H Control of Fuzzy Impulsive Systems with Quantized Feedback
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JOURNAL OF SOFTWARE, VOL. 4, NO. 5, JULY 2009
H f Control of Fuzzy Impulsive Systems with
Quantized Feedback Yingqi ZHAN) a, b a b
Department of Mathematics, Zhengzhou University, Zhengzhou, China College of Science, Henan University of Technology, Zhengzhou, China Email: [email protected]
Caixia LIU College of Science, Henan University of Technology, Zhengzhou, China Email: [email protected]
Abstract—This paper is concerned with the problem of f
H control of fuzzy nonlinear impulsive systems with quantized f
feedback. New results on the H feedback control are established for one class of fuzzy nonlinear uncertain impulsive systems and one class of fuzzy nonlinear impulsive systems with nonlinear uncertainties by choosing appropriately quantized strategies and applying Lyapunov function approach, respectively. Index Terms—-quantized feedback; fuzzy impulsive systems; Lyapunov functions;
H f control. I.
INTRODUCTION
Considerable efforts have been devoted to the study of quantized systems in recent years, for instance, see the papers in Brockett and Liberzon (2000), Ishii and Francis (2002, 2003), Liberzon (2003), Liu and Elia (2004), Ishii and Basar (2005), and the references therein. Among these results, mainly two approaches for studying control problems with quantized feedback are chosen, which are called static quantization policies (e.g., Delchamps, 1990; Elia and Mitter, 2001; Fu and Xie, 2005; Claudio De Persis, 2005) and dynamic quantization policies (e.g., Brockett and Liberzon, 2000; Tatikonda and Mitter, 2004). For the problem of H f control systems with quantized feedback, Guisheng Zhai, Xinkai Chen, Joe Imae and Tomoaki Kobayashi (2006) deal with analysis and design of H f feedback control systems with two quantized signals. Huijun Gao and Tongwen Chen (2008) present a new approach to quantized feedback control systems which, both single- and multiple-input cases considered, provide for stability and H f performance analysis as well as controller synthesis for discrete-time state-feedback control systems with logarithmic quantizers. The most significant feature is the utilization of a quantization dependent Lyapunov function. Francesca Ceragioli and Claudio De Persis (2007) discuss discontinuous stabilization of nonlinear systems with quantized and switching controls, i.e. considering the classical problem of stabilizing nonlinear systems in the case the © 2009 ACADEMY PUBLISHER
control laws take values in a discrete set. H f performance analysis of affine nonlinear systems is also given. Considerable attention, however, have been paid toward the study of fuzzy systems, for instance, see the papers T.Takagi and M.Sugeno (1985), Tong RM. (1977), Kazuo Tanaka, Takayuki Ikeda, and Hua O. Wang (1998) and the references therein. Among these results, a significant model is the T-S model proposed by T.Takagi and M.Sugeno (1985) which is described by fuzzy IF-THEN rules. From the point of view of system analysis, T-S model is appealing since the stability and performance characteristics of the system can be analyzed using a Lyapunov approach including the quadratic Lyapunov functions (e.g., E. Kim and H. Lee, 2000; Tanaka K, Ikeda T, and Hua O. Wang, 1998) and the nonquadratic ones (e.g., Thierry Marie Guerra and Laurent Vermeiren, 2004). In this paper, we concentrate on the problem of H f feedback control of two classes of T-S fuzzy impulsive systems via fuzzy quantized feedback. New results on the H f feedback control of one class of fuzzy nonlinear uncertain impulsive systems and one class of fuzzy nonlinear impulsive systems with nonlinear uncertainties are presented by choosing appropriately quantized strategies and applying the Lyapunov function approach, respectively. The paper is organized as follows. Section II gives the concept of quantizer, some notations and problem statement. Section III presents new results on H f feedback control of one class of fuzzy impulsive systems with non-quantized feedback. New results on analysis and design of H f control of one class of fuzzy nonlinear uncertain impulsive systems with quantized feedback are presented in Section IV. Section deals with analysis and design of H f control of one class of fuzzy nonlinear impulsive systems with nonlinear uncertainties with quantized feedback. Conclusions are presented in section VI. II.
PROBLEM
STATEMENT
In this section, some notations and definition of quantizer are introduced and the problem statement is given.
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445
Firstly, we give the definition of a quantizer with general form as in Liberzon, D. (2003), Ishii, H. and Francis, B.A. l
(2003). Let z R be the variable being quantized. A quantizer is defined as a piecewise constant function l
q : R l o D , where D is a finite subset of z R . This leads to a partition of z Rl into a finite number of quantization regions of the form {z R l : q ( z ) i}, i D. The quantization regions are not assumed to have any particular shapes. We assume that there exist positive real numbers M and ' such that the following conditions hold: z d M q ( z ) z d '; (1)
z ! M q ( z ) ! M '.
(2)
Throughout this paper, we denote by | | the standard Euclidean norm in the n-dimensional vector space R n , and denote by || || the corresponding induced matrix norm in
R nun . Condition (1) gives a bound on the quantization error when the quantizer does not saturate. Condition (2) provides a way to detect the possibility of saturation. We will refer to M and ' as the range of q and the quantization error, respectively. We also assume that { x : q ( x ) 0} for x in some neighborhood of the origin which is needed to preserve the origin as an equilibrium. In the control strategy to be developed below, we will use quantized measurements of the form as in Liberzon, D. (2003), Ishii, H. and Francis, B.A. (2003).
z
qP ( z ) : P q ( )
(3)
P
where P is an adjustable parameter, called the "zoom" variable, that is update at discrete instants of time. r
To be convenient, we denote ¦ i , j q ( x)
: hi ( x(t )) , wi : wi ( x(t )) , hi P q ( y)
hi P
1
Bk x(t ) , t
tk
0
x(t ) x0 x INPUT-OUTPUT form: r x (t ) ¦ i 1 wi [ Ai U ( Ai )]x(t ) [ Bi U ( Bi )]u (t ) Gi w(t )] r ¦ i 1 wi
¦
r
h {[ Ai U ( Ai )]x(t )
i 1 i
[ Bi U ( Bi )]u (t ) Gi w(t )} , t z tk
¦
z (t )
r i 1
wi Ci x(t )
¦
¦
'x
x(t0 ) wi
r
i
w 1 i
¦
r
¦
h Bik x(t ) : U (t , x(t )) , t
n j 1
tk
r
M ij ( x j (t )), ¦ir 1 wi ! 0, wi ! 0,
¦ ir 1 hi
,
1,
hi ! 0.
w 1 i
i
x(t ) [ x1 (t ), x1 (t ) xn (t )]T
where
(5)
h Ci x(t )
i 1 i
i 1 i
wi
hi
r
x0
is
state
vector,
u (t ) R m is control input, z (t ) R p is control output, Ai , Bi , Gi and , Ci are constant matrices and of appropriate dimensions,
U ( Ai ), U ( Bi ) are unknown disturbance matrix
which satisfy
U ( Ai ) Ei / i Fi ,|| U ( Bi ) ||d r0 , i {1, 2, r} where
(6)
/ i satisfy /Ti / i d I , i {1, 2, , r}
(7)
where
: ¦ir 1 ¦ rj 1 , hi
0 t0 t1 t2 tk , lim tk k of
f
and
: hi (qP ( x(t ))) and
'x |tk : x(tk ) x(tk )
: hi (qP ( y (t ))) .
The T-S fuzzy system, suggested by T.Takagi and M.Sugeno (1985), can represent a general class of nonlinear systems. It is based on "fuzzy partition" of input space and it can be viewed as the expansion of piecewise linear partition. Consider a uncertain impulsive nonlinear dynamic multiinput-multi-output system modeled by the T-S fuzzy system, which can be represented by the following forms: x IF-THEN form: Ri : IF x1 (t ) is M i1 and x2 (t ) is M i 2 xn (t ) is
M in
where x(tk )
lim x(tk h) and x(tk )
h o 0
lim x(tk h),
h o 0
x(tk ) , x0 Bk bI , 2 d b d 0 (k 1, 2,) Thus we can easily obtain U (tk , x(tk )) { bx(tk ).
