Adaptive sensor and actuator fault estimation for a class of …ncbs.knu.ac.kr/Publications/PDF-Files/VeluvoluKC15... · 2016-05-17 · SENSOR/ACTUATOR FAULT ESTIMATION FOR NONLINEAR

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. (2015)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/acs.2556

Adaptive sensor and actuator fault estimation for a classof uncertain Lipschitz nonlinear systems

Michael Defoort1, Kalyana C. Veluvolu2,*,† , Jagat J. Rath2 and Mohamed Djemai1

1LAMIH UMR CNRS 8201, UVHC 59313 Valenciennes, France2School of Electronics Engineering College of IT Engineering, Kyungpook National University, Daegu 702 701,

South Korea

SUMMARY

This paper deals with state and fault estimations for a class of uncertain Lipschitz nonlinear systems. Theproposed scheme combines a descriptor form observer and an adaptive sliding mode observer. The adaptivedescriptor sliding approach is proposed to jointly estimate the state and reconstruct the sensor fault whilerejecting the effect of uncertainties and Lipschitz nonlinearities. Using a Lyapunov analysis, the stabilitycondition is analyzed and the observer gains are designed such that the reduced–order system is practicallystable. Then, an adaptive super-twisting observer is derived to reconstruct the actuator faults. The mainfeature of the proposed adaptive scheme is that it does not overestimate the observer gains, which mitigateschattering in the presence of bounded uncertainties, actuator, and sensor faults with unknown boundaries.Simulation results for a robotic manipulator show the effectiveness of the proposed method. Copyright ©2015 John Wiley & Sons, Ltd.

Received 5 July 2013; Revised 29 January 2015; Accepted 12 March 2015

KEY WORDS: fault estimation; sliding mode observer; sensor fault

1. INTRODUCTION

Occurrence of faults can lead to critical situations for systems (instability, deterioration, etc.) andcan be extremely detrimental, not only to the equipment and surroundings but also to the humanoperator if they are not detected and isolated in time. Fault detection and isolation (FDI) and faulttolerant control have been widely investigated using various methods [1–6]. Observer-based FDItechniques rely on the estimation of outputs from measurements with the observer in order to detectthe fault. They are usually based on either the analysis of appropriate residuals (i.e., the comparisonbetween output estimations and the measurements) or an estimation of the fault variables. Mostof the existing frameworks on fault estimation can be combined with unknown input estimationproblem in the literature. For the case of linear systems, the unknown input estimation has been wellunderstood and several results are available [2, 7–10]. The work in [8] extended the earlier researchof [2] for the case of nonlinear Lipschitz bounded systems using a sliding mode observer (SMO).The work in [11] developed an observer for unknown input estimation that requires the evaluation ofoutput higher-order derivatives. However, in real-world applications, it is not easy to directly applythe existing schemes because of the presence of uncertainties, disturbances, and noise [12, 13].

Sliding mode observers have been developed to robustly estimate the system states through theconcept of sliding surface design and equivalent control. This discontinuous technique consists in

*Correspondence to: Kalyana C. Veluvolu, School of Electronics Engineering College of IT Engineering, KyungpookNational University Daegu, Daegu, South Korea 702701.

†E-mail: [email protected]

Copyright © 2015 John Wiley & Sons, Ltd.

M. DEFOORT ET AL.

constraining the system motion along manifolds of reduced dimensionality in the state space and isapplicable to a broad variety of practical applications. Robustness properties against various kindsof uncertainties such as parameter perturbations and external disturbances can be guaranteed.

Several SMO-based FDI approaches have been developed recently for linear systems and a classof nonlinear systems. The first-order SMOs in [14–16] are designed to directly tackle the actua-tor faults through the sliding mode terms via some transformations. Unfortunately, the realizationof first-order sliding mode implies the undesirable chattering phenomenon, that is, high-frequencyvibrations of the closed-loop system, which may excite unmodeled high-frequency dynamics,degrade the system performances, and lead to instability. Other techniques rely on a nonlinear coor-dinate transformation to obtain a form suite for the design of high-gain observer [17, 18]. The gainsare designed based on the inverse jacobian of the state transformation. However, this transformationis valid only locally and no sensor fault is considered. Recently, some SMOs have been proposedto reconstruct both the system state, actuator faults, and sensor faults. In [19, 20], actuator fault isreconstructed with the so-called equivalent output estimation error injection concept. Then, sensorfault is estimated via a low-pass filter and appropriate secondary SMO. Nevertheless, the sensorfault estimation is very sensitive to the filter parameters, and no uncertainty acting on the systemis considered. An SMO is proposed in [21, 22] to solve this problem for nonlinear systems with-out uncertainty assuming some conditions on the fault distribution matrix. Descriptor sliding modeapproaches are introduced in [5, 23] to simultaneously reconstruct state, sensor, and actuator faultsfor linear systems. In [24], a sliding mode unknown input observer is given for linear systems wherethe same additive perturbation affects both the dynamics and the output. In [25], a H1 SMO isderived for a class of uncertain nonlinear Lipschitz systems with sensor and actuator faults. Most ofthe mentioned SMOs require the knowledge of the upper bounds of the unknown input and uncer-tainties. Anyway, in practise, these bounds are usually difficult to obtain due to the complexity andunpredictability of the system uncertainties.

