Nonlinear Flexible Link State/Joint-fault Estimation Using TS Fuzzy Observers

Zahra ShamsElec. Eng. Dept.

Shahed UniversityTehran, Iran

Email: z.shams@shahed.ac.ir

S. SeyedtabaiiElec. Eng. Dept.

Shahed UniversityTehran, Iran

Email: stabaii@shahed.ac.ir

Abstract- In this paper the simultaneous estimation of the state/joint-fault (process fault) of a nonlinear single link flexible robot in a noisy environment and under sensor fault is investigated. Fault detection of nonlinear systems becomes more feasible when it is conducted over Takagi-Sugeno (TS) approximated fuzzy models. TS fuzzy model unknown input observer can estimate fault and states; however, the performance is application dependent and affected by the configuration of the algorithm. In this respect, a TS fuzzy model augmented by a proportional plus integral, for fault modeling, observer (AL1) and a reduced order TS fuzzy model observer (AL2) are lined up for the unmeasured signals estimation. The simulation results indicate that both algorithms run well in estimating the motor speed, motor angle, load speed and the sensor fault, however, there are obvious differences in how they handle the process fault and the load angle estimation. AL1 estimates the process fault with low variance than AL2 does, that possible false fault detection is also observed. Estimation of the load angle by AL2 suffers from bias while AL1 performs more accurately. Meanwhile, computation cost is the AL1 failing. As a result, AL1 is recognized more attractive and robust in estimating the state/joint-fault in noisy environments. The conclusion is validated through extensive simulations.

Keywords- Process fault; TS fuzzy observer; Robust observer; Unknown input observer;

1 INTRODUCTION

Fault is any abnormal deviation of system components’ behavior from their nominal’s, which may cause medium and/or long run system break down and investment loss. Industrial systems are all vulnerable to faults that may lead to degradation in performance and product quality. Single link flexible robot is no exception, which its joint may be damaged by wearing among other faults. Maintaining safe and lasting operation requires certain measures that can be fulfilled by automatic fault diagnosis and isolation (FDI) to prompt early warning.

Among fault diagnosis approaches, the quantitative model based techniques and in particular the observer based methods continue to gain attention in the fundamental and applied research fields [1]. The methods rely on analytical, rather than, measurement redundancies. In practice, due to model uncertainties, disturbances, perturbations and noises, accurate fault estimation, such as the fault size and shape, is a very challenging procedure [2] [3] [4].

Most nonlinear systems can be appropriately approximated by Takagi-Sugeno (TS) fuzzy models [5].

It is shown that if the state variables are used as the decision variables, the obtained fuzzy models represent a large class of nonlinear systems [6]. Therefore, the problem of fault detection in nonlinear systems is transformed to the fault analysis of TS fuzzy system models. In this respect, plenty of researches have been conducted. The diagnosis of nonlinear systems with state observers is a direct application of state estimation [2]. In [7] a robust observer with unknown input has been designed for continuous and discrete TS models subjected to disturbances. The authors in [8] and [9]are interested to the simultaneous estimation of the state and the unknown inputs in polynomial form for TS systems by considering the immeasurable premise variable as a disturbed system.

In [10], a fault tolerant control strategy is developed which compensates an actuator fault for TS systems. Actuator fault estimation has also been investigated in [11, 12], [13]. A proportional integral (PI) observers has been introduced in [14, 15] [16], which detects and estimates the size of actuator faults. The algorithms performance degrades in case of measurement noise [3]. Another technique for actuator

fault estimation for T–S fuzzy model has been reported in [17, 18].

In [19] the problem of sensor fault in automatic steering has been detailed. The sensor faults diagnosis of a bioreactor described by multiple models approach has been elaborated in [20]. A Robust fault detection observer is designed in [21] by using H2/H∞ formulation for discrete-time TS fuzzy system affected by sensor faults and unknown bounded disturbances.

Actuators and/or sensors fault estimation based on nonlinear state observer has been studied in [22] [21] where unknown input observer has been employed. Estimation of actuator and sensor faults is the subject of study in [23], where the faults are introduced as auxiliary variables to be estimated. A revised version of PI observer for detection of sensor and actuator faults has been detailed in [2].

