fault-reconstruction-based cascaded sliding mode observers...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 623426, 20 pagesdoi:10.1155/2012/623426

Research ArticleFault-Reconstruction-Based Cascaded SlidingMode Observers for Descriptor Linear Systems

Jinyong Yu,1 Guanghui Sun,1 and Hamid Reza Karimi2

1 Research Institute of Intelligent Control and Systems, Harbin Institute of Technology,Harbin 150001, China

2 Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway

Correspondence should be addressed to Jinyong Yu, [email protected]

Received 25 May 2012; Accepted 8 July 2012

Academic Editor: Zidong Wang

Copyright q 2012 Jinyong Yu et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper develops a cascaded sliding mode observer method to reconstruct actuator faultsfor a class of descriptor linear systems. Based on a new canonical form, a novel design methodis presented to discuss the existence conditions of the sliding mode observer. Furthermore, theproposed method is extended to general descriptor linear systems with actuator faults. Finally, theeffectiveness of the proposed technique is illustrated by a simulation example.

1. Introduction

With the development and applications of modern control techniques, the safety andreliability of control systems are becoming increasingly important. Therefore, the faultdiagnosis has become one of the most important techniques to ensure the safety andreliability of control systems [1, 2]. During the last two decades, many significant results havebeen obtained for the analysis and observer design of fault diagnosis of the regular systems,such as unknown input observers [3, 4], eigenstructure assignment method [5], H∞ filtering[6–9], parity space approach [10], and parameter identification approach [11].

Just like regular systems, the fault diagnosis for descriptor systems has recentlyattracted increasing attention due to their importance in real-world systems. In [12], aparametric approach is proposed to design unknown input observers to realize faultdetection of descriptor linear multivariable systems with unknown disturbances. By directlyidentifying parity space, a model-free approach for fault detection is developed, which canbe applied if the model of descriptor systems is unknown [13]. In [14], the factorization

2 Mathematical Problems in Engineering

approach for robust residual generation is extended to descriptor systems, and then a post-filter is added to ensure the robustness of fault diagnosis. In [15], H∞ filter is utilizedfor providing disturbance rejection and robustness properties of the fault detection andisolation schemes of linear time-invariant descriptor systems. In [16], several sufficientconditions of existence of unknown input observers are obtained for Takagi-Sugenodescriptor systems, which are affected by unknown inputs. Unfortunately, although thesemethods can successfully generate residuals, they fail to reconstruct fault signals.

Recently, fault reconstruction is a promising alternative for fault detection. Instead ofgenerating residuals, a number of methods, such as sliding mode observers (SMOs) [17–23], descriptor observer method [24–26], and PI observer [27–29], can be used to reconstructfault signals. The sliding mode control is employed in the situations including stateestimation and fault detection, since it is insensitive to matched uncertainties, nonlinearity,or disturbances [30]. Edwards et al. [17] firstly used the concept of the equivalent outputerror injection signals to reconstruct faults. Tan and Edwards [19] extended this work forrobust reconstruction of sensor and actuator faults by minimizing the effect of uncertainty onthe reconstruction in an L2 sense. Some well-studied works, aiming at reducing the systemconstraints associated with the results in [17, 19], have recently appeared in the literature[18, 20–23]. In order to relax the matching conditions, the cascaded sliding mode observermethod was proposed to deal with a class of systems with relative degree higher than one[20, 21]. In [22], the auxiliary outputs are defined such that the conventional sliding modeobserver in [17] can be used for systems without the observer matching condition. In orderto obtain those auxiliary outputs, high-order sliding-mode observers are constructed to act asexact differentiators using a super-twisting algorithm. Inspired by Floquet et al. [22], high-gain approximate differentiators and high-order sliding-mode robust differentiators wereproposed to generate auxiliary outputs for the design of sliding mode observers [18, 23].

