Abstract: Seesaw systems are high-order control systems with severe nonlinearity and strong coupling, making it difficult to achieve ideal control results using conventional methods. This paper, based on sliding mode control theory and combined with fuzzy control methods, designs a controller using a fuzzy sliding mode control strategy. This method retains the strong robustness of sliding mode control while ensuring and improving the quality of the sliding mode of the control system, and simultaneously eliminates system chattering. Computer simulation experiments demonstrate the effectiveness and feasibility of this control method.
Keywords: seesaw; sliding mode control; fuzzy control; chattering
Application Research of Seesaw System Based on Fuzzy Sliding Mode Control
Fan Zhi-yong, Zhang Jing-gang
(College of Electronic Information Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, China)
Abstract: The seesaw system are serious non-linear, strong-coupling of high-order control systems, it is very difficult to use conventional methods to obtain the desired control effect. In this paper, we use fuzzy control combining with sliding mode control theory designing a controller. This method not only retains the strong robustness of sliding mode control, but also guarantees and improves the quality of the sliding-mode, chattering is eliminated at the same time. Computer simulation results show that the control method is effective and feasible.
Key Words: seesaw; sliding mode control; fuzzy control; chattering;
1 Introduction
Generally speaking, most physical processes are characterized by complexity, high nonlinearity, susceptibility to external disturbances, and great uncertainty. It is very difficult to use traditional controllers to handle such systems. However, with the continuous development of control theory, intelligent control can achieve better control results than traditional control methods even without knowing the mathematical model of the system [1]. 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 disturbances, and overly complex models [2-5]. The seesaw system consists of a trolley, a DC servo motor, two potentiometers for measuring angle and position respectively, 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 balancing [6].
Due to the highly nonlinear and strongly coupled characteristics of seesaw systems, as well as the chattering problem in variable structure control, this paper introduces fuzzy sliding mode control algorithm into the system control to soften the control input. Using fuzzy control strategy not only ensures and improves the quality of the sliding mode of the control system, but also eliminates chattering in sliding mode control.
2. Mathematical Model of the Seesaw System
The schematic diagram of the seesaw system is shown in Figure (1).
Figure (1) Schematic diagram of the seesaw system
The parameters in the diagram are defined as follows:
The angle of inclination of the lever; X: position of the cart; d1 : height of the lever relative to the fulcrum 0.125 m; d2 : height of the center point of the lever relative to the fulcrum 0.058 m; Iw : moment of inertia 0.395 kg ·m² ; mb : mass of the cart 0.57 kg; mw : mass of the lever 3.6 kg; : gravitational acceleration 9.81 N/kg.
Define Lagrange operators
L=TU （1）
Where T is the kinetic energy of the system and U is the potential energy of the system. Taking the state variables as , in order to construct the Lagrange equations, we obtain respectively.
Substituting equation (4) into equations (2) and (3), we can obtain equations (5) and (6).
The expressions for the sum can be obtained from equations (5) and (6).
The system of equations (7) is the nonlinear state equation expression of the system.
3. Design of Fuzzy Sliding Mode Controller
Sliding mode variable structure control (SMD) has advantages such as fast response and strong robustness, and is widely used in nonlinear system control. However, SMD is prone to chattering, leading to eventual instability. Fuzzy sliding mode control (FSC) is an intelligent control method for effectively controlling complex objects under uncertain environments. It does not rely on the system model and is completely robust to disturbances, while retaining the advantages of both fuzzy control and SMD. The basic design method of fuzzy sliding mode control is to compensate for the influence of unmodeled dynamics by adjusting the control action through fuzzy logic during the approach phase of the sliding mode control system. The aim 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 magnitude of the control input to ensure that the approach conditions of sliding mode control are met. The principle of fuzzy sliding mode control is shown in Figure 1.
Figure 1. Schematic diagram of fuzzy sliding mode control
As shown in the figure, the fuzzy sliding mode control system consists of three parts: a switching function, a fuzzy controller, and the controlled object. The input to the sliding mode function is the system state variable, and the switching function is designed as s = C·X.
(1)
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 problem of rule explosion in high-order systems with multiple inputs. The change in control is used as the output of the sliding mode controller, which can make fuzzy sliding mode control a model-free control with less dependence on the controlled object [7].
Based on the principle of fuzzy control, a fuzzy set is defined.
Where PB, PM, PS, ZO, NS, NM, and NB represent positive large, positive medium, positive small, zero, negative small, negative medium, and negative large, respectively. The control table obtained under the condition of satisfying inequalities is shown in Table 1. The fuzzy rule used is...
Table 1 Fuzzy Control Rule Table
All the control rules in the table are designed based on satisfying the necessary and sufficient condition for achieving sliding mode[8], so the designed fuzzy sliding mode control system is stable.
4. Simulation Study
Define S, , whose universes of discourse are respectively
All fuzzy variables are selected from the normal distribution membership function.
(1) Take the value in the formula. The simulation results are shown in Figures 2-5.
Figure 2 Curve showing the change in the position of the trolley over time
Figure 3. Curve of lever angle changing over time
Figure 4. Switching function versus time curve
Figure 5. Curve of control law versus time
The simulation results above show that the controller designed using the scheme in this paper greatly accelerates the system response speed, effectively reduces the maximum deviation of the system, and can basically eliminate the chattering phenomenon of the system.
5. Conclusion
This paper introduces the working principle of a seesaw system and establishes its mathematical model. Addressing the chattering phenomenon present in conventional sliding mode control, fuzzy sliding mode control is introduced into the seesaw control system. Simulation results demonstrate that applying fuzzy sliding mode control to a seesaw system with strong coupling and nonlinear characteristics is feasible, and the controller designed using the sliding mode fuzzy control algorithm exhibits strong robustness.
References
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[2] Chia-Ju Wu.Quasi Time-Optimal PID Control of Multivariable Systems:A Seesaw Example[J] Journal of the Chinese Institute of Engineers, Vol.22, No.5,pp.617-625(1999)
[3] Lon-Chen Hung, Hung-Yuan Chung .Decoupled Control Using Neural Network-based Sliding-mode Controller for Nonlinear Systems[J] .Expert Systems with Applications 32 (2007) 1168–1182
[4] Jeng-HannLI, Tzuu-Hseng S. Li* and Ting-Han Ou. Design and Implementation of Fuzzy Sliding-Mode Controller for a Wedge Balancing System[J]. Journal of Intelligent and Robotic Systems 37: 285–306, 2003.
[5] Chun-Hsien Tsai, Hung-Yuan Chung. Neuro-Sliding Mode Control With Its Applications to Seesaw Systems. IEEE Transactions on Nural Networks, vol 5, No. 1, January 2004
[6] Wincon User Guide[M].522-530.2003
[7] Liu Jinkun. Matlab Simulation of Sliding Mode Variable Structure Control [M]. Beijing: Tsinghua University Press, 2005: 100-120.
[8] Liu Jinkun. MATLAB Simulation of Sliding Mode Variable Structure Control [M]. Beijing: Tsinghua University Press, 2005.
About the author: Fan Zhiyong, male, born in October 1985, is a master's student at Taiyuan University of Science and Technology. His research topic is "Intelligent Control of Seesaw Systems", and his research direction is advanced control theory and application.
Contact address: Room 633, Taiyuan University of Science and Technology, Shanxi Province, Postcode: 030024.
Contact number: 15834001536
Email: [email protected]
