on relay feedback and chaotic systems

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2001; 11:241}252 (DOI: 10.1002/rnc.573)

On relay feedback and chaotic systems

Akihiko Sugiki*,s, Shoshiro Hatakeyama and Katsuhisa Furuta

Department of Applied Systems Engineering, Tokyo Denki University, Ishizaka, Hatoyama, Saitama, Japan

SUMMARY

The feedback connection of a linear system with a memoryless nonlinearity is recognized as a candidate forchaos-generating structure. This paper proposes a chaotic nonlinear feedback system with a relay connectedto a biproper linear system. The proposed chaotic system is analysed using chaos criteria proposed byGenesio and Tesi. This paper also shows an equivalence relation between the proposed system and theBrockett chaotic system through the linear input}output transformation. Copyright ( 2001 John Wiley& Sons, Ltd.

KEY WORDS: chaos; relay feedback; Brockett's chaotic system; the linear input}output transformation

1. INTRODUCTION

In the control "eld, chaos has been studied implicitly from 1950s in sampled-data control systems[1], and explicitly recently. Chaotic signal has come to be used as reference signal for human-friendly machines such as the variation of wind direction for air conditioners. Its generation hasattracted attention. By combining a linear system and nonlinear feedback, several resultsconcerning chaotic signal generation have been provided [2}8]. Brockett has presented a linearsystem with a piecewise linear feedback element and analysed the system from a topologicalviewpoint for the generation of chaos [2], which may be the "rst theoretical paper showing thatthe chaotic behaviour can be generated by the combination of a higher-order linear system andnonlinear feedback. Vane\ c\ ek and C[ elikovskyH have studied the generation of chaos by the linearsystem with memoryless nonlinear element based on the root loci [5]. The nonlinear element inthe Brockett work is non-smooth, memoryless and has three zeros as shown in Figure. 1. On theother hand, the nonlinear element taking a three-position value like K, 0,!K is called relay andis commonly used in practice for control. The behaviour of linear systems with relay feedback hasbeen studied by the describing-function method for a long time; but, it has not been studied as towhen it produces chaotic signal. Moreover, Okabayashi and Furuta have shown an equivalencerelation between a piecewise-linear feedback system and a relay feedback system through the looptransformation in the design of a sliding mode controller [9]. In this paper, we study when the

*Correspondence to: A. Suguki, Department of Applied Systems Engineering, Tokyo Denki University, Ishizaka,Hatoyama, Saitama, Japan.

sE-mail: [email protected]

Copyright ( 2001 John Wiley & Sons, Ltd.

Figure 1. Nonlinear characteristics for generating chaos.

linear system with relay feedback generates chaos and its equivalence to the Brockett system. Therelay feedback system is shown to be transformed into the linear system having nonlinear elementwith piecewise linear characteristics in the feedback path through the linear input}outputtransformation. The relay feedback is analysed for generating chaos by the criteria of Genesio andTesi [7, 10, 12], and the class of appropriate parameters is given. The organization of the paper isas follows: Section 2 presents a chaotic nonlinear feedback system with a relay connected toa biproper linear system. Section 3 analyses the proposed system by means of the chaos criteriaproposed by Genesio and Tesi. Section 4 clari"es the relationship between the proposed systemand the Brockett chaotic system with a piecewise linear feedback element. Section 5 discusses thesimulated results. Section 6 concludes the results of the paper.

2. CHAOTIC RELAY FEEDBACK SYSTEM

Our idea to produce chaotic signals [11] is to use the biproper linear system

G(s)"s3#as2#bs!k

s3#as2#bs#2k

242 A. SUGIKI, S. HATAKEYAMA AND K. FURUTA

Copyright ( 2001 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2001; 11:241}252

Figure 2. Chaotic system with relay feedback.

and the output is fedback through the relay element

n3%-

(y)"G3, y'0

0, y"0

!3, y(0

as shown in Figure 2, where n3%-

(y) denotes the output of the relay to its input y. The parametersare assumed to be

a, b, k'0. (1)

A state model of the system is described as

CxR1

xR2

xR3D"C

0 1 0

0 0 1

!2k !b !aD Cx1

x2

x3D#C

0

0

!kD u

y"[3 0 0][x1, x

2, x

3]T#u, u"!n

3%-(y)

The model contains one algebraic loop with respect to u given by

u"!n3%-

(3x1#u)

Replacing the relay n3%-

by the saturation

n4!5

(y)"G3, y'd

ay, DyD)d

!3, y(!d

243ON RELAY FEEDBACK AND CHAOTIC SYSTEMS


BThe describing function-based criteria for the presence of chaos in nonlinear feedback systems include the following fourterms [7, 10, 12]:

f Existence of a stable limit cycle.f Existence of unstable equilibrium points.f Interaction between the limit cycle and the equilibrium points (the main condition).f Suitable "ltering e!ect in the systems.

