Fuzzy PDC-Based LQR Sliding Neural Network Control for Two-Wheeled Self-Balancing Cart
Abstract
:1. Introduction
2. Two-Wheeled Self-Balancing Cart Mathematical Model
- : the horizontal displacement of the center, O, of the chassis of the cart;
- : the mass of the vehicle body (kg);
- : the distance from center of mass to center of chassis (m);
- : the moment of inertia of the vehicle body when rotating around the center of mass ();
- : the angle between the vehicle body and the vertical direction (rad);
- : the wheel mass (kg);
- : the radius of the wheel (m);
- : the output torque of the right wheel motor ();
- : the output torque of the left wheel motor ();
- : the moment of inertia of the wheel ();
- : the gravity acceleration magnitude (9.8 ).
3. LQR Controller Design
4. Fuzzy PDC-Based Sliding Neural Network Design
4.1. Sliding Controller Design
4.2. Fuzzy PDC-Based LQR Controller Design
4.3. Neural Network Controller Design
- (a)
- Input Layer: Each node within this layer is related to the net input and net output values, which are, respectively, denoted as and .
- (b)
- Hidden Layer: Each node in this layer performs activation using the ReLU function, which is defined as [22,23,24,25,26,27,28]. The derivative of the ReLU is 1 when the input number is greater than zero; otherwise, it is equal to 0. The ReLU will be used as the activation function for the j-th node of the i-th input.
- (c)
- Output Layer: It consists of two nodes, one for LQR and the other for the sliding mode controller, which is responsible for performing the key operation of calculating the percentage of the two controllers. Among these nodes, we have a specific node labeled , which learns and memorizes the optimal percentage output by monitoring all input system states signals. This summation process is represented by the following equation:
- : The link weight denoting the strength of the output action corresponding to the output of the j-th column in relation to the k-th rule.
- : The j-th input to the node of layer.
- : The total number of output nodes.
- : The k-th column output of the ReNN controller.
5. Simulation Results
6. Experiment Results
7. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Mon, Y.-J. Fuzzy PDC-Based LQR Sliding Neural Network Control for Two-Wheeled Self-Balancing Cart. Electronics 2025, 14, 1842. https://doi.org/10.3390/electronics14091842
Mon Y-J. Fuzzy PDC-Based LQR Sliding Neural Network Control for Two-Wheeled Self-Balancing Cart. Electronics. 2025; 14(9):1842. https://doi.org/10.3390/electronics14091842
Chicago/Turabian StyleMon, Yi-Jen. 2025. "Fuzzy PDC-Based LQR Sliding Neural Network Control for Two-Wheeled Self-Balancing Cart" Electronics 14, no. 9: 1842. https://doi.org/10.3390/electronics14091842
APA StyleMon, Y.-J. (2025). Fuzzy PDC-Based LQR Sliding Neural Network Control for Two-Wheeled Self-Balancing Cart. Electronics, 14(9), 1842. https://doi.org/10.3390/electronics14091842