For simplicity it is assumed that, x(tk )
0 and
For the nonlinear plant represented by (4) or (5), we consider the fuzzy controller as follows: x IF-THEN form: Ri : IF x1 (t ) is M i1 and x2 (t ) is M i 2 xn (t ) is
M in
THEN
x (t ) [ Ai U ( Ai )]x(t ) [ Bi U ( Bi )]u (t )
Gi w(t ) , z (t ) Ci x(t )
'x
THEN
u (t )
t z tk
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(4)
Li qP ( x)
x INPUT-OUTPUT form:
(8)
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¦
u (t )
r i
q ( x)
h P [ Li qP ( x)] 1 i
(9)
The system (5) with (9) can be written the form of the T-S fuzzy control system as follows:
¦
x (t )
r
q ( x)
i, j
h h P {[ H ij U ( Ai ) U ( Bi ) L j ]x(t ) 1 i j
¦
z (t )
r i
w C x(t ) 1 i i
¦
r i
w 1 i
¦
r
x(t0 )
(10)
tk
³
0
z (t ) z t dt d J
2
³
f
0
w (t ) w t dt. T
J
x (t )
¦
h h j {[ H ij U ( Ai ) U ( Bi ) L j ]x(t )
i, j 1 i
Gi w(t )} , t z tk
is
1 PGi GiT P J H r02 T 1 T T 2 Ki PEi Ei P Fi Fi +Ki , j P Lj Lj ]
2O P CiT Ci H I
2
Ki
Ki , j
(15)
1 PGi GiT P J H 1 T r02 T T 2 Ki PEi Ei P Fi Fi +Ki , j P L j L j ]. CiT Ci H I
2
Ki
Ki , j
Theorem 3.2: Assume there exist nonnegtive constant H and J ( L2 gain d J ) such that Assumption 3.1 holds, then the closed-loop system (14) is exponentially stable with an H f norm J . In order to prove Theorem 3.2, we require the following Lemma 3.3: [3]
: Assume X , Y , Z is constant real matrix of T appropriate dimension, satisfying Y Y d I , then K ! 0 , Lemma 3.3
we have
1
K
XX T .
Proof of Theorem 3.2˖We consider the Lyapunov function (13)
V ( x) xT Px for the closed-loop system (14), the derivative of V ( x) along solutions of (14) in t (tk 1 , tk ] is candidate
computed as
V ( x)
¦
r
h h j {xT (t )( H ijT P PH ij ) x(t )
i, j 1 i
2 xT (t )( U ( Ai ) U ( Bi ) L j ) x(t )
2 xT (t ) PGi w(t )}
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1
Qij : [( Ai Bi L j )T P P ( Ai Bi L j )
(12)
The system (5) with (13) can be written the form of the T-S fuzzy control system as follows: r
, a
and a common positive definite
XYZ Z T Y T X T d K Z T Z
THEN
x INPUT-OUTPUT form: r u (t ) ¦ i 1 hi [ Li x(t )]
{L }
1
positive and satisfies
(11)
STATE FEEDBACK CONTROL
Li x(t )
r i i 1
= [ H P PH ij 2O P
In this section, for the nonlinear plant represented by (4) or (5), we consider the fuzzy non-quantized feedback controller as follows: x IF-THEN form: Ri : IF x1 (t ) is M i1 and x2 (t ) is M i 2 xn (t ) is M in
u (t )
r
T ij
. III.
Ai Bi L j .
P such that the sequence of matrices {Qij }ir, j
L2 gain
Definition 2.2: The uncertain impulsive system (10) is said to be exponentially stable with an H f norm J if (1) (Internal exponential stability) Systems (10) with w(t ) 0 ( t R ), the trivial solution (equilibrium point) is exponentially stable; (2) ( L2 gain d J ) Systems (10) have L2 gain less than or equal to
tk
r
less than or equal to J ( L2 gain d J ) if T
(14)
h Ci x(t )
, two sequences of constants {Ki }i 1 ,{Ki , j }i , j
gain less than or equal to J , we require the following definitions: Definition 2.1: Suppose J is given positive real number.
f
r
i 1 i
Consider H feedback control of system (14), we require the following Assumption 3.1: Assumption 3.1: Assume that there exist a positive constant
matrix
Our main aim is to find some sufficient condition which make the closed-loop system (10) exponentially stable and L2
The uncertain impulsive system (10) is said to have
w 1 i
sequence of matrices
Ai Bi L j .
H ij denotes H ij
i
¦
'x U (t , x(t )) bx(t ) , t
J !0
x0
where
¦
r
f
i 1 i
'x U (t , x(t )) bx(t ) , t
wi Ci x(t )
x(t ) x0 where H ij denotes H ij
P
h Ci x(t )
r i 1
0
x x Gi w(t ) [ Bi U ( Bi )]L j P[q ( ) ]} , t z tk
P
¦
z (t )
(16)
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By Lemma 3.3, we obtain
r
T
d ¦ i , j 1 hi h j { xT (t )Qij x(t )
T
2 x P U ( Ai ) x
2 x PEi / i FLi x 1 d Ki xT PEi EiT Px xT FiT Fi x
(J 2 H )[( w
Ki
d Ki , j xT P 2 x
1
Ki , j r02
Ki , j
r
xT LTj U T ( Bi ) U ( Bi ) L j x
xT LTj L j x
Let
H (t ) : V ( x) 2OV ( x) z T z J 2 wT w H ( xT x wT w)
d0 That is to say
V ( x) 2OV ( x) z T z J 2 wT w d H ( xT x wT w) d0, t (tk 1 , tk ]. Hence, T (tk 1 , tk ], integrating (17), we have
³
T
0
We have
¦
r i, j
h h {xT (t )( H ijT P PH ij 1 i j
2O P CiT Ci H I ) x(t ) 2 x (t )[ U ( Ai ) U ( Bi ) L j ]x(t ) T
2
T
2 x (t ) PGi w(t ) (J H ) w w}
0
³ ( z T z J 2 wT w)dt
2 xT (t ) PGi w(t ) (J 2 H ) wT w 1 1 (J 2 H )[( w 2 GiT Px)T ( w 2 GiT Px)] J H J H 1 2 xT PGi GiT Px J H
d0
By virtue of 'x |tk : x(tk ) x(tk )
bx(tk ), we have
x(tk ) bx(tk ) x(tk ) (1 b) x(tk ) and
xT (tk ) Px(tk ) xT (tk ) Px(tk ) xT (tk ) Px(tk ) (1 b 2 ) xT (tk ) Px(tk ) b(2 b)V ( x(tk )) Hence
³
Therefore, by (15), we obtain
H (t ) |(14) h h {xT (t )( H ijT P PH ij 1 i j
1 PGi GiT P) x(t ) 2 J H T 2 x (t )[ U ( Ai ) U ( Bi ) L j ]x(t ) 2O P CiT Ci H I
1 1 (J H )[( w 2 GiT Px)T ( w 2 G T Px)]} J H J H i 2
r
d ¦ i , j 1 hi h j {xT (t )( H ijT P PH ij 2O P 1 PGi GiT P Ki PEi EiT P 2 J H r2 1 Fi T Fi Ki , j P 2 0 LTj L j ) x(t ) CiT Ci H I
Ki
(J 2 H )[( w
(18)
0
V ( x(tk )) V ( x(tk ))
However
i, j
T V ( x) |(14) dt 2O ³ V ( x) |(14) dt
T
r
(17)
T
H (t ) |(14)
¦
1 1 GiT Px)T ( w 2 GiT Px)]} J H J H 2
d ¦ i , j 1 hi h j { xT (t )Qij x(t ) }
2 xT P U ( Bi ) L j x
d Ki , j xT P 2 x
447
Ki , j
1 1 GiT Px)T ( w 2 GiT Px)]} J H J H 2
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T
0
³
t1
0
V ( x) |(14) dt
t1 V ( x) |(14) dt ³ V ( x) |(14) dt 0
tk
T ³ V ( x) |(14) dt ³ V ( x) |(14) dt tk 1
tk
V ( x(t1 )) V ( x(0 )) V ( x(t2 )) V ( x(t1 )) V ( x(tk )) V ( x(tk1 )) V ( x(T )) V ( x(tk )) V ( x(t1 )) V ( x(t1 )) V ( x(tk )) V ( x(tk )) V ( x(T )) k
¦ i 1 b(2 b)V ( x(ti )) V ( x(T )) t0 Therefore, we can obtain
³
T
0
( z T z J 2 wT w)dt d 0
That is to say
(19)
448
³
T
0
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T
z T zdt d J 2 ³ wT wdt
H (t ) |(10)
(20)
0
¦
When w
0 , one has V ( x) 2OV ( x) d 0 , t (tk 1 , tk ].
(21)
Hence, we can easily achieve that system (14) is exponentially stable with an H f norm J . IV.
