Section 9.3 has presented the synchronization of chaotic systems, where Ai matrices of the fuzzy model should be the same as Ai matrices of the fuzzy reference model. This section presents chaotic model following control ŽCMFC , where. Aimatrices of the fuzzy model do not have to be the same as Ai matrices of the fuzzy reference model. Therefore, the CMFC is more difficult than the synchronization. In this section, the controlled objects are assumed to be chaotic systems. However, note that the CMFC can be designed for general nonlinear systems represented by T-S fuzzy models.
Consider a reference fuzzy model which represents a reference chaotic system.
Reference Rule i
Ž . Ž .
IFzR1 t is Ni1 and⭈⭈⭈and zR p t is Ni p,
Ž . Ž . Ž .
THENsxR t sD xi R t , is1, 2, . . . ,rR. 9.16
Ž . n
Assume that xR t gR and Ai/Di. The defuzzification process is given as
rR
sxRŽ .t s Ý®iŽzRŽ .t .D xi RŽ .t . Ž9.17.
is1
The CMFC can be regarded as nonlinear model following control for the
Ž . Ž . Ž . Ž .
reference fuzzy model 9.17 . Assume that e t sx t yxR t . Then, from Ž9.2 and 9.17 , we have. Ž .
r
seŽ .t s ÝhiŽzŽ .t .A xi Ž .t
is1
rR
yÝ®iŽzRŽ .t .D xi RŽ .t qBuŽ .t . Ž9.18.
is1
Consider two sub-fuzzy controllers to realize the CMFC:
Subcontroller A Control Rule i
Ž . Ž .
IFz t1 is Mi1 and⭈⭈⭈and z tp is Mi p,
Ž . Ž . Ž .
THEN uA t s yF xi t , is1, 2, . . . ,r. 9.19 Subcontroller B
Control Rule i
Ž . Ž .
IFzR1 t is Ni1 and⭈⭈⭈and zR p t is Ni p,
Ž . Ž . Ž .
THEN uB t sK xi R t , is1, 2, . . . ,rR. 9.20 The combination of the subcontroller A and the subcontroller B is repre- sented as
uŽ .t suAŽ .t quBŽ .t
r rR
s yÝhiŽzŽ .t .F xi Ž .t q Ý®iŽzRŽ .t .K xi RŽ .t . Ž9.21.
is1 is1
Ž . Ž .
By substituting 9.21 into 9.18 , the overall control system is represented as
r
seŽ .t s ÝhiŽzŽ . Žt . AiyBF xi. Ž .t
is1 rR
yÝ®iŽzRŽ . Žt . DiyBKi.xRŽ .t . Ž9.22.
is1
Ž .
THEOREM 32 The chaotic system represented by the fuzzy system 9.2 is
Ž .
exactly linearized®ia the fuzzy controller 9.21 if there exist the feedback gainsFi andKj such that
A yBF y A yBF T
Ž 1 1. Ž i i.4
=ŽA1yB F1.yŽAiyBFi.4s0, is2, 3, . . . ,r, Ž9.23. A yB F y D yBK T
Ž .
1 1 Ž j j. 4
=ŽA1yB F1.yŽDjyBKj. 4s0, js1, 2, . . . ,rR. Ž9.24.
Ž . Ž .
Then, the o®erall control system is linearized as sx t sGx t , where Gs A1yBF1sAiyBFisDjyBKj.
Proof. It is obvious that GsA1yBF1sAiyBFisDjyBKj if conditions Ž9.23 and 9.24 hold.. Ž .
An important remark is in order here.
Remark 30 The CMFC reduces to the synchronization problem when rsrR and AisDj for is1, . . . ,r and js1, . . . ,rR. The CMFC reduces to the stabilization problem when Dis0 and xRŽ .0 s0 for is1, . . . ,rR. There- fore, as mentioned above, the CMFC problem is more general and difficult than the stabilization and synchronization problems. In addition, the con- troller design described here can be applied not only to stabilization and synchronization but also to the CMFC in the same control framework.
