Harmonic balance method for time delay chaotic systems design

Transcription

1 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, Harmonic balance method for time delay chaotic systems design A. Buscarino, L. Fortuna, M. Frasca, G. Sciuto Laboratorio sui Sistemi Complessi, Scuola Superiore di Catania, Università degli Studi di Catania, Via S. Nullo 5/i, 955 Catania, Italy and Dipartimento di Ingegneria Elettrica Elettronica e Informatica, Facoltà di Ingegneria, Università degli Studi di Catania, viale A. Doria 6, 955 Catania, Italy M.G. Xibilia Diparimento di Scienze per l Ingegneria e per l Architettura, Facoltà di Ingegneria, Università degli Studi di Messina, Nuova Panoramica dello Stretto, 9866 Messina, Italy Abstract: Aim ofthis paper is to introduce a procedurebased on the harmonicbalance method forthedesignoftime delaychaoticsystems,startingfromasimplefeedbackschemeconsistingof a nonlinearity, a first-order integrator and a delay block. Through this approach, the conditions under which the considered scheme can show chaotic dynamics are discussed. Furthermore, guidelines to design circuital implementations of time delay chaotic circuits are presented. In particular, a series of n Bessel filters in cascade is used to implement the time delay. Following the proposed guidelines, a new chaotic circuit has been first designed and, then, implemented with off-the-shelf discrete components. Keywords: Chaotic behavior, harmonic balance techniques, nonlinear circuits, delay circuits, active filters.. INTRODUCTION The analysis of the conditions under which chaos can occurs in nonlinear dynamical systems is a topic of high interest. Such conditions, in fact, can be used as guidelines for the definition of new strategies for the design of new chaotic systems. In particular cases, e.g. when dealing with Lur e systems, it is possible to analytically derive approximate conditions for the existence of chaotic behavior. The procedure to obtain such conditions relies on the harmonic balance principle and has been introduced in [Genesio and Tesi (99)].Themethoddiscussedin[GenesioandTesi(99)] is based on the conjecture that the chaotic attractor is generated from the interaction between a stable predicted limit cycle and, at least, one unstable equilibrium point. Starting for this method, aim of this paper is to derive the approximate analytical conditions for the occurrence of chaos in a particular class of dynamical systems characterized by the presence of a time delay. The existence of a time delay in dynamical systems is usually considered as a source of instability [Xia et al. (9)]. However, in nonlinear systems, a time delay may allow the occurrence of complex dynamics and, in particular, chaos, since it makes the system infinite dimensional. Many mathematical models which include time delays and exhibit chaos have been presented in literature. In particular, high dimensional chaos induced by time delay in feedback systems has been reported in [Ikeda and Matsumoto(987)].Severalotherexamplesofchaoticbehavior in time delayed systems have been observed in purely mathematical models [Ghosh et al. (8),Zhang et al. (7)]. In the context of biological systems, a model with time delay for physiologicalcontrol systems has been introduced in [Mackey and Glass (977)]. Onthecontrary,fewexamplesofchaoticelectroniccircuits with time delay have been studied in literature [Pyragas and Tamaševičius (99),Zhang et al. (9)], and general procedures for the design of time delayed chaotic circuits are very difficult to be derived. In this paper, a procedure for the design of time delay systems in Lur e form is presented. The paper is organized as follows: in Section a general feedback scheme for time delayed chaotic system is introduced and the procedure based on the harmonic balance method for the system design is discussed. In Section the circuit implementation is discussed with particular attention to the design of the time delay block. In Section 4 the obtained experimental results are reported. Finally, Section 5 draws some concluding remarks. Copyright by the International Federation of Automatic Control (IFAC) 5

