Simulation of a Triple Inverted Pendulum Based on Fuzzy Control

The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms. This paper studies the use of fuzzy control method to study the stability control problem of a triple inverted pendulum system. By the linear model of the system, the feedback weight matrix of the LQR optimal control and the feedback parameters of the linear optimal control are designed to determine the parameters of the fuzzy controller. The simulation results show that the proposed method can achieve the stability control of the three stage inverted pendulum, and has good dynamic performance with simple parameter selection.


Introduction
The inverted pendulum system is a typical nonlinear, strong coupling, multivariable, naturally unstable system [1].In the control process, the inverted pendulum can effectively be used for studies key issues such as the quality of stability, robustness, mobility, tracking and that is why, it is the ideal model for testing various theories in control engineering [2] [3].So far, it has been used in classical control, modern control and intelligent control for stability analysis with techniques such as LQR control [4] [5] fuzzy control method [6], fuzzy neural network method [7] etc.When fuzzy control is used, the total number of fuzzy rules growth exponentially without the use of real-time processing [8].Through the fusion of multiple variables, the input variables of the fuzzy controller are reduced, diminishing the difficulty of the design and improving the real-time performance of the system.
In this paper, the design of a fuzzy controller based on the fusion structure for the triple inverted pendulum system is analyzed; the fusion weights of the multi-variables are based on the linear model.The control method The meaning of each parameter can refer to the literature [4].
The system status is , because the sensor can only measure the relative rotation angle of two connected swing rods, so the measurement is

Fuzzy Controller Structure Design
The fusion method is used to design a fuzzy controller for the triple inverted pendulum system.The eight input variables are divided into two groups: the direct quantity and the differential value and those 2 groups are a fused into two variables (see Figure 2).

Parameter Setting of Fuzzy Controller
Fuzzy controller for the above structure set the parameters as follows: 1) Determine the input and output domain For the E and EC domain can be selected as [−1, 1], the domain of Y is also selected as [−1, 1].
2) Determine the value of a variable membership function language For E, EC and Y, are selected 7 language values, respectively, as NB, NM, NS, ZE, PS, PM, PB.And the "trimf" membership function is adopted [10], in which the value of the core and the boundary value is the uniform distribution of Matlab by default.

Conclusion
In this paper, the used of fuzzy has facilitate the control of the triple inverted pendulum.Through the linear model of the system, the LQR optimal control feedback weight matrix was designed, and the fuzzy control parameters were designed based on the feedback parameters of the optimal linear control.The simulation results show that the method can realize the triple inverted pendulum system stability control and that fuzzy controller design method can also be applied to other multi variable and nonlinear system control problems.

Figure 2 .
Figure 2. Triple inverted pendulum fuzzy controller structure diagram system.

Table 1 .
Fuzzy rules of the fuzzy controller of the triple inverted pendulum.