Recently, adaptive sliding mode schemes have been proposed to avoid a gain overestimation.This gain adaptation induces a lower chattering phenomenon. A dynamic gain adaptation forfirst-order sliding mode has been given in [26] for a class of nonlinear systems with boundeduncertainties whose bounds are unknown. The adaptation algorithm ensures that the gain is not over-estimated, which leads to a reduction of chattering. Recently, it has been extended to second-ordersliding mode control [27]. Similarly to [26], the controller only guarantees the real 2�sliding mode,that is, the sliding surface and its first time derivative converge to a known neighborhood aroundthe origin.

Bounded-error or set-membership methods assume that the system is affected by a bounded error[28]. These methods allow the characterization of the set of state values that is consistent withthe measurement, the model structure and the prior knowledge of the error bounds [29]. The stateestimation based on bounded uncertainty assumption is called set-valued observer [30]. This schemehas been applied in several fault detection algorithms [31].

The objective of this paper is the robust state and fault estimation for uncertain Lipschitz non-linear systems with simultaneous unknown input and possible sensor fault. Here, the upper-boundsof the uncertainties, unknown inputs, and sensor faults are not a priori known. It should be high-lighted that the adaptive SMO design for uncertain nonlinear systems with actuator and sensorfaults with unknown boundaries remains an open problem. Through a series of transformations, theoriginal system is transformed into two subsystems. One subsystem is transformed into a descrip-tor form by considering the sensor fault and the uncertainties as supplementary state variables. Anadaptive descriptor sliding approach is proposed to jointly estimate the state and reconstruct thesensor fault while rejecting the effect of uncertainties and Lipschitz nonlinearities. The observergains are designed by solving some LMI conditions to guarantee the practical stability of the errordynamics. Then, using the adaptation scheme proposed in [27], an adaptive super-twisting observeris used to reconstruct the actuator faults. The contribution of our paper lies in the following fea-tures: (1) an accurate estimation of the state, the sensor and actuator faults is provided in spite ofthe presence of uncertainties and Lipschitz nonlinearities, (2) conditions on the observer gains aregiven in terms of LMI; and (3) the upper bounds of the uncertainties, sensor, and actuator faults

Copyright © 2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)DOI: 10.1002/acs

SENSOR/ACTUATOR FAULT ESTIMATION FOR NONLINEAR SYSTEMS

are not a priori known. Simulation results for a robotic manipulator show the effectiveness of theproposed method.

The paper is organized as follows. The problem statement is described in Section 2. In Section 3,the adaptive observer is presented to estimate the state, the sensor and actuator faults in spite ofthe presence of uncertainties and Lipschitz nonlinearities. An application to a robot-arm problem isprovided in Section 4. Section 5 concludes the paper.

1.1. Notation

Throughout this paper, �max.A/ (resp. �min.A/) denotes the maximum (resp. minimum) eigenvalueof a matrix A. For a symmetric matrix, A > 0means that A is a positive definite matrix. k:k denotesthe Euclidean norm of a vector.

For any vector x D Œx1; : : : xn� 2 Rn and any scalar a 2 R, we denote

sign.x/ D Œsign.x1/; : : : ; sign.xn/�T

jxja D diag .jx1ja; : : : ; jxnj

a/

dxca D jxjasign.x/

2. PROBLEM STATEMENT

Consider the following system²Px.t/ D Ax.t/C Bˆ.x.t/; u.t//CEfa.t/C��.t/y.t/ D Cx.t/CDfs.t/

(1)

where A 2 Rn�n, B 2 Rn�s , C 2 Rp�n, D 2 Rp�m2 , E 2 Rn�m1 , � 2 Rn�r with x 2M � Rn is the state vector, u 2 U � Rq (U is an admissible control set) is the control input andy 2 Y � Rp (Y is the output space) is the system output. The unknown function fa 2 Rm1 is theactuator fault. The signal � 2 Rr models the parameter uncertainties and/or disturbances. The sensorfault is fs 2 Rm2 . The function ˆ.x; u/ W RnCq 7! Rs is Lipschitz about x uniformly, that is,8x1; x2

kˆ.x1; u/ �ˆ.x2; u/k 6 Lˆkx1 � x2k (2)

where L� is the known Lipschitz constant.The objective of this paper is to design a fault estimation scheme for the uncertain system (1)

such that accurate estimates of x, fs and fa are provided. To facilitate the design of the observer,the following assumptions are required:

Assumption 1

(a) D, E, and � are full-column rank(b) The system matrix dimensions satisfy m1 Cm2 C r 6 p(c) For every complex number 0s0 with nonnegative real part: rank

��sIn � A E

C 0

��D nCm1.

