generating chaos via feedback control from a ... - Semantic Scholar
International Journal of Bifurcation and Chaos, Vol. 12, No. 10 (2002) 2283–2291 c World Scientific Publishing Company
GENERATING CHAOS VIA FEEDBACK CONTROL FROM A STABLE TS FUZZY SYSTEM THROUGH A SINUSOIDAL ∗ NONLINEARITY ZHONG LI and JIN BAE PARK Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea GUANRONG CHEN Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR, China YOUNG HOON JOO School of Electronics and Information Engineering, Kunsan National University, Kunsan, Chonbuk 573-701, Korea YOON HO CHOI School of Electronic and Mechanical Engineering, Kyonggi University, Kyonggi, 442-760, Korea Received February 23, 2001; Revised December 20, 2001
An approach is proposed for making chaotic a given stable Takagi–Sugeno (TS) fuzzy system 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 TS fuzzy system, where the chaos is in the sense of Li and Yorke. A numerical example is included to visualize the theoretical analysis and the controller design. Keywords: Chaos generation; TS fuzzy system; Lyapunov exponent; state feedback; parallel distributed compensation (PDC).
1. Introduction 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 more and more attention from the engineering and physics communities in recent years ∗
[Chen, 1998; Chen & Dong, 1998; Chen & Lai, 1996, 1997a, 1997b, 1998; Wang & Chen, 1999, 2000]. There are many practical reasons for chaos generation, since chaos has an impact on some novel time- and/or energy-critical applications. This includes high-performance circuits and devices (e.g. delta-sigma modulators and power converters), liquid mixing, chemical reactions, biological systems
This work was supported by the research project Brain Korea 21. 2283
2284 Z. Li et al.
(e.g. in the human brain, heart, and perceptual processes), secure information processing, and critical decision-making in political, economic and military events [Chen, 1998]. Interactions between fuzzy logic and chaos theory have been explored since 1990s. The explorations have been carried out mainly on fuzzy modeling of chaotic systems using the TS fuzzy model [Takagi & Sugeno, 1985; Wang et al., 1995, 1996a; Wang et al., 1996b; Tanaka et al., 1998; Joo et al., 1999; Lee et al., 2001], linguistic descriptions [Porto & Amato, 2000; Baglio et al., 1996], and suppression control of chaos via an LMI-based fuzzy control system design [Wang et al., 1996b; Tanaka et al., 1998]. In these investigations, to design a fuzzy 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 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. This paper is an attempt to study the converse, i.e. how to make a given stable TS fuzzy system chaotic by designing a suitable state feedback controller. In order to make a stable discrete-time TS fuzzy system chaotic, a natural and straightforward approach is to utilize the chaotic mechanism of stretching and folding to design a state feedback controller, which on one hand makes the originally stable TS fuzzy system orbitally divergent but on the other hand keeps the system state variables bounded. Among several candidates, the simple sinusoidal function is used to construct a state feedback controller, which can have arbitrarily small control gain as desired [Tang et al., 2001; Arena et al., 1996; Suykens & Vandswalle, 1993; Suykens et al., 1997]. 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 [Marotto, 1978] is then applied to show that the controlled system so designed is indeed chaotic in the mathematical sense of Li and Yorke [1975]. Finally, a simple example is included to verify and visualize the theory and the results of the paper.
2. Chaos Generator Design for a Discrete TS Fuzzy System 2.1. The TS fuzzy model The TS fuzzy model [Takagi & Sugeno, 1985; Wang et al., 1995, 1996a] is described by a set of fuzzy implications, 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 implication (rule) 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 xk (1) is Mi1 . . . and xk (n) is Min THEN xk+1 = Ai xk + uk ,
(1)
where xk = [xk (1), xk (2), . . . xk (n)]T , uk = [uk (1), uk (2), . . . uk (m)]T , i = 1, 2, . . . , r, in which r is the number of IF–THEN rules, Mij are fuzzy sets, and the equation xk+1 = Ai xk + uk is the output from the ith IF–THEN rule. Assume that Ai , i = 1, 2, . . . r, are n × n Schur stable matrices, that is, its spectral radius satisfies ρ(i) < 1. Now, given a pair of (xk , uk ), the final output of the fuzzy system is inferred by r X
xk+1 =
wi (k){Ai xk + uk }
i=1 r X
,
(2)
wi (k)
i=1
where wi (k) =
n Y
Mij (xk (j)) ,
j=1
in which Mij (xk (j)) is the degree of membership of xk (j) in Mij , with r X wi (k) > 0 i=1
wi (k) ≥ 0,
i = 1, 2, . . . r .
