New Condition of Stabilization of Uncertain Continuous Takagi-Sugeno Fuzzy System based on Fuzzy Lyapunov Function

Published Online April 2012 in MECS

Fuzzy control systems have experienced a big growth of industrial applications in the recent decades, because of their reliability and effectiveness. Many researches are investigated on the Takagi-Sugeno models [1-2] which can combine the flexible fuzzy logic theory and rigorous mathematical theory into a unified framework. Thus, it becomes a powerful tool in approximating a complex nonlinear system.

Two classes of Lyapunov functions are used to analysis these systems: quadratic Lyapunov functions and non-quadratic Lyapunov ones which are less conservative than first class. Many researches are investigated with non-quadratic Lyapunov functions [1-9].

In this paper, a new stability conditions for Takagi Sugeno uncertain fuzzy models based on the use of fuzzy Lyapunov function are presented. This criterion is expressed in terms of Linear Matrix Inequalities (LMIs) which can be efficiently solved by using various convex optimization algorithms [10]. The presented method is less conservative than existing results.

The organization of the paper is as follows. In section 2, we present the system description and problem formulation and we give some preliminaries which are needed to derive results. Section 3 will be concerned to stability analysis for T-S fuzzy systems. Section 4 concerns the proposed approach to stabilize a T-S fuzzy system with Parallel Distributed Compensation (PDC). Next, a new stabilization condition for uncertain system is given. Finally section 6 makes conclusion.

Notation : Throughout this paper, a real symmetric matrix 5 >  0 denotes S being a positive definite matrix. The superscript ‘‘T’’ is used for the transpose of a matrix.

• II.    System Description and Preliminaries

Consider an uncertain T-S fuzzy continuous model for a nonlinear system as follows:

IF z 1 ( t ) is M i 1 and ... and z p ( t ) is M p

THEN x ( t ) = ( A, +M i ) x ( t ) + ( B i +A B i ) u ( t )      (1)

i = 1,., r whereM j (i = 1,2,.,r, j = 1,2,.,p) is the fuzzy set and r is the number of model rules; x (t) еЖ” is the state vector, u (t )еЖ m is the input vector, Ai еЖ nx n , Bi еЖnxm , and z 1 (t),.,zp (t) are known premise variables. AAi and ABi are time-varying matrices representing parametric uncertainties in the plant model. These uncertainties are admissibly norm-bounded and structured.

The final outputs of the fuzzy systems are: r

• ^{x (} t ) = E hA ^{z (} t ) ) { ( A i-^{+A A} J ^{x (} t ⁾⁺ ( B ^{+A B} J ^{u (} t ) }      ⁽²⁾

i = 1

where

z (t ) = [z 1 (t )z 2 (t ).zp (t )]

h(z (t ))= Wi(z (t ^/Ew^z (t ))                  ,

= 1

p w i (z (t))=П Mij (z j (t)) for all t.

j = 1

The term M i 1 ( z j ( t ) ) is the grade of membership of ^z j ( t ) ^{in M}n

Since

r

2 W i ( z ( t ) ) x 0

i =1

W i ( ^{z (} t ) ) ^ 0,

Lemma 1 (Boyd et al. Schur complement [6])

r

2 h( z (t )) = i i =1

h i ^{( z (} t ) ) ^ o,

i = 1,2,™, r for all t.

i = 1,2, ™ , r

Given constant matrices    Q ₁, Q ₂ and Q 3   with

appropriate dimensions, where Q ₁ = Q ^T and Q ₂ = Q ^T ,

then

Q ₁ + Q T Q^{- 1} Q 3 ^ 0

if and only if

The time derivative of premise membership functions is given by:

Q 1 Q 3

* -Q

p 0 or

; 2 J

^-Q 2

*

Q 3

Q 1 j

p0

M z ⁽ t )) =

5 h i    d z ( t ) dx ( t )

d z ( t ) d x ( t ) dt

s

2UZ

I = 1

dx (t) x-— dt

We have the following property:

r

2 hk (z (t )) = 0(4)

к = 1

The PDC fuzzy controller is represented by

r

2w,(z (t^x (t)

u (t)=- r----------------=-2 hi- (z (t)) Fx (t)(5) 2 wi( z (t))

i = 1

The fuzzy controller design is to determine the local feedback gains ^{F i} in the consequent parts.