It is equivalent to the minimum phase condition. Furthermore, there exists ı > 0 such that

rank

��ıIn C A E �

C 0 0

��D nCm1 C r

(d) The detectability condition holds, that is, rank.CE/ D rank .E/.

Remark 1Assumption 1 enforces structural limitations and pertains to the dimensional aspects of the nonlinearsystem (1). One can notify that condition 1(b) can be relaxed if not all outputs are affected by sensorfaults or if all the outputs are affected by the same sensor fault.


M. DEFOORT ET AL.

Assumption 2It is assumed that the geometric condition Im.E/ \ Im.�/ D ¹0º holds. This assumption impliesthat the image/column space of matrices E and � are independent under any linear transformation.Roughly speaking, it induces a decoupling scheme between actuator faults and uncertainties.

Assumption 3The unknown functions fa, � , fs , and their first-time derivative are norm bounded. However, theupper bounds of the uncertainties, actuator, and sensor faults are not a priori known.

From Assumption 1, without loss of generality, we can assume the following geometric condition

associated withD, that is,D D

�0

D2

�whereD2 2 Rp�m1�m2 is full-column rank.E is partitioned

as E D

�E1E2

�such that E1 2 Rm1�m1 with rank.E1/ D m1. Let us introduce the nonsingular

transformation [25]

T1 D

�Im1 0

�E2E�11 In�m1

�(3)

CT �11 is partitioned as CT �11 D�C1 C2

�such that C1 D

�C11C12

�2 Rp�m1 and C2 D

�C21C22

�2

Rp�.n�m1/ with rank.C11/ D m1. Considering the nonsingular transformation matrices [25]

T D T2T1 ; T2 D

�Im1 C

�111 C21

0 In�m1

�; H D

�Im1 0

�C12C�111 Ip�m1

�(4)

we have

TAT �1 D

�A1 A2A3 A4

�; TE D

�E10

�; HCT �1D

�C11 0

0 C3

�; C3 D �C12C

�111 C21CC22

HD D

�0

D2

�; TB D

�B1B2

�; T� D

��1�2

�

From Assumption 2, using the transformation T , one gets Im.TE/ \ Im.T�/ D ¹0º. From thenonsingularity of E1, it follows that �1 D 0.

Therefore, system (1) in the new coordinates

�x1x2

�D T x,

�y1y2

�D Hy is

²Px1 D A1x1 C A2x2 C B1�.x1; x2; u/CE1fay1 D C11x1

(5)

²Px2 D A3x1 C A4x2 C B2�.x1; x2; u/C�2�y2 D C3x2 CD2fs

(6)

with �.x1; x2; u/ D ˆ�T �1

�x1 x2

�T; u

. From Equation (2), one can conclude that

k�.x1; Nx2; u/ � �.x1; x2; u/k 6 L�k Nx2 � x2k (7)

where L� is the corresponding Lipschitz constant.Based on the transformed system (5)–(6), an estimation scheme is proposed such that accurate

estimates of x, fs and fa are obtained.



3. SLIDING MODE OBSERVER DESIGN

The fault estimation strategy for system (1) is based on two steps. First, system (6) is transformedinto a descriptor form. An adaptive descriptor sliding approach is proposed to jointly estimate thestate x2 and reconstruct the sensor fault fs while rejecting the effect of uncertainties and Lipschitznonlinearities. The observer gains are designed by solving some linear matrix inequalities condi-tions to guarantee the practical stability of the error dynamics. At last, from system (5), using theadaptation scheme proposed in [27], an adaptive super-twisting observer is used to reconstruct theactuator faults fa.

3.1. Descriptor adaptive observer for state and sensor fault estimation

First, let us consider subsystem (6). It will be shown hereafter that it can be transformed into adescriptor form. Then, an adaptive descriptor sliding approach will be proposed to jointly estimatethe state x2 and reconstruct the sensor fault fs . Some conditions on the observer gains, based onlinear matrix inequalities, will be given to guarantee the practical stability of the error dynamics.