Although there are many achievements in regular systems, few results have beenreported to the descriptor case despite its importance in real-world systems. In [31, 32], thesliding mode observer method was employed to detect and isolate faults and to reconstructthe faults for descriptor systems. However, the uncertainty was not considered in theseresults. In [33], the sliding mode observer was proposed to minimize the effect of uncertainlyon the reconstruction of faults for descriptor systems. Unfortunately, the fault detection filterbased sliding mode observer has to satisfy the strict condition in [31–33], which severelylimits the applicability of these approaches for a wide range of practical systems.

Motivated by the above discussion, in this paper, we develop a cascaded slidingmode observer method to reconstruct actuator faults for a class of descriptor linear systems.The main contribution of this paper can be summarized as follows: (1) we present a novelcascaded sliding mode observer method to reconstruct actuator faults for a class of descriptorlinear systems; (2) in the design process, we remove this restrictive assumption and extendthe cascaded sliding mode observer approach of Tan et al. [20, 21] to descriptor systems; (3)a novel cascaded sliding mode observer is designed for reconstructing actuator faults for aclass of descriptor linear systems.

The paper is organized as follows. In Section 2, the problem is formulated, andappropriate coordinate transformations are introduced to exploit the system structure. InSection 3 the design algorithm of cascaded sliding mode observer for linear descriptorsystems is given. In Section 4, a design method of cascaded sliding mode observer and faultreconstruction for general descriptor systems are presented. In Section 5, an example is givento support the effectiveness of the proposed approach. Finally, the conclusions are drawn.

Mathematical Problems in Engineering 3

2. Problem Statement and System Analysis

Consider a descriptor linear system described by

Ex = Ax + Bu +Df

y = Cx,(2.1)

where x ∈ Rn is the state variable, u ∈ Rk is the input vector, y ∈ Rp is the output variable,and f ∈ Rq is unknown but bounded so that

∥∥f

∥∥ ≤ β, (2.2)

where the positive scalar β is known. The signal f models the actuator fault within the system.A ∈ Rm×n, B ∈ Rm×k, C ∈ Rp×n, and D ∈ Rm×q are known constant real matrices. Without lossof generality, it is assumed that rank(D) = q, rank(C) = p, and E is full row rank.

In [32], a sliding mode observer is given in the following form:

z = Fz + T1Bu +K1y +K2y +Gnυ

x = z + T2y

y = Cx,

(2.3)

where z ∈ Rn is the state vector of the SMO, x is the estimation of the state vector x, and υ isthe discontinuous output error injection vector defined by

υ =

⎧

⎪⎨

⎪⎩

−η P0ey∥∥P0ey

∥∥

ey /= 0

0 other,(2.4)

where ey = y − y, η > 0, F, T1, T2, K1, K2, Gn, and P0 are parameters to be designed.For the descriptor system (2.1), the sufficient conditions for the existence of the sliding

mode observer (2.3) are as follows:

rank[E DC 0

]

= n + q (2.5)

rank[sE −A D

C 0

]

= n + q, Re(s) ≥ 0. (2.6)


It is well known that condition (2.5) is quite restrictive and may not apply to a widerange of systems. In the following, we give two more relaxed conditions:

rank[EC

]

= n, (2.7)

rank[E DC 0

]

= n + l, (2.8)

where l ≤ q.Before presenting the main results, some lemmas are given as follows.

Lemma 2.1. If the conditions (2.7) and (2.8) hold, there exists a nonsingular matrix U such that

rank[E D1

C 0

]

= n + l, (2.9)

rank[E D2

C 0

]

= n, (2.10)

where [D1 D2] = DU, and D1 ∈ Rm×l, D2 ∈ Rm×(q−l).

Proof. if l is equal to q, the conclusion is obviously true. So the following is to prove the casethat l is less than q.

