- Research
- Open Access
Mittag-Leffler stabilization of fractional-order nonlinear systems with unknown control coefficients
- Xuhuan Wang^{1}Email author
https://doi.org/10.1186/s13662-018-1470-9
© The Author(s) 2018
- Received: 8 September 2017
- Accepted: 31 December 2017
- Published: 15 January 2018
Abstract
In this paper, we consider the problem of Mittag-Leffler stabilization of fractional-order nonlinear systems with unknown control coefficients. With the help of backstepping design method, the stabilizing functions and tuning functions are constructed. The controller is designed to ensure that the pseudo-state of the fractional-order nonlinear system converges to the equilibrium. The effectiveness of the proposed method has been verified by some simulation examples.
Keywords
- fractional order
- nonlinear systems
- backstepping
- adaptive control
- tuning function
1 Introduction
The concept of fractional differentiation appeared for the first time in a famous correspondence between L’Hospital and Leibniz, in 1695. Many mathematicians have further developed this area and we can mention the studies of Euler, Laplace, Abel, Liouville and Riemann. However, the fractional calculus remained for centuries a purely theoretical topic, with little if any connections to practical problems of physics and engineering. In recent years, the fractional calculus has been recognized as an effective modeling methodology for researchers [1]. As is well known, fractional calculus is a generalization of classical calculus to non-integer order. Compared with an integer-order system, a fractional-order system is a better option for engineering physics.
Fractional systems have been paid much attention to, for example, the fractional optimal control problems [2], stability analysis of Caputo-like discrete fractional systems [3, 4], fractional description of financial systems [5], fractional chaotic systems [6]. Especially, stabilization problem of fractional-order systems is a very interesting and important research topic. In recent years, more and more researchers and scientists have begun to address this problem [7–23]. With the help of the Lyapunov direct method, Mittag-Leffler stability and generalized Mittag-Leffler stability was studied [10, 20]. The Lyapunov direct method deals with the stability problem of fractional-order systems have been extended [11, 24]. Reference [25] studied the global Mittag-Leffler stability for a coupled system of fractional-order differential equations on network with feedback controls. Robust stability and stabilization of fractional-order interval systems with \(0<\alpha<1\) order have been studied [15]. Necessary and sufficient conditions on the asymptotic stability of the positivity continuous time fractional-order systems with bounded time-varying delays are investigated by the monotonic and asymptotic property [14]. The stabilization problem of a class of fractional-order chaotic systems have been addressed [12]. Pseudo-state stabilization problem of fractional-order nonlinear systems has attracted the attention of some researchers [7–9, 13, 16]. Moreover, any equilibrium of a general fractional-order nonlinear system described by either Caputo’s or Riemann-Liouville’s definition can never be finite-time stable was proved [26]. Finite-time fractional-order adaptive intelligent backstepping sliding mode control have been proposed to deal with uncertain fractional-order chaotic systems [27]. The time-optimal control problem for a class of fractional-order systems was proposed [28]. In addition, robust controller design problem for a class of fractional-order nonlinear systems with time-varying delays was investigated [29] and state feedback \(H_{\infty}\) control of commensurate fractional-order systems was studied [30].
Backstepping design method has been widely applied in stabilizing a general class of integer-order nonlinear systems. Backstepping design offers a choice of design tools for accommodation of uncertain nonlinearities [31]. It is well known that the backstepping design has been reported for nonlinear systems in strict-feedback form or triangular form [31–36]. Systematic design of globally stable and adaptive controllers for a class of parametric strict-feedback form are investigated by the backstepping design procedure [34]. The overparametrization and partial overparametrization problems were soon eliminated by elegantly introducing the tuning functions [33, 35]. On the other hand, with the aids of this frequency distribute model and the indirect Lyapunov method, the adaptive backstepping control of fractional-order systems were established [37–39]. As far as we know, there are few results on the generalization of backstepping into fractional-order systems. It was pointed out that the well-known Leibniz rule is not satisfied for fractional-order systems. Then an interesting question arises: when the states of system are not Leibniz rule, how to deal with the stabilization problem through design of tuning functions and adaptive feedback control law? So far, the stabilization problem of fractional-order nonlinear systems remains an open problem.
