Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 428143, 6 pages

http://dx.doi.org/10.1155/2014/428143

## Robust Stability Analysis of Nonlinear Fractional-Order Time-Variant Systems

School of Electric Power, South China University of Technology, Guangzhou, Guangdong 510641, China

Received 10 April 2014; Accepted 30 May 2014; Published 30 June 2014

Academic Editor: Ligang Wu

Copyright © 2014 Chen Caixue and Xie Yunxiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a stability theorem for a class of nonlinear fractional-order time-variant systems with fractional order by using the Gronwall-Bellman lemma. Based on this theorem, a sufficient condition for designing a state feedback controller to stabilize such fractional-order systems is also obtained. Finally, a numerical example demonstrates the validity of this approach.

#### 1. Introduction

Recently, fraction-order system or system containing fractional derivatives and integrals has been studied widely [1–5]. It was found that many systems in interdisciplinary fields could be elegantly described with the help of fractional derivatives and integrals, such as viscoelastic system, dielectric polarization, electrode-electrolyte polarization, and electromagnetic waves [2–4]. In fact, real word processes generally or most likely are fractional-order systems. Moreover, fractional-order controllers [6, 7] have so far been implemented to enhance the robustness and the performance of the closed loop control system.

The problem of stability is a very essential and crucial issue for control systems including fractional-order system. Very recently, the stability problem of fractional-order system has been investigated both from an algebraic and from an analytic point of view [8–11]. By analyzing the characteristic equation of the Jacobian matrix, an asymptotically stable critical was proposed in [12]. As a way of efficiently solving the robust stability and stabilization problem, the linear-matrix-inequality (LMI) approach was presented [13–15] that provided the sufficient condition and the designing method of stabilizing controllers for fractional-order system. Note that the existing LMI-based control methods for fractional-order system only focus on the linear system but not on the case of nonlinear system. To account this problem, based on the generalization of Gronwall-Bellman lemma, the analytical stability conditions and state feedback stabilization problem of nonlinear affine fractional-order system have been investigated in [16–18]. In [19], by using of Mittag-Leffler function, Laplace transform, and the generalized Gronwall inequality, a new sufficient condition ensuring local asymptotic stability and stabilization of a class of fractional-order nonlinear systems with fractional-order was proposed. Based on Lyapunov’s second method, a novel stability criterion for a class of nonlinear fractional differential system was presented in [20].

Motivated by the above mentioned works, the main purpose of this this paper is to consider the stability problem of a class of nonlinear fractional-order time-variant systems. The main contribution of this paper is as follows. First, the time-variant uncertainty was discussed for fractional-order nonlinear system. Second, by using Gronwall-Bellman lemma, a stability condition for such fractional-order time-variant systems is presented. The rest of this paper is organized as follows. In Section 2, the problem formulation and some preliminaries are presented. The main results are derived in Section 3. The efficiency of the approach is shown through an illustrative example in Section 4. Finally, some conclusions are drawn in Section 5.

Throughout this paper, denotes an -dimensional Euclidean space and is the set of all real matrices. The matrix norms are defined as . is the element of matrix , and , .

#### 2. Fractional Derivative and Preliminaries

In this paper, the following Caputo definition [1] is adopted for fractional derivative of order for function : where is an integer satisfying and is the well-known Gamma function. The Laplace transform of the Caputo fractional derivative (1) of order is where denotes the Laplace operator. Note that upon considering all the initial conditions to be zero, (2) can be reduced to

The two-parameter Mittag-Leffler function , which plays a very important role in the fractional calculus, is introduced as follows.

*Definition 1 (see [1]). *The two-parameter Mittag-Leffler function is defined by
The Laplace transform of the Mittag-Leffler function is
where .

To prove the main results in the next section, the following lemmas are needed.

Lemma 2 (see [1]). *If , is an arbitrary real number, is a constant satisfying , and is a real constant, then
*

For the -dimension matrix, one has the following lemma.