Generating Chaos Via Feedback Control 2285
Pr
ϕ (x)
By using hi (k)(= wi (k)/ i=1 wi (k)) instead of wi (k), Eq. (2) is rewritten as xk+1 =
r X
hi (k){Ai xk + uk }
i=1
=
σ
" r X
−σ
− 2σ
0 σ
2σ
x
#
hi (k)Ai xk + uk .
−σ
(3)
i=1
Fig. 1.
Note that r X hi (k) = 1 i=1
hi (k) ≥ 0,
(4)
The sinusoidal function used for chaos generation.
the controller (6) and (7 ), are uniformly bounded by a constant, σ(1 − α)−1 .
i = 1, 2, . . . r The solution of the controlled system (3) can be written as
Proof.
in which hi (k) can be regarded as the normalized weight of the IF–THEN rules.
xk =
#k
hi (k)Ai
The chaotification problem is to design a control input sequence, {uk }∞ k=0 , with an arbitrarily small magnitude, σ > 0, namely, for all k = 1, 2, . . . r ,
(5)
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:
+
k−1 X
" r X
j=1
i=1
#k−1−j
hi (k)Ai
i=1
#k−1−j " r k−1
X X
+ hi (k)Ai kuj k
j=1
T
where xk = [xk (1), xk (2), . . . , xk (n)]T , β is a constant, and ϕ : < → < is a continuous sinusoidal function defined by (see Fig. 1)
ϕ(x) = σ sin
π x . σ
(7)
Obviously, |ϕ(x)| ≤ σ for any x ∈ 0 such that g is differentiable with all eigenvalues of g0 (x) exceeding the unity in absolute value for all x ∈ B(x∗ ; r); (ii) There exists a point x0 ∈ B(x∗ ; r), with x0 6= x∗ , such that for some positive integer m, gM (x0 ) = x∗ and det((gm )0 (x0 )) 6= 0. Theorem 1 [Marotto, 1978].
If system (12) has a snap-back repeller then the system is chaotic in the sense of Li and Yorke, namely, (i) There exists a positive integer n such that for every integer p ≥ n, system (12) has p-periodic points (ii) There exist a scrambled set (an uncountable invariant set S containing no periodic points) such that (a) g(S) ⊂ S, (b) for every y ∈ S and any periodic point x of (12),
wi (k)
lim sup kgk (x) − gk (y)k > 0 ,
i=1
k→∞
(c) for every x, y ∈ S with x 6= y,
3. Verification of Chaos in the Controlled TS Fuzzy System In this section, the Marotto theorem is first reviewed, and then applied to prove that the controlled systems (3) and (6) are chaotic in the sense of Li and Yorke.
3.1. The Marotto theorem
k→∞
(iii) there exists an uncountable subset S0 of S such that for any x, y ∈ S0 , lim inf kgk (x) − gk (y)k = 0 .
Consider a general n-dimensional discrete-time autonomous system of the form xk+1 = g(xk ) ,
lim sup kgk (x) − gk (y)k > 0 ;
(12)
where g is a C 1 nonlinear map. Let gt denote the t times of compositions of g with itself. A point, x∗ , is said to be a p-periodic point of g if gp (x∗ ) = x∗ but gt (x∗ ) 6= x∗ for 1 ≤ t < p. If p = 1, that is, g(x∗ ) = x∗ , x∗ is called a fixed point. Let g0 (x) and det(g0 (x)) be the Jacobian of g at the point x and its determinant, respectively, and
k→∞
3.2. Verification of chaos in the controlled TS fuzzy system The theoretical result of the controller design is summarized as follows. Suppose that hi (k), i = 1, 2, . . . , r, are continuously differentiable in the neighborhood of the fixed point, x∗ = 0, of the controlled system (3). Then, there exists a positive constant β such that if Theorem 2.
Generating Chaos Via Feedback Control 2287
β > β, then the controlled TS fuzzy systems (3) and (6) are chaotic in the sense of Li and Yorke. Proof.
The controlled systems (3) and (6) are
xk+1 =
r X
hi (k){Ai xk + uk }
i=1
=
" r X
4. A Simulation Example
#
hi (k)Ai xk + Φ(βxk ) ≡ g(xk ) . (13)
i=1
Obviously, x∗ = 0 is a fixed point of (13), which is now proven to be a snap-back repeller. Differentiating (13) at this fixed point yields
"
# r
X
kg0 (0)k = hi (k)Ai + πβI .
i=1
To conclude, if β > β ≡ max{(1 + α)/π, (3σ)/2, β1 , β2 }, then x0 = 0 is a snap-back repeller of the map g defined in (13), so the controlled systems (3) and (6) are chaotic in the sense of Li and Yorke.