Most of the works use the Lyapunov stability theory and some the L2 optimization technique to formulate the stability conditions for the state observers synthesis in term linear matrix inequalities constraints [24].

In this paper, the problem of joint-fault (process fault) and state estimation of a single link flexible robot under sensor fault, process noise and noisy measurements are discussed. Two algorithms are designed which follow the line of the design used in the PI observer (AL1) [25] and the reduced order robust observer (AL2) [3] methods. Both algorithms successfully estimate the sensor fault and almost in a same way the states one to three. Performance of AL1 in detecting the process fault is more robust than AL2 as lower estimation variance is obtained. The same occurs with the estimate of the state 4 where bias in the estimate with AL2 is observed. Therefore, it is concluded that the PI fault model embedded in AL1 pays off in the estimation of the flexible robot joint-fault, of course, by increasing the computation costs.

In section 2, the robot dynamics and its TS fuzzy model is described. Two algorithms of TS fuzzy model process fault/states estimators are detailed in section 3. In section 4, the simulation results in detecting robot joint-fault/state estimation is described and lastly conclusion comes in section 5.

2 FLEXIBLE LINK ROBOT DYNAMICS

Flexible manipulators are used for picking up and displacing a load to a specific location. It has several advantages over rigid Robots including faster response, lower energy consumptions, lighter weight, smaller actuators, less overall cost, more maneuverable and transportable, and higher payload to robot weight ratio [26]. The system consists of a flexible arm mounted on a motor shaft. Fig. 1 shows the details of such a system. Due to the complexity of multiple links robots studies are limited to single-link manipulators [26].

The flexible link robot dynamic equations are given by, [27]

{θm(t )=ωm( t )

ωm( t )= kJm

(θl( t )−θm( t ))− bJ m

ωm( t )+kτ

J mu( t )

θl (t )=ωl ( t )

ωl( t )=− kJ l

(θl ( t )−θm( t ) )−mghJ l

sin(θl ( t ))(1)

where θ, ω, l and m represent the shaft angle, the shaft speed, load side and motor side, respectively.

Fig. 1. Flexible link joint Robot [10]

2.1 Flexible link robot TS Fuzzy modelMost nonlinear physical systems can be accurately

modeled by Takagi-Sugeno (TS) fuzzy systems [5]. The TS fuzzy set appoximates a nonlinear system by a combination of several linear local dynamics where each is represented by one of the fuzzy implications. Thus, the global behavior is obtained by the contribution of each of the linear sub-models. It is argued that if a nonlinear system state variables are used as the fuzzy decision variables, the fuzzy model is capable of replicating a large class of nonlinear systems [6]. Thus, it is important to employ observer based fault detection on these models which it would be applicable to many nonlinear systems [28].

A TS Fuzzy model of a noisy system subject to process and sensor faults is expressed by,

(2)

where x(t)∈n is the state, u(t)∈p1 is the input vector, w(t)∈p2 is the exogenous disturbance, y(t)∈q is the measured output, fa(t) is the actuator and/or process and fs is the sensor faults. Ai, Bi, Fi, C, F, D and Hi are constant real matrices of appropriate dimensions. The μi(x) is a membership function of the state variables, x. It is defined by,

where r represents the local models index. In case of (1) the state variables are,

x=[θm ωm θl ωl ]T

(3)

and the nonlinear term sin(θl(t)) can be modeled appropriately with TS fuzzy model with r=2. The joint-fault in (1) is expressed by,

− ΔbJ m

ωm( t )=Fi f a ( t )

in (2).

3 TS FUZZY MODEL STATE/ FAULT ESTIMATORS

In this section, two TS fuzzy model observers are modified to get compatible with the faulty system described by (2) which is perturbed by joint-fault, sensor fault and process and measurement noises.