The open-loop system is given by the equation (6)

r x (t )=2 Mz (t))(Ai+^AJ x (t)             (6)

i = 1

By substituting (5) into (2), the closed-loop fuzzy system can be represented as:

rr x (t)=22h (z (t))hj (z (t)){Aa. -BmF }x (t) (7) i =1 j =1

where A _A i = A i + A A , and B _A , = B i + A B i

Assumption 1

The time derivative of the premises membership function is upper bounded such that h_k | <  ф _к , for к = 1, ™ , r , where, ф _к , к = 1, ™ , r are given positive constants.

Assumption 2

The matrices denote the uncertainties in the system and take the form of

[ A A i. A B , ] = DF ( t ) [ E A E B^

where DE and E are known constant matrices ,AiBi and F (t) is an unknown matrix function satisfying :

FT (t) F (t )< I, Vt where I is an appropriately dimensioned identity matrix.

Lemma 2 (Peterson and Hollot [8])

Let Q = Q^T , H , E and F ( t ) satisfying F^T ( t ) F ( t ) < I are appropriately dimensional matrices then the following inequality

Q + HF ( t ) E + E^TF^T ( t ) H^T p 0

is true, if and only if the following inequality holds for any £ f 0

Q + £ ^{- 1} HH^T + £ E^tE p 0

• III.    Basic Stability Conditions

Consider the open-loop system (8). r x (t )=2 h(z (t ))Ax (t)                  (8)

i = 1

This section gives a new condition for stability of the unforced T-S fuzzy system by using the Lyapunov theory.

Theorem 1 [11]

Under assumption 1 and for 0 <  £ <  1 , the Takagi Sugeno fuzzy system (8) is stable if there exist positive definite symmetric matrices P_k , к = 1,2, ™ , r , matrix R = R^T such that the following LMIs hold.

Pk + R x 0, к e{1,™,r}

Pj + MR x 0, j e{1,™,r} рф +1 {AT (Pj+ mr)+(Pj+ fr)A.

2(11)

+ A^T ( p . + ^ R ) + ( P i + ^ R ) A j } x 0, i <  j

r where i, j = 1,2, ™, r and Рф = 2 фк (pk+ R) and к =1

Ц = 1 - £

• IV.    Stabilization with PDC Controller

Consider the closed-loop system without uncertainties which can be rewritten as

x ⁽ t ) = 1 h ( z ⁽ t ) ) h . ( z ⁽ t ) ) G i x ⁽ t )

' =1                         (           1              (12)

+ 211 h. (z (t)) hj( z (t Wj^ jx (t), =1 . j j                                              I         2 I where

G j = A . - BF and Ga = A, - B . F .

In this section we define a fuzzy Lyapunov function and then consider stability conditions.

Theorem 2

Under assumption 1 and for given 0 < e < 1 , the Takagi-Sugeno system (12) is stable if there exist positive definite symmetric matrices Pk, k = 1,2,., r , and R, matrices F1,.,Fr such that the following LMIs hols.

Pk + R x0,   k e{1,.,r}

Pj + pR > 0, j = 1,2,.,r(14)

Рф + GT (Pk + pR)+(Pk + pR)g..} j 0,

• i ,    k e { 1, . , r }

f G + G .V ,           , f G + G .]

I2j (Pk + pR) + (Pk + pR)f2jj 0, for i, j, k = 1,2,., r such that i j j where

k = 1

Proof

Let consider the Lyapunov function in the following form:

r

V ( x ( t ) ) = 1 h k ( z ( t ) ) V k ( x ( t ) )               (17)

k = 1

with

Vk (x (t)) = xT (t)(Pk + pR)x (t),     k = 1,2,.,r where

Pk = PT,R = RT, 0 < e < 1,p = 1 -e, and (Pk + pR) > 0, k = 1,2,., r

The time derivative of V (x (t)) with respect to t along the trajectory of the system (12) is given by:

rr

V ( ^x ( t ) ) = 1 ^h k ( ^z ( t ) ) V k ( x ( t ) ) + 1 h k ( z ( t ) ) V k ( x ( t ) ) k = 1                                      k = 1