To visualize the theoretical analysis and design, an example is included here for illustration. First, consider a nonchaotic discrete-time TS fuzzy model, given by "
Rule 1 : IF xk is M1 , THEN "
(14)
= A1
0
If β > (1 + α)/π, then kg0 (0)k > 1. By the continuity of g0 (x) in the neighborhood of the fixed point, there exists a small positive constant, r, such that when x ∈ B(x∗ ; r), kg0 (x)k > 1. Therefore, the Gerschgorin theorem [Golub & van Loan, 1983] implies that all eigenvalues of g0 (x) exceed the unity in absolute values for all x ∈ B(x0 , r). Next, it is shown that there exists a point, x0 ∈ B(x∗ , r), such that g2 (x0 ) = 0 = x∗ and (g2 (x0 ))0 6= 0. Indeed, it is easy to see that if β > 3α/2, then there exist x1 = [σ/2β, . . . , σ/2β]T and x2 = [3σ/2β, . . . , 3σ/2β]T , such that g(x1 ) > 0 and g(x2 ) < 0 . Therefore, by the Mean Value Theorem in Calculus, there exists a point, x1 < x∗1 < x2 , such that g(x∗1 ) = 0. ˜ = [r, r, · · · r]T . Then, it is clear that there Let x exists a constant β1 > 0 such that if β > β1 , then g(0) = 0 < x∗1 and g(˜ x) > x∗1 . Using the Mean Value Theorem again, one concludes that there exists a point, x0 ∈ B(x∗ ; r), such that g(x0 ) = x∗1 . Therefore, g2 (x0 ) = g(x∗1 ) = 0 . On the other hand, there exists a constant β2 > 0 such that g0 (x∗1 ) < 0, for cos(π/σ βx∗1 ) < 0. Therefore, (g2 )0 (x0 ) = g0 (x∗1 )g0 (x0 ) 6= 0 .
xk
+ uk "
Rule 2 : IF xk is M2 , THEN = A2
xk
#
yk+1
#
yk
"
xk+1
xx+1
#
yk+1
#
+ uk
yk
where "
A1 =
d
0.3
1
0
#
"
,
A2 =
−d
0.3
1
0
#
,
xk ∈ [−d, d] and d > 0, with membership functions
1 xk M1 = 1− 2 d
,
1 xk M2 = 1+ 2 d
.
The controlled TS fuzzy system is described as follows: xk+1 =
r X i=1 r X
hi (k){Ai xk + uk } =
r X
hi (k)Ai xk + uk .
i=1
π = hi (k)Ai xk + σ sin βxk . σ i=1 In the simulation, the magnitude of the control input is arbitrarily chosen to be σ = 0.1. Thus, kuk k∞ < σ, and β can also be regarded as a control parameter. Without control, the TS fuzzy model is stable, as shown in Fig. 2. When β takes values of 0.25, 0.4, 0.45, 0.5 and 1.3, the phase portrait diagrams, time wave diagrams, and bifurcation diagrams are shown in Figs. 3–15, respectively. These numerical results verify the theoretical analysis
2288 Z. Li et al. 0.35
0.154
0.3
0.152 0.15
0.25
beta=0.25
0.148 x(k)
y(k)
0.2
0.146
0.15
0.144 0.1
0.142 0.05
0
0.14
0
0.02
0.04
Fig. 2.
0.06
0.08
0.1 x(k)
0.12
0.14
0.16
0.18
0.138
0.2
0
The stable TS fuzzy system.
100
Fig. 3.
0.28
200
300 k
400
500
600
Periodic orbits at β = 0.25.
0.17 0.16
0.27
beta=0.25
0.15 0.14
beta=0.4
0.26 x(k)
y(k)
0.13 0.12 0.25 0.11 0.1
0.24
0.09 0.23
0
100
Fig. 4.
200
300 k
400
500
0.08
600
Periodic orbits at β = 0.25.
0
100
Fig. 5.
0.26
200
300 k
400
500
600
Period-doubling bifurcation at β = 0.4.
0.18
0.24
0.16
0.22 0.14 0.2
beta=0.4 0.12
beta=0.45
x(k)
y(k)
0.18 0.16
0.1
0.14 0.08 0.12 0.06
0.1 0.08
0
Fig. 6.
100
200
300 k
400
500
Period-doubling bifurcation at β = 0.4.
600
0.04
0
Fig. 7.
100
200
300 k
400
500
600
Special period-doubling bifurcation at β = 0.45.
