Ijetr041842

International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869 (O) 2454-4698 (P), Volume-5, Issue-1, May 2016

BPI Observer-based Fault Tolerant Control Design for a Class of Fuzzy Bilinear Systems Dhikra Saoudi  Abstract— In this paper, a new methodology of Bilinear Proportional Integral (BPI) observer-based Fault Tolerant Control (FTC) for a class of nonlinear systems with bilinear terms is proposed. Nonlinear systems are represented by the Takagi-Sugeno (T-S) Fuzzy Bilinear Model (FBM). The objective is to design a stable and robust fault-tolerant controller based on a T-S fuzzy BPI observer. The proposed BPI observer estimates simultaneously the states of the system and the occurring time varying faults with the final goal to adapt control law strategies in order that the system remains stable even in fault case. Further, the stability conditions of the designed BPI observer and the state feedback control are analysed with Lyapunov theory to ease the design of FTC. The gains of the BPI observer and the state feedback control law are obtained by solving linear matrix inequalities (LMI) under equality constraints. Moreover, we proposed some sufficient conditions for robust BPI observer-based fault tolerant control of the FBM with parametric uncertainties. Finally, an academic system is presented to illustrate the robustness of the proposed controller with respect to uncertainties.

Index Terms—T-S fuzzy bilinear model, Bilinear proportional integral observer, Observer-based control, Fault tolerant control, Lyapunov function, Parametric uncertainty, LMI.

controller designs based on fuzzy observers for T-S fuzzy systems. Nonetheless, in [17], [18], the authors have proposed sufficient design conditions for robust stabilization of T-S fuzzy models with parametric uncertainties based on state estimation. However, to the best of our knowledge, no previous study has investigated the problem of an BPI observer-based FTC for T-S fuzzy bilinear models and a robust stabilization for uncertain T-S fuzzy models.

In this paper, we consider the fault-tolerant control problem for the fuzzy bilinear systems with parametric uncertainties. Fuzzy BPI observer is designed to estimate simultaneously the states and the occurring time varying faults. Based on the fault estimation, the fault-tolerant controller is designed to ensure that the closed-loop system with faults is asymptotically stable.

This paper is organized as follows. In section II, the considered structure of the T-S fuzzy bilinear system is presented. In Section III, the problem statement is described.

In Section IV, the problem of observer-based control for the FBM is developed. In section V, the robust control of the FBM with parametric uncertainties is proposed. An example is provided to show the effectiveness of the proposed design in the subject of section VI. II. T-S FUZZY BILINEAR SYSTEMS MODELING

I. INTRODUCTION The Bilinear systems are a special class of nonlinear systems, where the dynamics are jointly linear in the state and input variables, that attract the interest of researchers [1], [2].

Moreover, there are many engineering applications as well as models in biology, ecology, nuclear engineering are naturally described by bilinear systems. Furthermore, many nonlinear systems can be adequately approximated bilinear systems.

As a special extension, fuzzy bilinear systems based on the T-S fuzzy model with bilinear rule consequence have attracted the interest of researchers [3]-[5]. It is proved in these papers that often nonlinear behaviors can be approximated by T-S bilinear multimodel description. Some topics of control are extended to T-S FBM, such as stability and stabilization in [6], [7], observers and state estimation in [8]-[10], faults detection and fault tolerant control [9]-[11].

Fault Tolerant Control have become challenging problems in the area of modern control theory. FTC allows having a control loop that fulfils its objectives when faults appear.

Thus great interest has been accorded to the fault tolerant controllers for nonlinear systems in the literature [12]-[14].

Moreover, it is noted that all of the aforementioned assume that the system states are measured, which is not true in many control systems and real applications. For this reason, observer-based fuzzy controllers were considered in many researches. In [15], [16], the authors have studied the Dhikra SAOUDI, LSA - Laboratoire des Systèmes Avancés, Polytechnic High School of Tunisia, University of Carthage, BP 743, 2078 La Marsa, Tunisia.

