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

Abstract

This paper presents a stability theorem for a class of nonlinear fractional-order time-variant systems with fractional order 𝛼  (1<𝛼<1) 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 𝛼  (1<𝛼<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 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, ℝ^𝑛 denotes an 𝑛-dimensional Euclidean space and ℝ^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices. The matrix norms are defined as ‖𝐴‖=∑^𝑛_𝑖,𝑗=1|𝑎_𝑖𝑗|(𝑖,𝑗=1,2,…,𝑛). 𝑎_𝑖,𝑗 is  the  element  of  matrix 𝐴, and lim_𝑡→∞𝐴(𝑡)=(lim_𝑡→∞𝑎_𝑖,𝑗(𝑡))_𝑛×𝑛,  lim_𝑡→∞𝐴(𝑡)=(lim_𝑡→∞𝑎_𝑖,𝑗(𝑡))_𝑛×𝑛,(𝑖,𝑗=1,2,…,𝑛).

2. Fractional Derivative and Preliminaries

In this paper, the following Caputo definition [1] is adopted for fractional derivative of order 𝛼 for function 𝑓(𝑡): 𝐷^𝛼𝑓(𝑡)=𝑑^𝛼𝑓(𝑡)𝑑𝑡^𝛼=1Γ(𝛼−𝑚)^𝑡₀𝑓^(𝑚)(𝜏)(𝑡−𝜏)^{𝛼+1−𝑚}𝑑𝜏, (1) where 𝑚 is an integer satisfying 𝑚−1<𝑎≤𝑚 and Γ(⋅) is the well-known Gamma function. The Laplace transform of the Caputo fractional derivative (1) of order 𝛼 is ^∞₀𝐷^𝛼𝑓(𝑡)𝑒^−𝑠𝑡𝑑𝑡=𝑠^𝛼𝐿𝑓(𝑡)−^𝑚−1_𝑘=0𝑠^{−𝛼−𝑘+1}𝑓^(𝑘)(𝑡)_𝑡=0, (2) where 𝑠∈ℂ denotes the Laplace operator. Note that upon considering all the initial conditions to be zero, (2) can be reduced to 𝐿𝐷^𝛼𝑓(𝑡)=𝑠^𝛼𝐿𝑓(𝑡). (3)

The two-parameter Mittag-Leffler function 𝐸_𝛼,𝛽(𝑧), 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 𝐸_𝛼,𝛽(𝑧)=^∞_𝑘=0𝑧^𝑘Γ𝛼𝑘+𝛽,𝛼>0,𝛽>0. (4) The Laplace transform of the Mittag-Leffler function is ^∞𝑒^−𝑠𝑡𝑡^{𝛼𝑘+𝛽−1}𝐸^(𝑘)_𝛼,𝛽𝛼𝑡^𝛼𝑑𝑡=𝑘!𝑠^𝛼−𝛽(𝑠^𝛼−𝛼)^𝑘+1,Re(𝑠)>|𝛼|^1/𝛼, (5) where 𝐸^(𝑘)_𝛼,𝛽=(𝑑^𝑘/𝑑𝑡^𝑘)𝐸_𝛼,𝛽.

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

Lemma 2 (see [1]). If 𝛼<2, 𝛽 is an arbitrary real number, 𝜂 is a constant satisfying 𝛼𝜋/2<𝜂<min{𝜋,𝜋𝛼}, and 𝑀>0 is a real constant, then 𝐸_𝛼,𝛽(𝑧<𝑀)|𝑧|(𝑧|𝑧|1+,𝜂≤arg)≤𝜋,>0. (6)

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

Lemma 3 (see [17]). If 𝐴∈ℂ^𝑛×𝑛 and 𝛼<2, 𝛽 is an arbitrary real number, 𝜂 is such that 𝛼𝜋/2<𝜂<min{𝜋,𝜋𝛼} and 𝐶>0 is a real constant, then 𝐸_𝛼,𝛽(𝐴<𝑀)1+‖𝐴‖,𝜂≤arg𝜆_𝑖(𝐴)≤𝜋,𝑖=1,2,…,𝑛, (7) where 𝜆_𝑖(𝐴) denotes the 𝑖th eigenvalue of matrix 𝐴 and ‖𝐴‖ denotes the 𝑙₂-norm.