3.1 AL1: PI observer designTo estimate the states and faults of (2), the output

is filtered by a stable matrix

and it is reformulated using a new state variable, r(t) as follows [25],

{r ( t )=∑i=1

r

μ i( A 1i r ( t )+B1i u( t )+F1 i f ( t )+H1 i w( t ))

q( t )=C1 r (t )

r ( t )=[ x ( t )yf ( t )] , f ( t )=[ f a ( t )

f s( t ) ] , A 1 i=[ A i oTC −T ]

F 1 i=[ F i o0 TF ] , B 1i=[B i

0 ] , H 1i =[ H i

0 ] , C 1 =[0 I q ]

where the matrices F1i are full column rank. If it is assumed that the fault is in polynomial form of k-1 degree and their kth derivatives are bounded, i.e. it obeys the following model,

A proportional integral observer is set up to estimate the states x, yf and the fault and its derivatives as given below,

{^r ( t )=∑

i=1

r

μi ( r )( A 1i r ( t )+B1i u( t )+F1i f ( t )+ K pi (q( t )− q( t )))+gr ( t )

q ( t )=C1 r ( t )

^f ( t )=∑i=1

r

μi( r )K li(q( t )−q( t ))+ f 1 ( t )+g f ( t )

^f j( t )=∑i=1

r

μi ( r ) K lij (q ( t )−q ( t ))+ f j+1( t )+g fj( t ) for j :1 . ..k−1

(6)

where KPi, KIi and KjIi are the proportional and integral

gains, respectively. The variables gr(t) and gf(t), gfj(t) are introduced in order to compensate the influence of the immeasurable decision variables. For a matter of simplicity, the system (4) and the observer (6) is restructured to the following compact form,

{¯r ( t )=∑i=1

r

μ i( A 2i r ( t )+B2i u( t )+H2i w( t ) )+J . f k( t )

q( t )=C2 r ( t )

{¯r ( t )=∑i=1

r

μ i( A 2i¯r ( t )+B2i u( t )+K i ( q( t )−¯q( t ) ))+g( t )

¯q( t )=C2¯r ( t )

r ( t )=[r ( t )f ( t )f 1( t )⋯f k−1( t )] , ¯r ( t )=[ r ( t )

f ( t )f 1 ( t )⋯f k−1( t )] , g( t )=[gr ( t )

g f ( t )g f 1( t )⋯g fk−1 ( t )] , J=[

000⋮I n f

]H2 i=[

H1 i

00⋮0

] , A2i=[A1 i F1 i 0 0 ⋯ 00 0 Inv

0 ⋯ 0

0 0 0 I nv⋯ 0

⋯ ⋯ ⋯ ⋯ ⋯ I nv

0 0 0 0 0 0] , B2 i =[B1 i

00⋯0

]K i=[

K pi

K li

K li1

⋯K li

k−1] ,C2=[C1 0 0 ⋯ 0 ]

where nf = nfa + nfs and Inf is an identity matrix.The observer estimation error,

e ( t )= r ( t )−¯r ( t ) , e y( t )=q( t )−¯q( t ) (9)

is expressed by,

(10)

The estimation error e(t) is asymptotically stable and L2

stability is guaranteed [25] , if there exist a matrix P>0, matrices Ni >0 and the positive scalars and 0 for each fuzzy implication that satisfies the following LMI condition under minimized μ,

[ϕi PJ PH 2i P δ1 . I I¿ −μ . I 0 0 0 0¿ ¿ −μ . I 0 0 0¿ ¿ ¿ −λ . I 0 0¿ ¿ ¿ ¿ −λ0 I 0¿ ¿ ¿ ¿ ¿ −μI

]<0

ϕi=(PA2 i−N iC2 )+( PA2 i−N i C2)T

(11)

Due to ϵ(t), z(t) and g(t) some residues remain on the estimation as (10) indicates. The gain of the PI observer for each fuzzy implication is given by,

K i=P−1 . N i

3.2 AL2: Reduced states robust state/fault observerBy introducing a new state variable x from the

augmentation of x with fs, a new arrangement for TS fuzzy model (2) is derived which is expressed by [3],

{Q ¯x ( t )=∑i=1

r

μ i( A 1i x( t )+B1 iu ( t )+F1 i f a ( t )+ H1i w ( t )+ E1 f s( t ))

y ( t )=C1 x( t )

x ( t )=[ xT ( t ) , f sT ( t ) ]T , Q=[I n 0

0 0q×m ] , A1 i=[A i 00 −F ] ,

F1i=[Fi

0q×p1 ] , B1 i=[Bi

0q×p ] , H1 i=[H i

0q×d ] , E1=[0n×m

F ] , C1=[ C F ]

