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Stabilization of a class of fractional-order nonautonomous systems using quadratic Lyapunov functions
- Quan Xu^{1, 2, 3}Email authorView ORCID ID profile,
- Shengxian Zhuang^{3},
- Xiaohui Xu^{2, 4},
- Chang Che^{3} and
- Yankun Xia^{5}
https://doi.org/10.1186/s13662-017-1459-9
© The Author(s) 2018
- Received: 4 September 2017
- Accepted: 28 December 2017
- Published: 15 January 2018
Abstract
In this paper, we aim to solve the stabilization problem for a large class of fractional-order nonautonomous systems via linear state feedback control and adaptive control. By constructing quadratic Lyapunov functions and utilizing a new property for Caputo fractional derivative we derive some sufficient conditions for the global asymptotical stabilization of a class of fractional-order nonautonomous systems. We give two illustrative examples to validate the effectiveness of the theoretical results.
Keywords
- fractional calculus
- state feedback control
- adaptive control
- quadratic Lyapunov functions
- fractional-order nonautonomous systems
1 Introduction
Fractional calculus, as a mathematical tool dealing with derivatives and integrals of arbitrary orders, has played a central role in physics [1], differential and integral equations [2], signal processing [3], human relationships [4], image encryption [5], thermal conductivity [6], electrical circuits [7], dynamical models [8], nonlinear control systems [9], complex networks [10], and so on. One of very important areas of application is nonlinear control systems. In particular, stability analysis and stabilization are of theoretical and practical importance for control systems, certainly including fractional-order systems. The early studies on the stability of fractional-order systems mainly concentrated on the linear cases, and many well-known results have been obtained. However, the stability analysis of fractional-order nonlinear systems remains an open problem. More details on the development of stability of fractional-order systems can be found in [11–15].
Recently, the stability and stabilization problem of a class of fractional-order nonlinear systems has attracted increasing interest of scholars [16–20]. In these literatures, it has been shown that most of the well-known chaotic systems can be modeled as this kind of fractional-order nonlinear systems (so-called semilinear systems), where the nonlinear vector field can be separated into linear and nonlinear parts. Moreover, the Mittag-Leffler function, the Laplace transform, and the Gronwall lemma are the main techniques used to prove the stability. It was shown that these techniques are neither simple nor straightforward and the proofs of the corresponding theorems are very complex. It should be also noted that the stability results in [16–19] are local, and all of the stability results in [16–20] are valid only for fractional-order autonomous systems. However, as we know, the stability results for fractional-order nonautonomous systems, including chaotic and nonchaotic systems, are still relatively few.
In integer-order nonautonomous systems, the Lyapunov direct method is an effective way to analyze the stability of a system. Motivated by the application of fractional calculus in nonlinear systems, the Lyapunov direct method has been extended to fractional-order systems by Li et al. [21, 22]. Similarly, Baleanu et al. [23] and Wu et al. [24] extended the theorem to fractional-order functional systems and fractional-order discrete systems. The fractional Lyapunov method generalizes the idea that the stability condition is derived by constructing a suitable Lyapunov function and calculating its fractional derivative. However, it is not an easy task to apply this method in the stability analysis of a general fractional-order nonlinear system. In [25], it has been demonstrated that fractional derivatives of noninteger orders cannot satisfy the Leibniz rule. So far, there are no techniques available to calculate the fractional derivative of a general composite Lyapunov function.
Recently, some efforts have been devoted to application of the fractional Lyapunov direct method in stability analysis of fractional-order systems. Especially, in [26], a simple Lyapunov function \(V= \lambda^{T} x\), \(\lambda>0\) has been proposed to solve the stabilization problem for fractional-order linear positive systems. By constructing some suitable stochastic Lyapunov functions Agarwal et al. [27] established some sufficient conditions for two types of stability of stochastic differential equations. In [28], based on the frequency distributed fractional integrator model, a fractional-order system is transformed into an equivalent integer-order system, and then similar stability results as those for integer-order systems are obtained by using quadratic Lyapunov functions. Hu et al. [29] considered an integer-order derivative instead of the fractional-order derivative of a Lyapunov function to prove the revised Lyapunov stability theorems. Nevertheless, the proposed Lyapunov functions [21–24, 26–29] are valid only for some fractional-order systems with special characteristics.