Generating Chaos Via Feedback Control 2289 0.35
0.18
0.3
0.16 beta=0.5 0.14
0.25 0.12 x(k)
y(k)
0.2
0.15
0.1
beta=0.45 0.08
0.1 0.06 0.05
0
0.04
0
Fig. 8.
100
200
300 k
400
500
0.02
600
0
100
Special period-doubling bifurcation at β = 0.45.
Fig. 9.
200
300 k
400
500
600
Period-8 bifurcation at β = 0.5.
0.25
0.3
0.2
beta=1.3
0.25
0.15 0.2
0.1 0.05 beta=0.5
y(k)
y(k)
0.15
0.1
0 -0.05 -0.1
0.05
-0.15 0
-0.2 -0.05
0
100
Fig. 10.
200
300 k
400
500
-0.25 -0.2
600
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
x(k)
Period-8 bifurcation at β = 0.5.
Fig. 11.
Phase portrait with some structure.
0.3
0.2
0.2
0.15 0.1
beta=1.3
0.1
beta=1.3
0.05 y(k)
x(k)
0 0
-0.1 -0.05
-0.2 -0.1
-0.3
-0.15 -0.2
-0.4 0
100
200
Fig. 12.
300 k
400
k − xk diagram.
500
600
0
100
200
Fig. 13.
300 k
400
k − yk diagram.
500
600
2290 Z. Li et al. 0.2
0.3
0.15
0.2
0.1 0.1 0.05
y(k)
x(k)
0 0
-0.1 -0.05 -0.2
-0.1
-0.3
-0.15 -0.2 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.4 0.1
0.2
0.3
beta
Fig. 14.
Bifurcation diagram for β − xk .
and the design of the chaos generator developed in this paper.
5. Conclusions An approach is developed in this paper for designing a simple state-feedback controller of arbitrarily small magnitude that can drive an originally stable TS fuzzy system chaotic. The controller 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. To our knowledge, except for a couple of related publications [Baglio, et al., 1996; Porto & Amato, 2000], this paper is the first attempt in the development of using fuzzy systems to systematically generate chaos. Many chaotic systems, such as the Lorenz and the Henon systems, can be represented by TS fuzzy models. Hence, this design method of making a stable TS fuzzy system chaotic provides a means to further explore the interaction between fuzzy control systems theory and chaos theory, which has great potential in future engineering applications of chaos.
References Arena, P., Baglio, S., Fortuna, L. & Manganaro, G. [1996] “Generation of n-double scrolls via cellular neural networks,” Int. J. Circuit Th. Appl. 24, 241–252. Baglio, S., Fortuna, L. & Manganaro, G. [1996] “Design of fuzzy iterators to generate chaotic time series with
0.4
0.5
0.6
0.7
0.8
0.9
1
beta
Fig. 15.
Bifurcation diagram for β − yk .
assigned Lyapunov exponent,” Electron. Lett. 32(4), 292–293. Chen, G. & Lai, D. [1996] “Feedback control of Lyapunov exponents for discrete-time dynamical systems,” Int. J. Bifurcation and Chaos 6(7), 1341–1349. Chen, G. & Lai, D. [1997a] “Anticontrol of chaos via feedback,” Proc. IEEE Conf. Decision and Control (San Diego, CA), pp. 10–12. Chen, G. & Lai, D. [1997b] “Making a dynamical system chaotic: Feedback control of Lyapunov exponents for discrete-time dynamical systems,” IEEE Trans. Circuits Syst. I 44(3), 250–253. Chen, G. [1998] “Chaos: Control and anticontrol,” IEEE Circuits Syst. Soc. Newslett. 9(1), 1–5. Chen, G. & Dong, X. [1998] From Chaos to Order: Methodologies, Perspective and Applications (World Scientific, Singapore). Chen, G. & Lai, D. [1998] “Feedback anticontrol of chaos,” Int. J. Bifurcation and Chaos 8(7), 1585–1590. Golub, G. H. & Van Loan, C. F. [1983] Matrix Computations (Johns Hopkins University Press, Baltimore, MD). Joo, Y. H., Shieh, L. S. & Chen, G. [1999] “Hybrid state-space fuzzy model-based controller with dualrate sampling for digital control of chaotic systems,” IEEE Trans. Fuzzy Syst. 7(4), 394–408. Lee, H. J., Park, J. B. & Chen, G. [2001] “Robust fuzzy control of nonlinear systems with parametric uncertainties,” IEEE Trans. Fuzzy Syst. 9(2), 369–399. Li, T. Y. & Yorke, J. A. [1975] “Period three implies chaos,” Amer. Math. Monthly 82, 481–485. Marotto, F. R. [1978] “Snap-back repellers imply chaos in
GENERATING CHAOS VIA FEEDBACK CONTROL FROM A STABLE TS FUZZY SYSTEM THROUGH A SINUSOIDAL ∗ NONLINEARITY ZHONG LI and JIN BAE PARK Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea GUANRONG CHEN Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR, China YOUNG HOON JOO School of Electronics and Information Engineering, Kunsan National University, Kunsan, Chonbuk 573-701, Korea YOON HO CHOI School of Electronic and Mechanical Engineering, Kyonggi University, Kyonggi, 442-760, Korea Received February 23, 2001; Revised December 20, 2001
An approach is proposed for making chaotic a given stable Takagi–Sugeno (TS) fuzzy system 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 TS fuzzy system, where the chaos is in the sense of Li and Yorke. A numerical example is included to visualize the theoretical analysis and the controller design. Keywords: Chaos generation; TS fuzzy system; Lyapunov exponent; state feedback; parallel distributed compensation (PDC).