Because C11 is invertible, from subsystem (5), one gets x1 D C�111 y1. Hence, subsystem (6) canbe rewritten as follows:²

Px2 D A3C�111 y1 C A4x2 C B2�

C�111 y1; x2; u

�C�2�

y2 D C3x2 CD2fs(8)

Because Assumption 1(c) holds, the pair .A4; C3/ is detectable. This preliminary assumption isrequired to estimate x2 from subsystem (6). System (8) can take the following descriptor form´

T PNx D NA4 Nx C NB1y1 C NB2�C�111 y1; x2; u

�C N� N�

y2 D NC Nx(9)

where the extended state is Nx D�xT2 �T .D2fs/

T�T

and

NA4 D

24A4 0 0

0 �ıIr 0

0 0 �Ip�m1

35 ; NB1 D

24A3C�1110

0

35 ; NB2 D

24B200

35 ; NC D

�C3 0 Ip�m1

�

T D

24In�m1 ı�1�2 00 Ir 0

0 0 0

35 ; N� D

24ı�1�2 0

Ir 0

0 D2

35 ; N� D

�ı� C P�fs

�

Let us first introduce the following two lemmas that will be used in the derivation of the adaptiveobserver.

Lemma 1 ([5])There exists a matrix L1 such that the matrix S D T C L1 NC is nonsingular. Furthermore, it holdsthat NCS�1L1 D Ip�m1 and NA4S�1L1 D �G with G D

�0 0 Ip�m1

�T.

Lemma 2 ([5])

If rank

��ıIn�m1 C A4 �2

C3 0

��D n � m1 C r , there is a gain matrix L2 such that the matrix

S�1. NA4 � L2 NC/ is Hurwitz.

Using Assumption 1(c), one can easily show that, rank

��ıIn�m1 C A4 �2

C3 0

��D n � m1 C r .

Hence, one can apply Lemma 2.From Assumption 3, it follows that N� is bounded by an a priori unknown constant, that is, N� 6 �.

This constant is not a priori known.


M. DEFOORT ET AL.

Let us introduce the descriptor SMO²S P D . NA4 � L2 NC/´ �Gy2 C NB1y1 C NB2�.C

�111 y1; Ox2; u/C k

NB2M.y2 � NC ONx/C N��1. Ne/ONx D ´C S�1L1y2

(10)

where ´ is an auxiliary variable, ONx DhOxT2O�T Of Ts

iTis the estimation of Nx and the corresponding

estimation error is

Ne D ONx � Nx (11)

The term NB2�C�111 y1; Ox2; u

�C k NB2M.y2 � NC ONx/ removes the effects of the Lipschitz nonlineari-

ties, where matrix M and constant k will be defined hereafter. �1. Ne/ is the adaptive robust term toeliminate the effect of disturbances N� , defined by

�1 D �ˇ sign .s/ (12)

using the sliding surface

s D N NC Ne (13)

The design of matrix N will be described hereafter. The gain adaptation algorithm is represented asfollows:

P D

²˛ksk sign .ksk � / for ˇ > for ˇ 6 (14)

with ˇ.0/ > 0, ˛ > 2, 0 < < 1 and > 0 being small.

Remark 2In Equation (14), the parameter is introduced for guaranteeing only positive value for ˇ. In thesequel, without loss of generality and also for the sake of clarity, only the case ˇ > is consideredfor discussion and proof.

Lemma 3With the classical first-order sliding mode term (12), the adaptive gain ˇ, defined in (14), has anupper bound ˇ� for all t > 0 with ˇ� > �.

The proof is similar as in [26] and is not presented here.

Theorem 1For system (9), under Assumptions 1–3, consider the adaptive observer (10)–(14). If there arepositive definite matrices P and Q such that²

NBT2 S�TP D M NC

N�TS�TP D N NC(15)

PS�1. NA4 � L2 NC/C . NA4 � L2 NC/TS�TP 6 �Q (16)

and a parameter k such that

k >L2�

2�min.Q/(17)

Then, the estimation error is practically stable (i.e. final and uniform boundedness). It means that allestimation error trajectories converge to a neighborhood of the origin.



ProofUsing Lemma 1, Equation (10) yields

SPONx D S P C L1 Py2DNA4 � L2 NC

�´ �Gy2 C NB1y1 C NB2�

C�111 y1; Ox2; u

�� k NB2M NC Ne C N��1 C L1 NC PNx

D . NA4 � L2 NC/ ONx C L2 NC Nx C NB1y1 C NB2�C�111 y1; Ox2; u

�� k NB2M NC Ne C N��1 C L1 NC PNx

From Equation (9), adding L1 NC PNx to both sides, one gets

S PNx D NA4 Nx C NB1y1 C NB2�C�111 y1; x2; u

�C N� N� C L1 NC PNx

The dynamic behavior of the error trajectories is

PNe D S�1NA4 � L2 NC

�Ne C NB2

�C�111 y1; Ox2; u

��

C�111 y1; x2; u

�� k NB2M NC Ne C N�

�1 � N�

��(18)

Consider the Lyapunov function candidate

V D NeTP Ne C1

2.ˇ � ˇ�/2 (19)

where P is a positive definite matrix which satisfies conditions (15)–(16).The time derivative along the state trajectories

