Robust observer based Fault Tolerant Tracking Control for T–S uncertain systems subject to sensor and actuator faults

Accepted Manuscript Robust observer based Fault Tolerant Tracking Control for T-S uncertain systems subject to sensor and actuator faults Salama Makni, Maha Bouattour, Ahmed El Hajjaji, Mohamed Chaabane PII: DOI: Reference:

S0019-0578(18)30457-9 https://doi.org/10.1016/j.isatra.2018.11.022 ISATRA 2970

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ISA Transactions

Received date : 16 May 2018 Revised date : 4 November 2018 Accepted date : 16 November 2018 Please cite this article as: S. Makni, M. Bouattour, A.E. Hajjaji et al. Robust observer based Fault Tolerant Tracking Control for T-S uncertain systems subject to sensor and actuator faults. ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.11.022 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Robust observer based Fault Tolerant Tracking Control for T-S uncertain systems subject to sensor and actuator faults Salama Makni STA laboratory, Sfax university, Gabes university, Gabes 6029, Tunisia Maha Bouattoura , Ahmed El Hajjajib , Mohamed Chaabanea b 33

a 3038,

STA laboratory, Sfax university, Tunisia Rue St Leu Amiens 80000, MIS laboratory, France

I Fully

documented templates are available in the elsarticle package on CTAN. author Email address: [email protected] (Ahmed El Hajjaji)

∗ Corresponding

Preprint submitted to Journal of ISA Transactions

May 15, 2018

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This paper addresses the fault estimation and the Fault Tolerant Tracking Control problems (FTTC) for a class of uncertain perturbed Takagi–Sugeno (T–S) fuzzy systems subject to simultaneously sensor and actuator faults. The main challenges in this study is that sensor faults, actuator faults, external disturbances and uncertainties are taken into simultaneous account in an unified framework. A robust adaptive observer is designed to estimate unmeasured system states and both sensor and actuator fault vectors simultaneously using descriptor approach and LMI formulation. The developed idea is, then, extended to Fault Tolerant Tracking Control design not only to stabilize the closed-loop system, but also to drive system to track reference model states despite the presence of actuator and sensor faults. To reduce the conservatism, all the design conditions are formulated in a convex optimization problem under LMI constraints. Illustration and comparison results are shown to prove the effectiveness of the proposed method.

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Robust observer based Fault Tolerant Tracking Control for T-S uncertain systems subject to sensor and actuator faults

Abstract This paper investigates the sensor and actuator fault (SAF) estimations and the fault tolerant tracking control (FTTC) problem for uncertain systems represented by TakagiSugeno (T-S) fuzzy model affected by Unknown Bounded Disturbance (UBD). A robust descriptor adaptive observer based FTTC is firstly designed to simultaneously estimate the unmeasurable system states and SAF vectors and then to track reference trajectories. Sufficient design conditions are developed in terms of Linear Matrix Inequalities (LM Is). The gains of both observer and fault tolerant controller are computed by solving a set of LMI constraints in single step. Robustness against external disturbances is analysed using the H∞ performance index to attenuate its effect on the tracking error for bounded reference inputs. Finally, simulation results are illustrated by considering three types of actuator faults. Moreover, two comparisons are presented to show the mentioned process effectiveness. Keywords: SAF, robust descriptor adaptive observer based FTTC, T-S fuzzy models, UBD, LM Is, comparisons.

1. Introduction Real-world systems are generally nonlinear. One way to address the observer or the controller design issues for nonlinear systems, consists of using the T-S Fuzzy models [1, 2, 3, 4, 5, 6]. Systems can be affected by an unknown input such sensor or actuator faults. So, active fault tolerant controls have been developed to compensate the fault effects and stabilize the whole states. In addition, driving the system output toward the desired trajectories, even in the presence of faults, is among the control objectives. Preprint submitted to Journal of ISA Transactions

October 23, 2018

However, there are few studies based on the T-S fuzzy model which consider both tracking control and fault estimation problems. Fault Detection and Isolation FDI problems have been studied to distinguish between faults, disturbances and uncertainty effects in [7]. Many techniques and approaches have been used for FDI problems. Among these techniques, we can mentioned adaptive observers [1, 8, 9], fuzzy descriptor approach [1, 7, 10, 11], sliding mode observers [12, 13] and linear matrix inequality LM I approach [14, 15, 16, 17]. In literature, the FTTC for T-S systems subject both SAF has not been fully investigated if we consider the works ([18, 19, 20, 21, 22] and references therein). Thus, in [19], a PI observer has been designed to estimate only the state and actuator faults for FTTC of doubly fed induction generators, but authors have considered the actuator fault as a bounded signal. In [18], an integral sliding mode control technique has been applied to compensate the actuator fault effect. Nevertheless, authors have considered a specific form of external disturbances and the chattering issue persists. Indeed, authors in [23] have considered the stabilization problem for nonlinear systems subject to actuator faults, disturbances and uncertainties where an adaptive controller has been designed to compensate the actuator fault effects. However, the sensor fault has not been considered. In [22], an higher order sliding mode observer has been designed to estimate only the sensor faults for FTTC of electrical vehicles. Whereas, this type of observer is more complex and the controller gains have been set beforehand. In addition, in [24], authors have designed a descriptor observer for the sensor fault estimation and a fault tolerant tracking control of vehicle lateral dynamics. But, the Lyapunov matrix has been considered in a specific form. Indeed, in [25], a fault tolerant control based on sliding mode observer has been developed for uncertain linear systems where only sensor faults have been considered. Compared with single fault problem, multiple fault ones are more general. Since in many practical systems, actuator and sensor faults may occur simultaneously. So, researchers should consider SAF under unified framework. Using the H∞ performance, an observer based fault tolerant control has been designed for systems affected by both SAF [26]. Moreover, authors in [27, 28] have developed a fault tolerant control for systems with SAF. Whereas, the tracking problem is not investigated in [26, 27, 28]. Furthermore, to stabilize closed loop systems and to track 2