155

The T–S fuzzy bilinear models based on the T-S fuzzy model with bilinear rule consequence are defined by extending the T-S fuzzy ordinary model. Similar to [19], the fuzzy bilinear model can be represented by the following fuzzy if-then rules: Ri : if 1 (t ) is Fi1 and and  g (t ) is Fig x(t )  Ai x(t )  Bi u (t )  N i x (t ) u(t) then    y (t )  Cx(t ) where Ri denotes the i th fuzzy rule i  1,

(1) , r , r is the

number of if-then rules, i (t ) are the premise variables which can be measurable or not measurable, and Fij  j (t )  are fuzzy set. x(t )  n is the state vector, u (t )   is the control input, and y (t )   p is the system output. The matrices Ai , Bi , Ni , C are known matrices. Then, the overall FBM can be described as follows: r   x(t )   hi ( (t ))  Ai x(t )  Bi u (t )  N i x(t ) u(t)   i 1  y (t )  Cx(t ) 

(2)

where hi (.) verify the following properties:  r  hi ( (t ))  1 i  1, 2,  i 1 0  h ( (t ))  1 i 

, r

(3)

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BPI Observer-based Fault Tolerant Control Design for a Class of Fuzzy Bilinear Systems In the remainder of the paper, the following lemmas are used.

Lemma 1: For any matrices A and B with appropriate dimensions, the following property holds for any positive scalar  : AT B  BT A   AT A   1 BT B

(4)

Lemma 2: Given a scalar   0 and a symmetric positive definite matrix P , the following inequality holds: 2 xT y   1 xT Px   yT P 1 y ,

x, y n

(5)

Lemma 3: Let us consider P a positive definite matrix and Q a full column rank matrix. It follows that the matrix QPQT is a positive definite matrix.

III. PROBLEM STATEMENT Let us consider the T-S fuzzy bilinear models containing the actuator faults which can be rewritten in the following form: r   Ai x(t )  Bi u (t )   x(t )   hi ( (t ))   (6)  N x ( t ) u ( t )  G f ( t )  i 1 i i     y (t )  Cx(t )

 is a given scalar, and Di  1n , i  1, to be determined.

, r are vectors

In the following section, we will be developing the BPI observer-based control problem for the T-S FBM. IV. STABILIZING FAULT TOLERANT CONTROL FOR FBM In order to estimate both state and faults of FBM, we propose a BPI observer and make an analysis of the error system, from which we provides a design method of a fuzzy observer for the fuzzy bilinear system (2). The considered BPI observer is described by the following equation: r   H i z (t )  Li y (t )  J i u (t )   z (t )   hi  (t )    ˆ i 1    M i u (t ) y (t )  i f (t )   (8)  xˆ(t )  z (t )  Ey (t )   yˆ (t )  Cxˆ (t )  r  fˆ (t )   hi  (t )   i  yˆ (t )  y (t )   i 1  where z (t ) n , xˆ(t )  n , yˆ(t )   p and fˆ (t )   f are the observer state vector, the estimated state vector, the estimated output vector and the estimated faults vector respectively. H i , M i , Li , J i , i ,  i and E

where f (t ) is an actuator fault.

are unknown matrices of the BPI observer of appropriate dimensions to be determined.

Assumption 1: The fault f (t ) satisfies f (t )  1 and the

Note that the premise variables do not depend on the state variables estimated by a BPI observer.

first time derivative of f (t ) with respect to time is norm bounded i.e. f (t )   2 and 0  1   , 0   2   .

The main objective of FTC design is to find a control law u (t ) such that the closed-loop system remains stable despite

the presence of actuator faults. Then, it is necessary to estimate the states and faults. In this paper, a BPI observer is utilized for estimate both the states and the actuator faults with the final goal to adapt control law strategies such that the system remains stable even in actuator fault case. Following the design concept in [1], the fuzzy control law for FBM is formulated as follows: r  Di xˆ(t ) u (t )   hi  (t )  i 1 1  xˆ T (t ) DiT Di xˆ (t ) r (7)   hi  (t )   sin ˆi (t ) i 1 r

  hi  (t )   Di xˆ (t ) cos ˆi (t )

Let us consider the state and fault estimation errors defined respectively by: ex  t   x  t   xˆ  t 

(9)

e f  t   f  t   fˆ  t 

(10)

From (8) and (9), the state estimation error given as follows: ex  t   Tx  t   z  t 

(11)

where T  I n + EC .