1. Introduction 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 more and more attention from the engineering and physics communities in recent years ∗
[Chen, 1998; Chen & Dong, 1998; Chen & Lai, 1996, 1997a, 1997b, 1998; Wang & Chen, 1999, 2000]. There are many practical reasons for chaos generation, since chaos has an impact on some novel time- and/or energy-critical applications. This includes high-performance circuits and devices (e.g. delta-sigma modulators and power converters), liquid mixing, chemical reactions, biological systems
This work was supported by the research project Brain Korea 21. 2283
2284 Z. Li et al.
(e.g. in the human brain, heart, and perceptual processes), secure information processing, and critical decision-making in political, economic and military events [Chen, 1998]. Interactions between fuzzy logic and chaos theory have been explored since 1990s. The explorations have been carried out mainly on fuzzy modeling of chaotic systems using the TS fuzzy model [Takagi & Sugeno, 1985; Wang et al., 1995, 1996a; Wang et al., 1996b; Tanaka et al., 1998; Joo et al., 1999; Lee et al., 2001], linguistic descriptions [Porto & Amato, 2000; Baglio et al., 1996], and suppression control of chaos via an LMI-based fuzzy control system design [Wang et al., 1996b; Tanaka et al., 1998]. In these investigations, to design a fuzzy 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 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. This paper is an attempt to study the converse, i.e. how to make a given stable TS fuzzy system chaotic by designing a suitable state feedback controller. In order to make a stable discrete-time TS fuzzy system chaotic, a natural and straightforward approach is to utilize the chaotic mechanism of stretching and folding to design a state feedback controller, which on one hand makes the originally stable TS fuzzy system orbitally divergent but on the other hand keeps the system state variables bounded. Among several candidates, the simple sinusoidal function is used to construct a state feedback controller, which can have arbitrarily small control gain as desired [Tang et al., 2001; Arena et al., 1996; Suykens & Vandswalle, 1993; Suykens et al., 1997]. 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 [Marotto, 1978] is then applied to show that the controlled system so designed is indeed chaotic in the mathematical sense of Li and Yorke [1975]. Finally, a simple example is included to verify and visualize the theory and the results of the paper.
2. Chaos Generator Design for a Discrete TS Fuzzy System 2.1. The TS fuzzy model The TS fuzzy model [Takagi & Sugeno, 1985; Wang et al., 1995, 1996a] is described by a set of fuzzy implications, 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 implication (rule) 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 xk (1) is Mi1 . . . and xk (n) is Min THEN xk+1 = Ai xk + uk ,
(1)
where xk = [xk (1), xk (2), . . . xk (n)]T , uk = [uk (1), uk (2), . . . uk (m)]T , i = 1, 2, . . . , r, in which r is the number of IF–THEN rules, Mij are fuzzy sets, and the equation xk+1 = Ai xk + uk is the output from the ith IF–THEN rule. Assume that Ai , i = 1, 2, . . . r, are n × n Schur stable matrices, that is, its spectral radius satisfies ρ(i) < 1. Now, given a pair of (xk , uk ), the final output of the fuzzy system is inferred by r X
xk+1 =
wi (k){Ai xk + uk }
i=1 r X
,
(2)
wi (k)
i=1
where wi (k) =
n Y
Mij (xk (j)) ,
j=1
in which Mij (xk (j)) is the degree of membership of xk (j) in Mij , with r X wi (k) > 0 i=1
wi (k) ≥ 0,
i = 1, 2, . . . r .