The equation (18) can be rewritten as,

r

V ( x ( t ) ) = x^T ( t ) | 1 h k ( ^z ( t ) ) ( P k + p R ) I x ( t )

• V k =1

Г

+x (t)|Xhk (z (t))(Pk + pR)Ix (t) V k =1/

(

+x (t)|Xhk (z (t))(Pk + pR)Ix (t) V k =1/

By substituting (12) into (19), we obtain,

V ( x ( t ) ) = Y 1 ( x , z ) + Y 2 ( x , z ) + Y ₃ ( x , z ) (20) where

r

^Y 1 ( x , z ) = x^T ( t ) | X ^h _k ( z ( t ) ) • ( ^p _k + p ^R ) I x ( t ) ⁽²¹⁾

rr

^Y 2 ( ^x , ^z ) = x^T ( t ) 11 h k ( ^z ( t ) ) h i ² ( ^z ( t ) ) k = 1 i = 1

^x { G^T ( P_k + p R ) + ( P_fc + p R ) G_H } x ( t )

rr

^Y 3 ( x , z ) = x ( t j 111 h k ( ^z ( t ) ) h i ( ^z ( t ) ) h j ( ^z ( t ) ) k = 1 i = 1 U j j

Г G j + G ji

T

( P k + p R ) + ( P k + p R )M

Then, based on assumption 1, an upper bound of

Y 1 ( x , z ) obtained as:

r

Y 1 ( x , z ) < 1 ф _к • x ( t f ( P_k + p R ) x ( t )              ⁽²⁴⁾

k = 1

Based on (4), it follows that

r

1 h_k ( z ( t ) ) e R = R = 0 where R is any symmetric k = 1

matrix of proper dimension.

Adding R to (24), then r

• Y 1 ( x , z ) < 1 ф k • x ( t f ( P k + R ) x ( t )           ⁽²⁵⁾

k = 1

Then,

r

• V ( x ( t ) ) < 1 ф k X ^T ( t )( P k + R > ( t ) k =1

+ Y ₂ ( x , z ) + Y ₃ ( x , z )

If (15) and (16) holds, the time derivative of the fuzzy Lyapunov function is negative. Consequently, we have rr

V (x (t))

V k=1 i=1

x{(GiT (Pk + mR) + (Pk + mR)-,,)} r             rr

+2 Фк ((Pk + R)) +222 hk (z (t)) h, (z (t)) hj (z (t))

к =1                           k=1 i=1 i ч j

Ug.- +G-. Y                  [G-+G-. 1Y

x iV | (Pk + MR ) + (Pk + MR Y j"^ fl

VIIVJ x (t)

ч 0

and the closed loop fuzzy system (12) is stable. This is complete the proof.

V. Robust Stability Condition with PDC Controller

Consider the closed-loop system (7). A sufficient robust stability condition is given follow.

• Theorem 3

Under assumption 1, and assumption 2 and for given 0 < e< 1, the Takagi-Sugeno system (7) is stable if there exist positive definite symmetric matrices P_k,k = 1,2,.,r , andR, matrices F1,.,F_r such that the following LMIs hols.

	p. + R >- Pj + mR	0, - 0,	k e {1,.,r} j = 1,2,., r	(26) (27)
[ф1	(Pk + MR ) Dai	(Pk	+MR)DM"
*	-AI	0	ч 0
*	*	-AI		(28)
,k	e{l,., r}

Ф1 = p. + -,’ (P. + mR)+(P. + mR)-,,

+ A(Pk + mR )[ETE_m +(EbF )’ Eb,F, ]

Ф2 (Pk + MR )(D,+ Dj ) * -AI **

(Pk + MR)(^D.,+ D_bJ) 0

-AI

ч 0

for i, j, k = 1,2,., r such that i ч j with

f - + -                    - + -

Ф2 =V2) (Pk + MR) + (Pk + MR )V2

+ A (Pk + MR)[(E, + Ej )’ (E, + Ej )

