. Firstly, with the aid of fuzzy approximation principle, the fuzzy system structure identification problem will be taken into account based on regularization theory. Next, the well-posedness of a class of nonlinear systems, which can be approximated by some function approximators, are focused on. Especially, for a particular optimal control problem, the converging character of the sequences in the solution space against to approximate model errors will be studied intensively. Furthermore, the relationship between such converging character and the existence of the related robust controller will be considered as well. The methods of constructing regularization functional are to be explored, and the convergence theorem will be established. To get the specific algorithms, the Cauchy problems will be solved by using finite differential methods. Finally, the problems of well-posedness and how to make the process to be well-posed by using regularization theory will be discussed deeply for the process of designing optimal controller from the information in the given nonlinear system's historical input-output data based on T-S model. The studies in this project will extend issues of the nonlinear system data-driven optimal control method from the well-posedness aspect, and can complete the necessary theoretical basis as well.

拟研究基于Takagi-Sugeno(T-S)模糊模型的，以数据驱动为背景的，非线性系统最优控制问题的适定性与适定化的问题。首先，结合模糊逼近原理，研究基于正则化理论的T-S模糊系统辨识方法。其次，研究可用某种函数逼近器逼近的一类非线性系统最优控制问题的适定性。特别是深入研究相对逼近模型误差，某一特定最优控制问题解空间中函数序列的收敛特性。探讨这种收敛性与相关鲁棒控制器存在性之间的关系。研究正则化泛函的构造方法。建立收敛定理。构造有限差分法求解相应的Cauchy问题，得到具体算法。最后，研究基于T-S模糊模型的，从目标非线性系统的历史输入输出数据到模糊最优控制器这一整体过程的适定性，以及基于正则化理论的适定化方法。本项目研究有利于从问题的适定性角度丰富基于数据驱动的非线性系统最优控制方法的研究内容，并完善必要的理论基础。

基于数据驱动的控制问题是一个反问题。在研究中发现，类似于其它的反问题，此类问题中也存在着适定性的问题。解决反问题当中适定性问题的有效方法是正则化方法。而Tikhonov正则化方法是提出最早、理论最完整的一套方法。在实际的应用中，正则化解的精度同正则化参数的选取密切相关。但已有的正则化参数选取方法总是不能够满足要求。因此，本项目一个重要的课题就是正则化参数的选取问题。得到一种在离散问题中行之有效的正则化参数计算方法。此外，利用Tikhonov正则化方法，研究了T-S模糊系统辨识问题。利用正则化方法，研究了离散系统和离散广义系统的基于数据驱动的迭代学习控制（跟踪控制）问题等。研究了与项目相关的动态神经网络系统的稳定性问题，非线性系统的容错控制问题，可扩展大系统的稳定性问题等。在这些问题中均得到了一些较好的结论和结果。