QUANTIZED FEEDBACK CONTROL
Considering the H f feedback control of nonlinear uncertain fuzzy impulsive systems (5) with quantized controller law (9), we have the following result: Theorem 4.1: Assume there exist nonnegtive constant H and J ( L2 gain d J ) such that Assumption 3.1 holds; Moreover, assume that for arbitrary fixed V ! 0 , chosen large enough such that
M is
M ! 4 x ' (1 V )
(22)
where
4x : 2
O
1 PGi GiT P) x(t ) J H T 2 x (t )[ U ( Ai ) U ( Bi ) L j ]x(t ) 2O P CiT Ci H I
1 1 GiT Px)T ( w 2 GiT Px)] J H J H x x 2 xT (t ) P[ Bi U ( Bi )]L j P[q( ) ]}
(J 2 H )[( w
P
d ¦ i , j 1 hi h j { xT (t )Qij x(t ) x x 2 xT (t ) P[ Bi U ( Bi )]L j P[q ( ) ]
P
V ( x) xT Px for the closed-loop system (10), the derivative of V ( x) along solutions of (10) in t (tk 1 , tk ] is computed as q ( x)
h h j P {xT (t )( H ijT P PH ij ) x(t )
i, j 1 i
2 xT (t )( U ( Ai ) U ( Bi ) L j ) x(t )
(23)
2 xT (t ) PGi w(t )
x x 2 xT (t ) P[ Bi U ( Bi )]L j P[q ( ) ]}
P
H (t ) |(10) :
¦
h hj
i, j 1 i
{x (t )( H P PH ij
T i
2O P C Ci H I ) x(t ) 2 xT (t )[ U ( Ai ) U ( Bi ) L j ]x(t ) 2 xT (t ) PGi w(t ) (J 2 H ) wT w x x 2 xT (t ) P[ Bi U ( Bi )]L j P[q ( ) ]}
P
Therefore, by (15), we obtain
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P
P
{O | x | (| x | 4 x P')}
According to (22), for any nonzero x , we can find a positive scalar P such that
4 x (1 V ) P' d| x |d M P
(24)
This is also true in the case of x 0 , where we set P as an extreme case and consider the output of the quantizer as zero. Hence, there exists a control strategy for P , such that H (t ) |(10) d 0 . Repeating the approach of the above proof of Theorem 3.2, we can achieve the fuzzy impulsive system (10) exists a control strategy for P such that the closed-loop system (10) is exponentially stable with an H f norm J . V.
T ij
qP ( x )
O | x | (| x | 4 x P')
We have T
1 1 GiT Px)T ( w 2 GiT Px)]} J H J H
q ( x)
d ¦ i , j 1 hi h j
candidate
qP ( x )
P
2
P
r
Proof of Theorem 4.1˖We consider the Lyapunov function
r
P
x x 2 xT (t ) P[ Bi U ( Bi )]L j P[q ( ) ]}
Then there exists a control strategy P which depends on the state, and makes the closed-loop system (10) exponentially stable with an H f norm J .
P
2
r
r
|| B ||: max{|| Bi ||: i 1, 2, , r}.
r
2
d ¦ i , j 1 hi h j P { xT (t )Qij x(t )
|| L ||: max{|| L j ||: j 1, 2, , r}
¦
q ( x)
h h j P {xT (t )( H ijT P PH ij
i, j 1 i
(J 2 H )[( w
|| P |||| L || (|| B || r0 )
O : min{O (Qij ) : i, j 1, 2, , r}
V ( x)
r
FUZZY NONLINEAR IMPULSIVE SYSTEMS WITH NONLINEAR UNCERTAINTIES
Consider a nonlinear dynamic multi-input-multi-output system with nonlinear uncertainty modeled by the T-S fuzzy system, which can be represented by the following forms: x IF-THEN form: Ri : IF x1 (t ) is M i1 and x2 (t ) is M i 2 xn (t ) is
M in THEN
x (t )
Ai x(t ) Bi u (t )
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Gi w(t ) fi ( x, t ) , z (t ) Ci x(t ) 'x Bk x(t ) , t tk
449
t z tk
i
0
x(t )
¦
i 1
wi Ci x(t )
¦
¦
'x 0
x(t ) wi
r
w 1 i
r
j 1
r i 1
¦
¦
,
r i 1
hi
tk
¦
x(t0 ) where
wi
(27)
where g (t ) is known continuous function. We will give the sufficient conditions of the H f control (26). For the nonlinear plant represented by (25) or (26), we consider the fuzzy controller as follows: x IF-THEN form: Ri : IF x1 (t ) is M i1 and x2 (t ) is M i 2 xn (t ) is M in
Li x(t ) Li qP ( x)
r qP ( x ) i 1 i
h
[ Li qP ( x)]
i, j
h h [( Ai Bi L j ) x(t ) Gi w(t ) fi ( x, t )] 1 i j h h j [ H ij x(t ) Gi w(t ) fi ( x, t )] , t z tk
i, j 1 i
i
w 1 i
¦
r
h Ci x(t )
i 1 i
tk
Ai Bi L j .
{Ki }ir 1 , , a sequence of
two sequences of constants
matrices
{Li }ir 1 and a common positive definite matrix P
(t )}r is positive and such that the sequence of matrices {Q ij i, j 1 satisfies
(t ) : [( A B L )T P P( A B L ) Q i i j i i j ij 2O P CiT Ci 1 (H g (t )) I Ki P 2
Ki
1 PGi GiT P] J H 2
T ij
[ H P PH ij 2O P CiT Ci
(H
1
Ki
g (t )) I Ki P 2
(34)
1 PGi GiT P]. J H 2
(30)
(31)
the closed-loop system (32) is exponentially stable with an H f norm J . Proof of Theorem 5.1˖We consider the Lyapunov function
V ( x) xT Px for the closed-loop system (32), the derivative of V ( x) along solutions of (32) in t (tk 1 , tk ] is candidate
computed as
V ( x)
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r
Theorem 5.1: Assume there exist nonnegtive constant H and J ( L2 gain d J ) such that Assumption 5.1 holds, then
The system (26) with (30) or (26) with (31) can be written the form of the T-S fuzzy control system as follows: r
wi Ci x(t )
(33)
J !0,
(29)
or
¦ ¦
r i 1
P
Consider H f feedback control of system (33), we require the following Assumption 5.1: Assumption 5.1: Assume that there exist a positive constant
(28)
x INPUT-OUTPUT form: r u (t ) ¦ i 1 hi [ Li x(t )]
¦
{H ij x(t ) Gi w(t ) ,
H ij denotes H ij
or
u (t )
P
x0
THEN
r
hh
i, j 1 i
qij ( x ) j
'x U (t , x(t )) bx(t ) , t
hi ! 0.