In classic Lyapunov theory, the quadratic form is one of the most commonly used Lyapunov functions for general integer-order nonlinear systems. Motivated by this, Aguila-Camacho et al. [30] and Duarte-Mermoud et al. [31] introduced two new inequalities for estimating the Caputo fractional derivative of a quadratic function and a common quadratic function, respectively. Thus, wecan try to prove the stability for general fractional-order nonlinear systems by using common quadratic Lyapunov functions.
Quite recently, Liu et al. [32] studied the stability of a class of fractional nonlinear systems using the fractional Lyapunov direct method and a new lemma proposed in [30, 31]. In this paper, we aim to solve the stabilization problem for such fractional-order systems via linear state feedback control and adaptive control.
The main contributions of this paper are as follows. First, constructing quadratic Lyapunov functions and using a new property for the Caputo derivative, we respectively investigate the stabilization of a class of fractional-order nonautonomous systems via state feedback control and adaptive control. Then, we derive some sufficient conditions for the global asymptotical stabilization. Note that using the existing results in [16–20], we may draw a wrong conclusion about asymptotic stability for our nonautonomous model. Moreover, asymptotic stability analysis of fractional-order nonautonomous systems is more difficult. This technical difficulty can be overcome by the fractional Lyapunov direct method [21, 22] and a new property for the Caputo derivative. Further, we also prove that quadratic Lyapunov functions are always valid for our model. Differing from [16–20], our results can only be applied not only to fractional-order autonomous systems, but also to fractional-order nonautonomous systems. In practice, we can easily realize the stabilization of this kind of fractional-order systems based on our criteria.
The remainder of this paper is organized as follows. In Section 2, we introduce some useful definitions, properties, and preliminaries for fractional calculus and fractional-order systems. In Section 3, we give three stabilization criteria for a class of fractional-order nonautonomous systems. In Section 4, we provide three numerical examples to illustrate the effectiveness of the theoretical results. Finally, we give some conclusions and further work in Section 5.
Notions
Let \(\mathbb{ R}= ( -\infty,+\infty ) \), \(\mathbb{R}^{n}\) be the n-dimensional Euclidean space, and let \(\mathbb{R}^{m \times n}\) be the space of \(m \times n\) real matrices; \(\Vert x \Vert = \sqrt{x^{T} x} \) is the two-norm of a vector x, \(\Vert P \Vert = \sqrt{ \lambda_{\max} ( P^{T} P )}\) is the two-norm of a matrix P, and \(\lambda_{\min} ( \cdot )\) (\(\lambda_{\max} (\cdot ) \)) denotes the minimum (maximum) eigenvalue of the corresponding matrix.
2 Preliminaries
In this paper, we consider the Caputo definition of fractional derivative, which is most popular in engineering applications.
Definition 2.1
([33])
Definition 2.2
([33])
Property 2.1
([33])
\(I_{t}^{\alpha} D_{t}^{\alpha} f ( t ) = f ( t ) -f ( t_{0} ) \), \(\forall t \geq t_{0}\).
For the stability analysis of system (2.4), the fractional Lyapunov direct method has been proposed [21, 22], which is stated in Lemma 2.1.
Remark 2.1
From [22] we know that system (2.4) has the same equilibrium points as the integer-order system \(\dot{x} ( t ) = f ( t,x ) \).
Definition 2.3
A continuous function \(\gamma :[0,t]\rightarrow [0,+\infty]\) is said to belong to class \(\mathcal{K}\) if it is strictly increasing and \(\gamma ( 0 ) =0\).
Lemma 2.1
Remark 2.2
Obviously, condition (a) of Lemma 2.1 is naturally satisfied if we choose \(V= x^{T} Px\) as a Lyapunov function.
According to [31], calculating the fractional derivative of the product of two functions implies evaluating an infinite sum. Obviously, this is not an easy task. This is also the main reason that fractional theory, especially fractional Lyapunov direct theory, is not as popular as it should be. Recently, a new property for the Caputo derivative is stated in Lemma 2.2 [31], which can facilitate estimating the fractional derivative of a common quadratic Lyapunov function.
Lemma 2.2
([31])
Definition 2.4
([22])
3 Main results
In this section, by constructing quadratic Lyapunov functions we derive some stabilization criteria for a class of fractional-order nonautonomous systems based on the fractional Lyapunov direct method and other inequality techniques.