2 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, Fig.. Feedback scheme of the time delay chaotic circuit.. HARMONIC BALANCE METHOD FOR THE DESIGN OF TIME DELAYED CHAOTIC SYSTEMS Let us consider a feedback scheme of a time delay system with the minimum number of block needed to observe a chaotic behavior in a dynamical circuit with a single state variable: a nonlinearity, a RC circuit, and a time delay block. The nonlinearity is needed, since chaos is a peculiar characteristic of nonlinear systems; the RC circuit implements the dynamics of the single state variable; and the presence of the delay makes the system infinite dimensional, allowing the onset of chaotic behavior even in a system with a single state variable. The proposed minimal system can be summarized in the feedback scheme shown in Fig.. The dynamics of the system shown in Fig. can be expressed in terms of dimensionless equations as follows: ẋ(t) = k( ax(t) bh(x(t τ))) () where x(t) R is the circuit state variable, h(x) : R R is the nonlinear function, τ R + is the time delay, k is a scaling factor, and a and b are system parameters. b represents the gain multiplying the nonlinearity, while ka is the pole of the RC circuit. System () represents a nonlinear system in the so-called Lur e form. Lur e systems are nonlinear feedback system of the simplest structure formed by a dynamical linear part L(s) and a feedback nonlinear part N. In our case L(s) = ke sτ s+ka and N = bh(x). Following [Genesio and Tesi (99)], in order to observe chaos in a Lur e system the existence of a stable predicted limit cycle (PLC) and a separate unstable equilibrium point should be verified. Furthermore, the interaction between the unstable equilibrium point and the stable predicted limit cycle has to be assessed. Finally, it is required that the linear part of the Lur e system has filtering properties, which in our case is always satisfied, since L(s) = ke sτ s+ka. As concerns the condition on the PLC, it can be checked as usually by considering, due to piece-wise linear (PWL) nonlinearities considered in this paper, limit cycle solutions of the type y (t) = Asin(ωt) and approximating the nonlinearity N with the corresponding static describing functions [Atherton (98)] N. The amplitude A of y satisfies the following equations: +L(jω)N(A) = () where s = jω has been considered in L(s). Solving Eqs. () means to consider the intersections between the curve L(jω) and the curve /N(A) in the complex plane. Each intersection corresponds to a limit cycle with a given frequency and amplitude. The stability properties of the predicted limit cycle y can be derived by applying the limit cycle criterion [Slotine and Li (99)]: if the points near the intersection, corresponding to the considered limit cycle, along the curve /N(A) for increasing values of A are not encircled by the curve L(jω), then the limit cycle is stable. Otherwise the limit cycle is unstable. As concerns the existence of separate equilibrium points, sinceinthefollowingpwlnonlinearitieswillbeused,each equilibrium point E i can be analyzed taking into account the following system: ẋ(t) = k( ax(t) b x(t τ)+u) () which represents the dynamics of the system () in the PWL region to which the equilibrium point under examination belongs, i.e., b = bm i where m i is the slope of the PWL nonlinearity in the region of the equilibrium point E i. u is a constant input. For system () two different types of asymptotic stability can be defined [Niculescu et al. (998)]: delay-independent anddelay-dependentstability.delay-independent stability holdsifthesystem()isstableforallthevaluesofτ R +. Delay-dependent stability holds if the system () is stable for some values of τ and unstable for other values of τ. The stability properties of system () can be derived by studying the transcendental characteristic equation s + ka+kbe sτ = associatedto it. In particular,to checkthe stability properties of the generic equilibrium point E i the following criteria [Niculescu et al. (998)] can be applied: () the equilibrium point E i is delay-independent stable if and only if a+b > and a b ; () the equilibrium point E i is delay-dependent stable if and only if b > a. The equilibrium point E i is stable for τ τ = cos ( a/b) k (b a ). Finally, the existence of chaos requires the interaction between the stable predicted limit cycle and the unstable equilibrium point E. In