# Robust Stability and Stabilization of Interval Uncertain Descriptor Fractional-Order Systems with the Fractional-Orderα: The 1≤α<2 Case

Authors: Yuanhua Li; Heng Liu; Hongxing Wang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Stability and stabilization of fractional-order interval system is investigated. By adding parameters to linear matrix inequalities, necessary and sufficient conditions for stability and stabilization of the system are obtained. The results on stability check for uncertain FO-LTI systems with interval coefficients of dimensionn only need to solve one 4n-by-4n LMI. Numerical examples are presented to shown the effectiveness of our results.


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## Body

## 1. Introduction

During the last two decades, the study of fractional-order control systems has received more and more attention. As a generalization of the traditional calculus, the fractional calculus have found many applications in viscoelastic systems, robotics, finance, and so on ([1–4], etc.). Studying on fractional-order calculus has become an active research field.Stability analysis is a basic problem in control theory. For Caputo fractional derivative-based linear systems, the stability results were formulated with the fractional-orderα belonging to 0<α<1 and 1≤α<2, in [5–7] and so on. In [8], the stability issue of interval fractional-order linear time-invariant (FO-LTI) systems was first presented and discussed. The stability of single-input single-output FO-LTI systems was further discussed based on an experimentally verified Kharitonov-like procedure [9].Robust stability analysis was carried out for FO-LTI interval system with fractional commensurate order1≤α<2 based on the maximum eigenvalue of a Hermitian matrix by applying Lyapunov inequality in [10] and then further discussed in [7].The uncertain FO-LTI systems have been wildly studied. In [11], the robust stability problem was discussed based on the ranges of the corresponding interval eigenvalues by applying the matrix perturbation theory. In [12], a new and effectively robust stability checking method was first proposed for FO-LTI interval uncertain systems in terms of LMIs, and an analytical design of the stabilizing controllers for fractional-order dynamic interval systems was given. Note that the above-mentioned results on stability check for uncertain FO-LTI systems with interval coefficients of dimension n need to solve one n2+2n×n2+2n LMI. Therefore, it is valuable to seek some simple necessary and sufficient conditions for checking robust stability of uncertain FO-LTI systems with interval coefficients. With the above motivation, based on the results of [12], the robust stability and stabilization problems of uncertain FO-LTI interval systems with the fractional-order α belonging to 1≤α<2 are further investigated in this paper.This paper is organized as follows: in Section2, we present some preliminaries results on the fractional derivative, the linear algebra and the matrix theory. In Section 3, we study the problems of the stability and stabilization of uncertain FO-LTI systems with interval coefficients in terms of LMIs. In Section 4, numerical examples are presented to illustrate our proposed results. Finally, Section 5 concludes this work.Notations. Throughout this paper, Rm×n stands for the set of m by n matrices with real entries. The symbols MT, SymM, and In stand for the transpose of M, the expression MT+M, and the identity matrix of order n, respectively. The symbol en∈Rn is used to denote the row vector with the kth element being 1, i=1,…,n; that is, (1)en=1,1,…,1∈Rn.The symbol ⊗ is the Kronecker product of two matrices and (2)A⊗CT=AT⊗CT.The symbol · will be used in some matrix expressions to indicate a symmetric structure; that is, if matrices H1=H1H∈Rm×m and H2=H2H∈Rn×n were given, then(3)H1·LH2=H1LTLH2.Let N=diaga1,a2,…,an; consider the symbol (4)N1/2=diag⁡a1,a2,…,an,in which ai≥0, i=1,…,n.


