基于最小学习参数算法的随机非线性系统的模糊控制研究

Stochastic nonlinear control theory is always one of the hot topics in the automatic control area. In this project, for the stochastic nonlinear systems with input time delays, unknown dead zone and unknown control directions, based on the quadratic Lyapunov functions, we will design the fuzzy adaptive tracking controllers by combining the fuzzy logic systems, the backstepping technique, the dynamic surface control, the hyperbolic tangent functions, the state transformation, the integral mean value theorem, the Nussbaum-type functions and the minimal learning parameters algorithm. These controllers such that all the signals in the closed loop system are bounded in probability and the tracking error can converge to a small residual set around the origin in the mean square sense. And these controllers contain as few adjusted parameters as possible by re-parameterization, therefore, the computational burden can be decreased and the system running efficiency can be improved. It should be pointed out that during the stability analysis process of the stochastic nonlinear systems by using Lyapunov stability theory, the existence of the second order Hessian term such that the quartic Lyapunov functions other than the classical quadratic Lyapunov functions are used to analyze the stability of the stochastic systems in the most existing results. In this project, the long-standing obstacle for stochastic systems control by quadratic Lyapunov function is overcome by combining the fuzzy logic systems and the hyperbolic tangent functions.

随机非线性控制理论一直是自动化控制领域研究的热点课题之一。本项目基于二次Lyapunov函数，结合模糊逻辑系统、递推技术、动态面控制技术、双曲正切函数、状态坐标变换、积分中值定理、Nussbaum类型函数以及最小学习参数算法为具有输入时滞、具有死区以及具有未知控制方向的随机非线性系统设计模糊自适应跟踪控制器。该类控制器使得闭环系统中的所有信号在概率意义下都是有界的，跟踪误差在均方意义下收敛到原点的某个小的剩余集内。且通过重新参数化使得控制器中包含最少的自适应参数，从而有效减小算法的运算量，提高系统的运行效率。需要指出的是对随机非线性系统作基于Lyapunov函数的分析设计时，由于二阶Hessian项的出现导致现有文献大多采用四次Lyapunov函数而非二次Lyapunov函数，本项目将利用双曲正切函数以及模糊逻辑系统克服使用传统二次Lyapunov函数研究随机非线性系统稳定性所遇到的困难。

非线性控制理论一直是自动化控制领域研究的热点课题之一。本项目结合模糊逻辑系统、递推技术、动态面控制技术、双曲正切函数、状态坐标变换、积分中值定理、二次Lyapunov函数以及最小学习参数算法为三类非线性系统设计了模糊自适应控制器且对三类闭环系统均进行了稳定性证明，共发表或录用了4篇论文。具体内容如下：结合状态观测器和最小学习参数算法为具有未知分布时变时延和类反斜线滞后的非线性系统设计出了输出反馈控制器，且控制器中自适应参数个数只有一个；分别为非线性参数化n阶非线性系统设计出了只含有一个自适应参数和含有n个自适应参数的模糊自适应跟踪控制器；为具有输入时变时延的随机非线性系统设计出了模糊自适应跟踪控制器。在控制器设计过程中，尽量结合最小学习参数算法设计出含有最少参数的控制器以求减小算法的运算量，提高系统的运行效率。针对随机非线性系统本项目结合双曲正切函数和经典的二次Lyapunov函数对闭环系统的稳定性进行分析，从而克服了长久以来必须选择四次Lyapunov函数证明随机非线性系统稳定性的困难。