Caputo Barrier Functions and Their Applications to the Safety, Safety-and-Stability, and Input-to-State Safety of a Class of Fractional-Order Systems
Abstract
:1. Introduction
1.1. Motivation
1.2. Contributions
- (1)
- We demonstrate the possibility of transferring the reciprocal barrier function (RBF) and the zeroing barrier function (ZBF) to Caputo fractional-order systems (CFOS), and we also propose Caputo RBF and Caputo ZBF. Based on two innovative Caputo BFs, we systematically derive the state safety criteria for fractional-order nonlinear dynamic systems. Our established state-safety theorems provide rigorous guarantees that all system states will remain within an known available state set, given that the initial conditions adhere to the set constraint.
- (2)
- On the basis of the theorem of asymptotical stability with guaranteed safety for CFOSs [29], we further propose the theorems of uniformly asymptotical stability with guaranteed safety and exponential stability with guaranteed safety for CFOSs. These two theorems demonstrate the possibility and solvability of achieving the synchronization of safety and uniformly asymptotical stability (or exponential stability).
- (3)
- We constructed a new description for the definition of Caupto input-to-state safety. The inspiration for this new description comes from the final product of the Caputo reciprocal barrier function proof process. The core inequality of our proposed definition of the Caputo input-to-state safety not only provides a unified representation of safety and ISSf for CFOSs, but it also facilitates the design and derivation of ISSf controllers. Then, under the definition of Caputo ISSf, using Caputo reciprocal barrier function and Caputo zeroing barrier function, we establish two ISSf criteria that can be directly applied to design Caputo ISSf controllers.
1.3. Organizations
2. Caputo Barrier Functions
2.1. Caputo RBF
2.2. Caputo ZBF
2.3. Comparison
3. Caputo Stability with Guaranteed Caputo Safety
3.1. Uniformly Asymptotic Stability with Guaranteed Caputo Safety
3.2. Exponential Stability with Guaranteed Caputo Safety
4. Caputo Input-to-State Safety
5. Conclusions
- (1)
- While CRBFs ensure the forward invariance of the set for CFOSs, thereby guaranteeing system safety, a significant limitation arises: the computational complexity inherent in fractional-order differentiation makes CRBFs less tractable for real-world safety control applications compared to their integer-order counterparts. This shortcoming stems from the nonlocal nature of fractional derivatives, which complicates barrier function implementation under Caputo dynamics.
- (2)
- We renewed the description of ISSf based on [15] and then established two theorems of ISSf via using Caputo RBF and Caputo ZBF, respectively, which can be used to more conveniently achieve the analysis and control of Caputo ISSf.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Prajna, S.; Rantzer, A. On the necessity of barrier certificates. IFAC Proc. Vol. 2005, 38, 526–531. [Google Scholar] [CrossRef]
- Kong, H.; Song, X.; Han, D.; Gu, M.; Sun, J. A New Barrier Certificate for safety verification of hybrid systems. Comput. J. 2014, 57, 1033–1045. [Google Scholar]
- Zhu, Z.; Chai, Y.; Yang, Z.; Huang, C. Exponential-alpha safety criteria of a class of dynamic systems with barrier functions. IEEE-CAA J. Autom. Sin. 2022, 9, 1939–1951. [Google Scholar]
- Dai, L.; Gan, T.; Xia, B.; Zhan, N. Barrier certificates revisited. J. Symb. Comput. 2017, 80, 62–86. [Google Scholar]
- Ames, A.; Grizzle, J.; Tabuada, P. Control barrier function based quadratic programs with application to adaptive cruise control. In Proceedings of the 2014 IEEE 53rd Annual Conference on Decision and Control (CDC), Los Angeles, CA, USA, 15–17 December 2014. [Google Scholar]
- Ames, A.; Xu, X.; Grizzle, J.; Tabuada, P. Control barrier function based quadratic programs for safety critical systems. IEEE Trans. Autom. Control 2017, 62, 3861–3876. [Google Scholar]
- Xu, X.; Tabuada, P.; Grizzle, J.; Ames, A. Robustness of control barrier functions for safety critical control. IFAC Pap. Online 2015, 48, 54–61. [Google Scholar]
- Sogokon, A.; Ghorbal, K.; Tan, Y.; Platzer, A. Vector barrier certificates and comparison systems. Lect. Notes Comput. Sci. 2018, 10951, 418–437. [Google Scholar]
- Zhu, Z.; Chai, Y.; Yang, Z. A novel kind of sufficient conditions for safety criterion based on control barrier function. Sci. China Inf. Sci. 2021, 64, 199205. [Google Scholar]
- Zhu, Z.; Chai, Y.; Yang, Z.; Huang, C. Safety criteria based on barrier function under the framework of boundedness for some dynamic systems. Sci. China Inf. Sci. 2022, 65, 122203. [Google Scholar]
- Wisniewski, R.; Sloth, C. Converse Barrier Certificate Theorems. IEEE Trans. Autom. Control 2016, 61, 1356–1361. [Google Scholar]
