# Fault-Reconstruction-Based Cascaded Sliding Mode Observers for Descriptor Linear Systems

Authors: Jinyong Yu; Guanghui Sun; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

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


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## Body

## 1. Introduction

With the development and applications of modern control techniques, the safety and reliability of control systems are becoming increasingly important. Therefore, the fault diagnosis has become one of the most important techniques to ensure the safety and reliability of control systems [1, 2]. During the last two decades, many significant results have been 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 recently attracted increasing attention due to their importance in real-world systems. In [12], a parametric approach is proposed to design unknown input observers to realize fault detection of descriptor linear multivariable systems with unknown disturbances. By directly identifying parity space, a model-free approach for fault detection is developed, which can be applied if the model of descriptor systems is unknown [13]. In [14], the factorization 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 utilized for providing disturbance rejection and robustness properties of the fault detection and isolation schemes of linear time-invariant descriptor systems. In [16], several sufficient conditions of existence of unknown input observers are obtained for Takagi-Sugeno descriptor systems, which are affected by unknown inputs. Unfortunately, although these methods can successfully generate residuals, they fail to reconstruct fault signals.Recently, fault reconstruction is a promising alternative for fault detection. Instead of generating 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 reconstruct fault signals. The sliding mode control is employed in the situations including state estimation 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 output error injection signals to reconstruct faults. Tan and Edwards [19] extended this work for robust reconstruction of sensor and actuator faults by minimizing the effect of uncertainty on the reconstruction in an L2 sense. Some well-studied works, aiming at reducing the system constraints 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 observer method 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 mode observer in [17] can be used for systems without the observer matching condition. In order to obtain those auxiliary outputs, high-order sliding-mode observers are constructed to act as exact differentiators using a super-twisting algorithm. Inspired by Floquet et al. [22], high-gain approximate differentiators and high-order sliding-mode robust differentiators were proposed to generate auxiliary outputs for the design of sliding mode observers [18, 23].Although there are many achievements in regular systems, few results have been reported to the descriptor case despite its importance in real-world systems. In [31, 32], the sliding mode observer method was employed to detect and isolate faults and to reconstruct the faults for descriptor systems. However, the uncertainty was not considered in these results. In [33], the sliding mode observer was proposed to minimize the effect of uncertainly on the reconstruction of faults for descriptor systems. Unfortunately, the fault detection filter based sliding mode observer has to satisfy the strict condition in [31–33], which severely limits the applicability of these approaches for a wide range of practical systems.Motivated by the above discussion, in this paper, we develop a cascaded sliding mode 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 novel cascaded sliding mode observer method to reconstruct actuator faults for a class of descriptor linear systems; (2) in the design process, we remove this restrictive assumption and extend the 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 a class of descriptor linear systems.The paper is organized as follows. In Section2, the problem is formulated, and appropriate coordinate transformations are introduced to exploit the system structure. In Section 3 the design algorithm of cascaded sliding mode observer for linear descriptor systems is given. In Section 4, a design method of cascaded sliding mode observer and fault reconstruction for general descriptor systems are presented. In Section 5, an example is given to support the effectiveness of the proposed approach. Finally, the conclusions are drawn.


