Synchronization certificates for fractional-order neural networks often assume the full multivector state is measured. In practice only a partial output is available—the full multivector states of a subset of neurons—and the controller must run on an estimate. We give an output-feedback theory for leader-following synchronization of fractional-order Takagi–Sugeno fuzzy Clifford-valued neural networks in which each follower measures only some of its neurons, runs a local Luenberger-type fractional observer to reconstruct the rest, and pins the estimate to the leader (certainty equivalence). The analysis has three parts. First, a detectability certificate: a per-neuron M-matrix test on the observer-error energies under which the observer error decays, in which the output-injection gain raises the damping of the measured neurons only; the gain is essential for feasibility—without it the certificate is infeasible (\(a_o = (-2.34, -2.15, 1.67)\) vs. \((2.86, 3.06, 1.67)\))—and the simulator shows the open-loop error failing to synchronize (stalling at 0.80) while the gain drives it to \(2.6 \times 10^{-4}\). Second, a cascade certificate: the observer error is autonomous and drives the tracking error through a coupling \(\theta^{-1}\|H\|^2\); a triangular \(2\times 2\) fractional comparison gives joint synchronization at the rate \(\min(\text{observer}, \text{tracking})\), which is observer-limited here (1.27). Third, the price of output feedback, made quantitative: the coupling forces a tracking-rate loss \(\theta\) (the “separation” is weak, not clean), and the residual is \(14\times\) larger than full-state feedback; under a bounded sensor attack both ultimate errors scale as \(\bar{d}^2\), amplified by the control gain \(\|H\|^2\)—so high-gain pinning that speeds tracking also magnifies sensor-attack sensitivity. Selected inequalities and certificates are numerically audited, and the associated certified–empirical gap is measured and reported.
- Open Access
- Article
Observer-Based Output-Feedback Synchronization of Fractional-Order Fuzzy Clifford-Valued Neural Networks: A Cascade Certificate and the Quantified Price of Output Feedback
- Grienggrai Rajchakit
Author Information
Received: 14 Jul 2026 | Revised: 06 Aug 2026 | Accepted: 20 Oct 2026 | Published: 23 Oct 2026
Abstract
Keywords
output-feedback synchronization | fractional-order neural networks | Cliffordvalued neural networks | Takagi–Sugeno fuzzy systems | fractional-order observer | sensorfault robustness
References
- 1.
Li, Y.; Chen, Y.Q.; Podlubny, I. Stability of Fractional-Order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag-Leffler Stability. Comput. Math. Appl. 2010, 59, 1810–1821. https://doi.org/10.1016/j.camwa.2009.08.019.
- 2.
Ye, H.; Gao, J.; Ding, Y. A Generalized Gronwall Inequality and Its Application to a Fractional Differential Equation. J. Math. Anal. Appl. 2007, 328, 1075–1081. https://doi.org/10.1016/j.jmaa.2006.05.061.
- 3.
Tuan, H.T.; Trinh, H. A Qualitative Theory of Time Delay Nonlinear Fractional-Order Systems. SIAM J. Control Optim. 2020, 58, 1491–1518. https://doi.org/10.1137/19m1299797.
- 4.
Bao, H.; Park, J.H.; Cao, J. Synchronization of Fractional-Order Complex-Valued Neural Networks with Time Delay. Neural Netw. 2016, 81, 16–28. https://doi.org/10.1016/j.neunet.2016.05.003.
- 5.
Rajchakit, G.; Chanthorn, P.; Kaewmesri, P.; et al. Global Mittag-Leffler Stability and Stabilization Analysis of Fractional-Order Quaternion-Valued Memristive Neural Networks. Mathematics 2020, 8, 422. https://doi.org/10.3390/math8030422.
- 6.
Aguila-Camacho, N.; Duarte-Mermoud, M.A.; Gallegos, J.A. Lyapunov Functions for Fractional Order Systems. Commun. Nonlinear Sci. Numer. Simul. 2014, 19, 2951–2957. https://doi.org/10.1016/j.cnsns.2014.01.022.
- 7.
Duarte-Mermoud, M.A.; Aguila-Camacho, N.; Gallegos, J.A.; et al. Using General Quadratic Lyapunov Functions to Prove Lyapunov Uniform Stability for Fractional Order Systems. Commun. Nonlinear Sci. Numer. Simul. 2015, 22, 650–659. https://doi.org/10.1016/j.cnsns.2014.10.008.
- 8.
Kaslik, E.; Sivasundaram, S. Nonlinear Dynamics and Chaos in Fractional-Order Neural Networks. Neural Netw. 2012, 32, 245–256. https://doi.org/10.1016/j.neunet.2012.02.030.
- 9.
Matignon, D. Stability Results for Fractional Differential Equations with Applications to Control Processing. In Proceedings of the Computational Engineering in Systems Applications, Lille, France, 9–12 July 1996; pp. 963–968.
- 10.
Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999.
- 11.
Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006.
- 12.
Diethelm, K. The Analysis of Fractional Differential Equations; Springer: Berlin, Germany, 2010.
- 13.
Buchholz, S.; Sommer, G. On Clifford Neurons and Clifford Multi-Layer Perceptrons. Neural Netw. 2008, 21, 925–935. https://doi.org/10.1016/j.neunet.2008.03.004.
- 14.