desired trajectories, a SAF estimation and a FTTC design are needed (see [21, 26] for linear systems). The objective of this work is to design a robust observer to estimate both unmeasurable states and SAF and then a FTTC to track the desired trajectories for uncertain T-S systems in presence of SAF and external disturbances. Firstly, a robust adaptive observer is designed for state and SAF estimation for T-S fuzzy systems with structured uncertainties and norm bounded disturbances subject to time varying faults. Then, a FTTC is developed to compensate the actuator fault effect and not only to stabilize the closed-loop system, but also to ensure the tracking of reference model states. To reduce the tracking error for bounded reference input and to minimize the disturbance effects, the H∞ performance index is considered. To summarise the advantages of our work. First of all, we investigate the problem of both sensor and actuator fault estimation under one unified framework for uncertain nonlinear systems with external disturbances. Whereas, many researchers have considered only sensor faults [22, 24] or actuator faults [18, 23]. Then, we have studied the fault tolerant tracking control which is more complex and difficult than fault tolerant control without tracking as [22, 26, 29]. Moreover, sufficient design conditions of observer and controller gains able to track the desired trajectories in the presence of sensor and actuator faults and both disturbance and uncertainties, are formulated in an unified optimisation problem under LMI constraints which can be solved in single step. Whereas, in [29, 30, 31, 32] a two step solving algorithm is needed. Authors in [20, 30, 33, 34] assume that the actuator faults are constant. Whereas, in this work, we consider the time varying actuator faults. Moreover, in [17, 24, 35, 36], authors consider a diagonal Lyapunov matrix. Nevertheless, in this paper, we consider a full Lyapunov matrix to reduce the conservatism. Two comparisons with recent existing results are also given to prove the efficiency of our method. This paper is structured as follows. The system description is given in section 2. In section 3, an observer based FTTC is developed to track desired trajectories. In Section 4, we present simulation results to illustrate the proposed method where the effectiveness is shown via two comparisons. Finally, section 5 concludes the paper.

3

Notations: In this paper, we note that P T and P −1 are respectively the transpose and the inverse of matrix P . sym(P ) = P + P T . In a matrix, ∗ denotes the transposed

element in the symmetric position. 2. System description

Let’s consider the following fault free fuzzy model as a reference model : x(t) ˙ =

r X

µi (ξ(t))(Ai x(t) + Bi u(t))

i=1

(1)

y(t) = Cx(t) where x(t) ∈ Rn , u(t) ∈ Rm and y(t) ∈ Rp represent state vector, input vector and

the output vector, respectively. r is the number of submodels and ξ(t) is the premise

variable vector assumed measurable. Ai , Bi and C are constant real matrices and µi are the weighting functions which satisfy : r X

µi (ξ(t)) = 1 and µi (ξ(t)) > 0 f or i = 1, . . . , r.

(2)

i=1

Generally speaking, engineering systems can be affected by SAF, uncertainties and UBD. So, we should consider these malfunctions in one unified framework. Consequently, we define the next faulty system : x˙ f (t) =

r P

˜i uf (t)) + Da fa (t) + Bd d(t) µi (ξ(t))(A˜i xf (t) + B

i=1

(3)

yf (t) = Cxf (t) + Ds fs (t) ˜i = Bi + ∆Bi , fs (t) ∈ Rs , fa (t) ∈ Ra and d(t) ∈ Rnd where A˜i = Ai + ∆Ai , B

represent sensor fault, actuator fault and UBD vectors, respectively. Bd , Da and Ds

are constant real matrices. We note that Ds is a full rank column. It is proposed that the derivative of actuator fault vector is norm bounded with respect to time by a finite value fm . ∆Ai and ∆Bi are the structured uncertainty matrices in the following form   h i h i Fa (t) h i  Eai Ebi ∆Ai ∆Bi = Mai Mbi  Fb (t) 4

where Mai , Eai , Mbi and Ebi are real known constant matrices of appropriate dimensions, Fa (t) ∈ Rl and Fb (t) ∈ Rl are time varying matrix functions satisfying: Fa (t)T Fa (t) ≤ I, Fb (t)T Fb (t) ≤ I, ∀t In the following section, we propose an observer based fault tolerant tracking control not only to estimate state and SAF but also to track the desired trajectories. Assumption 1

Bi is a column full rank matrix and C is a row full rank matrix

i = 1..r. Assumption 2 In this work, we assume that rank(CBi ) = nu , i = 1..r These latter are reasonable assumptions for the control system. Three Lemmas are used in this paper. Lemma 1 [37] For matrices X, Y and P = P T > 0, we have the next inequality : X T Y + Y T X 6 X T P X + Y T P −1 Y Lemma 2 [38] Given matrices M , E and F (t) with appropriate dimensions such that F (t)T F (t) ≤ I. Then we have, for any  > 0 : M F (t)E + E T F (t)T M T ≤ M M T + −1 E T E Lemma 3 [39] Given matrix X = X T and matrix Π < 0 where X T ΠX < 0, then there exists a positive scalar ν such that : X T ΠX 6 −2νX − ν 2 Π−1 Remark 1.

In [17], authors have designed only an adaptive observer for the estima-

tion of sensor and actuator faults. Whereas, in this work, we have designed an observer based fault tolerant tracking control to both estimate sensor and actuator fault and drive state to desired trajectories in the presence of uncertainties and using the H∞ performance. Indeed, we have used a full Lyapunov function to reduce the conservatism.

5

3. Observer based Fault Tolerant Tracking Control To estimate unmeasurable systems states and sensor fault vectors simultaneously, we consider the descriptor approach and the augmented state system. Then, we develop an algorithm for the actuator fault estimation. For this, equation (3) becomes : r P ¯x ¯i uf (t)) + D ¯ a fa (t) + B ¯d d(t) + N xs (t) E ¯˙ f (t) = µi (ξ(t))(A¯i x ¯f (t) + B i=1

yf (t) = C¯ x ¯f (t) = C0 x ¯f (t) + xs (t)

(4)

where 

In

0

0

0

h

C

Ip 

¯= E

C¯ =



N =

0 Ip





 , A¯i =  i

, C0 = 

¯d =  , B 

and xs (t) = Ds fs (t), x ¯f (t) = 

h

0

C 

Bd 0

A˜i

xf (t)

0 −Ip i 0 ,





¯i =  , B 

¯a =  , D

Da 0



˜i B 0



,

(5)



.

 xs (t) xs (t) were supposed as an auxiliary state of (4). After that, we design the following fuzzy adaptive descriptor observer : E z(t) ˙ =

r X

¯1i uf (t)) + D ¯ a fˆa (t) µi (ξ(t))(Fi z(t) + B

i=1

ˆ¯f (t) = z(t) + Lyf (t) x fˆs (t) = (DsT Ds )−1 DsT

h

0

Ip

˙ ¯ e¯˙ x (t) + e¯x (t)) fˆa (t) = αΓC( f f

i

(6) ˆ x ¯f (t)

ˆ¯f (t) is the estimated vector of (4) and z(t) is an auxiliary state vector with where x h iT ¯1i = B T 0 B . i

ˆ We denote e¯xf (t) = x ¯f (t) − x ¯f (t), α is a symmetric positive definite learning rate matrix and E, Fi , L and Γ are the gains to be determined.

