A kind of stable adaptive fuzzy control of nonlinear system is implemented based on variable universe method proposed firstly in [1]. First of all, the basic structure of variable universe adaptive fuzzy controllers is briefly introduced. Then the contraction-expansion factor which is a key tool of variable universe method is defined by means of integral regulation idea, and then a kind of adaptive fuzzy controllers is designed by using such contraction-expansion factor. The simulation on first order nonlinear system is done, as a result, its simulation effect is quite good in comparison with the corresponding result in [5, 6]. Secondly, it is proved that the variable universe adaptive fuzzy control is asymptotically stable by use of Lyapunov theory. The simulation on second order nonlinear system shows that its simulation effect is also quite good in comparison with the corresponding result in [5]. Besides, a useful tool, called symbolic factor, is proposed, which may be of universal significance. It can greatly reduce the setting time and enhance the robustness of the system.

This paper focuses on the control problem of the quadruple inverted pendulum by variable universe adaptive fuzzy control that is firstly proposed in [3-5]. First, the mathematical model on the quadruple inverted pendulum is described and its controllability is verified. Then, an efficient controller on the quadruple inverted pendulum is designed by using variable universe adaptive fuzzy control theory. At last, the simulation of the quadruple inverted pendulum is well shown in detail. Besides, the experiment results on the hardware systems, i.e., real object systems, on a single inverted pendulum, a double inverted pendulum and a triple inverted pendulum are briefly introduced.

The internal relations between fuzzy controllers and PID controllers are rewealed. First, it is pointed out that a fuzzy controller with one input and one output is just a piecewise p controller. Then it is proved that a fuzzy controller with two inputs and one output is just a piecewise PD (or I) controller with interaction between P and D (or PI). At last, the conclusion that a fuzzy controller with three inputs and one output is just a piecewise PID controller with interacting among P, I and D is given. Moreover, a kind of difference scheme of fuzzy controllers is designed.

A new idea, output-back fuzzy logic systems, is proposed in the paper. It is proved that output-back fuzzy logic systems must be equivalent to feedback neural networks. After the notion of generalized fuzzy logic systems is defined, which contains at least a typical fuzzy logic systems and an output-back fuzzy logic systems, one important conclusion is drawn that generallzed fuzzy logic systems are almost equivalent to neural networks.

A kind of modeling for fuzzy control systems is first proposed bere, which is called modeling method based on fuzzy inference (MMFI), It should be regarded as the third modeling method that is different from two well-known modelling methods, that is, the first modeling method, mechanism modeling method (MMM), and the second modelling method, system identification modeling method (SIMM). This method can, based on the interpolation mechanism on fuzzy logic system, transfer a group of fuzzy inference rules describing a practice system into a kind of nonlinear differential equation with variable coefficients, called HX equations, so that the mathematical model of the system can be obtained. This means that we solve the difficult problem of how to get a model represented as differential equations on a complicated or fuzzy control system.