Lemma 4 (Gronwall-Bellman lemma [21]). If 𝑥(𝑡)≤ℎ(𝑡)+^𝑡_𝑡0𝑘(𝑠)𝑥(𝑠)𝑑𝑠,𝑡∈𝑡₀,𝑇, (8) where all the functions involved are continuous on [𝑡₀,𝑇), 𝑇≤∞, and 𝑘(𝑡)≥0, then 𝑥(𝑡) satisfies 𝑥(𝑡)≤ℎ(𝑡)+^𝑡_𝑡0ℎ(𝑠)𝑘(𝑠)exp^𝑡_𝑠𝑘(𝑢)𝑡𝑑𝑢𝑑𝑠,𝑡∈₀,𝑇. (9)

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

Consider the 𝑛-dimensional nonlinear fractional-order time-variant system described by the following form: 𝐷^𝛼𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐵𝑢(𝑡), (10) where 0≤𝛼<1,  𝐴(𝑡)∈ℝ^𝑛×𝑛 is a time-variant regular matrix, and 𝑓(𝑡,𝑥(𝑡)) is a nonlinear function of state 𝑥(𝑡), which is Lebesgue measurable with respect to 𝑡 on (0,∞). Let 𝑓(𝑡,0)≡0, then 𝑥(𝑡)=0 is a zero solution of nonlinear fractional-order time-variant system (10) with 𝑢(𝑡)=0.

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 𝑢(𝑡)=0, assume that(i)lim_𝑡→∞𝐴(𝑡)=𝐴,lim_𝑡→∞𝐴(𝑡)=𝐴,  and ‖𝐴−𝐴‖<𝑏;(ii)𝑓(𝑡,0)=0, and lim_{‖𝑥‖→0}(‖𝑓(𝑡,𝑥(𝑡))‖/‖𝑥(𝑡)‖)=0;(iii)|arg(spec(𝐴))|>𝑎𝜋/2, 𝛼𝜌(𝐴)>1,
then the zero solution of system (10) is asymptotically stable.