These have been given not from the rigorous de"nition of chaos, but enables one to easily estimate a range of parametervalues leading to chaos for a given nonlinear feedback system.

we can realize the explicit control input

u"G3, !3a(1#a)~1x

1'3

!3a(1#a)~1x1, D!3a(1#a)~1x

1D)3

!3, !3a(1#a)~1x1(!3

where a ("3/d) is assumed large enough.

3. CHAOS CANDIDATE REGION OF PARAMETERS

This section analyses the proposed relay feedback system by the chaos criteria of Genesio andTesi, which are based on the describing function method [7, 10, 12]. In the proposed system, thelimit cycle

y( (t)"AK #BK sinu( t"3tnk

#

12(ab#k)costn(2k!ab)

sinJbt

t"G!n/2, AK (!BK

arcsin(AK /BK ), DAK /BK D)1

n/2, AK 'BK(2)

is obtained from the dual-input describing function method de"ned as

G(0)N0(AK , BK )"!1, G ( ju( )N

1(AK , BK )"!1

where N0

and N1

are the dual-input describing function given by

N0(A,B)"

1

2nA Pn

~nn3%-

(A#B sin q) dq"6tnA

N1(A, B)"

1

nB Pn

~nn3%-

(A#B sin q)sin q dq"12 cost

nB

Equilibrium points of the proposed system are

E*"[0 0 0]T, E`,~"[$1.5 0 0]T.

The main condition of the criteriaB is formulated as

DAK !E*D)BK (3)



Figure 3. Chaos candidate region.

under the assumptionBK '0 (4)

Condition (3) always holds when Equation (2) has the form of a limit cycle. Assumptions (1) and(4) lead to the parameter condition

2k!ab'0 (5)

This inequality gives a possible parameter region in which the proposed system is chaotic, asshown in Figure 3 for k"1.8.

Here, the dotted region denotes a chaotic region obtained by simulation. The other threeconditions can be veri"ed by using a computer (see Reference [12] for further details). Figure 4shows chaotic attractors generated by the proposed system by selecting points ¸, M, and N insidethe chaotic area (5).

4. LINEAR INPUT}OUTPUT TRANSFORMATION

In this section, we show the relationship between the proposed relay feedback system and theBrockett chaotic system, which is described by the linear system

S (s)"1

s3#as2#bs



Figure 4. Chaotic attractors from the proposed relay feedback system with d"0.01, the initial state[2 0 !1.2]T, and the parameters indicated in Figure 3.

and the nonlinear feedback

n"3

(y)"Gky, Dy D)1

!2ky#3k sgn(y), 1(Dy D)3

!3k sgn(y), 3(Dy D

as shown in Figure 5. The parameters chosen for the generation of the chaos are

a"1, b"1.25, k"1.8 (6)

Since the characteristics of the nonlinear element n"3

in Dy D'3 were introduced in order tobound the trajectory, we can introduce the system composed of the linear system S(s) and thenonlinear element

n1-(y)"G

ky, Dy D)1

!2ky#3k sgn(y), 1(Dy D(7)

by restricting the initial condition. This system is shown in Figure 6(c).



Figure 5. Chaotic system considered by Brockett.

Figure 6. Conversion of nonlinear characteristics by the linear input-output transformation.

The system we are considering is a linear system P(s) having the nonlinear feedback element

n45(y)"G

3k

dy, DyD)d

3k sgn(y), d(Dy D(8)

This system is shown in Figure 6(a).This section shall prove that the above-mentioned system is equivalent to the Brockett system

(with n1-) when

P(s)"[c1!S (s)][1#c

2S (s)]~1

c1"

1!dk

, c2"2k (9)



Figure 7. Conversion of the simpli"ed Brockett nonlinearity and the original one (k"1.8).

Note that letting dP0 in the system of Figure 6(a) yields the proposed system. Consider thelinear transformation

!u1-#c

1y1-"u

45!y

45!c

2u1-"y

1-

(10)

as illustrated in Figure 6(b). From Equations (10) and (8), by expressing y"u45, y

45"n(y), the

following relations are derived for Du45D)d, that is, y

45"(3k/d)u

45:

y1-"

!c2#3k/d

1#(3k/d)c1

u1-

for u45'd, that is, y

45"3k,

y1-"!c

2u1-!3k



Figure 8. Chaotic trajectory from the Brockett system with the parameters (6) and the initial state[2 0 !1.2]T.

for u45(!d, that is, y

45"!3k,

y1-"!c

2u1-#3k.