+ (E_biFj + EbjF )’ (EbFj + EbjF )’ ]

r

m=1 -e, and P. = 2 .k ( p. +R)

k =1

Proof

Let consider the Lyapunov function in the following form:

r

V (x (t )) = 2 hk (z (t ))Vk (x (t)) (30) k =1

with

V (x (t )) = xT (t)(Pk + mR )x (t), k = 1,2,.,r where

Pk = p. ,R = RT, 0 < e < 1,M = 1 -e, and (Pk + mR ) - 0, k = 1,2,., r

The time derivative of V (x (t)) with respect to t along the trajectory of the system (12) is given by:

rr

V (x (t ))=2 hk (z (t ))Vk (x (t ))+2 hk (z (t ))V‘k (x (t)) k =1                                     k =1

The equation (31) can be rewritten as,

r

V (x (t ))=xT (t )| 2 hk (z (t))(Pk+ mr ) lx (t)

V k =1

• + x1 T (t )|2hk (z (t))(Pk + MR )| x (t)

V k =1

• +x’ (t )|2hk (z( t))(Pk + MR Я x( t)

V k =1

By substituting (7) into (32), we obtain,

V (x (t)) = Y1 (x ,z ) + Y2 (x,z ) + Y3 (x ,z )

where

• Y, (x,z ) = xT (t)f2hk (z (t))■ (Pk + MR)!x (t)

V k =1

rr

Y2 (x, z )= xT (t )22 hk (z (t)) h,2 (z (t))

k =1 ,=1

x {-,T (Pk + MR ) + (Pk + MR )- u } x (t )

rr

+ x’ (t )22hk (z (t ))hi (z (t )) k =1 ,=1

xK to, Dbi 1[Aa   0 JEa   1ЪPk + MR)

]VL a 6,JL 0    Ab, ]L-' .1 ]) k

+ (Pk + MR )f[ Da Db,^a   0 JEa   111x (t)

V L 0   A b i ]L Eb Ft ])

where   -j = A, -B,Fj   ,   -,, = A, -BiFi rr

^Y(x, z)= x⁽t T EEE hk⁽z⁽t)) hi⁽z⁽t)) h⁽z⁽t)) k =1 i =1 i x j

x (t)

]TФk (Pk + R)+G' (Pk + HR) + (Pk + HR)Gii k =1

+2-¹ (Pk + HR )[ Da.  Dbi]

rr

+ x⁽t )^rEEE hk⁽z⁽t)) hi⁽z⁽t)) hj⁽z⁽t)) k =1 i =1 i x j

+л Г e,.

—(EbiF)' 1

(Pk + hR ) x 0

( Га . о "|Г^

x| [ Da-         О(Pk + ^R )

Г L ^{0 л}ь. J L^-EbiFj A)

(            Гл     0 1ГE .    1)1

+ (P_k + hR)[D   D₆,    ai a       ^Jx(t)

⁽k V ai ]L 0   Лbi JL— EbiFj JJj ⁽)

by Schur complement, we obtain,

Ф1 (Pk + HR )Da.

*           -XI

**

(Pk + HR ) Dbi

-XI

with

x 0

rr

⁺x⁽t^)T EEE hk (z⁽t)) hi (z⁽t)) hj (z⁽t)) k =1 i =1 i x j

Ф1 = Pф + Gi' (Pk + HR) + (Pk + HR) G_tt

( Г_Л0 1Г E     1)'

x^J ГD   D_b 1 a a       (P_k + HR)

■ aj 4 ^{0   л}bJL— EbjFi J)⁽k )

+ P_k + hR )⁽ГD„  D_b^Г^'   ^{0 1}Ej Hx (t)

⁽k V a bj^JL ^{0   л}bj JL-EbjFi JJ

+ X( Pk + HR )L E'Eai +(Eb.F.1 EbiF-1

(G-+G- )'                 (G-+G-.