PV 6 NeT�PS�1. NA4 � L2 NC/C

NA4 � L2 NC

�TS�TP

Ne

C2 NeTPS�1 NB2�C�111 y1; Ox2; u

��

C�111 y1; x2; u

�� kM NC Ne

�C2 NeTPS�1 N�

�1 � N�

�C .ˇ � ˇ�/˛ksk sign .ksk � /

6 �NeTQ Ne C 2L�kNekkM NC Nek � 2kkM NC Nek2C2 NeT NC TN T

�ˇ sign

N NC Ne

�� N�

�C .ˇ � ˇ�/˛kN NC Nek sign

kN NC Nek �

�If kN NC Nek < , PV is sign indefinite. Therefore, the asymptotic closed-loop stability cannot be

guaranteed. Nevertheless, let us consider the case that kN NC Nek > . Then, it yields

PV 6 �NeTQ Ne C 2L�kNekkM NC Nek � 2kkM NC Nek2C2�kN NC Nek � 2ˇkN NC Nek C 2ˇ�kN NC Nek � 2ˇ�kN NC Nek C .ˇ � ˇ�/˛kN NC Nek

6 �NeTQ Ne C 2L�kNekkM NC Nek � 2kkM NC Nek2C2.� � ˇ�/kN NC Nek � jˇ � ˇ�jkN NC Nek.�2C ˛/

Because ˛ > 2 and from Lemma 3, one can obtain

PV 6 ��min.Q/kNek2 C 2L�kNekkM NC Nek � 2kkM NC Nek2

6 ��kNek kM NC Nek

� ��min.Q/ �L��L� 2k

� �kNek

kM NC Nek

�

From condition (17), matrix

��min.Q/ �L��L� 2k

�> 0, it can be concluded that the estimation error

trajectories converge to a neighborhood of the origin (i.e. final and uniform boundedness). Thus, wecomplete the proof. �

Remark 3The convergence of the error dynamics given in Theorem 1 in the absence of measurement noise isestablished to a given bound around the origin, thus ensuring uniform practical stability [32]. In thepresence of noise, the convergence of the estimation error can be established consequently within asmall bound depending on the size of noise and the sampling time.


M. DEFOORT ET AL.

Remark 4To establish the convergence of error dynamics as shown in Theorem 1, two linear equality matricesobtained from (15) need to be solved. Similarly to [5], the solution of these linear equality matricesis equivalent to the following minimization problem. Find the smallest �1 > 0 and �2 > 0 such that�

��1INBT2 S

�TP �M NC/T�

� �I

�< 0 and

��2I

N�TS�TP �N NC

�T� �I

�< 0 (20)

Remark 5The real estimation of the sensor fault fs is given by .DT

2 D2/�1DT

2Ofs

3.2. Adaptive super-twisting observer for actuator fault estimation

In the following, the proposed adaptive sliding mode algorithm design will be incorporated in orderto estimate the actuator fault. Let us consider subsystem (5). The following nonlinear adaptiveobserver can be used

POx1 D A1 Ox1 C A2 Ox2 C B1�. Ox1; Ox2; u/C �2 (21)

�2 is the robust adaptive sliding mode term designed as

�2 D �k1d Ox1 � x1c12 � k2

Z t

0

sign . Ox1 � x1/ (22)

where the adaptive gains satisfy [27]

Pk1 D

²�1 sign .k Ox1 � x1k � "1/ ; if k1 > ˛m�2 ; if k1 6 ˛m

k2 D �3k1

(23)

where �1, �2, �3, "1 are arbitrary positive constants. The parameter ˛m is an arbitrarily small positiveconstant.

Lemma 4System (5)–(6) satisfying Assumptions 1–3 is driven to the following sliding manifold°Ox1 � x1 D POx1 � Px1 D 0

±in finite time and remains on it.

ProofLet us define the estimation error as

e1 D Ox1 � x1D Ox1 � C

�111 y1

The error dynamics is as follows

Pe1 D A1e1 C A2. Ox2 � x2/C B1.�. Ox1; Ox2; u/ � �.x1; x2; u//C �2 �E1fa (24)

From [27], one can conclude that the adaptive super-twisting controller (22) ensures the establish-ment of a real 2�sliding mode with respect to e1. �

Corollary 1Provided that Assumptions 1–3 are satisfied. Consider the adaptive observer (10)–(14) and(21)–(23). Then, as t !1, we have kA2. Ox2�x2/CB1.�.x1; Ox2; u/��.x1; x2; u//k ! 0. Undersuch condition, the actuator fault can be reconstructed as

Ofa.t/ D �E�11 k2

Z t

0

sign . Ox1 � x1/ (25)