## 2. Problem Statement and System Analysis

Consider a descriptor linear system described by(2.1)Ex˙=Ax+Bu+Dfy=Cx,
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
(2.2)∥f∥≤β,
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 loss of 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:
(2.3)z˙=Fz+T1Bu+K1y+K2y+Gnυx^=z+T2yy^=Cx^,
where z∈Rn~ is the state vector of the SMO, x^ is the estimation of the state vector x, and υ is the discontinuous output error injection vector defined by
(2.4)υ={-ηP0ey∥P0ey∥ey≠00other,
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:
(2.5)rank[EDC0]=n+q(2.6)rank[sE-ADC0]=n+q,Re(s)≥0.It is well known that condition (2.5) is quite restrictive and may not apply to a wide range of systems. In the following, we give two more relaxed conditions:
(2.7)rank[EC]=n,(2.8)rank[EDC0]=n+l,
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
(2.9)rank[ED1C0]=n+l,(2.10)rank[ED2C0]=n,
where [D1D2]=DU, and D1∈Rm×l, D2∈Rm×(q-l).Proof.
ifl is equal to q, the conclusion is obviously true. So the following is to prove the case that l is less than q.
Obviously, there exists a nonsingular matrixU1 so that DU1=[D1D-2] and (2.9) hold.
Then,(2.11)rank[EDC0]=rank[ED1D-2C00]=n+l.
So there exists a matrix Y=[Y1Y2]   so that
(2.12)[D-20]=[ED1C0][Y1Y2].
Thus, we have  D-2=EY1+D1Y2 and CY1=0.
Setting(2.13)U2=[I-Y20I]
and  U=U1U2, we have
(2.14)rank[ED2C0]=rank[EEY1CCY1]=rank[EC]=n.Lemma 2.2.
If the following conditions(2.15)rank[ED1C0]=n+l,rank[sE-AD1C0]=n+l,Re(s)≥0
hold, there exist two nonsingular matrices P and Q such that
(2.16)PEQ=[0E12In~-p~E22],PAQ=[A11A12A21A22]PD1=[DI0],CQ=[0Ip],
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=[0Il]T∈R(m-n+p)×l, and the subblock A11   has the structure
(2.17)A11=[A111A112],
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, and U such that
(2.18)PEQ=[0E12In-pE22],PAQ=[A11A12A21A22](2.19)PB=[B1B2],CQ=[0Ip](2.20)PDU=[D1100D22],
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=[0Il]T∈R(m-n+p)×l, D22∈R(n-p)×(q-l), and the subblock A11 has the structure
(2.21)A11=[A111A112],
where A111∈R(m-n+p-l)×(n-p), A112∈Rl×(n-p), and (A21,A111) is detectable.Proof .
By Lemma2.1, there exists a nonsingular matrix U such that (2.9) and (2.10) hold, where DU=[D1D2].
Obviously,(2.22)rank[sE-AD1C0]=n+l,Re(s)≥0.
By Lemma2.2, there exist two nonsingular matrices P and Q such that (2.18) and (2.19) hold and
(2.23)PD1=[D110].
SettingPD2=[D21D22], we have
(2.24)rank[ED2C0]=rank[P00I][ED2C0][Q00I]=rank[0E12D21In-pE22D220Ip0]=rank(D21)+n.
Combining (2.10) and (2.24), we have rank(D21)=0. Obviously, D21=0.By Lemma2.3, it can be assumed without loss of generality that system (2.1) has the following form:
(2.25)[0E12In-pE22][x˙1x˙2]=[A11A12A21A22][x1x2]+[B1B2]u+[D110]f1+[0D22]f2y=x2,
where x=[x1Tx2T]T,x1∈Rn-p, x2∈Rp and
(2.26)f→Uf=[f1Tf2T]T.The descriptor system (2.25) may be considered as the system with the fault f1 and the disturbance f2. Using the fault reconstruction method in [33], the fault f1 can be reconstructed and the L2 gain from the f2 to reconstruction error of fault f1 can be minimized. But the fault f2 and the state x1 cannot be estimated. Inspired by Tan et al. [20, 21], the cascaded sliding mode 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
(3.1)z˙=(T1A-K1C)z+T1Bu+K1y+FT2y+Gnυx^=z+T2yy^=x^2,