(13)

Adding K1iy to both sides of (13) yields,

{¯x( t )=∑i=1

r

μi( A 2 i x ( t )+B2i u( t )+F2 i f a( t )+H 2 i w( t )+E2 i f s( t )+G1i−¿

¿ K1 i y (t )) ¿ y ( t )=C1 x ( t )=C 0¯ x ( t )+Ff s( t )

A2 i=Q1 i−¿

¿ A1i , B2 i=Q1 i−

¿

¿ B1 i , E2 i =Q1i−

¿

¿ E1 , H2 i=Q 1i−

¿

¿ H1i , F2i =Q1 i−

¿

¿ F1 i ¿ Q1 i

−¿

¿=[ I 0

−F−C F− K 12 i−1 ] , K12i =diag ( λi 1 , .. . , λ iq) ,Q1 i

−¿

¿ K 1i =[0F−]¿ ¿¿

¿¿

¿

(14)

It is shown [3]that under the defined conditions, there is a transformation matrix, T that transforms (14) to the following equation [4],

T=[C 1¿T

C 1 ]−1

, x (t )=Tr ( t )

{ r ( t )=∑i=1

r

μ i( A 3i r ( t )+B3i u( t )+F3i f a ( t )+ H3i w ( t )+ E3 i f s ( t )+N y ( t ))

y ( t )=C1 Tr ( t )=[0q×(n+m−q ) Iq ]r ( t )

A3 i=T −1 A2 i T , B3i=T−1 B2i , E3 i=T−1 E2 i , H 3i=T−1 H 2i ,

N=T −1 Q1 i

−¿¿ K1 i , F3 i=T−1 F2 i , N=T−1[ 0

F− ]

¿

(15)

Now, (15) is decomposed into two set of variables, already measured, r2, and those to be estimated, r1, as follows,

{ r1 ( t )=∑i=1

r

μi( A 11 i r1 ( t )+ A 12 i r2( t )+B31 iu ( t )+F31 i f a( t )+H31 i w( t )+E31i f s ( t )+N 1 y ( t ))

r2 ( t )=∑i=1

r

μi ( A 21 i r1( t )+ A 22 i r2( t )+B32i u( t )+F32i f a ( t )+ H32i w( t )+E32 i f s( t )+N2 y ( t ))

y ( t )=[ 0q×(n+m−q) Iq ] [r1( t )r2( t ) ]=r 2( t )

r1( t )∈ℜn+m−q ,r 2( t )∈ℜq , r ( t )=[r1 ( t )T r2( t )T ]T

[ A11i A12 i

A21 i A22 i ]=A3i , [B31 i

B32 i ]=B3 i ,[ E31 i

E32 i ]=E3 i ,[H31 i

H32 i ]=H3i , [N1

N 2 ]=N ,[F31 i

F32 i ]=F3 i

(16)

By considering the following equations for fs,

(17)

a reduced order estimator is designed for r1 and fa as follows [3],

{r 1( t )=z1( t )+∑

k[( N1+K2 k−K2 k N2 ) y ( t )]

f a( t )=z2( t )+∑k

[( K3 k−K3 k N2 ) y (t )]

z1 ( t )=∑i , j , k

r

[ ( A 11i−K 2 j A 21i )z1( t )+(B31 i−K2 j B32i )u ( t )+ (F31 i−K 2 j F32 i ) f a(t )

+(( A 11i−K2 j A 21 i )( N1+K 2k−K2 k N2)+( A 12 i−K2 j A 22 i )) y ( t )]

z2 ( t )=∑i , j , k

r

[−K3 j F 32 i z2( t )−K3 j A21 i r1( t )−K3 j B32 i u( t )

+(−K3 j F 32i ( K3 k−K3 k N2)−K3 j A 22 i ) y ( t )]