ProofLemma 4 implies that system (5)–(6) is driven to the sliding surface ¹e1 D Pe1 D 0º in finite timeand remains on it. Therefore, on sliding mode, the equivalent error dynamics of (24) becomes

0 D A2. Ox2 � x2/C B1.�.x1; Ox2; u/ � �.x1; x2; u// � k2

Z t

0

sign . Ox1 � x1/ ds �E1fa (26)

Because � is Lipschitz and from Theorem 1, the following holds. As t ! 1, we have kA2. Ox2 � x2/C B1.�.x1; Ox2; u/ � �.x1; x2; u//k ! 0. This concludes the proof. �

4. APPLICATION TO FLEXIBLE JOINT ROBOT ARM

We consider the laboratory model of a single-link flexible joint robot [33], shown in Figure 1 anddefined by the following nonlinear differential equations:

P m D !mP!m D

kJm. l � m/ �

SmJm!m C

K�Jmu

P l D !l

P!l D �kJl. l� m/ �

SlJl!l �

mgbJl

sin. l/

(27)

where m and l are the angular rotations of the motor and the link respectively, with !m and !lbeing their angular velocities, Jm D 0:0037 kgm2 represents the inertia of the DC motor, Jl D0:093 kgm2 the inertia of the controlled link, k D 0:18 Nm=rad the elastic constant, m D 0:21 kgthe link mass, g D 9:81 m=s2 the acceleration due to gravity, K� D 0:08 Nm=V the amplifier gainand Sm D 0:0083 Nm=V (resp. Sl ) the viscous friction coefficient of the motor (of the link resp.).

In order to evaluate the effectiveness of the proposed method, we consider the followingperturbations and actuator fault for the nominal model rewritten in the following form withx D

�x1 x2 x3 x4

�D� m !m l !l

�T

Px D

264

0 1 0 0

�k=Jm �B=Jm k=Jm 0

0 0 0 1

k=Jl 0 �k=Jl 0

375 x C

26640 0K�Jm

0

0 0

0 �mgbJl

3775�

u

sin.x3/

�C

264e1e2e3e4

375 fa C

2640

0

0

�1

375 � (28)

In this example, the viscous friction coefficient Sl is assumed to be unknown. It generates the pertur-bation � D Sl

Jlwl . The actuator fault fa 2 R and its distribution matrix are selected for illustration

purpose and to highlight the effectiveness of the proposed strategy in reconstruction. The actuatorfault distribution vector elements are chosen to be e1 D 1:2, e2 D 0:4, e3 D 0:8, e4 D 0:2. Themeasured output vector is

y D Cx CDfs (29)

Figure 1. Flexible joint robotic system.


M. DEFOORT ET AL.

where fs 2 R represents the sensor fault. Here, C D

241 0 0 00 1 0 0

0 0 0 1

35 and D D

24 0

0:1

0:2

35.

Hence, the dynamical system (28) can be represented in the form of (1). Setting ı D 0:5, it isclear that Assumptions 1–2 are satisfied.

Using transformation (4), system (28) in the new coordinates

�x1x2

�D T x,

�y1y2

�D Hy is

described by (5)–(6) with

A1 D 0:4167; A2 D�1 0 0

�; A3 D

24�5:1623�0:21530:0918

35 ; A4 D

24�2:6599 48:6486 0�0:9167 0 1

�0:1667 �1:9355 0

35

B1 D�0 0

�; B2 D

2421:6216 0

0 0

0 �3:3227

35 ; E1 D 1:2; �2 D

24001

35

C11 D 1; C3 D

�1 0 0

0 0 1

�; D2 D

�0:1

0:2

�

For illustration purpose, we choose

fs D

²0:1 sin.20t/ if t 6 5 (i.e. no sensor fault)0:1 sin.20t/C 1 sin.5.t � 5// if t > 5 (i.e. sensor fault)

0 5 10 15 20−15

−10

−5

0

5

10

t(s)

Sta

te a

nd it

s es

timat

e

(a)

0 5 10 15 20−6

−4

−2

0

2

4

6

8

t(s)

Sta

te a

nd it

s es

timat

e

(b)

0 5 10 15 20−4

−2

0

2

4

6

8

10

t(s)

Sta

te e

stim

atio

n er

ror

(c)

0 5 10 15 20−6

−4

−2

0

2

t(s)

Sta

te e

stim

atio

n er

ror

(d)

Figure 2. (a)–(d) State estimation with the proposed sliding mode observer and state estimation error in thepresence of measurement noise.



and

fa D

8<:0:5 sin.10t/ if t 6 10 (i.e. actuator fault)�0:25C 2 sin.t � 10/ if 16:3 > t > 1 (i.e. actuator fault)0 if t > 16:3 (i.e. no actuator fault)

From Lemmas 1–2, one can choose L1 and L2 as follows:

L1 D

2666664

0 0

0 0

0 0

0 0

0:5 0

0 0:5

3777775 ; L2 D

2666664

9:61 6:98

�1:78 1:65

5:10 12:63

3:44 4:49

3:06 0:53

0:53 1:84

3777775

From Theorem 1, one can obtain matrices P , Q, M , and N by solving the LMI conditions (16),(20) using the LMI-toolbox in Matlab. The sliding mode gains are set as ˛ D 1000, D 0:0001 and D 0:0001, �1 D 0:5, �2 D 0:4, "1 D 0:0001, ˛m D 0:1.

