Design of seesaw control system based on fuzzy sliding mode control

The seesaw system is a typical control system that is more complex than the inverted pendulum system and closer to practical applications. It has the characteristics of severe nonlinearity, strong coupling, sensitivity to interference, and overly complex models [2-5]. The seesaw system consists of a trolley, a DC servo motor, two potentiometers for measuring angle and position, and a seesaw triangle. The mechanism for balancing the seesaw is to use the movement of the trolley in the seesaw system to achieve the purpose of balance [6].

Due to the high nonlinearity and strong coupling of the seesaw system and the chattering problem of variable structure control, this paper introduces the fuzzy sliding mode control algorithm into the system control to soften the control quantity. The use of fuzzy control strategy can not only ensure and improve the quality of the sliding mode of the control system, but also eliminate the chattering phenomenon in the sliding mode control.

2 Mathematical model of seesaw system

The schematic diagram of the seesaw system is shown in Figure (1).

Figure (1) Schematic diagram of seesaw system

The parameters in the figure are defined as follows:

The tilt angle of the lever; X: the position of the trolley; d1: the height of the lever relative to the fulcrum 0.125m; d2: the height of the center point of the lever relative to the fulcrum 0.058m; Iw: moment of inertia 0.395kg.m2; mb: the mass of the trolley 0.57kg; mw: the mass of the lever 3.6kg;: acceleration due to gravity 9.81N/kg.

Define the Lagrangian operator

L = TU (1)

Where T is the kinetic energy of the system and U is the potential energy of the system. Take the state variable as , to construct the Lagrangian equation, respectively solve

Substituting equation (4) into equations (2) and (3), we can obtain equations (5) and (6):

The expressions of and can be obtained respectively through equations (5) and (6)

Equation group (7) is the nonlinear state equation expression of the system.

3 Design of fuzzy sliding mode controller

Sliding mode variable structure control has the advantages of fast response speed and strong robustness, and is widely used in nonlinear system control. However, sliding mode control is prone to cause system chattering, resulting in the ultimate instability of the system. Fuzzy sliding mode control is an intelligent control method for effectively controlling complex objects under uncertain environments. It does not rely on the model of the system and is completely robust to interference, while maintaining the advantages of fuzzy control and sliding mode control. The basic design method of fuzzy sliding mode control is to compensate for the influence of unmodeled dynamics through fuzzy logic adjustment control in the approaching stage of the sliding mode control system. Its purpose is to improve the quality of the control system, reduce the time to reach the sliding surface, and reduce chattering. In this paper, fuzzy control rules are used to adjust the size of the control input to ensure that the sliding mode control arrival conditions are met. The principle of fuzzy sliding mode control is shown in Figure 1.

Figure 1 Schematic diagram of fuzzy sliding mode control

As can be seen from the figure, the fuzzy sliding mode control system consists of three parts, namely the switching function, the fuzzy controller, and the controlled object. The input of the sliding mode function is the system state variable, and the switching function is designed as s=C·X

The input of the fuzzy controller is the switching function and its rate of change, which can effectively reduce the number of fuzzy rules and solve the rule explosion problem in high-order systems with multiple inputs. The control change is used as the output of the sliding mode controller, which makes the fuzzy sliding mode control a model-free control with a low degree of dependence on the controlled object [7].

According to the fuzzy control principle, fuzzy sets are defined.

Among them, PB, PM, PS, ZO, NS, NM, and NB represent positive large, positive medium, positive small, zero, negative small, negative medium, and negative large, respectively. Under the condition of satisfying the inequality, the control table obtained is shown in Table 1. The fuzzy rules used are

Table 1 Fuzzy control rules table

All the control rules in the table are designed based on satisfying the necessary and sufficient conditions for achieving sliding mode control [8], so the designed fuzzy sliding mode control system is stable.

4 Simulation study

Figure 2 Curve of the car position changing with time

Figure 3 Curve of lever angle changing with time

Figure 4 Switching function versus time curve

Figure 5 Control law variation curve over time

It can be seen from the above simulation results that the controller designed using the scheme in this paper greatly speeds up the response speed of the system, and can effectively reduce the maximum deviation of the system, and the chattering phenomenon of the system can be basically eliminated.

5 Conclusion

This paper introduces the working principle of the seesaw system and establishes a mathematical model of the seesaw system. Aiming at the chattering phenomenon existing in conventional sliding mode control, the fuzzy sliding mode control method is introduced into the seesaw control system. Through simulation, it can be seen that it is feasible to apply fuzzy sliding mode control to the seesaw system with strong coupling and nonlinear characteristics, and the controller designed using the sliding mode fuzzy control algorithm has strong robustness.