f iT ( x, t ) fi ( x, t ) d xT g (t ) x, i {1, 2, , r}
x (t )
r
x x f i ( x, t ) Bi L j P[q( ) ]} , t z tk z (t )
and impulsive properties are the same as system (4). We consider f i ( x, t ) satisfies
u (t )
q ( x)
P
wi ! 0, wi ! 0,
1,
tk
h h j P {( Ai Bi L j ) x(t ) Gi w(t ) fi ( x, t )
(26)
fi ( x, t ) is nonlinear component, the other variables
u (t )
r
¦ M ij ( x j (t )), ¦
(32)
h Ci x(t )
x x Bi L j P[q( ) ]}
h Ci x(t )
r i 1
r
i 1 i
i, j 1 i
i 1 i
h Bik x(t ) : U (t , x(t )) , t
n
¦
x (t )
x0
¦
where
i
¦
w 1 i
P
i 1 i
wi
hi
r
i
¦
or
Gi w(t ) fi ( x, t )] , t z tk z (t )
r
x0
h [ Ai x(t ) Bi u (t ) 1 i
r
wi Ci x(t )
'x U (t , x(t )) bx(t ) , t
x(t ) x0 x INPUT-OUTPUT form: r x (t ) ¦ i 1 wi [ Ai x(t ) Biu (t ) Gi w(t ) fi ( x, t )] r ¦ i 1 wi
¦
r i 1
¦
(25)
0
r
¦
z (t )
¦
r
h h j {xT (t )( H ijT P PH ij ) x(t )
i, j 1 i
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2 xT (t ) Pfi ( x, t ) 2 xT (t ) PGi w(t )}
(35)
V ( x)
¦
By Lemma 3.3, we obtain T
T
2
2 x (t ) Pfi ( x, t ) d x [Ki P
1
Ki
r
q ( x)
h h j P {xT (t )( H ijT P PH ij ) x(t )
i, j 1 i
2 xT (t ) Pfi ( x, t ) 2 xT (t ) PGi w(t ) x x 2 xT (t ) PBi L j P[q( ) ]}
g (t ) I ]x
P
Let
H (t ) : V ( x) 2OV ( x) z T z J 2 wT w H ( xT x wT w)
H (t ) : V ( x) 2OV ( x) z T z J 2 wT w H ( xT x wT w) We have
H (t ) |(32)
H (t ) |(33)
r
d ¦ i , j 1 hi h j {xT (t )( H ijT P PH ij 2O P CiT Ci 1
Ki
g (t )) I Ki P 2
(J 2 H )[( w
¦
r i, j
r
d ¦ i, j
1 PGi GiT P ) x (t ) J H 2
2O P CiT Ci (H
1 1 GiT Px)T ( w 2 GiT Px )]} J H J H
x x 2 xT (t ) PBi L j P[q( ) ]}
(t )x(t ) h h { xT (t )Q ij 1 i j
P
such that (38)
Therefore, by (34), we obtain
H (t ) |(33)
d0 Hence repeating the approach of the above proof of Theorem 3.2, we can achieve the system (26) with controller law (30) exists a control strategy for P such that the closedloop system (32) is exponentially stable with an H norm J . Theorem 5.2: Assume there exist nonnegtive constant H and J ( L2 gain d J ) such that Assumption 5.1 holds; f
Moreover, assume that for arbitrary fixed V ! 0 , chosen large enough such that
M is (36)
where
x : 2T 4 O (t )) : i, j 1, 2, , r ; t t 0} O : min{O (Q ij
T : max{|| PBi L j ||: i, j 1, 2, , r}. Then there exists a control strategy P which depends on the state, and makes the closed-loop system (33) exponentially stable with an H f norm J . Proof of Theorem 5.2˖We consider the Lyapunov function
V ( x) x Px for the closed-loop system (33), the derivative of V ( x) along solutions of (33) in t (tk 1 , tk ] is computed as
P
x (1 V ) P' d| x |d M P 4
(t )x(t )} h h { x (t )Q ij 1 i j
candidate
P
According to (36), we can find a positive scalar
T
T
1
g (t )) I Ki P 2 ]x(t ) Ki 2 xT (t ) PGi w(t ) (J 2 H ) wT w
2
x ' (1 V ) M !4
q ( x)
d ¦ i , j 1 hi h j P {xT (t )[ H ijT P PH ij
1 1 (J 2 H )[( w 2 GiT Px)T ( w 2 G T Px )]} J H J H i r
P
Let
Therefore, by (34), we obtain
(H
(37)
r
q ( x)
d ¦ i , j 1 hi h j P {xT (t )[ H ijT P PH ij 2O P CiT Ci (H
1
g (t )) I Ki P 2
Ki
(J 2 H )[( w
1 PGi GiT P]x(t ) J H 2
1 1 GiT Px )T ( w 2 G T Px )] J H J H i 2
x x 2 xT (t ) PBi L j P[q( ) ]}
P
¦
r
P
q ( x) (t ) x(t ) h h j P { xT (t )Q ij
i, j 1 i
(J 2 H )[( w
1 1 GiT Px )T ( w 2 G T Px )] J H J H i 2
x x 2 xT (t ) PBi L j P[q( ) ]}
P
r
qP ( x )
d ¦ i , j 1 hi h j
P
(t ) x(t ) { xT (t )Q ij
x x 2 xT (t ) PBi L j P[q( ) ]}
P
P
x P' ) d 0 d O | x | (| x | 4 Hence, there exists a control strategy for P , such that H (t ) |(33) d 0 . Repeating the approach of the above proof of Theorem 3.2, we can achieve the fuzzy impulsive system (33)
© 2009 ACADEMY PUBLISHER
JOURNAL OF SOFTWARE, VOL. 4, NO. 5, JULY 2009
exists a control strategy for
P
such that the closed-loop f
system (33) is exponentially stable with an H norm VI.
451
J.
CONCLUSIONS
In this paper, we discuss the problem of H f control of two classes of fuzzy nonlinear impulsive systems with quantized feedback. New results on the H f control problems are established for one class of fuzzy nonlinear uncertain impulsive systems and one class of fuzzy nonlinear impulsive systems with nonlinear uncertainties by choosing appropriately quantized strategies, respectively. ACKNOWLEDGMENTS The research of this paper was supported by the National Natural Science Foundation of P.R. China under Grant No. 60874006, by the National Natural Science Foundation of P.R. China (Mathematics Tianyuan Foundation) under Grant No. 10826078, and by Foundation of Henan University of Technology under Grant No. 07XJC013. The authors are very grateful to all the anonymous reviewers and the editors for their helpful comments and suggestions. REFERENCES [1] Delchamps, D. F., “Stabilizing a linear system with quantized state feedback,” IEEE Transactions on Automatic Control, vol. 35, pp. 916-924, August 1990. [2] Wong, W. S. and Brockett, R. W., “Systems with finite communication bandwidth constraints. II. Stabilization with limited information feedback,” IEEE Transactions on Automatic Control, vol. 44, pp. 1049-1053, May 1999. [3] Carlos E. de Souza and Xi Li, “Delay-dependent robust H f control of uncertain linear state-delayed systems,” Automatica, vol. 35, pp. 1313-1321, 1999. [4] Brockett, R. W. and Liberzon, D., “Quantized feedback stabilization of linear systems,” IEEE Transactions on Automatic Control, vol. 45, pp. 1279-1287, July 2000. [5] Elia, N. and Mitter, S. K., “Stabilization of linear systems with limited information,” IEEE Transactions on Automatic Control, vol. 46, pp.1384-1400, Sept. 2001. [6] Liberzon, D., “Hybrid feedback stabilization of systems with quantized signals,” Automatica, vol 39, pp. 1543-1554, Sept. 2003. [7] Ishii, H. and Francis, B. A.., “Quadratic stabilization of sampled-data systems with quantization,” Automatica, vol. 39, pp. 1793-1800, Oct. 2003. [8] Fagnani, F. and Zampieri, S., “Stability analysis and synthesis for scalar linear systems with a quantized feedback,” IEEE Transactions on Automatic Control, vol. 48, pp. 1569-1584, Sept. 2003.
© 2009 ACADEMY PUBLISHER
[9] Liu, J.and Elia, N., “Quantized feedback stabilization of nonlinear affine systems,” International Journal of Control, vol. 77, pp. 239-149, March 2004. [10] Tatikonda, S. and Mitter, S., “Control under communication constraints,” IEEE Transactions on Automatic Control, vol. 49, pp.1056-1068, July 2004. [11] Ling, Q. and Lemmon, M. D., “Stability of quantized control systems under dynamic bit assignment,” IEEE Transactions on Automatic Control, vol. 50, pp. 734-1740, May 2005. [12] Fu, M., and Xie, L., “The sector bound approach to quantized feedback control,” IEEE Transactions on Automatic Control, vol. 50, pp. 1698-1711, Nov. 2005. [13] Claudio De Persis, “On feedback stabilization of nonlinear systems under quantization,” Proceedings of the 44th IEEE CDC, 2005, 7698-7703. [14] Guisheng Zhai; Xinkai Chen; Joe Imae; Tomoaki Kobayashi, f “Analysis and Design of H Feedback Control Systems with Two Quantized Signals,” Proceedings of the 2006 IEEE International Conference on ICNSC, pp. 346-350, April 2006. [15] Francesca Ceragioli and Claudio De Persis, “Discontinuous stabilization of nonlinear systems: Quantized and switching controls,” Systems and Control Letters, Vol. 56, pp. 461-473, 2007. [16] Huijun Gao and Tongwen Chen, “A new approach to quantized feedback control systems,” Automatica, vol. 44, pp.534-542, Feb. 2008. [17] Zhang Ying-qi, Liu Cai-xia, Mu Xiao-wu, “Stability Analysis and Observer Design for Lipschitz Nonlinear Neutral-type Delay System with Uncertainties,” Henan Science, Vol. 27, pp. 6-10, Jan. 2009. [18] T.Takagi and M.Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans, Syst.,Man, Cybern., vol . SMC-15, pp.116-132, Jan. 1985. [19] Y.Y. Cao and P.M. Frank, “Analysis and synthesis of nonlinear time delay systems via fuzzy control approach,” IEEE Trans. Fuzzy Syst., vol 8, pp. 200-211, Feb. 2000. [20] E.Kim and H.Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 523-534, May 2000. [21] T. Taniguchi, K. Tanaka, H. Ohtake, H.W. Wang, “Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems,” IEEE Trans. Fuzzy Syst., 2001, pp. 525-538. [22] Kazuo Tanaka, Takayuki Ikeda, and Hua O. Wang, “Fuzzy Regulators and Fuzzy Observers: Relaxed Stability Conditions and LMI-Based Designs,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 250-265, Feb. 1998. [23] Thierry Marie Guerra, Laurent Vermeiren, “LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form,” IEEE Trans. Syst., Man, Cybern. B, Cybern, vol .40, pp. 823-829, 2004. [24] Tong RM., “A control engineering review of fuzzy systems,” Automatica, Vol .13, pp.559-568, 1977. [25] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets Syst., vol. 45, pp. 135-156, Feb.1992.
