# Fault Tolerant Control for Interval Fractional-Order Systems with Sensor Failures

Authors: Xiaona Song; Hao Shen
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

The problem of robust fault tolerant control for continuous-time fractional-order (FO) systems with interval parameters and sensor faults of0<α<2 has been investigated. By establishing sensor fault model and state observer, an observer-based FO output feedback controller is developed such that the closed-loop FO system is asymptotically stable, not only when all sensor components are working well, but also in the presence of sensor components failures. Finally, numerical simulation examples are given to illustrate the application of the proposed design method.


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## Body

## 1. Introduction

Fault tolerant control research and their application to a wide range of industrial and commercial processes have been the subjects of intensive investigations over the past two decades [1, 2]. Since unexpected faults or failures may result in substantial damage, much effort has been devoted to the fault tolerant control for various systems, such as active fault tolerant control for T-S fuzzy systems [3], reliable controller design for linear systems [4], robust satisfactory fault tolerant control of discrete-time systems [5], fault tolerant controller design for singular systems [6], and observer based fault-tolerant control for networked control systems [7]. On the other hand, fractional-order (FO) systems have attracted increasing interests, mainly due to the fact that many real-world physical systems are better characterized by FO differential equation [8–13]. The stability analysis of FO systems has been widely investigated, and there have been many stability results related to the continuous-time FO systems [14–20] and discrete-time FO systems [21]. In particular, in terms of linear matrix inequality, the stability condition has been given for continuous-time FO systems of order 0<α<1 in [18] and of order 1≤α<2 in [20]. For FO-LTI systems with interval parameters, the stability and the controllability problems have been addressed for the first time in [22] and [23], respectively.Recently, for the FO controller design problem, many authors have done some valuable works [24–26] and applied them to control a variety of dynamical processes, including integer-order and FO systems, so as to enhance the robustness and performance of the control systems. While for interval FO systems, in [27, 28], authors have investigated the stabilization problem of 0<α<1 and 1≤α<2, respectively. However, the above papers dealt with state feedback control design that requires all state variables to be available. In many cases, this condition is too restrictive. So it is meaningful to control the FO systems via output feedback controller design method, and the observer-based output feedback controller design method is one of the available choices. Moreover, to the best of our knowledge, few results have been obtained for observer-based FO output feedback controller design of FO systems with interval parameters; and sensor faults, which motivates this present study.This paper investigates the observer-based FO output feedback controller design for the FO systems with interval parameters and sensor faults, the purpose is to design the observer-based FO output feedback control law such that the resulting closed-loop FO system is stable for the order0<α≤1 and 1≤α<2, respectively. Explicit expression of the desired observer-based FO output feedback controller is given.Notations. Throughout this paper, for real symmetric matrices X and Y, the notation X≥Y (resp., X>Y) means that the matrix X-Y is positive semidefinite (resp., positive definite). The notation MT represents the transpose of the matrix M.  In×n denotes the n×n identity matrix. In symmetric block matrices, “*” is used as an ellipsis for terms induced by symmetry. Matrices, if not explicitly stated, are assumed to have appropriate dimensions. Sym(X) denotes the expression X+XT. ⊗ stands for the Kronecker products.