In this paper, we investigate the Mittag-Leffler stabilization problem of a class of fractional-order nonlinear systems. Compared with the existing results, the main contributions of this paper are as follows: (1) The backstepping design is extended to fractional-order nonlinear systems with unknown control coefficients, and an adaptive control scheme with tuning functions is proposed. It is proved that the stabilization problem of fractional-order nonlinear systems can be solved by the designed control scheme. (2) The Mittag-Leffler stabilization problem is achieved using a systematic design procedure and without any growth restriction on nonlinearities. (3) The controller is designed to ensure that the pseudo-state of the fractional-order system convergence to the equilibrium. (4) Successfully overcoming the difficulty of the fractional-order system without the Leibniz rule, and the tuning function is constructed to avoid overparameterization.
The remainder of this paper is organized as follows: Section 2 the problem formulation, some necessary concepts and some necessary lemmas are given. In Section 3, as the main part of this note, an adaptive controller and tuning functions are designed by using the backstepping method for fractional-order nonlinear systems. In Section 4, two numerical simulations are provided to illustrate the effectiveness of the proposed results. Finally, Section 5 concludes this study.
2 Problem formulation and preliminary results
Remark 1
It is worth pointing out that if let the unknown constants \(b_{i}=1\) (\(i=1,\ldots,n\)) and \(q=1\) in (1), the systems (1) reduces to the well-known parametric strict-feedback system. Moreover, if \(b_{i}=1\) (\(i=1,\ldots,n\)) and \(\varphi_{0}(x)\) is a constant, then system (1) will become the parametric strict-feedback form of fractional-order nonlinear system.
Definition 1
([40])
Remark 2
- (1)
If C is a constant, then \(D^{q} C=0\).
- (2)
\(D^{q}(\mu g(t)+\omega h(t))=\mu D^{q} g(t)+\omega D^{q} h(t)\),
- (3)Leibniz rule:$$D^{q}\bigl( g(t)h(t)\bigr)=\sum_{r=0}^{\infty} \frac{\Gamma(q+1)}{\Gamma(r+1)\Gamma (q-r+1)}D^{q-r}g(t)D^{r}h(t). $$
Remark 3
The well-known Leibniz rule \(D^{q}(fg) = (D^{q} f)g + f(D^{q} g)\) is not satisfied for differentiation of non-integer orders.
Lemma 1
([13])
Lemma 2
([7, 11] (Mittag-Leffler stability))
Lemma 3
([24])
Remark 4
In the case when \(x(t)\in R^{n}\), Lemma 3 is still valid. That is, \(\alpha\in(0,1)\) and \(t \geq t_{0}\), \(\frac {1}{2}D^{\alpha}x^{T}(t)x(t)\leq x^{T}(t)D^{\alpha}x(t)\). In addition, let \(x(t)\in R\) be a real continuously differentiable function. Then, for any \(p = 2^{n} , n \in N\), \(D^{\alpha}x^{p}(t)\leq px^{p-1}(t)D^{\alpha}x(t)\), where \(0<\alpha<1 \) (see [7]).
3 Backstepping design
In this section, we will give the backstepping design procedure of fractional-order systems.
Theorem 1
Proof
The design procedure is recursive. Its ith-order subsystem is stabilized with respect to a Lyapunov function \(V_{i}\) by the design of a stabilizing function \(\alpha_{i}\) and a tuning function \(\tau_{i}\). The update law for the parameter estimate θ̂ and the feedback control u are designed at the final step.
Unless \(z_{i}=0\), we have \(D^{q}V_{n}\leq0\), thus there exists a class-K function \(\gamma_{3}\) such that \(D^{q}V_{n}\leq-\gamma_{3}( \Vert \eta \Vert )\).
According to Lemma 2, the z-system is asymptotically Mittag-Leffler stable. □
Remark 5
In this paper, we constructed the virtual controllers and tuning functions to deal with the fractional stabilization problem, the backstepping technique has been extended to fractional-order systems. It should be noted that the Mittag-Leffler stability implies asymptotic stability [11]. Therefore, the Lyapunov direct method can be applied to obtain the asymptotical stability of the closed-loop system.
4 Simulation results
In this section, two examples are given to verify the effectiveness of the proposed scheme.