A 11i =A 11 i+F31 i T21 , A 12i =A 12 i +F31 i T 22

A 21 i=A 21 i+F31 i T21 , A 22i =A 22i +F31i T 22

∑i

,∑i , j

, ∑i , j ,k

, denote ∑i=1

g

μi , ∑i=1

g

∑j=1

g

μ i μ j , ∑i=1

g

∑j=1

g

∑k=1

g

μi μ j μk

(18)

The observer estimation error is denoted by,

(19)

It is shown [3][4] that If there exist a symmetric positive-definite matrix P, and matrices Mi >0 (for each fuzzy implication) and j=1, 2… g that satisfies the following linear matrix inequalities,

(20)

where

(21)

Then, the estimation error has H∞ performance level γ1

as expressed below,

(22)

with the following observer gain:

4 SIMULATION RESULTS

Consider a given single flexible link robot defined by (1) and modeled with the following TS fuzzy equations (2),

A1=[0 1 0 0−48.6 −1. 25 48 . 6 00 0 0 119 . 5 0 −22 . 83 0 ] , A2=[0 1 0 0

−48. 6 −1 .25 48. 6 00 0 0 119 .5 0 −18. 77 0 ]

B1=[021 .600 ] , B2=B1 , F1=[0500 ] , F2=F1 , H i=[1111 ] ,

F=[1000 ] , C=[100

010

001

000 ]

{μ1( s( t ))=s( t )+0 .21721.2172

μ2( s( t ))=1−s (t )1 .2172

s( t )=sin( x3) /x3

The system is equipped with three sensors measuring θm, θl and ωm. It is derived by u(t)=sin(t) and is affected by Gaussian noise of variance 0.1 both in the process and the measurement sections. The initial states of the system is x=[1 0 3 0]T and the initial estimator values are assigned at x=[0.5 0.5 0.5 0.5]T.

State/joint-fault estimationIn the first test, a 10% joint-fault applied at the 15

seconds of the test to the system. The calculated observer parameters of AL1 and AL2 are given in Table 1 and 2.

TABLE 1. Gains of observe AL1 for the two sub-models of TS fuzzy system

μ=0.40 =2.8*107 0=7.6*106

i 1 2

Kpi

6.56 *1011 2.73*1011 4.56*1011 7.37*1011 3.64*1011 7.28*10

6.33*1011 2.74*1011 4.38*1011 7.09*1011 3.86*1011 6.61*10

6.56*1011 2.72*1011 4.58*1011 7.37*1011 3.57*1011 7.39*10

6.50*1011 2.69*1011 4.81*1011 7.30*1011 3.38*1011 7.55*10

1.06*109 3.79*108 1.38*109 1.2*109 3.01*108 1.35*10

-4.73*105 8.13*108 -2.97*108 -2.4*107 8.27*108 -9.70*10

1.91*108 -2.5*108 8.67*108 2.39*108 -2.80*108 8.56*108

Kli -6.30*1010 1.57*1011 -3.30*1011 -7.80*1010 1.65*1011 -3.30*1011

Kli1 -6.15*1011 1.43*1012 -3.00*1012 -7.50*1011 1.42*1012 -2.90*1012

Kli2 -3.31*1012 6.47*1012 -1.50*1013 -4.00*1012 6.91*1012 -1.50*1013

Kli3 -9.14*1012 1.60*1013 -3.80*1013 -1.10*1013 1.72*1013 -3.80*1013

TABLE 2. Gains of observer AL2 for the two sub-models of TS fuzzy system

i 1 2

K4i

0 -0.24 2.52 0 -0.24 2.521

0 0.03 -0.78 0 0.03 -0.78

0 0.13 1.56 0 0.13 1.56

The performance of the algorithms in the detection of the joint-fault has been depicted in Fig.2. As it is shown, both algorithms sense the occurred fault, however, AL1 estimation is less noisy than what of AL2. In case of AL2, false detection of the process fault is also observed. If one assumes that the bandwidth of the joint-fault is very low, then by simple filtering of the results accurate fault estimation is achieved.