By taking into account (2) and (8), the dynamics of the state estimation error and the closed-loop system with the state feedback control law defined in (7) becomes after some calculations:

i 1

 H i ex (t )  TAi  Li C  H iT  x (t )    ex  t    hi  (t )    TN i  M i C  x (t )u (t )  TBi  J i  u (t )  i 1     TGi  i  f (t )  i e f (t ) 

where:

r

sin ˆi (t )  cos ˆi (t ) 

Di xˆ(t ) 1  xˆ (t ) DiT Di xˆ (t ) T

(12)

1 1  xˆ T (t ) DiT Di xˆ (t )

r r   x(t )   Bi D j cos ˆj ex (t )  x(t )   hi  (t ) h j  (t )   ij   G f (t )  i 1 j 1  i 

   ˆi    ,   2 2

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International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869 (O) 2454-4698 (P), Volume-5, Issue-1, May 2016 (13)

Then, the state x (t ) of the system, the state estimation error ex (t ) and the fault estimation error e f (t ) are bounded. The

where:  ij  Ai   N i sin ˆ j   Bi D j cos ˆj

Hence, if the i 1, 2, , r

following

(14)

conditions

hold

true

(15)

TNi  M i C  0

(16)

TBi  J i  0

(17)

Then, the equation of the observing error becomes ex  t    hi  (t )   H i ex (t )   i f (t )  i e f (t )  r

(18)

i 1

with i  TGi  i From (15) and using T  I n + EC , we get H i  TAi  Ki C

(19)

with Ki  H i E  Li

(20)

The goal is to design a feedback control law such that the system remains stable even if a fault occurs. A result is summarized in the following theorem.

Theorem 1: If there exist symmetric positive definite matrices Q , 1 ,  2 , 3 , P1 and P2 , matrices Wi , Vi Yi , Z i , Ri , U i , S and positive scalars  ,  ,   0 such that LMIs (21) subject to linear equality constraints (22), (23) are satisfied i  1 r , j  1 r . *

*

*

*

*

I

*

*

*

*

0

I

*

*

*

0



*

*

0

E  P21S , J i  P21 Ri , M i  P21U i , H i   I n  EC  Ai  P21Wi C , i  P21Yi ,

(25)

Li  P21Wi  H i E ,  i  P31Z i , Di  Vi Q 1

TAi - Li C - H iT  0

 LM 1  NQ i   BV i j  Q  T  i j    BV  0  0   0 

gains of the observer and the state feedback control law are given by:

1 3

0

0

0

2  Q

*

0

0

0

0

2  I

0

0

0

I

0

0

0

0

0

I

*  * *  * *   * *  0 * *  * *   LM 2 *   LM 3  1 2  *

(21)

(22)

U i C   P2  SC  Ni

(23)

LM 2   P2  SC  Ai  AiT  P2  SC   WiC  WiC    11 T

T

V (t )  xT (t ) P1 x(t )  eTx (t ) P2ex (t )  eTf (t ) P3e f (t ) (26)

where P1 , P2 and P3 are symmetric and positive definite matrices with appropriate dimensions. Stability condition for the estimation error yields that the time derivative of (26) should be negative definite. The derivative of V (t ) with respect to time yields:  xT (t ) P1 x(t )  xT (t ) P1 x(t )   T  T V (t )   hi  (t ) h j  (t )  ex (t ) P2ex (t )  ex (t ) P2ex (t )  i 1 j 1  T  T e f (t ) P3e f (t )  e f (t ) P3e f (t )  r

r

(27) that becomes by using equations (18) and (13):  xT  ij x  eTx i ex  2 xT PG  1 i f   T ˆ r r   2  x P1 Bi D j cos  j ex  V (t )   hi  (t ) h j  (t )    T T i 1 j 1 2ex P2  i f  2ex P2i e f    T T 2e f P3 f  2e f P3 iCex    (28)

where:

Ri   P2  SC  Bi

LM 1  AQ  QAiT   2 I i

Proof: In the framework of a FTC by state-feedback, a new procedure is proposed such that the closed-loop system and the convergence of the state and fault estimation errors becomes asymptotically robust stable by using a Lyapunov function-based design approach. Let us consider a Lyapunov function V (t ) depending on x(t ) , ex (t ) and e f (t ) defined by:

ij  Tij P1  P1ij

(29)

i  HiT P2  P2 Hi

(30)

By using the Assumption (1) and applying Lemma (2) for three terms of the above inequality (28), it follows that:

(24)

LM 3  YiT  Zi C

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BPI Observer-based Fault Tolerant Control Design for a Class of Fuzzy Bilinear Systems







2eTx P2  i f   1eTx 1ex   f T Ti P2 11 P2  i f   e 1ex   1 T x

t 0

1





2e P3 f   1eTf  2 e f   f T P3 21 P3 f T f

  e  2e f   1 T f

(31)





which can be also bounded by: (39)

i, j

  1 xT  3 x   3

If the following inequality holds:

with:

r

r

 h  (t ) h  (t )  

 1  12 max  Ti P2 11 P2  i 

i

i 1 j 1

 2   22 max  P3 21 P3 

(32)

 3  12 max  GiT P131 PG 1 i

j

ij

0

(40)

Then, we can obtain that V (t )   X a (t )   2

Let us define the scalar  the maximum value of the sum of all constant coefficients obtained in the equalities (32):

  max 1   2   3 

(33)

Then, the time derivative of the Lyapunov function (28) is bounded as follows: T x

set according to Lyapunov stability theory and lie in it. This set is smaller as the constant  converges to zero.

r

r

r

Y   hi  (t ) h j  (t )  Yij

where Yi and Yij are given matrices. By using these notations, the inequality (40) becomes:

i 1 j 1

1      *

2   0 4   

where the augmented vector X a and the matrix  ij are

with:

respectively given by:

1   ij   13

(36)

  P1 Bi D j cos ˆj i   11 *

0 P2i    i C 

T

 1 2

(43)

i 1 j 1

(35)

 x(t )    X a (t )   ex (t )   e f (t )   

(42)

i 1

The above inequality (34) can be reformulated as follows:

V (t )  X (t ) hi  (t ) h j  (t )  ij X a (t )  

It follows that V (t )  0 if  X a (t ) 2   , t  0 . Then, we can deduce that the state x (t ) , the state estimation error ex (t ) and the fault estimation error e f (t ) converge to a small

Y   hi  (t )  Yi

(34)

r

(41)

Let us introduce the following notations for the sake of simplicity:

 x  ij x  e i ex    T 2  x P1 Bi D j cos ˆj ex  r r     V (t )   hi  (t ) h j  (t )  2eTx P2i e f  2eTf P3 i Cex  i 1 j 1  1 T  1 T  ex 1ex   e f  2 e f   1 T     x 3 x    T

 ij   13  ij   *  * 

(38)



i 1 j 1

  min min  ij 



1 T 2 x PG x  3 x   f T GiT P1 31 PG 1 i f  1 i f

r



r

2

T

T a

r

  min min   hi  (t ) h j  (t )   ij 

  P3   

(45)

2      P1 Bi D j cos ˆj

   11 4   i *  (47)

(44)

0 

(46)

P2i    i C  P3    1 2  T

Let us consider a symmetric matrix  defined as: (37)

 P11

 

 0

Let us define the positive scalar 

 P11 0  , 1   1   0

0  I

(48)

By using Lemma (3), and post and pre-multiplying the inequality (44) by  , we can obtain that:

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International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869 (O) 2454-4698 (P), Volume-5, Issue-1, May 2016  P111 P11  * 

2 P11 1  0 4 1 1 

(49)

4 The term 1 1 can be replaced by considering the

following inequality which holds for any scalar  such that

(56)

                0 4

1 T

4



1

4



1

(50)



1



4 4  1 1  21   2 



1

(51)

Then from the above inequality (51) and with the Schur Complement, it follows that the inequality (49) holds if: P  P  *   *  1 1

1  1

1

P  1 1 1

0   I   0 4   

2

2 1 *

(52)

P111 P11  P11 AiT  Ai P11   N i P11 sin ˆj   P11 N iT sin ˆj T   Bi D j P11 cos ˆj   P11  Bi D j  cos ˆj   1 P113 P11

(53) Applying lemma (1), the previous equation can be rewritten as: P111 P11  P11 AiT  Ai P11  P11 N iT N i P11   2 I P

B D 

T

i

j

The parameter uncertainties considered norm-bounded and presented by the form

 Ai where

Bi i

Ni  

i

Fi (t )  E1i

here

E3i 

E2i

1

1 1

appropriate dimensions, and Fi (t ) is an unknown matrix function satisfying Fi T (t ) Fi (t )  I , in which I is the identity matrix of appropriate dimension.

In order to describe the dynamic of BPI observer (8), the state and fault estimation errors is defined by (9), (10). Then, from (56) and (8), we have:  H i ex (t )  TAi  Li C  H iT  x (t )    r   TN i  M i C  x (t )u (t )  TBi  J i  u (t )  ex  t    hi  (t )    i 1  T Ai x (t )  T Bi u (t )  T N i x (t )u (t )    TGi  i  f (t )  i e f (t )   

(54)

1 3 1

(58)

From the inequality (54) and the matrices (46), (47), using the notations (42), (43), replacing H i by (19) and T  I n  EC and by introducing the following variable changes: Q  P11 , D j  V j Q , Wi  P2 K i , Yi  P2i Ri  P2 J i , U i  P2 M i , Z i  P3  i , S  P2 E

(57)

, E1i , E2i , E3i are known real constant matrices of

Bi D P   P  P 1 j 1

are

The feedback controller and the state of BPI observer for T-S FBM with parametric uncertainties (56) is the same as that for (6).

Now, the term P111 P11 of (52) can be rewritten as:

1 1

bilinear model with actuator faults: r    Ai  Ai  x(t )   Bi  Bi  u (t )   x(t )   hi ( (t ))      N  N  x(t )u (t )  G f (t )   i 1 i i i    y ( t )  Cx ( t ) 

   ij   ij  x(t )  Gi f (t )  r r  x(t )   hi  (t ) h j  (t )       Bi  Bi  D j cos ˆ j ex (t )  i 1 j 1  

(59)

(55) where:

we can obtain the inequalities given in theorem (1) by applying the Schur's complement [20] to (52) under equality constraints (22), (23). This completes the proof of the theorem.

 ij  Ai   N i sin ˆj   Bi D j cos ˆj

(60)

 ij  Ai  N i sin ˆj  Bi D j cos ˆj

(61)

This section is devoted to the design of a BPI observer-based control for T-S FBM. The system parameters used are known but in reality the system parameters may either be uncertain or time-dependent. Moreover, the problem of stabilization remains a key problem in the study of uncertain T-S fuzzy control systems and their controller designs. The study of BPI observer-based control for T-S FBM with parametric uncertainties will be subject the next section.