Generating Chaos Via Feedback Control 2285
Pr
ϕ (x)
By using hi (k)(= wi (k)/ i=1 wi (k)) instead of wi (k), Eq. (2) is rewritten as xk+1 =
r X
hi (k){Ai xk + uk }
i=1
=
σ
" r X
−σ
− 2σ
0 σ
2σ
x
#
hi (k)Ai xk + uk .
−σ
(3)
i=1
Fig. 1.
Note that r X hi (k) = 1 i=1
hi (k) ≥ 0,
(4)
The sinusoidal function used for chaos generation.
the controller (6) and (7 ), are uniformly bounded by a constant, σ(1 − α)−1 .
i = 1, 2, . . . r The solution of the controlled system (3) can be written as
Proof.
in which hi (k) can be regarded as the normalized weight of the IF–THEN rules.
xk =
#k
hi (k)Ai
The chaotification problem is to design a control input sequence, {uk }∞ k=0 , with an arbitrarily small magnitude, σ > 0, namely, for all k = 1, 2, . . . r ,
(5)
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:
+
k−1 X
" r X
j=1
i=1
#k−1−j
hi (k)Ai
i=1
#k−1−j " r k−1
X X
+ hi (k)Ai kuj k
j=1
T
where xk = [xk (1), xk (2), . . . , xk (n)]T , β is a constant, and ϕ : < → < is a continuous sinusoidal function defined by (see Fig. 1)
ϕ(x) = σ sin
π x . σ
(7)
Obviously, |ϕ(x)| ≤ σ for any x ∈ 0 such that g is differentiable with all eigenvalues of g0 (x) exceeding the unity in absolute value for all x ∈ B(x∗ ; r); (ii) There exists a point x0 ∈ B(x∗ ; r), with x0 6= x∗ , such that for some positive integer m, gM (x0 ) = x∗ and det((gm )0 (x0 )) 6= 0. Theorem 1 [Marotto, 1978].
If system (12) has a snap-back repeller then the system is chaotic in the sense of Li and Yorke, namely, (i) There exists a positive integer n such that for every integer p ≥ n, system (12) has p-periodic points (ii) There exist a scrambled set (an uncountable invariant set S containing no periodic points) such that (a) g(S) ⊂ S, (b) for every y ∈ S and any periodic point x of (12),
wi (k)
lim sup kgk (x) − gk (y)k > 0 ,
i=1
k→∞
(c) for every x, y ∈ S with x 6= y,
3. Verification of Chaos in the Controlled TS Fuzzy System In this section, the Marotto theorem is first reviewed, and then applied to prove that the controlled systems (3) and (6) are chaotic in the sense of Li and Yorke.
3.1. The Marotto theorem
k→∞
(iii) there exists an uncountable subset S0 of S such that for any x, y ∈ S0 , lim inf kgk (x) − gk (y)k = 0 .
Consider a general n-dimensional discrete-time autonomous system of the form xk+1 = g(xk ) ,
lim sup kgk (x) − gk (y)k > 0 ;
(12)
where g is a C 1 nonlinear map. Let gt denote the t times of compositions of g with itself. A point, x∗ , is said to be a p-periodic point of g if gp (x∗ ) = x∗ but gt (x∗ ) 6= x∗ for 1 ≤ t < p. If p = 1, that is, g(x∗ ) = x∗ , x∗ is called a fixed point. Let g0 (x) and det(g0 (x)) be the Jacobian of g at the point x and its determinant, respectively, and
k→∞
3.2. Verification of chaos in the controlled TS fuzzy system The theoretical result of the controller design is summarized as follows. Suppose that hi (k), i = 1, 2, . . . , r, are continuously differentiable in the neighborhood of the fixed point, x∗ = 0, of the controlled system (3). Then, there exists a positive constant β such that if Theorem 2.
Generating Chaos Via Feedback Control 2287
β > β, then the controlled TS fuzzy systems (3) and (6) are chaotic in the sense of Li and Yorke. Proof.
The controlled systems (3) and (6) are
xk+1 =
r X
hi (k){Ai xk + uk }
i=1
=
" r X
4. A Simulation Example
#
hi (k)Ai xk + Φ(βxk ) ≡ g(xk ) . (13)
i=1
Obviously, x∗ = 0 is a fixed point of (13), which is now proven to be a snap-back repeller. Differentiating (13) at this fixed point yields
"
# r
X
kg0 (0)k = hi (k)Ai + πβI .
i=1
To conclude, if β > β ≡ max{(1 + α)/π, (3σ)/2, β1 , β2 }, then x0 = 0 is a snap-back repeller of the map g defined in (13), so the controlled systems (3) and (6) are chaotic in the sense of Li and Yorke.