equations it can be expressed as follows: A η E (4) where η represents a term which include the degree of approximation introduced by the harmonic balance method. In order to verify the conditions for the existence of chaos in Lur e systems, thus, Eqs. () and (4) have to be solved and the equilibrium point should satisfy neither the stability criterion ) nor criterion ). This can be done by a proper choice of the nonlinearity h(x) and of the parameters a, b and τ of system (). In particular, the time delay τ has an important role since it may change the stability properties of delaydependent stable equilibrium points. It may also affect the Nyquist diagram of L(s) and thus the intersections 5

3 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, with the describing function of the nonlinearity, so that in this case the time delay acts on the conditions of the existence of the limit cycle and on its interaction with the equilibrium point. In general, for a given nonlinearity and for fixed values of a and b, one obtains a range of values of τ, i.e., τ min τ τ max for which the system exhibits a chaotic behavior. The outlined procedure defines a class of systems with the feedbackschemeoffig.andchaoticbehavior.inthenext Section, we illustrate how this procedure can be applied for the design of a time delayed chaotic circuit.. CIRCUIT IMPLEMENTATION The circuit implementation is based on operational amplifier (OP AMP) blocks. A first block implements the RC circuit. The time constant of the block has been chosen equal to RC = ms, which scales the dynamics of system () by a factor equal to k =. A second block implements the PWL nonlinearity of the circuit. PWL nonlinearities are typically found in many chaotic circuits, like in the Chua s circuit [Fortuna et al. (9)],andthereforedesignstrategiesfortheirimplementation are widely studied. In particular, for the design of thisblock,theapproachdescribedin[fortunaetal.(9)] has been followed, which exploits the saturation working region of operational amplifiers to obtain a PWL inputoutput characteristics with tracts with different slopes. Concerning the third block, the time delay to be implemented is in the order of magnitude of milliseconds. This has been obtained by using a cascade of n low-pass Bessel filters. Considering a scaling factor k = means that τ is in the order of magnitude of milliseconds and thus its implementation requires the definition of an appropriate strategy. The idea underlying our approach is to use a cascade of n Bessel filters. The transfer function of an ideal time delay e τs is characterized by a flat magnitude for any frequency, and an linearly decaying phase Φ(ω) = ωτ. Bessel low-pass filters are linear filters with a maximally flat magnitude and a maximally linear phase response [Daryanani (976)], and thus can be used to implement a time delay block. A Bessel low-pass filter with transfer function H(s) introduces a time delay up to the db frequency equal to τ i = dφ(ω) dω. Our idea is to use a Bessel filter introducing a givendelay τ i and to obtain the desired delay τ with n blocks in cascade according to: τ nτ i (5) Inthisway,thetime delaycanbeeasilytunedbychanging the number n of filters in cascade. In particular, second orderfiltersimplementedbyusingthesallen-keytopology [Daryanani (976)] shown in Fig. have been used, where each filter is characterized by the following transfer function: H(s) = +C (R +R )s+c C R R s (6) Fig.. Schematics of the Sallen Key low pass active filter implementing a low-pass Bessel filter. The values of the components of this filter have been choseninordertorealizeabesselfilterwithdb frequency equal to f c khz and taking into account off-the-shelf componentvalues.thetime delayintroducedbythisfilter in the band up to f c can be calculated as τ i = dφ(ω) dω. For the values of the components chosen τ i.ms. Larger delays are realized by taking into account a cascade of n filters. In the circuit implemented, the following nonlinearity has been considered: π x if x < π h(x) = π x if x < π π x+ if x > π (7) In this case, h(x) given by Eq. (7) can be approximated by [Gelb and Velde (968)]: where N(A) = 4b ( π ) πa f b A πa if γ < f(γ) = π (sin γ +γ γ ) if γ if γ > Considering η =.5, the conditions for the existence