## 2. Preliminaries and Problem Formulation

In this paper, we adopt the following Caputo definition for fractional derivative, which allows utilization of initial values of classical integer-order derivatives with known physical interpretations [12, 29]:
(1)Dαf(t)=dαf(t)dtα=1Γ(n-α)∫0t‍f(n)(τ)dτ(t-τ)α+1-n,
where n is an integer satisfying n-1<α≤n, f(t) is a continuous function, and Γ(α) is the Euler gamma function given by
(2)Γ(α)=∫0∞‍να-1e-νdν.Considering the following fractional-order (FO) LTI systems with interval parameters:(3)Dαx(t)=Ax(t)+Bu(t),0<α<2,(4)y(t)=C0x(t),(5)x(t)=x0,
where α is the time fractional derivative order. x(t)∈Rn is the state, u(t)∈Rm is the control input, and y(t)∈Rs is the measured output. The system matrices C are known real constant matrices with appropriate dimensions; A, B are interval uncertainties in the sense that
(6)A∈AI=[A_,A¯]={[aij]:a_ij≤aij≤a¯ij,1≤i,j≤n},B∈BI=[B_,B¯]={[bij]:b_ij≤bij≤b¯ij,1≤i≤n,1≤j≤m},
where A_=[a_ij]n×n, A¯=[a¯ij]n×n  satisfy a_ij≤a¯ij for all 1≤i,j≤n,  and B_=[b_ij]n×m,  B¯=[b¯ij]n×m  satisfy b_ij≤b¯ij for all 1≤i≤n,1≤j≤m.To deal with the uncertain interval, we introduce the following notations:(7)A0=12(A_+A¯),ΔA=12(A¯-A_)={χij}n×n,B0=12(B_+B¯),ΔB=12(B¯-B_)={Ϝij}n×m.
It can be seen that all elements of ΔA and ΔB are nonnegative, so we can define(8)DA=[χ11e1n⋯χ1ne1n⋯χn1enn⋯χnnenn]n×n2,EA=[χ11e1n⋯χ1nenn⋯χn1e1n⋯χnnenn]n2×nT,DB=[Ϝ11e1n⋯Ϝ1pe1n⋯Ϝn1enn⋯Ϝnmenn]n×(nm),EB=[Ϝ11e1m⋯Ϝ1pemm⋯Ϝn1e1m⋯Ϝnmemm](nm)×mT,where ekn∈ℝn and ekm∈ℝm denote the column vectors with the kth element being 1 and all the others being 0. Also, denote
(9)HA={diag(ρ11,…,ρ1n,…,ρn1,…,ρnn)∈ℝn2×n2,|ρij|≤1,i,j=1,…,nℝn2×n2},HB={diag(σ11,…,σ1m,…,σn1,…,σnm)∈ℝ(nm)×(nm),|σij|≤1,i=1,…,n,j=1,…,m}.
Here, for AI and BI, we have the following lemma.Lemma 1 (see [27, 28]).
Let(10)AJ={A=A0+DAFAEA∣FA∈HA},BJ={B=B0+DBFBEB∣FB∈HB}.
Then AI=AJ,BI=BJ.To investigate the fault tolerant control problem in the event of sensor failures, the fault model should be established first.Letting sensor faults function matrix as [30]
(11)W=diag(w1,w2,…,ws),
where 0≤wil≤wi≤wiu<∞,  and wil<1,wiu≥1,  i=1,2,…,s are known real constants, W≠0.Now, introducing the following matrices(12)W0=diag(w01,w02,…,w0s),J=diag(j1,j2,…,js),|L|=diag(|l1|,|l2|,…,|ls|),
where w0i=(wil+wiu)/2, li=(wi-w0i)/w0i, ji=(wiu-wil)/(wiu+wil), (i=1,2,…,s), then one can obtain
(13)W=W0(I+L),|L|≤J≤I.
Now, we consider the following observer-based FO output feedback controller for the FO system (3)-(4):
(14)Dαx^(t)=A0x^(t)+B0u(t)+G[Wy(t)-y^(t)],(15)y^(t)=C0x^(t),(16)u(t)=Kx^(t),
where G and K are constant matrices to be determined.Define the observer error as(17)e(t)=x(t)-x^(t).
From (3), (4), (14)–(16), and Lemma 1, the following dynamic equations of state and error can be obtained:
(18)Dαx(t)=[A0+DAFAEA+(B0+DBFBEB)K]x(t)-(B0+DBFBEB)Ke(t),(19)Dαe(t)=[DAFAEA+DBFBEBK+G(I-W)C0]x(t)+(A0-GC0-DBFBEBK)e(t).Combining (18) and (19) yields the following augmented FO systems:
(20)Dαx¯(t)={A¯0+DABFABEAB}x¯(t)=A^x¯(t),
where
(21)x¯(t)=[x(t)e(t)],A^=A¯0+DABFABEAB,A¯0=[A0+B0K-B0KG(I-W)C0A0-GC0],DAB=[DADBDADB],FAB=[FA00FB],EAB=[EA0EBK-EBK].
Our objective is to find a systematic way to determine G and K, given α, A0,  B0,  C0, DA,  DB, EA, and EB such that the closed-loop system is stable. Note that α here is in the range of 0 to 2, which is never covered in the literature in terms of observer-based FO output feedback stabilization problem.


## 3. Main Results

In this section, we give a solution to the stability analysis and the observer-based fractional-order (FO) output feedback control problems formulated in the previous part. We first give the following results which will be used in the proof of our main results.Lemma 2 (see [17]).
LetA be a real matrix. Then, Dαx(t)=Ax(t) is asymptotically stable if and only if |arg(spec(A))|>πα/2, where spec(A) is the spectrum of all eigenvalues of A.Lemma 3 (see [31]).
LetA∈Rn×n be a real matrix and θ=π-πα/2. Then |arg(spec(A))|>πα/2, where 1≤α<2, if and only if there exists P>0 such that
(22)[(AP+PAT)sinθ(AP-PAT)cosθ*(AP+PAT)sinθ]<0.Lemma 4 (see [32]).
For any matricesX and Y with appropriate dimensions, the following holds:
(23)XTY+YTX≤1ɛXTX+ɛYTY,
for any ɛ>0.Now, we are in a position to present a solution to the stability analysis and observer-based FO output feedback control problem.