To visualize the theoretical analysis and design, an example is included here for illustration. First, consider a nonchaotic discrete-time TS fuzzy model, given by "
Rule 1 : IF xk is M1 , THEN "
(14)
= A1
0
If β > (1 + α)/π, then kg0 (0)k > 1. By the continuity of g0 (x) in the neighborhood of the fixed point, there exists a small positive constant, r, such that when x ∈ B(x∗ ; r), kg0 (x)k > 1. Therefore, the Gerschgorin theorem [Golub & van Loan, 1983] implies that all eigenvalues of g0 (x) exceed the unity in absolute values for all x ∈ B(x0 , r). Next, it is shown that there exists a point, x0 ∈ B(x∗ , r), such that g2 (x0 ) = 0 = x∗ and (g2 (x0 ))0 6= 0. Indeed, it is easy to see that if β > 3α/2, then there exist x1 = [σ/2β, . . . , σ/2β]T and x2 = [3σ/2β, . . . , 3σ/2β]T , such that g(x1 ) > 0 and g(x2 ) < 0 . Therefore, by the Mean Value Theorem in Calculus, there exists a point, x1 < x∗1 < x2 , such that g(x∗1 ) = 0. ˜ = [r, r, · · · r]T . Then, it is clear that there Let x exists a constant β1 > 0 such that if β > β1 , then g(0) = 0 < x∗1 and g(˜ x) > x∗1 . Using the Mean Value Theorem again, one concludes that there exists a point, x0 ∈ B(x∗ ; r), such that g(x0 ) = x∗1 . Therefore, g2 (x0 ) = g(x∗1 ) = 0 . On the other hand, there exists a constant β2 > 0 such that g0 (x∗1 ) < 0, for cos(π/σ βx∗1 ) < 0. Therefore, (g2 )0 (x0 ) = g0 (x∗1 )g0 (x0 ) 6= 0 .
xk
+ uk "
Rule 2 : IF xk is M2 , THEN = A2
xk
#
yk+1
#
yk
"
xk+1
xx+1
#
yk+1
#
+ uk
yk
where "
A1 =
d
0.3
1
0
#
"
,
A2 =
−d
0.3
1
0
#
,
xk ∈ [−d, d] and d > 0, with membership functions
1 xk M1 = 1− 2 d
,
1 xk M2 = 1+ 2 d
.
The controlled TS fuzzy system is described as follows: xk+1 =
r X i=1 r X
hi (k){Ai xk + uk } =
r X
hi (k)Ai xk + uk .
i=1
π = hi (k)Ai xk + σ sin βxk . σ i=1 In the simulation, the magnitude of the control input is arbitrarily chosen to be σ = 0.1. Thus, kuk k∞ < σ, and β can also be regarded as a control parameter. Without control, the TS fuzzy model is stable, as shown in Fig. 2. When β takes values of 0.25, 0.4, 0.45, 0.5 and 1.3, the phase portrait diagrams, time wave diagrams, and bifurcation diagrams are shown in Figs. 3–15, respectively. These numerical results verify the theoretical analysis
2288 Z. Li et al. 0.35
0.154
0.3
0.152 0.15
0.25
beta=0.25
0.148 x(k)
y(k)
0.2
0.146
0.15
0.144 0.1
0.142 0.05
0
0.14
0
0.02
0.04
Fig. 2.
0.06
0.08
0.1 x(k)
0.12
0.14
0.16
0.18
0.138
0.2
0
The stable TS fuzzy system.
100
Fig. 3.
0.28
200
300 k
400
500
600
Periodic orbits at β = 0.25.
0.17 0.16
0.27
beta=0.25
0.15 0.14
beta=0.4
0.26 x(k)
y(k)
0.13 0.12 0.25 0.11 0.1
0.24
0.09 0.23
0
100
Fig. 4.
200
300 k
400
500
0.08
600
Periodic orbits at β = 0.25.
0
100
Fig. 5.
0.26
200
300 k
400
500
600
Period-doubling bifurcation at β = 0.4.
0.18
0.24
0.16
0.22 0.14 0.2
beta=0.4 0.12
beta=0.45
x(k)
y(k)
0.18 0.16
0.1
0.14 0.08 0.12 0.06
0.1 0.08
0
Fig. 6.
100
200
300 k
400
500
Period-doubling bifurcation at β = 0.4.
600
0.04
0
Fig. 7.
100
200
300 k
400
500
600
Special period-doubling bifurcation at β = 0.45.