## 2. Preliminaries

Throughout the paper, only the Caputo definition is used. The following Caputo definition is adopted for fractional derivatives of orderα of function f(t) [13]: (5)Dαft=1Γα-m∫0tfmτt-τα-m+1dτ,m-1≤α<mwith m∈Z, α∈R+, where Γ(·) is the Gamma function:(6)Γz=∫0∞e-ttz-1dt.Consider the following FO-LTI interval system:(7)Dαxt=Axt+But,where α is the fractional commensurate order, xt∈Rn and ut∈Rp stand for the state vector and control input, respectively, and the system matrices A and B are interval uncertain in the sense that (8)A∈AI=A_,A¯=aijn×n:a_ij≤aij≤a¯ij,1≤i,j≤n,B∈BI=B_,B¯=bijn×p:b_ij≤bij≤b¯ij,1≤i≤n,1≤j≤p,where A_, A¯, B_, and B¯ are given matrices.To take into account the stability [14], we introduce the following definition.Definition 1.
The fractional-order interval system (7) is said to be asymptotically stabilizable via linear state-feedback control if there exists a state-feedback controller u(t)=Kx(t) such that the closed-loop system (9)Dαft=A+BKxtis asymptotically stable.Denote(10)A0=12A_+A¯,ΔA=12A¯-A_=γijn×n,B0=12B_+B¯,ΔB=12B¯-B_=βijn×p.To handle the interval uncertain, the following notations are introduced: (11)
D
A
=
diag
⁡
γ
11
,
…
,
γ
1
n
,
…
,
γ
n
1
,
…
,
γ
n
n
n
2
×
n
2
,
H
A
=
diag
⁡
δ
11
,
…
,
δ
1
n
,
…
,
δ
n
1
,
…
,
δ
n
n
∈
R
n
2
×
n
2
,
δ
i
j
≤
1
,
i
,
j
=
1
,
…
,
n
,
D
B
=
diag
⁡
β
11
,
…
,
β
1
p
,
…
,
β
n
1
,
…
,
β
n
p
n
p
×
n
p
,
H
B
=
diag
⁡
η
11
,
…
,
η
1
p
,
…
,
η
n
1
,
…
,
η
n
p
∈
R
n
p
×
n
p
,
η
i
j
≤
1
,
i
=
1
,
…
,
n
,
j
=
1
,
…
,
p
.Lemma 2.
Lettij>0,i,j=1,…,n: (12)
T
A
1
=
diag
⁡
t
11
,
…
,
t
1
n
,
…
,
t
n
1
,
…
,
t
n
n
n
2
×
n
2
,
T
A
2
=
T
A
1
-
1
,
τ
i
j
>
0
,
i
=
1
,
…
,
n
,
j
=
1
,
…
,
p
,
T
B
1
=
diag
⁡
τ
11
,
…
,
τ
1
p
,
…
,
τ
n
1
,
…
,
τ
n
p
n
p
×
n
p
,
T
B
2
=
T
B
1
-
1
,
(13)
A
J
=
A
=
A
0
+
I
n
⊗
e
n
T
A
1
D
A
F
A
T
A
2
e
n
T
⊗
I
n
,
F
A
∈
H
A
,
(14)
B
J
=
B
=
B
0
+
I
n
⊗
e
p
T
B
1
D
B
F
B
T
B
2
e
n
T
⊗
I
p
,
F
B
∈
H
B
.Then AJ=AI, BJ=BI.Proof.
SinceTA1, FA, DA, and TA2 are all diagonal, it is easy to check that (15)TA1DAFATA2=DAFA,In⊗en=en0⋯00en⋯0⋮⋮⋱⋮00⋯en,TA1DAFATA2enT⊗En=DAFAInIn⋮In=diag⁡δ11γ11,…,δ1nγ1ndiag⁡δ21γ21,…,δ2nγ2n⋮diag⁡δn1γn1,…,δnnγnn,endiag⁡δi1γi1,…,δinγin=δi1γi1δi2γi2⋯δi2γin1×n,i=1,…,n.It follows that (16)In⊗enDAFAenT⊗En=en0⋯00en⋯0⋮⋮⋱⋮00⋯endiag⁡δ11γ11,…,δ1nγ1ndiag⁡δ21γ21,…,δ2nγ2n⋮diag⁡δn1γn1,…,δnnγnn=endiag⁡δ11γ11,…,δ1nγ1nendiag⁡δ21γ21,…,δ2nγ2n⋮endiag⁡δn1γn1,…,δnnγnn=δ11γ11δ12γ12⋯δ1nγ1nδ21γ21δ22γ22⋯δ2nγ2n⋮⋮⋱⋮δn1γn1δn2γn2⋯δnnγnn.Thus, AI⊆AJ.
Noting that the above proof is reversible, it is easy to know thatAJ⊆AI. Therefore, AJ=AI.
In the same way, we haveBJ=BI.Lemma 3 (see [7, 15]).
LetA∈Rn×n be a deterministic real matrix without uncertainty. Then, a necessary and sufficient condition for the asymptotical stability of Dαxt=Axt is (17)arg⁡specA>π2α,where specA is the spectrum of all eigenvalues of A.Lemma 4 (see [16]).
LetA∈Rn×n be a real matrix. Then argspecA>(π/2)α where 1≤α<2, if and only if there exists P>0 such that (18)AP+PATsin⁡θ·PAT-APcos⁡θAP+PATsin⁡θ<0,where θ=π-α(π/2).Lemma 5 (see [17]).
For any matricesX and Y with appropriate dimensions, we have (19)XTY+YTX≤εXTX+ε-1YTY,∀ε>0.Lemma 6 (see [18]).
LetX, Y, and Z be real matrices of suitable dimensions. Then, for any x∈Rn, (20)max⁡xTXFYx2:FTF≤I=xTXXTxxTYTYx.Lemma 7 (see [18]).
LetX, Y, and Z be n×n symmetric matrices such thatX≥0, Y<0, and Z≥0. Furthermore, assume that xTYx2-4xTXxxTZx>0 for all nonzero x∈Rn. Then, there exists a constant λ>0 such that (21)λ2X+λY+Z<0.