6

Now, we define the next fault tolerant tracking control : uf (t) =u(t) +

r X j=1

=u(t) +

r X j=1

µj (ξ(t))Kj (ˆ xf (t) − x(t)) − fˆa (t)

(7)

¯ j (x ˆ µj (ξ(t))K ¯f (t) − x ¯(t)) − fˆa (t)

¯ j = [Kj 0]. Kj , (j=1..r), are the controller gains to be where x ¯(t) = [x(t)T 0]T and K determined. Remark 2.

Observer (6) based fault tolerant tracking control (7) has been designed

for nonlinear uncertain system. First of all, the adaptive descriptor observer can estimate both state and faults accurately. After that, a control law that accommodates of the actuator fault effect, is proposed to ensure the tracking of the desired reference trajectories. The block scheme of FTTC system is given in Fig.1. The estimation of both actuator faults and system states are required for control law (7). An adaptive descriptor observer (6) is designed to estimate these vectors.

Figure 1: Tracking Fault Tolerant Controller design scheme

Remark 3.

We have investigated the problem of nonlinear systems which are

affected by both sensor and actuator faults simultaneously. Whereas, many researchers have considered only sensor faults [22, 24] or actuator faults [18, 23]. 7

Replacing (7) in (3), we obtain : r P r P

x˙ f (t) =

˜i u(t) + Bd d(t) µi (ξ(t))µj (ξ(t))(A˜i xf (t) + B

i=1 j=1

(8)

˜i Kj x ˜i Kj x(t) + Da fa (t) − B ˜i fˆa (t)) +B ˆf (t) − B Knowing that efa (t) = fa (t) − fˆa (t) is the actuator fault estimation error, equation (8)

can be rewritten as follows : x˙ f (t) =

r P r P

˜i Kj )xf (t) µi (ξ(t))µj (ξ(t)) (A˜i + B

i=1 j=1

˜ +B ˜i Kj x(t) − B ˜i K ¯ j e¯x (t) + B ˜di d(t) ˜i ef (t) −B a f




(9)

˜ = [d(t)T fa (t)T u(t)T ]T ˜di = [Bd (Da − B ˜i ) B ˜i ] and d(t) such that B The tracking error is defined as follows: et (t) = xf (t) − x(t)

Then, from (1) and (9), the tracking error dynamic can be formulated as follows : e˙ t (t) =

r P r P

˜i Kj )et (t) µi (ξ(t))µj (ξ(t)) (Ai + B

i=1 j=1


˜ ˜i K ¯ j e¯x (t) + B ˜i ef (t) + B ‡ d(t) +∆Ai xf (t) − B a f i

(10)

˜i ) ∆Bi ]. where Bi‡ = [Bd (Da − B After that, we have :

ˆ¯f (t) − LC¯ x ˆ z(t) = x ¯f (t) = x ¯f (t) − LC0 x ¯f (t) − Lxs (t)

(11)

By substituting (11) and (7) into equation (6), we get : ˆ¯˙ f (t) − ELC¯ x Ex ¯˙ f (t) =

r P r P

i=1 j=1

ˆ µi (ξ(t))µj (ξ(t)) Fi (x ¯f (t) − LC0 x ¯f (t) − Lxs (t))


¯1i K ¯jx ¯1i u(t) − B ¯1i K ¯jx ¯1i fˆa (t) + D ¯ a fˆa (t) ˆ +B ¯f (t) + B ¯(t) − B (12)

Subtracting (12) from (4) yields : r P r P ¯ + ELC) ¯ x ˆ¯˙ f (t) = (E ¯˙ f (t) − E x µi (ξ(t))µj (ξ(t)) (A¯1i + Fi LC0 )¯ xf (t) i=1 j=1

¯2i K ¯jx ˆ ˆ −Fi x ¯f (t) + (N + Fi L)xs (t) + A¯2i xf (t) + B ¯f (t)
¯2i K ¯jx ¯2i fˆa (t) + B ¯2i u(t) + D ¯ a ef (t) + B ¯d d(t) −B ¯(t) − B a (13)

8

where



A¯1i = 

Ai

0



, A¯2i =

h

0 −Ip Equation (13) can be rewritten as follows : E e¯˙ xf (t) =

r P r P

i=1 j=1

∆ATi

0

iT

¯2i = and B

h

∆BiT

0

iT

¯2i K ¯ j )¯ ¯2i Kj et (t) + A¯2i x µi (ξ(t))µj (ξ(t)) (Fi − B exf (t) + B ¯f (t)

˜ ¯a + B ¯2i )ef (t) + B † d(t) +(D a i




(14)

such that :

We note : Bi† =

h

¯d B

¯2i −B

¯ + ELC¯ E=E

(15)

Fi = A¯1i + Fi LC0

(16)

N = −Fi L. i

(17)

¯2i B

By substituting (17) into (16), we get :  Fi = 

Ai

0

−C

−Ip



.

(18)

From equations (18) and (17), we have :         0 −Ai 0 L1 −Ai L1  =   =  Ip C Ip L2 CL1 + L2 By identification, we get :



L=

0 Ip

 

(19)

Based on [40] and to satisfy condition (15), a solution is given as follows :   I 0  E= HC H

(20)

where H ∈ Rp is a full rank matrix which is non singular and will be calculated. Then, we have :



E −1 = 

I

0

−C

H −1

9

 

(21)

So, error dynamic (14) can be rewritten as follows : e¯˙ xf (t) =

r P r P

i=1 j=1

˜ ¯ i xf (t) + G ¯ i efa (t) + J¯i d(t)) µi (ξ(t))µj (ξ(t))(S¯ij e¯xf (t) + T¯ij et (t) + R (22)

such that :



¯j) =  ¯2i K S¯ij = E −1 (Fi − B

Ai − ∆Bi Kj

0

−CAi + C∆Bi Kj − H −1 C 



∆Bi Kj

¯2i Kj =  T¯ij = E −1 B

 −C∆Bi Kj   ∆Ai −1 ¯ i = E A¯2i =   R −C∆Ai  Da + ∆Bi −1 ¯ i = E (D ¯a + B ¯2i ) =  G −CDa − C∆Bi  Bd −∆Bi ∆Bi J¯i = E −1 Bi† =  −CBd C∆Bi −C∆Bi

The error dynamic of actuator fault is given as follows :

such that

−H −1

 

(23)

(24)

(25)  

(26)



(27)




˙ ¯ aT P e¯˙ x (t) + e¯x (t) e˙ fa (t) = f˙a (t) − fˆa (t) = f˙a (t) − αD f f

(28)

(29)

¯ aT P = ΓC¯ D We define P as a Lyapunov matrix with the following form   P1 P2  P = P2T P3

(30)

Define the next vector v as : v(t) = Z

h

et (t)T

xf (t)T

e¯xf (t)T

efa (t)T

iT

(31)

where Z = diag(Z1 , Z2 , Z3 , Z4 ) is a pre-specified weight matrix with appropriate dimension with Z3 = diag(Z31 , Z32 ). Theorem 1.