- Ratschan, S. Converse Theorems for Safety and Barrier Certificates. IEEE Trans. Autom. Control 2018, 63, 2628–2632. [Google Scholar]
- Romdlony, M.; Jayawardhana, B. Stabilization with guaranteed safety using control lyapunov-barrier function. Automatica 2016, 66, 39–47. [Google Scholar]
- Romdlony, M.; Jayawardhana, B. On the new notion of Input-to-State Safety. In Proceedings of the 2016 IEEE 55th Conference on Decision and Control (CDC), Las Vegas, NV, USA, 12–14 December 2016. [Google Scholar]
- Kolathaya, S.; Ames, A. Input-to-State Safety with Control Barrier Functions. IEEE Control Syst. Lett. 2019, 3, 108–113. [Google Scholar]
- Prajna, S.; Jadbabaie, A.; Pappas, G. Stochastic safety verification using barrier certificates. In Proceedings of the 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No. 04CH37601), Nassau, Bahamas, 14–17 December 2004. [Google Scholar]
- Glotfelter, P.; Cortés, J.; Egerstedt, M. Nonsmooth barrier functions with applications to multi-robot systems. IEEE Control Syst. Lett. 2017, 1, 310–315. [Google Scholar]
- Borrmann, U.; Wang, L.; Ames, A.; Egerstedt, M. Control barrier certificates for safe swarm behavior. IFAC Pap. Online 2015, 48, 68–73. [Google Scholar]
- Wang, L.; Ames, A.; Egerstedt, M. Safety barrier certificates for collisions-free multirobot systems. IEEE Trans. Robot. 2017, 33, 661–674. [Google Scholar]
- Wang, L.; Ames, A.; Egerstedt, M. Safety barrier certificates for heterogeneous multi-robot systems. In Proceedings of the 2016 American Control Conference (ACC), Boston, MA, USA, 6–8 July 2016. [Google Scholar]
- Ma, H.; Ren, H.; Zhou, Q.; Li, H.; Wang, Z. Observer-based neural control of n-link flexible-joint robots. IEEE Trans. Neural Netw. Learn. Syst. 2024, 35, 5295–5305. [Google Scholar]
- Ren, H.; Zhang, C.; Ma, H.; Li, H. Cloud-based distributed group asynchronous consensus for switched nonlinear cyber-physical systems. IEEE Trans. Ind. Inform. 2025, 21, 693–702. [Google Scholar]
- Zhang, C.; Yang, H.; Jiang, B. Fault Estimation and Accommodation of Fractional-Order Nonlinear, Switched, and Interconnected Systems. IEEE Trans. Cybern. 2022, 52, 1443–1453. [Google Scholar]
- Makhlouf, A.B.; Naifar, O. On the Barbalat lemma extension for the generalized conformable fractional integrals: Application to adaptive observer design. Asian J. Control 2023, 25, 563–569. [Google Scholar]
- Yu, J.; Hu, C.; Jiang, H. α-stability and α-synchronization for fractional-order neural networks. Neural Netw. 2012, 35, 82–87. [Google Scholar] [CrossRef]
- Delavari, H.; Baleanu, D.; Sadati, J. Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 2012, 67, 2433–2439. [Google Scholar] [CrossRef]
- Kilbas, A.; Srivastava, H.; Trujillo, J. Theory and Applications of Fractional Differentiable Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Zhu, Z.; Huang, P.; Zhang, X.; Chai, Y.; Song, Z. First attempt of barrier functions for Caputo’s fractional-order nonlinear dynamical systems. Sci. China Inf. Sci. 2022, 66, 179205. [Google Scholar] [CrossRef]
- Zhu, Z.; Liu, N.; Zhang, X.; Song, Z.; Chai, Y. A Novel Attempt of Barrier Function to the Safety of Caputo’s fractional-order systems. In Proceedings of the 2022 8th International Conference on Control, Decision and Information Technologies (CoDIT 22), Istanbul, Turkey, 17–20 May 2022. [Google Scholar]
- Khalil, H. Nonlinear Systems, 3rd ed.; Prentice Hall: Englewood Cliffs, NJ, USA, 2002. [Google Scholar]
- Bihari, I. A generalization of a lemma of bellman and its application to uniqueness problems of differentiable equations. Acta Math. Acad. Sci. Hung. 1956, 7, 81–94. [Google Scholar] [CrossRef]
- Slotine, J.; Li, W. Applied Nonlinear Control; Prentice Hall: Englewood Cliffs, NJ, USA, 1991. [Google Scholar]
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Zhu, Z.; Shen, B.; Yao, L.; Chai, Y.; Song, Z. Caputo Barrier Functions and Their Applications to the Safety, Safety-and-Stability, and Input-to-State Safety of a Class of Fractional-Order Systems. Mathematics 2025, 13, 1215. https://doi.org/10.3390/math13081215
Zhu Z, Shen B, Yao L, Chai Y, Song Z. Caputo Barrier Functions and Their Applications to the Safety, Safety-and-Stability, and Input-to-State Safety of a Class of Fractional-Order Systems. Mathematics. 2025; 13(8):1215. https://doi.org/10.3390/math13081215
Chicago/Turabian StyleZhu, Zheren, Bingbing Shen, Le Yao, Yi Chai, and Zhihuan Song. 2025. "Caputo Barrier Functions and Their Applications to the Safety, Safety-and-Stability, and Input-to-State Safety of a Class of Fractional-Order Systems" Mathematics 13, no. 8: 1215. https://doi.org/10.3390/math13081215
APA StyleZhu, Z., Shen, B., Yao, L., Chai, Y., & Song, Z. (2025). Caputo Barrier Functions and Their Applications to the Safety, Safety-and-Stability, and Input-to-State Safety of a Class of Fractional-Order Systems. Mathematics, 13(8), 1215. https://doi.org/10.3390/math13081215