Obviously, there exists a nonsingular matrixU1 so thatDU1 = [D1 D2] and (2.9) hold.Then,

rank[E DC 0

]

= rank

[

E D1 D2

C 0 0

]

= n + l. (2.11)

So there exists a matrix Y =[Y1Y2

]

so that

[

D2

0

]

=[E D1

C 0

][Y1

Y2

]

. (2.12)

Thus, we have D2 = EY1 +D1Y2 and CY1 = 0.Setting

U2 =[I −Y2

0 I

]

(2.13)

and U = U1U2, we have

rank[E D2

C 0

]

= rank[E EY1

C CY1

]

= rank[EC

]

= n. (2.14)


Lemma 2.2. If the following conditions

rank[E D1

C 0

]

= n + l,

rank[sE −A D1

C 0

]

= n + l, Re(s) ≥ 0

(2.15)

hold, there exist two nonsingular matrices P and Q such that

PEQ =[

0 E12

In−p E22

]

, PAQ =[A11 A12

A21 A22

]

PD1 =[DI

0

]

, CQ =[

0 Ip]

,

(2.16)

where E12 ∈ R(m−n+p)×p, E22 ∈ R(n−p)×p, A11 ∈ R(m−n+p)×(n−p), A12 ∈ R(m−n+p)×p, A21 ∈ R(n−p)×(n−p),A22 ∈ R(n−p)×p, DI = [0 Il]

T ∈ R(m−n+p)×l, and the subblock A11 has the structure

A11 =[A111

A112

]

, (2.17)

in which A111 ∈ R(m−n+p−l)×(n−p), A112 ∈ Rl×(n−p), and the pair (A21, A111) is detectable.It can be established easily by Lemma 2 in [33], and hence the proof is omitted.

Lemma 2.3. If the conditions (2.6), (2.7), and (2.8) hold, there exist nonsingular matrices P ,Q, andU such that

PEQ =[

0 E12

In−p E22

]

, PAQ =[A11 A12

A21 A22

]

(2.18)

PB =[B1

B2

]

, CQ =[

0 Ip]

(2.19)

PDU =[D11 00 D22

]

, (2.20)

where E12 ∈ R(m−n+p)×p, E22 ∈ R(n−p)×p, A11 ∈ R(m−n+p)×(n−p), A12 ∈ R(m−n+p)×p, A21 ∈ R(n−p)×(n−p),A22 ∈ R(n−p)×p, B1 ∈ R(m−n+p)×k, B2 ∈ R(n−p)×k, D11 = [0 Il]

T ∈ R(m−n+p)×l, D22 ∈ R(n−p)×(q−l), andthe subblock A11 has the structure

A11 =[A111

A112

]

, (2.21)

where A111 ∈ R(m−n+p−l)×(n−p), A112 ∈ Rl×(n−p), and (A21, A111) is detectable.


Proof . By Lemma 2.1, there exists a nonsingular matrix U such that (2.9) and (2.10) hold,where DU = [D1 D2].

Obviously,

rank[sE −A D1

C 0

]

= n + l, Re(s) ≥ 0. (2.22)

By Lemma 2.2, there exist two nonsingular matrices P and Q such that (2.18) and(2.19) hold and

PD1 =[D11

0

]

. (2.23)

Setting PD2 =[D21D22

]

, we have

rank[E D2

C 0

]

= rank[P 00 I

][E D2

C 0

][Q 00 I

]

= rank

⎡

⎣

0 E12 D21

In−p E22 D22

0 Ip 0

⎤

⎦

= rank(D21) + n.

(2.24)

Combining (2.10) and (2.24), we have rank(D21) = 0. Obviously, D21 = 0.

By Lemma 2.3, it can be assumed without loss of generality that system (2.1) has thefollowing form:

[0 E12

In−p E22

][x1

x2

]

=[A11 A12

A21 A22

][x1

x2

]

+[B1

B2

]

u

+[D11

0

]

f1 +[

0D22

]

f2

y = x2,

(2.25)

where x = [xT1 xT

2 ]T, x1 ∈ Rn−p, x2 ∈ Rp and

f −→ Uf =[

fT1 fT

2

]T. (2.26)

The descriptor system (2.25)may be considered as the systemwith the fault f1 and thedisturbance f2. Using the fault reconstructionmethod in [33], the fault f1 can be reconstructedand the L2 gain from the f2 to reconstruction error of fault f1 can be minimized. But the faultf2 and the state x1 cannot be estimated. Inspired by Tan et al. [20, 21], the cascaded slidingmode observer is applied to estimate both the state x and fault f in the following.


