Abstract
This paper presents a low gain feedback design for fractional-order linear systems. A family of linear state feedback laws, parameterized in a positive low gain parameter \(\varepsilon \), is constructed through eigenstructure assignment. Under the assumption that the open-loop system is not exponentially unstable, the peak value of the low gain feedback for any given initial condition can be made arbitrarily small by decreasing the value of the low gain parameter toward zero. To establish this property, we derive an explicit asymptotic expansion of high-order derivatives of the Mittag-Leffler function and obtain upper bounds on their values. This salient feature of the low gain feedback design allows us to achieve semi-global stabilization of fractional-order linear systems subject to actuator saturation. The results in this paper extend the corresponding results for integer-order linear systems.
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
References
Aghayan, Z.S., Alfi, A., Machado, J.T.: Robust stability analysis of uncertain fractional order neutral-type delay nonlinear systems with actuator saturation. Appl. Math. Model. 90, 1035–1048 (2021)
Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2951–2957 (2014)
Bandyopadhyay, B., Kamal, S.: Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach. Springer (2015)
Bernstein, D.S., Michel, A.N.: A chronological bibliography on saturating actuators. Int. J. Robust Nonlinear Control 5(5), 375–380 (1995)
Bingi, K., Ibrahim, R., Karsiti, M.N., Hassan, S.M., Harindran, V.R., et al.: Fractional-Order Systems and PID Controllers. Springer (2020)
Bongulwar, M.R., Patre, B.M.: Design of FOPID controller for fractional-order plants with experimental verification. Int. J. Dyn. Control 6(1), 213–223 (2017)
Braaksma, B.L.J.: Asymptotic expansions and analytic continuations for a class of Barnes-integrals. Compos. Math. 15, 239–341 (1936)
Chevrié, M., Sabatier, J., Farges, C., Malti, R.: \({H}_2\)-norm of a class of fractional transfer functions suited for modeling diffusive phenomena. In: 2015 American Control Conference (ACC), pp. 2199–2204. IEEE (2015)
Chuang, L., Kai, C., Junguo, L., Rongnian, T.: Stability and stabilization analysis of fractional-order linear systems subject to actuator saturation and disturbance. IFAC-PapersOnLine 50(1), 9718–9723 (2017)
Erdélyi, A.: Higher Transcendental Functions. McGraw Hill Book Co (1955)
Farges, C., Moze, M., Sabatier, J.: Pseudo-state feedback stabilization of commensurate fractional order systems. Automatica 46(10), 1730–1734 (2010)
Fox, C.: The asymptotic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. s2–27(1), 389–400 (1928)
Fuller, A.: In-the-large stability of relay and saturating control systems with linear controllers. Int. J. Control 10(4), 457–480 (1969)
Gao, Z., Liao, X.: Integral sliding mode control for fractional-order systems with mismatched uncertainties. Nonlinear Dyn. 72(1), 27–35 (2013)
Garra, R., Garrappa, R.: The Prabhakar or three parameter Mittag-Leffler function: Theory and application. Commun. Nonlinear Sci. Numer. Simul. 56, 314–329 (2018)
Garrappa, R., Popolizio, M.: Computing the matrix Mittag-Leffler function with applications to fractional calculus. J. Sci. Comput. 77(1), 129–153 (2018)
Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.: Mittag-Leffler Functions. Related Topics and Applications. Springer, Berlin Heidelberg (2020)
Hu, T., Lin, Z.: Control Systems with Actuator Saturation: Analysis and Design. Birkhauser Boston Inc., Boston (2001)
Joshi, M.M., Vyawahare, V.A.: Constrained model predictive control for linear fractional-order systems with rational approximation. J. Appl. Nonlinear Dyn. 8(1), 35–53 (2019)
Kaczorek, T.: Fractional positive continuous-time linear systems and their reachability. Int. J. Appl. Math. Comput. Sci. 18(2), 223–228 (2008)
Karami-Mollaee, A., Tirandaz, H., Barambones, O.: On dynamic sliding mode control of nonlinear fractional-order systems using sliding observer. Nonlinear Dyn. 92(3), 1379–1393 (2018)
Li, Y., Chen, Y., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965–1969 (2009)
Li, Y., Chen, Y., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59(5), 1810–1821 (2010)
Lim, Y.H., Oh, K.K., Ahn, H.S.: Stability and stabilization of fractional-order linear systems subject to input saturation. IEEE Trans. Autom. Control 58(4), 1062–1067 (2012)
Lin, Z.: Low Gain Feedback. Springer (1999)
Lin, Z., Fang, H.: On asymptotic stabilizability of linear systems with delayed input. IEEE Trans. Autom. Control 52(6), 998–1013 (2007)