### 3.1. The Case of1≤α<2

First, we will present a solution to the stability analysis for FO systems (20) with order 1≤α<2.Theorem 5.
Given the controller gain matrixK and the observer gain matrix G, the system (20) with order 1≤α<2 and θ=π-πα/2 is robustly asymptotically stable for any sensor faults if there exists real symmetric positive definite matrix P and scalar constants ϵi>0, (i=1,2), such that
(24)[Γ1Γ2Γ3Γ4Γ5*-ϵ1I000**-ϵ2I00***-ϵ1I0****-ϵ2I]<0,
where
(25)Γ1=[(A¯1P+PA¯1T)sinθ(A¯1P-PA¯1T)cosθ*(A¯1P+PA¯1T)sinθ],Γ2=[ϵ1(sinθ)GwJϵ1(cosθ)GwJ-ϵ1(cosθ)GwJϵ1(sinθ)GwJ],Γ3=[ϵ2(sinθ)DABϵ2(cosθ)DAB-ϵ2(cosθ)DABϵ2(sinθ)DAB],Γ4=[C¯0P00C¯0P]T,Γ5=[EABP00EABP]T,A¯1=[A0+B0K-B0KG(I-W0)C0A0-GC0],Gw=[0-GW0],C¯0=[C00].Proof.
The FO-LTI interval system (3)-(4) is asymptotically stable for any sensor faults if the FO system Dαx¯(t)=A^x¯(t) is asymptotically stable. This is equivalent to that there exists a symmetric positive definite matrix P∈ℝn×n, such that
(26)[(A^P+PA^T)sinθ(A^P-PA^T)cosθ*(A^P+PA^T)sinθ]=[(A¯0P+PA¯0T)sinθ(A¯0P-PA¯0T)cosθ*(A¯0P+PA¯0T)sinθ]+Sym{[(DABFABEABP)sinθ(DABFABEABP)cosθ-(DABFABEABP)cosθ(DABFABEABP)sinθ]}=[(A¯1P+PA¯1T)sinθ(A¯1P-PA¯1T)cosθ*(A¯1P+PA¯1T)sinθ]+Sym{[(GwLC¯0P)sinθ(GwLC¯0P)cosθ-(GwLC¯0P)cosθ(GwLC¯0P)sinθ]}+Sym{[(DABFABEABP)sinθ(DABFABEABP)cosθ-(DABFABEABP)cosθ(DABFABEABP)sinθ]}<0.
From Lemma 4, one can have
(27)Sym{[(GwLC¯0P)sinθ(GwLC¯0P)cosθ-(GwLC¯0P)cosθ(GwLC¯0P)sinθ]}=Sym{[(sinθ)Gw(cosθ)Gw(-cosθ)Gw(sinθ)Gw][L00L]=Sym×[C¯0P00C¯0P]}≤ϵ1[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]×[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]T+ϵ1-1[C¯0P00C¯0P]T[C¯0P00C¯0P],(28)Sym{[(DABFABEABP)sinθ(DABFABEABP)cosθ-(DABFABEABP)cosθ(DABFABEABP)sinθ]}=Sym{[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB][FAB00FAB]×[EABP00EABP]}≤ϵ2[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]×[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]T+ϵ2-1[EABP00EABP]T[EABP00EABP].
Substituting (27) and (28) into (26), one can get
(29)[(A^P+PA^T)sinθ(A^P-PA^T)cosθ*(A^P+PA^T)sinθ]≤[(A¯1P+PA¯1T)sinθ(A¯1P-PA¯1T)cosθ*(A¯1P+PA¯1T)sinθ]+ϵ1[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]×[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]T+ϵ2[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]×[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]T+ϵ1-1[C¯0P00C¯0P]T[C¯0P00C¯0P]+ϵ2-1[EABP00EABP]T[EABP00EABP].
Taking (29) into account and using the Schur complement of (24), one obtains
(30)[(A^P+PA^T)sinθ(A^P-PA^T)cosθ*(A^P+PA^T)sinθ]<0.
It follows from the above inequality (30) and Lemma 3 that |arg(spec(A^))|>απ/2. Therefore, by Lemma 2, the FO-LTI system (20) is asymptotically stable. This completes the proof.The observer-based FO output feedback control problem for FO systems (20) with order 1≤α<2 is presented in the following theorem.Theorem 6.
Given positive scalar constantsηi(i=1,2),  the FO system (20) with order 1≤α<2 and θ=π-πα/2 is asymptotically stable if there exist the matrices X1>0, K,  G~;  the following conditions are satisfied:
(31)[Ω1Ω2Ω3Ω4Ω5*-η1-1I000**-η2-1I00***-η1I0****-η2I]<0,
where
(6)Λ1=[A0+A0T+B0K+KTB0T-B0K+C0T(I-W0)TG~TG~(I-W0)C0-KTB0TX1A0+A0TX1-G~C0-C0TG~T],Λ2=[A0-A0T+B0K-KTB0T-B0K-C0T(I-W0)TG~TG~(I-W0)C0+KTB0TX1A0-A0TX1-G~C0+C0TG~T],Ω1=[(sinθ)Λ1(cosθ)Λ2*(sinθ)Λ1],Λ3=[0-G~W0J],Ω2=[η1(sinθ)Λ3η1(cosθ)Λ3-η1(cosθ)Λ3η1(sinθ)Λ3],Λ4=[DADBX1DAX1DB],Ω3=[η2(sinθ)Λ4η2(cosθ)Λ4-η2(cosθ)Λ4η2(sinθ)Λ4],Λ5=[C0T0],Ω4=[Λ500Λ5],Ω5=[EABT00EABT].
Furthermore a desired FO observer-based output feedback controller is given in the form of (14) with parameter as follows:
(33)G=X1-1G~.Proof.
The FO-LTI system (20) is asymptotically stable. It follows from Theorem 5 that this is equivalent to that there exist a symmetric positive definite matrix P∈ℝn×n and positive scalar constants ηi, (i=1,2) such that
(34)[(A^P+PA^T)sinθ(A^P-PA^T)cosθ*(A^P+PA^T)sinθ]≤[(A¯1P+PA¯1T)sinθ(A¯1P-PA¯1T)cosθ*(A¯1P+PA¯1T)sinθ]+η1[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]×[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]T+η2[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]×[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]T+η1-1[C¯0P00C¯0P]T[C¯0P00C¯0P]+η2-1[EABP00EABP]T[EABP00EABP]<0.
Introducing the following nonsingular matrix
(35)X=[I00X1].
Let
(36)P=X-1;
then, by some calculation, we have
(37)P>0.
Now, pre- and postmultiplying the inequality in (34) by diag(X,X) and diag(X,X), respectively, set XG=G~;  then we have
(38)Ω1++η1Ω2Ω2T+η2Ω3Ω3T+η1-1Ω4Ω4T+η2-1Ω5Ω5T<0.
Inequality (38) is equivalent to (31) by the Schur complement. This completes the proof.