The initial conditions for plant and estimator are x D�3 2:75 0:75 1:5

�and Ox D

�0 0 0 0

�.

Recalling the transformation x D T �1�xT1 xT2

�T, each component of the state x in the original

coordinate is written as x D�x.1/ x.2/ x.3/ x.4/

�T. Similarly, each component of the estimated

state Ox D T �1�OxT1 Ox

T2

�Tis denoted as Ox D

�Ox.1/ Ox.2/ Ox.3/ Ox.4/

�T. To test the robustness of the

proposed design, a random noise is added to the output measurements.Figure 2 shows the corresponding tracking performance. Despite the presence of faults and per-

turbation, which creates large oscillations in the robot arm motion, the observer is able to track thesystem state (Figure 2) with high accuracy. We see that the state estimation based on the adaptiveobserver (10)–(14) and (21)–(23) is satisfactory event if there are some faults.

0 5 10 15 20−4

−2

0

2

4

6

8

t(s)

Sen

sor

faul

t and

its

estim

ate

(a)

0 5 10 15 20−0.5

0

0.5

1

1.5

t(s)

Per

turb

atio

n an

d its

est

imat

e

(b)

0 5 10 15 20−10

−5

0

5

10

t(s)

Act

uato

r fa

ult a

nd it

s es

timat

e

(c)

Figure 3. (a)–(c) Sensor fault, perturbation, and actuator fault estimations in the presence of measurementnoise.


M. DEFOORT ET AL.

The sensor fault signal fs and the perturbation term � are reconstructed from the adaptive observer(10)–(14) and are depicted in Figure 3(a)–(b). The actuator fault signal fa is reconstructed from theadaptive super-twisting observer (21)–(23) and is depicted in Figure 3(c).

These simulation results highlight that the proposed strategy, described in Theorem 1 and Corol-lary 1, manages to accomplish the objective (state and fault estimations) with good performancesin terms of convergence speed, steady-state error and robustness with respect to perturbations. Theproposed reconstruction method not only provides when the actuators or sensors are faulty but alsogives the shape and magnitude of faults.

5. CONCLUSION

This paper considers state and fault estimations for a class of uncertain Lipschitz nonlinear systems.First, an adaptive descriptor sliding approach is proposed to jointly estimate the state and recon-struct the sensor fault while rejecting the effect of uncertainties and Lipschitz nonlinearities. Using aLyapunov analysis, the stability condition is analyzed and the observer gains are designed such thatthe reduced–order system is practically stable. Then, an adaptive super-twisting observer is derivedto reconstruct the actuator faults. The effectiveness of the proposed strategy is illustrated throughsimulations to solve the fault and state reconstruction for a robotic arm.

REFERENCES

1. Frank P. Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy-a survey and somenew results. Automatica 1990; 26:459–474.

2. Chen J, Patton RJ. Robust Model-Based Fault Diagnosis for Dynamic Systems. Kluwer Press: USA, 1999.3. Efimov D, Zolghadri A, Raissi T. Actuator fault detection and compensation under feedback control. Automatica

2011; 47(8):1699–1705.4. Yang H, Jiang B, Cocquempot V, Lu L. Supervisory fault tolerant control with integrated fault detection and isolation:

A switched system approach. Applied Mathematics and Computer Science 2012; 22(1):87–97.5. Liu M, Shi P. Sensor fault estimation and tolerant control for Ito stochastic systems with a descriptor sliding mode

approach. Automatica 2013; 49:1242–1250.6. Xiao B, Hu Q, Friswell MI. Active fault-tolerant attitude control for flexible spacecraft with loss of actuator

effectiveness. International Journal of Adaptive Control and Signal Processing 2013; 27:925–943.7. Hou M, Muller PC. Design of observers for linear systems with unknown inputs. IEEE Trans Automatic Control

1992; 37(6):871–875.8. Pertew AM, Marquez HJ, Zhao Q. H-infinity synthesis of unknown input observers for non-linear lipschitz systems.