Proof. By the assumption (i), it yields lim_𝑡→∞𝑎_𝑖𝑗(𝑡)=𝑎_𝑖𝑗,lim_𝑡→∞𝑎_𝑖𝑗(𝑡)=𝑎_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛, (11) that is, for any given 0<𝜀<(𝑏−‖𝐴−𝐴‖)/2𝑛², there exists 𝑡_𝑖𝑗≥𝑡₀>0, when 𝑡≥𝑡_𝑖𝑗, it has 𝑎_𝑖𝑗(𝑡)<𝑎_𝑖𝑗𝑎+𝜀,_𝑖𝑗(𝑡)>𝑎_𝑖𝑗−𝜀. (12) Setting 𝑇₁=max{𝑡_𝑖𝑗∣𝑖,𝑗=1,2,…,𝑛}, when 𝑡≥𝑇₁, then 𝑎_𝑖𝑗−𝜀<𝑎_𝑖𝑗(𝑡)<𝑎_𝑖𝑗+𝜀. (13) As a result, 𝑎_𝑖𝑗−𝜀<𝑎_𝑖𝑗<𝑎_𝑖𝑗+𝜀. (14) Without loss of generality, assume that |arg(spec(𝐴))|>𝑎𝜋/2, then from (13) and (14), it obtains 𝑎_𝑖𝑗(𝑡)−𝑎_𝑖𝑗<𝑎_𝑖𝑗−𝑎_𝑖𝑗+2𝜀<𝑎_𝑖𝑗−𝑎_𝑖𝑗+2𝜀,𝑡≥𝑇₁. (15) It follows that, ^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗(𝑡)−𝑎_𝑖𝑗<^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗−𝑎_𝑖𝑗+^𝑛_𝑖,𝑗=12𝜀,𝑡≥𝑇₁, (16) that is 𝐴(𝑡)−𝐴<𝐴−𝐴+2𝑛²𝜀<𝑏,𝑡≥𝑇₁. (17) On the other hand, by the assumption (ii) of Theorem 5, for any given 𝜀₀>0, there exists 𝑇₂≥𝑡₀>0, when 𝑡≥𝑇₂, it has 𝑓(𝑡,𝑥(𝑡))<𝜀₀‖𝑥(𝑡)‖. (18) Rewriting system (10) with 𝑢(𝑡)=0 as 𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+𝐴(𝑡)−𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)), (19) taking Laplace transform on the system (19), it obtains 𝑋(𝑠)=𝐼𝑠^𝛼−𝐴⁻¹𝑠^𝛼−1𝑥(0)+𝐿𝐴(𝑡)−𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)). (20) Then, taking Laplace inverse transform for (20) by using the inverse Laplace transform formula of the Mittag-Leffler function, it yields 𝑥(𝑡)=𝐸_𝛼,1𝐴𝑡^𝛼𝑥(0)+^𝑡₀(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝜏)^𝛼×𝐴(𝜏)−𝐴𝑥(𝜏)+𝑓(𝜏,𝑥(𝜏))𝑑𝜏. (21) Taking 𝑇^∗=max{𝑇₁,𝑇₂}, when 𝑡≥𝑇^∗, we have ‖𝑥(𝑡)‖≤𝐸_𝛼,1𝐴𝑡^𝛼𝑥(0+‖)‖_𝑡0‖(𝑡−𝜏)‖^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝜏)^𝛼×𝐴(𝜏)−𝐴‖𝑥(𝜏)‖+𝑓(𝜏,𝑥(𝜏))𝑑𝜏. (22) Substituting (17) and (18) into (22), it results ‖𝑥(𝑡)‖≤𝐸_𝛼,1𝐴𝑡^𝛼𝑥(0+‖)‖_𝑡0𝐶₁‖(𝑡−𝜏)‖^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝜏)^𝛼‖𝑥(𝜏)‖𝑑𝜏, (23) where 𝐶₁=𝑏+𝜀₀.
