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the future work, we will investigate how to suppress the spatial instability by the delayed-feedback control [8].
Index Terms—Chaotifying, continuous-time TS fuzzy system, discretization, state feedback.
REFERENCES
I. INTRODUCTION
[1] K. Kaneko, Theory and Applications of Coupled Map Lattices. Chichester, U. K.: Wiley, 1993. , “Spatial period-doubling in open flow,” Phys. Lett. A, vol. 111, [2] no. 7, pp. 321–325, 1985. [3] R. J. Deissler and K. Kaneko, “Velocity-dependent Lyapunov exponents as a measure of chaos for open-flow systems,” Phys. Lett. A, vol. 119, no. 8, pp. 397–402, 1987. [4] F. H. Willeboordse and K. Kaneko, “Pattern dynamics of a coupled map lattice for open flow,” Physica D, vol. 86, no. 3, pp. 428–455, 1995. [5] K. Konishi, M. Hirai, and H. Kokame, “Decentralized delayed-feedback control of a coupled map model for open flow,” Phys. Rev. E, vol. 58, no. 3, pp. 3055–3059, 1998. [6] A. Yamaguchi, “On the mechanism of spatial bifurcations in the open flow system,” Int. J. Bifurcation Chaos, vol. 7, no. 7, pp. 1529–1538, 1997. [7] K. Konishi, H. Kokame, and K. Hirata, “Stability of steady state in one-way open coupled map lattices,” Phys. Lett. A, vol. 263, no. 4/6, pp. 307–314, 1999. , “Delayed-feedback control of spatial bifurcations and chaos in [8] open flow models,” Phys. Rev. E, vol. 62, no. 1, pp. 384–388, 2000. , “Spatiotemporal stability of one-way open coupled nonlinear sys[9] tems,” Phys. Rev. E, vol. 62, no. 5, pp. 6383–6387, 2000. [10] G. A. Johnson, M. Löcher, and E. R. Hunt, “Stabilized spatiotemporal waves in a convectively unstable open flow system: Coupled diode resonators,” Phys. Rev. E, vol. 51, no. 3, pp. 1625–1628, 1995. [11] T. Kapitaniak, L. O. Chua, and G. Q. Zhong, “Experimental hyperchaos in coupled Chua’s circuits,” IEEE Trans. Circuits Syst. I, vol. 41, pp. 499–503, July 1994. [12] M. P. Kennedy, “Robust op amp realization of Chua’s circuit,” Frequenz, vol. 46, no. 3/4, pp. 66–80, 1992. [13] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory. Upper Saddle River, NJ: Prentice-Hall, 1992.
Chaotifying Continuous-Time TS Fuzzy Systems via Discretization Zhong Li, Jin Bae Park, and Young Hoon Joo
Abstract—An approach is proposed for making a given stable continuous-time Takagi-Sugeno (TS) fuzzy-system chaotic, by first discretizing it and then using state feedback control of arbitrarily small magnitude. The feedback controller chosen among several candidates is a simple sinusoidal function of the system states, which can lead to uniformly bounded state vectors of the controlled system with positive Lyapunov exponents, and satisfy the chaotic mechanisms of stretching and folding, thereby yielding chaotic dynamics. This approach is mathematically proven for rigorous generation of chaos from a stable continuous-time TS fuzzy system, where the generated chaos is in the sense of Li and Yorke. A numerical example is included to visualize the theoretical analysis and the controller design.
Manuscript received March 23, 2001; revised June 4, 2001. This work was supported by the Research Project Brain Korea 21. This paper was recommended by Associate Editor C. K. Tse. Z. Li and J. B. Park are with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea (e-mail:
[email protected];
[email protected]). Y. H. Joo is with the School of Electronic and Information Engineering, Kunsan National University, Kunsan, Chonbuk 573-701, Korea (e-mail:
[email protected]). Publisher Item Identifier S 1057-7122(01)08988-7.