where z∈Rn is the state vector of the SMO, x^=[x^1Tx^2T]T with x^1∈Rn-p and x^2∈Rp is the estimation of the state vector x, Gn=[0I]T,T1 and T2 are defined by
(3.2)T1=[Z1In-pZ20](3.3)T2=[0Ip]-T1[E12E22](3.4)Z1=[Z110],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 discontinuous output error injection vector defined by:
(3.5)υ={-ηP2e2∥P2e2∥e2≠00other,e2=x^2-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.Definee=x^-x as the state estimation error, the following estimation error dynamic is obtained:
(3.6)e˙1=(A21+Z1A11)e1+(Z1A12+A22-K11)e2+D22f2e˙2=Z2A11e1+(Z2A12-K12)e2-Z2D11f1+υ,
where e=[e1Te2T]T,e1∈Rn-p, K1=[K11TK12T]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, thene2=e˙2=0, and the error equation becomes
(3.7)e˙1=(A21+Z1A11)e1+D22f2Z2+υeq=-A11e1+D11f1,
where Z2+ 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
(3.8)υeq=-ηP2e2∥P2e2∥+δ,
where δ is a small positive constant.The remaining system freedom can be used to estimate the statex1 and reconstruct the fault f2. Equation (3.7) can be rewritten as
(3.9)e˙1=(A21+Z1A11)e1+D22f2(3.10)D11⊥Z2+υeq=-A111e1,
where D11⊥=[Im+p-n-l0].For anyA111, there exists a nonsingular matrix W so that WA111=[A^111T0]T, where A^111∈Rp^×(n-p) is full row rank. We have
(3.11)[Ip^0]WD11⊥Z2+υeq=-A^111e1.The system (3.9) and (3.11) may be considered as the linear system with the q-l faults, the n-p states f1 and the p^ outputs. Using the sliding mode observer design method for the linear system in [17], we can design a secondary sliding mode observer to estimate e1 and f2 if the following conditions hold:
(3.12)rank(A^111D22)=rank(D22)rank[sI-(A21+Z1A11)D22A^1110]=n-p+q-l,Re(s)≥0.
Obviously, (3.12) are equivalent to
(3.13)rank(A111D22)=rank(D22)(3.14)rank[sI-(A21+Z1A11)D22A1110]=n-p+q-l,Re(s)≥0.
Combined with (3.4), (3.14) is equivalent to
(3.15)rank[sI-A21D22A1110]=n-p+q-l,Re(s)≥0.From the above analysis, if the conditions (3.13) and (3.15) satisfy, there exists a cascaded 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 thee^1 and f^2 are the estimations of e1 and f2, respectively. Then, the reconstruction signal of the fault f1 is described by
(3.16)f^1=A112e^1+[0Il]Z2+υeq,
and the estimation of the state x1 is described by
(3.17)x^1-e^1→x1.The reconstruction of fault is described by(3.18)f^=U-1[f^1Tf^2T]TEquations (3.13) and (3.15) are the sufficient conditions for the existence of the cascaded sliding mode observer, but these cannot be checked using the parameters of the original system (2.1). Now, for system (2.1), sufficient conditions for the existence of the cascaded 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 conditions hold:
(3.19)rank[EC]=n(3.20)rank[EAD00E0DC0000C00]=n+q+rank[EDC0],(3.21)rank[sE-ADC0]=n+q,Re(s)≥0.Proof.
Ifl is equal to q, the conclusion is obviously true. So, the following is to prove the case that l is less than q.
Substituting (2.8) and (2.25) into (3.20), we have
(3.22)rank[EAD00E0DC0000C00]=rank[0E12A11A12D11000In-pE22A21A220D2200000E1200D11000In-pE12000D220Ip000000000Ip0000]=n+p+l+rank[A11D110In-p0D22]=n+p+l+rank[A11100A112Il0In-p0D22]=n+p+2l+rank[In-pD22A1110]=2n+q+l.
So we have(3.23)rank[In-pD22A1110]=n-p+q-l.
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 Reconstruction for General Descriptor Systems

