Structure analysis and function evaluation of a kind of fuzzy PID controllers

In this paper, a kind of fuzzy PID (Proportional integral and derivate) controllers is discussed, which has 3 input variables (error, difference of error, sum of error) and one output variable; triangular fully-overlapped symmetric membership function for input variables; singleton equally-spaced membership function for the output variable; linear control rules; Sum-Product inference method; and Center of area (COA) defuzzification algorithm. The paper consists of three main parts. In the first part, the structure properties of fuzzy PID controllers are studied. The explicit expression of this kind of fuzzy PID controllers is derived. It is proved that the analytical structure of fuzzy PID controllers is the sum of a global three-dimensional multi-level relay and a local nonlinear controller. When the number of fuzzy sets tends to infinity, the local nonlinear controller will disappear, and the degree of nonlinearity of the fuzzy PID controller becomes zero. In the second part, the function properties of fuzzy PID controllers are studied. It is revealed that the fuzzy PID controller is a variable gain nonlinear PID controller; so linear PID controllers can be regarded as a special example of fuzzy PID controllers. Moreover, they are equivalent to the sum of three fuzzy controllers with one-to-one mapping; so they do not suffer from some weaknesses such as composed-action, input coupling, etc. Based on these theoretical results, a systematic design approach of fuzzy PID control systems is proposed and demonstrated by 2 simulation examples in the third part of this paper. It is shown that the proposed fuzzy PID controller not only has good structure and function characteristics, but also can be very simply and quickly designed; therefore, it is very suitable for a wide range of applications.