In contrast to the main stream of ordering or suppressing chaos, the opposite direction of making a nonchaotic dynamical system chaotic or retaining the existing chaos of a chaotic system, known as “chaotification” (or sometimes, “anticontrol”), has attracted continuous attention from the engineering and physics communities in recent years [1]–[8]. There are many practical reasons for chaos generation, for instance, chaos has an impact on some novel time- and/or energy-critical applications. Specific examples include high-performance circuits and devices (e.g., delta-sigma modulators and power converters), liquid mixing, chemical reactions, biological systems (e.g., in the human brain, heart, and perceptual processes), secure information processing, and critical decision-making in political, economic and military events [1]. Some systematic and rigorous approaches have been developed to chaotify general discrete-time systems [3]–[8], which inspire us to extend these technologies to some special systems. Interactions between fuzzy logic and chaos theory have been explored since the 1990s. The explorations have been carried out mainly on fuzzy modeling of chaotic systems using the TS model [9]–[15], linguistic descriptions [16], [17], and suppression control of chaos via an LMI-based fuzzy control system design [12], [13]. In these investigations, to design a fuzzy-model-based controller the underlying chaotic systems are represented by TS fuzzy models. Some classical control techniques such as the LMI-based design methodologies are then employed to find feedback gains of the fuzzy-model-based controllers that can satisfy some specifications such as stability, decay rate, and constraints on the control input and output of the overall fuzzy control systems. In our previous work, approaches were proposed to make discrete-time TS fuzzy systems chaotic by designing a suitable state feedback controller, which can have arbitrarily small control gain [18]. This controller, although with an arbitrarily small maximum magnitude, is capable of making the Lyapunov exponents of the controlled systems strictly positive while keeping all the system states uniformly bounded, thereby obtaining chaotic dynamics in the controlled systems. The Marotto theorem [19] is applied to show that the controlled system is indeed chaotic in the mathematical sense of Li and Yorke [20]. However, the question of how to chaotify a continuous-time TS fuzzy system is still open. This paper attempts to provide a solution to this problem. In order to utilize previous research results to make a nonchaotic or even stable continuous-time TS fuzzy system chaotic, a natural and straightforward approach is to convert it to a discrete-time version, that is, to discretize the continuous-time TS fuzzy system. Conversion of the continuous-time controller to an equivalent digital controller is known as digital redesign. Digital redesign techniques were first considered in [21], and then developed by many others [14], [22]–[24]. Here, this discretization technique is adopted for discretization of a continuous-time TS fuzzy system. Then, among several candidates, a simple sinusoidal function is used to construct a state feedback controller, and the controlled discretized TS fuzzy system is proven to be chaotic in the sense of Li and Yorke. Finally, a simple example is included to verify and visualize the theory and the results of the paper. II. DISCRETIZATION OF THE CONTINUOUS-TIME TS FUZZY MODELS We first formulize the continuous-time TS fuzzy model, then give its discretization.