3. Design of Cascaded Sliding Mode Observer

The primary sliding mode observer for system (2.25) is

z = (T1A −K1C)z + T1Bu +K1y + FT2y +Gnυ

x = z + T2y

y = x2,

(3.1)

where z ∈ Rn is the state vector of the SMO, x = [xT1 xT

2 ]T with x1 ∈ Rn−p and x2 ∈ Rp is the

estimation of the state vector x, Gn = [0 I]T , T1 and T2 are defined by

T1 =[Z1 In−pZ2 0

]

(3.2)

T2 =[0Ip

]

− T1

[E12

E22

]

(3.3)

Z1 =[

Z11 0]

, (3.4)

Z1 ∈ R(n−p)×(m−n+p), Z11 ∈ R(n−p)×(m−n+p−q), Z2 ∈ Rp×(m−n+p) is full rank, υ is the discontinuousoutput error injection vector defined by:

υ =

⎧

⎨

⎩

−η P2e2‖P2e2‖ e2 /= 0

0 other,(3.5)

e2 = x2 − x2, η > 0, Z11, Z2, K1, F, and P2 are parameters to be designed.In [33], it is shown that for an appropriate choice of observer parameters an ideal

sliding motion takes place on S = {(e1, e2) | e2 = 0} in finite time.Define e = x − x as the state estimation error, the following estimation error dynamic

is obtained:

e1 = (A21 + Z1A11)e1 + (Z1A12 +A22 −K11)e2 +D22f2

e2 = Z2A11e1 + (Z2A12 −K12)e2 − Z2D11f1 + υ,(3.6)

where e = [eT1 eT2 ]T , e1 ∈ Rn−p, K1 = [KT

11 KT12]

T , K11 ∈ R(n−p)×p, and K12 ∈ Rp×p.Assuming the primary sliding mode observer has been designed, and that a sliding

motion has been achieved, then e2 = e2 = 0, and the error equation becomes

e1 = (A21 + Z1A11)e1 +D22f2

Z+2υeq = −A11e1 +D11f1,

(3.7)


where Z+2 is the generalized inverse matrix of Z2, υeq is the equivalent output error injection

term that can be approximated to any degree of accuracy by replacing (3.8)with

υeq = −η P2e2‖P2e2‖ + δ

, (3.8)

where δ is a small positive constant.The remaining system freedom can be used to estimate the state x1 and reconstruct the

fault f2. Equation (3.7) can be rewritten as

e1 = (A21 + Z1A11)e1 +D22f2 (3.9)

D⊥11Z

+2υeq = −A111e1, (3.10)

where D⊥11 = [ Im+p−n−l 0 ].

For any A111, there exists a nonsingular matrix W so that WA111 = [ AT111 0 ]T , where

A111 ∈ Rp×(n−p) is full row rank. We have

[

Ip 0]

WD⊥11Z

+2υeq = −A111e1. (3.11)

The system (3.9) and (3.11)may be considered as the linear systemwith the q− l faults,the n − p states f1 and the p outputs. Using the sliding mode observer design method for thelinear system in [17], we can design a secondary sliding mode observer to estimate e1 and f2if the following conditions hold:

rank(

A111D22

)

= rank(D22)

rank

[

sI − (A21 + Z1A11) D22

A111 0

]

= n − p + q − l, Re(s) ≥ 0.(3.12)

Obviously, (3.12) are equivalent to

rank(A111D22) = rank(D22) (3.13)

rank[sI − (A21 + Z1A11) D22

A111 0

]

= n − p + q − l, Re(s) ≥ 0. (3.14)

Combined with (3.4), (3.14) is equivalent to

rank[sI −A21 D22

A111 0

]

= n − p + q − l, Re(s) ≥ 0. (3.15)

From the above analysis, if the conditions (3.13) and (3.15) satisfy, there exists acascaded sliding mode observer for the descriptor system (2.1).

Next, the fault reconstruction method based cascaded sliding mode observer is given.


