fault detection and isolation for nonlinear systems via

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Journal Code: Article ID Dispatch: 09.08.14 CE: Ada, Emma Jane M.R N C 3 2 3 2 No. of Pages: 23 ME:

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control (2014)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.3232

Fault detection and isolation for nonlinear systems viahigh-order-sliding-mode multiple-observer‡

H. Ríos1, J. Davila2, L. Fridman1,*,† and C. Edwards3

1National Autonomous University of Mexico, Department of Control Engineering and Robotics, Division of ElectricalEngineering, Engineering Faculty, C.P. 04510, México, Federal District, Mexico

2National Polytechnic Institute, Section of Graduate Studies and Research, ESIME-UPT, C.P. 07340, México, FederalDistrict, Mexico

3College of Engineering, Mathematics and Physical Sciences; University of Exeter; North Park Road, Exeter,EX4 4QF, UK

SUMMARY

In this paper, fault detection

Q1

and isolation problems are studied for a certain class of nonlinear systems.Under some structural conditions, multiple high-order sliding-mode observers are proposed. The value ofthe equivalent output injection is used for detecting faults and the multiple-model approach for isolatingparticular faults in the system. The proposed method provides fast detection and isolation of actuator andplant faults. Simulation results support the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.

Received 5 June 2013; Revised 17 June 2014; Accepted 16 July 2014

KEY WORDS: high-order sliding-mode observers; fault detection; fault isolation

1. INTRODUCTION

1.1. Antecedents and motivations

Our society strongly depends on the availability and correct functioning of complex technologicalprocesses. Often, key processes need to be monitored to prevent improper operation that can damageimportant components and, in critical cases, to avoid fatal consequences. The main purpose of faultdetection and isolation (FDI) schemes is to indicate that something is wrong (i.e., to generate analarm when a fault occurs) and then determine which subsystem or component has developed a faultor failure (i.e., identifying its location).

Improvements in modeling techniques have allowed the use of model-based FDI schemes. Suchmethods have been considered as an effective approach for FDI in theory and in practice (see, e.g.,[2, 3] and [4]). In particular, the most commonly employed schemes are observer based, where theinformation generated by the observer is used to deduce information about the faults.

In residual generation schemes, the output error between the system and the observer is analyzedto form a residual. When the system is fault free, the residual should be small (approximately zero);however, when a fault occurs, it should become discernably different from zero. An extensive listof publications, describing different FDI methods based on residuals, exists: see, for example, thesurvey in [5], fault detection filters in [6], and the tutorial books [2, 3].

*Correspondence to: L. Fridman, National Autonomous University of Mexico, Department of Control Engineering andRobotics, Division of Electrical Engineering, Engineering Faculty, C.P. 04510, México, Federal District, Mexico.

†E-mail: [email protected]‡A preliminary version of this work has been presented in [1].

Copyright © 2014 John Wiley & Sons, Ltd.

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While, initially, work was directed toward linear systems, serious efforts in the last decade havebeen directed toward nonlinear systems. Observer-based FDI schemes that incorporate differenttheoretical approaches have been developed. For example, the work in [7–9], and [10], describesdifferential geometric, algebraic, and feedback control approaches that have been proposed for faultdetection. Techniques based on an adaptive estimation approach have also been successfully appliedfor fault detection, residual generation, and fault tolerant control, see, for example, [11–15] and [16].In terms of sliding-mode theory, the most recent contributions have been presented in [17–21] and[22]. Most of this work formulates the unknown input problem as an actuator or sensor fault recon-struction problem. In particular, high-order-sliding-mode (HOSM) techniques are applied in [23] todecrease the level of uncertainty in a linear-parameter-varying system, which leads to the improve-ment of set-membership estimates generated by an interval observer. This approach is applied tofault detection by verifying the consistency between the output trajectory and its estimated domain.In [24], an FDI scheme based on HOSM techniques for a three tanks system is proposed. TheHOSM observer design allows the estimation of the magnitude of a relative degree one actuatorfaults. A residual generation approach, using the estimated parameters, inputs, outputs, and theirestimated derivatives obtained by means of the HOSM differentiator [25], is presented in [26]. Allthe approaches in the three papers described earlier are restricted to only one type of fault with afixed relative degree. In the work described in this paper, we exploit the qualities associated with theHOSM methods given by [25], and add a multi-model approach that allows us to isolate faults evenif the relative degree vector changes as a consequence of the faults.

The multiple-model based approach has also attracted much attention in the last decade. Thereare many advantages of using a multiple-model approach for fault diagnosis in complex systems(see, for example [27, 28] and [29]). The main reason is because it is often impractical to representall possible system failure behavior by a single model. Moreover, because the number of failuremodes is finite, it is possible to employ a control algorithm that has been specifically designed foreach fault scenario. Of course, some applications may require a large number of modes to repre-sent all possible faults, which results in high computational costs. Nevertheless, the multiple-modelapproach has been applied to many FDI problems in aerospace systems, including the aircraft flightcontrol systems [30, 31] and INAs [32].

The main idea of the multiple-model approach for FDI is as follows: a model set must be createdthat contains all the different plausible fault conditions in the system in the form of local modelsor sub-models. In addition, the model set usually includes the nominal fault-free model. Faultsare identified by estimating which of the possible local model is most valid using multiple-modelestimation algorithms. When there are no faults present in the system, the nominal model willbe valid. In case of an a priori considered fault, one of the other models in the model set willbecome valid.

1.2. Main contribution

Motivated in the succeeding text, and taking into account that multiple-model based FDI schemeshave already been successfully used in many applications, the main contribution of this paper is asfollows:

� An FDI approach that allows us to estimate the system state in finite time, and solve the FDIproblem, even if the relative degree vector of the output w.r.t. the fault changes within thepossible faulty scenarios.

To fulfill the aforementioned goal, the following methodology has been proposed in thismanuscript:

1. A HOSM multiple-observer is used for estimating the state of a class of nonlinear systems infinite time.

2. The equivalent output injection of the HOSM observers is exploited to solve the fault detectionproblem.

3. A multiple-model approach is employed to solve the fault isolation problem for such systems.

Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control (2014)DOI: 10.1002/rnc

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FAULT DETECTION AND ISOLATION FOR NONLINEAR SYSTEMS 3

This means that a new methodology that consists of a combination between a HOSM multiple-observer and a multiple-model approach is proposed. Such a methodology exploits the qualitiesassociated with the HOSM methods to detect faults, and the multi-model approach allows us toisolate them even if the relative degree vector changes as a consequence of the faults. This con-trasts with existing methods, most of which are restricted to only one type of fault with a fixedrelative degree.

1.3. Structure of the paper

The paper has the following structure. Section 2 deals with the problem statement. In Section 3,the preliminaries are presented. Section 4 describes the design of the HOSM multiple-observer andthe main assumptions of the proposed approach, while in Section 5, the FDI scheme is presented.A simulation example is given in Section 6, and finally, some concluding remarks are given inSection 7.