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IF x1 (t) is Mi1 . . . and xn (t) is Min
A. The Continuous-Time TS Fuzzy Model The TS fuzzy model is described by a set of fuzzy implications [9]–[11], which characterize local relations of the system in the state space. The main feature of the TS model is to express the local dynamics of each fuzzy rule (implication) by a linear state-space system model, and the overall fuzzy system is then modeled by fuzzy “blending” of these local linear system models through some suitable membership functions. Specifically, a general TS fuzzy system is described as follows: Rule i: IF x1 (t) is Mi1 . . . and xn (t) is Min
Ai x(t) + Bi u(t)
THEN x_ (t) =
where
THEN
can be converted to the following model: Discrete-time Plant Rule i:
IF x1 (k) is Mi1 . . . and xn (k) is Min
x(t) = [x1 (t); x2(t); . . . ; xn u(t) = [u1 (t); u2(t); . . . ; um(t)]T
i = 1; 2; . . . ; r , in which r is the number of IF-THEN rules, Mij are fuzzy sets, and the equation x_ (t) = Ai x(t) + Bi u(t) is the output
where
2
Gi = exp(AiTs ) = I + Ai Ts + Ai2 T2!s + 1 1 1 T Hi = exp(Ai )Bi d = (Gi 0 I)A0i 1Bi and Ts is the sampling time. Proof: The exact solution of (3) at t = kTs + Ts , where Ts is the sampling period, is given by
x(kTs + Ts )
A
Bu
n j =1
Mij (xj );
in which Mij (xj ) is the degree of membership of xj in Mij , with r wi > 0; i = 1; 2; . . . r: i=1 wi 0; By using i (=wi = ri=1 wi ) instead of wi , (2) is rewritten as r _ (t) = i f i (t) + i (t)g i=1 r r = i i (t) + i i (t): (3) i=1 i=1 Note that r i = 1; i = 1; 2; . . . r: (4) i=1 i 0;
x
Ax
Bu
A x
B u
in which i (k) can be regarded as the firing strength of the IF-THEN rules. B. Discretization of the Continuous-Time TS Fuzzy Model
There are a few methods for discretizing a linear time-invariant (LTI) continuous-time system. Unfortunately, these discretization methods cannot be directly applied to the discretization of the continuous-time TS fuzzy models, since the defuzzified system is not LTI but linear time-varying. It is very difficult to obtain the state transition matrix for discretization. The recently developed theorem gives a rigorous mathematical foundation for the discretization of a continuous-time TS fuzzy model [24]. Theorem 1: The continuous-time TS fuzzy model of the form Continuous-time Plant Rule i:
i (kTs + Ts );
i=1 kT +T
+
kT r
2
where
wi =
r
=8
x u
Ax
(5)
0
from the ith IF-THEN rule. Assume that i ; i = 1; 2; . . . r , are n 2 n Hurwitz stable matrices, that is, their eigenvalues having sufficiently negative real parts. Now, given a pair of ( (t); (t)), the final output of the fuzzy system is inferred by r _ (t) = i=1 wi f ir (tw) + i (t)g (2) i=1 i
x
x(k + 1) = Gi x(k) + Hiu(k)
THEN
(1)
(t)]T
x_ (t) = Ai x(t) + Biu(t)
i=1
r
8
i=1
r i=1
x(kTs)
i (kTs )
i (kTs + Ts );
>0
r i=1
i ( )
(i Bi ) u( ) d
(6)
where 8( ri=1 i (kTs + Ts ); ri=1 i (kTs )) = 9( ri=1 i (kTs + Ts ))9( ri=1 i (kTs )) is the state transition matrix of (3), and 9 is the fundamental matrix of the uncontrolled TS fuzzy system (with (t) = 0), and is nonsingular for all t. For a sufficiently small Ts , the input u(t) can be regarded approximately as a piecewise constant over the integration interval, namely, (t) = (kTs ) for kTs t < kTs + Ts . Then, (6) can be rewritten as
u
u
u
x(kTs + Ts) = Gx(kTs) + H u(kTs)
(7)
where
G = 8 H =
r
i=1 kT +T
kT
2
i (kTs + Ts );
r i=1
r
8
i=1
r i=1
i (kTs )
i (kTs + Ts );
r i=1
i ( )
(i Bi ) d :