Example 1
Example 2
5 Conclusions
The problem of Mittage-Leffler stabilization has been investigated for a class of fractional-order nonlinear systems with the unknown control coefficients. The backstepping design scheme is extended to fractional-order systems, and an adaptive control law is proposed with fractional-order update laws to achieve an asymptotical Mittag-Leffler stabilization for the close-loop system, and the tuning function is constructed to avoid overparameterization. Finally, the effectiveness of the proposed method has been verified by some simulation examples.
Declarations
Acknowledgements
The work is supported in part by the National Natural Science Foundation of China (No. 11661065) and the Scientific Research Fund of Jiangxi Provincial Education Department (GJJ171135, GJJ151264, GJJ151267, GJJ161261).
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares to have no conflicts of interest.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Zhou, Y: Advances in fractional differential equations. III. Comput. Math. Appl. 64(10), 2965 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Baleanu, D, Jajarmi, A, Hajipour, M: A new formulation of the fractional optimal control problems involving Mittag-Leffler nonsingular kernel. J. Optim. Theory Appl. 175(3), 718-737 (2017) MathSciNetView ArticleGoogle Scholar
- Baleanu, D, Wu, GC, Bai, YR, Chen, FL: Stability analysis of Caputo-like discrete fractional systems. Commun. Nonlinear Sci. Numer. Simul. 48, 520-530 (2017) MathSciNetView ArticleGoogle Scholar
- Baleanu, D, Wu, GC, Zeng, SD: Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solitons Fractals 102, 99-105 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Hajipour, M, Jajarmi, A, Baleanu, D: An efficient non-standard finite difference scheme for a class of fractional chaotic systems. J. Comput. Nonlinear Dyn. 13(2), 021013 (2017) View ArticleGoogle Scholar
- Jajarmi, A, Hajipour, M, Baleanu, D: New aspects of the adaptive synchronization and hyperchaos suppression of a financial model. Chaos Solitons Fractals 99, 285-296 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Ding, DS, Qi, DL, Wang, Q: Non-linear Mittag-Leffler stabilisation of commensurate fractional-order non-linear systems. IET Control Theory Appl. 9(5), 681-690 (2015) MathSciNetView ArticleGoogle Scholar
- Ding, DS, Qi, DL, Meng, Y, Xu, L: Adaptive Mittag-Leffler stabilization of commensurate fractional-order nonlinear systems. In: Proceedings of 53rd IEEE CDC, pp. 6920-6926 (2014) Google Scholar
- Ding, DS, Qi, DL, Peng, JM, Wang, Q: Asymptotic pseudo-state stabilization of commensurate fractional-order nonlinear systems with additive disturbance. Nonlinear Dyn. 81, 667-677 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Li, Y, Chen, YQ, Podlubny, I: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965-1969 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Li, Y, Chen, YQ, 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) MathSciNetView ArticleMATHGoogle Scholar
- Shukla, MK, Sharma, BB: Stabilization of a class of fractional order chaotic systems via backstepping approach. Chaos Solitons Fractals 98, 56-62 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Wang, Q, Zhang, JL, Ding, DS, Qi, DL: Adaptive Mittag-Leffler stabilization of a class of fractional order uncertain nonlinear systems. Asian J. Control 18(6), 2343-2351 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Shen, J, Lam, J: Stability and performance analysis for positive fractional-order systems with time-varying delays. IEEE Trans. Autom. Control 61(9), 2676-2681 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Lu, JG, Chen, YQ: Robust stability and stabilization of fractional-order interval systems with the fractional order α: the \(0<\alpha<1\) case. IEEE Trans. Autom. Control 55(1), 152-158 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Farges, C, Moze, M, Sabatier, J: Pseudo-state feedback stabilization of commensurate fractional order systems. Automatica 46(10), 1730-1734 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Li, Y, Chen, YQ, Podlubny, I: Reply to “Comments on ’Mittag-Leffler stability of fractional order nonlinear dynamic systems’ [Automatica 45(8) (2009) 1965-1969]”. Automatica 75, 330 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Chen, LP, He, YG, Chai, Y, Wu, RC: New results on stability and stabilization of a class of nonlinear fractional-order systems. Nonlinear Dyn. 75, 633-641 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Chen, LP, Wu, RC, Cao, JD, Liu, JB: Stability and synchronization of memristor-based fractional-order delayed neural networks. Neural Netw. 71, 37-44 (2015) View ArticleGoogle Scholar