Remark 3.1
In general, a differential equation is called semilinear if it consists of the sum of a well-understood linear part plus a lower-order nonlinear part. It is well known that many real systems in engineering and science can be modeled as semilinear systems [34]. Therefore, our model can describe a large class of fractional-order physical systems, including linear systems with nonlinear perturbation, chaotic systems, and so on. Our major aim is to show how a simple state feedback controller can be designed to stabilize this type of systems if the nonlinear part satisfies a constraint.
Assumption 3.1
For a nonlinear function g, \(g ( t,0 ) =0\) and there exists a constant \(\varepsilon>0\) such that \(\Vert g ( t,x ) \Vert \leq\varepsilon \Vert x \Vert \) for \(x\in \mathbb{R}^{n}\) and \(t\in[t_{0},+\infty]\).
Remark 3.2
In fact, Assumption 3.1 is the so-called global Lipschitz condition. It seems that this assumption is not easy to satisfy in general nonlinear systems. However, as stated in Remark 3.1, our model has potential applications in robust stabilization of linear systems with nonlinear perturbation and chaotic control and synchronization. According to Remark 3 in [35], if a system is Lipschitz, then it can be made contracting by means of a simple (static) state feedback, implying the possibility of designing simple observers/controllers. Fortunately, since the chaotic is bounded, we can easily verify that many well-known chaotic systems, such as Chua’s circuit, Arneodo’s system, Lorenz’s system, Duffing’s oscillator, Chen’s system, etc., all satisfy this assumption.
Quite recently, in [32], conditions for the asymptotical stability of system (3.1) have been derived by using the fractional Lyapunov direct method and Lemma 2.2.
Lemma 3.1
([32])
In this paper, we further study the stabilization of system (3.1) via linear state feedback control and adaptive control.
Theorem 3.1
Proof
Remark 3.3
It should be noted that the feasible set Y and Z satisfying (3.4) can be solved by using the LMI Toolbox in Matlab software. Thus, a feedback gain matrix K can be further obtained by \(K = Z Y^{-1}\). However, without any constraint on K, the feedback controller to stabilize a huge dimension system may be relatively complex and difficult to apply in practice.
Theorem 3.2
Suppose that Assumption 3.1 holds. If the feedback gain matrix K is selected such that all eigenvalues of Ã satisfy \(\operatorname{Re} \lambda ( \tilde{A} ) < 0\) and the solution P of the equation \(P \tilde{A} + \tilde{A}^{T} P =-I \) satisfies \(1-2 \varepsilon\lambda_{\max} ( P ) >0\), then the controlled system (3.3) is globally asymptotically stable about its equilibrium point, where I denotes the identity matrix.
Proof
Next, the fractional adaptive law is designed to tune the feedback gain matrix K of system (3.3). For simplification and without loss of generality, we assume that \(K=\operatorname{diag}( k_{1} (t), k_{2} (t),\ldots, k _{n} (t) )\).
Theorem 3.3
Proof
Recall now that, for \(0<\alpha\leq\beta<1 \) and \(\theta> 0\), \(E _{\alpha} ( - \theta t^{\alpha} ) \) and \(t^{\beta-1} E _{\alpha, \beta} ( -\theta t^{\alpha} ) \) are both completely monotonic (see [22]). As a result, from (3.26) we obtain \(\lim_{{t}\rightarrow+\infty} x ( t ) ^{T} x ( t ) =0\), namely \(\lim_{{t}\rightarrow+\infty} x ( t ) =0\). Considering (3.13), this implies that \(\lim_{t\rightarrow+\infty} D_{t}^{\alpha} k_{i} =0\). Noting now that the Caputo fractional derivative of a constant is always zero, we immediately conclude that \(k_{i}\) (\(i =1,2,\ldots, n \)) converges to a finite constant. Consequently, system (3.3) is stabilized by the adaptive law (3.13). The proof is completed. □
Remark 3.4
It should be noted that the Lyapunov (3.14) is the function of two variables \(x ( t ) \) and \(k_{i} ( t ) \), and \(D_{t}^{\alpha} V \leq- x ( t ) ^{T} x ( t ) \). To prove the convergence of \(x ( t ) \), condition (2.5) of Lemma 2.1 is not satisfied. In other words, the fractional Lyapunov direct method is not sufficient for proving the asymptotical stability of system (3.3) with adaptive law (3.13). It is well known that LaSalle’s invariance principle, Barbalat’s lemma, and other mathematical techniques can be used to solve the adaptive stability problem of integer-order nonlinear systems. However, these tools cannot be directly used in the fractional-order case. In this paper, the adaptive stability problem of fractional-order systems has been settled by utilizing the fractional Lyapunov function method combined with fractional inequality techniques, the Mittag-Leffler function, and the Laplace transform.