In addition to the process fault, the algorithms also estimate the system states from the noisy measured outputs. As Fig. 3, 4 and 5 indicate, the three of the states are estimated accurately by both algorithms and no substantial differences exist. However, there exist problems with the estimation of the state 4, the load angle, by the second algorithm, where the biased estimate is obtained. Hence, it is concluded that AL1 arrangement is more robust than AL2 in estimating the simultaneous state/joint-fault of a single link flexible robot in a noisy environments. However, the problem with AL1 is more computation demands.

time (s)0 5 10 15 20 25 30 35 40

d b

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

db(AL2))db(AL1)db

Fig. 2. The Plant fault and its estimates by AL1 and AL2

time (s)0 5 10 15 20 25 30 35 40

stat

e 1

0

5

10

15

20

25

x1x1(AL2)x1(AL1)

Fig. 3. The Sates 1 and their estimates by AL1 and AL2

time (s)0 5 10 15 20 25 30 35 40

stat

e 2

-10

-5

0

5

10

15x2x2(AL2)x2(AL1)

Fig. 4. The Sates 2 and their estimates by AL1 and AL2

time (s)0 5 10 15 20 25 30 35 40

stat

e 3

0

5

10

15

20

25

x3x3(AL2)x3(AL1)

Fig. 5. The Sates 3 and their estimates by AL1 and AL2

time (s)0 5 10 15 20 25 30 35 40

stat

e 4

-8

-6

-4

-2

0

2

4

6

8

10x4x4(AL2)x4(AL1)

Fig. 6. The Sates 4 and their estimates by AL1 and AL2

State/joint-fault estimation under sensor faultOnce more, the performances of the algorithms are

evaluated in simultaneous estimation of the state/process-fault in the presence of sensor fault. The system is derived and affected by noise as it is done in the previous test. Moreover a 0.25sin(500t) disturbance is applied to the measurement part. A combined signal is introduced as the sensor fault shown in Fig 7. The calculated parameters of the two algorithms are given in Table 3 and 4.

TABLE 3. Gains of observer AL1 for the two sub-models of TS fuzzy system

μ=0.51 =1.7*107 0=6.5*106

i 1 2

5.01 *1012 2.73*1012 4.56*1012 4.81*1012 3.11*1012 4.48*1012

4.89*1012 2.74*1012 4.38*1012 4.69*1012 3.11*1012 4.38*1012

5.03*1012 2.72*1012 4.58*1012 4.83*1012 3.11*1012 4.58*1012

Kpi 5.19*1012 2.69*1012 4.81*1012 4.98*1012 3.09*1012 4.73*1012

4.62*109 2.37*109 4.31*109 4.43*109 2.72*109 4.23*109

1.65*109 2.48*109 2.97*108 1.65*109 2.58*109 2.75*108

4.98*1012 1.60*109 5.37*109 4.76*109 2.00*109 5.28*109

Kli -5.90*1012 9.57*1010 -8.5*1011 -5.60*1011 4.43*1010 -8.47*1010

3.39*108 2.11*109 -1.2*109 3.43*108 2.10*109 -1.20*109

Kli1 -7.10*1012 1.03*1011 -1.00*1013 -6.80*1012 4.04*1012 -1.02*1013

-1.40*1010 2.40*1010 -3.70*1010 -1.30*1010 2.24*1010 -3.74*1010

Kli2 -3.00*1013 4.54*1012 -4.31*1013 -2.90*1013 1.88*1012 -4.36*1013

-5.30*1011 1.17*1010 -7.90*1011 -5.00*1011 6.99*1011 -7.88*1011

Kli3 -6.10*1013 9.07*1012 -8.80*1013 -5.80*1013 3.70*1012 -8.70*1013

-2.20*1012 6.21*1011 -3.4*1012 -2.19*1012 4.23*1011 -3.30*1012

TABLE 4. Gains of observer AL2 for the two sub-models of TS fuzzy system

i 1 2

0 -0.24 2.52 0 -0.24 2.521

K4i 0 0.03 -0.78 0 0.03 -0.78

0 0.13 1.56 0 0.13 1.56

Both algorithms detect the sensor fault as it has been depicted in Fig. 7. Their strength in capturing the process fault in the presence of sensor fault has been portrayed in Fig. 8. Again, AL1 is more robust in estimating the process fault as lower variance is resulted. Again false detection of the process fault by AL2 is noticed.