of chaos can be satisfied for a =, b = 5 and different values of τ. In fact, for a = and b = 5, the system has three unstable equilibrium points: E =, E = E = π ( π). In particular, E and E are unstable for all τ, while E is delay-dependent stable with τ =.5ms. The system behavior with respect to τ is now briefly discussed in terms of the conditions for the existence of chaos.forsmall τ (τ <.5ms),the Nyquist plot ofl(jω) doesnotintersectthecurve /N(A)asshowninFig.4(a) (obtained for τ =.4ms). Consequently, the system in Eq. () with nonlinearity (7) does not exhibit a limit cycle. The dynamic behavior is characterized by a stable equilibrium point, in fact E is delay-dependent stable with τ =.5ms. Increasing τ (.5ms < τ <.65ms), E becomes unstable and the two curves intersect giving rise to a stable predicted limit cycle whose amplitude can bederivedsolvingeq.().thelimitcycleande beginsto interact for τ.65ms. The Nyquist plot corresponding (8) (9) 54

4 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, x τ [ms] Fig.. Numerical bifurcation diagram of system () with nonlinearity (7) with respect to time delay τ. Other parameters are a =, b = 5. to this last case is shown in Fig. 4(b). Eq. () with τ =.65ms, a =, and b = 5 leads to a stable predicted limit cycle with amplitude A =.4. Moreover, this set of parameters satisfies condition (4) with either E = E or E = E, demonstrating that the limit cycle generated by E interacts with these two unstable equilibrium points. For this set of parameters, the system behaves chaotically, asconfirmedbythemaximumlyapunovexponentλ max.6, calculated with the procedure discussed in [Zhang et al. (7)]. In summary for increasing values of τ first a stable equilibrium point E is obtained, then this point becomes unstableandthesystemdynamicbehaviorischaracterized by a stable limit cycle (not interacting with E ). Further increasing the value of the time delay leads to interaction between the predicted limit cycle and E. This conclusion can be also derived by taking into account the amplitude (as a function of τ) of the predicted limit cycle, as shown in Fig. 5: for small values of τ the amplitude A of the PLC is below the interaction threshold, but monotonically increase, for increasing values of τ, starting to interact with the unstable equilibrium points for τ.65ms. The system behavior can be summarized in the bifurcation diagram reported in Fig. In order to implement a given time delay τ =.65ms which guarantees the existence of chaos, n = 8 blocks can be used. Let us now consider the transfer function of an ideal time delay: G(s) = e sτ () and let us comparethe Bode diagramofh 8 (s) with G (s), as shown in Fig. 6. The frequency response of H 8 (s) is a good approximation of that of G (s) in the frequency range of the circuit dynamics. The complete circuit implementing the proposed example is shown in Fig. 7. The circuit follows the scheme based on three blocks. The block performing the integration of the state variable is realized through OP-AMP U (algebraic adder) and a RC circuit acting as an integrator. The nonlinear stage is constituted by two OP-AMPs, namely U and U. U is connected in the non inverting configuration with resistors R and R chosen as R R = V sat V, in order to ensure that the output of U saturates when the input is, in absolute value, greater that V = π.57v. Moreover,being U connected in the inverting configuration, R and R 5 are suitably fixed in order to Im Im Re.5.5 (a) Re (b) Fig. 4. Graphical solution of Eq. () for different values of τ. The continuous line refers to the function /N(A), while the dotted line is the Nyquist plot of L(jω) for (a) τ =.4ms, and (b) τ =.65ms. A(τ) τ [ms] Fig. 5. Amplitude of the predicted limit cycle as a function of τ. The dotted curve indicates the right hand term of Eq. (4). obtain the PWL nonlinearity of Eq. (7) when added to U output. The circuit is governed by the following equation: Ẋ(T) = R 8 C ( X(T)+ R 7 R 6 R 5 R X(T τ ) R 7 R 4 g) () 55