### 3.2. The Case of0<α≤1

In this subsection, first we will present a solution to the stability analysis for FO systems (20) with order 0<α≤1.Theorem 7.
Given the controller gain matrixK and the observer gain matrix G, the FO system (20) with order 0<α≤1 is robustly asymptotically stable for any sensor faults if there exist two real symmetric positive definite matrices Pk1∈ℝn×n, k=1,2, two skew-symmetric matrices Pk2∈ℝn×n, k=1,2, and scalar constants ɛ1ij>0, ɛ2ij>0(i,j=1,2), such that
(39)[Σ11Σ12Σ13*-Σ220**-Σ33]<0,(40)[P11P12-P12P11]>0,[P21P22-P22P21]>0,
where(41)
A
¯
1
=
[
A
0
+
B
0
K
-
B
0
K
G
(
I
-
W
0
)
C
0
A
0
-
G
C
0
]
,
G
w
=
[
0
-
G
W
0
]
,
C
¯
0
=
[
C
0
0
]
,
Σ
11
=
∑
i
=
1
2
‍
∑
j
=
1
2
‍
{
Sym
[
Θ
i
j
⊗
(
A
¯
1
P
i
j
)
]
+
ɛ
1
i
j
[
I
2
⊗
(
G
w
L
)
(
G
w
L
)
T
]
+
ɛ
2
i
j
(
I
2
⊗
D
A
B
D
A
B
T
)
}
,
Σ
12
=
[
I
2
⊗
(
C
¯
0
P
11
)
T
I
2
⊗
(
C
¯
0
P
12
)
T
I
2
⊗
(
C
¯
0
P
21
)
T
I
2
⊗
(
C
¯
0
P
22
)
T
]
,
Σ
13
=
[
I
2
⊗
(
E
A
B
P
11
)
T
I
2
⊗
(
E
A
B
P
12
)
T
I
2
⊗
(
E
A
B
P
21
)
T
I
2
⊗
(
E
A
B
P
22
)
T
]
,
Σ
22
=
diag
(
ɛ
111
,
ɛ
112
,
ɛ
121
,
ɛ
122
)
⊗
I
2
n
,
Σ
33
=
diag
(
ɛ
211
,
ɛ
212
,
ɛ
221
,
ɛ
222
)
⊗
I
2
n
,
Θ
11
=
[
sin
(
π
2
α
)
-
cos
(
π
2
α
)
cos
(
π
2
α
)
sin
(
π
2
α
)
]
,
Θ
12
=
[
cos
(
π
2
α
)
sin
(
π
2
α
)
-
sin
(
π
2
α
)
cos
(
π
2
α
)
]
,
Θ
21
=
[
sin
(
π
2
α
)
cos
(
π
2
α
)
-
cos
(
π
2
α
)
sin
(
π
2
α
)
]
,
Θ
22
=
[
-
cos
(
π
2
α
)
sin
(
π
2
α
)
-
sin
(
π
2
α
)
-
cos
(
π
2
α
)
]
.Proof.
Suppose that (39)-(40) hold. It follows from (20) that
(42)∑i=12‍∑j=12‍Sym{Θij⊗(A^Pij)}=∑i=12‍∑j=12‍Sym{Θij⊗(A¯1Pij)+Θij⊗(GwLC¯0Pij)=∑i=12‍∑j=12‍Sym+Θij⊗(DABFABEABPij)}.
Note that (13) and Lemma 1 imply
(43)LLT≤J2,FABFABT≤I.
It follows from (43) that
(44)(I2⊗L)(I2⊗L)T≤J2,(I2⊗FAB)(I2⊗FAB)T≤I.
Note that ΘijΘijT=I2(i,j=1,2); it follows from (44) and Lemma 4 that for any real scalars ɛ1ij>0  and ɛ2ij>0(i,j=1,2)(45)Sym{Θij⊗(GwLC¯0Pij)}=Sym{(Θij⊗Gw)(I2⊗L)(I2⊗C¯0Pij)}≤ɛ1ij(Θij⊗Gw)(I2⊗L)(I2⊗L)T(Θij⊗Gw)T+ɛ1ij-1(I2⊗C¯0Pij)T(I2⊗C¯0Pij)≤ɛ1ij[I2⊗(GwJ)(GwJ)T]+ɛ1ij-1(I2⊗C¯0Pij)T(I2⊗C¯0Pij),(46)Sym{Θij⊗(DABFABEABPij)}=Sym{(Θij⊗DAB)(I2⊗FAB)(I2⊗EABPij)}≤ɛ2ij(Θij⊗DAB)(I2⊗FAB)(I2⊗FAB)T(Θij⊗DAB)T+ɛ2ij-1(I2⊗EABPij)T(I2⊗EABPij)≤ɛ2ij(I2⊗DABDABT)+ɛ2ij-1(I2⊗EABPij)T(I2⊗EABPij).
Substituting (45) and (46) into (42), one has
(47)∑i=12‍∑j=12‍Sym{Θij⊗(A^Pij)}≤∑i=12‍∑j=12‍{Sym[Θij⊗(A¯1Pij)]+ɛ1ij[I2⊗(GwJ)(GwJ)T]≤∑i=12‍∑j=12‍+ɛ2ij(I2⊗DABDABT)}+∑i=12‍∑j=12‍{ɛ1ij-1(I2⊗C¯0Pij)T(I2⊗C¯0Pij)+∑i=12‍∑j=12‍+ɛ2ij-1(I2⊗EABPij)T(I2⊗EABPij)}.
Taking (47) into account and using the Schur complement of (39), one obtains
(48)∑i=12‍∑j=12‍Sym{Θij⊗(A^Pij)}<0.
It follows from the above inequality (48) and Theorem 1 of [27] that the uncertain FO-LTI interval system with 0<α≤1 is asymptotically stable.The observer-based FO output feedback control problem for FO systems (20) with order 0<α≤1 is presented in the following theorem.Theorem 8.
Given positive scalar constantsμ1i, μ2i(i=1,2), The FO system (20) with order 0<α≤1 is asymptotically stable if there exist the matrices G~1,  K and symmetric matrix Y1>0; the following condition is satisfied:
(49)[∑i=12‍Sym[Θi1⊗A¯2]Ξ1Ξ2Ξ3Ξ4*-Ξ5000**-Ξ600***-Ξ70****-Ξ8]<0,
where
(50)A¯2=[A0+B0K-B0KG~1(I-W0)C0Y1A0-G~1C0],Λ¯3=[0-G~1W0J],Λ¯4=[DADBY1DAY1DB],Ξ1=[I2⊗Λ¯3I2⊗Λ¯3],Ξ2=[I2⊗Λ¯4I2⊗Λ¯4],Ξ3=[I2⊗C¯0TI2⊗C¯0T],Ξ4=[I2⊗EABTI2⊗EABT],Ξ5=diag(μ11-1⊗I2n,μ12-1⊗I2n),Ξ6=diag(μ21-1⊗I2n,μ22-1⊗I2n),Ξ7=diag(μ11⊗I2n,μ12⊗I2n),Ξ8=diag(μ21⊗I2n,μ22⊗I2n).
Furthermore a desired observer-based FO output feedback controller is given in the form of (14) with parameter as follows:
(51)G=Y1-1G~1.Proof.
The FO-LTI system (20) is asymptotically stable. It follows from Theorem 7 that this is equivalent to that there exist two real symmetric positive definite matrices Pk1∈ℝn×n, k=1,2, and two skew-symmetric matrices Pk2∈ℝn×n, k=1,2, such that
(52)∑i=12‍∑j=12‍Sym{Θij⊗(A^Pij)}<0.
By setting P11=P21=Q, P12=P22=0 in (52), we can get that if
(53)∑i=12‍Sym{Θi1⊗(A^Q)}<0,
the FO-LTI system (20) is asymptotically stable. Similar to the proof of Theorem 7, (53) is equivalent to that there exist a symmetric positive definite matrix Q∈ℝn×n and positive real scalars μ1i and μ2i(i=1,2)  such that
(54)∑i=12‍Sym{Θi1⊗(A^Q)}≤∑i=12‍{Sym[Θi1⊗(A¯1Q)]+μ1i[I2⊗(GwJ),(I2⊗GwJ)]T≤∑i=12‍+μ2i(I2⊗DAB)(I2⊗DAB)T}+∑i=12‍{μ1i-1(I2⊗C¯0Q)T(I2⊗C¯0Q)+∑i=12‍+μ2i-1(I2⊗EABQ)T(I2⊗EABQ)(I2⊗C¯0Q)T}<0.
Introducing the following nonsingular matrix
(55)Y=[I00Y1].
Let
(56)Q=Y-1>0.
Now, pre- and postmultiplying the inequality in (54) by I2⊗Y and I2⊗Y, respectively, set YG=G~1;  then we have
(57)∑i=12‍{Sym[Θi1⊗(YA¯1)]+μ1i[I2⊗(Λ¯3),(I2⊗Λ¯3)]T∑i=12‍+μ2i(I2⊗XDAB)(I2⊗XDAB)T[I2⊗(Λ¯3),(I2⊗Λ¯3)]T}+∑i=12‍{μ1i-1(I2⊗C¯0)T(I2⊗C¯0)+∑i=12‍+μ2i-1(I2⊗EAB)T(I2⊗EAB)(I2⊗C¯0)T}<0.
Inequality (57) is equivalent to (49) by the Schur complement. This completes the proof.