For positive scalars 1 , 2 and α, observer (6) based control law (7) en-

sures the tracking trajectories and fulfils the H∞ performance, if there exist symmetric 10

matrices P1 > 0, P3 > 0, Q1 > 0, Q2 > 0 and R > 0, invertible matrix W , matrices P2 , Γ and Kj such that the following optimization problem is hold minimize σ

subject to

Πii < 0

for i = 1 . . . r

2 Πii + Πij + Πji < 0 r−1

for j = 1 . . . r

(32)

(33)

if i 6= j (34)

¯ aT P = ΓC¯ D where



Πij =  



Ω2i

     =    

Ψ†ij

∗

Ω3ij

Ψ7

  Ψ†ij =  Ψ2i  Ψ4ij

Ψ‡11i

∗

Ω2i

∗

Ψ3 Ψ5 ∗

 

∗



  ∗   Ψ6 ∗

∗

∗

∗

∗

∗

0

Ψ‡22

0

0

Ψ‡33

0

0

Ψ‡43

Ψ‡44

BiT Q1

BiT Q2

Ψ‡53

−DaT C T P3

∗

Ψ11i ‡ = sym(ATi Q1 ) + Z1T Z1 Ψ22i ‡ = sym(ATi Q2 ) + Z2T Z2

T Ψ33 ‡ = −sym(P2 C) + Z31 Z31

Ψ43i ‡ = P2T Ai − P3 CAi

T Ψ44 ‡ = −sym(P3 ) + Z32 Z32

Ψ53 ‡ = DaT C T P2T

11

∗ Ψ55 ‡

          

Ψ55 ‡ = −sym(DaT P2 CDa ) + R + Z4T Z4  BdT Q1 BdT Q2   Ψ2i =  DaT Q1 − BiT Q1 DaT Q2 − BiT Q2  0 BiT Q2 Ψ3 = −diag(σI, σI, σI), σ = γ 2  0 0 P2T    0 0 −W C + C    0 0 0 Ψ4ij =    0 0 −W C    Q1 Q2 0  Bi Kj 0 −Bi Kj Ψ5 = zeros(6, 3) Ψ6 = −diag(I, I, I, I, I, I) h i Ω3ij = Ω4ij Ω5 Ω4ij



0    MT Q  1 ai 1 =  E Kj bi  TQ 2 Mbi 1

Ω5 = zeros(4, 6)

0

0

BdT P2 − BdT C T P3

0

0

0

0

−P2T Da

0



  0   0



  −CDa     0    0    0  0

−W P3 −W + I 0 0

Eai

0

0

0

0

0

0

TQ 1 Mai 2

0

T P − M T CT P Mai 2 3 ai

0

0

0

0

0

−Ebi Kj

0

Ebi

0

−Ebi

Ebi

T P − M T CT P Mbi 2 3 bi

0

0

0

0

TQ 2 Mbi 2

0

      

Ψ7 = −diag(1 I, 1 I, 2 I, 2 I)

Then, adaptive observer (6) can realize xf (t), et (t), e¯xf (t) and efa (t) uniformly bounded. α is a symmetric positive definite learning rate matrix and Z1 , Z2 , Z3 and Z4 are prespecified weight matrices. Proof. We consider the Lyapunov function below : V (t) = e¯xf (t)T P e¯xf (t) + et (t)T Q1 et (t) + xf (t)T Q2 xf (t) + efa (t)T α−1 efa (t), where P , Q1 and Q2 are symmetric positive matrices, such that the following inequality ˜ <0 V˙ (t) + v(t)T v(t) − γ 2 d˜T (t)d(t)

12

(35)

is satisfied. ˜ = sym(e¯˙ x (t)T P e¯x (t)) + sym(e˙ t (t)T Q1 et (t)) V˙ (t) + v(t)T v(t) − γ 2 d˜T (t)d(t) f f +sym(e˙ fa (t)T α−1 efa (t)) + sym(x˙ f (t)T Q2 xf (t)) ˜ +v(t)T v(t) − γ 2 d˜T (t)d(t) from Lemma 1, for positive matrix R = RT , we obtain the following inequality :
T efa (t)T α−1 f˙a (t) + efa (t)T α−1 f˙a (t) 6 efa (t)T Refa (t) + δ 2 where δ = fm λmax (α−1 R−1 α−1 ).

Using equations (9), (10), (22), (28) and taking into consideration equation (31), we get : ˜ 6 V˙ (t) + v(t)T v(t) − γ 2 d˜T (t)d(t) with

Π1ij





   =  

Πij = 

r X r X

µi (ξ(t))µj (ξ(t))θ(t)T Πij θ(t) + δ

i=1 j=1

(36)

Π1ij

∗

Π2i

Π3

T ˜T T sym(AT i Q1 + Kj Bi Q1 ) + Z1 Z1



<0

∗

˜ ∆AT i Q1 + Q2 Bi Kj

˜T Q2 ) + diag(Z T Z2 ) sym(A 2 i

¯TB ˜ T Q1 + P T¯ij −K j i

¯TB ˜ T Q2 + P R ¯i −K j i

˜ T Q1 B i

−

Π2i =

˜ T Q2 − D ¯T PR ¯i B a i

¯ T P T¯ij D a

h

Bi‡ T Q1

˜ T Q2 B di

J¯iT P

Π3 = −γ 2 I

¯a −J¯iT P D

i

∗

∗

∗

∗

π33

∗

π43

π44

      

and T π33 = sym(S¯ij P ) + diag(Z3T Z3 )

¯ aT P + G ¯T P − D ¯ aT P S¯ij π43 = −D i

¯ aT P G ¯ i ) + R + Z T Z4 π44 = −sym(D 4 h iT ˜ T θ(t) = et (t)T xTf (t) e¯xf (t)T efa (t)T d(t) ¯ a , P and C¯ by their parameters in (29), we get P1 = P2 C. Then, replacing Replacing D ¯i, G ¯ i , J¯i and P by their parameters (23), (24), (25), (26), (27) and (30) we S¯ij , T¯ij , R obtain : Πij = Ψ†ij + Φ†ij 13

(37)

where



Ψ†ij =  

Ψ1ij

Ψ1ij

∗

Ψ2i

Ψ3



Ψ11ij

   Q2 Bi Kj   =  −KjT BiT Q1    0  BiT Q1 

 ∗

∗

∗

∗

Ψ22 −KjT BiT Q2

∗

∗

∗

Ψ33

∗

0

Ψ43i

T sym(−P3 H −1 ) + Z32 Z32

∗

BiT Q2

Ψ53

Ψ54

BdT Q2

0

DaT Q2 − BiT Q2

BdT P2 − BdT C T P3

0

0

0

0

BdT Q1

  Ψ2i =  DaT Q1 − BiT Q1  0

BiT Q2

Ψ3 = −diag(σI, σI, σI)