Lemma 3 (see [17]). *If and , is an arbitrary real number, is such that and is a real constant, then
**
where denotes the th eigenvalue of matrix and denotes the -norm.*

Lemma 4 (Gronwall-Bellman lemma [21]). *If
**
where all the functions involved are continuous on , , and , then satisfies
*

#### 3. Stability Analysis of Nonlinear Fractional-Order Time-Variant Systems

Consider the -dimensional nonlinear fractional-order time-variant system described by the following form: where , is a time-variant regular matrix, and is a nonlinear function of state , which is Lebesgue measurable with respect to on . Let , then is a zero solution of nonlinear fractional-order time-variant system (10) with .

Based on the aforementioned definition and lemmas, we first give out the stability theorem for nonlinear fractional-order time-variant system (10).

Theorem 5. *For the nonlinear fractional-order system (10) with , assume that*(i)*, and ;*(ii)*, and ;*(iii)*, ,**
then the zero solution of system (10) is asymptotically stable.*

*Proof. *By the assumption (i), it yields
that is, for any given , there exists , when , it has
Setting , when , then
As a result,
Without loss of generality, assume that , then from (13) and (14), it obtains
It follows that,
that is
On the other hand, by the assumption (ii) of Theorem 5, for any given , there exists , when , it has
Rewriting system (10) with as
taking Laplace transform on the system (19), it obtains
Then, taking Laplace inverse transform for (20) by using the inverse Laplace transform formula of the Mittag-Leffler function, it yields
Taking , when , we have
Substituting (17) and (18) into (22), it results
where .

Since the eigenvalues of satisfy the assumption (iii) of Theorem 5, then from Lemma 3 there exists a real constant such that

Substituting (24) into (23), it gives

Setting
and using Gronwall-Bellman lemma, it follows from (25) that
The integral in (27) equals the sum of the two parts
Since and when , we obtain
Similarly,

From (28) to (30), and with , (27) gives
Now for arbitrarily small , it can be proved that
which implies stability of the zero solution. This completes the proof.

In particular, if , the following corollary is easily derived.

Corollary 6. *For the nonlinear fractional-order system (10) with , assume that*(i)* and , ;*(ii)*, and ,** then the zero solution of system (10) is asymptotically stable.*

Based on Corollary 6 and according to the stability theory of linear fractional-order system and the pole placement technique of the linear control theory (see, e.g., [16, 17, 22]), it is easy to get the stabilization theorem of nonlinear fractional-order system (10).

Corollary 7. *Assume that *(i)*;*(ii)*, and ,**and if is selected to, such that, , then the nonlinear fractional-order system (10) can be stabilized by the state feedback controller .*

#### 4. Numerical Example

Consider the following fractional-order time-variant Lorenz system:

where . Lorenz system (33) can be rewritten as where and

Setting , , , , . To simulate the fractional order systems, we use Oustaloup’s recursive poles/zeros filter [3] (an integer order system) of order three to approximate the fractional operator , which has an error about dB in the frequency range rad/s to rad/s. We use this recursive poles/zeros filter of order three to approximate the fractional operator. The initial value of the system . The commensurate fractional-order Lorenz system (34) is chaotic without the controller, which is shown in Figure 1.

Note that and To stabilize the fractional-order system (33), we select , which makes and , by the eigenvalues of being . So the conditions of Corollary 7 are satisfied. Thus, the fractional-order system (33) can be stabilized by the state feedback controller . The simulation result (Figure 2) shows that it is asymptotically stable and its states converge to zero, which shows that the obtained theoretic results are feasible and efficient for the nonlinear fractional-order system.

#### 5. Conclusion

In this paper, based on the Gronwall-Bellman lemma and the property of fractional calculus, a stabilization theorem of a class of nonlinear fractional-order time-variant systems with fractional order has been proven theoretically. Furthermore, the sufficient condition for designing a state feedback controller to stabilize such fractional-order systems was also obtained. Finally, a numerical example demonstrates the validity of this approach.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research is supported by National Natural Science Foundation of China (61104072) and Research Fund of Hunan Provincial Education Department (14JJ2073).

