仿射非线性系统的微分平坦及相关问题研究

Based on differential geometric and differential algebric methods for nonlinear systems, the current project propose to study the characterization problems of differential flatness for nonlinear control-affine systems and related topics such as orbital feedback linearization, orbital flatness, maximal feedback linearization with internal stability, etc. First, by using on the dynamic extensions method, we investigate the feedback linearization of two-input control-affine systems by k-fold prolongation and give the the checkable necessary and sufficient geometric conditions. Based on these results, we will investigate the characterization problems of differential flatness for two-input control-affine systems. Secondly, we will characterize multi-input control-affine systems that are feedback equivalent to the linear form at any nominal point (equilibrium or not), derive the checkable necessary and sufficient conditions and the algorithm to find the corresponding time-scaling functions. Moreover, by studying the relation between orbital feedback linearization and orbital flatness, we will characterize all the orbital flat outputs for nonlinear single-input control systems and investigate the characterization of orbital flatness for nonlinear multi-input control-affine systems. Finally, we plan to study the maximal feedback linearization with internal stability for nonlinear single-input control systems and find the corresponding maximal flat output. Furthermore, we will study the maximal feedback linearization with internal stability for nonlinear sinlg-input control-affine systems that are orbitally flat and also the orbitally maximal feedback linearization with internal stability for for nonlinear sinlg-input control-affine systems.

本项目利用非线性系统微分几何与微分代数方法，开展仿射非线性系统的微分平坦及相关问题研究。首先，利用非线性系统的动态延展方法，研究双输入仿射非线性系统k阶动态延展反馈线性化的充要条件，进而研究双输入仿射非线性系统的微分平坦性判定。其次，研究多输入仿射非线性系统（在平衡点和非平衡点）轨道反馈线性化问题，给出可验证的充分必要条件和时间尺度变换函数的构造算法。在此基础上，研究轨道反馈线性化与轨道微分平坦之间的关系，得到单输入仿射非线性系统轨道平坦输出的算法，并研究多输入仿射非线性系统的轨道微分平坦性判定。最后，研究单输入仿射非线性系统基于内动态稳定性的最大线性化的问题，给出最大平坦输出的构造算法；研究单输入仿射轨道微分平坦系统的最大线性化及零动态稳定性和单输入仿射非线性系统的轨道最大线性化及零动态稳定性。

本项目利用非线性系统微分几何与微分代数方法，开展了仿射非线性系统的微分平坦及相关问题研究。首先，研究了具有三角形式漂移项的多链式系统的反馈等价性与微分平坦性、多输入仿射非线性系统的一阶动态延展反馈线性化及微分平坦性，并分别给出了微分平坦输出的计算方法，研究了具有最小differential weight的多输入微分平坦系统的规范型、多输入仿射非线性系统的一阶动态退化反馈线性化问题；然后，在前期研究基础上，研究了一类单输入仿射非线性系统的轨道微分平坦性、仿射非线性系统的微分平坦与对称性的关系、三维非线性系统的与对称群相合的微分平坦输出等问题；此外，研究了单输入仿射非线性系统的最大线性化及零动态稳定性，对单输入仿射非线性系统进行了分类，并利用理论结果研究了在四维拉格朗日系统中的应用；最后，研究了控制系统相关的方程解的存在性及迭代逼近以及线性奇异时滞系统的能观性及观测器设计问题。