## 3.1. The Case of1≤α<2

First, we will present a solution to the stability analysis for FO systems (20) with order 1≤α<2.Theorem 5.
Given the controller gain matrixK and the observer gain matrix G, the system (20) with order 1≤α<2 and θ=π-πα/2 is robustly asymptotically stable for any sensor faults if there exists real symmetric positive definite matrix P and scalar constants ϵi>0, (i=1,2), such that
(24)[Γ1Γ2Γ3Γ4Γ5*-ϵ1I000**-ϵ2I00***-ϵ1I0****-ϵ2I]<0,
where
(25)Γ1=[(A¯1P+PA¯1T)sinθ(A¯1P-PA¯1T)cosθ*(A¯1P+PA¯1T)sinθ],Γ2=[ϵ1(sinθ)GwJϵ1(cosθ)GwJ-ϵ1(cosθ)GwJϵ1(sinθ)GwJ],Γ3=[ϵ2(sinθ)DABϵ2(cosθ)DAB-ϵ2(cosθ)DABϵ2(sinθ)DAB],Γ4=[C¯0P00C¯0P]T,Γ5=[EABP00EABP]T,A¯1=[A0+B0K-B0KG(I-W0)C0A0-GC0],Gw=[0-GW0],C¯0=[C00].Proof.
The FO-LTI interval system (3)-(4) is asymptotically stable for any sensor faults if the FO system Dαx¯(t)=A^x¯(t) is asymptotically stable. This is equivalent to that there exists a symmetric positive definite matrix P∈ℝn×n, such that
(26)[(A^P+PA^T)sinθ(A^P-PA^T)cosθ*(A^P+PA^T)sinθ]=[(A¯0P+PA¯0T)sinθ(A¯0P-PA¯0T)cosθ*(A¯0P+PA¯0T)sinθ]+Sym{[(DABFABEABP)sinθ(DABFABEABP)cosθ-(DABFABEABP)cosθ(DABFABEABP)sinθ]}=[(A¯1P+PA¯1T)sinθ(A¯1P-PA¯1T)cosθ*(A¯1P+PA¯1T)sinθ]+Sym{[(GwLC¯0P)sinθ(GwLC¯0P)cosθ-(GwLC¯0P)cosθ(GwLC¯0P)sinθ]}+Sym{[(DABFABEABP)sinθ(DABFABEABP)cosθ-(DABFABEABP)cosθ(DABFABEABP)sinθ]}<0.
From Lemma 4, one can have
(27)Sym{[(GwLC¯0P)sinθ(GwLC¯0P)cosθ-(GwLC¯0P)cosθ(GwLC¯0P)sinθ]}=Sym{[(sinθ)Gw(cosθ)Gw(-cosθ)Gw(sinθ)Gw][L00L]=Sym×[C¯0P00C¯0P]}≤ϵ1[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]×[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]T+ϵ1-1[C¯0P00C¯0P]T[C¯0P00C¯0P],(28)Sym{[(DABFABEABP)sinθ(DABFABEABP)cosθ-(DABFABEABP)cosθ(DABFABEABP)sinθ]}=Sym{[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB][FAB00FAB]×[EABP00EABP]}≤ϵ2[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]×[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]T+ϵ2-1[EABP00EABP]T[EABP00EABP].
Substituting (27) and (28) into (26), one can get
(29)[(A^P+PA^T)sinθ(A^P-PA^T)cosθ*(A^P+PA^T)sinθ]≤[(A¯1P+PA¯1T)sinθ(A¯1P-PA¯1T)cosθ*(A¯1P+PA¯1T)sinθ]+ϵ1[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]×[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]T+ϵ2[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]×[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]T+ϵ1-1[C¯0P00C¯0P]T[C¯0P00C¯0P]+ϵ2-1[EABP00EABP]T[EABP00EABP].
Taking (29) into account and using the Schur complement of (24), one obtains
(30)[(A^P+PA^T)sinθ(A^P-PA^T)cosθ*(A^P+PA^T)sinθ]<0.
It follows from the above inequality (30) and Lemma 3 that |arg(spec(A^))|>απ/2. Therefore, by Lemma 2, the FO-LTI system (20) is asymptotically stable. This completes the proof.The observer-based FO output feedback control problem for FO systems (20) with order 1≤α<2 is presented in the following theorem.Theorem 6.
Given positive scalar constantsηi(i=1,2),  the FO system (20) with order 1≤α<2 and θ=π-πα/2 is asymptotically stable if there exist the matrices X1>0, K,  G~;  the following conditions are satisfied:
(31)[Ω1Ω2Ω3Ω4Ω5*-η1-1I000**-η2-1I00***-η1I0****-η2I]<0,
where
(6)Λ1=[A0+A0T+B0K+KTB0T-B0K+C0T(I-W0)TG~TG~(I-W0)C0-KTB0TX1A0+A0TX1-G~C0-C0TG~T],Λ2=[A0-A0T+B0K-KTB0T-B0K-C0T(I-W0)TG~TG~(I-W0)C0+KTB0TX1A0-A0TX1-G~C0+C0TG~T],Ω1=[(sinθ)Λ1(cosθ)Λ2*(sinθ)Λ1],Λ3=[0-G~W0J],Ω2=[η1(sinθ)Λ3η1(cosθ)Λ3-η1(cosθ)Λ3η1(sinθ)Λ3],Λ4=[DADBX1DAX1DB],Ω3=[η2(sinθ)Λ4η2(cosθ)Λ4-η2(cosθ)Λ4η2(sinθ)Λ4],Λ5=[C0T0],Ω4=[Λ500Λ5],Ω5=[EABT00EABT].
Furthermore a desired FO observer-based output feedback controller is given in the form of (14) with parameter as follows:
(33)G=X1-1G~.Proof.
The FO-LTI system (20) is asymptotically stable. It follows from Theorem 5 that this is equivalent to that there exist a symmetric positive definite matrix P∈ℝn×n and positive scalar constants ηi, (i=1,2) such that
(34)[(A^P+PA^T)sinθ(A^P-PA^T)cosθ*(A^P+PA^T)sinθ]≤[(A¯1P+PA¯1T)sinθ(A¯1P-PA¯1T)cosθ*(A¯1P+PA¯1T)sinθ]+η1[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]×[(sinθ)GwJ(cosθ)GwJ(-cosθ)GwJ(sinθ)GwJ]T+η2[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]×[(sinθ)DAB(cosθ)DAB-(cosθ)DAB(sinθ)DAB]T+η1-1[C¯0P00C¯0P]T[C¯0P00C¯0P]+η2-1[EABP00EABP]T[EABP00EABP]<0.
Introducing the following nonsingular matrix
(35)X=[I00X1].
Let
(36)P=X-1;
then, by some calculation, we have
(37)P>0.
Now, pre- and postmultiplying the inequality in (34) by diag(X,X) and diag(X,X), respectively, set XG=G~;  then we have
(38)Ω1++η1Ω2Ω2T+η2Ω3Ω3T+η1-1Ω4Ω4T+η2-1Ω5Ω5T<0.
Inequality (38) is equivalent to (31) by the Schur complement. This completes the proof.