∗

          

Ψ55 (38)

0



  0  (39)  0 (40)

Ψ11ij = sym(ATi Q1 + KjT BiT Q1 ) + Z1T Z1 Ψ22i = sym(ATi Q2 ) + Z2T Z2 T Ψ33 = sym(−C T H −T P2T ) + Z31 Z31

Ψ43i = −H −T P2T + P2T Ai − P3 CAi − P3 H −1 C Ψ53 = −DaT C T P2T + DaT P2 H −1 C

Ψ55 = R + Z4T Z4

Ψ54 = −DaT C T P3 + DaT P2 H −1

and σ = γ 2 Then

Φ†ij = sym(X4 F T Y4 ) + sym(X5 F T Y5 ) such that h iT X4 = 0 Eai 0 0 0 0 0 0 h iT X5 = Ebi Kj 0 −Ebi Kj 0 Ebi 0 −Ebi Ebi h i T T Y4 = Mai Q1 Mai Q2 0 Y44 0 0 0 0 h i T T Y5 = Mbi Q1 Mbi Q2 0 Y54 0 0 0 0 14

(41)

where T T T Y44 = Mai P2 − Mai C P3 T T T Y54 = Mbi P2 − Mbi C P3

We define :



0

∗

   0 0   Λ= 0 0    0 0  0 0



∗

∗

∗

∗

∗

∗

P2 C

0

−P2 CDa

0

P3

0

−DaT P2 C

0

DaT P2 CDa

         

Adding and subtracting sym(Λ) to equation (38), Ψ1ij can be rewritten as follows (42)

Ψ1ij = Ω1ij + Ω2i Ω1ij = sym(X1 Y1 ) + sym(X2 Y2 ) + sym(X3 Y3 ) 

Ω2i

∗

∗

∗

∗

∗

∗

∗

∗

∗

0

Ψ‡22

0

0

Ψ‡33

0

0

Ψ‡43

Ψ‡44

BiT Q1

BiT Q2

Ψ‡53

−DaT C T P3

iT −P2T Da h iT X2 = 0 0 0 P3 0 h iT X3 = Q1 Q2 0 0 0 h i Y1 = 0 0 (−H −1 C + C) −H −1 −CDa h i Y2 = 0 0 −H −1 C (−H −1 + I) 0 h i Y3 = Bi Kj 0 −Bi Kj 0 0 X1 =

h

     =    

Ψ‡11i

0

0

P2T

0

Ψ11i ‡ = sym(ATi Q1 ) + Z1T Z1 Ψ22i ‡ = sym(ATi Q2 ) + Z2T Z2

T Ψ33 ‡ = −sym(P2 C) + Z31 Z31

Ψ43i ‡ = P2T Ai − P3 CAi

T Ψ44 ‡ = −sym(P3 ) + Z32 Z32

15

∗ Ψ55 ‡

          

Ψ53 ‡ = DaT C T P2T

Ψ55 ‡ = −sym(DaT P2 CDa ) + R + Z4T Z4

Then, applying Lemma 2 into equation (41) and Schur complement, we get : 

Πij =  



Ψ4ij

      =      

∗

Ω3ij

Ψ7

Ω2i

  Ψ†ij =  Ψ2i  Ψ4ij

∗ Ψ3 Ψ5

P2T

0

0

−W C + C

−W

0

0

0

P3

0

0

−W C

−W + I

Q1

Q2

0

0

0

0

0

Bi Kj 0 −Bi Kj Ψ5 = zeros(6, 3) Ψ6 = −diag(I, I, I, I, I, I) W = Hh−1 i Ω3ij = Ω4ij Ω5 Ω4ij

Ψ†ij



0    MT Q  1 ai 1 =  E Kj bi  TQ 2 Mbi 1

0

 

∗

(43) 

  ∗   Ψ6

−P2T Da



  −CDa     0    0    0  0

Eai

0

0

0

0

0

0

TQ 1 Mai 2

0

T P − M T CT P Mai 2 3 ai

0

0

0

0

0

−Ebi Kj

0

Ebi

0

−Ebi

Ebi

T P − M T CT P Mbi 2 3 bi

0

0

0

0

TQ 2 Mbi 2

0

Ω5 = zeros(4, 6)

      

Ψ7 = −diag(1 I, 1 I, 2 I, 2 I)

Applying Lemma 3, if (32) and (33) hold, then there exists a positive scalar τ such that ˜ < −τ kθ(t)k2 + δ. V˙ (t) + v(t)T v(t) − γ 2 d˜T (t)d(t)

It follows that V˙ (t) < 0 for τ kθ(t)k2 > δ, which means that θ(t) converges to a small set ℵ = {θ(t)|kθ(t)k2 6

δ τ}

according to Lyapunov stability theory. Therefore,

estimation errors of both state and faults are uniformly ultimately bounded. End Proof  Remark 4.

In [41], authors are interested to sensor fault estimation for uncertain 16

nonlinear systems. Whereas, in this paper, the problem of both sensor and actuator faults estimation is discussed for nonlinear systems with uncertainties and disturbances. Remark 5.

In [17, 35], a diagonal Lyapunov matrix has been proposed and in [42],

almost of the off diagonal elements of the Lyapunov matrix have been assumed to be zero. These methods may be restrictive. However, in our work, the Lyapunov matrix is supposed to be full to increase the slack variables and thus to obtain less conservative conditions. Remark 6.

Authors in [20, 30] have considered constant actuator fault. Whereas, in

this paper, we have considered time varying actuator fault signals. Remark 7.

In [29, 30, 32, 43], the computation of the observer gains and the

controller gains is done separately. However, in this work, the resolution of LM Is is done only in one step which is less restrictive. 4. Simulation results In this part, we illustrate the validity of the observer based fault tolerant control in trajectory tracking context. For this, three types of actuator faults have been considered to show the effectiveness of our method. Moreover, regarding the benefit of our method to the method proposed in [35] in term of conservatism reduction will be compared. Furthermore, an other comparison with the method developed in [42] will be presented to prove that our method is more general and upgrade the results given in [42]. So, let’s consider  the next theoretical T-S system  in the form of (3)  with parameters : 0 −1 −0.5 −2 1 1         A1 =  −1 −3 2.5  , A2 =  0 −3 3      1 1 −2 0.5 1 −3       1 1 0             B1 =  1  , B2 =  0.5  , Bd =  0.1        0.8 1 0   0.6   h i   C = 1 0 0.5 , Ds = 1, Da =  0    0 17