From these relations and the parameters (9), it is con"rmed that the characteristics of thepiecewise linear element n

1-is obtained. Conversely, it can be shown that the saturation n

45is

obtained from the piecewise linear element n1-

by the transformation (10) with the parameters (9).Consequently, the transformation between n

1-and n

45is uniquely determined. Its illustration is

given in Figure 7 (left). Note that the Brockett original system S(s) with the nonlinearity nbr

can beconverted into P(s) with the following nonlinear element:

n452

(y)"G3k

dy, Dy D)d

3k sgn(y), d(Dy D)6

2ky!9k sgn(y), 6(Dy D

through the proposed transformation. Its illustration is given in Figure (right).



E The following state model is used:

CxR1

xR2

xR3D"C

0 1 0

0 0 1

0 !1.25 !1D Cx1

x2

x3D#C

0

0

3D u

y"[1/3 0 0][x1

x2

x3]T.

In the analysis of the Brockett system by the criteria of Genesio and Tesi, since the dual-inputdescribing function of a relay is calculated much easier than piecewise linear one, the transformedsystem to the relay feedback should be studied. It is noted that the linear input}outputtransformation enables one to easily analyse the Brockett system by the chaos criteria. Practic-ally, analysing the proposed system yields chaos-generating parameters (6), which were demon-strated in the Brockett paper. Brockett's chaotic attractorE of Figure 8 is also observed in theproposed relay feedback system (the left trajectory of Figure 4).

5. DISCUSSIONS

Chaotic behaviour in the proposed system depends on its initial state. Selecting the initial state[3 0!1.2]T produces the chaos as shown in Figure 9. For large initial states, the proposedsystem dose not converge any attractors (go to in"nity), and the linear system P(s) with thenonlinear element n

1-2generates a limit cycle. On the other hand, chaos is always generated for

large initial states by the system composed of the P(s) and the following nonlinear feedbackelement:

n453

(y)"G3ky!15k sgn(y), DyD'6

3k sgn(y), d(DyD)6

3k

dy, DyD)d

and the corresponding piecewise linear element by the proposed transformation is

n1-3

(y)"Gk

4y!

15k

4sgn(y), Dy D'3

!2ky#3k sgn(y), 1(Dy D)3

ky, Dy D)1

as illustrated in Figure 10. The initial dependency of the chaotic trajectory may imply that thechaos criteria should include a condition concerning initial states.

As shown in Figure 11, the root loci show some invariant characteristics by the change of theconsidered nonlinear elements, which implies that the root locus plays an important role tocomprehend chaos-generating mechanism. The signi"cance of the root locus in the synthesis ofchaos has been reported in the literature [5].



Figure 9. Chaotic trajectory of the initial state [3 0!1.2]T.

Figure 10. Nonlinear elements to generate chaos for large initial states.



Figure 11. Root loci of chaotic models with the parameters (6): (left) the Brockett system, (middle) theproposed system, (right) both systems.

There exists a slight di!erence between the chaos of Figure 4(left) and Figure 8. This is possiblydue to sampling e!ect associated with numerical computation, because the di!erence can beeliminated by selecting sampling periods small enough. This is an interesting phenomenon causedby the relay feedback design of the Brockett system.

6. CONCLUSION

The chaotic relay feedback system has been proposed, and its equivalence to the Brockett chaoticsystem was clari"ed by the linear input}output transformation. It was also shown that the chaoscondition based on the describing function is easily veri"ed by the transformation to the relayfeedback.

REFERENCES

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3. Baillieul J, Brockett RW, Washburn RB. Chaotic motion in nonlinear feedback systems. IEEE ¹ransactions onCircuits and Systems 1980; 27(11):990}997.

4. Matsumoto T. Chaos in electric circuit. Proceedings of the IEEE 1987; 75(8):1033}1057.5. Vane\ c\ ek A, Cx elikovskyH S. Chaos synthesis via root locus. IEEE ¹ransactions on Circuits and Systems-I 1994;

41(1):59}60.6. Amrani D, Atherton DP. Designing autonomous relay systems with chaotic motion. Proceedings of the 28st IEEE

Conference on Decision and Control 1989; 512}517.7. Genesio R, Tesi A. Chaos prediction in nonlinear feedback system. IEE Proceedings-D 1991; 138(4):313}320.8. Anishchenko VS. Dynamical Chaos * Models, Experiments, and Applications.=orld Scienti,c: Singapore, 1995.9. Okabayashi R, Furuta K. A note on sliding mode control systems from the viewpoint of the Lur'e prob-

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on relay feedback and chaotic systems

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