4 Illustrative examples
In this section, we give three illustrative examples to validate the theoretical results and use the predictor-corrector method [36] for numerical simulations.
Example 1
Example 2
5 Conclusions
In this paper, we propose some stabilization criteria for a large class of fractional-order nonautonomous systems, for which quadratic Lyapunov functions are always valid. The proposed stability criteria can be seen as an extension and improvement of the existing results in the literatures [16–20, 32]. It has been also shown that our proofs of the theorems are simple and straightforward, and the stability conditions are more convenient for testing. Our future work includes the applications of the fractional Lyapunov direct method and fractional inequality techniques in the stabilization of more general fractional-order nonlinear systems.
Declarations
Acknowledgements
This research has been supported by the ‘Chunhui Plan’ Cooperative Research for Ministry of Education (Grant No. Z2016133), the Scientific Research Foundation of the Education Department of Sichuan Province (Grants Nos. 16ZB0163, 17ZA0364), the Key Scientific Research Fund Project of Xihua University (Grant No. Z17124), the National Natural Science Foundation of China (Grant No. 11402214), the Open Research Fund of Key Laboratory of Automobile Engineering of Sichuan Province (Grant No. szjj2016-017), the Open Research Subject of Artificial Intelligence Key Laboratory of Sichuan Province (Grant No. 2017RYJ03), the Open Research Fund of Key Laboratory of Automobile Measurement and Control & Safety of Sichuan Province (Grant No. szjj2017-074), and the Open Research Fund of Key Laboratory of Numerical Simulation of Sichuan Province (Grant No. 2017KF004).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
- Valdes-Parada, F, Ochoa-Tapia, J, Alvarez-Ramirez, J: Effective medium equations for fractional Fick’s law in porous media. Physica A 373, 339-353 (2007) View ArticleGoogle Scholar
- Singh, J, Kumar, D, Sunil Kumar, R: An efficient computational approach for time-fractional Rosenau-Hyman equation. Neural Comput. Appl. (2017, in press) Google Scholar
- Magin, R, Ortigueira, MD, Podlubny, P, Trujillo, J: On the fractional signals and systems. Signal Process. 91(3), 350-371 (2011) View ArticleMATHGoogle Scholar
- Singh, J, Kumar, D, Al Qurashi, M, Baleanu, D: A novel numerical approach for a nonlinear fractional dynamical model of interpersonal and romantic relationships. Entropy 19, Article ID 375 (2017) View ArticleGoogle Scholar
- Wu, GC, Baleanu, D: Image encryption technique based on fractional chaotic time series. J. Vib. Control 22, 2092-2099 (2016) MathSciNetView ArticleGoogle Scholar
- Kumar, D, Singh, J, Baleanu, D: A fractional model of convective radial fins with temperature-dependent thermal conductivity. Rom. Rep. Phys. 69(1), Article ID 103 (2017) Google Scholar
- Yang, XJ, Machado, JAT, Cattani, C, Gao, F: On a fractal LC-electric circuit modeled by local fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 47, 200-206 (2017) View ArticleGoogle Scholar
- Singh, J, Kumar, D, Al Qurashi, M, Baleanu, D: A new fractional model for giving up smoking dynamics. Adv. Differ. Equ. 2017, Article ID 88 (2017) MathSciNetView ArticleGoogle Scholar
- Luo, Y, Chen, YQ: Fractional order [proportional derivative] controller for a class of fractional order systems. Automatica 45(10), 2446-2450 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Xu, Q, Zhuang, SX, Liu, SJ, Xiao, J: Decentralized adaptive coupling synchronization of fractional-order complex-variable dynamical networks. Neurocomputing 186, 119-126 (2016) View ArticleGoogle Scholar
- Li, CP, Zhang, FR: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193(1), 27-47 (2011) View ArticleGoogle Scholar
- Wu, GC, Baleanu, D, Zeng, SD: Finite-time stability of discrete fractional delay systems: gronwall inequality and stability criterion. Commun. Nonlinear Sci. Numer. Simul. 57, 229-308 (2017) MathSciNetGoogle 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