## 3. Main Results

In this section, by adding parameters into linear matrix inequalities, necessary and sufficient conditions for stability and stabilization of the system are obtained. Those results are generalization of the main theorems in [12].Theorem 8.
Let1≤α<2. The uncertain FO-LTI interval system (7) with controller ut is asymptotically stable if and only if there exist some symmetric positive definite matrix P∈Rn×n and a real scalar constant ε>0 such that (22)Γ=Γ11···Γ21Γ22··Ω2P0-εIn·0Ω2P0-εIn<0,where (23)θ=π-απ2,Γ11=Γ22=A0P+PA0Tsin⁡θ+εΩ1,Γ21=PA0T-A0Pcos⁡θ,Ω1=diag⁡∑i=1nt1i2γ1i2,…,∑i=1ntni2γni2n×n,Ω2=diag⁡∑i=1nti1-2,…,∑i=1ntin-2n×n.Proof.
It is easy to check that(24)In⊗enTA1DAIn⊗enTA1DAT=diag⁡∑i=1nt1i2γ1i2,…,∑i=1ntni2γni2.
DenoteΠ=TA2enT⊗In; then we see that (25)Π=diag⁡t11-1,…,t1n-1,…,tn1-1,…,tnn-1In⋮In=diag⁡t11-1,…,t1n-1⋮diag⁡tn1-1,…,tnn-1,ΠTΠ=diag⁡∑i=1nti1-2,…,∑i=1ntin-2n×n,(26)ΠP00ΠPTΠP00ΠP=PΠTΠP00PΠTΠP=PΠTΠ1/2ΠTΠ1/2P00PΠTΠ1/2ΠTΠ1/2P=ΠTΠ1/2P00ΠTΠ1/2PT·ΠTΠ1/2P00ΠTΠ1/2P,where P∈Rn×n is an arbitrary positive define matrix.
By applying (13), we have(27)
A
P
+
P
A
T
sin
⁡
θ
·
P
A
T
-
A
P
cos
⁡
θ
A
P
+
P
A
T
sin
⁡
θ
=
A
0
P
+
P
A
0
T
sin
⁡
θ
·
P
A
0
T
-
A
0
P
cos
⁡
θ
A
0
P
+
P
A
0
T
sin
⁡
θ
+
S
ym
⁡
I
n
⊗
e
n
T
A
1
D
A
F
A
T
A
2
e
n
T
⊗
I
n
P
sin
⁡
θ
I
n
⊗
e
n
T
A
1
D
A
F
A
T
A
2
e
n
T
⊗
I
n
P
cos
⁡
θ
-
I
n
⊗
e
n
T
A
1
D
A
F
A
T
A
2
e
n
T
⊗
I
n
P
cos
⁡
θ
I
n
⊗
e
n
T
A
1
D
A
F
A
T
A
2
e
n
T
⊗
I
n
P
sin
⁡
θ
=
A
0
P
+
P
A
0
T
sin
⁡
θ
·
P
A
0
T
-
A
0
P
cos
⁡
θ
A
0
P
+
P
A
0
T
sin
⁡
θ
+
S
ym
⁡
I
n
⊗
e
n
T
A
1
D
A
sin
⁡
θ
I
n
⊗
e
n
T
A
1
D
A
cos
⁡
θ
-
I
n
⊗
e
n
T
A
1
D
A
cos
⁡
θ
I
n
⊗
e
n
T
A
1
D
A
sin
⁡
θ
F
A
0
0
F
A
T
A
2
e
n
T
⊗
I
n
P
0
0
T
A
2
e
n
T
⊗
I
n
P
=
Z
+
S
ym
⁡
M
F
A
^
N
,in which (28)Z=A0P+PA0Tsin⁡θ·PA0T-A0Pcos⁡θA0P+PA0Tsin⁡θ,M=In⊗enTA1DAsin⁡θIn⊗enTA1DAcos⁡θ-In⊗enTA1DAcos⁡θIn⊗enTA1DAsin⁡θ,FA^=FA00FA,N=TA2enT⊗InP00TA2enT⊗InP.
Sufficiency. Suppose that there exists a symmetric positive definite matrix P∈Rn×n such that (22) holds. By applying (26), (27), and Lemma 5, we have (29)
Z
+
S
ym
⁡
M