JOURNAL OF SOFTWARE, VOL. 4, NO. 5, JULY 2009
H f Control of Fuzzy Impulsive Systems with
Quantized Feedback Yingqi ZHAN) a, b a b
Department of Mathematics, Zhengzhou University, Zhengzhou, China College of Science, Henan University of Technology, Zhengzhou, China Email: [email protected]
Caixia LIU College of Science, Henan University of Technology, Zhengzhou, China Email: [email protected]
Abstract—This paper is concerned with the problem of f
H control of fuzzy nonlinear impulsive systems with quantized f
feedback. New results on the H feedback control are established for one class of fuzzy nonlinear uncertain impulsive systems and one class of fuzzy nonlinear impulsive systems with nonlinear uncertainties by choosing appropriately quantized strategies and applying Lyapunov function approach, respectively. Index Terms—-quantized feedback; fuzzy impulsive systems; Lyapunov functions;
H f control. I.
INTRODUCTION
Considerable efforts have been devoted to the study of quantized systems in recent years, for instance, see the papers in Brockett and Liberzon (2000), Ishii and Francis (2002, 2003), Liberzon (2003), Liu and Elia (2004), Ishii and Basar (2005), and the references therein. Among these results, mainly two approaches for studying control problems with quantized feedback are chosen, which are called static quantization policies (e.g., Delchamps, 1990; Elia and Mitter, 2001; Fu and Xie, 2005; Claudio De Persis, 2005) and dynamic quantization policies (e.g., Brockett and Liberzon, 2000; Tatikonda and Mitter, 2004). For the problem of H f control systems with quantized feedback, Guisheng Zhai, Xinkai Chen, Joe Imae and Tomoaki Kobayashi (2006) deal with analysis and design of H f feedback control systems with two quantized signals. Huijun Gao and Tongwen Chen (2008) present a new approach to quantized feedback control systems which, both single- and multiple-input cases considered, provide for stability and H f performance analysis as well as controller synthesis for discrete-time state-feedback control systems with logarithmic quantizers. The most significant feature is the utilization of a quantization dependent Lyapunov function. Francesca Ceragioli and Claudio De Persis (2007) discuss discontinuous stabilization of nonlinear systems with quantized and switching controls, i.e. considering the classical problem of stabilizing nonlinear systems in the case the © 2009 ACADEMY PUBLISHER
control laws take values in a discrete set. H f performance analysis of affine nonlinear systems is also given. Considerable attention, however, have been paid toward the study of fuzzy systems, for instance, see the papers T.Takagi and M.Sugeno (1985), Tong RM. (1977), Kazuo Tanaka, Takayuki Ikeda, and Hua O. Wang (1998) and the references therein. Among these results, a significant model is the T-S model proposed by T.Takagi and M.Sugeno (1985) which is described by fuzzy IF-THEN rules. From the point of view of system analysis, T-S model is appealing since the stability and performance characteristics of the system can be analyzed using a Lyapunov approach including the quadratic Lyapunov functions (e.g., E. Kim and H. Lee, 2000; Tanaka K, Ikeda T, and Hua O. Wang, 1998) and the nonquadratic ones (e.g., Thierry Marie Guerra and Laurent Vermeiren, 2004). In this paper, we concentrate on the problem of H f feedback control of two classes of T-S fuzzy impulsive systems via fuzzy quantized feedback. New results on the H f feedback control of one class of fuzzy nonlinear uncertain impulsive systems and one class of fuzzy nonlinear impulsive systems with nonlinear uncertainties are presented by choosing appropriately quantized strategies and applying the Lyapunov function approach, respectively. The paper is organized as follows. Section II gives the concept of quantizer, some notations and problem statement. Section III presents new results on H f feedback control of one class of fuzzy impulsive systems with non-quantized feedback. New results on analysis and design of H f control of one class of fuzzy nonlinear uncertain impulsive systems with quantized feedback are presented in Section IV. Section deals with analysis and design of H f control of one class of fuzzy nonlinear impulsive systems with nonlinear uncertainties with quantized feedback. Conclusions are presented in section VI. II.
PROBLEM
STATEMENT
In this section, some notations and definition of quantizer are introduced and the problem statement is given.
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445
Firstly, we give the definition of a quantizer with general form as in Liberzon, D. (2003), Ishii, H. and Francis, B.A. l
(2003). Let z R be the variable being quantized. A quantizer is defined as a piecewise constant function l
q : R l o D , where D is a finite subset of z R . This leads to a partition of z Rl into a finite number of quantization regions of the form {z R l : q ( z ) i}, i D. The quantization regions are not assumed to have any particular shapes. We assume that there exist positive real numbers M and ' such that the following conditions hold: z d M q ( z ) z d '; (1)
z ! M q ( z ) ! M '.
(2)
Throughout this paper, we denote by | | the standard Euclidean norm in the n-dimensional vector space R n , and denote by || || the corresponding induced matrix norm in
R nun . Condition (1) gives a bound on the quantization error when the quantizer does not saturate. Condition (2) provides a way to detect the possibility of saturation. We will refer to M and ' as the range of q and the quantization error, respectively. We also assume that { x : q ( x ) 0} for x in some neighborhood of the origin which is needed to preserve the origin as an equilibrium. In the control strategy to be developed below, we will use quantized measurements of the form as in Liberzon, D. (2003), Ishii, H. and Francis, B.A. (2003).
z
qP ( z ) : P q ( )
(3)
P
where P is an adjustable parameter, called the "zoom" variable, that is update at discrete instants of time. r
To be convenient, we denote ¦ i , j q ( x)
: hi ( x(t )) , wi : wi ( x(t )) , hi P q ( y)
hi P
1
Bk x(t ) , t
tk
0
x(t ) x0 x INPUT-OUTPUT form: r x (t ) ¦ i 1 wi [ Ai U ( Ai )]x(t ) [ Bi U ( Bi )]u (t ) Gi w(t )] r ¦ i 1 wi
¦
r
h {[ Ai U ( Ai )]x(t )
i 1 i
[ Bi U ( Bi )]u (t ) Gi w(t )} , t z tk
¦
z (t )
r i 1
wi Ci x(t )
¦
¦
'x
x(t0 ) wi
r
i
w 1 i
¦
r
¦
h Bik x(t ) : U (t , x(t )) , t
n j 1
tk
r
M ij ( x j (t )), ¦ir 1 wi ! 0, wi ! 0,
¦ ir 1 hi
,
1,
hi ! 0.
w 1 i
i
x(t ) [ x1 (t ), x1 (t ) xn (t )]T
where
(5)
h Ci x(t )
i 1 i
i 1 i
wi
hi
r
x0
is
state
vector,
u (t ) R m is control input, z (t ) R p is control output, Ai , Bi , Gi and , Ci are constant matrices and of appropriate dimensions,
U ( Ai ), U ( Bi ) are unknown disturbance matrix
which satisfy
U ( Ai ) Ei / i Fi ,|| U ( Bi ) ||d r0 , i {1, 2, r} where
(6)
/ i satisfy /Ti / i d I , i {1, 2, , r}
(7)
where
: ¦ir 1 ¦ rj 1 , hi
0 t0 t1 t2 tk , lim tk k of
f
and
: hi (qP ( x(t ))) and
'x |tk : x(tk ) x(tk )
: hi (qP ( y (t ))) .
The T-S fuzzy system, suggested by T.Takagi and M.Sugeno (1985), can represent a general class of nonlinear systems. It is based on "fuzzy partition" of input space and it can be viewed as the expansion of piecewise linear partition. Consider a uncertain impulsive nonlinear dynamic multiinput-multi-output system modeled by the T-S fuzzy system, which can be represented by the following forms: x IF-THEN form: Ri : IF x1 (t ) is M i1 and x2 (t ) is M i 2 xn (t ) is
M in
where x(tk )
lim x(tk h) and x(tk )
h o 0
lim x(tk h),
h o 0
x(tk ) , x0 Bk bI , 2 d b d 0 (k 1, 2,) Thus we can easily obtain U (tk , x(tk )) { bx(tk ).