Assuming that the secondary sliding mode observer has been designed and the e1 andf2 are the estimations of e1 and f2, respectively. Then, the reconstruction signal of the fault f1is described by

f1 = A112e1 +[

0 Il]

Z+2υeq, (3.16)

and the estimation of the state x1 is described by

x1 − e1 −→ x1. (3.17)

The reconstruction of fault is described by

f = U−1[

f T1 f T

2

]T(3.18)

Equations (3.13) and (3.15) are the sufficient conditions for the existence of thecascaded sliding mode observer, but these cannot be checked using the parameters of theoriginal system (2.1). Now, for system (2.1), sufficient conditions for the existence of thecascaded sliding mode observer can be given by Theorem 3.1.

Theorem 3.1. There exists a cascaded sliding mode observer for system (2.1) if the following condi-tions hold:

rank[EC

]

= n (3.19)

rank

⎡

⎢⎢⎣

E A D 00 E 0 DC 0 0 00 C 0 0

⎤

⎥⎥⎦

= n + q + rank[E DC 0

]

, (3.20)

rank[sE −A D

C 0

]

= n + q, Re(s) ≥ 0. (3.21)

Proof. If l is equal to q, the conclusion is obviously true. So, the following is to prove the casethat l is less than q.


Substituting (2.8) and (2.25) into (3.20), we have

rank

⎡

⎢⎢⎣

E A D 00 E 0 DC 0 0 00 C 0 0

⎤

⎥⎥⎦

= rank

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

0 E12 A11 A12 D11 0 0 0In−p E22 A21 A22 0 D22 0 00 0 0 E12 0 0 D11 00 0 In−p E12 0 0 0 D22

0 Ip 0 0 0 0 0 00 0 0 Ip 0 0 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

= n + p + l + rank[A11 D11 0In−p 0 D22

]

= n + p + l + rank

⎡

⎣

A111 0 0A112 Il 0In−p 0 D22

⎤

⎦

= n + p + 2l + rank[In−p D22

A111 0

]

= 2n + q + l.

(3.22)

So we have

rank[In−p D22

A111 0

]

= n − p + q − l. (3.23)

By Lemma 1 in [32], (3.13) holds.Substituting (2.18), (2.19), and (2.20) into (3.21), we can obtain (3.15).

4. Cascaded Sliding Mode Observer Design and Fault Reconstructionfor General Descriptor Systems

In Section 2, it is assumed that E is full row rank. In the following, it is discussed that E isrank deficient. Let r := rank(E) ≤ min{m,n}.

Now, since rank(E) = r, there exists a regular matrix P ∗ such that (2.1) is restrictedsystem equivalent to

E∗x = A∗x + B∗u +D∗f

y1 = −B1u = A1x +D1f

y = Cx,

(4.1)


where P ∗E =[E∗0

]

, P ∗A =[A∗

A1

]

, P ∗B =[B∗

B1

]

, P ∗D =[D∗

D1

]

, E∗ ∈ Rr×n, A∗ ∈ Rr×n, B∗ ∈ Rr×k,

D∗ ∈ Rr×p, A1 ∈ R(m−r)×n, B1 ∈ R(m−r)×k, and D1 ∈ R(m−r)×q.First passing the output y1 through a nonsingular matrix P∗ so that

P∗y1 =

{

y11(t) = A11x

y12(t) = A12x +D1f,(4.2)

where D1 ∈ Rp×q is row full rank.Consider a new state xf ∈ Rp which is a filtered version of y12 satisfying

xf = −Afxf +Afy12, (4.3)

where −Af ∈ Rp×p is a stable (filter) matrix.Equations (4.1), (4.2), and (4.3) can be combined to form an augmented state-space

system with order n as follows:

Ea˙x = Aax + Bau +Daf

ya = Cax,(4.4)

where Ea =[E∗ 00 I

]

, Aa =[

A∗ 0AfA12 −Af

]

, Ba =[B∗0

]

, Da =[

D∗AfD1

]

, Ca =[A11 0

C 00 I

]

, x = [xT xTf]T , ya =

[yT11 yT xT

f ]T , n = n + p, and p = p +m − r.