F
A
^
N
≤
Z
+
ε
M
M
T
+
ε
-
1
N
T
N
=
Γ
11
·
Γ
21
Γ
22
+
1
ε
Π
T
Π
1
/
2
P
0
0
Π
T
Π
1
/
2
P
T
·
Π
T
Π
1
/
2
P
0
0
Π
T
Π
1
/
2
P
<
0
.By using the Schur complement of (22), one obtains (30)AP+PATsin⁡θ·PAT-APcos⁡θAP+PATsin⁡θ<0.It follows from Lemma 4 that argspecA>π/2. Therefore, by Lemma 3, the uncertain FO-LTI interval system (7) is asymptotically stable.
Necessity. Suppose that the uncertain FO-LTI interval system (7) is asymptotically stable. Then, argspecA>π/2. It follows from Lemma 4 that there exists a symmetric positive definite matrix P such that (31)AP+PATsin⁡θ·PAT-APcos⁡θAP+PATsin⁡θ<0.
By using Lemma2 and after some calculations, one can obtain from (27) that (32)
Z
+
S
ym
⁡
M
F
A
^
N
<
0
.
Therefore, for allς1,ς2∈Rn, ς=ς1Tς2TT≠0, (33)ςTZς<-2ςTMFA^Nς;that is, (34)ςTZς<-2max⁡ςTMFA^NςFA^TFA^≤I≤0.
Consequently, given anyς∈R2n and ς≠0, we have (35)ςTZς2>4max⁡ςTMFA^NςFA^TFA^≤I.
Applying Lemma6, we obtain (36)ςTZς2>4ςTMMTςςTNTNς.
It follows by Lemma7 that there exists a constant ε>0 such that (37)Z+ε-1NTN+εMMT<0.
So we derive that(38)Z+εMMT+ε-1NTN=Γ11·Γ21Γ22+1εΠTΠ1/2P00ΠTΠ1/2PT·ΠTΠ1/2P00ΠTΠ1/2P<0.
Applying the well-known Schur complement yields (22).
This ends the proof.Next, let us establish a stabilization result.Theorem 9.
Let1≤α<2. The uncertain FO-LTI interval system (7) is asymptotically stable if and only if there are a matrix X∈Rp×n, a symmetric positive definite matrix P∈Rn×n, and two real scalars εi>0i=1,2, such that (39)Γ1=Γ111·····Γ211Γ221····Ω2P0-ε1In···0Ω2P0-ε1In··Ω4X000-ε2Ip·0Ω4X000-ε2Ip<0,where (40)θ=π-απ2,Γ111=Γ221=A0P+PA0T+B0X+B0XTsin⁡θ+ε1Ω1+ε2Ω3,Γ211=PA0T-A0P+B0XT-B0Xcos⁡θ,Ω1=diag⁡∑i=1nt1i2γ1i2,…,∑i=1ntni2γni2n×n,Ω2=diag⁡∑i=1nti1-2,…,∑i=1ntin-2n×n,Ω3=diag⁡∑i=1pτ1i2η1i2,…,∑i=1pτni2ηni2n×n,Ω4=diag⁡∑i=1nτi1-2,…,∑i=1nτip-2p×p.Moreover, the robustly asymptotically stabilizing state-feedback gain matrix is given by (41)K=XP-1.Remark 10.
When one takesTA1=In2 and TA2=Inp, it is easy to obtain equivalence of Theorem 8 and [12, Theorem 1] and Theorem 9 and [12, Theorem 2], respectively.


