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

Authors: Chen Caixue; Xie Yunxiang
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper presents a stability theorem for a class of nonlinear fractional-order time-variant systems with fractional orderα  (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.


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## Body

## 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 Section2, 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,Rn denotes an n-dimensional Euclidean space and Rn×m is the set of all n×m real matrices. The matrix norms are defined as ∥A∥=∑i,j=1n‍|aij|(i,j=1,2,…,n). ai,j is  the  element  of  matrix A, and lim⁡-t→∞A(t)=(lim⁡-t→∞ai,j(t))n×n,  lim⁡_t→∞A(t)=(lim⁡_t→∞ai,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):
(1)Dαf(t)=dαf(t)dtα=1Γ(α-m)∫0tf(m)(τ)(t-τ)α+1-mdτ,
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
(2)∫0∞Dαf(t)e-stdt=sαL{f(t)}-∑k=0m-1‍s-α-k+1f(k)(t)|t=0,
where s∈C denotes the Laplace operator. Note that upon considering all the initial conditions to be zero, (2) can be reduced to
(3)L{Dαf(t)}=sαL{f(t)}.The two-parameter Mittag-Leffler functionEα,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(4)Eα,β(z)=∑k=0∞‍zkΓ(αk+β),α>0,β>0.
The Laplace transform of the Mittag-Leffler function is
(5)∫∞e-sttαk+β-1Eα,β(k)(αtα)dt=k!sα-β(sα-α)k+1,Re(s)>|α|1/α,
where Eα,β(k)=(dk/dtk)Eα,β.To prove the main results in the next section, the following lemmas are needed.Lemma 2 (see [1]).
Ifα<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
(6)|Eα,β(z)|<M1+|z|,η≤|arg⁡(z)|≤π,|z|>0.For then-dimension matrix, one has the following lemma.Lemma 3 (see [17]).
IfA∈Cn×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
(7)∥Eα,β(A)∥<M1+∥A∥,η≤|argλi(A)|≤π,kkkkkkkkkkkkkkkkkkkkkkkkkki=1,2,…,n,
where λi(A) denotes the ith eigenvalue of matrix A and ∥A∥ denotes the l2-norm.Lemma 4 (Gronwall-Bellman lemma [21]).
If(8)x(t)≤h(t)+∫t0tk(s)x(s)ds,t∈[t0,T),
where all the functions involved are continuous on [t0,T), T≤∞, and k(t)≥0, then x(t) satisfies
(9)x(t)≤h(t)+∫t0t‍h(s)k(s)exp⁡[∫stk(u)du]ds,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkt∈[t0,T).