If the conditions (15), i 1, 2, , r , we get:

V. ROBUST CONTROL OF THE FUZZY BILINEAR MODEL WITH

By using (7), we obtain:

(16),

(17)

are

 H i ex (t )  T Ai x(t )  T Biu (t )    ex  t    hi  (t )   T Ni x(t )u (t )  i 1   TG   f (t )   e (t )  i i f   i  r

satisfied

(62)

UNCERTAINTIES

In this section, we shall consider a uncertain nonlinear system which can be described by the following T-S fuzzy

159

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BPI Observer-based Fault Tolerant Control Design for a Class of Fuzzy Bilinear Systems r r  1i ex (t )  i2 f (t )  ex  t    hi  (t ) h j  (t )   3   i x(t )  i e f (t )  i 1 j 1  

(63)

where : 1i  H i  T Bi D j cos ˆj

(64)

i2  TGi  i

(65)

3i  T Ai  T Bi D j cos ˆj  T N i sin ˆj

(66)

Hereafter, we give the following theorem which presents sufficient conditions such that the closed-loop fuzzy system (59) is globally asymptotically stable in the presence of actuator faults and which gives the gains of the observer and the state feedback control law.

Theorem 2: If there exist symmetric positive definite matrices Q , 1 ,  2 , 3 , P1 and P2 , matrices Wi , Vi Yi , Z i , Ri , U i , S and positive scalars  ,  ,  ,   0 such that LMIs (67) subject to linear equality constraints (68), (69) are satisfied i  1 r , j  1 r .  LM 1  NQ  i  BV i j  Q   E1i Q   E2iV j  EQ  3i  LM 4  EV  2i j  LM 5     BV i j   0

*

*

*

I

*

*

0 I

*

0

0 31

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

*  *  * * * * * * * *   * * * * * * * *   I * * * * * * *   0  I * * * * * *  0 0 0  I * * * * *   0 0 0  I * * * *  0 0 0 0   1 I * * *   0 0 0 0 0   1I * *   0 0 0 0 0 0 LM 6 *   0 0 0 0 0 0 0 LM 7  *

*

**

*

*

*

*

*

*

*

*

*

*

*

  1 E1Ti E1i  E2i D j   E3i LM 6    Ti   E2i D j   1  2 2 T i   21 *  LM 7     LM 3   *   LM 2   1  LM 3   2 





*

*

*

*

 I

*

*

*

0

 I

*

*

0

0

 I

*

0

0

0

  1 I

0

0

0

0

*   *  *   *  *     1 I  

Then, the state x (t ) of the system, the state estimation error ex (t ) and the fault estimation error e f (t ) are bounded. The gains of the observer and the state feedback control law are given by: E  P21S , J i  P21 Ri , M i  P21U i , H i   I n  EC  Ai  P21Wi C , i  P21Yi ,

(71)

Li  P21Wi  H i E ,  i  P31Z i , Di  Vi Q 1

Proof: In order to prove the stability of the closed-loop system and the convergence of the state and fault estimation errors, sufficient conditions are derived using Lyapunov function (26). Then, the derivative of V (t ) with respect to time yields:  xT   ij   ij  x  eTx ij ex    T  2 xT PG  1 i f  2ex P2i e f r r     V (t )   hi  (t ) h j  (t )  2  xT P1  Bi  Bi  D j cos ˆj ex  i 1 j 1   T 2 T 3  2ex P2  i f  2ex P2  i x   2eT P f  2eT P  Ce  f 3 f 3 i x     (72)

(67) where:

Ri   P2  SC  Bi

(68) ij  P1ij  Tij P1

U i C   P2  SC  Ni where:

(69) ij  P1ij  Tij P1 (74)

LM 1  AQ  QAiT   2 I i



T

T

LM 3  YiT  Zi C LM 5 

T i

T

1  2   I  EC  T i

T



 P2 H i  T Bi D j cos ˆj

 I  EC  2



ij  H iT   T Bi D j  cos ˆj P2

LM 2   P2  SC  Ai  AiT  P2  SC   WiC  WiC    11 LM 4  

(73)



(75)

T

By using (31), the time derivative of V (t ) (72) is bounded as follows:

160

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International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869 (O) 2454-4698 (P), Volume-5, Issue-1, May 2016  xT   ij   ij  x  eTx ij ex    2  xT P1  Bi  Bi  D j cos ˆj ex  r r   V (t )   hi  (t ) h j  (t )  2eTx P2i e f  2eTx P2 i3 x  i 1 j 1  T  1 T 2e f P3 i Cex   ex 1ex   1 T  1 T   e  e   x  x   f 2 f 3   (76)