Since the eigenvalues of 𝐴 satisfy the assumption (iii) of Theorem 5, then from Lemma 3 there exists a real constant 𝐶₂>0 such that 𝐸_𝛼,1𝐴𝑡^𝛼≤𝐶₂1+𝐴𝑡^𝛼,𝐸_𝛼,1𝐴(𝑡−𝜏)^𝛼≤𝐶₂1+𝐴(𝑡−𝜏)^𝛼. (24)
Substituting (24) into (23), it gives ‖𝑥(𝑡≤𝐶)‖²1+𝐴𝑡^𝛼‖𝑥(0)‖+𝐶₁𝐶₂^𝑡₀‖(𝑡−𝜏)‖^𝛼−11+𝐴(𝑡−𝜏)^𝛼‖𝑥(𝜏=𝐶)‖𝑑𝜏₂1+𝐴𝑡^𝛼‖𝑥(0)‖+𝐶₁𝐶₂^𝑡₀‖(𝑡−𝜏)‖^𝛼−11+𝐴(𝑡−𝜏)^𝛼‖𝑥(𝜏)‖𝑑𝜏. (25)
Setting ℎ(𝑡)=𝐶₂1+𝐴𝑡^𝛼‖𝑥(0)‖,𝑘(𝑡)=(𝑡−𝜏)^𝛼−11+𝐴(𝑡−𝜏)^𝛼, (26) and using Gronwall-Bellman lemma, it follows from (25) that ‖𝑥(𝑡≤𝐶)‖²1+𝐴𝑡^𝛼‖𝑥(0)‖+𝐶₁𝐶₂×^𝑡₀‖𝑥(0)‖1+𝐴𝜏^𝛼⋅(𝑡−𝜏)^𝛼−11+𝐴(𝑡−𝜏)^{𝛼1−1/(𝛼‖𝐴‖)}𝑑𝜏. (27) The integral in (27) equals the sum of the two parts ₀^𝑡/2‖𝑥(0)‖1+𝐴𝜏^𝛼⋅(𝑡−𝜏)^𝛼−11+𝐴(𝑡−𝜏)^{𝛼1−1/(𝛼‖𝐴‖)}+𝑑𝜏_𝑡𝑡/2‖𝑥(0)‖1+𝐴𝜏^𝛼⋅(𝑡−𝜏)^𝛼−11+𝐴(𝑡−𝜏)^{𝛼1−1/(𝛼‖𝐴‖)}𝑑𝜏. (28) Since 𝛼<1 and (𝑡−𝜏)≥𝜏 when 𝜏∈[0,𝑡/2], we obtain ₀^𝑡/2‖𝑥(0)‖1+𝐴𝜏^𝛼⋅(𝑡−𝜏)^𝛼−11+𝐴(𝑡−𝜏)^{𝛼1−1/(𝛼‖𝐴‖)}≤𝑑𝜏_𝑡𝑡/2‖𝑥(0)‖1+𝐴𝜏^𝛼⋅𝜏^𝛼−11+𝐴𝜏^{𝛼1−1/(𝛼‖𝐴‖)}𝑑𝜏. (29) Similarly, ^𝑡_𝑡/2‖𝑥(0)‖1+𝐴𝜏^𝛼⋅(𝑡−𝜏)^𝛼−11+𝐴(𝑡−𝜏)^{𝛼1−1/(𝛼‖𝐴‖)}≤𝑑𝜏_𝑡𝑡/2‖𝑥(0)‖1+𝐴(𝑡−𝜏)^𝛼−1⋅(𝑡−𝜏)^𝛼−11+𝐴(𝑡−𝜏)^{𝛼1−1/(𝛼‖𝐴‖)}𝑑𝜏𝑡−𝜏=𝑠₀^𝑡/2‖𝑥(0)‖𝑠^𝛼−11+𝐴𝑠^𝛼1+𝐴𝑠^{𝛼1−1/(𝛼‖𝐴‖)}𝑑𝑠. (30)
From (28) to (30), and with 𝑎𝜌(𝐴)>1, (27) gives ‖𝑥(𝑡≤𝐶)‖²‖𝑥(0)‖1+𝐴𝑡^𝛼+2𝐶₁𝐶₂₀^𝑡/2‖𝑥(0)‖𝑠^𝛼−11+𝐴𝑠^{𝛼2−1/(𝛼‖𝐴‖)}=𝑑𝑠2𝐶₁𝐶₂‖𝑥(0)‖𝛼𝐴+𝐶−1²‖𝑥(0)‖1+𝐴𝑡^𝛼+2𝐶₁𝐶₂‖𝑥(0)‖1−𝛼𝐴1+𝐴(𝑡^𝛼/2)^{1−1/(𝛼‖𝐴‖)}. (31) Now for arbitrarily small 𝛿>0, it can be proved that ‖𝑥(0⟹‖𝑥(𝑡<)‖<𝛿)‖2𝐶₁𝐶₂𝛿𝛼𝐴+𝐶−1²𝛿1+𝐴𝑡^𝛼+2𝐶₁𝐶₂𝛿1−𝛼𝐴1+𝐴(𝑡^𝛼/2)^{1−1/(𝛼‖𝐴‖)}, (32) which implies stability of the zero solution. This completes the proof.