1.4. Notation

The set <C is defined as <C D ¹& 2 < W & > 0º. Denote by L1, the set of all inputs � that satisfyk�k < 1. With reference to a scalar function h with a vector argument x defined in an open set˝ 2 <n such that h.x/ W <n ! <, define dh.x/ D @h.x/

@[email protected]/@x1� � � @h.x/

@xn

i. Also define

Lf .x/h.x/ D dh.x/f .x/ as the Lie derivative of h.x/ along f .x/, and the k � th Lie derivative asLkf .x/

h.x/ D d.Lk�1f .x/

h.x//f .x/. Finally, the function d�cr is defined as d�cr D j�jr sign.�/.

2. PROBLEM STATEMENT

Consider the following nonlinear system subject to faults

Px.t/ D f .x.t//C Bu.t/C F�.x.t//!�.t/; � D 1; : : : ; q; (1)

y.t/ D h.x.t//; .t/ D y.t/C v.t/; (2)

where x.t/ 2 X � <n is the state vector, u.t/ 2 U � <m is the control input vector, y.t/ 2Y � <p is the output, and v.t/ 2 <p is a Lebesgue-measurable sampling noise. The quantity .t/represents the available signal obtained from real-time measurements. The vector field f .x.t// andthe function h.x.t// D

�h1.x.t// � � � hp.x.t//

�Tare assumed to be sufficiently smooth, and the

matrix B D ŒB1 � � � Bm� 2 <n�m is a known constant matrix.The fault scenarios are considered to be known a priori, and it is assumed that their effects can

be modeled as additive terms of the form:

F�.x.t//!�.t/ 2 F ; where F D®F1.x.t//!1.t/; : : : ; Fq.x.t//!q.t/

¯; (3)

that is, there exist q different possible fault cases represented by q faulty components F�.x.t//!�.t/that can change the system properties (plant faults) and/or the dynamical input properties of thesystem (actuator faults). Assume that !�.t/ 2 <p and that the p vector fields of F�.x.t//, thatis, F�.x.t// D

�F�;1.x.t//; : : : ; F�;p.x.t//

�, are smooth almost everywhere. It is assumed that

all the faults under consideration allow the existence and uniqueness of solutions to the wholesemi-axis t > 0.

The aim of this paper is to detect and isolate the presence of faults in system (1), even whenthe faults do not appear always in the same state equations. To achieve this aim, a bank of HOSMobservers is proposed.


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3. PRELIMINARIES

Consider system (1) in the fault-free case, that is, when F�.x.t//!�.t/ D 0. The following relativedegree definition is introduced [33].

Definition 1The system output y.t/ has a (vector) relative degree .r1; : : : ; rp/ at a point x0, if

d�Lkf .x/

hi .x/�Bj D 0; 8j D 1; : : : ; m; 8k < ri � 1; 8i D 1; : : : ; p;

d�Lri�1

f .x/hi .x/

�Bj ¤ 0; for at least one 1 6 j 6 m; 8i D 1; : : : ; p:

(4)

for all x in a neighborhood of x0.

3.1. High-order-sliding-mode differentiator

The results presented in this section have to be understood in a component-wise sense. In order toestimate the derivatives of a signal y.t/ based on the measurements .t/ D y.t/Cv.t/, where v.t/is a bounded Lebesgue-measurable signal, the following HOSM differentiator [25] is introduced:

P#1 D #2 � ˛1M1r d#1 � .t/c

r�1r ;

P#i D #iC1 � ˛iM1

r�iC1

l#i � P#i�1

k r�ir�iC1

; i D 2; : : : ; r � 1;

P#r D �˛rMl#r � P#r�1

k0;

(5)

where M is an upper bound of y.r/.t/ and the set of gains¹˛kºrkD1 is chosen recursively and suffi-

ciently large. In particular, according to [25], one possible choice is ˛6 D 1:1, ˛5 D 1:5, ˛4 D 2,˛3 D 3, ˛2 D 5, and ˛1 D 8 (for the case when r 6 6).Q2 The following theorem that is taken from[34], and describes the properties of the HOSM differentiator in the presence of noise and sampletime, is introduced.

Theorem 1Consider the HOSM differentiator (5) of order r � 1, with proper parameters ˛k . Letˇ

y.r/.t/ˇ< M; jv.t/j 6 �vM�r ; (6)

where � is a positive parameter. Suppose that is sampled with a possibly variable time step�s > 0, and �s 6 �� , with �v , �� being some positive constants. Then for any positive constants�1; �2; : : : ; �r and any � , 0 < � < �1, there exist �v , �� , �t > 0, such that if the inequality

j#1 � .t/j 6 � M�r (7)

holds at all sampling instants within the finite-time interval of the length �t� , then starting from thebeginning of this interval, the inequalitiesˇ

#k � y.k�1/.t/

ˇ6 �kM�r�kC1; k D 1; : : : ; r (8)

hold and are kept forever.

Remark 1Note that with discrete sampling, the DEs are replaced by their Euler approximations. Furthermore,the differentiator (5) provides the best possible asymptotic accuracy in the presence of input noises[25]. Reliable derivatives/estimates are only available after a finite-time transient �t� .


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The HOSM differentiator given in (5) provides the derivatives up to order r � 1 by means of

the vector # , that is, Œ#1 #2 : : : #r �T D

hOy.t/ POy.t/ : : : Oy.r�1/.t/

iT, where

®Oy.k/

¯r�1kD0

are theestimated derivative signals. Notice that in the absence of noise and sample time, the HOSMdifferentiator (5) ensures the exact estimation of the derivatives in finite time.

4. HIGH-ORDER-SLIDING-MODE MULTIPLE-OBSERVER DESIGN

In this section, different fault cases that can appear in the system (1) are studied. Then, the conditionsrequired to design the bank of HOSM observers are established for each case.

4.1. Fault-free case

First consider the fault-free case. Then, system (1) takes the following structure:

Px.t/ D f .x.t//C Bu.t/;

y.t/ D h.x.t//; .t/ D y.t/C v.t/:(9)

Define the mapping ˆ.x.t// and its corresponding Jacobian matrix as follows:

ˆ.x.t// D

266666666666666666664

h1.x.t//

Lf .x.t//h1.x.t//

:::

Lr1�1f .x.t//

h1.x.t//

:::

hp.x.t//

Lf .x.t//hp.x.t//

:::

Lrp�1

f .x.t//hp.x.t//

377777777777777777775

)@ˆ.x.t//

@x.t/D

266666666666666666664

dh1.x.t//

dLf .x.t//h1.x.t//

:::

dLr1�1f .x.t//

h1.x.t//

:::

dhp.x.t//

dLf .x.t//hp.x.t//

:::

dLrp�1

f .x.t//hp.x.t//

377777777777777777775

: (10)

Assumption 1Assume that relative degree condition (4) is satisfied by system (9), and the Jacobian matrix in (10)

is such that rank

�@ˆ.x.t//

@x.t/

�D n, 8x 2 X .