Lin, Z., Saberi, A.: Semi-global exponential stabilization of linear systems subject to input saturation via linear feedbacks. Syst. Control Lett. 21(3), 225–239 (1993)
Liu, H., Pan, Y., Cao, J., Wang, H., Zhou, Y.: Adaptive neural network backstepping control of fractional-order nonlinear systems with actuator faults. IEEE Trans. Neural Netw. Learn. Syst. 31(12), 5166–5177 (2020)
Lu, J.G., Chen, G.: Robust stability and stabilization of fractional-order interval systems: an LMI approach. IEEE Trans. Autom. Control 54(6), 1294–1299 (2009)
Lu, J.G., Chen, Y.Q.: Robust stability and stabilization of fractional-order interval systems with the fractional order \(\alpha \): The \(0<\alpha <1\) case. IEEE Trans. Autom. Control 55(1), 152–158 (2009)
Luo, D., Wang, J., Shen, D., Fečkan, M.: Iterative learning control for fractional-order multi-agent systems. J. Franklin Inst. 356(12), 6328–6351 (2019)
Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific (2010)
Mainardi, F., Spada, G.: Creep, relaxation and viscosity properties for basic fractional models in rheology. Eur. Phys. J. Special Top. 193(1), 133–160 (2011)
Matignon, D.: Stability properties for generalized fractional differential systems. ESAIM Proc. 5, 145–158 (1998)
Matignon, D., d‘Andréa Novel, B.: Some results on controllability and observability of finite-dimensional fractional differential systems. Comput. Eng. Syst. Appl. 2, 952–956 (1996)
Meneses, H., Arrieta, O., Padula, F., Vilanova, R., Visioli, A.: PI/PID control design based on a fractional-order model for the process. IFAC-PapersOnLine 52(1), 976–981 (2019)
Oustaloup, A.: La Commande CRONE: Commande Robuste d‘Ordre non Entier. Hermès, Paris (1991)
Paris, R.B.: Exponentially small expansions in the asymptotics of the Wright function. J. Comput. Appl. Math. 234(2), 488–504 (2010)
Petráš, I.: Novel fractional-order model predictive control: State-space approach. IEEE Access 9, 92769–92775 (2021)
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press (1999)
Rhouma, A., Bouani, F., Bouzouita, B., Ksouri, M.: Model predictive control of fractional order systems. J. Comput. Nonlinear Dyn. 9(3), 310111–310117 (2014)
Ross, B.: Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974, vol. 457. Springer (2006)
Sabatier, J., Moze, M., Farges, C.: LMI stability conditions for fractional order systems. Comput. Math. Appl. 59(5), 1594–1609 (2010)
Shahri, E.S.A., Alfi, A., Machado, J.T.: Lyapunov method for the stability analysis of uncertain fractional-order systems under input saturation. Appl. Math. Model. 81, 663–672 (2020)
Sussmann, H., Sontag, E., Yang, Y.: A general result on the stabilization of linear-systems using bounded controls. IEEE Trans. Autom. Control 39(12), 2411–2425 (1994)
Sussmann, H.J., Yang, Y.: On the stabilizability of multiple integrators by means of bounded feedback controls. In: Proceedings of the 30th IEEE Conference on Decision and Control, pp. 70–72. IEEE (1991)
Tarbouriech, S., Garcia, G.: Control of Uncertain Systems with Bounded Inputs. Springer (1997)
Teel, A.R.: Global stabilization and restricted tracking for multiple integrators with bounded controls. Syst. Control Lett. 18(3), 165–171 (1992)
Wang, J., Shao, C., Chen, Y.Q.: Fractional order sliding mode control via disturbance observer for a class of fractional order systems with mismatched disturbance. Mechatronics 53, 8–19 (2018)
Wang, R., YunNing, Z., Chen, Y., Chen, X., Lei, X.: Fuzzy neural network-based chaos synchronization for a class of fractional-order chaotic systems: an adaptive sliding mode control approach. Nonlinear Dyn. 100(2), 1275–1287 (2020)
Wang, X., Xu, B., Shi, P., Li, S.: Efficient learning control of uncertain fractional-order chaotic systems with disturbance. IEEE Trans. Neural Netw. Learn. Syst (2020)
Wright, E.M.: The asymptotic expansion of integral functions defined by Taylor series. Philosoph Trans. R Soc. London Ser. A Math. Phys. Sci. 238(795), 423–451 (1940)
Xu, X., Liu, L., Feng, G.: Stabilization of linear systems with distributed infinite input delays: a low gain approach. Automatica 94, 396–408 (2018)
Zhang, X., Huang, W.: Robust \({H}_{\infty }\) adaptive output feedback sliding mode control for interval type-2 fuzzy fractional-order systems with actuator faults. Nonlinear Dyn. 104(1), 537–550 (2021)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Xu, J., Lin, Z. Low gain feedback for fractional-order linear systems and semi-global stabilization in the presence of actuator saturation. Nonlinear Dyn 107, 3485–3504 (2022). https://doi.org/10.1007/s11071-021-07084-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-021-07084-w