0.1

0.3

0

0.15

0.2





0.1

0.2

0.2



      0 , Ma2 =  0.1 0.25 0.1     0.1 0.2 0.15 0.15 0.1 0.2     0.15 0 0 0         Ea1 =  0 −0.15 0.1 , Mb1 =  0.25      0.2 0.15 0 −0.2     0.2 −0.15 0 0         Ea2 =  0 −0.15 0 , Mb2 =  0.25      0.2 0 0 −0.1 Eb1 = 0.2, Eb2 = 0.1   Ma1 =  

Fa (t) = 0.3(sin(t))2 , Fb (t) = 0.1cos(t)sin(t) The two weighting functions are given as follows : µ1 (t) =

(1−tanh(0.5−sin(cos(0.5t)t))) 2

µ2 (t) = 1 − µ1 (t).

d(t) = exp(−0.8t) and u(t) = 0.15sin(1.5t)

Applying Theorem 1, for Z1 = Z2 = 0.01I3 , Z31 = 0.0001I3 , Z32 = 0.1I2 and Z4 =0.1, we minimize σ to 0.1588 and  we get the next feasible solutions 1 0 0 0      0 1 0 0    E=   0 0 1 0    0.5496 0 0.2748 0.5496     0 −1 −0.5 0 −2 1 1 0          −1 −3 2.5  0 −3 0  3 0      F1 =  , F2 =    1  0.5 1 1 −2 0  −3 0      −1 0 −0.5 −1 −1 0 −0.5 −1 h iT L= 0 0 0 1 , Γ = 0.4999 h i K1 = −0.0073 0 −0.0037 h i K2 = 10−3 0.2716 0.0009 0.1369

18

4.1. Exponential actuator fault case The sensor fault have the following structure   1.5sin(t − 1), 5 ≤ t < 12 fs (t) =  0, else and the considered actuator fault affecting the system at 8 ≤ t < 20 is given by   5.2(1 − 1.5sin(t − 8))exp(−0.5t), 8≤ t <20 fa (t) =  0, else Applying Theorem 1, observer (6) based control law (7), for α = 600, not only esti-

mates and converges the state to the desired trajectories but also estimate SAF vectors. In Fig 2., we present the faulty system states, their estimates and reference model states. The fault tolerant tracking control signal and the input vector of the nominal system are given in Fig 3. Sensor fault signal fs (t) and its estimate is given in Fig 4. We present in Fig 5. the signal of actuator fault fa (t) and its estimate. These figures illustrate clearly that the mentioned observer is able to estimate both SAF accurately. Despite the presence of disturbance and norm bounded uncertainties, we get a satisfied estimation quality. Moreover, we distinguish three important parts. In the first one, at 5≤ t <8, we have applied only sensor fault. Whereas, at 8≤ t <12, both SAF have been considered and in the third part, at 12≤ t <20, we have applied the actuator fault. We remark that even in the presence of both SAF, the proposed observer can estimate unmeasurable system states and faults accurately and the controller can compensate the actuator fault effects and ensures the tracking between the reference model and the faulty system states.

19

0.2

x1f x1r x ˆ1f

0 −0.2

0

5

10

15

20

25

0.2

x2f x2r x ˆ2f

0 −0.2

0

5

10

15

20

25

0.2

x3f x3r x ˆ3f

0 −0.2

0

5

10

15

20

25

Time (s)

Figure 2: Exponential actuator fault: Faulty system states, their estimates and reference model states

0.2

u uf

0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 −0.3

0

5

10

15

20

25

Time (s)

Figure 3: Exponential actuator fault: Nominal and FTTC signals

2

fs fˆs

1.5

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

Time (s)

Figure 4: Exponential actuator fault: Sensor fault fs and its estimate

20

25

0.2

fa fˆa 0.15

0.1

0.05

0

−0.05

0

5

10

15

20

25

Time (s)

Figure 5: Exponential actuator fault: Actuator fault fa and its estimate

4.2. Time varying actuator fault case The actuator fault occurring when 14 ≤ t < 25 is defined by   0.007 + (0.005/25)t, 14≤ t <25 fa (t) =  0, else And the  sensor fault have the following form  2sin(t − 10), 10 ≤ t < 20 fs (t) =  0, else The simulation results are given by Figs 6-9.

Even if the actuator fault signal is considered as a time varying one, we obtain a good estimation quality and the controller compensate the actuator fault effect and guarantee the state convergence to desired trajectories. 0.2

x1f x1r x ˆ1f

0 −0.2

0

5

10

15

20

25

0.2

x2f x2r x ˆ2f

0 −0.2

0

5

10

15

20

25

0.2

x3f x3r x ˆ3f

0 −0.2

0

5

10

15

20

25

Time (s)

Figure 6: Time varying actuator fault: Faulty system states, their estimates and reference model states

21

0.2

u uf

0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2

0

5

10

15

20

25

Time (s)

Figure 7: Time varying actuator fault: Nominal and FTTC signals

2.5

fs fˆs

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2

0

5

10

15

20

25

Time (s)

Figure 8: Time varying actuator fault: Sensor fault fs and its estimate

0.08

fa fˆa

0.06

0.04

0.02

0

−0.02

−0.04

0

5

10

15

20

25

Time (s)

Figure 9: Time varying actuator fault: Actuator fault fa and its estimate

22

4.3. Conservatism comparison In order to show the benefits of our method, we have compared our method with the proposed one in [35] in terms of conservatism reduction. For this, we study the feasibility [0.7 1.2] as follows :  b ∈  0.7] and  fields for a ∈ [−0.1 b a −1 −0.5         A1 =  −1 −3 2.5 , B1 =  1      0.8 1 1 −2 1.2

1.1

b

1

0.9

0.8

0.7 −0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

a

Figure 10: Feasibility fields: [35] in (red-triangle), Theorem 1 in (blue-circle)

The feasibility fields are indicated as follows : the red triangle for the proposed method in [35] and the blue circle for the suggested method in this paper. In Fig 10, we show the feasibility fields via applying Theorem 1 and Theorem given in [35]. As illustrated in Fig 10, the controller can not be synthesized for a = 0.04 and b = 0.9 from Theorem in [35] when a solution exists with Theorem 1 and the observer and controller gains are given as follows   :    E=   

1

0

0

0

0

1

0

0

     0 0 1 0   0.5480 0 0.2740 0.5480    0.04 −1 −0.5 0 −2        −1 −3 2.5  0 0  , F2 =  F1 =      1  0.5 1 −2 0     −1 0 −0.5 −1 −1 23

1

1

−3

3

1

−3

0

−0.5

0



  0    0   −1

L=

h

K1 =

0 h

0

0

1

iT

, Γ = 0.4993 i 0 −0.0028

−0.0056 h K2 = 10−3 0.0077

0.0002 0.0042

i

Fig 10 shows that our feasibility field is more important than this obtained using method of [35]. We can conclude that our method lead to relaxed and less conservative results. Remark 8.