Remark 2The fulfillment of Assumption 1 implies local observability of system (1), that is, it is observable8x 2 X . Notice that if Assumption 1 is satisfied, then the mapping ˆ.x/ is a local diffeomorphismon X . Moreover, if X D <n, then ˆ.x.t// is a global diffeomorphism (see, for example [33]), andthen the system in (1) is observable 8x 2 <n.

Suppose that Assumption 1 is satisfied. Then, it is possible to design the following observer:

PNx.t/ D f . Nx.t//C Bu.t/C

�@ˆ. Nx.t//

@ Nx.t/

��1�;

Ny.t/ D h. Nx.t//;

(11)

with the estimated state vector Nx.t/ 2 <n, the estimated output Ny.t/ 2 <p , and the correction term� 2 <n that will be described in the sequel. The solutions of (11) are understood in the Filippovsense [35], to provide the possibility of using discontinuous signals in the � term. These solutions


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coincide with the usual solutions, when the right-hand sides are continuous. It is also assumedthat all the considered correction terms allow the existence and extension of solutions to the wholesemi-axis t > 0.

Let the following assumption be satisfied.

Assumption 2There is a set of known constants ¹Miº

piD1 > 0 such that the following inequalities are satisfied:

ˇ°Lrif . Nx/

hi . Nx/ � Lrif .x/

hi .x/±piD1

ˇ< ¹Miº

piD1 : (12)

Remark 3From a physical point of view (for example, in mechanical systems), Assumption 2 implies theknowledge of an upper bound on the acceleration and a bound for the initial estimation error. This isnot particularly restrictive because most physical signals are bounded, and, usually, it is possible toestimate a priori with a reasonable level of accuracy the initial conditions of the system trajectories.Notice that Assumption 2 is satisfied if the system (1) is bounded-input bounded-state; see, forexample, [36].

The HOSM differentiator in (5) is used as auxiliary dynamics. The differentiator-based auxiliarydynamics takes the following form for each output:

P#1;1 D #1;2 � ˛1;1M1r1

1

˙ey1.t/

˘ r1�1r1 ;

P#1;j D #1;jC1 � ˛1;jM1

r1�iC1

1

l#1;j � P#1;j�1

k r1�i

r1�iC1 ; i D 2; : : : ; r1 � 1;

P#1;r1 D �˛1;r1M1

l#1;r1 �

P#1;r1�1

k0;

:::

P#p;1 D #p;2 � ˛p;1M1rpp

˙eyp .t/

˘ rp�1rp ;

P#p;j D #p;jC1 � ˛p;jM1

rp�iC1

p

l#p;j � P#p;j�1

k rp�i

rp�iC1; i D 2; : : : ; rp � 1;

P#p;rp D �˛p;rpMp

l#p;rp �

P#p;rp�1

k0; (13)

where®eyi .t/

¯piD1D ¹ Nyi .t/ � i .t/º

piD1 are the output errors. Then, the correction term is taken

from (13) as

� D

26666666666666666666664

�˛1;1M1r1

1

˙ey1.t/

˘ r1�1r1

�˛1;jM1

r1�iC1

1

l#1;j � P#1;j�1

k r1�i

r1�iC1 ; i D 2; : : : ; r1 � 1

�˛1;r1M1

l#1;r1 �

P#1;r1�1

k0:::

�˛p;1M1rpp

˙eyp .t/

˘ rp�1rp

�˛p;jM1

rp�iC1

p

l#p;j � P#p;j�1

k rp�i

rp�iC1; i D 2; : : : ; rp � 1

�˛p;rpMp

l#p;rp �

P#p;rp�1

k0

37777777777777777777775

: (14)


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The constants ¹MiºpiD1 are chosen such that Assumption 2 is satisfied, and

®˛i;j

¯p;riiD1;jD1

arechosen as in (5). Notice that the HOSM-based controllers are powerful tools when used in observa-tion schemes, given that the chattering effects are reduced to ‘computational chattering’, which isharmless to the system.

Taking into account the previous explanations, the following lemma can be stated.

Lemma 1Let the observer in (11) with the correction term in (14) be applied to system (9) and supposeAssumption 1 is satisfied. Then, provided the gains ¹Miº

piD1 are selected such that Assumption

2 holds and the parameters®˛i;k

¯p;riiD1;kD1

are chosen according to [25], the state estimation errorconverges to a ˇ�neighborhood of the origin, dependent on the amplitude of the noise, in finitetime, that is, kx.t/ � Nx.t/k 6 ˇ, for ˇ > 0 and 8t > t�§.

Proof

Denote°Ny.j�1/i .t/ � yi

.j�1/.t/±p;riiD1;jD1

D®Mii;j .t/

¯p;riiD1;jD1

. Then, because of the auxiliary

dynamics in (13), the output observation error dynamics takes the following form:

M1 P1;1 DM11;2 � ˛1;1M1r1

1 dM11;1 � v1.t/cr1�1

r1 ;

M1 P1;j DM11;jC1 � ˛1;jM1

r1�jC1

1

˙M11;j �M1 P1;j�1

˘ r1�j

r1�jC1 ; j D 2; : : : ; r1 � 1;

M1 P1;r1 D ‰1 . Nx; x; u/ � ˛1;r1M1

˙M11;r1 �M1 P1;r1�1

˘0;

:::

Mp Pp;1 DMpp;2 � ˛p;1M1rpp

˙Mpp;1 � vp.t/

˘ rp�1rp ;

Mp Pp;j DMpp;jC1 � ˛p;jM1

rp�jC1

p

˙Mpp;j �Mp Pp;j�1

˘ rp�j

rp�jC1 ; j D 2; : : : ; rp � 1;

Mp Pp;rp D ‰p . Nx; x; u/ � ˛p;rpMp

˙Mpp;rp �Mp Pp;rp�1

˘0;

(15)

where ¹‰i . Nx; x; u/ºpiD1 D

°Lrif . Nx/


hi .x/±piD1

. Dividing by Mi (15), the followingdifferential inclusion is obtained

P1;1 D 1;2 � ˛1;1 d1;1 � v1.t/=M1cr1�1

r1 ;

P1;j D 1;jC1 � ˛1;j˙1;j � P1;j�1

˘ r1�j

r1�jC1 ; j D 2; : : : ; r1 � 1;

P1;r1 2 Œ�1; 1� � ˛1;r1˙1;r1 � P1;r1�1

˘0;

:::

Pp;1 D p;2 � ˛p;1˙p;1 � vp.t/=Mp

˘ rp�1rp ;

Pp;j D p;jC1 � ˛p;j˙p;j � Pp;j�1

˘ rp�j

rp�jC1 ; j D 2; : : : ; rp � 1;

Pp;rp 2 Œ�1; 1� � ˛p;rp˙p;rp � Pp;rp�1

˘0;

(16)

where the last row of each inclusion block follows from Assumption 2, that is, ¹‰i . Nx; x; u/ºpiD1 2

¹Œ�Mi ;Mi �ºpiD1. Therefore, from Theorem 1, there exist parameters

®˛i;j

¯p;riiD1;jD1

and t� > 0 suchthat for all t > t�

§t� is the time when the observer (11)–(14) has converged to a ˇ�neighborhood, defined by the constants®�i;k

¯p;riiD1;kD1

,Mi , and the amplitude of the noise.