Both algorithms successfully estimate the state 1 to the state 3, however, the estimation of the load angle is challenging by AL2 where bias in estimation occurs as it has been shown in Fig. 9.

time (s)0 5 10 15 20 25 30 35 40

Sens

or F

ault

-0.2

0

0.2

0.4

0.6

0.8

1

1.2fsfs(AL2)fs(AL1)

Fig. 7. The Sensor fault and its estimates by AL1 and AL2

time (s)0 5 10 15 20 25 30 35 40

d b

-1

-0.5

0

0.5

1

1.5

2

db(AL2)db(AL1)db

Fig. 8. The Plant fault and its estimates by AL1 and AL2

time (s)0 5 10 15 20 25 30 35 40

stat

e 4

-8

-6

-4

-2

0

2

4

6

8

10x4x4(AL2)x4(AL1)

Fig. 9. The Sates 4 and their estimates by AL1 and AL2

5 CONCLUSION

In this paper, the state/joint-fault estimation of a single link flexible robot in a noisy environment and under sensor fault is investigated. Two unknown input TS Fuzzy model observers are employed in dealing with the task. The performance of the algorithms is application dependent and highly affected by their configurations. Both algorithms perform well in

estimating the sensor fault and three of the states, where almost similar results is attained, however, there is contrasts’ in their outcome in case of the process fault and the load angle estimation. AL1 estimates the process fault with less variance than AL2 where false detection is also observed. AL2 gives biased estimate of the load angle while AL1 performs more accurately. Eventually, AL1 looks fitter for the state/joint- fault estimation than AL2 in a noisy environment.

ACKNOWLEDGEMENT

This work has been partially supported by the research department of Shahed University, Tehran, Iran.

REFERENCE

[1] R. J. Patton, P. M. Frank, and R. N. Clark, Issues of fault diagnosis for dynamic systems: Springer Science & Business Media, 2013.

[2] T. Youssef, H. R. Karimi, and M. Chadli, "Faults Diagnosis based on Proportional Integral Observer for TS Fuzzy Model with Unmeasurable Premise Variable," IEEE International Conference on Fuzzy Systems 2014.

[3] J. Han, H. Zhangn, Y. Wang, and X. Liu, "Robust state/fault estimation and fault tolerant control for T–S fuzzy systems with sensor and actuator faults," Journal of the Franklin Institute vol. 353, pp. 615–641, 2016.

[4] Z. Ke, J. Bin, and S. Peng, "Observer-based Fault Estimation and Accommodation for Dynamic Systems," Springer-Verlag, Berlin, Germany, 2013.

[5] T. Takagi and M. Sugeno, "Fuzzy identification of systems and its applications to modeling and control," IEEE Trans. Syst. Man Cybernetic SMC, vol. 15, pp. 116-132, 1985.

[6] J. Yoneyama, "H1 filtering for fuzzy systems with immeasurable premise variables: an uncertain system approach," Fuzzy Sets Syst, vol. 160 (12) pp. 1738–1748, 2009.

[7] M. Chadli and H. R. Karimi, "Robust observer design for unknown inputs Takagi–Sugeno models," IEEE Transactions on Fuzzy Systems, vol. 21, pp. 158-164, 2013.

[8] J. L. Z. Lendek, T. M. Guerra, R. Babuska, and B. De Schutter, "Adaptive observers for TS fuzzy systems with unknown polynomial inputs," Fuzzy Sets and Systems, vol. 161, pp. 2043–2065, 2010.

[9] D. Ichalal, B. Marx, J. Ragot, and D. Maquin "Simultaneous state and unknown inputs estimation with PI and PMI observers for Takagi-Sugeno model with unmeasurable premise variables," 17th Mediterranean Conference on Control and Automation, 2009.

[10] D. Ichalal, B. Marx, J. Ragot, and D. Maquin, "Observer based actuator fault tolerant control for nonlinear Takagi-Sugeno systems: an LMI approach," in Control & Automation (MED), 2010 18th Mediterranean Conference on, 2010, pp. 1278-1283.