The exact evaluation of the state-transition matrix 8(1; 1) is very difficult, if not impossible, since the continuous-time TS fuzzy model (3) is time-varying. To solve this problem, we select a set of discrete-time points, kTs , such that ri=1 i (t) i and ri=1 i (t) i r (kT ) can be approximated by constant matrices s i and i=1 i r (kT ) , respectively, over each interval [ kT ; kT i s i s s + Ts ]. i=1 Then, a set of difference equations can be used to describe the discrete-time TS fuzzy model at each kTs [25]. and H have the folIn the time interval kTs t < kTs + Ts ; G lowing representations:
A
B
G (kTs ) = exp H (kTs ) =
r
i=1 kT +T
kT
i (kTs )Ai Ts
exp
r i=1
i (kTs )Ai
A
B
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r
2 (kTs + Ts 0 ) r
= exp
i=1
r
2
i=1
01
i (kTs )Ai
x
1) For the uncontrolled system (3). The origin, that is, 0 = 0, is obviously the fixed point of the uncontrolled system (3), and its Jacobian at the origin is ri=1 i ( 0 ) i . The characteristic equation is
i (kTs )Bi d
i (kTs )Ai Ts r
x A
0I i (kTs )Bi :
A
A
A
A
x) 0 I )x01 = I + (1=2!)x + (1=3!)x2 + 1 1 1 ; one also
H (kTs ) =
+T
kT kT
exp
r i=1
2 (kTs + Ts 0 ) r
= exp
2 =
2
r i=1
i=1
r i=1 r i=1 r
r i=1
i=1
i (kTs )Bi Ts =
where
r i=1
01
i=1
r i=1
Hi = Bi Ts
0
i (kTs )Bi
n
( 0 ij ) = 0
j =1
A G
I
A
r
i Gi
A x A
G
r i=1 r
G
i kGi k i maxfkGi kg = maxfkGi kg
i=1
=def < 1: r i=1
i Gi
kx(k)k kx(k)k:
By the contraction mapping theorem, it is concluded that the uncontrolled discretized system (5) is stable in the neighborhood of the origin.
i (kTs )Bi
III. CONTROLLER DESIGN FOR CHAOTIFYING CONTINUOUS-TIME TS FUZZY MODELS After converting the continuous-time TS fuzzy system to a discrete-time counterpart, we can design a controller for the discretized TS fuzzy system to make it chaotic. The chaotification problem for the discretized system (5) is to design a control input sequence, f (k)g1 k=0 , with an arbitrarily small magnitude, > 0, namely
i (kTs )Hi
u
ku(k)k ;
Denote kTs by k , one has (5). If the subsystems are stable in each local subspace, and i ( (t)); i = 1; 2; . . . ; r, are continuously differential in the neighborhood of the origin, the local stability of the overall system is described as the following theorem. Theorem 2 (Local Stability Theorem): In (1), if i ; i = 1; 2; . . . r , are all n 2 n Hurwitz stable matrices, then the uncontrolled system (3) (with (t) = 0) and uncontrolled discretized system (5) (with (k) = 0) are stable in the neighborhood of the origin. Proof: We first note that i ; i = 1; 2; . . . r , are all n 2 n Hurwitz stable matrices, that is, all of their eigenvalues have sufficiently negative real parts.
x
A
u
A
i (x0 )
A
kx(k + 1)k
exp(Ai )Bi d = (Gi 0 I)Ai01Bi :
u
i=1
i (x0 )(I 0 Ai )
Thus
0I
2
T
i=1 r
I
Gi = exp(Ai Ts) = I + Ai Ts + Ai2 T2!s + 1 1 1 I + Ai Ts and
A
i (kTs )Bi d
01
r
i=1
r
i (x0 )Ai =
where ij ; j = 1; 2; . . . n, are the eigenvalues of i ; i = 1; 2; . . . r , and all have sufficiently negative real parts. If 0, then j 0 ri=1 i i j > 0. So j 0 ri=1 i i j = 0 holds only when < 0. This means that ri=1 i ( 0 ) i is a Hurwitz stable matrix, hence the uncontrolled system (3) is stable in the neighborhood of the origin. 2) For the uncontrolled discretized system (5). Since i ; i = 1; 2; . . . r , are Hurwitz stable matrices, obviously, i = exp( i Ts ) are Schur stable matrices, that is, ( i ) < 1, there exists a certain norm, k 1 k, such that k i k < 1. From (4) and the convexity of the matrix norm, it follows that
i=1
i (kTs )Ai Ts + O TS2 i (kTs )Ai
r
=
i (kTs )Ai
i (kTs )Ai Ts
i (kTs )Ai
I 0
(8)
i=1 i=1 For a sufficiently small sampling period Ts > 0, and by using a power series expansion, one has r G (kTs ) = exp i (kTs ) i Ts i=1 r = I + i (kTs ) i Ts + O Ts2 i=1 r i (kTs )(I + i Ts ) i=1 r r i (kTs ) exp( i Ts ) = i (kTs )Gi i=1 i=1