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ˇ®i;j .t/

¯p;riiD1;jD1

ˇ6®�i;jMi�

ri�jC1¯p;riiD1;jD1

; (17)

for some positive constants®�i;k

¯p;riiD1;kD1

. Then, according to [25], each element of the dynamicsin (16) converges to a region close to zero that is proportional to a certain order of the noise, infinite time¶. Notice that each i;k.t/ is bounded by �i;kMi�

ri�kC1, in other words, i;k.t/ is insidea neighborhood of radius �i;kMi�

ri�kC1. Then, according to Assumption 1, it is possible to find aˇ�neighborhood of x, that is,

N.x; ˇ/ D ¹ Nx.t/ 2 <nj kx.t/ � Nx.t/k 6 ˇº : (18)

It is clear that the value of ˇ depends on the constants®�i;k

¯p;riiD1;kD1

and on the way in which

the elements of � are combined due to the matrix

�@ˆ. Nx.t//

@ Nx.t/

��1. Nevertheless, ˇ can always be

calculated because it depends on the parameters of the HOSM differentiator and on the amplitudeof the noise. �

In order to know when the observer converges, there exist some analytical reaching time estima-tions (see, e.g., [37] and [38]), for the case ri D 2, that is, for the super-twisting algorithm [39].Nevertheless, such estimations are very crude, so that in practice, it is better to obtain its value inthe following way.

According to [34], to detect when the observer converges to the ˇ�neighborhood, proportionalto certain order of the noise, it is sufficient to verify that the following inequalities are satisfiedˇ

eyi .t/ˇ6 �i;1Mi�

ri ; 8t 2�0; �ti �

�; 8i D 1; : : : ; p: (19)

It is usual to estimate the constants �i;1 and �t through simulation. Therefore, when all the outputerrors eyi .t/ are inside its respective error band �i;1Mi�

ri during the time interval�0; �ti �

�, the

observer will have converged, and t� will be the time at which all these inequalities are satisfied,that is,

t� D max8iD1;:::;p

�ti �: (20)

4.2. Plant fault case

Consider the case when system (1) has a plant fault, that is, F�.x.t//!�.t/ D �f�.x.t//, with� 2 <

n�n known distribution matrices and f�.x.t// known faulty vector fields. Then, system (1)takes the following structure

Px.t/ D f�;�.x.t//C Bu.t/;

y.t/ D h.x.t//; .t/ D y.t/C v.t/;(21)

where f�;�.x.t// D f .x.t// C �f�.x.t//. Let Assumption 1 be satisfied with the new vectorfield f�;�.x.t//. Then, it is possible to design an observer of the form

PNx.t/ D f�;�. Nx.t//C Bu.t/C

�@ˆ. Nx.t//

@ Nx.t/

��1�;

Ny.t/ D h. Nx.t//;

(22)

where the correction term � is designed according to (14) with a new set of gains®Mi;f�;�

¯piD1

satisfying the following assumption.

¶Because of the discontinuous nature of the proposed high-order-sliding-mode (HOSM) observer, the solution of thedifferential inclusion (16) is understood in the Filippov sense [35]. These solutions coincide with the usual solutions,when the right-hand sides are continuous. For the particular employed HOSM algorithm, the convergence of a uniquesolution is given by Reference [25] based on homogeneity properties.


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Assumption 3There is a set of known constants

®Mi;f�;�

¯piD1

> 0 such that the following inequalities are satisfied:

ˇ°Lrif�;�. Nx/

hi . Nx/ � Lrif�;�.x/

hi .x/±piD1

ˇ<®Mi;f�;�

¯piD1

: (23)

Then, according to Lemma 1, the state estimation error converges to a ˇf�;��neighborhood, pro-portional to a certain order of the noise, in finite time, that is, kx.t/ � Nx.t/k 6 ˇf�;� , for ˇf�;� > 0and 8t > t�

f�;�

||.

4.3. Actuator fault case

Consider the case when system (1) has an actuator fault, that is, F�.x.t//!�.t/ D Ba�u.t/, wherethe Ba� are known matrices. Then, system (1) takes the following structure

Px.t/ D f .x.t//C B�;�u.t/;

y.t/ D h.x.t//; .t/ D y.t/C v.t/;(24)

where B�;� D B C Ba� . Let Assumption 1 be satisfied for the new input matrix B�;� . Then, it ispossible to design the observer

PNx.t/ D f . Nx.t//C B�;�u.t/C

�@ˆ. Nx.t//

@ Nx.t/

��1�;

Ny.t/ D h. Nx.t//;

(25)

where the correction term � is designed according to (14) with a new set of gains®Mi;B�;�

¯piD1

satisfying the following assumption.

Assumption 4There are known constants

®Mi;B�;�

¯piD1

> 0 such that the following inequalities are satisfied:

ˇ°Lrif . Nx/


hi .x/±piD1

ˇ<®Mi;B�;�

¯piD1

: (26)

Therefore, according to Lemma 1, the state estimation error converges to a ˇB�;��neighborhood,proportional to certain order of the noise, in finite time, that is, kx.t/ � Nx.t/k 6 ˇB�;� ,for ˇB�;� > 0 and 8t > t�B�;�

**.Arguing as in Sections 4.1–4.3, it is possible to design the following bank of q C 1 HOSM

observers:

PNx�.t/ D f�. Nx�.t//C B�u.t/C

�@ˆ�. Nx�.t//

@ Nx�.t/

��1��;

Ny�.t/ D h. Nx�.t//;

(27)

with estimated state vectors Nx�.t/ 2 <n, estimated outputs Ny�.t/ 2 <p , � 2 ƒ D ¹0; 1; : : : ; qº,and the correction terms �� 2 <n. The structure for f� and B� depends on the kind of faultsconsidered††, as will be seen later in the simulation example.

||t�f�;�

is the time when the observer (22)–(14) has converged to a ˇf�;��neighborhood, defined by the constants®�i;j

¯p;riiD1;jD1

,®Mi;f�;�

¯piD1

, and the amplitude of the noise. Notice that t�f�;�

can be calculated by (20).**t�B�;�

is the time when the observer (25)–(14) has converged to a ˇB�;��neighborhood, defined by the constants®�i;k

¯p;riiD1;kD1

,®Mi;B�;�

¯piD1

, and amplitude of the noise. Notice that t�B�;�

can be calculated by (20).††The observer � D 0 is for the fault-free case.