## 3.2. The Case of0<α≤1

In this subsection, first we will present a solution to the stability analysis for FO systems (20) with order 0<α≤1.Theorem 7.
Given the controller gain matrixK and the observer gain matrix G, the FO system (20) with order 0<α≤1 is robustly asymptotically stable for any sensor faults if there exist two real symmetric positive definite matrices Pk1∈ℝn×n, k=1,2, two skew-symmetric matrices Pk2∈ℝn×n, k=1,2, and scalar constants ɛ1ij>0, ɛ2ij>0(i,j=1,2), such that
(39)[Σ11Σ12Σ13*-Σ220**-Σ33]<0,(40)[P11P12-P12P11]>0,[P21P22-P22P21]>0,
where(41)
A
¯
1
=
[
A
0
+
B
0
K
-
B
0
K
G
(
I
-
W
0
)
C
0
A
0
-
G
C
0
]
,
G
w
=
[
0
-
G
W
0
]
,
C
¯
0
=
[
C
0
0
]
,
Σ
11
=
∑
i
=
1
2
‍
∑
j
=
1
2
‍
{
Sym
[
Θ
i
j
⊗
(
A
¯
1
P
i
j
)
]
+
ɛ
1
i
j
[
I
2
⊗
(
G
w
L
)
(
G
w
L
)
T
]
+
ɛ
2
i
j
(
I
2
⊗
D
A
B
D
A
B
T
)
}
,
Σ
12
=
[
I
2
⊗
(
C
¯
0
P
11
)
T
I
2
⊗
(
C
¯
0
P
12
)
T
I
2
⊗
(
C
¯
0
P
21
)
T
I
2
⊗
(
C
¯
0
P
22
)
T
]
,
Σ
13
=
[
I
2
⊗
(
E
A
B
P
11
)
T
I
2
⊗
(
E
A
B
P
12
)
T
I
2
⊗
(
E
A
B
P
21
)
T
I
2
⊗
(
E
A
B
P
22
)
T
]
,
Σ
22
=
diag
(
ɛ
111
,
ɛ
112
,
ɛ
121
,
ɛ
122
)
⊗
I
2
n
,
Σ
33
=
diag
(
ɛ
211
,
ɛ
212
,
ɛ
221
,
ɛ
222
)
⊗
I
2
n
,
Θ
11
=
[
sin
(
π
2
α
)
-
cos
(
π
2
α
)
cos
(
π
2
α
)
sin
(
π
2
α
)
]
,
Θ
12
=
[
cos
(
π
2
α
)
sin
(
π
2
α
)
-
sin
(
π
2
α
)
cos
(
π
2
α
)
]
,
Θ
21
=
[
sin
(
π
2
α
)
cos
(
π
2
α
)
-
cos
(
π
2
α
)
sin
(
π
2
α
)
]
,
Θ
22
=
[
-
cos
(
π
2
α
)
sin
(
π
2
α
)
-
sin
(
π
2
α
)
-
cos
(
π
2
α
)
]
.Proof.
Suppose that (39)-(40) hold. It follows from (20) that
(42)∑i=12‍∑j=12‍Sym{Θij⊗(A^Pij)}=∑i=12‍∑j=12‍Sym{Θij⊗(A¯1Pij)+Θij⊗(GwLC¯0Pij)=∑i=12‍∑j=12‍Sym+Θij⊗(DABFABEABPij)}.
Note that (13) and Lemma 1 imply
(43)LLT≤J2,FABFABT≤I.
It follows from (43) that
(44)(I2⊗L)(I2⊗L)T≤J2,(I2⊗FAB)(I2⊗FAB)T≤I.
Note that ΘijΘijT=I2(i,j=1,2); it follows from (44) and Lemma 4 that for any real scalars ɛ1ij>0  and ɛ2ij>0(i,j=1,2)(45)Sym{Θij⊗(GwLC¯0Pij)}=Sym{(Θij⊗Gw)(I2⊗L)(I2⊗C¯0Pij)}≤ɛ1ij(Θij⊗Gw)(I2⊗L)(I2⊗L)T(Θij⊗Gw)T+ɛ1ij-1(I2⊗C¯0Pij)T(I2⊗C¯0Pij)≤ɛ1ij[I2⊗(GwJ)(GwJ)T]+ɛ1ij-1(I2⊗C¯0Pij)T(I2⊗C¯0Pij),(46)Sym{Θij⊗(DABFABEABPij)}=Sym{(Θij⊗DAB)(I2⊗FAB)(I2⊗EABPij)}≤ɛ2ij(Θij⊗DAB)(I2⊗FAB)(I2⊗FAB)T(Θij⊗DAB)T+ɛ2ij-1(I2⊗EABPij)T(I2⊗EABPij)≤ɛ2ij(I2⊗DABDABT)+ɛ2ij-1(I2⊗EABPij)T(I2⊗EABPij).
Substituting (45) and (46) into (42), one has
(47)∑i=12‍∑j=12‍Sym{Θij⊗(A^Pij)}≤∑i=12‍∑j=12‍{Sym[Θij⊗(A¯1Pij)]+ɛ1ij[I2⊗(GwJ)(GwJ)T]≤∑i=12‍∑j=12‍+ɛ2ij(I2⊗DABDABT)}+∑i=12‍∑j=12‍{ɛ1ij-1(I2⊗C¯0Pij)T(I2⊗C¯0Pij)+∑i=12‍∑j=12‍+ɛ2ij-1(I2⊗EABPij)T(I2⊗EABPij)}.