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

Consider then-dimensional nonlinear fractional-order time-variant system described by the following form:
(10)Dαx(t)=A(t)x(t)+f(t,x(t))+Bu(t),
where 0≤α<1,  A(t)∈Rn×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(11)lim⁡¯t→∞aij(t)=a-ij,lim⁡_t→∞⁡aij(t)=a_ij,(i,j=1,2,…,n),
that is, for any given 0<ε<(b-∥A--A_∥)/2n2, there exists tij≥t0>0, when t≥tij, it has
(12)aij(t)<a-ij+ε,aij(t)>a_ij-ε.
Setting T1=max⁡⁡{tij∣i,j=1,2,…,n}, when t≥T1, then
(13)a_ij-ε<aij(t)<a-ij+ε.
As a result,
(14)a_ij-ε<a-ij<a-ij+ε.
Without loss of generality, assume that |arg⁡(spec(A-))|>aπ/2, then from (13) and (14), it obtains
(15)|aij(t)-a-ij|<|a-ij-a_ij+2ε|<|a-ij-a_ij|+2ε,(t≥T1).
It follows that,
(16)∑i,j=1n|aij(t)-a-ij|<∑i,j=1n|a-ij-a_ij|+∑i,j=1n2ε,(t≥T1),
that is
(17)∥A(t)-A-∥<∥A--A_∥+2n2ε<b,(t≥T1).
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
(18)∥f(t,x(t))∥<ε0∥x(t)∥.
Rewriting system (10) with u(t)=0 as
(19)Dαx(t)=A-x(t)+(A(t)-A-)x(t)+f(t,x(t)),
taking Laplace transform on the system (19), it obtains
(20)X(s)=(Isα-A-)-1(sα-1x(0)kkkkkkkkkkkk+L{(A(t)-A-)x(t)+f(t,x(t))}).
Then, taking Laplace inverse transform for (20) by using the inverse Laplace transform formula of the Mittag-Leffler function, it yields
(21)x(t)=Eα,1(A-tα)x(0)+∫0t(t-τ)α-1Eα,α(A-(t-τ)α)×((A(τ)-A-)x(τ)+f(τ,x(τ)))dτ.
Taking T*=max⁡{T1,T2}, when t≥T*, we have
(22)∥x(t)∥≤∥Eα,1(A-tα)∥∥x(0)∥+∫0t‍∥(t-τ)∥α-1∥Eα,α(A-(t-τ)α)∥×(∥(A(τ)-A-)∥∥x(τ)∥+∥f(τ,x(τ))∥)dτ.
Substituting (17) and (18) into (22), it results
(23)∥x(t)∥≤∥Eα,1(A-tα)∥∥x(0)∥+∫0tC1∥(t-τ)∥α-1∥Eα,α(A-(t-τ)α)∥∥x(τ)∥dτ,
where C1=b+ε0.
Since the eigenvalues ofA- satisfy the assumption (iii) of Theorem 5, then from Lemma 3 there exists a real constant C2>0 such that
(24)∥Eα,1(A-tα)∥≤C21+∥A-tα∥,∥Eα,1(A-(t-τ)α)∥≤C21+∥A-(t-τ)α∥.
Substituting (24) into (23), it gives
(25)∥x(t)∥≤C21+∥A-tα∥∥x(0)∥+C1C2∫0t‍∥(t-τ)∥α-11+∥A-(t-τ)α∥∥x(τ)∥dτ=C21+∥A-∥tα∥x(0)∥+C1C2∫0t‍∥(t-τ)∥α-11+∥A-∥(t-τ)α∥x(τ)∥dτ.
Setting(26)h(t)=C21+∥A-tα∥∥x(0)∥,k(t)=(t-τ)α-11+∥A-∥(t-τ)α,
and using Gronwall-Bellman lemma, it follows from (25) that
(27)∥x(t)∥≤C21+∥A-∥tα∥x(0)∥+C1C2×∫0t‍∥x(0)∥1+∥A-∥τα·(t-τ)α-1(1+∥A-∥(t-τ)α)1-1/(α∥A-∥)dτ.
The integral in (27) equals the sum of the two parts
(28)∫0t/2‍∥x(0)∥1+∥A-∥τα·(t-τ)α-1(1+∥A-∥(t-τ)α)1-1/(α∥A-∥)dτ+∫t/2t‍∥x(0)∥1+∥A-∥τα·(t-τ)α-1(1+∥A-∥(t-τ)α)1-1/(α∥A-∥)dτ.
Since α<1 and (t-τ)≥τ when τ∈[0,t/2], we obtain
(29)∫0t/2‍∥x(0)∥1+∥A-∥τα·(t-τ)α-1(1+∥A-∥(t-τ)α)1-1/(α∥A-∥)dτ≤∫t/2t‍∥x(0)∥1+∥A-∥τα·τα-1(1+∥A-∥τα)1-1/(α∥A-∥)dτ.
Similarly,
(30)∫t/2t‍∥x(0)∥1+∥A-∥τα·(t-τ)α-1(1+∥A-∥(t-τ)α)1-1/(α∥A-∥)dτ≤∫t/2t‍∥x(0)∥1+∥A-∥(t-τ)α-1·(t-τ)α-1(1+∥A-∥(t-τ)α)1-1/(α∥A-∥)dτt-τ=s__∫0t/2‍∥x(0)∥sα-1(1+∥A-∥sα)(1+∥A-∥sα)1-1/(α∥A-∥)ds.
From (28) to (30), and with aρ(A-)>1, (27) gives
(31)∥x(t)∥≤C2∥x(0)∥1+∥A-∥tα+2C1C2∫0t/2‍∥x(0)∥sα-1(1+∥A-∥sα)2-1/(α∥A-∥)ds=2C1C2∥x(0)∥α∥A-∥-1+C2∥x(0)∥1+∥A-∥tα+2C1C2∥x(0)∥(1-α∥A-∥)(1+∥A-∥(tα/2))1-1/(α∥A-∥).
Now for arbitrarily small δ>0, it can be proved that
(32)∥x(0)∥<δ⟹∥x(t)∥<2C1C2δα∥A-∥-1+C2δ1+∥A-∥tαkkkkkkkk+2C1C2δ(1-α∥A-∥)(1+∥A-∥(tα/2))1-1/(α∥A-∥),
which implies stability of the zero solution. This completes the proof.In particular, iflim⁡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 Corollary6 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:(33)Dαx1=a(t)(x2-x1)+u(t)Dαx2=bx1-x1x3-c(t)x2+u(t)Dαx3=x1x2-dx3,where0≤α<1. Lorenz system (33) can be rewritten as
(34)Dαx(t)=A(t)x(t)+f(t,x(t))+B(t)u(t),
where x(t)=[x1(t),x2(t),x3(t)]T and
(35)A(t)=(-a(t)a(t)0b-c(t)000-d),B(t)=(111),f(t,x(t))=(0-x1(t)x3(t)x1(t)x2(t)).Settinga(t)=9+sin⁡t, 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/s0.9, 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 withu(t)=0.Note thatf(t,0)=0 and
(36)lim⁡∥x∥→0⁡∥f(t,x(t))∥∥x(t)∥=lim⁡∥x∥→0⁡(x1(t)x2(t))2+(-x1(t)x3(t))2x12(t)+x22(t)+x32(t)≤lim⁡∥x∥→0⁡(x1(t)x2(t))2+(-x1(t)x3(t))2x12(t)=lim⁡∥x∥→0⁡x22(t)+x32(t)=0.
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 inputu(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 orderα(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.



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