Generating Chaos Via Feedback Control 2289 0.35
0.18
0.3
0.16 beta=0.5 0.14
0.25 0.12 x(k)
y(k)
0.2
0.15
0.1
beta=0.45 0.08
0.1 0.06 0.05
0
0.04
0
Fig. 8.
100
200
300 k
400
500
0.02
600
0
100
Special period-doubling bifurcation at β = 0.45.
Fig. 9.
200
300 k
400
500
600
Period-8 bifurcation at β = 0.5.
0.25
0.3
0.2
beta=1.3
0.25
0.15 0.2
0.1 0.05 beta=0.5
y(k)
y(k)
0.15
0.1
0 -0.05 -0.1
0.05
-0.15 0
-0.2 -0.05
0
100
Fig. 10.
200
300 k
400
500
-0.25 -0.2
600
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
x(k)
Period-8 bifurcation at β = 0.5.
Fig. 11.
Phase portrait with some structure.
0.3
0.2
0.2
0.15 0.1
beta=1.3
0.1
beta=1.3
0.05 y(k)
x(k)
0 0
-0.1 -0.05
-0.2 -0.1
-0.3
-0.15 -0.2
-0.4 0
100
200
Fig. 12.
300 k
400
k − xk diagram.
500
600
0
100
200
Fig. 13.
300 k
400
k − yk diagram.
500
600
2290 Z. Li et al. 0.2
0.3
0.15
0.2
0.1 0.1 0.05
y(k)
x(k)
0 0
-0.1 -0.05 -0.2
-0.1
-0.3
-0.15 -0.2 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.4 0.1
0.2
0.3
beta
Fig. 14.
Bifurcation diagram for β − xk .
and the design of the chaos generator developed in this paper.
5. Conclusions An approach is developed in this paper for designing a simple state-feedback controller of arbitrarily small magnitude that can drive an originally stable TS fuzzy system chaotic. The controller 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. To our knowledge, except for a couple of related publications [Baglio, et al., 1996; Porto & Amato, 2000], this paper is the first attempt in the development of using fuzzy systems to systematically generate chaos. Many chaotic systems, such as the Lorenz and the Henon systems, can be represented by TS fuzzy models. Hence, this design method of making a stable TS fuzzy system chaotic provides a means to further explore the interaction between fuzzy control systems theory and chaos theory, which has great potential in future engineering applications of chaos.
References Arena, P., Baglio, S., Fortuna, L. & Manganaro, G. [1996] “Generation of n-double scrolls via cellular neural networks,” Int. J. Circuit Th. Appl. 24, 241–252. Baglio, S., Fortuna, L. & Manganaro, G. [1996] “Design of fuzzy iterators to generate chaotic time series with
0.4
0.5
0.6
0.7
0.8
0.9
1
beta
Fig. 15.
Bifurcation diagram for β − yk .
assigned Lyapunov exponent,” Electron. Lett. 32(4), 292–293. Chen, G. & Lai, D. [1996] “Feedback control of Lyapunov exponents for discrete-time dynamical systems,” Int. J. Bifurcation and Chaos 6(7), 1341–1349. Chen, G. & Lai, D. [1997a] “Anticontrol of chaos via feedback,” Proc. IEEE Conf. Decision and Control (San Diego, CA), pp. 10–12. Chen, G. & Lai, D. [1997b] “Making a dynamical system chaotic: Feedback control of Lyapunov exponents for discrete-time dynamical systems,” IEEE Trans. Circuits Syst. I 44(3), 250–253. Chen, G. [1998] “Chaos: Control and anticontrol,” IEEE Circuits Syst. Soc. Newslett. 9(1), 1–5. Chen, G. & Dong, X. [1998] From Chaos to Order: Methodologies, Perspective and Applications (World Scientific, Singapore). Chen, G. & Lai, D. [1998] “Feedback anticontrol of chaos,” Int. J. Bifurcation and Chaos 8(7), 1585–1590. Golub, G. H. & Van Loan, C. F. [1983] Matrix Computations (Johns Hopkins University Press, Baltimore, MD). Joo, Y. H., Shieh, L. S. & Chen, G. [1999] “Hybrid state-space fuzzy model-based controller with dualrate sampling for digital control of chaotic systems,” IEEE Trans. Fuzzy Syst. 7(4), 394–408. Lee, H. J., Park, J. B. & Chen, G. [2001] “Robust fuzzy control of nonlinear systems with parametric uncertainties,” IEEE Trans. Fuzzy Syst. 9(2), 369–399. Li, T. Y. & Yorke, J. A. [1975] “Period three implies chaos,” Amer. Math. Monthly 82, 481–485. Marotto, F. R. [1978] “Snap-back repellers imply chaos in