and by (exp( has
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for all k
= 0; 1; 2; . . . ;
(9)
such that the controlled system (3) becomes chaotic. For this purpose, among several possible candidates, the simple sinusoidal function is used to construct the control input as follows:
u(k) = 8( x(k))
['( x1 (k)); '( x2 (k)); . . . ; '( xn(k))]T
where
(10)
x(k) = [x1 (k); x2(k); . . . ; xn(k)]T ; is a constant, and ' :
< ! < is a continuous sinusoidal function defined by (see Fig. 1)
'(x) = sin x : (11) Obviously, j'(x)j for any x 2 0 such that g is differentiable with all eigenvalues of g 0 (x) exceeding the unity in absolute value for all x 2 B (x3 ; r); 2) There exists a point x0 2 B (x3 ; r), with x0 6= x3 , such that for some positive integer m; g m (x0 ) = x3 and det((g m )0 (x0 )) 6= 0. Theorem 4 [19]: If system (14) has a snap-back repeller then the system is chaotic in the sense of Li and Yorke, namely, 1) There exists a positive integer n such that for every integer p n, system (12) has p-periodic points. 2) There exist a scrambled set (an uncountable invariant set S containing no periodic points) such that a) g (S ) S b) for every y 2 S and any periodic point x of (14)
k010j =
k
lim sup kg (x) 0 g (y)k > 0 k
kx(0)k
010j
01
j =0
c)
lim sup kg (x) 0 g (y)k > 0: k
ku(j )k
x; y
k
k
2 S0 ;
S
such that for any
lim!1 inf kg (x) 0 g (y)k = 0: k
k
k
(13) B. Verification of Chaos in the Controlled TS Fuzzy System
This means that the sinusoidal function folds an expanding trajectory back toward the origin when the trajectory becomes too large in magnitude, thus bounding the controlled system trajectory globally. On the other hand, it will be shown in the next section that if is chosen to be large enough, the controller designed above can lead all eigenvalues of the controlled system Jacobian, at every time step, exceeds the unity in absolute value. Consequently, it can be proven that all the Lyapunov exponents of the controlled system are strictly positive, so that the system trajectory is locally expanding in all directions. The combination of these two effects, stretching and folding, will then yield complex chaotic dynamics within the bounded region of the controlled system trajectories. IV. VERIFICATION OF CHAOS IN THE CONTROLLED TS FUZZY MODEL In this section, the Marotto theorem is first reviewed, and then applied to prove that the controlled system (5), (10) and (11) is chaotic in the sense of Li and Yorke. A. The Marotto Theorem Consider a general n-dimensional discrete-time autonomous system of the form
x(k + 1) = g(x(k))
!1
3) There exists an uncountable subset S0 of
k010j ku(j )k
1 1 0 :
k
!1 for every x; y 2 S with x 6= y k
(14)
The theoretical result of the controller design is summarized as follows. Theorem 5: Suppose that i (k); i = 1; 2; . . . ; r , are continuously differentiable in the neighborhood of the fixed point, x3 = 0, of the controlled system (5), (10) and (11). Then, there exists a positive constant such that if > , then the controlled TS fuzzy system (5), (10) and (11) is chaotic in the sense of Li and Yorke. Proof: The controlled system (5), (10) and (11) is
x(k + 1) =
=
r i=1
i fGi x(k) + u(k)g
r i=1
i Gi x(k) + 8( x(k)) g(x(k)): (15)
Obviously, x3 = 0 is a fixed point of (15), which is now proven to be a snap-back repeller. Differentiating (15) at this fixed point yields
kg0 (0)k =
r i=1
+ I :
i Gi 0
(16)
If > (1+ )= , then kg 0 (0)k > 1. By the continuity of g 0 (x) in the neighborhood of the fixed point, there exists a small positive constant, r , such that when x 2 B (x3 ; r); kg0 (x)k > 1.
