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Hua O. Wang and Kazuo Tanaka

46

H.O. Wang and K. Tanaka

control problems besides stabilization. In this chapter, we develop a uniﬁed

approach to address some of these problems including stabilization, synchronization, and chaotic model following control (CMFC) for chaotic systems.

The uniﬁed approach is based on the Takagi–Sugeno (TS) fuzzy modeling

and the associated parallel distributed compensation (PDC) control design

methodology [17]. In this framework, a nonlinear dynamical system is ﬁrst

approximated by the TS fuzzy model. In this type of fuzzy model, local dynamics in diﬀerent state space regions are represented by linear models. The

overall model of the system is achieved by fuzzy “blending” of these linear

models. The control design is carried out based on the fuzzy model. For each

local linear model, a linear feedback control is designed. The resulting overall

controller, which is nonlinear in general, is again a fuzzy blending of each

individual linear controller. This control design scheme is referred to as the

PDC technique in the literature [17]. More importantly, it has been shown in

[17] that the associated stability analysis and control design can be aided by

convex programming techniques for LMIs.

In this chapter, for chaos control, a cancellation technique (CT) is presented as a main result for stabilization of chaotic systems. The CT also plays

an important role in the synchronization and the CMFC. Two cases are considered in the synchronization. The ﬁrst one deals with the feasible case of the

cancellation problem. The other one addresses the infeasible case of the cancellation problem. Furthermore, the CMFC problem, which is more diﬃcult

than the synchronization problem, is discussed using the CT method. One of

the most important aspects is that the approach described here can be applied not only to stabilization and synchronization but also to the CMFC in

the same control framework. That is, it is a rather uniﬁed approach to a class

of chaos control problems. In fact, the stabilization and the synchronization

discussed here can be regarded as a special case of the CMFC. Simulation

results demonstrate the utility of the uniﬁed design approach.

2 Fuzzy Modeling of Chaotic Systems

To utilize the LMI-based fuzzy system design techniques, we start with representing chaotic systems using TS fuzzy models. In this regard, the techniques

described in [17] are employed to construct fuzzy models for chaotic systems.

In the following, a number of typical chaotic systems with the control input

term added are represented in the TS modeling framework.

Lorenz’s equation with input term

x˙ 1 (t) = −ax1 (t) + ax2 (t) + u(t) ,

x˙ 2 (t) = cx1 (t) − x2 (t) − x1 (t)x3 (t) ,

x˙ 3 (t) = x1 (t)x2 (t) − bx3 (t) ,

Fuzzy Modeling and Control of Chaotic Systems

47

where a, b, and c are constants and u(t) is the input term. Assume that

x1 (t) ∈ [−d d] and d > 0. Then, we can have the following fuzzy model

which exactly represents the nonlinear equation under x1 (t) ∈ [−d d]:

˙

IF x1 (t) is M1 THEN x(t)

= A1 x(t) + Bu(t) ,

˙

IF x1 (t) is M2 THEN x(t) = A2 x(t) + Bu(t) ,

Rule 1 :

Rule 2 :

T

where x(t) = [x1 (t) x2 (t) x3 (t)] ,

−a a

0

−a a

A1 = c −1 −d , A2 = c −1

0

d −b

0 −d

1

B = 0,

0

M1 (x1 (t)) =

1

2

1+

x1 (t)

d

,

M2 (x1 (t)) =

0

d ,

−b

1

2

1−

x1 (t)

d

.

Here a = 10, b = 8/3, c = 28, and d = 30.

Rossler’s equation with input term

x˙ 1 (t) = −x2 (t) − x3 (t) ,

x˙ 2 (t) = x1 (t) + ax2 (t) ,

x˙ 3 (t) = bx1 (t) − {c − x1 (t)}x3 (t) + u(t) ,

where a, b and c are constants. Assume that x1 (t) ∈ [c − d c + d] and

d > 0. Then, we obtain the following fuzzy model which exactly represents

the nonlinear equation under x1 (t) ∈ [c − d c + d]:

˙

IF x1 (t) is M1 THEN x(t)

= A1 x(t) + Bu(t) ,

˙

IF x1 (t) is M2 THEN x(t)

= A2 x(t) + Bu(t) ,

Rule 1 :

Rule 2 :

T

where x(t) = [x1 (t) x2 (t) x3 (t)] ,

0 −1 −1

0

0 , A2 = 1

A1 = 1 a

b 0 −d

b

0

B = 0,

1

M1 (x1 (t)) =

1

2

1+

c − x1 (t)

d

,

−1 −1

a

0 ,

0

d

M2 (x1 (t)) =

1

2

1−

c − x1 (t)

d

.

48

H.O. Wang and K. Tanaka

Here a = 0.34, b = 0.4, c = 4.5, and d = 10.

Duﬃng forced-oscillation model

x˙ 1 (t) = x2 (t)

x˙ 2 (t) = −x31 (t) − 0.1x2 (t) + 12 cos(t) + u(t)

Assume that x1 (t) ∈ [−d

fuzzy model as well:

d] and d > 0. Then, we can have the following

˙

= A1 x(t) + Bu∗ (t) ,

IF x1 (t) is M1 THEN x(t)

˙

IF x1 (t) is M2 THEN x(t)

= A2 x(t) + Bu∗ (t) ,

Rule 1:

Rule 2:

where x(t) = [x1 (t) x2 (t)] and u∗ (t) = u(t) + 12 cos(t),

T

A1 =

0

0

1

,

−0.1

B=

0

,

1

M1 (x1 (t)) = 1 −

A2 =

x21 (t)

,

d2

0

−d2

1

−0.1

M2 (x1 (t)) =

,

x21 (t)

.

d2

Here d = 50.