Therefore the LMI-based methodology represents a unified approach to the problem of controlling chaos.
If B is a nonsingular matrix, the error system is exactly linearized and
y1Ž . y1Ž .
stabilized using FisB GyAi and KisB GyDi . However, the assumption that Bis a nonsingular matrix is very strict. On the other hand, if B is not a nonsingular matrix, Theorem 32 can be utilized by the approxima- tion CT technique. The LMI conditions can be derived from Theorem 32 in the same way as described in Section 9.2.
Note that G is not always a stable matrix even if the conditions of Theorem 32 hold. From Theorem 32 and the stability conditions, we define the following design problems:
Stable Fuzzy Controller Design Using the CT: CFS minimize
X,S,M1,M2, . . . ,Mr
subject to X)0,)0,S)0, I S
)0, S I
yA Xi qBMiyXATi qMiTBT)0, is1, 2, . . . ,r,
Ž . Ž .4T
S A X1 yBM1 y A Xi yBMi
)0, ŽA X1 yBM1.yŽA Xi yBMi.4 I
is2, 3, . . . ,r,
Ž . T
S A X1 yBM1 yŽD Xj yBNj.4
)0,
ŽA XyBM .y D XyBN I
1 1 Ž j j.4
js1, 2, . . . ,rR, where XsPy1, M1sF X1 , MisF X, andi NjsK X.j
Stable Fuzzy Controller Design Using the CT: DFS minimize
X,S,M1,M2, . . . ,Mr
subject to X)0,)0, S)0, I S
)0, S I
T T
X XAiyM Bi
)0, is1, 2, . . . ,r, A Xi yBMi X
Ž . Ž .4T
S A X1 yBM1 y A Xi yBMi
)0, ŽA X1 yBM1.yŽA Xi yBMi.4 I
is2, 3, . . . ,r,
Ž . T
S A X1 yBM1 yŽD Xj yBNj.4
)0,
ŽA XyBM .y D XyBN I
1 1 Ž j j.4
js1, 2, . . . ,rR, where XsPy1, M1sF X1 , MisF X, andi NjsK X.j
Decay Rate Fuzzy Controller Design Using the CT: CFS maximize␣
X,S,M1,M2, . . . ,Mr
minimize
X,S,M1,M2, . . . ,Mr
subject to X)0,)0,␣)0,S)0, I S
)0, S I
yA Xi qBMiyXATiqMiTBTy2␣X)0, is1, 2, . . . ,r,
Ž . Ž .4T
S A X1 yBM1 y A Xi yBMi
)0, ŽA X1 yBM1.yŽA Xi yBMi.4 I
is2, 3, . . . ,r,
Ž . T
S A X1 yBM1 yŽD Xj yBNj.4
)0,
ŽA XyBM .y D XyBN I
1 1 Ž j j.4
js1, 2, . . . ,rR, where XsPy1, M1sF X1 , MisF X, andi NjsK X.j
Decay Rate Fuzzy Controller Design Using the CT: DFS minimize␣
X,S,M1,M2, . . . ,Mr
minimize
X,S,M1,M2, . . . ,Mr
subject to X)0,)0, 0F␣-1,S)0, I S
)0, S I
T T
␣X XAiyM Bi
)0, is1, 2, . . . ,r, A Xi yBMi X
Ž . Ž .4T
S A X1 yBM1 y A Xi yBMi
)0, ŽA X1 yBM1.yŽA Xi yBMi.4 I
is2, 3, . . . ,r,
Ž . T
S A X1 yBM1 yŽD Xj yBNj.4
)0,
ŽA XyBM .y D XyBN I
1 1 Ž j j.4
js1, 2, . . . ,rR, where XsPy1, M1sF X1 , MisF X, andi NjsK X.j
Remark 31 In the LMIs, if all elements in  ⭈S are near zero, that is,
 ⭈Sf0, the cancellation problems for decay rate fuzzy controller designs are feasible. In this case, GsA1yBF1sAiyBFisDjyBKj ᭙i,j, and G is a stable matrix.