[11] Y.Y. Xu, S.C. Tong, and Y.M. Li, "Adaptive fuzzy fault-tolerant output feedback control of uncertain nonlinear systems with actuator faults based on dynamic surface technique," J. Frankl. Inst, vol. 350, pp. 1768–1786, 2013.

[12] Y.Y. Xu, S.C. Tong, and Y.M. Li, "Prescribed performance fuzzy adaptive fault-tolerant control of non-linear systems with actuator faults,," IET Control Theory Appl, vol. 8 (6), pp. 420–431, 2014.

[13] X.D. Zhu and Y. Q. Xia, "Hinfinit descriptor fault detection filter design for T-S fuzzy discrete-time systems," J. Frankl. Inst, vol. 351, pp. 5358–5375, 2014.

[14] K. Zhang, B. Jiang, and M. Staroswiecki, "Dynamic output feedback-fault tolerant controller design for Takagi–Sugeno fuzzy systems with actuator faults," IEEE Trans. Fuzzy Syst., vol. 18(1), pp. 194–201, 2010.

[15] B. Jiang, K. Zhang, and P. Shi, "Integrated fault estimation and accommodation design for discrete-time Takagi–Sugeno fuzzy systems with actuator faults," IEEE Trans. Fuzzy Syst., vol. 19(2), pp. 291–304, 2011.

[16] S.J. Huang and G.H. Yang, "Fault tolerant controller design for T–S fuzzy systems with time-varying delay and actuator faults: a k-step-fault-estimation approach," IEEE Trans. Fuzzy Syst., vol. 22 (6), pp. 1526–1540, 2014.

[17] S.K. Nguang, P. Shi, and S.X. Ding, "Delay-dependent fault estimation for uncertain time-delay nonlinear systems: an LMI approach," Int. J. Robust Nonlinear Control, vol. 16(18), pp. 913–933, 2006.

[18] B. Jiang and F.N. Chowdhury, "Fault estimation and accommodation for linear MIMO discrete-time systems," IEEE Trans. Control Syst. Technol, vol. 13(3), pp. 493–499, 2005.

[19] A. Akhenak, M. Chadli, J. Ragot, and D. Maquin, "Design of observers for Takagi-Sugeno fuzzy models for Fault Detection and Isolation," 7th IFAC Symposium on Fault Detection, Super-vision and Safety of Technical Processes, SAFE-PROCESS, vol. 09, 2009.

[20] R. Orjuela, B. Marx , J. Ragot , and D. Maquin "Diagnosis of nonlinear systems by multiple models approach," 6th International Conference Francophone of Automatic, CIFA, 2010.

[21] M. Chadli, A. Abdo, and S. X. Ding, "H−/H∞ fault detection filter design for discrete-time Takagi–Sugeno fuzzy system.," Automatica, vol. 49, pp. 1996–2005, 2013.

[22] B. Marx, D. Koenig, and J. Ragot "Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis," IET Control Theory and Applications, vol. 1, pp. 1487–1495, 2007.

[23] M. Bouattour, M. Chadli, A. El Hajjaji, and M. Chaabane, "State and Faults Estimation for T-S Models and Application to Fault Diagnosis," 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, SAFEPROCESS, vol. 09, 2009.

[24] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory," Studies In Applied Mathematics, 1994.

[25] T. Youssef, M. Chadli, H. Karimi, and R. Wang, "Actuator and sensor faults estimation based on proportional integral observer for TS fuzzy model," Journal of the Franklin Institute, 2016.

[26] S. F. A. Latip, A. R. Husain, A. Sarayati, and S. H. A. Razak, "A review on fault-tolerant control for single-link flexible manipulator system," in Applied Mechanics and Materials, 2012, pp. 2389-2393.

[27] Q. Jia, W. Chen, Y. Zhang, and H. Li, "Fault reconstruction for Takagi–Sugeno fuzzy systems via learning observers," International Journal of Control, vol. 89, pp. 564-578, 2016.

[28] T. Youssef, M. Chadli, H. R. Karimi, and M. Zelmat, "Design of unknown inputs proportional integral observers for TS fuzzy models," Neurocomputing, vol. 123, pp. 156-165, 2014.