Our objective is to find more relaxed and less conservative results.

For this, we propose to compare our method with the presented approach in [42]. The Lyapunov matrix in [42] had a special structure where almost of the off diagonal elements are fixed to zeros. However, in our work, we present a general form of the Lyapunov matrix. Furthermore, we propose a full matrix and all the LM Is elements have been computed in one step. As a result, if we apply Theorem 1 in [42], we find infeasible LM Is. Whereas, via applying Theorem 1 in our paper, we have obtained feasible LM Is. Moreover, we have obtained a good estimation quality of state and both sensor and actuator fault. Indeed, the fault tolerant control track state to the desired trajectories. As conclusion, we prove the efficiency of our method and we can conclude that our method upgrade the result given in [42]. 4.4. Comparison results To illustrate the benefits of our method, we have compared our approach with the proposed one in [35]. For this, we consider the previous example with the following sensor and  actuator faults  0.015sin(1.5t), 8≤ t <18 fa (t) =  0, else   1.5sin(t − 1), 5 ≤ t < 12 fs (t) =  0, else Solving LM I (55) in [35], we get the following observer and control gains: h iT L11 = 282.0704 152.1386 183.7622 , h iT , L12 = 269.416 145.6937 175.8247

L21 = 205.9129, L22 = 203.1795 h i h K1 = 0.0130 −0.0620 0.0412 , K2 = −0.0563

K1f = 0.0054, K2f = −0.0074

24

−0.0010 0.0743

i

We carry out the simulation to compare our fault estimation and fault tolerant tracking control method with the method in [35]. Authors in [35] have considered the same fault vector f (t) for both sensors and actuators. Whereas, we have considered two different fault vectors. Indeed, in practical applications, sensor faults are often different from actuator faults. The simulation results are illustrated in Figs 11-13. We show in Fig 13 the sensor fault signal fs (t) and its estimate. We can see that our observer is able to estimate sensor fault vector accurately. Figs 11 and 12 show the comparison results of tracking, state and actuator fault estimation. From Fig 12, it is worth pointing out that our observer can estimate the actuator fault vector better than the proposed observer in [35]. Noting that the fault tolerant tracking control depend on actuator fault estimation, Fig 11 shows that our control law is not only able to compensate the actuator fault effect but also ensures without any problem the tracking of the desired trajectories. However, by using the method of [35], the actuator fault effect is more important and the control law can not drive faulty state to track the nominal one in presence of faults. 0.2

x1f x ˆ 1f x1f [37] x1r

0 −0.2

0

5

10

15

20

0.2 0 −0.2

0

5

10

15

20

0.2

25

x3f x ˆ 3f x3 [37] x3r

0 −0.2

25

x2f x ˆ 2f x2f [37] x2r

0

5

10

15

20

25

Time (s)

Figure 11: Faulty system states of our method, their estimates, Faulty system states of [35] and reference model states

25

0.08

fa fˆa fˆa [37]

0.06

0.04

0.02

0

−0.02

−0.04

0

5

10

15

20

25

Time (s)

Figure 12: The comparison of actuator fault estimation

2

fs fˆs

1.5 1 0.5 0 −0.5 −1 −1.5 −2

0

5

10

15

20

25

Time (s)

Figure 13: Sensor fault fs and its estimate

5. Conclusion In this work, a robust adaptive descriptor observer based fault tolerant tracking control is designed for T-S fuzzy nonlinear systems subject to SAF and uncertainties. The mentioned observer can estimate unmeasurable system states and SAF vectors simultaneously. Moreover, the developed control law ensures the tracking of the desired trajectories in spite the presence of SAF, uncertainties and external disturbances. The tracking conditions are formulated in terms of LM Is using an appropriate Lyapunov function and the H∞ performance index. Finally, simulation results are given to show the efficiency of the proposed method. In addition, two comparisons with recent works are given to prove that our method is efficient, more general and less conservative.

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References [1] S. Makni, M. Bouattour, A. E. Hajjaji, M. Chaabane, H_/H∞ observer design for nonlinear T-S fuzzy systems subject to sensor and actuator faults, in: 2017 18th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA), 2017, pp. 296–301. [2] A. Benzaouia, A. E. Hajjaji, Advanced Takagi–Sugeno Fuzzy Systems, Springer International Publishing, 2014. [3] S. Li, H. Wang, A. Aitouche, Y. Tian, N. Christov, Actuator fault and disturbance estimation using the T-S fuzzy model, IFAC-PapersOnLine 50 (1) (2017) 15722 – 15727, 20th IFAC World Congress. [4] K. Tanaka, H. O. Wang, Fuzzy control systems design and analysis, A Linear Matrix Inequality Approach., Wiley Interscience, 2002. [5] K. Sun, S. Sui, S. Tong, Fuzzy adaptive decentralized optimal control for strict feedback nonlinear large-scale systems, IEEE Transactions on Cybernetics 48 (4) (2018) 1326–1339. [6] S. Tong, K. Sun, S. Sui, Observer-based adaptive fuzzy decentralized optimal control design for strict-feedback nonlinear large-scale systems, IEEE Transactions on Fuzzy Systems 26 (2) (2018) 569–584. [7] Y. Lu, Y. Song, Descriptor observer based approach for adaptive reconstruction of measurement noises and sensor faults in uncertain nonlinear systems, in: The 27th Chinese Control and Decision Conference (2015 CCDC), 2015, pp. 2764– 2769. [8] B. Jiang, M. Staroswiecki, V. Cocquempot, Fault accommodation for nonlinear dynamic systems, IEEE Transactions on Automatic Control 51 (9) (2006) 1578– 1583. [9] A. Xu, Q. Zhang, Nonlinear system fault diagnosis based on adaptive estimation, Automatica 40 (7) (2004) 1181 – 1193. 27