International Journal of Control 2005; 78(15):1155–1165.9. Hui S, Zak SH. Observer design for systems with unknown inputs. International Journal of Applied Maths and

Computer Science 2005; 15(4):431–446.10. Van gorp J, Defoort M, Veluvolu KC, Djemai M. Hybrid sliding mode observer for switched linear systems with

unknown inputs. Journal Franklin Institute 2014; 351(7):3987–4008.11. Vijayaraghavan K, Rajamani R, Bokor J. Quantitative fault estimation for a class of non-linear systems. International

Journal of Control 2007; 80(1):64–74.12. Floquet T, Edwards C, Spurgeon SK. On sliding mode observer for systems with unknown inputs. International

Journal of Adaptive Control and Signal Processing 2007; 21(8):638–656.13. Yan XG, Edwards C. Fault estimation for single output nonlinear systems using an adaptive sliding mode estimator.

IET Control Theory and Applications 2008; 2(10):841–850.14. Chen W, Chowdhury FN. A synthesized design of sliding-mode and Luenberger observers for early detection of

incipient faults. International Journal of Adaptive Control and Signal Processing 2010; 24(12):1021–1035.15. Veluvolu KC, Kim MY, Lee D. Nonlinear Sliding Mode High-Gain Observers for Fault Diagnosis and Estimation.

International Journal of Systems Science 2011; 42(7):1065–1074.16. Veluvolu KC, Defoort M, Soh YC. High-Gain Observer with Sliding Mode for Nonlinear State Estimation and Fault

Reconstruction. Journal Franklin Institute 2014; 351(4):1995–2014.17. Barbot JP, Floquet T. Iterative higher order sliding mode observer for nonlinear systems with unknown inputs.

Dynamics of Continuous, Discrete and Impulsive Systems 2010; 17(6):1019–1033.18. Rios H, Davila J, Fridman L. High-order sliding mode observers for nonlinear autonomous switched systems with

unknown inputs. Journal Franklin Institute 2012; 349(10):2975–3002.19. Edwards C, Spurgeon SK, Patton RJ. Sliding mode observers for fault detection and isolation. Automatica 2000;

36(4):541–553.20. Tan CP, Edwards C. Sliding mode observers for detection and reconstruction of sensor faults. Automatica 2002;

38:1815–1821.



21. Dimassi H, Loria A, Belghith S. A new secured transmission scheme based on chaotic synchronization viasmooth adaptive unknown-input observers. Communications in Nonlinear Science and Numerical Simulation 2012;17(9):3727–3739.

22. Yang J, Zhu F, Sun X. State estimation and simultaneous unknown input and measurement noise reconstruction basedon associated observers. International Journal of Adaptive Control and Signal Processing 2013; 27(10):846–858.

23. Lee DJ, Park Y, Park YS. Robust H1 sliding mode descriptor observer for fault and output disturbance estimationof uncertain systems. IEEE Trans Automatic Control 2012; 57(11):2928–2934.

24. Fridman L, Davila J, Levant A. High-order sliding-mode observation for linear systems with unknown inputs.Nonlinear Analysis: Hybrid Systems 2010; 5(2):189–2005.

25. Raoufi R, Marquez HJ, Zinober AS. H1 sliding mode observers for uncertain nonlinear Lipschitz systems withfault estimation synthesis. International Journal Robust and Nonlinear Control 2010; 20(16):1785–1801.

26. Plestan F, Shtessel Y, Brgeault V, Poznyak A. New methodologies for adaptive sliding mode control. InternationalJournal of Control 2010; 83:1907–1919.

27. Shtessel Y, Taleb M, Plestan F. A novel adaptive-gain supertwisting sliding mode controller: Methodology andapplication. Automatica 2012; 48:759–769.

28. Milanese M, Novara C. Nonlinear Set Membership prediction of river flow. Systems and Control Letters 2004; 53:1–39.

29. Bravo J, Alamo T, Redondo M, Camacho E. An algorithm for bounded-error identification of nonlinear systemsbased on dc functions. Automatica 2008; 44:437–444.

30. Shamma J, Tu K. Set-valued observers and optimal disturbance rejection. IEEE Transactions on Automatic Control1999; 44:253–264.

31. Blesa J, Puig V, Saludes J. Identification for passive robust fault detection using zonotope-based set-membershipapproaches. International Journal of Adaptive Control and Signal Processing 2011; 25:788–812.

32. Traore D, Plestan F, Glumineau A, Leon J. Sensorless induction motor: high order sliding mode controller andadaptive interconnected observeer. IEEE Transactions on Industrial Electronics 2008; 55(11):3818–3827.

33. Koshkouei AJ, Zinober ASI. Sliding mode state observation for non-linear systems. International Journal of Control2004; 77(2):118–127.


Documents

Adaptive sensor and actuator fault estimation for a class of …ncbs.knu.ac.kr/Publications/PDF-Files/VeluvoluKC15... · 2016-05-17 · SENSOR/ACTUATOR FAULT ESTIMATION FOR NONLINEAR