Taking (47) into account and using the Schur complement of (39), one obtains
(48)∑i=12‍∑j=12‍Sym{Θij⊗(A^Pij)}<0.
It follows from the above inequality (48) and Theorem 1 of [27] that the uncertain FO-LTI interval system with 0<α≤1 is asymptotically stable.The observer-based FO output feedback control problem for FO systems (20) with order 0<α≤1 is presented in the following theorem.Theorem 8.
Given positive scalar constantsμ1i, μ2i(i=1,2), The FO system (20) with order 0<α≤1 is asymptotically stable if there exist the matrices G~1,  K and symmetric matrix Y1>0; the following condition is satisfied:
(49)[∑i=12‍Sym[Θi1⊗A¯2]Ξ1Ξ2Ξ3Ξ4*-Ξ5000**-Ξ600***-Ξ70****-Ξ8]<0,
where
(50)A¯2=[A0+B0K-B0KG~1(I-W0)C0Y1A0-G~1C0],Λ¯3=[0-G~1W0J],Λ¯4=[DADBY1DAY1DB],Ξ1=[I2⊗Λ¯3I2⊗Λ¯3],Ξ2=[I2⊗Λ¯4I2⊗Λ¯4],Ξ3=[I2⊗C¯0TI2⊗C¯0T],Ξ4=[I2⊗EABTI2⊗EABT],Ξ5=diag(μ11-1⊗I2n,μ12-1⊗I2n),Ξ6=diag(μ21-1⊗I2n,μ22-1⊗I2n),Ξ7=diag(μ11⊗I2n,μ12⊗I2n),Ξ8=diag(μ21⊗I2n,μ22⊗I2n).
Furthermore a desired observer-based FO output feedback controller is given in the form of (14) with parameter as follows:
(51)G=Y1-1G~1.Proof.
The FO-LTI system (20) is asymptotically stable. It follows from Theorem 7 that this is equivalent to that there exist two real symmetric positive definite matrices Pk1∈ℝn×n, k=1,2, and two skew-symmetric matrices Pk2∈ℝn×n, k=1,2, such that
(52)∑i=12‍∑j=12‍Sym{Θij⊗(A^Pij)}<0.
By setting P11=P21=Q, P12=P22=0 in (52), we can get that if
(53)∑i=12‍Sym{Θi1⊗(A^Q)}<0,
the FO-LTI system (20) is asymptotically stable. Similar to the proof of Theorem 7, (53) is equivalent to that there exist a symmetric positive definite matrix Q∈ℝn×n and positive real scalars μ1i and μ2i(i=1,2)  such that
(54)∑i=12‍Sym{Θi1⊗(A^Q)}≤∑i=12‍{Sym[Θi1⊗(A¯1Q)]+μ1i[I2⊗(GwJ),(I2⊗GwJ)]T≤∑i=12‍+μ2i(I2⊗DAB)(I2⊗DAB)T}+∑i=12‍{μ1i-1(I2⊗C¯0Q)T(I2⊗C¯0Q)+∑i=12‍+μ2i-1(I2⊗EABQ)T(I2⊗EABQ)(I2⊗C¯0Q)T}<0.
Introducing the following nonsingular matrix
(55)Y=[I00Y1].
Let
(56)Q=Y-1>0.
Now, pre- and postmultiplying the inequality in (54) by I2⊗Y and I2⊗Y, respectively, set YG=G~1;  then we have
(57)∑i=12‍{Sym[Θi1⊗(YA¯1)]+μ1i[I2⊗(Λ¯3),(I2⊗Λ¯3)]T∑i=12‍+μ2i(I2⊗XDAB)(I2⊗XDAB)T[I2⊗(Λ¯3),(I2⊗Λ¯3)]T}+∑i=12‍{μ1i-1(I2⊗C¯0)T(I2⊗C¯0)+∑i=12‍+μ2i-1(I2⊗EAB)T(I2⊗EAB)(I2⊗C¯0)T}<0.
Inequality (57) is equivalent to (49) by the Schur complement. This completes the proof.