In Section2, it is assumed that E is full row rank. In the following, it is discussed that E is rank deficient. Let r:=rank(E)≤min{m,n}.Now, sincerank(E)=r, there exists a regular matrix P* such that (2.1) is restricted system equivalent to
(4.1)E*x˙=A*x+B*u+D*fy1=-B-1u=A-1x+D-1fy=Cx,
where P*E=[E*0],P*A=[A*A-1], P*B=[B*B-1],P*D=[D*D-1],E*∈Rr×n, A*∈Rr×n, B*∈Rr×k, D*∈Rr×p, A-1∈R(m-r)×n,B-1∈R(m-r)×k, and D-1∈R(m-r)×q.First passing the outputy1 through a nonsingular matrix P* so that
(4.2)P*y1={y11(t)=A11xy12(t)=A12x+D1f,
where D1∈Rp-×q is row full rank.Consider a new statexf∈Rp- which is a filtered version of y12 satisfying
(4.3)x˙f=-Afxf+Afy12,
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:
(4.4)Eax~˙=Aax~+Bau+Dafya=Cax~,
where Ea=[E*00I],  Aa=[A*0AfA12-Af], Ba=[B*0], Da=[D*AfD1], Ca=[A110C00I], x~=[xTxfT]T, ya=[y11TyTxfT]T, n~=n+p-, and p~=p+m-r.Obviously, the matrixEa is full row rank so that the cascaded sliding mode observer can be designed using the method in Section 3.In the following, the existence conditions of the cascaded sliding mode observer for general descriptor systems are given by Theorem4.1.Theorem 4.1.
There exists a cascaded sliding mode observer for system (2.1) with rank-deficient E if the following conditions hold:
(4.5)rank[EAD0C0]=m+p(4.6)rank[EAD0E00C0]=n+rank[ED](4.7)rank[EA0D000EA0D000E00D0C000000C000]=n+q+rank[EAD00E0D0C00],(4.8)rank[sE-ADC0]=n+q,Re(s)≥0.Proof.
Define a nonsingular matrix as follows:(4.9)P1=diag([I00P*]P*,[I00P*]P*,I).
We have(4.10)rankP*[ED]=rank[E*D*0D-1]=r+p-(4.11)rank[EAD0E00C0]=rankP1[EAD0E00C0]=rank[A1100Ip-E*0C0]+r,(4.12)rank[EaCa]=rank[E*00Ip-A110C00Ip-]=rank[A1100Ip-E*0C0].
Combining (4.6), (4.10), (4.11), and (4.12), we have
(4.13)rank[EaCa]=n~.
Define a nonsingular matrix as follows:(4.14)P2=diag([I00P*]P*,[I00P*]P*,[I00P*]P*,I,I).
We have
(4.15)rankP2[EA0D000EA0D000E00D0C000000C000]=r+p-+rank[E*A*D*00A12D100E*0D*000D1C000A110000C000A1100](4.16)rank[EaAaDa00Ea0DaCa0000Ca00]=2p-+rank[E*A*D*00A12D100E*0D*000D1A11000C0000A11000C00],(4.17)rankP1[EAD00E0D0C00]=rank[EAD00A11000A12D100E0D000D10C*00]=rank[A12D10E0D00D1A1100C*00]+r=rank[EaDaCa0]+r.
Combining (4.7), (4.15), (4.16), and (4.17), we have
(4.18)rank[EaAaDa00Ea0DaCa0000Ca00]=n~+q+rank[EaDaCa0].
Define a nonsingular matrix as follows:(4.19)P3=diag([I00P*]P*,I).
We have(4.20)rank[sE*-A*D*C*0]=rankP3[sE*-A*D*C0]=rank[sE-AD-A110-A12D1C*0]=n+q;
thus,
(4.21)rank[sE*-A*0D*-A1100-A120D1C*000Ip-0]=n+q+p-.
Define(4.22)P4=[I000000Af0s+Af0I000000I00000I],
Then,(4.23)rank[sE*-A*0D*-A1100-A120D1C*000Ip-0]=rankP4[sE*-A*0D*-A1100-AfA120AfD1C*000Ip-0]=rank[sE*-A*0D*-AfA12s+AfAfD1-A1100C*000Ip-0]=rank[sEa-AaDaCa0].
Hence,(4.24)rank[sEa-AaDaCa0]=n~+q,Re(s)≥0.