Obviously, the matrix Ea is full row rank so that the cascaded sliding mode observercan be designed using the method in Section 3.

In the following, the existence conditions of the cascaded sliding mode observer forgeneral descriptor systems are given by Theorem 4.1.

Theorem 4.1. There exists a cascaded sliding mode observer for system (2.1) with rank-deficient E ifthe following conditions hold:

rank[E A D0 C 0

]

= m + p (4.5)

rank

⎡

⎣

E A D0 E 00 C 0

⎤

⎦ = n + rank[

E D]

(4.6)

rank

⎡

⎢⎢⎢⎢⎢⎣

E A 0 D 0 00 E A 0 D 00 0 E 0 0 D0 C 0 0 0 00 0 C 0 0 0

⎤

⎥⎥⎥⎥⎥⎦

= n + q + rank

⎡

⎣

E A D 00 E 0 D0 C 0 0

⎤

⎦, (4.7)

rank[sE −A D

C 0

]

= n + q, Re(s) ≥ 0. (4.8)


Proof. Define a nonsingular matrix as follows:

P1 = diag([

I 00 P∗

]

P ∗,[I 00 P∗

]

P ∗, I)

. (4.9)

We have

rankP ∗[E D]

= rank

[

E∗ D∗

0 D1

]

= r + p (4.10)

rank

⎡

⎣

E A D0 E 00 C 0

⎤

⎦ = rankP1

⎡

⎣

E A D0 E 00 C 0

⎤

⎦

= rank

⎡

⎢⎢⎣

A11 00 IpE∗ 0C 0

⎤

⎥⎥⎦+ r,

(4.11)

rank[Ea

Ca

]

= rank

⎡

⎢⎢⎢⎢⎢⎣

E∗ 00 Ip

A11 0C 00 Ip

⎤

⎥⎥⎥⎥⎥⎦

= rank

⎡

⎢⎢⎣

A11 00 IpE∗ 0C 0

⎤

⎥⎥⎦. (4.12)

Combining (4.6), (4.10), (4.11), and (4.12), we have

rank[Ea

Ca

]

= n. (4.13)

Define a nonsingular matrix as follows:

P2 = diag([

I 00 P∗

]

P ∗,[I 00 P∗

]

P ∗,[I 00 P∗

]

P ∗, I, I)

. (4.14)


We have

rankP2

⎡

⎢⎢⎢⎢⎢⎣

E A 0 D 0 00 E A 0 D 00 0 E 0 0 D0 C 0 0 0 00 0 C 0 0 0

⎤

⎥⎥⎥⎥⎥⎦

= r + p + rank

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E∗ A∗ D∗ 00 A12 D1 00 E∗ 0 D∗

0 0 0 D1

C 0 0 0A11 0 0 00 C 0 00 A11 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4.15)

rank

⎡

⎢⎢⎣

Ea Aa Da 00 Ea 0 Da

Ca 0 0 00 Ca 0 0

⎤

⎥⎥⎦

= 2p + rank

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E∗ A∗ D∗ 00 A12 D1 00 E∗ 0 D∗

0 0 0 D1

A11 0 0 0C 0 0 00 A11 0 00 C 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (4.16)

rankP1

⎡

⎣

E A D 00 E 0 D0 C 0 0

⎤

⎦ = rank

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

E A D 00 A11 0 00 A12 D1 00 E 0 D0 0 0 D1

0 C∗ 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

= rank

⎡

⎢⎢⎢⎢⎢⎣

A12 D1 0E 0 D0 0 D1

A11 0 0C∗ 0 0

⎤

⎥⎥⎥⎥⎥⎦

+ r

= rank[Ea Da

Ca 0

]

+ r.