## 4. Numerical Examples

Example 1 (see [12]).
Consider the robust stability of the following uncertain FO-LTI interval system:(42)Dαxt=Axt,where α=1.6 and A∈AI=[A_,A¯]: (43)A_=-1.50.51.00.4-2.50.90.2-1.3-2.0,A¯=-1.050.651.30.7-2.11.30.35-1.05-1.75.Taking TA12=0.001∗I9, a feasible solution of(22) is as follows: (44)P=0.17090.0299-0.05010.02990.10700.0050-0.05010.00500.0872,ε=4.8484.Let(45)A=-1.250.551.150.55-2.2510.27-1.18-1.85and let the initial conditions be x1(0)=2, x2(0)=-1, and x3(0)=1. Time response of the state variables is depicted in Figure 1.Figure 1
Time response of the state variables.In the following example, we have shown the effectiveness of our results by choosing different parameters.Example 2 (see [12]).
Consider the robust stability of the following uncertain FO-LTI interval system:(46)Dαxt=Axt+But,where α=1.8 and A∈AI=[A_,A¯], B∈BI=[B_,B¯] with (47)A_=-1.950.350.7-1.3-3.90.7-0.65-1.95-3.25,B_=0.40.50.9,A¯=-1.050.651.3-0.7-2.11.3-0.35-1.05-1.75,B¯=0.50.61.(I)
TakingTA12=0.1×I9 and TB12=0.1×I3, a feasible solution of (39) is as follows: (48)P=19.0307-4.87263.6969-4.87269.9174-4.47833.6969-4.47839.8415,ε1=23.2647,X=4.43930.1546-27.5631,ε2=23.4538.Finally, the asymptotically stabilizing state-feedback gain matrix is obtained as (49)K=0.5980-1.3300-3.6306.(II)
TakingTA12=I9 and TB12=I3, we have (50)P=10-10×0.0509-0.03340.0506-0.03340.10020.04050.05060.04050.3195,ε1=8.5956×10-10,X=10-9×-0.2978-0.3742-0.7021,ε2=2.7808×10-10,(51)K=-116.5937-78.82926.4958.(III)
TakingTA12=4×I9 and TB12=4×I3, we have (52)P=10-11×0.0374-0.00910.0146-0.00910.06640.02640.01460.02640.1502,ε1=-2.7962×10-12,X=10-10×-0.2460-0.3091-0.5579,ε2=8.3122×10-11,(53)K=-68.4565-47.1044-22.2054.
Let(54)A=-1.50.451-1-31-0.5-1.5-2∈AI,B=0.450.550.95∈BI,and let the initial conditions be x1(0)=2, x2(0)=-1, and x3(0)=1, and let K be as in (49), (51), and (53), respectively. Time response of the state variables is depicted in Figures 2–4, respectively.Figure 2
Time response of the state variables whenK is as in (49).Figure 3
Time response of the state variables whenK is as in (51).Figure 4
Time response of the state variables whenK is as in (53).Remark 11.
Applying [12, Theorem 2 (28)] to Example 2 gives the same results as in Example 2 (II).


## 5. Conclusion

In this paper, the robust asymptotical stability of fractional-order interval systems with the fractional-orderα belonging to 1≤α<2 has been studied. The results on stability check for uncertain FO-LTI systems with interval coefficients of dimension n only need to solve one 4n-by-4n LMI. LMI stability conditions for fractional systems are proposed. Numerical examples have shown the effectiveness of our results. To the best of our knowledge, the idea of introducing free parameters is used for the first time to derive an analytical design of the stabilizing controllers for fractional-order dynamic interval systems. Working towards relaxing the requirements for the knowledge of system uncertainties and applying the proposed control methods to fractional-order nonlinear systems while maintaining the simplicity of the controller design are our further investigation directions.



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*Source: 606048-2015-09-16.xml*