-i^--j^ I (Pk + HR) + (Pk + HR )|-^ Ij I ²)                          Г ²JJ

I (r                          лai + лa

II

I L ai a bi bj j 0        ЛА.+Л, l\                                 L

Then, based on assumption 1, an upper bound of

Y1 (x,z ) obtained as:

r

^Y1 (x, z )<E ^ф ■^x (t 1 (^p_k + HR)^x (t)    ⁽³⁷⁾

k =1

Based on (4), it follows that

r

E hik (z (t)) sR = R = 0 where R is any symmetric k =1

matrix of proper dimension.

Adding R to (34), then

r

^Y1 (x, z )<E ^ф ■^x (t) (Pk^{+ R} )^x (t) ⁽³⁸⁾

k =1

Then,

r

V (x (⁴ ))<E *kx^T (⁴ )(^Pk^{+ R} )x (⁴ )^{+ Y}2 (x , z )^{+ Y}3 (^x, z) k =1

If

r

E Фk (Pk + R)+G' (Pk + HR) + (Pk + HR) Gii k =1

[ Dai Dbi]

A .

a

л bi

E . a

- EbiFj

T

I (Pk + HR)

E . -vE .

ai aj

-EbiFj -EbjFi

T

(Pk + HR) + (Pk + HR)

(,-                           -,Гл„.+л„,

X Го +D D.-a-D.-'i ai ^aj l L ai aj bi bj j 0

^Лbi^+Лbj

X

Ea. + Ea

-EbiFj -EbjFi

x 0

Then, based on Lemma 2, an upper bound of

Y1 (x ,z ) obtained as:

(G-+G- )'                (G-+G- )

-i^^ I (Pk + HR) + (Pk + HR )Г-i2^J г                  п Г D' + DT

+X-' (Pk + H^R )L ^Dai+ ^Dj ^Dbi+ DbjKT _nT _ ^Dbi^{+ D}

+XL(Eai+ Eajj (-EbFj-EbjFi j j

Ea + Eaj

.-EbiFj -EbjFi

(Pk + HR ) x 0

+ (Pk + HR )l[Dai

Db]

A .

a

Лы

E,.

L— EbiFj

x 0

by Schur complement, we obtain,

Then, based on Lemma 2, an upper bound of

Y1 (x,z ) obtained as:

Ф2

*

*

(P_k + HR )(Dai+ Dj )

-XI

*

(Pk + HR )(Dbi + Dbj )

-XI

x0

with

(G + G Y                 f G + G 1

Ф2 = ^2J (Pk + MR)+(P + MR)^ 'j₂ji j

+ 4P + MR )[(Ea+ E, )T (Ea + E₄ ) + (EbF + E^F )T (E_tiF_y + E^F )T ]

If (28) and (29) holds, the time derivative of the fuzzy Lyapunov function is negative. Consequently, we have V (x (t)) ^ 0 and the closed loop fuzzy system (7) is stable. This is complete the proof.

И. Numerical Example

Consider the following T-S fuzzy system:

r

x( t )=E ^h-( z (t ^Ax (t)                 ⁽³⁹⁾

=1

with: r = 2

the premise functions are given by:

1 + sinx, (t)                    1 -sinx.(t)

h (x 1 (t))=     2^ ;h2(x 1 (t))=     2^ ;

Г-5 -41        Г-2 -41

• A, =             ; A, =            ;

• ¹    L-1 -2 ]     ²L 20 -2_]

37.7864 26.8058        98.5559 28.757726.8058 36.2722 ² 28.7577 22.9286

It is assumed that x 1 (t )| < -2. For ^11 = 0, ^₁₂= 0.5, ^₂₁= -0.5, and ^₂₂ = 0, we obtain

; p₂ =

W. CONCLUSION

This paper provided a new condition for the stability and stabilization of Takagi-Sugeno fuzzy systems in terms of a combination of the LMI approach and the use of non-quadratic Lyapunov function as Fuzzy Lyapunov function.

In addition, a new condition of stability of uncertain system is given for Takagi-Sugeno fuzzy systems by the use of proposed fuzzy Lyapunov function.

Acknowledgment

The authors would like to thank the anonymous reviewers for their careful reading of this paper and for their helpful comments. This work was supported by the National High Technology Research and Development Program of China under grant no. 2006AA060101.