In particular, if lim_𝑡→∞𝐴(𝑡)=𝐴, the following corollary is easily derived.

Corollary 6. For the nonlinear fractional-order system (10) with 𝑢(𝑡)=0, assume that(i)lim_𝑡→∞𝐴(𝑡)=𝐴 and |arg(spec(𝐴))|>𝑎𝜋/2, 𝛼𝜌(𝑎)>1;(ii)𝑓(𝑡,0)=0, and lim_{‖𝑥‖→0}(‖𝑓(𝑡,𝑥(𝑡))‖/‖𝑥(𝑡)‖)=0, 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)lim_𝑡→∞𝐴(𝑡)=𝐴;(ii)𝑓(𝑡,0)=0, and lim_{‖𝑥‖→0}(‖𝑓(𝑡,𝑥(𝑡))‖/‖𝑥(𝑡)‖)=0,and if 𝐾 is selected to, such that, |arg(spec(𝐴+𝐵𝐾))|>𝑎𝜋/2, then the nonlinear fractional-order system (10) can be stabilized by the state feedback controller 𝑢(𝑡)=𝐾𝑥(𝑡).

4. Numerical Example

Consider the following fractional-order time-variant Lorenz system: 𝐷^𝛼𝑥₁=𝑎(𝑡)𝑥₂−𝑥₁(𝑡)𝐷+𝑢_𝛼𝑥₂=𝑏𝑥₁−𝑥₁𝑥₃−𝑐(𝑡)𝑥₂(𝑡)𝐷+𝑢_𝛼𝑥₃=𝑥₁𝑥₂−𝑑𝑥₃, (33)

where 0≤𝛼<1. Lorenz system (33) can be rewritten as 𝐷^𝛼𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐵(𝑡)𝑢(𝑡), (34) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)]^𝑇 and 𝐴(𝑡)=(𝑡)𝑎(𝑡)0(𝑡)0(𝑡)=111,𝑓((𝑡=0−𝑎𝑏−𝑐00−𝑑,𝐵𝑡,𝑥))−𝑥₁(𝑡)𝑥₃(𝑡)𝑥₁(𝑡)𝑥₂(𝑡). (35)

Setting 𝑎(𝑡)=9+sin𝑡, 𝑏=28, 𝑐(𝑡)=−8, 𝑑=8/3, 𝛼=0.90. 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 1/𝑠^0.9, which has an error about 2 dB in the frequency range 𝜔=10⁻² rad/s to 𝜔=10² rad/s. We use this recursive poles/zeros filter of order three to approximate the fractional operator. The initial value of the system [(𝑥₁(0),𝑥₂(0),𝑥₃(0)]=(0.22,0.24,0.12). The commensurate fractional-order Lorenz system (34) is chaotic without the controller, which is shown in Figure 1.

Figure 1 

Time response of the selected systems with 𝑢(𝑡)=0.

Note that 𝑓(𝑡,0)=0 and lim_{‖𝑥‖→0}𝑓(𝑡,𝑥(𝑡))‖𝑥(𝑡)‖=lim_{‖𝑥‖→0}(𝑥₁(𝑡)𝑥₂(𝑡))²+(−𝑥₁(𝑡)𝑥₃(𝑡))²𝑥²₁(𝑡)+𝑥²₂(𝑡)+𝑥²₃(𝑡)≤lim_{‖𝑥‖→0}𝑥₁(𝑡)𝑥₂(𝑡)²+−𝑥₁(𝑡)𝑥₃(𝑡)²𝑥²₁(𝑡)=lim_{‖𝑥‖→0}𝑥²₂(𝑡)+𝑥²₃(𝑡)=0. (36) To stabilize the fractional-order system (33), we select 𝐾=1−5−1, which makes ‖𝐴−𝐴‖=4 and |arg(𝐴+𝐵𝐾)|>𝛼𝜋/2, by the eigenvalues of (𝐴+𝐵𝐾) being (−20.3516,−8.1380,−0.1771). So the conditions of Corollary 7 are satisfied. Thus, the fractional-order system (33) can be stabilized by the state feedback controller 𝑢(𝑡)=𝐾𝑥(𝑡). 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.

Figure 2 

Time response of the selected systems with the control input 𝑢(𝑡)=𝐾𝑥(𝑡).

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 𝛼(0≤𝛼<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.

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).