1ij  Ai P11  N i P11 sin ˆ j  Bi D j P11 cos ˆj T  P11AiT   P11N iT sin ˆ j   P11  Bi D j  cos ˆj T  2ij  T Bi D j P21 cos ˆj   P21 T Bi D j  cos ˆj T  3ij   Bi D j P11 cos ˆj  P21 T Ai  T T   P21 T Bi D j  cos ˆ j   P21 T N i  sin ˆj

The equation ij can be reformulated as follows:

where  is defined by (33).

 ij  1ij   ij2   ij3

The above inequality (76) can be reformulated as follows: r

r

V (t )  X aT (t ) hi  (t ) h j  (t )  ij X a (t )  

(77)

i 1 j 1

where the augmented vector X a is defined by (36) and the matrix ij is given by:  1ij   ij   * * 

ij2 ij   11 *

(85)

0    ij3  1   2 

where: 0  1ij   * * 

0 0   0  ,  ij2   * * 0   

11 ij 0 *

 1ij    *  * 

 ij21 0 *

0  0 , 0  

(87)

0  0 0 

0  2ij

3 ij

(78)

with

(86)

*

T T 1 1 ˆ 11 ij  P2 T Ai    P2 T Bi D j  cos  j

1ij  ij  ij   13

(79)

 

    P1  Bi  Bi  D j cos ˆj  i3 2 ij

T

(88)

  P21 T N i  sin ˆj T

 ij21   Bi D j P11 cos ˆj

P2

(80)

Using the uncertainties structure and by using the separation lemma 1 [21], we obtain

3ij  P2i   i C  P3 T

(81)

From (91), we have: ij  ij  ij

(82)

Tij1   ij   * * 

0  0 0 

0 2 ij

T

*

(89)

where: where  ij is defined by (37) and :





Tij1  1  2  2 T i Fi Fi T  1ij   ij   *  * 

 3ij  2ij *

0  0 0 

T i

T T   P11 DTj E2Ti E2i D j P11

+ 1 P11 E1Ti E1i P11   1 P11 DTj E2Ti E2i D j P11 + 1 P11 E3Ti E3i P11   2  1T i Fi Fi T

T i

(90)

TT

Tij2   1 P21 E1Ti E1i P21   1 P21 E3Ti E3i P21

(83)

+ 1 P21 DTj E2Ti E2i D j P21   1  2 i Fi Fi T





+ 1  2  2  i Fi Fi T

with: 1ij   ij T  2ij   P2T Bi D j cos ˆj   T Bi D j  P2 cos ˆj

 

3ij    P1Bi D j cos ˆj  i3

T

(84)

Multiplying the previous equations (84) on the left and right

by P11 and P21 , we can rewrite (84) as:

161

  P21 DTj E2Ti E2i D j P21

From (82) and (89), we have: 1   *

P2

T i

T i

2  1   4     *

2   0 4   

(91)

where 1ij , ij2 , ij4 are defined by (45)-(47) respectively and 1ij , ij2 , ij4 are given by:

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BPI Observer-based Fault Tolerant Control Design for a Class of Fuzzy Bilinear Systems 1ij  Tij1

(92)

   0 0 2 ij

T ij4   *

2 ij

0  0

Using (49), (51) and with the Schur Complement, it follows that the inequality (91) holds if: 2 2  P111 P11  1 P11 1   0 0    4 *  0 0  (93)  0  * * 2 1  I   4  * * *   

D1  3.343 0.098 1.254 , D2  0.041 1.008 0.254

9.022 4.533 9.022 0 E  5.976 8.965 , M 1  M 2  5.976 0  2.046 3.569 3.046 0  0.688 1.056  504.851 752.266  L1  103 *  0.399 0.584  , L2   656.529 987.282  1.567 2.344   165.730 249.618

J1   18.166 9.956 4.461 , J 2  47.110 27.882 13.229 T

T

 51.288 10.218 77.669  23.644 46.592 0.148    H1   29.304 31.204 32.793 , H 2   53.635 17.602 25.086 112.873 80.082 15.084   14.954 1.058 9.543 

1  2   4.511 2.988 1.523

T

We calculate each terms of inequality (93) and by considering the variable changes (55), we can obtain the inequalities given in theorem (2) by applying the Schur's complement [20] to (93) under equality constraints (68), (69).