Hitzer, E.; Nitta, T.; Kuroe, Y. Applications of Clifford’s Geometric Algebra. Adv. Appl. Clifford Algebr. 2013, 23, 377–404. https://doi.org/10.1007/s00006-013-0378-4.
- 15.
Brandstetter, J.; van den Berg, R.; Welling, M.; et al. Clifford Neural Layers for PDE Modeling. In Proceedings of the International Conference on Learning Representations (ICLR), Kigali, Rwanda, 1–5 May 2023.
- 16.
Aouiti, C.; Touati, F. Global Dissipativity of Clifford-Valued Multidirectional Associative Memory Neural Networks with Mixed Delays. Comput. Appl. Math. 2020, 39, 310. https://doi.org/10.1007/s40314-020-01367-5.
- 17.
Zhao, N.; Qiao, Y. Global Exponential Synchronization of Clifford-Valued Memristive Fuzzy Neural Networks with Delayed Impulses. Cogn. Comput. 2024, 16, 671–681. https://doi.org/10.1007/s12559-023-10221-9.
- 18.
Rajchakit, G.; Sriraman, R.; Boonsatit, N.; et al. Global Exponential Stability of Clifford-Valued Neural Networks with Time-Varying Delays and Impulsive Effects. Adv. Differ. Equ. 2021, 2021, 208. https://doi.org/10.1186/s13662-021-03367-z.
- 19.
Tatar, N.E. Fractional Halanay Inequality and Application in Neural Network Theory. Acta Math. Sci. 2019, 39, 1605–1618.
- 20.
Thinh, L.V.; Tuan, H.T. A Generalized Fractional Halanay Inequality and Its Applications. SIAM J. Control Optim. 2025, 63, 916–935. https://doi.org/10.1137/24m1680945.
- 21.
Yu, W.; Chen, G.; Lü, J. On Pinning Synchronization of Complex Dynamical Networks. Automatica 2009, 45, 429–435. https://doi.org/10.1016/j.automatica.2008.07.016.
- 22.
Olfati-Saber, R.; Murray, R.M. Consensus Problems in Networks of Agents with Switching Topology and Time-Delays. IEEE Trans. Autom. Control 2004, 49, 1520–1533. https://doi.org/10.1109/tac.2004.834113.
- 23.
Takagi, T.; Sugeno, M. Fuzzy Identification of Systems and Its Applications to Modeling and Control. IEEE Trans. Syst. Man Cybern. 1985, SMC-15, 116–132. https://doi.org/10.1109/tsmc.1985.6313399.
- 24.
Tanaka, K.; Wang, H.O. Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach; Wiley: New York, NY, USA, 2001. https://doi.org/10.1002/0471224596.
- 25.
Bao, H.; Cao, J. State Estimation of Fractional-Order Neural Networks with Time Delay. In Proceedings of the 2017 Chinese Automation Congress (CAC), Jinan, China, 20–22 October 2017; pp. 1573–1577. https://doi.org/10.1109/cac.2017.8243018.
- 26.
Li, R.; Gao, X.; Cao, J.; et al. Dissipativity and Exponential State Estimation for Quaternion-Valued Memristive Neural Networks. Neurocomputing 2019, 363, 236–245. https://doi.org/10.1016/j.neunet.2019.07.036.
- 27.
Zouari, F.; Ibeas, A.; Boulkroune, A.; et al. Adaptive Neural Output-Feedback Control for Nonstrict-Feedback Time-Delay Fractional-Order Systems with Output Constraints and Actuator Nonlinearities. Neural Netw. 2018, 105, 256–276. https://doi.org/10.1016/j.neunet.2018.05.014.
- 28.
Bounemeur, A.; Chemachema, M. Adaptive Fuzzy Fault-Tolerant Control for a Class of Nonlinear Systems under Actuator Faults: Application to an Inverted Pendulum. Int. J. Robot. Control Syst. 2021, 1, 102–115. https://doi.org/10.31763/ijrcs.v1i2.306.
- 29.
Bounemeur, A.; Chemachema, M.; Bouzina, S. Fuzzy Fault-Tolerant Control Applied on Two Inverted Pendulums with Nonaffine Nonlinear Actuator Failures. Int. J. Robot. Control Syst. 2023, 3, 144–160. https://doi.org/10.31763/ijrcs.v3i2.917.
- 30.
Bounemeur, A.; Chemachema, M. Optimal Adaptive Fuzzy Fault-Tolerant Control Applied on a Quadrotor Attitude Stabilization Based on Particle Swarm Optimization. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 2024, 238, 704–719. https://doi.org/10.1177/09596518231199212.
- 31.
Abdelhamid, B.; Mohamed, C. Robust Fuzzy Adaptive Fault-Tolerant Control for a Class of Second-Order Nonlinear Systems. Int. J. Adapt. Control Signal Process. 2025, 39, 15–30. https://doi.org/10.1002/acs.3916.
- 32.
Sun, Z.Z.; Wu, X. A Fully Discrete Difference Scheme for a Diffusion-Wave System. Appl. Numer. Math. 2006, 56, 193–209. https://doi.org/10.1016/j.apnum.2005.03.003.
- 33.
Jiang, S.; Zhang, J.; Zhang, Q.; et al. Fast Evaluation of the Caputo Fractional Derivative and Its Applications to Fractional Diffusion Equations. Commun. Comput. Phys. 2017, 21, 650–678. https://doi.org/10.4208/cicp.oa-2016-0136.
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