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Therefore, the Gerschgorin theorem [26] implies that all eigenvalues of g 0 (x) exceed the unity in absolute value for all x 2 B (x0 ; r). Next, it is shown that there exists a point, x0 2 B (x3 ; r), such that 2 g (x0 ) = 0 = x3 and (g2 (x0 ))0 6= 0. Indeed, it is easy to see that if > (3 =2), then there exist x1 = [(=2 ); . . . ; (=2 )]T and x2 = [(3=2 ); . . . ; (3=2 )]T , such that
g(x1 ) > 0
and
g(x2 ) < 0:
Therefore, by the mean value theorem in Calculus, there exists a point, x1 < x31 < x2 , such that g(x31 ) = 0. ~ = [r; r; . . . r]T . Then, it is clear that there exists a constant Let x 1 > 0 such that if > 1 , then
g(0) = 0 < x31
and
g(~x) > x31 :
Fig. 2.
Trajectory of the TS fuzzy Lorenz system.
Fig. 3.
Trajectory of the discretized TS fuzzy Lorenz system.
Using the mean value theorem again, one concludes that there exists a point, x0 2 B (x3 ; r), such that g (x0 ) = x31 . Therefore
g2 (x0 ) = g(x31 ) = 0: On the other hand, there exists a constant 2 for cos((= ) x13 ) < 0. Therefore
> 0 such that g0 (x31 ) < 0,
(g 2 )0 (x0 ) = g 0 (x31 )g 0 (x0 ) 6= 0: To conclude, if > maxf(1 + )=; (3 )=2; 1 ; 2 g, then x0 = 0 is a snap-back repeller of the map g defined in (15), so the controlled system (5), (10) and (11) is chaotic in the sense of Li and Yorke. V. A SIMULATION EXAMPLE To visualize the theoretical analysis and design, an example is included here for illustration. Consider a continuous-time TS fuzzy system, which is the fuzzy model of the Lorenz equation, given by Rule 1: IF x1 (t) is M1 ; x (t) d x1 (t) THEN dt 2 x3 (t) Rule 2: IF x1 (t) is M2 ; x (t) d x1 (t) THEN dt 2 x3 (t) where
A1 =
0d r
d
01
x1 min 0d d A2 = r 01 0 x1 max 0
= A1
x1 (t) x2 (t) x3 (t)
= A2
x1 (t) x2 (t) x3 (t)
0
0x1 min 0b
Rule 1: IF x1 (k) is M1 ; x1 (k + 1) THEN x2 (k + 1) x3 (k + 1) Rule 2: IF x1 (k) is M2 ; x1 (k + 1) THEN x2 (k + 1) x3 (k + 1) where
G1 =
0
0x1 max 0b
and membership functions are
M1 =
In terms of Theorem 1, the discretized TS fuzzy model of the continuous-time TS fuzzy Lorenz system is obtained as follows:
0x1 + x1 max M2 = x1 0 x1 min : x1 max 0 x1 min x1 max 0 x1 min
where Mi ; i = 1; 2, are positive semi-definite for all x1 2 [x1 min ; x1 max ], and d; r , and b are parameters. With the parameter choice (d; r; b) = (10; 28; 8=3) and initial value (0.1, 0.1, 0.1), the trajectory of the TS fuzzy system of the Lorenz system is shown in Fig. 2.