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Remark 4Notice that to design the observers, it is only necessary to calculate the inverse matrices

�@ˆ. Nx/@ Nx

��1,

not the inverse transformations ˆ�1. N/.

Remark 5It is important to note that the vector relative degree can change because of the different faults.However, while Assumption 1 is satisfied, it will always be possible to design an observer thatconverges in finite time.

Remark 6Notice that in the absence of noise and sampling, observers (27) ensure the exact estimation of thecorresponding state in finite time.

Notice that if the observability mappings defined in (10) are equal for every �, it is possible todesign only one observer (selecting correctly the gains M�i of the correction term‡‡) that will becapable of estimating the state of the system, with an error proportional to certain order of the noise,independently of the faults that occur. However, this only solves the problem of state estimation, notthe problem of FDI that is the topic of this paper.

On the other hand, when a bank of observers is designed, irrespective of whether the observabilitymappings are equal or different, it is possible that more than one of the q C 1 observers estimatesthe state. Nevertheless, the equivalent output injection will be different for each of them; this is thekey point in order to solve the FDI problem, and will be discussed in the sequel.

5. FAULT DETECTION AND ISOLATION

One of the main features of sliding-mode observers is the possibility of unknown input identificationusing the equivalent output injection (see, e.g., [40] and [41]). Here, the equivalent output injectionwill be used as the residual to achieve fault detection.

Define the output estimation error as

°Ny.j�1/

�i.t/ � yi

.j�1/.t/±p;riiD1;jD1

D®�i ;j .t/

¯p;riiD1;jD1

; 8� 2 ƒ: (28)

Suppose that the active scenario is the fault-free case, and therefore, the output error dynamicsbetween system (9) and observer � D 0 is given by

P01;1 D 01;2 � ˛1;1M1r1

01

˙01;1 � v1.t/

˘ r1�1r1 ;

P01;j D 01;jC1 � ˛1;jM1

r1�jC1

01

˙01;j � P01;j�1

˘ r1�j

r1�jC1 ; j D 2; : : : ; r1 � 1;

P01;r1 D Lr1f . Nx0/

h1. Nx0/ � Lr1f .x/

h1.x/ � ˛1;r1M01

˙01;r1 � P01;r1�1

˘0;

:::

P0p ;1 D 0p ;2 � ˛p;1M1rp

0p

˙0p ;1 � vp.t/

˘ rp�1rp ;

P0p ;j D 0p ;jC1 � ˛p;jM1

rp�jC1

0p

˙0p ;j � P0p ;j�1

˘ rp�j

rp�jC1 ; j D 2; : : : ; rp � 1;

P0p ;rp D Lrpf . Nx0/

hp. Nx0/ � Lrpf .x/

hp.x/ � ˛p;rpM0p

˙0p ;rp � P0p ;rp�1

˘0:

(29)

‡‡For this purpose, it is necessary to check the corresponding assumption of boundedness for every � (Assumption 2, 3,or 4, depending on the scenario); then, the maximum value of Mi is selected such that it satisfies all assumptions ofboundedness.


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Remark 7Notice that the output error dynamics between system (9) and the other observers does not nec-essarily have any particular structure. However, if the observability mapping is the same forevery �, the output error dynamics will have the same structure as (29). Nevertheless, the terms°Lrif . Nx�/

hi . Nx�/ � Lrif .x/

hi .x/±piD1

will be different, for all � ¤ 0, because of the system structure.

In steady state, all the terms®0i ;ri .t/

¯piD1

are directly affected by the discontinuous correc-

tion terms®�0i ;ri .t/

¯piD1D°˙0i ;ri � P0i ;ri�1

˘0±piD1

, that is, they are zero in the ‘average’ sense.

Therefore, the equivalent output injection principle can be exploited.The expressions for

®0i ;ri .t/

¯piD1

are given by


h1. Nx0/ � Lr1f .x/

h1.x/ � ˛1;r1M01�01;r1.t/;


h2. Nx0/ � Lr2f .x/

h2.x/ � ˛2;r2M02�02;r2.t/;

:::



hp.x/ � ˛p;rpM0p�0p ;rp .t/:

(30)

At the moment in which the state estimation is reached, the discontinuous terms®�0i ;ri .t/

¯piD1

take the value of the equivalent output injection, and, for the noise-free case; they are equal to zero,that is,

0 D �˛1;r1M01�eq01;r1

.t/;

0 D �˛2;r2M02�eq02;r2

.t/;

:::

0 D �˛p;rpM0p�eq0p ;rp

.t/:

(31)

In this work, the functions �eq0i ;ri .t/ in (31), which need to be applied to dynamics (29) afterreaching the sliding surfaces i;ri .t/ D 0, to ensure that the output error trajectories stays on thesurfaces thereafter, are called the equivalent output injections (for details, see [42]).

The following equivalent output injection properties can be established.

1. The equivalent output injections describe the ‘average’ effect of the high-frequency terms°˙0i ;ri � P0i ;ri�1

˘0±piD1

.

2. The equivalent output injections provide estimations of the ‘disturbance’ terms°Lrif . Nx0/

hi . Nx0/ � Lrif .x/

hi .x/±piD1

.

Hence, taking into account the effect of noise and the statements of Theorem 1, (31) simplifies to

�01;r1C1M01�2 >

ˇ˛1;r1M01�

eq01;r1

.t/ˇ;

�02;r2C1M02�2 >

ˇ˛2;r2M02�

eq02;r2

.t/ˇ;

:::

�0p ;rpC1M0p�2 >

ˇ˛p;rpM0p�

eq0p ;rp

.t/ˇ:

(32)


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Then, the equivalent output injections satisfy the following inequalities:

ˇ�eq01;r1

.t/ˇ6 �01;r1C1

˛1;r1�2; 8t > t�

ˇ�eq02;r2

.t/ˇ6 �02;r2C1

˛2;r2�2; 8t > t�

:::ˇ�eq0p ;rp

.t/ˇ6�0p ;rpC1

˛p;rp�2; 8t > t�:

(33)

In accordance with the previous explanations, only the equivalent output injection of observer � D 0will satisfy the inequalities (33). In general, the output error dynamics generated by the observersassociated with � ¤ 0 will not have the structure (29), because the vector relative degree couldchange between the different scenarios, and Assumptions 2, 3, and 4 may be only satisfied for therespective scenario. Consequently, the trajectories that are associated with the observers � ¤ 0 willbe different from those that the system produces in the active scenario; thus, the equivalent outputinjection will not satisfy (33).