For simplicity it is assumed that, x(tk )
0 and
For the nonlinear plant represented by (4) or (5), we consider the fuzzy controller as follows: x IF-THEN form: Ri : IF x1 (t ) is M i1 and x2 (t ) is M i 2 xn (t ) is
M in
THEN
x (t ) [ Ai U ( Ai )]x(t ) [ Bi U ( Bi )]u (t )
Gi w(t ) , z (t ) Ci x(t )
'x
THEN
u (t )
t z tk
© 2009 ACADEMY PUBLISHER
(4)
Li qP ( x)
x INPUT-OUTPUT form:
(8)
446
JOURNAL OF SOFTWARE, VOL. 4, NO. 5, JULY 2009
¦
u (t )
r i
q ( x)
h P [ Li qP ( x)] 1 i
(9)
The system (5) with (9) can be written the form of the T-S fuzzy control system as follows:
¦
x (t )
r
q ( x)
i, j
h h P {[ H ij U ( Ai ) U ( Bi ) L j ]x(t ) 1 i j
¦
z (t )
r i
w C x(t ) 1 i i
¦
r i
w 1 i
¦
r
x(t0 )
(10)
tk
³
0
z (t ) z t dt d J
2
³
f
0
w (t ) w t dt. T
J
x (t )
¦
h h j {[ H ij U ( Ai ) U ( Bi ) L j ]x(t )
i, j 1 i
Gi w(t )} , t z tk
is
1 PGi GiT P J H r02 T 1 T T 2 Ki PEi Ei P Fi Fi +Ki , j P Lj Lj ]
2O P CiT Ci H I
2
Ki
Ki , j
(15)
1 PGi GiT P J H 1 T r02 T T 2 Ki PEi Ei P Fi Fi +Ki , j P L j L j ]. CiT Ci H I
2
Ki
Ki , j
Theorem 3.2: Assume there exist nonnegtive constant H and J ( L2 gain d J ) such that Assumption 3.1 holds, then the closed-loop system (14) is exponentially stable with an H f norm J . In order to prove Theorem 3.2, we require the following Lemma 3.3: [3]
: Assume X , Y , Z is constant real matrix of T appropriate dimension, satisfying Y Y d I , then K ! 0 , Lemma 3.3
we have
1
K
XX T .
Proof of Theorem 3.2˖We consider the Lyapunov function (13)
V ( x) xT Px for the closed-loop system (14), the derivative of V ( x) along solutions of (14) in t (tk 1 , tk ] is candidate
computed as
V ( x)
¦
r
h h j {xT (t )( H ijT P PH ij ) x(t )
i, j 1 i
2 xT (t )( U ( Ai ) U ( Bi ) L j ) x(t )
2 xT (t ) PGi w(t )}
© 2009 ACADEMY PUBLISHER
1
Qij : [( Ai Bi L j )T P P ( Ai Bi L j )
(12)
The system (5) with (13) can be written the form of the T-S fuzzy control system as follows: r
, a
and a common positive definite
XYZ Z T Y T X T d K Z T Z
THEN
x INPUT-OUTPUT form: r u (t ) ¦ i 1 hi [ Li x(t )]
{L }
1
positive and satisfies
(11)
STATE FEEDBACK CONTROL
Li x(t )
r i i 1
= [ H P PH ij 2O P
In this section, for the nonlinear plant represented by (4) or (5), we consider the fuzzy non-quantized feedback controller as follows: x IF-THEN form: Ri : IF x1 (t ) is M i1 and x2 (t ) is M i 2 xn (t ) is M in
u (t )
r
T ij
. III.
Ai Bi L j .
P such that the sequence of matrices {Qij }ir, j
L2 gain
Definition 2.2: The uncertain impulsive system (10) is said to be exponentially stable with an H f norm J if (1) (Internal exponential stability) Systems (10) with w(t ) 0 ( t R ), the trivial solution (equilibrium point) is exponentially stable; (2) ( L2 gain d J ) Systems (10) have L2 gain less than or equal to
tk
r
less than or equal to J ( L2 gain d J ) if T
(14)
h Ci x(t )
, two sequences of constants {Ki }i 1 ,{Ki , j }i , j
gain less than or equal to J , we require the following definitions: Definition 2.1: Suppose J is given positive real number.
f
r
i 1 i
Consider H feedback control of system (14), we require the following Assumption 3.1: Assumption 3.1: Assume that there exist a positive constant
matrix
Our main aim is to find some sufficient condition which make the closed-loop system (10) exponentially stable and L2
The uncertain impulsive system (10) is said to have
w 1 i
sequence of matrices
Ai Bi L j .
H ij denotes H ij
i
¦
'x U (t , x(t )) bx(t ) , t
J !0
x0
where
¦
r
f
i 1 i
'x U (t , x(t )) bx(t ) , t
wi Ci x(t )
x(t ) x0 where H ij denotes H ij
P
h Ci x(t )
r i 1
0
x x Gi w(t ) [ Bi U ( Bi )]L j P[q ( ) ]} , t z tk
P
¦
z (t )
(16)
JOURNAL OF SOFTWARE, VOL. 4, NO. 5, JULY 2009
By Lemma 3.3, we obtain
r
T
d ¦ i , j 1 hi h j { xT (t )Qij x(t )
T
2 x P U ( Ai ) x
2 x PEi / i FLi x 1 d Ki xT PEi EiT Px xT FiT Fi x
(J 2 H )[( w
Ki
d Ki , j xT P 2 x
1
Ki , j r02
Ki , j
r
xT LTj U T ( Bi ) U ( Bi ) L j x
xT LTj L j x
Let
H (t ) : V ( x) 2OV ( x) z T z J 2 wT w H ( xT x wT w)
d0 That is to say
V ( x) 2OV ( x) z T z J 2 wT w d H ( xT x wT w) d0, t (tk 1 , tk ]. Hence, T (tk 1 , tk ], integrating (17), we have
³
T
0
We have
¦
r i, j
h h {xT (t )( H ijT P PH ij 1 i j
2O P CiT Ci H I ) x(t ) 2 x (t )[ U ( Ai ) U ( Bi ) L j ]x(t ) T
2
T
2 x (t ) PGi w(t ) (J H ) w w}
0
³ ( z T z J 2 wT w)dt
2 xT (t ) PGi w(t ) (J 2 H ) wT w 1 1 (J 2 H )[( w 2 GiT Px)T ( w 2 GiT Px)] J H J H 1 2 xT PGi GiT Px J H
d0
By virtue of 'x |tk : x(tk ) x(tk )
bx(tk ), we have
x(tk ) bx(tk ) x(tk ) (1 b) x(tk ) and
xT (tk ) Px(tk ) xT (tk ) Px(tk ) xT (tk ) Px(tk ) (1 b 2 ) xT (tk ) Px(tk ) b(2 b)V ( x(tk )) Hence
³
Therefore, by (15), we obtain
H (t ) |(14) h h {xT (t )( H ijT P PH ij 1 i j
1 PGi GiT P) x(t ) 2 J H T 2 x (t )[ U ( Ai ) U ( Bi ) L j ]x(t ) 2O P CiT Ci H I
1 1 (J H )[( w 2 GiT Px)T ( w 2 G T Px)]} J H J H i 2
r
d ¦ i , j 1 hi h j {xT (t )( H ijT P PH ij 2O P 1 PGi GiT P Ki PEi EiT P 2 J H r2 1 Fi T Fi Ki , j P 2 0 LTj L j ) x(t ) CiT Ci H I
Ki
(J 2 H )[( w
(18)
0
V ( x(tk )) V ( x(tk ))
However
i, j
T V ( x) |(14) dt 2O ³ V ( x) |(14) dt
T
r
(17)
T
H (t ) |(14)
¦
1 1 GiT Px)T ( w 2 GiT Px)]} J H J H 2
d ¦ i , j 1 hi h j { xT (t )Qij x(t ) }
2 xT P U ( Bi ) L j x
d Ki , j xT P 2 x
447
Ki , j
1 1 GiT Px)T ( w 2 GiT Px)]} J H J H 2
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T
0
³
t1
0
V ( x) |(14) dt
t1 V ( x) |(14) dt ³ V ( x) |(14) dt 0
tk
T ³ V ( x) |(14) dt ³ V ( x) |(14) dt tk 1
tk
V ( x(t1 )) V ( x(0 )) V ( x(t2 )) V ( x(t1 )) V ( x(tk )) V ( x(tk1 )) V ( x(T )) V ( x(tk )) V ( x(t1 )) V ( x(t1 )) V ( x(tk )) V ( x(tk )) V ( x(T )) k
¦ i 1 b(2 b)V ( x(ti )) V ( x(T )) t0 Therefore, we can obtain
³
T
0
( z T z J 2 wT w)dt d 0
That is to say
(19)
448
³
T
0
JOURNAL OF SOFTWARE, VOL. 4, NO. 5, JULY 2009
T
z T zdt d J 2 ³ wT wdt
H (t ) |(10)
(20)
0
¦
When w
0 , one has V ( x) 2OV ( x) d 0 , t (tk 1 , tk ].
(21)
Hence, we can easily achieve that system (14) is exponentially stable with an H f norm J . IV.
