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

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.


Introduction
Recently, fraction-order system or system containing fractional derivatives and integrals has been studied widely [1][2][3][4][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][3][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][9][10][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 linearmatrix-inequality (LMI) approach was presented [13][14][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 fractionalorder 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][17][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 fractionalorder nonlinear systems with fractional-order  (1 <  < 2) 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 fractionalorder nonlinear system.Second, by using Gronwall-Bellman lemma, a stability condition for such fractional-order timevariant 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, R  denotes an -dimensional Euclidean space and R × is the set of all  ×

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  − 1 <  ≤  and Γ(⋅) is the well-known Gamma function.The Laplace transform of the Caputo fractional derivative (1) of order  is where  ∈ C 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.
To prove the main results in the next section, the following lemmas are needed.
Lemma 2 (see [1]).If  < 2,  is an arbitrary real number,  is a constant satisfying /2 <  < min{, }, and  > 0 is a real constant, then For the -dimension matrix, one has the following lemma.

Stability Analysis of Nonlinear Fractional-Order Time-Variant Systems
Consider the -dimensional nonlinear fractional-order timevariant system described by the following form: where 0 ≤  < 1, () ∈ R × is a time-variant regular matrix, and (, ()) is a nonlinear function of state (), which is Lebesgue measurable with respect to  on (0, ∞).
Based on the aforementioned definition and lemmas, we first give out the stability theorem for nonlinear fractionalorder time-variant system (10).Theorem 5.For the nonlinear fractional-order system (10) with () = 0, assume that then the zero solution of system (10) is asymptotically stable.
Since the eigenvalues of  satisfy the assumption (iii) of Theorem 5, then from Lemma  . ( The integral in (27) equals the sum of the two parts . ( Since  < 1 and ( − ) ≥  when  ∈ [0, /2], we obtain Similarly, From ( 28) to (30), and with () > 1, (27) gives Now for arbitrarily small  > 0, it can be proved that which implies stability of the zero solution.This completes the proof.

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  (0 ≤  < 1) 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.