A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control
Abstract
:1. Introduction
2. Fractional Differential-Algebraic Equations
3. Transformation Based on Sliding Mode Control Theory
- Based on nonlinear control theory, g(x) should satisfy the necessary and sufficient condition of the controllability for the nonlinear system Equation (4), namely
- In order to obtain control vector u, Js g(x) should be invertible, where Js is the Jacobi matrix of s(x).
4. The Numerical Algorithm
5. Violation Corrections
6. Results
6.1. An Example with Exact Solutions
6.2. Comparisons with Other Methods
6.3. The Effect of Violation Corrections
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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No. | Method | Maximum Difference (%) | |||
---|---|---|---|---|---|
h = 0.01 | h = 0.001 | h = 0.0005 | h = 0.0001 | ||
1 | The numerical method [26] | 36.47 | 4.56 | 2.31 | 0 |
2 | The predictor-corrector method [20] | 36.94 | 4.57 | 2.31 | 0 |
3 | The present method | 11.93 | 0.85 | 0.066 | −0.70 |
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Tai, Y.; Chen, N.; Wang, L.; Feng, Z.; Xu, J. A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control. Mathematics 2020, 8, 1134. https://doi.org/10.3390/math8071134
Tai Y, Chen N, Wang L, Feng Z, Xu J. A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control. Mathematics. 2020; 8(7):1134. https://doi.org/10.3390/math8071134
Chicago/Turabian StyleTai, Yongpeng, Ning Chen, Lijin Wang, Zaiyong Feng, and Jun Xu. 2020. "A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control" Mathematics 8, no. 7: 1134. https://doi.org/10.3390/math8071134
APA StyleTai, Y., Chen, N., Wang, L., Feng, Z., & Xu, J. (2020). A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control. Mathematics, 8(7), 1134. https://doi.org/10.3390/math8071134