4.4 相平面分析方法
使用相平面分析技术有助于描述和理解低阶模糊控制系统的动态行为，故相平面分析方法被用于分析一些模糊系统的稳定性［60－62］，但这种技术只限于二维规则结构的模糊系统。
4.5 描述函数方法
描述函数方法可用于预测极限环的存在、频率、幅度和稳定性。通过建立模糊控制器与多值继电控制器的关系，描述函数方法可用于分析模糊控制系统的稳定性［31］。另外，指数输入的描述函数技术也能用于调查模糊控制系统的暂态响应［63］。虽然描述函数方法能用于SISO和MISO模糊控制器以及某些非线性对象模型，但不能用于三输入及以上的模糊控制器。由于这种方法一般都用于非线性系统中确定周期振荡的存在性，因此只是一种近似方法。
4.6 圆稳定性判据方法
圆判据可用于分析和再设计一个模糊控制系统。使用扇区有界非线性的概念，一般化的奈魁斯特(圆)稳定性判据可用于分析SISO和MIMO模糊系统的稳定性［62］，并且扩展圆判据可用于推导一类简单模糊PI控制系统稳定性的充分条件［63］。由于圆判据要求比较严格，Furutani提出一种移动的波波夫判据，用于分析模糊控制系统的稳定性。
Small gain theory is a very general tool for continuous and discrete systems in nonlinear control theory. Based on the analytical structure of fuzzy controller and the nonlinear nature of object and fuzzy controller, some scholars have established sufficient conditions for the stability of bounded input and bounded output (Bibo) of Mamdani fuzzy PI [59], PD [9], PID [14] and a class of simple and typical TS [20,21] fuzzy control systems by using small gain theory; It is proved that using nonlinear fuzzy PI controller instead of conventional PI controller does not affect the stability at the equilibrium point. Because these stability results are based on the structure of the controller, they are more non conservative than those whose analytical structure of the fuzzy controller is unknown.4.4 phase plane analysis method
The use of phase plane analysis technology is helpful to describe and understand the dynamic behavior of low-order fuzzy control systems. Therefore, phase plane analysis method is used to analyze the stability of some fuzzy systems [60-62], but this technology is limited to fuzzy systems with two-dimensional regular structure.
4.5 description function method
The description function method can be used to predict the existence, frequency, amplitude and stability of limit cycles. By establishing the relationship between fuzzy controller and multivalued relay controller, the description function method can be used to analyze the stability of fuzzy control system [31]. In addition, the description function technique of exponential input can also be used to investigate the transient response of fuzzy control system [63]. Although the description function method can be used in SISO and miso fuzzy controllers and some nonlinear object models, it can not be used in fuzzy controllers with three inputs or more. Since this method is generally used to determine the existence of periodic oscillation in nonlinear systems, it is only an approximate method.
4.6 criteria and methods of circular stability
The circle criterion can be used to analyze and redesign a fuzzy control system. Using the concept of sector bounded nonlinearity, the generalized Nyquist (circle) stability criterion can be used to analyze the stability of SISO and MIMO fuzzy systems [62], and the extended circle criterion can be used to derive sufficient conditions for the stability of a class of simple fuzzy PI control systems [63]. Due to the strict requirements of circle criterion, Furutani proposed a moving Popov criterion to analyze the stability of fuzzy control system.
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