(4.17)

Combining (4.7), (4.15), (4.16), and (4.17), we have

rank

⎡

⎢⎢⎣

Ea Aa Da 00 Ea 0 Da

Ca 0 0 00 Ca 0 0

⎤

⎥⎥⎦

= n + q + rank[Ea Da

Ca 0

]

. (4.18)

Define a nonsingular matrix as follows:

P3 = diag([

I 00 P∗

]

P ∗, I)

. (4.19)


We have

rank[sE∗ −A∗ D∗

C∗ 0

]

= rankP3

[sE∗ −A∗ D∗

C 0

]

= rank

⎡

⎢⎢⎣

sE −A D−A11 0−A12 D1

C∗ 0

⎤

⎥⎥⎦

= n + q;

(4.20)

thus,

rank

⎡

⎢⎢⎢⎢⎢⎣

sE∗ −A∗ 0 D∗

−A11 0 0−A12 0 D1

C∗ 0 00 Ip 0

⎤

⎥⎥⎥⎥⎥⎦

= n + q + p. (4.21)

Define

P4 =

⎡

⎢⎢⎢⎢⎢⎣

I 0 0 0 00 0 Af 0 s +Af

0 I 0 0 00 0 0 I 00 0 0 0 I

⎤

⎥⎥⎥⎥⎥⎦

, (4.22)

Then,

rank

⎡

⎢⎢⎢⎢⎢⎣

sE∗ −A∗ 0 D∗

−A11 0 0−A12 0 D1

C∗ 0 00 Ip 0

⎤

⎥⎥⎥⎥⎥⎦

= rankP4

⎡

⎢⎢⎢⎢⎢⎣

sE∗ −A∗ 0 D∗

−A11 0 0−AfA12 0 AfD1

C∗ 0 00 Ip 0

⎤

⎥⎥⎥⎥⎥⎦

= rank

⎡

⎢⎢⎢⎢⎢⎣

sE∗ −A∗ 0 D∗

−AfA12 s +Af AfD1

−A11 0 0C∗ 0 00 Ip 0

⎤

⎥⎥⎥⎥⎥⎦

= rank[sEa −Aa Da

Ca 0

]

.

(4.23)

Hence,

rank[sEa −Aa Da

Ca 0

]

= n + q, Re(s) ≥ 0. (4.24)

Combining (4.12), (4.16), and (4.24), we get the conclusion by Theorem 3.1.


Corollary 4.2. There exists a cascaded sliding mode observer for linear system if the following condi-tions hold:

rank[CAD CDCD 0

]

= rank(CD) + rank(D), (4.25)

rank[sI −A D

C 0

]

= n + q, Re(s) ≥ 0. (4.26)

Proof. Obviously, since E = I for linear systems, (4.5), (4.6), and (4.8) hold. Hence, thefollowing is to prove that (4.7) holds.

In [21], the canonical form of the linear system is given as follows:

A =[A1 A2

A3 A4

]

, C =[

0 T]

, D =[D1 00 D2

]

, A3 =[A3a A3b

A3c A3d

]

D1 =[D11

0

]

, D2 =[

0D22

]

,

(4.27)

where A1 ∈ R(n−p)×(n−p), A3a ∈ R(p−l)×(q−l), T ∈ Rp×p is orthogonal, D1 ∈ R(n−p)×l, D11 ∈R(q−l)×(q−l), and D22 ∈ Rl×l are invertible.

Substituting (4.27) into (4.25), we obtain

rank[T 00 T

][CAD CDCD 0

]

= rank(A3aD11) + 2l. (4.28)

Combining (4.25) and (4.28), we obtain

rank(A3a) = q − l. (4.29)

Substituting (4.27) into (4.7), we obtain

rank

⎡

⎢⎢⎢⎢⎢⎣

E A 0 D 0 00 E A 0 D 00 0 E 0 0 D0 C 0 0 0 00 0 C 0 0 0

⎤

⎥⎥⎥⎥⎥⎦

= n + rank

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

In−p 0 A1 A2 D1 0 0 00 Ip A3 A4 0 D2 0 00 0 In−p 0 0 0 D1 00 0 0 Ip 0 0 0 D2

0 Ip 0 0 0 0 0 00 0 0 Ip 0 0 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

= 2n + p + rank

⎡

⎣

A3 D2 0 0In−p 0 D1 00 0 0 D2

⎤

⎦


= 3n − q + l + 2 rank(D22) + rank(A3a) + rank(D11)

= 3n + l + q,

rank

⎡

⎣

E A D 00 E 0 D0 C 0 0

⎤

⎦ = rank

⎡

⎣

In−p 0 D1 00 Ip 0 D2

0 Ip 0 0

⎤

⎦ + n

= 2n + l.