Henon mapping model

x1 (t + 1) = −x21 (t) + 0.3x2 (t) + 1.4 + u(t) ,

x2 (t + 1) = x1 (t) .

Assume that x1 (t) ∈ [−d d] and d > 0. The following equivalent fuzzy model

can be constructed as well:

Rule 1:

IF x1 (t) is M1 THEN x(t + 1) = A1 x(t) + Bu∗ (t) ,

Rule 2:

IF x1 (t) is M2 THEN x(t + 1) = A2 x(t) + Bu∗ (t) ,

where x(t) = [x1 (t) x2 (t)] and u∗ (t) = u(t) + 1.4,

T

A1 =

d

1

B=

1

,

0

0.3

,

0

M1 (x1 (t)) =

Here d = 30.

1

2

A2 =

1−

x1 (t)

d

−d

1

,

0.3

0

,

M2 (x1 (t)) =

1

2

1+

x1 (t)

d

.

Fuzzy Modeling and Control of Chaotic Systems

49

In all cases above, the fuzzy models exactly represent the original systems.

As shown in [17], the TS fuzzy model is a universal approximator for nonlinear

dynamical systems. Other chaotic systems can be approximated by the TS

fuzzy models.

The fuzzy models above have the common B matrix in the consequent

parts and x1 (t) in the premise parts. In this chapter, all the fuzzy models

are assumed to be the common B matrix case, i.e., the fuzzy model (1) is

considered.

P lant Rule i:

If z1 (t) is Mi1 and · · · and zp (t) is Mip ,

then sx(t) = Ai x(t) + Bu(t),

i = 1, 2, . . . , r ,

(1)

where p = 1 and z1 (t) = x1 (t). Equation (1) is represented by the defuzziﬁcation form

r

i=1

sx(t) =

wi (z(t)) {Ai x(t) + Bu(t)}

r

i=1 wi (z(t))

r

hi (z(t)) {Ai x(t) + Bu(t)} ,

=

(2)

i=1

˙

where sx(t) denotes x(t)

and x(t + 1) for continuous-time fuzzy systems

(CFS) and discrete-time fuzzy systems (DFS), respectively. In the fuzzy models above for chaotic systems, z(t) = z1 (t) = x1 (t).

Remark 1. The fuzzy models above have a single input. We can also consider multi-inputs case. For instance, we may consider Lorenz’s equation with

multi-inputs:

x˙ 1 (t) = −ax1 (t) + ax2 (t) + u1 (t) ,

x˙ 2 (t) = cx1 (t) − x2 (t) − x1 (t)x3 (t) + u2 (t) ,

x˙ 3 (t) = x1 (t)x2 (t) − bx3 (t) + u3 (t) .

Same as before, we can derive the the following fuzzy model to exactly represents the nonlinear equation under x1 (t) ∈ [−d d]:

Rule 1:

Rule 2 :

˙

IF x1 (t) is M1 THEN x(t)

= A1 x(t) + Bu(t) ,

˙

= A2 x(t) + Bu(t) ,

IF x1 (t) is M2 THEN x(t)

where u(t) = [u1 (t) u2 (t) u3 (t)]T and x(t) = [x1 (t) x2 (t) x3 (t)]T ,

(3)

50

H.O. Wang and K. Tanaka

−a a

0

A1 = c −1 −d ,

0

d −b

1 0 0

B = 0 1 0,

0 0 1

M1 (x1 (t)) =

1

2

1+

x1 (t)

d

−a a

A2 = c −1

0 −d

,

M2 (x1 (t)) =

0

d ,

−b

1

2

1−

x1 (t)

d

.

This fuzzy model with three inputs is used as a design example later in this

chapter.

3 Stabilization

Two techniques for the stabilization of chaotic systems (or nonlinear systems)

are presented in this section. We ﬁrst consider the common stabilization problem followed by a so-called cancellation technique (CT). In particular, the CT

plays an important role in synchronization and CMFC, which are discussed

in Sects. 4 and 5, respectively.

3.1 Stabilization via Parallel Distributed Compensation

Equation (4) shows the PDC controller for the fuzzy models given in Sect. 2.

Rule 1:

Rule 2 :

IF x1 (t) is M1 THEN u(t) = −F 1 x(t) ,

IF x1 (t) is M2 THEN u(t) = −F 2 x(t) .

(4)

Please note that the chaotic systems under consideration in the previous

section are represented (coincidentally) by simple TS fuzzy models with two

rules. Therefore the following PDC fuzzy controller also has only two rules:

u(t) = −

2

i=1

wi (z(t))F i x(t)

2

i=1

wi (z(t))

2

=−

hi (z(t))F i x(t) .

(5)

i=1

By substituting (5) into (2), we have

r

hi (z(t)) Ai − BF i x(t) ,

sx(t) =

(6)

i=1

where r = 2. We recall stable and decay rate fuzzy controller designs for

CFS and DFS cases, where the following conditions are simpliﬁed due to the

common B matrix case. These design conditions are all given for the general

TS model with r number of rules.