Example 25 Let us consider the fuzzy model for Lorenz’s equation with three inputs. The parameters are set as follows:
Rule 1
IFx t1Ž .is M1,
Ž . Ž . Ž .
THEN x˙t sA x1 t qBu t . Rule 2
IFx t1Ž .is M2,
Ž . Ž . Ž .
THEN x˙t sA x2 t qBu t .
Ž . w Ž . Ž . Ž .xT Here, x t s x t1 x t2 x t3 ,
y0.5⭈a 0.5⭈a 0 A1s 2⭈c y1 yd ,
0 d y0.5⭈b
y0.5⭈a 0.5⭈a 0
A2s 2⭈c y1 d ,
0 yd y0.5⭈b
1 0 0
Bs 0 1 0 ,
0 0 1
1 x t1Ž . 1 x t1Ž .
M x t1Ž 1Ž ..s2 ž1q d /, M2Žx t1Ž ..s 2ž1y d /. Consider the following reference fuzzy model:
Reference Rule 1
IFx1RŽ .t is N1,
Ž . Ž .
THEN x˙R t sD x1 R t . Reference Rule 2
IFx1RŽ .t is N2,
Ž . Ž .
THEN x˙R t sD x2 R t .
Ž . w Ž . Ž . Ž .xT
Here, xR t s xR1 t xR2 t xR3 t ,
ya a 0 ya a 0 D1s c y1 yd , D2s c y1 d ,
0 d yb 0 yd yb
1 xR1Ž .t 1 xR1Ž .t
N x1Ž R1Ž .t .s 2ž1q d /, N2ŽxR1Ž .t .s 2ž1y d /,
Ž . w x
where xR1t g yd d. The stable fuzzy controller design using the CT is feasible. Figures 9.21 and 9.22 show the control result, where the control input is added at t)10 sec. It can be seen that the designed fuzzy controller
Ž . Ž .
realizes chaotic model following control, that is, e t1 ™0, e t2 ™0, and e t3Ž .™0.
Ž . Fig. 9.21 Control result 1 Example 25 .
Example 26 Let us consider the fuzzy model for Rossler’s equation with the input term. The parameters are set as follows:
Rule 1
IFx t1Ž .is M1,
Ž . Ž . Ž .
THEN x˙t sA x1 t qBu t .
Ž . Fig. 9.22 Control result 2 Example 25 .
Rule 2
IFx t1Ž .is M2,
Ž . Ž . Ž .
THEN x˙t sA x2 t qBu t .
Ž . w Ž . Ž . Ž .xT Here, x t s x t1 x t2 x t3 ,
0 y1 y1 0 y1 y1
A1s 1 a 0 , A2s 1 a 0 ,
0.5⭈b 0 yd 0.5⭈b 0 d
0 Bs 0 ,
1
1 2⭈cyx t1Ž . M x t1Ž 1Ž ..s 2ž1q d /,
1 2⭈cyx t1Ž . M2Žx t1Ž ..s 2ž1y d /. Consider the following reference fuzzy model:
Reference Rule 1
IFx1RŽ .t is N1,
Ž . Ž .
THEN x˙R t sD x1 R t . Reference Rule 2
IFx1RŽ .t is N2,
Ž . Ž .
THEN x˙R t sD x2 R t .
Ž . w Ž . Ž . Ž .xT
Here, xR t s xR1 t xR2 t xR3 t ,
0 y1 y1 0 y1 y1
D1s 1 a 0 , D2s 1 a 0 ,
b 0 yd b 0 d
1 cyxR1Ž .t N x1Ž R1Ž .t .s 2ž1q d /,
1 cyxR1Ž .t N2ŽxR1Ž .t .s 2ž1y d /,
Ž . w x
where xR1t g cyd cqd. The stable fuzzy controller design using the CT is feasible. Figures 9.23 and 9.24 show the control result, where the control input is added at t)30 sec. The designed fuzzy controller realizes chaotic model following control.