- Yu, JM, Hu, H, Zhou, SB, Lin, XR: Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems. Automatica 49, 1798-1803 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Trigeassou, JC, Maamri, N, Sabatier, J, Oustaloup, A: A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91, 437-445 (2011) View ArticleMATHGoogle Scholar
- Ke, YQ, Miao, CF: Mittag-Leffler stability of fractional-order Lorenz and Lorenz-family systems. Nonlinear Dyn. 83, 1237-1246 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Zhao, YG, Wang, YZ, Zhang, XF, Li, HT: Feedback stabilisation control design for fractional order non-linear systems in the lower triangular form. IET Control Theory Appl. 10(9), 1061-1068 (2016) MathSciNetView ArticleGoogle Scholar
- Aguila-Camacho, N, Duarte-Mermoud, MA, Gallegos, JA: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2951-2957 (2014) MathSciNetView ArticleGoogle Scholar
- Li, HL, Hu, C, Jiang, YL, Zhang, L, Teng, ZD: Global Mittag-Leffler stability for a coupled system of fractional-order differential equations on network with feedback controls. Neurocomputing 214, 233-241 (2016) View ArticleGoogle Scholar
- Shen, J, Lam, J: Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica 50(2), 547-551 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Bigdeli, N, Ziazi, HA: Finite-time fractional-order adaptive intelligent backstepping sliding mode control of uncertain fractional-order chaotic systems. J. Franklin Inst. 354, 160-183 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Wei, YH, Du, B, Cheng, SS, Wang, Y: Fractional order systems time-optimal control and its application. J. Optim. Theory Appl. 174(1), 122-138 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Hua, CC, Zhang, T, Li, YF, Guan, XP: Robust output feedback control for fractional order nonlinear systems with time-varying delays. IEEE/CAA Journal of Automatica Sinica 3(4), 477-482 (2016) MathSciNetView ArticleGoogle Scholar
- Shen, J, Lam, J: State feedback \(H_{\infty}\) control of commensurate fractional-order systems. Int. J. Syst. Sci. 45(3), 363-372 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Krstić, M, Kanellakopoulos, I, Kokotović, PV: Nonlinear and Adaptive Control Design. Wiley, New York (1995) MATHGoogle Scholar
- Ding, ZT: Adaptive control of non-linear systems with unknown virtual control coefficients. Int. J. Adapt. Control Signal Process. 14(5), 505-517 (2000) View ArticleMATHGoogle Scholar
- Krstić, M, Kanellakopoulos, I, Kokotović, PV: Adaptive nonlinear control without overparametrization. Syst. Control Lett. 19, 177-185 (1992) MathSciNetView ArticleMATHGoogle Scholar
- Kanellakopoulos, I, Kokotović, PV, Morse, AS: Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans. Autom. Control 36, 1241-1253 (1991) MathSciNetView ArticleMATHGoogle Scholar
- Chen, WT, Xu, CX: Adaptive nonlinear control with partial overparametrization. Syst. Control Lett. 44(1), 13-24 (2001) MathSciNetView ArticleMATHGoogle Scholar
- Krstić, M, Kokotović, PV: Control Lyapunov functions for adaptive nonlinear stabilization. Syst. Control Lett. 26(1), 17-23 (1995) MathSciNetView ArticleMATHGoogle Scholar
- Wei, YH, Chen, YQ, Liang, S, Wang, Y: A novel algorithm on adaptive backstepping control of fractional order systems. Neurocomputing 165, 395-402 (2015) View ArticleGoogle Scholar
- Wei, YH, Tse, PW, Yao, Z, Wang, Y: Adaptive backstepping output feedback control for a class of nonlinear fractional order systems. Nonlinear Dyn. 86, 1047-1056 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Sheng, D, Wei, YH, Cheng, SS, Shuai, JM: Adaptive backstepping control for fractional order systems with input saturation. J. Franklin Inst. 354, 2245-2268 (2017) MathSciNetView ArticleGoogle Scholar
- Arshad, M, Lu, DC, Wang, J: \((N +1)\)-dimensional fractional reduced differential transform method for fractional order partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 48, 509-519 (2017) MathSciNetView ArticleGoogle Scholar