In particular, when the output error dynamics of another observer, for example, observer � D 1,has the structure (29), both observers � D 0 and � D 1 have the same observability mapping.Hence, the expressions for

®1i ;ri .t/

¯piD1

will be given by


h1. Nx1/ � Lr1f .x/

h1.x/ � ˛1;r1M11�11;r1.t/;


h2. Nx1/ � Lr2f .x/

h2.x/ � ˛2;r2M12�12;r2.t/;

:::



hp.x/ � ˛p;rpM1p�1p ;rp .t/:

(34)

If the gains®M1i

¯piD1

are designed such that°Lrif . Nx1/

hi . Nx1/ � Lrif .x/

hi .x/±piD16®M1i

¯piD1

is

satisfied, then the observer associated with � D 1 will estimate the state with an error proportionalto a certain order of the noise. In this way, (34) can be rewritten as follows:

�11;r1C1M11�2 >

ˇLr1f . Nx1/

h1. Nx1/ � Lr1f .x/

h1.x/ˇ

„ ƒ‚ …j"11 j

Cˇ˛1;r1M11�

eq11;r1

.t/ˇ;

�12;r2C1M12�2 >

ˇLr2f . Nx1/

h2. Nx1/ � Lr2f .x/

h2.x/ˇ

„ ƒ‚ …j"12 j

Cˇ˛2;r2M12�

eq12;r2

.t/ˇ;

:::

�1p ;rpC1M1p�2 >

ˇLrpf . Nx1/


hp.x/ˇ

„ ƒ‚ …j"1p j

Cˇ˛p;rpM1p�

eq1p ;rp

.t/ˇ;

(35)

where®"1i¯piD1

are the differences in the model corresponding to the observer associatedwith � D 1. The following lemma establishes the class of faults that can be detected and isolatedfor the particular case in which the observability mappings are equal for two different values of �.


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Lemma 2Let the bank of HOSM observers (27) be applied to the system in (1)–(3). For all � ¤ suchthat �; 2 ƒ, with the property that their corresponding observability mappings are the same,if the trajectories

®"�i¯piD1

and®"�i¯piD1

are sufficiently large out of their active scenario, then

the faults � and can be detected and isolated by means of �eq� D��eq�1;r1 ; � � � ; �

eq�p ;rp

�Tand

�eq� D

��eq�1;r1 ; � � � ; �

eq�p ;rp

�T.

ProofIt is clear that if the scenarios � and have the same observability mapping, they will have thestructure (29), and the only difference will be given by the terms

®"�i¯piD1

and®"�i¯piD1

. Then, ifthe difference between the models is sufficiently large out of their active scenario, then, supposingthat the active scenario is �, the equivalent output injections satisfy the following inequalities

ˇ�eq�1;r1.t/

ˇ6 ��1;r1C1

˛1;r1�2 <

��1;r1C1

˛1;r1�2 C

1

˛1;r1M11

ˇ"�1ˇ;

ˇ�eq�2;r2.t/

ˇ6 ��2;r2C1

˛2;r2�2 <

��2;r2C1

˛2;r2�2 C

1

˛2;r2M12

ˇ"�2ˇ;

:::ˇ�eq�p ;rp .t/

ˇ6��p ;rpC1

˛p;rp�2 <

��p ;rpC1

˛p;rp�2 C

1

˛p;rpM1p

ˇ"�p

ˇ:

(36)

Notice that the terms that depend on the amplitude of the noise have the same order, that is, O.�2/;thus, the order of the terms

®"�i¯piD1

has to be discernably larger in order to detect and isolate thefaults � and . In this way, the faults will be distinguishable using the information in �eq� and �eq� . �

In the other case, when the observability mappings are different, inequalities (36) are satisfiedtrivially.

Now, the relations (33) will be exploited to establish a residual evaluation. The overall power ofeach element of the equivalent output injections will be taken as a scalar residual, that is,

¹r�.t/ºq

�D0D°�eq

�1;r1.t/2 C �

eq

�2;r2.t/2 C � � � C �

eq

�p ;rp.t/2

±q�D0

: (37)

Theoretically, if condition (36) is satisfied under statements given by Lemma 2 or for differentobservability mappings, a simple threshold-based decision criterion could be applied to the signalsr�.t/ in order to achieve fault detection. Such a detection rule could be represented as follows:

O� D arg min�r�.t/: (38)

However, the signals r�.t/, � ¤ ��, can occasionally cross the zero value when the active scenario is��, which would make the decision (38) unreliable. Therefore, because of the previous explanationand to measurement noise, the average power of the signals r�.t/ is taken over a suitable receding-horizon time interval of finite length, that is,

¹R�.t/ºq

�D0D

²1

T

Z t

t��T

r�.�/d�

³q�D0

; (39)

whereT is the width of the time interval used§§. Then, the fault detection rule (FDR) is establishedas the value of � for which R�.t/ is minimum, that is,

§§The width �T varies according to the kind of system, to the faults, or even to the inputs. However, it can be definedas the minimum time interval for which (40) has a unique O� for all possible �.


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O� D arg min�R�.t/; (40)

with O� representing the active scenario. Therefore, it is possible to detect and isolate the fault in thesystem according to the residuals from each observer. Obviously, each residual is sensitive, that is,satisfies condition (33) or (36), only to the corresponding fault. Hence, it is possible to distinguishbetween the occurrence of each fault due to the characteristics of the equivalent output injection.

Remark 8Notice that it will only be possible to detect and isolate faults just after the observer for the activescenario has converged, that is, after time t�, t�

f�;�, or t�B�;� , depending on the case.

Theoretically, the equivalent output injection is the result of infinite switching frequency in thediscontinuous term ��. However, the realization of the observer produces finite high-frequencyswitching, making necessary the application of a filter.

Define the following equivalent output injection estimator of ��eq :

�� PN��eq D �� N��eq ; (41)

where each �� is designed according to [43], that is, �s � �� 1 for all � D 0; : : : ; q, where �sis the sample time. One possible choice is �� D

p�s . Then, the equivalent output injections can be

used to detect and isolate the faults.Notice that the filter in (41) can be designed to estimate the equivalent output injection and also

to avoid to a certain extent the effect of noise (as is illustrated in the simulations in the sequel).

5.1. Discussion: faults not belonging to F and uncertainties

For the case of a fault not belonging to F , if it satisfies the relative degree condition (4), and allthe assumptions for the plant or actuator fault case, such an observer will be able to estimate thestates. However, in this case, every residual may be influenced by such a fault, indicating that it isnot contained in F . In this way, fault detection may still be achievable, but not isolation.

If the new fault does not satisfy the relative degree condition in (4), for a plant or actuator fault,no observer will estimate the state. However, the fault detection may still be possible due to theproperties of the equivalent output injection, again, indicating that such a fault is not contained inthe fault set F .

For the uncertainty case, a detailed study is formally necessary because the uncertainties directlyaffect, and possibly in different forms, the proposed residual. Also, the process for the residualdesign would have to change drastically or it would be necessary to undertake a study, for example,a statistical one, of the uncertainty effects on the residuals, and thus, to establish some thresholdsthat allow the realization of the FDI process. However, in order to show that the proposed approachcan work under certain uncertainties, simulation results are shown using exactly the same approachdescribed in this section.