[10] M. Bouattour, M. Chadli, A. E. Hajjaji, M. Chaabane, Estimation of state, actuator and sensor faults for T-S models, in: 49th IEEE Conference on Decision and Control (CDC), 2010, pp. 1613–1618. [11] K. Tanaka, H. Ohtake, H. O. Wang, A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions, IEEE Transactions on Fuzzy Systems 15 (3) (2007) 333–341. [12] M. Liu, P. Shi, Sensor fault estimation and tolerant control for itô stochastic systems with a descriptor sliding mode approach, Automatica 49 (5) (2013) 1242– 1250. [13] Z. Qiao, T. Shi, Y. Wang, Y. Yan, C. Xia, X. He, New sliding-mode observer for position sensorless control of permanent-magnet synchronous motor, IEEE Transactions on Industrial Electronics 60 (2) (2013) 710–719. [14] C. Sun, F. Wang, X. He, Robust fault estimation for takagi–sugeno nonlinear systems with time-varying state delay, Circuits, Systems, and Signal Processing CSSP 34 (2) (2015) 641–661. [15] X.-J. Wei, Z.-J. Wu, H. R. Karimi, Disturbance observer-based disturbance attenuation control for a class of stochastic systems, Automatica 63 (2016) 21–25. [16] S. Boyd, L. E. Gaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM., 1994. [17] S. Makni, M. Bouattour, M. Chaabane, A. E. hajjaji, Robust adaptive observer design for fast fault estimation for nonlinear T-S fuzzy systems using descriptor approach, in: 6th International Conference on Systems and Control (ICSC), 2017, pp. 249–254. [18] J. Qin, Q. Ma, H. Gao, W. X. Zheng, Fault-Tolerant Cooperative Tracking Control via Integral Sliding Mode Control Technique, IEEE/ASME Transactions on Mechatronics 23 (1) (2018) 342–351.

28

[19] S. Abdelmalek, L. Barazane, A. Larabi, An advanced robust fault-tolerant tracking control for a doubly fed induction generator with actuator faults, Turkish Journal of Electrical Engineering and Computer Sciences 25 (2) (2017) 1346–1357. [20] T. Bouarar, B. Marx, D. Maquin, J. Ragot, Trajectory tracking fault tolerant controller design for Takagi-Sugeno systems subject to actuator faults, in: International Conference on Communications, Computing and Control Applications (CCCA), 2011, pp. 1–6. [21] K. Indriawati, T. Agustinah, A. Jazidie, Robust Observer-Based Fault Tolerant Tracking Control for Linear Systems with Simultaneous Actuator and Sensor Faults: Application to a DC Motor System, International Review on Modelling and Simulations (IREMOS) 8 (4) (2015) 410–417. [22] S. K. Kommuri, M. Defoort, H. R. Karimi, K. C. Veluvolu, A Robust ObserverBased Sensor Fault-Tolerant Control for PMSM in Electric Vehicles, IEEE Transactions on Industrial Electronics 63 (12) (2016) 7671–7681. [23] M. Bataghva, M. Hashemi, Adaptive sliding mode synchronisation for fractionalorder non-linear systems in the presence of time-varying actuator faults, IET Control Theory Applications 12 (3) (2018) 377–383. [24] S. Aouaouda, M. Chadli, M. Boukhnifer, H. Karimi, Robust fault tolerant tracking controller design for vehicle dynamics: A descriptor approach, Mechatronics 30 (2015) 316–326. [25] M. Assidi, A. B. Brahim, S. Dhahri, F. B. Hmida, Sensor fault tolerant control for uncertain linear systems, in: 17th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA), 2016, pp. 249–254. [26] L. T. H, L. C. Peng, N. Saeid, R. R. G, Observer-Based H∞ Fault-Tolerant Control for Linear Systems With Sensor and Actuator Faults, IEEE Systems Journal (99) (2018) 1–10.

29

[27] Z. Cong, J. M. Imad, Fault-tolerant controller design with a tolerance measure for systems with actuator and sensor faults, in: American Control Conference (ACC), 2017, pp. 4129–4134. [28] Y. Fuqiang, L. Mingjian, Fault tolerant control for T-S fuzzy systems with simultaneous actuator and sensor faults, in: 29th Chinese Control And Decision Conference (CCDC), 2017, pp. 7354–7359. [29] J. Han, H. Zhang, Y. Wang, 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 353 (2) (2016) 615–641. [30] M. Bouattour, M. Chadli, A. E. Hajjaji, M. Chaabane, State and faults estimation for T-S models and application to fault diagnosis, IFAC Proceedings Volumes 42 (8) (2009) 492 – 497. [31] Q. Jia, W. Chen, Y. Zhang, H. Li, Fault reconstruction and fault-tolerant control via learning observers in Takagi-Sugeno fuzzy descriptor systems with time delays, IEEE Transactions on Industrial Electronics 62 (6) (2015) 3885–3895. [32] M. Sami, R. J. Patton, Active fault tolerant control for nonlinear systems with simultaneous actuator and sensor faults, International Journal of Control, Automation and Systems 11 (6) (2013) 1149–1161. [33] Z. Gao, H. Wang, Descriptor observer approaches for multivariable systems with measurement noises and application in fault detection and diagnosis, Systems and Control Letters 55 (4) (2006) 304–313. [34] D. Ichalal, B. Marx, J. Ragot, D. Maquin, New fault tolerant control strategies for nonlinear Takagi-Sugeno systems, International Journal of Applied Mathematics and Computer Science (2012) 197–210. [35] T. Bouarar, B. Marx, D. Maquin, J. Ragot, Fault-tolerant control design for uncertain Takagi-Sugeno systems by trajectory tracking: a descriptor approach, IET Control Theory and Applications 7 (14) (2013) 1793–1805.

30

[36] I. H. Brahim, D. Mehdi, M. Chaabane, Robust fault detection for uncertain T-S fuzzy system with unmeasurable premise variables: Descriptor approach, International Journal of Fuzzy Systems 20 (2) (2018) 416–425. [37] B. Jiang, J. L. Wang, Y. C. Soh, An adaptive technique for robust diagnosis of faults with independent effects on system outputs, International Journal of Control 75 (11) (2002) 792–802. [38] K. Zhou, P. P. Khargonekar, Robust stabilization of linear systems with normbounded time-varying uncertainty, Systems & Control Letters 10 (1) (1988) 17– 20. [39] H. Gassara, A. E. Hajjaji, M. Chaabane, Adaptive fault estimation design for TS fuzzy systems with interval time varying delay, in: 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011), 2011, pp. 334–339. [40] Z. Gao, X. Shi, S. X. Ding, Fuzzy state/disturbance observer design for T-S fuzzy systems with application to sensor fault estimation, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 38 (3) (2008) 875–880. [41] S. Aouaouda, M. Chadli, H. R. Karimi, Robust static output-feedback controller design against sensor failure for vehicle dynamics, IET Control Theory Applications 8 (9) (2014) 728–737. [42] S. Aouaouda, T. Bouarar, O. Bouhali, Fault tolerant tracking control using unmeasurable premise variables for vehicle dynamics subject to time varying faults, Journal of the Franklin Institute 351 (9) (2014) 4514 – 4537. [43] D. Kharrat, H. Gassara, A. E. Hajjaji, M. Chaabane, Adaptive observer and fault tolerant control for takagi-sugeno descriptor nonlinear systems with sensor and actuator faults, International Journal of Control, Automation and Systems 16 (3) (2018) 972–982.

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