QUANTIZED FEEDBACK CONTROL
Considering the H f feedback control of nonlinear uncertain fuzzy impulsive systems (5) with quantized controller law (9), we have the following result: Theorem 4.1: Assume there exist nonnegtive constant H and J ( L2 gain d J ) such that Assumption 3.1 holds; Moreover, assume that for arbitrary fixed V ! 0 , chosen large enough such that
M is
M ! 4 x ' (1 V )
(22)
where
4x : 2
O
1 PGi GiT P) x(t ) J H T 2 x (t )[ U ( Ai ) U ( Bi ) L j ]x(t ) 2O P CiT Ci H I
1 1 GiT Px)T ( w 2 GiT Px)] J H J H x x 2 xT (t ) P[ Bi U ( Bi )]L j P[q( ) ]}
(J 2 H )[( w
P
d ¦ i , j 1 hi h j { xT (t )Qij x(t ) x x 2 xT (t ) P[ Bi U ( Bi )]L j P[q ( ) ]
P
V ( x) xT Px for the closed-loop system (10), the derivative of V ( x) along solutions of (10) in t (tk 1 , tk ] is computed as q ( x)
h h j P {xT (t )( H ijT P PH ij ) x(t )
i, j 1 i
2 xT (t )( U ( Ai ) U ( Bi ) L j ) x(t )
(23)
2 xT (t ) PGi w(t )
x x 2 xT (t ) P[ Bi U ( Bi )]L j P[q ( ) ]}
P
H (t ) |(10) :
¦
h hj
i, j 1 i
{x (t )( H P PH ij
T i
2O P C Ci H I ) x(t ) 2 xT (t )[ U ( Ai ) U ( Bi ) L j ]x(t ) 2 xT (t ) PGi w(t ) (J 2 H ) wT w x x 2 xT (t ) P[ Bi U ( Bi )]L j P[q ( ) ]}
P
Therefore, by (15), we obtain
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P
P
{O | x | (| x | 4 x P')}
According to (22), for any nonzero x , we can find a positive scalar P such that
4 x (1 V ) P' d| x |d M P
(24)
This is also true in the case of x 0 , where we set P as an extreme case and consider the output of the quantizer as zero. Hence, there exists a control strategy for P , such that H (t ) |(10) d 0 . Repeating the approach of the above proof of Theorem 3.2, we can achieve the fuzzy impulsive system (10) exists a control strategy for P such that the closed-loop system (10) is exponentially stable with an H f norm J . V.
T ij
qP ( x )
O | x | (| x | 4 x P')
We have T
1 1 GiT Px)T ( w 2 GiT Px)]} J H J H
q ( x)
d ¦ i , j 1 hi h j
candidate
qP ( x )
P
2
P
r
Proof of Theorem 4.1˖We consider the Lyapunov function
r
P
x x 2 xT (t ) P[ Bi U ( Bi )]L j P[q ( ) ]}
Then there exists a control strategy P which depends on the state, and makes the closed-loop system (10) exponentially stable with an H f norm J .
P
2
r
r
|| B ||: max{|| Bi ||: i 1, 2, , r}.
r
2
d ¦ i , j 1 hi h j P { xT (t )Qij x(t )
|| L ||: max{|| L j ||: j 1, 2, , r}
¦
q ( x)
h h j P {xT (t )( H ijT P PH ij
i, j 1 i
(J 2 H )[( w
|| P |||| L || (|| B || r0 )
O : min{O (Qij ) : i, j 1, 2, , r}
V ( x)
r
FUZZY NONLINEAR IMPULSIVE SYSTEMS WITH NONLINEAR UNCERTAINTIES
Consider a nonlinear dynamic multi-input-multi-output system with nonlinear uncertainty modeled by the T-S fuzzy system, which can be represented by the following forms: x IF-THEN form: Ri : IF x1 (t ) is M i1 and x2 (t ) is M i 2 xn (t ) is
M in THEN
x (t )
Ai x(t ) Bi u (t )
JOURNAL OF SOFTWARE, VOL. 4, NO. 5, JULY 2009
Gi w(t ) fi ( x, t ) , z (t ) Ci x(t ) 'x Bk x(t ) , t tk
449
t z tk
i
0
x(t )
¦
i 1
wi Ci x(t )
¦
¦
'x 0
x(t ) wi
r
w 1 i
r
j 1
r i 1
¦
¦
,
r i 1
hi
tk
¦
x(t0 ) where
wi
(27)
where g (t ) is known continuous function. We will give the sufficient conditions of the H f control (26). For the nonlinear plant represented by (25) or (26), we consider the fuzzy controller as follows: x IF-THEN form: Ri : IF x1 (t ) is M i1 and x2 (t ) is M i 2 xn (t ) is M in
Li x(t ) Li qP ( x)
r qP ( x ) i 1 i
h
[ Li qP ( x)]
i, j
h h [( Ai Bi L j ) x(t ) Gi w(t ) fi ( x, t )] 1 i j h h j [ H ij x(t ) Gi w(t ) fi ( x, t )] , t z tk
i, j 1 i
i
w 1 i
¦
r
h Ci x(t )
i 1 i
tk
Ai Bi L j .
{Ki }ir 1 , , a sequence of
two sequences of constants
matrices
{Li }ir 1 and a common positive definite matrix P
(t )}r is positive and such that the sequence of matrices {Q ij i, j 1 satisfies
(t ) : [( A B L )T P P( A B L ) Q i i j i i j ij 2O P CiT Ci 1 (H g (t )) I Ki P 2
Ki
1 PGi GiT P] J H 2
T ij
[ H P PH ij 2O P CiT Ci
(H
1
Ki
g (t )) I Ki P 2
(34)
1 PGi GiT P]. J H 2
(30)
(31)
the closed-loop system (32) is exponentially stable with an H f norm J . Proof of Theorem 5.1˖We consider the Lyapunov function
V ( x) xT Px for the closed-loop system (32), the derivative of V ( x) along solutions of (32) in t (tk 1 , tk ] is candidate
computed as
V ( x)
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r
Theorem 5.1: Assume there exist nonnegtive constant H and J ( L2 gain d J ) such that Assumption 5.1 holds, then
The system (26) with (30) or (26) with (31) can be written the form of the T-S fuzzy control system as follows: r
wi Ci x(t )
(33)
J !0,
(29)
or
¦ ¦
r i 1
P
Consider H f feedback control of system (33), we require the following Assumption 5.1: Assumption 5.1: Assume that there exist a positive constant
(28)
x INPUT-OUTPUT form: r u (t ) ¦ i 1 hi [ Li x(t )]
¦
{H ij x(t ) Gi w(t ) ,
H ij denotes H ij
or
u (t )
P
x0
THEN
r
hh
i, j 1 i
qij ( x ) j
'x U (t , x(t )) bx(t ) , t
hi ! 0.