#### References

- I. Podlubny,
*Fractional Differential Equations*, Academic Press, New York, NY, USA, 1999. View at MathSciNet - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific Publishing, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - M. D. Ortigueira and J. A. T. Machado, “Fractional signal processing and applications,”
*Signal Processing*, vol. 83, no. 11, pp. 2285–2286, 2003. View at Publisher · View at Google Scholar · View at Scopus - J. Wang, Y. Zhou, and W. Wei, “Optimal feedback control for semilinear fractional evolution equations in Banach spaces,”
*Systems & Control Letters*, vol. 61, no. 4, pp. 472–476, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1063–1077, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - H.-F. Raynaud and A. Zergaïnoh, “State-space representation for fractional order controllers,”
*Automatica*, vol. 36, no. 7, pp. 1017–1021, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - S. Ladaci, J. J. Loiseau, and A. Charef, “Fractional order adaptive high-gain controllers for a class of linear systems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 13, no. 4, pp. 707–714, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - C. Hwang and Y.-C. Cheng, “A numerical algorithm for stability testing of fractional delay systems,”
*Automatica*, vol. 42, no. 5, pp. 825–831, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - H.-S. Ahn and Y. Chen, “Necessary and sufficient stability condition of fractional-order interval linear systems,”
*Automatica*, vol. 44, no. 11, pp. 2985–2988, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Li, Y. Q. Chen, and I. Podlubny, “Stability of fractional-order nonlinear dynamic systems: lyapunov direct method and generalized Mitta-Leffler stability,”
*Computers and Mathematics with Applications*, vol. 24, pp. 1429–1468, 2009. View at Google Scholar - C. Yakar, “Fractional differential equations in terms of comparison results and Lyapunov stability with initial time difference,”
*Abstract and Applied Analysis*, vol. 2010, Article ID 762857, 16 pages, 2010. View at Publisher · View at Google Scholar - G. Sun, M. Wang, L. Huang, and L. Shen, “Generating multi-scroll chaotic attractors via switched fractional systems,”
*Circuits, Systems, and Signal Processing*, vol. 30, no. 6, pp. 1183–1195, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - M. S. Tavazoei and M. Haeri, “A note on the stability of fractional order systems,”
*Mathematics and Computers in Simulation*, vol. 79, no. 5, pp. 1566–1576, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - J. Sabatier, M. Moze, and C. Farges, “LMI stability conditions for fractional order systems,”
*Computers & Mathematics with Applications*, vol. 59, no. 5, pp. 1594–1609, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - J.-G. Lu and G. Chen, “Robust stability and stabilization of fractional-order interval systems: an LMI approach,”
*IEEE Transactions on Automatic Control*, vol. 54, no. 6, pp. 1294–1299, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - I. N'Doye, M. Zasadzinski, N. E. Radhy, and A. Bouaziz, “Stabilization of a class of nonlinear affine fractional-order systems using generalizations of Bellman-Gronwall lemma,” in
*Proceedings of the 17th Mediterranean Conference on Control and Automation*, pp. 324–330, Makedonia Palace, Thessaloniki, Greece, June 2009. - X. J. Wen, Z. M. Wu, and J. G. Lu, “Stability analysis of a class of non-linear fractional-order systems,”
*IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 55, no. 11, pp. 1178–1183, 2008. View at Google Scholar - M. P. Lazarević and A. M. Spasić, “Finite-time stability analysis of fractional order time-delay systems: gronwall's approach,”
*Mathematical and Computer Modelling*, vol. 49, no. 3-4, pp. 475–481, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - L. Chen, Y. He, Y. Chai, and R. Wu, “New results on stability and stabilization of a class of nonlinear fractional-order systems,”
*Nonlinear Dynamics*, vol. 75, no. 4, pp. 633–641, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - X.-F. Zhou, L.-G. Hu, S. Liu, and W. Jiang, “Stability criterion for a class of nonlinear fractional differential systems,”
*Applied Mathematics Letters*, vol. 28, pp. 25–29, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - G. S. Jones, “Fundamental inequalities for discrete and discontinuous functional equations,”
*Journal of the Society for Industrial and Applied Mathematics*, vol. 12, no. 1, pp. 43–57, 1964. View at Google Scholar · View at MathSciNet - I. N. 'doye, M. Zasadzinski, M. Darouach, and N. Radhy, “Robust controller design for linear fractional-order systems with nonlinear time-varying model uncertainties,” in
*Proceedings of the 17th Mediterranean Conference on Control and Automation*, pp. 821–826, Makedonia Palace, Thessaloniki, Greece, June 2009.