3.3 模糊控制器是非线性增益规划控制器
典型和复杂的各类TS模糊控制器，从结构上已被证明是非线性增益规划器［20，21］。这些TS模糊控制器由多个梯形(或三角形)输入模糊集、带有线性后项的TS模糊规则、Zadeh模糊逻辑AND操作和重心解模糊器构成。与常规增益规划器在不同操作点带有不同常数增益的线性控制器不同的是：非线性增益规划器的增益随着被控系统的输出而不断变化。这些证明不仅弥补了以往一些学者对模糊控制器与增益控制器之间关系的简单说明，而且从另一方面解释了模糊控制器在处理非线性问题中的有效性。
3.4 模糊控制器与多值继电控制器的关系
Kickert和Mamdani揭示了模糊控制器与多值继电控制器的关系。一类简单的模糊控制器，其输入-输出特性具有多值继电特性，故可看作多值继电控制器［31］。Ying［3］证明了采用两个输入变量、多个三角形输入模糊集、线性控制规则、均匀分布的独点输出模糊集、不同推理方法和重心解模糊器的Mamdani模糊控制器，是一个全局的两维多值继电控制器和一个局部的非线性PI控制器之和［3］。这些结果被一般化到采用非均匀分布的多个三角形输入模糊集的SISO、采用非线性控制规则的SISO和MIMOMamdani模糊控制器［4－6］。根据模糊控制器与多值继电控制器的关系，可用经典控制理论中描述函数的方法来分析和设计模糊控制系统，并确保其稳定性。
The control input on the zero diagonal of the two-dimensional control rule set is zero. In working principle, fuzzy controller is similar to sliding mode variable structure controller [10, 24-28]. Wu and Liu expressed the fuzzy control as a kind of variable structure control, and the sliding mode is used to determine the best parameter value in the fuzzy control rules [29]. If the rule set of variable structure type is adopted, the fuzzy controller has semantic and quantitative variable structure characteristics. For two-dimensional and three-dimensional fuzzy controllers, its specific mathematical expression has been derived [30]. Compared with the usual sliding mode control, fuzzy control has stronger robustness, and the variable structure characteristics of fuzzy controller help people to design robust and stable fuzzy controller.3.3 fuzzy controller is a nonlinear gain planning controller
Typical and complex TS fuzzy controllers have been structurally proved to be nonlinear gain planners [20, 21]. These TS fuzzy controllers are composed of multiple trapezoidal (or triangular) input fuzzy sets, TS fuzzy rules with linear afterterms, Zadeh fuzzy logic and operation and barycentric defuzzifier. Different from the linear controller with different constant gains at different operating points of the conventional gain planner, the gain of the nonlinear gain planner changes continuously with the output of the controlled system. These proofs not only make up for the simple explanation of the relationship between fuzzy controller and gain controller, but also explain the effectiveness of fuzzy controller in dealing with nonlinear problems.
3.4 relationship between fuzzy controller and multivalued relay controller
Kickert and Mamdani revealed the relationship between fuzzy controller and multivalued relay controller. A kind of simple fuzzy controller, whose input-output characteristics have multivalued relay characteristics, can be regarded as multivalued relay controller [31]. Ying [3] proved that the Mamdani fuzzy controller using two input variables, multiple triangular input fuzzy sets, linear control rules, uniformly distributed single point output fuzzy sets, different reasoning methods and center of gravity defuzzifier is the sum of a global two-dimensional multivalued relay controller and a local nonlinear PI controller [3]. These results are generalized to SISO with non uniformly distributed triangular input fuzzy sets, SISO with nonlinear control rules and mimomamdani fuzzy controller [4-6]. According to the relationship between fuzzy controller and multivalued relay controller, the method of describing function in classical control theory can be used to analyze and design fuzzy control system and ensure its stability.
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