f iT ( x, t ) fi ( x, t ) d xT g (t ) x, i {1, 2, , r}
x (t )
r
x x f i ( x, t ) Bi L j P[q( ) ]} , t z tk z (t )
and impulsive properties are the same as system (4). We consider f i ( x, t ) satisfies
u (t )
q ( x)
P
wi ! 0, wi ! 0,
1,
tk
h h j P {( Ai Bi L j ) x(t ) Gi w(t ) fi ( x, t )
(26)
fi ( x, t ) is nonlinear component, the other variables
u (t )
r
¦ M ij ( x j (t )), ¦
(32)
h Ci x(t )
x x Bi L j P[q( ) ]}
h Ci x(t )
r i 1
r
i 1 i
i, j 1 i
i 1 i
h Bik x(t ) : U (t , x(t )) , t
n
¦
x (t )
x0
¦
where
i
¦
w 1 i
P
i 1 i
wi
hi
r
i
¦
or
Gi w(t ) fi ( x, t )] , t z tk z (t )
r
x0
h [ Ai x(t ) Bi u (t ) 1 i
r
wi Ci x(t )
'x U (t , x(t )) bx(t ) , t
x(t ) x0 x INPUT-OUTPUT form: r x (t ) ¦ i 1 wi [ Ai x(t ) Biu (t ) Gi w(t ) fi ( x, t )] r ¦ i 1 wi
¦
r i 1
¦
(25)
0
r
¦
z (t )
¦
r
h h j {xT (t )( H ijT P PH ij ) x(t )
i, j 1 i
450
JOURNAL OF SOFTWARE, VOL. 4, NO. 5, JULY 2009
2 xT (t ) Pfi ( x, t ) 2 xT (t ) PGi w(t )}
(35)
V ( x)
¦
By Lemma 3.3, we obtain T
T
2
2 x (t ) Pfi ( x, t ) d x [Ki P
1
Ki
r
q ( x)
h h j P {xT (t )( H ijT P PH ij ) x(t )
i, j 1 i
2 xT (t ) Pfi ( x, t ) 2 xT (t ) PGi w(t ) x x 2 xT (t ) PBi L j P[q( ) ]}
g (t ) I ]x
P
Let
H (t ) : V ( x) 2OV ( x) z T z J 2 wT w H ( xT x wT w)
H (t ) : V ( x) 2OV ( x) z T z J 2 wT w H ( xT x wT w) We have
H (t ) |(32)
H (t ) |(33)
r
d ¦ i , j 1 hi h j {xT (t )( H ijT P PH ij 2O P CiT Ci 1
Ki
g (t )) I Ki P 2
(J 2 H )[( w
¦
r i, j
r
d ¦ i, j
1 PGi GiT P ) x (t ) J H 2
2O P CiT Ci (H
1 1 GiT Px)T ( w 2 GiT Px )]} J H J H
x x 2 xT (t ) PBi L j P[q( ) ]}
(t )x(t ) h h { xT (t )Q ij 1 i j
P
such that (38)
Therefore, by (34), we obtain
H (t ) |(33)
d0 Hence repeating the approach of the above proof of Theorem 3.2, we can achieve the system (26) with controller law (30) exists a control strategy for P such that the closedloop system (32) is exponentially stable with an H norm J . Theorem 5.2: Assume there exist nonnegtive constant H and J ( L2 gain d J ) such that Assumption 5.1 holds; f
Moreover, assume that for arbitrary fixed V ! 0 , chosen large enough such that
M is (36)
where
x : 2T 4 O (t )) : i, j 1, 2, , r ; t t 0} O : min{O (Q ij
T : max{|| PBi L j ||: i, j 1, 2, , r}. Then there exists a control strategy P which depends on the state, and makes the closed-loop system (33) exponentially stable with an H f norm J . Proof of Theorem 5.2˖We consider the Lyapunov function
V ( x) x Px for the closed-loop system (33), the derivative of V ( x) along solutions of (33) in t (tk 1 , tk ] is computed as
P
x (1 V ) P' d| x |d M P 4
(t )x(t )} h h { x (t )Q ij 1 i j
candidate
P
According to (36), we can find a positive scalar
T
T
1
g (t )) I Ki P 2 ]x(t ) Ki 2 xT (t ) PGi w(t ) (J 2 H ) wT w
2
x ' (1 V ) M !4
q ( x)
d ¦ i , j 1 hi h j P {xT (t )[ H ijT P PH ij
1 1 (J 2 H )[( w 2 GiT Px)T ( w 2 G T Px )]} J H J H i r
P
Let
Therefore, by (34), we obtain
(H
(37)
r
q ( x)
d ¦ i , j 1 hi h j P {xT (t )[ H ijT P PH ij 2O P CiT Ci (H
1
g (t )) I Ki P 2
Ki
(J 2 H )[( w
1 PGi GiT P]x(t ) J H 2
1 1 GiT Px )T ( w 2 G T Px )] J H J H i 2
x x 2 xT (t ) PBi L j P[q( ) ]}
P
¦
r
P
q ( x) (t ) x(t ) h h j P { xT (t )Q ij
i, j 1 i
(J 2 H )[( w
1 1 GiT Px )T ( w 2 G T Px )] J H J H i 2
x x 2 xT (t ) PBi L j P[q( ) ]}
P
r
qP ( x )
d ¦ i , j 1 hi h j
P
(t ) x(t ) { xT (t )Q ij
x x 2 xT (t ) PBi L j P[q( ) ]}
P
P
x P' ) d 0 d O | x | (| x | 4 Hence, there exists a control strategy for P , such that H (t ) |(33) d 0 . Repeating the approach of the above proof of Theorem 3.2, we can achieve the fuzzy impulsive system (33)
© 2009 ACADEMY PUBLISHER
JOURNAL OF SOFTWARE, VOL. 4, NO. 5, JULY 2009
exists a control strategy for
P
such that the closed-loop f
system (33) is exponentially stable with an H norm VI.
451
J.
CONCLUSIONS
In this paper, we discuss the problem of H f control of two classes of fuzzy nonlinear impulsive systems with quantized feedback. New results on the H f control problems are established for one class of fuzzy nonlinear uncertain impulsive systems and one class of fuzzy nonlinear impulsive systems with nonlinear uncertainties by choosing appropriately quantized strategies, respectively. ACKNOWLEDGMENTS The research of this paper was supported by the National Natural Science Foundation of P.R. China under Grant No. 60874006, by the National Natural Science Foundation of P.R. China (Mathematics Tianyuan Foundation) under Grant No. 10826078, and by Foundation of Henan University of Technology under Grant No. 07XJC013. The authors are very grateful to all the anonymous reviewers and the editors for their helpful comments and suggestions. REFERENCES [1] Delchamps, D. F., “Stabilizing a linear system with quantized state feedback,” IEEE Transactions on Automatic Control, vol. 35, pp. 916-924, August 1990. [2] Wong, W. S. and Brockett, R. W., “Systems with finite communication bandwidth constraints. II. Stabilization with limited information feedback,” IEEE Transactions on Automatic Control, vol. 44, pp. 1049-1053, May 1999. [3] Carlos E. de Souza and Xi Li, “Delay-dependent robust H f control of uncertain linear state-delayed systems,” Automatica, vol. 35, pp. 1313-1321, 1999. [4] Brockett, R. W. and Liberzon, D., “Quantized feedback stabilization of linear systems,” IEEE Transactions on Automatic Control, vol. 45, pp. 1279-1287, July 2000. [5] Elia, N. and Mitter, S. K., “Stabilization of linear systems with limited information,” IEEE Transactions on Automatic Control, vol. 46, pp.1384-1400, Sept. 2001. [6] Liberzon, D., “Hybrid feedback stabilization of systems with quantized signals,” Automatica, vol 39, pp. 1543-1554, Sept. 2003. [7] Ishii, H. and Francis, B. A.., “Quadratic stabilization of sampled-data systems with quantization,” Automatica, vol. 39, pp. 1793-1800, Oct. 2003. [8] Fagnani, F. and Zampieri, S., “Stability analysis and synthesis for scalar linear systems with a quantized feedback,” IEEE Transactions on Automatic Control, vol. 48, pp. 1569-1584, Sept. 2003.
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[9] Liu, J.and Elia, N., “Quantized feedback stabilization of nonlinear affine systems,” International Journal of Control, vol. 77, pp. 239-149, March 2004. [10] Tatikonda, S. and Mitter, S., “Control under communication constraints,” IEEE Transactions on Automatic Control, vol. 49, pp.1056-1068, July 2004. [11] Ling, Q. and Lemmon, M. D., “Stability of quantized control systems under dynamic bit assignment,” IEEE Transactions on Automatic Control, vol. 50, pp. 734-1740, May 2005. [12] Fu, M., and Xie, L., “The sector bound approach to quantized feedback control,” IEEE Transactions on Automatic Control, vol. 50, pp. 1698-1711, Nov. 2005. [13] Claudio De Persis, “On feedback stabilization of nonlinear systems under quantization,” Proceedings of the 44th IEEE CDC, 2005, 7698-7703. [14] Guisheng Zhai; Xinkai Chen; Joe Imae; Tomoaki Kobayashi, f “Analysis and Design of H Feedback Control Systems with Two Quantized Signals,” Proceedings of the 2006 IEEE International Conference on ICNSC, pp. 346-350, April 2006. [15] Francesca Ceragioli and Claudio De Persis, “Discontinuous stabilization of nonlinear systems: Quantized and switching controls,” Systems and Control Letters, Vol. 56, pp. 461-473, 2007. [16] Huijun Gao and Tongwen Chen, “A new approach to quantized feedback control systems,” Automatica, vol. 44, pp.534-542, Feb. 2008. [17] Zhang Ying-qi, Liu Cai-xia, Mu Xiao-wu, “Stability Analysis and Observer Design for Lipschitz Nonlinear Neutral-type Delay System with Uncertainties,” Henan Science, Vol. 27, pp. 6-10, Jan. 2009. [18] T.Takagi and M.Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans, Syst.,Man, Cybern., vol . SMC-15, pp.116-132, Jan. 1985. [19] Y.Y. Cao and P.M. Frank, “Analysis and synthesis of nonlinear time delay systems via fuzzy control approach,” IEEE Trans. Fuzzy Syst., vol 8, pp. 200-211, Feb. 2000. [20] E.Kim and H.Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 523-534, May 2000. [21] T. Taniguchi, K. Tanaka, H. Ohtake, H.W. Wang, “Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems,” IEEE Trans. Fuzzy Syst., 2001, pp. 525-538. [22] Kazuo Tanaka, Takayuki Ikeda, and Hua O. Wang, “Fuzzy Regulators and Fuzzy Observers: Relaxed Stability Conditions and LMI-Based Designs,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 250-265, Feb. 1998. [23] Thierry Marie Guerra, Laurent Vermeiren, “LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form,” IEEE Trans. Syst., Man, Cybern. B, Cybern, vol .40, pp. 823-829, 2004. [24] Tong RM., “A control engineering review of fuzzy systems,” Automatica, Vol .13, pp.559-568, 1977. [25] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets Syst., vol. 45, pp. 135-156, Feb.1992.