G2 =
1 0 dTs
rTs
0 1 0 dTs
rTs 0
= G1
x1 (k) x2 (k) x3 (k)
= G2
x1 (k) x2 (k) x3 (k)
dTs
0
dTs
0
1 0 Ts
0x1 min Ts x1 min Ts 1 0 bTs 1 0 Ts
0x1 max Ts x1 max Ts 1 0 bTs
Fig. 3 shows the trajectory of the discrete-time version of the continuous-time TS fuzzy Lorenz model, with Ts = 0.004 s. It shows that the overall shape of the trajectory is very similar to that in Fig. 2. When the parameters are chosen as d = 200, r = {-}40, b = 200, the eigenvalues of Ai are –132.1267, –68.8733, and –200.0000, so they are Hurwitz stable matrices. Hence, by Theorem 2, the continuous- and discrete-time TS fuzzy Lorenz system are stable. Then anticontroller is used as described in (10)–(11) to make the discretized TS fuzzy Lorenz
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Fig. 4. Period-doubling bifurcation at
Fig. 6.
Bifurcation diagram for
0x
.
Fig. 7.
Bifurcation diagram for
0x
.
= 0:6.
are shown in Figs. 4–9, respectively. These numerical results verify the theoretical analysis and the design of the chaotifying controller developed in this paper. VI. CLOSING REMARKS Fig. 5.
Chaotic time-wave diagrams.
system chaotic, the controlled TS fuzzy Lorenz system is described as follows:
x(k + 1) =
r i=1 r
= i=1 r
= i=1
G x(k) + u(k)g
i f
i
i
G x(k) + u(k)
i
G x(k) + sin
i
i
x
(k) :
In the simulation, the magnitude of the control input is arbitrarily chosen to be = 0:1. Thus, kuk k < , and can also be regarded as a control parameter. When takes values of 0.6 and 2.1, the time wave diagrams, phase portrait diagrams, and bifurcation diagrams
1
In this paper, an approach was developed to drive an originally stable continuous-time TS fuzzy system chaotic. A method was proposed to discretize the continuous-time TS fuzzy system first, then a simple state-feedback controller of arbitrarily small magnitude is designed, which is simple since only a sinusoidal function is used in the design. And yet the result is mathematically rigorous since it has been proven that the designed controlled system indeed becomes chaotic in the sense of Li and Yorke. It is interesting to note that, instead of the sinusoidal function, the mod-operation controller can also make the stable discrete-time TS fuzzy system chaotic. Besides, we should also note that the proposed approach is available for any fuzzy system subject to with continuously differentiable membership functions (MF). Other fuzzy systems, such as that with triangular MF, need specified definition to satisfy the conditions in Theorem 5. In addition, for the choice of sampling time Ts , the smaller it is, the more exact for discretization but the more time-consuming for control. So Ts should be chosen as large as possible under
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[5] [6] [7] [8] [9] [10] [11] [12] [13] Fig. 8. Bifurcation diagram for
0x
.
[14] [15] [16] [17] [18] [19] [20]
Fig. 9. Chaotic phase portrait at
= 2:1.
[21] [22] [23]
the guarantee of accuracy of discretization. Furthermore, this approach is easy to be utilized to practical digital control. To our knowledge, this paper is the first attempt to systematically control a continuous-time TS fuzzy system to become chaotic on purpose. Many chaotic systems can be represented by TS fuzzy models. Hence, this design method of making a stable continuous-time TS fuzzy system chaotic provides a means to further explore the interaction between fuzzy control system theory and chaos theory, which has great potential in future engineering applications of chaos. REFERENCES [1] G. Chen, “Chaos: Control and anticontrol,” IEEE Circuits Syst. Soc. Newslett., vol. 9, pp. 1–5, Mar. 1998. [2] G. Chen and X. Dong, From Chaos to Order: Methodologies, Perspective and Applications. Singapore: World Scientific, 1998. [3] G. Chen and D. Lai, “Feedback control of Lyapunov exponents for discrete-time dynamical systems,” Int. J. Bifurcation Chaos, vol. 6, no. 7, pp. 1341–1349, 1996. [4] , “Anticontrol of chaos via feedback,” Proc. IEEE Conf. Decision and Control, pp. 10–12, 1997.
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