6. SIMULATION EXAMPLE

Consider the mass-spring-damper system, depicted in Figure 1.F1The sets of DEs that describe the system behavior are the following:

m1 Rp1 C .k1 C k2/ p1 � k2p2 D Fu; (42)

m2 Rp2 C .k2 C k3/ p2 C b tanh. Pp2/ � k2p1 D 0; (43)


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Col

orO

nlin

e,B

&W

inPr

int

Figure 1. Block diagram of mass-spring-damper system.

Table I. Mass-springer-damper parameters.

Parameter Value

m1 1:28 kgm2 1:05 kgk1 450 N=mk2 175 N=mk3 450 N=mb 15 Ns=m

where p1 and p2 describe the positions; Fu is the input force acting on the system; m1 and m2 arethe masses; k1, k2, and k3 represent the spring constants; and b is the dry friction coefficient. Then,systems (42)–(43) can be represented in the form of (9) as follows:

2664Px1

Px2

Px3

Px4

3775 D

26666664

x2

�k1 C k2

m1x1 C

k2

m1x3

x4k2

m2x1 �

k2 C k3

m2x3 �

b

m2tanh.x4/

37777775C

266664

0

1

m10

0

377775u; (44)

y D�x1 x3

�T; (45)

where x D Œ p1 Pp1 p2 Pp2 �T is the state vector, u D Fu is the input force, and y is the measurableoutput. The parameters of the mass-springer-damper system are shown in Table I. T1

Simulations have been carried out in the MATLAB Simulink environment, with the Euler dis-cretization method, a sampling time �s D 0:001 s, and the real initial conditions x.0/ D Œ0; 0; 0; 0�T .In the succeeding text, state observers are designed to estimate the velocities.

6.1. Fault-free observer

First, the fault-free observer (� D 0) will be designed. The Jacobian matrix defined in (10) has thefollowing structure:

@ˆ0. Nx0/

@ Nx0D

2641 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

375 :

It is clear that rank�@ˆ0. Nx0/@ Nx0

�D 4 for all Nx0 2 X � <4 and therefore Assumption 1 is satisfied.

Then, the fault-free observer takes the following form:


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26664PNx01PNx02PNx03PNx04

37775 D

26666664

Nx02

�k1 C k2

m1Nx01 C

k2

m1Nx03

Nx04k2

m2Nx01 �

k2 C k3

m2Nx03 �

b

m2tanh. Nx04/

37777775C

266664

0

1

m10

0

377775uC

�@ˆ0. Nx0/

@ Nx0

��1�0; (46)

Ny0 D�Nx01 Nx03

�T; (47)

with observer initial conditions Nx0.0/ D Œ0:1; 0:1; 0:1; 0:1�T . The correction term is calculated using

the following auxiliary dynamics:

P#01;1 D #01;2 � ˛1;1M12

01

ley01 .t/

k 12

;

P#01;2 D �˛1;2M01

l#01;2 �

P#01;1

k0;

P#02;1 D #2;2 � ˛02;1M12

02

ley02 .t/

k 12

;

P#02;2 D �˛2;2M02

l#02;2 �

P#02;1

k0;

(48)

where ey01 D Nx01 � x1 and ey02 D Nx03 � x3 and the parameters ˛1;1 D ˛2;1 D 1:5 and ˛1;2 D˛2;2 D 1:1. Based on each acceleration, that is, Px2 and Px4, and the initial condition error, the gainsM01 D 1 and M02 D 0:5 are sufficiently large constants such that Assumption 2 is satisfied foreach output. Then, the correction term takes the following form:

�0 D

2666666664

�˛1;1M12

01

ley01 .t/

k 12

�˛1;2M01

l#01;2 �

P#01;1

k0�˛2;1M

12

02

ley02 .t/

k 12

�˛2;2M02

l#02;2 �

P#02;1

k0

3777777775: (49)

6.2. Actuator fault observer

Consider the case when an actuator fault occurs in system (44)–(45) and the actuator is working at10% of its capability, that is, F1.x/!1 D Ba1u. Then, the dynamics has the following structure:

264Px1Px2Px3Px4

375 D

2666664

x2

�k1 C k2

m1x1 C

k2

m1x3

x4k2

m2x1 �

k2 C k3

m2x3 �

b

m2tanh.x4/

3777775C

26664

00:1

m10

0

37775u; (50)

y D�x1 x3

�T; (51)

The observability mapping for systems (50)–(51) is the same as that for the system (44)–(45), andthe Jacobian matrix takes the same structure, that is, @ˆ1. Nx1/

@ Nx1D @ˆ0. Nx0/

@ Nx0.


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Thus, Assumption 1 is satisfied, and the actuator fault observer takes the following form


3775 D

2666664

Nx12

�k1 C k2

m1Nx11 C

k2

m1Nx13

Nx14k2

m2Nx11 �

k2 C k3

m2Nx13 �

b

m2tanh. Nx14/

3777775C

26664

00:1

m10

0

37775uC

�@ˆ1. Nx1/

@ Nx1

��1�1; (52)

Ny1 D�Nx11 Nx13

�T; (53)


the auxiliary dynamics in (48) replacing ey01 by ey11 D Nx11 � x1 and ey03 by ey13 D Nx13 � x3.Q3Choosing M1;f�;1 D 1 and M2;f�;1 D 0:5 is sufficiently large to ensure that Assumption 4 is

satisfied for each output.

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.02

0.04

0.06

0.08

0.1

Time [sec]

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0.5

1

Time [sec]

0 0.1 0.2 0.3 0.4 0.5−1.5

−1−0.5

00.5

Real State 1Estimated State 01Estimated State 11Estimated State 21

2

02

12

22

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−0.015

−0.01

−0.005

0

0.005

0.01

Time [sec]

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−0.05

0

0.05

0.1

0.15

0.2

Time [sec]

0 0.1 0.2 0.3 0.4 0.5

−1

−0.5

0

0.5

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

3

03

13

23

4

04

14

24

Real StateEstimated StateEstimated StateEstimated State



Figure 2. Real and state estimation trajectories. It is clear that each observer estimates its state in the corre-sponding scenario. Nevertheless, observers � D 1 and � D 2 clearly maintain the estimation in spite of thefault occurrence, while observer � D 0 loses accuracy of estimation when an actuator fault happens due to

the corresponding gain M1 that is not able to support such a fault.


