The Variable Spaces in Fuzzy Control Systems

Fuzzy Control is a new control method emerged in 1970's and its fundaments are fuzzy set theory and fuzzy approximation theory.

For the fuzzy control method, firstly, the input variables and the output variables (or state variables) of the controlled systems in the continuous variable spaces are divided into some fuzzy domains and then the membership functions on each fuzzy domain of the fuzzy variables are established. Based on this, the variable state (or value) of the controlled systems in the practical variable spaces are transformed as the membership value on each fuzzy domain by the membership functions, that is, the values of the input and output variables in the continuous spaces will be represented with the discrete fuzzy domains and their membership values. On the contrary, the discrete fuzzy domain and its membership value by the domain membership functions can approximate the continuous variable values and are transformed as the approximating values of the real variables in the continuous-changing spaces.

If the fuzzy domain is abstracted as the concept 'symbol' in artificial intelligence and pattern recognition fields and the membership value of each domain is regard as the value of the symbolic variable, there exist three variable spaces in the representation of the variables and the dynamics of the controlled systems by the fuzzy relations and fuzzy approximations as follows.

1) the continuous real variable spaces. The variable states (or value) of the controlled systems are represented as the continuous-changing real value in the space.

2) the discrete symbolic variable space. The variable states are represented as the discrete-changing value of the symbolic variables in the space, by the fuzzy domains and the corresponding membership values.

3) the continuous approximating variable spaces. The variable states are represented as the continuous-changing approximating value in the space, based on the fuzzy domains and the corresponding membership functions and membership values.

In fact, the system analysis and control design of the fuzzy control systems, such as, kinematics analysis, stability analysis, control law design, and so on, are switched and carried on among three variable spaces based on fuzzy relations and fuzzy approximations. As we know, maybe there isn't exist the analytic system analysis and design control methods for the general nonlinear dynamic systems(see blog article “Some notes on the control problems for general nonlinear systems”[http://blog.sciencenet.cn/blog-3343777-1083712.html]). Therefore, based on the fuzzy relations and fuzzy approximation, or the multi-variable function approximation with the more strictly and deeply theoretic fundamentals, how to construct the representations of the input and output variables in the symbolic space and approximation space and how to establish the system analysis and control design methods switched freely among three variable spaces are important and open problems for the general nonlinear systems. It is important here that these establishing methods are with strictly theoretic analysis and high-quality control design, such as, stability analysis and design, controllability/observability analysis, performance index analysis and design, and so on.