This completes the proof of the theorem.

VI. NUMERICAL EXAMPLE To illustrate the effectiveness of the proposed method, an example is considered. The dynamical FBM is given by the following data: 0  2 1  5   2 0 0  1        A1   1 3 0  , B1  1.2  , N1   0 0 0  , G1  0   2  2.5  0 0 1  1  1 8

1   2   1.523 2.988

The simulation results are shown in figures (1)-(4). The time-varying actuator fault and its estimate are depicted on Figure 1. It appears clearly the good estimation of this actuator fault. Furthermore, the BPI observer provides the fault estimation which errors are illustrated in the figures 2.

One can shows the fault estimation error is zero-mean except at time 1s, 7s and 9s that suits abrupt changes of the fault.

1  2 0 4  2 0 0  1        A2   0.5 2 0  , B2   2  , N 2   0 0 0  , G1  0   2  0   0 0 1  1  1 1 1 0 1  C  1 1 0 

Uncertainties are also defined by the following matrices: 0.1 0 0  0 0.5 0  1 0  0 0 0.1 , E  0 0 0.3 , E  0.5 , E  0  1   11   21   31   0 0.1 0  0 0 0   0  0 0.1 0 0  0 0.5 0  1 0  0 0 0.1 , E  0 0 0.6  , E  0.5 , E  0  2   12   22   32   0 0.1 0  0 0 0   0  0 The weighting functions are defined by:

0 0 0 0  0 0  0 0 0 0  0 0 

Fig 1. Evolution of the fault and its estimate

  x  5 2  h1  x(t )   exp  1  1   , h2  x(t )   1  h1  x(t )   2 2    

The additive actuator fault signal is defined as follows : 0 t  1s  0.5sin 0.5 1s  t  7 s    f (t )   0.3 7 s  t  9s   0 t  9s

Thus, the resolution of the conditions of theorem 2 leads to the following controller and observer gain matrices respectively with   0.05

162

Fig 2. Fault estimation error

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International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869 (O) 2454-4698 (P), Volume-5, Issue-1, May 2016 REFERENCES [1]

[2] [3]

[4]

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Fig 3. The output y1 (t ) of the system: nominal output, output without FTC and output with FTC

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Fig 4. The output y2 (t ) of the system: nominal output, output without FTC and output with FTC The figures (3-4) compares the outputs of the nominal model, the outputs of the faulty system without FTC and the outputs with FTC. One can see that the trajectory of the system follows the trajectory of the reference model, even in fault case. It is clear that the proposed strategy is robust with respect to the actuator additive fault f(t) and with parametric uncertainties. Moreover, the BPI observer-based FTC scheme provides good results in the presence of norm-bounded parametric uncertainties, as shown in the figures.

VII. CONCLUSION This paper proposes a state feedback fault tolerant control law methodology for T-S fuzzy bilinear systems. A T-S fuzzy BPI observer is designed for the proposed FTC strategy in order to simultaneously estimate time varying faults and state variables. This method aims to adapt the control law in order to compensate the effect of the time-varying faults by maintaining the stability of the system. The stability conditions of the designed observer and the state-feedback control has been provided and solved through a set of Linear Matrix Inequalities under equality constraints. Moreover, sufficient conditions for controller design based on state estimation for robust stabilization of T-S FBM with parametric uncertainties has been proposed. Finally, the performance of the proposed BPI observer based FTC approach has been given to an example in order to illustrate the validity of this method.

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