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6.3. Actuator + plant fault observer

Consider the case when a plant fault occurs in systems (50)–(51) and the spring effect k1 is no longerpresent, that is, F2.x/!2 D Ba1uC2f2.x/. Then, the dynamics has the following structure:

264Px1Px2Px3Px4

375 D

2666664

x2

�k2

m1x1 C

k2

m1x3

x4k2

m2x1 �

k2 C k3

m2x3 �

b

m2tanh.x4/

3777775C

26664

00:1

m10

0

37775u; (54)

y D�x1 x3

�T; (55)

Now, an actuator + plant fault observer (� D 2) will be designed. The observability mapping forsystems (54)–(55) is the same as that for systems (50)–(51). The Jacobian matrix also takes the samestructure, that is, @ˆ2. Nx2/

@ Nx2D @ˆ1. Nx1/

@ Nx1:

Col

orO

nlin

e,B

&W

inPr

int

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

1

2

a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

5

10 x 10−3

b)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−1

0123

Time [sec]

c)

0 1 2

Fault−FreeActuator Fault Actuator + Plant Fault

9.9 9.95 10 10.05 10.1 10.15 10.2

0

0.5

1

1.5

2

d)

9.9 9.95 10 10.05 10.1 10.15 10.2−1

0

1

2

3

Time [sec]

e)= 10[ ]

Actuator + Plant Fault

Figure 3. Residuals and fault detection rule. a/ Residuals R0, R1, and R2. b/ A zoom in a zero neighbor-hood of the residuals. c/ Fault detection rule (� estimation). d/ Zoom at the moment of actuator + plant

fault in the residuals. e/ Zoom at the moment of the actuator + plant fault in the fault detection rule.


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Therefore, Assumption 1 is satisfied and the actuator + plant fault observer takes the followingform:


3775 D

2666664

Nx22

�k2

m1Nx21 C

k2

m1Nx23

Nx24k2

m2Nx21 �

k2 C k3

m2Nx23 �

b

m2tanh. Nx24/

3777775C

26664

00:1

m10

0

37775uC

�@ˆ1. Nx2/

@ Nx2

��1�2; (56)

Ny2 D�Nx21 Nx23

�T; (57)


the auxiliary dynamics in (48) replacing ey01 by ey21 D Nx21 � x1 and ey03 by ey23 D Nx23 � x3.Choosing M1;B�;2 D 1 and M2;B�;2 D 0:5 is sufficiently large to ensure that Assumption 3 issatisfied for each output.

The parameter �� is the same for every observer: specifically �� D 0:06 for � D 0; 1; 2: Thesimulation develops in the following way: the system is fault free during the first 5 s after whichthe first actuator fault happens (whereby the actuator works at only 10% of its capability); then,at 10 sec, the effect of spring 1 is removed when the actuator + plant fault occurs. The results aredepicted in Figures 2–5. F2–F5

The residuals generated by the observers can be seen in Figure 3. It is clear that only one residualis almost equal to zero when the corresponding fault is acting on the system. Therefore, the faultwill be detected and isolated implementing the FDR scheme given in (38) with T D 0:01. Thecorresponding results are shown in graph c/ of Figure 3. Finally, in graphs d/ and e/, a zoom at themoment when the actuator + plant fault has occurred is shown. It is possible to see that there existsa time delay (td � 0:025Œsec�) after the fault occurs. Of course, this time delay is due to the filterused to estimate the equivalent output injection. However, this time could be arbitrarily reduced bytaking different values for the parameter ��.

Next, the same simulation scenario is repeated, taking into account the fact that the outputcontains measurement noise (Figure 4).

Col

orO

nlin

e,B

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inPr

int

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0.01

0.02

0.03

0.04

−5

0

5

10

15x 10−3

Time [sec]

1( )1( )

2( )2( )

Figure 4. Output with and without measurement noise. The noisy signal is generated by a sum of sinusoidalsignals of high frequency and amplitude of 3 � 10�3.


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inPrint

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.5

1

1.5

a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−1

0

1

2

3

Time [sec]

c)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0.02

0.04

0.06

b)

0 1 2

Fault−Free Actuator + Plant FaultActuator Fault

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.5

1

1.5

d)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−1

0123

Time [sec]

f)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0.02

0.04

0.06

e)

0 1 2

Actuator Fault Actuator + Plant FaultFault−Free

Figure 5. Residuals and fault detection rule with noise. a/ Residuals R0, R1, and R2, �� D 0:06. b/ Azoom in a zero neighborhood of the residuals, �� D 0:06. c/ Fault detection rule, �� D 0:06. d/ ResidualsR0, R1, and R2, �� D 0:0775. e/ A zoom in a zero neighborhood of the residuals, �� D 0:0775. f / Fault

detection rule, �� D 0:0775.

The noisy residuals generated by the observers can be seen in Figure 5. In graphs a/, b/, andc/, the effect of the noise on the residuals is shown using the same filter parameter �� as before.It is clear that the noise affects the FDR producing false alarms (see graph c/). Nevertheless, theperformance can be improved if the filter parameter �� is modified taking into account some noiseinformation (shown in graphs d/, e/, and f /).

In order to show that the proposed approach is robust again certain uncertainties, consider thefollowing uncertainty mass-spring-damper system:

264Px1Px2Px3Px4

375 D

2666664

x2

�k1 C k2

m1x1 C

k2

m1x3

x4k2

m2x1 �

k2 C k3

m2x3 �

b

m2tanh.x4/

3777775C

26664

01

m10

0

37775uC

2640 0 0 0

20 0 10 0

0 0 0 0

10 0 20 2

375 x;


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Col

orO

nlin

e,B

&W

inPr

int

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

10

20

a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−1

0123

Time [sec]

c)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

1

2

b)

0 1 2

Fault−Free

Actuator FaultActuator + Plant Fault

Figure 6. Residuals and fault detection rule with noise and uncertainties. a/ Residuals R0, R1, and R2,�� D 0:06. b/ A zoom in a zero neighborhood of the residuals, �� D 0:06. c/ Fault detection rule,

�� D 0:06.

where the last term represents parameter uncertainty. The simulation is developing in the sameway, which was described previously. For this simulation, the observer gains are chosen as follows:M01 D 10 and M02 D 5 for the fault free, M1;f�;1 D 10 and M2;f�;1 D 5 for the actuator fault,M1;B�;2 D 10 and M2;B�;2 D 5 for the actuator + plant fault, and the filter constant �� D 0:06 foreach observer. Then, the FDR (38) is implemented with T D 0:1. The corresponding results aredepicted by Figure 6. It is easy to see that even in the uncertainty case, it is still possible to solve F6the FDI problem with the proposed approach. Q4Notice that there may exist such uncertainties, highenough, that spoil the proposed approach. Therefore, it is necessary to realize a more detailed studyof the FDI problem under uncertainties.

7. CONCLUSIONS

A combination of HOSM observers and a multiple-model approach is proposed to solve the FDIproblem for a certain class of nonlinear systems. Under structural conditions, and based on themain features of sliding-mode observers, the value of the equivalent injection is used as a residualgenerator for detecting particular faults acting on the system. The isolation problem is solved using amultiple-model approach. The fast convergence of the HOSM methods provide fast FDI. The effectof Lebesgue-measurable noise in the outputs has been studied, and their corresponding effects weredescribed. The workability of the proposed methodology has been illustrated in simulations.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support from PAPIIT 17211; CONACyT 56819, 151855,and CVU 270504; FONCICyT 93302; SIP-IPN; and CDA-IPN.

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2. Chen J, Patton RJ. Robust Model-Based Fault Diagnosis for Dynamic Systems. Kluwer Acedmic Publishers: Boston,MA, 1999.


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