## 4. Simulation

Consider the fault tolerant control problem for the fractional-order (FO) systems (3)-(4) of order 0<α<2 with the following parameters
(58)α=0.8,A0=[-100.10.01],B0=[1.8001],C0=[10.101],DA=[0.0010.00100000.0010.001],DB=[0.0020.00200000.0020.002],EA=[0.00100.001000.00100.001]T,EB=[0.00200.002000.00200.002]T,wl=[0.2000.5],wu=[1.2001.5].
The purpose is to design a observer-based FO output feedback control law such that the closed-loop system is stable in the event of sensor failure. Now, we choose
(59)η1=0.1,η2=0.1,
and the initial state x0=[-0.10.1]T.  Then, using Matlab Linear Matrix Inequality (LMI) Control Toolbox to solve the LMI (49), we can obtain the solution as follows:
(60)Y1=103×[0.0215-0.2798-0.2798-2.7749],G=[-0.1054-0.03880.0102-0.0002],K=[-16.2663-0.0800-0.0144-27.9000].
While, for the FO system (3)-(4) with the following parameters:
(61)α=1.6,A0=[-0.1000.01],B0=[-500-5],C0=[10.101],DA=[0.0010.00100000.0010.001],DB=[0.0020.00200000.0020.002],EA=[0.00100.001000.00100.001]T,EB=[0.00200.002000.00200.002]T,wl=[0.2000.5],wu=[1.2001.5].
Then, we choose
(62)μ11=0.1,μ12=0.1,μ21=0.1,μ22=0.1,
and the initial state x0=[0.1-0.1]T. Also using Matlab LMI Control Toolbox to solve the LMI (31), we can obtain the solution as follows:
(63)X1=[38.63690.78500.7850-434.9420],(64)G=[0.01090.00070.00020.0008],(65)K=[0.5454-0.00120.00430.5240].
With the observer-based FO output feedback controller, the closed-loop system is stable. The state response x1(t) of the FO systems of order 0<α≤1 and 1≤α<2 are given in Figures 1 and 2, respectively, while the corresponding control input u1(t) are shown in Figures 3 and 4. The state response x2(t) of the FO systems of order 0<α≤1 and 1≤α<2 are given in Figures 5 and 6, respectively. While Figures 7 and 8 show the corresponding control input u2(t).Figure 1
State responsex1(t)  (0<α≤1).Figure 2
State responsex1(t)  (1≤α<2).Figure 3
Control inputu1(t)  (0<α≤1).Figure 4
Control inputu1(t)  (1≤α<2).Figure 5
State responsex2(t)  (0<α≤1).Figure 6
State responsex2(t)  (1≤α<2).Figure 7
Control inputu2(t)  (0<α≤1).Figure 8
Control inputu2(t)  (1≤α<2).From these simulation results, it can be seen that the designed observer-based FO output feedback controller ensures the asymptotic stability of the FO systems of0<α<2 in the event of the sensor faults.


## 5. Conclusion

The problem of fault tolerant control for fractional-order (FO) systems with uncertain interval parameters and sensor faults is studied. By establishing sensor fault model and state observer, an observer-based FO output feedback controller, which stabilizes the FO systems of0<α<2 in the event of some sensor failures, is given. Finally, numerical simulation results show that the proposed method is effective.



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*Source: 836743-2013-09-12.xml*