- Aghababa, MP: A Lyapunov-based control scheme for robust stabilization of fractional chaotic systems. Nonlinear Dyn. 78, 2129-2140 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Wu, GC, Baleanu, D, Xie, HP: Chaos synchronization of fractional chaotic maps based on the stability condition. Physica A 460, 374-383 (2016) MathSciNetView ArticleGoogle Scholar
- Wen, XJ, Wu, MZ, Lu, JG: Stability analysis of a class of nonlinear fractional-order systems. IEEE Trans. Circuits Syst. II, Express Briefs 55(11), 1178-1182 (2008) View ArticleGoogle Scholar
- Tavazoei, MS: Comments on “Stability analysis of a class of nonlinear fractional order systems”. IEEE Trans. Circuits Syst. II, Express Briefs 56(6), 519-520 (2009) View ArticleGoogle Scholar
- Chen, L, Chai, Y, Wu, R, Yang, J: Stability and stabilization of a class of nonlinear fractional-order systems with Caputo derivative. IEEE Trans. Circuits Syst. II, Express Briefs 59(9), 602-606 (2012) View ArticleGoogle Scholar
- Zhang, R, Tian, G, Yang, S, Cao, H: Stability analysis of a class of fractional order nonlinear systems with order lying in \((0,2)\). ISA Trans. 56, 102-110 (2015) View ArticleGoogle Scholar
- Chen, L, He, Y, Chai, Y, Wu, R: New results on stability and stabilization of a class of nonlinear fractional-order systems. Nonlinear Dyn. 75(4), 633-641 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Li, Y, Chen, YQ, Podlubny, I: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 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
- Baleanu, D, Ranjbar, A, Sadati, S, Delavari, R, Abdeljawad, T, Gejji, V: Lyapunov-Krasovskii stability theorem for fractional systems with delay. Rom. J. Phys. 56(5-6), 636-643 (2011) MathSciNetMATHGoogle Scholar
- Wu, GC, Baleanu, D, Luo, WH: Lyapunov functions for Riemann-Liouville-like fractional difference equations. Appl. Math. Comput. 314, 228-236 (2017) MathSciNetGoogle Scholar
- Tarasov, VE: No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 18(11), 2945-2948 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Benzaouia, A, Hmamed, A, Mesquine, F, Benhayoun, M, Tadeo, F: Stabilization of continuous-time fractional positive systems by using a Lyapunov function. IEEE Trans. Autom. Control 59(8), 2203-2208 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Agarwal, R, Hristova, S, O’Regan, D: Lyapunov functions and strict stability of Caputo fractional differential equations. Adv. Differ. Equ. 2015, 346 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Trigeassou, JC, Maamri, N, Sabatier, J, Oustaloup, A: A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91(3), 437-445 (2011) View ArticleMATHGoogle Scholar
- Hu, JB, Lu, GP, Zhang, SB, Zhao, LD: Lyapunov stability theorem about fractional system without and with delay. Commun. Nonlinear Sci. Numer. Simul. 20(3), 905-913 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Aguila-Camacho, N, Duarte-Mermoud, MA, Gallegos, J: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2951-2957 (2014) MathSciNetView ArticleGoogle Scholar
- Duarte-Mermoud, MA, Aguila-Camacho, N, Gallegos, J, Castro-Linares, R: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22(1-3), 650-659 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Liu, S, Jiang, W, Li, X, Zhou, XF: Lyapunov stability analysis of fractional nonlinear systems. Appl. Math. Lett. 51, 13-19 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
- Agarwal, RP, Belmekki, M, Benchohra, M: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differ. Equ. 2009, Article ID 981728 (2009) MathSciNetMATHGoogle Scholar
- DeLellis, P, Bernardo, M, Russo, G: On QUAD, Lipschitz, and contracting vector fields for consensus and synchronization of networks. IEEE Trans. Circuits Syst. I, Regul. Pap. 58(3), 576-583 (2011) MathSciNetView ArticleGoogle Scholar
- Diethelm, K, Ford, NJ, Freed, AD: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3-22 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Petráš, I: Fractional Order Nonlinear Systems Modeling, Analysis and Simulation. Springer, Berlin (2011) View ArticleMATHGoogle Scholar