(4.30)

Obviously, (4.7) holds.

Remark 4.3. For the linear system, the rank conditions (4.25) and (4.26) are identical to theones in [21], it is obvious that the conclusion of the paper is more general compared with[21].

5. Simulation

A machine infinite bus system linear model is described as follows [29]:

x1 = x4 x2 = x5 x3 = x6

x4 =1

M1(u1 − Y12V1V2(x1 − x2) − Y15V1V5(x1 − x7) −D2x4)

x5 =1

M2(u2 − Y21V1V2(x2 − x1) − Y25V2V5(x2 − x7) −D2x5)

x6 =1

M3(u3 − Y34V3V4x3 − Y35V3V5(x3 − x7) −D3x6)

0 = Pch − Y51V5V1(x7 − x1) − Y52V5V2(x7 − x2)

− Y53V5V3(x7 − x3) − Y54V5V4x7,

(5.1)

where x1, x2, x3, and x7 are the generator angles, x4, x5, and x6 are the generator speeds. u1,u2, and u3 are the mechanical power, Pch is unknown load, the nominal values of inertia M1,M2 andM3, of dampingD1,D2, andD3, of admittance Y15, Y25, Y35, Y51, Y52, Y53, and Y54 andof potential V1, V2, V3, V4, and V5 are shown in

M1 = 0.014 M2 = 0.026 M3 = 0.02 D1 = 0.057 D2 = 0.15D3 = 0.11 Y15 = 0.5 Y25 = 1.2 Y34 = 0.7 Y45 = 1Y34 = 0.7 Y12 = 1 Vi = 1 (i = 1–5).

(5.2)

It is assumed that the available measurements are the generator angles x1, x2, x3, andx7. In order to illustrate the effectiveness of the design method, it is assumed that there existfaults on the actuator u1 − u3. It is easy to verify that the existence conditions of sliding mode


0 2 4 6 8 10

0

0.2

Time (s)

−0.2

−0.4

−0.6

−0.8

−1

ꉱf1

f1

Figure 1: Fault signal f1 and its reconstruction signal f1.

observer in [32] do not hold, but the existence conditions of cascaded sliding mode observerhold.

In the following simulation, the cascaded sliding mode observer in Section 3 isdesigned to reconstruct the actuator faults.

Considering system (5.1) affected by the inputs u1 = 1, u2 = 1, and u3 = 2 + sin(5t), theunknown load Pch = sin(t) and an uncertain admittance

Yij = Yij + ΔYij , (5.3)

where ΔYij = δij sin(ωijt), |δij | < 0.1, |ωij | < 1rd/s, i = 1, . . . , 5, j = 1, . . . , 5.Figures 1, 2, and 3 show faults and reconstruction signals. Although there exists

unknown input and parameter uncertainty in the system, the cascaded slidingmode observerfaithfully reconstructs the faults.

6. Conclusions and Future Works

This paper proposes a fault reconstruction method for a class of descriptor systems usingcascaded sliding mode observer. The method can effectively relax the restrictions on theexistence of a sliding mode observer, which allows the applicability of our proposed methodto a wider range of systems. In our future work, the proposed actuator fault reconstructionschemes can be extended to some sensor fault reconstruction problems by using a suitableoutput filtering technique. Another interesting future research topic is to extend the currentresults to fault estimation of nonlinear systems based on T-S fuzzy models [34–36].


0 2 4 6 8 10Time (s)

0

0.2

−0.2

−0.4

−0.6

−0.8

−1

ꉱf2

f2


0 2 4 6 8 10Time (s)

ꉱf3

f3

1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

−1


Acknowledgment

This work was supported by the Fundamental Research Funds for the Central Universitiesunder Grant no. HIT.NSRIF.2012031.


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