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 conditions hold:(4.25)rank[CADCDCD0]=rank(CD)+rank(D),(4.26)rank[sI-ADC0]=n+q,Re(s)≥0.Proof.
Obviously, sinceE=I for linear systems, (4.5), (4.6), and (4.8) hold. Hence, the following is to prove that (4.7) holds.
In [21], the canonical form of the linear system is given as follows:
(4.27)A=[A1A2A3A4],C=[0T],D=[D100D2],A3=[A3aA3bA3cA3d]D1=[D110],D2=[0D22],
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
(4.28)rank[T00T][CADCDCD0]=rank(A3aD11)+2l.
Combining (4.25) and (4.28), we obtain
(4.29)rank(A3a)=q-l.
Substituting (4.27) into (4.7), we obtain
(4.30)rank[EA0D000EA0D000E00D0C000000C000]=n+rank[In-p0A1A2D10000IpA3A40D20000In-p000D10000Ip000D20Ip000000000Ip0000]=2n+p+rank[A3D200In-p0D10000D2]=3n-q+l+2rank(D22)+rank(A3a)+rank(D11)=3n+l+q,rank[EAD00E0D0C00]=rank[In-p0D100Ip0D20Ip00]+n=2n+l.
Obviously, (4.7) holds.Remark 4.3.
For the linear system, the rank conditions (4.25) and (4.26) are identical to the ones 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]:
(5.1)x˙1=x4x˙2=x5x˙3=x6x˙4=1M1(u1-Y12V1V2(x1-x2)-Y15V1V5(x1-x7)-D2x4)x˙5=1M2(u2-Y21V1V2(x2-x1)-Y25V2V5(x2-x7)-D2x5)x˙6=1M3(u3-Y34V3V4x3-Y35V3V5(x3-x7)-D3x6)0=Pch-Y51V5V1(x7-x1)-Y52V5V2(x7-x2)-Y53V5V3(x7-x3)-Y54V5V4x7,
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 and M3, of damping D1, D2, and D3, of admittance Y15, Y25, Y35, Y51, Y52, Y53, and Y54 and of potential V1, V2, V3, V4, and V5 are shown in
(5.2)M1=0.014M2=0.026M3=0.02D1=0.057D2=0.15D3=0.11Y15=0.5Y25=1.2Y34=0.7Y45=1Y34=0.7Y12=1Vi=1(i=1–5).It is assumed that the available measurements are the generator anglesx1, x2, x3, and x7. In order to illustrate the effectiveness of the design method, it is assumed that there exist faults on the actuator  u1–u3. It is easy to verify that the existence conditions of sliding mode observer in [32] do not hold, but the existence conditions of cascaded sliding mode observer hold.In the following simulation, the cascaded sliding mode observer in Section3 is designed to reconstruct the actuator faults.Considering system (5.1) affected by the inputs u1=1, u2=1, and u3=2+sin(5t), the unknown load Pch=sin(t) and an uncertain admittance
(5.3)Yij=Yij+ΔYij,
where ΔYij=δijsin(ωijt), |δij|<0.1, |ωij|<1rd/s,i=1,…,5,j=1,…,5.Figures1, 2, and 3 show faults and reconstruction signals. Although there exists unknown input and parameter uncertainty in the system, the cascaded sliding mode observer faithfully reconstructs the faults.Figure 1
Fault signalf1 and its reconstruction signal f^1.Figure 2
Fault signalf2 and its reconstruction signal f^2.Figure 3
Fault signalf3 and its reconstruction signal f^3.


## 6. Conclusions and Future Works

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



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*Source: 623426-2012-09-20.xml*