5 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, Mag [db] 5 5 delayed circuits with the feedback scheme as in Fig.. In thiscase,thenonlinearityh(x)iscontinuous(inparticular, sinusoidal). 4. EXPERIMENTAL RESULTS Ph [deg] 5 G(s)=e τ s H 8 (s) 4 ω [rad/s] Fig. 6. Magnitude and phase Bode diagrams of G(s) and H 8 (s). where g = ( X(T τ ) + π X(T τ ) + π ), and τ is the time delay realized in the circuit with n = 8 Bessel filters in cascade. Eqs. () with choice of the components values reported in Fig. 7 match Eqs. () with nonlinearity (7) and parameters a = and b = 5. It could be noted that choosingh(x) as defined in Eqs. (7), the system () could be viewed as a PWL approximation of the Ikeda time delay model, described by the following state equation: ẋ = k( ax(t) bsinx(t τ)) () where x represents the lag of the phase of the electric field across an optical bistable resonator, a is the relaxation coefficient, b is the intensity of the incident light, and τ is theround-triptimesofthelightintheresonator[ikedaand Matsumoto (987)]. Under this perspective, the proposed circuit implements a PWL approximation of the Ikeda chaotic system (). Moreover,the Ikeda model represents anotherexampleofasystembelongingtotheclassoftime Fig. 7. Circuit implementing the proposed example. The following components are used: R = R = R 8 = kω, R = 4.kΩ, R 4 = 4.kΩ, R 5 =.74kΩ, R 6 =.74kΩ, R 7 = 4.9kΩ, R 9 = R = R = R =... = R = R = kω, C = µf, C = C 4 =... = C i =... = C 4 = nf, and C = C 5 =... = C i+ =... = C 5 = nf. The supply voltage is V sup = ±9V. The circuit described in Section has been implemented with off-the-shelf discrete components. The time delay block of the circuit has been implemented by taking into account a maximum number n = of filters in cascade, so that the effective time delay introduced can be tuned and it was possible to carry out an analysis with respect to different values of n. Waveforms have been acquired by usingani USB655dataacquisitionboard,with sampling frequency f s = khz. The behavior of the circuit for the nominal values of the parameters (in particular, for n = 8) is shown in Fig. 8(b). For these values the circuit exhibits a chaotic dynamics as predicted by the theoretical and numerical analysis discussed above. The behavior with respect to different time delays, obtained by varying the number of Sallen-Key cascaded filters has been then characterized. Different behaviors have been observed, as reported in Fig. 8. In particular, increasing n bifurcations from limit cycles to chaotic attractors can be observed. Other examples of the behavior with respect to different values of n are reported in Fig CONCLUSIONS In this paper, a general procedure for the design of chaotic time delay continuous-time systems in Lur e form has been investigated. Analytical conditions under which chaotic behavior can be observed in simple system with a minimal feedback scheme have been derived through an approach based on the harmonic balance method. The introduced procedure can be used to assist the design and implementationofnonlineartime delaychaoticcircuits.in particular,itgivesamethodologyforthedesignofthenonlinearity and of the system parameters. This methodology hasbeenthenappliedtodefineanexample,whichhasbeen first theoretically and numerically analyzed, and, then, implemented and experimentally investigated, confirming the suitability of the approach. The onset of chaos has been experimentally observed, and the time delay reveals itself as an interesting bifurcation parameter: small delays induce periodic oscillations, while chaotic behavior occurs for larger delays. Although the procedure introduced is general, the use of PWL nonlinearities simplifies the implementation stage and makes possible the implementation of the whole circuits by only using simple off-the-shelf circuital components like resistors, capacitors and operational amplifiers. A key issue of the implementation is the design of the circuitry providing the suitable time delay needed to observe a chaotic behavior. The proposed delay circuit is a modular circuit, composed by n Bessel filters in cascade, approximating an ideal delay. The approximation of an ideal delay through a cascade of multiple second order filters allowed to design an efficient and simple circuitry which avoids the drawback of usual 56

6 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September,.5.5 X(T) [V] X(T) [V] X(T τ) [V] (a) X(T τ) [V] (b) X(t) 4 4 X(t τc) (c) Fig. 8. Experimental results. Behavior of the circuit in the phase plane X(T τ ) X(T ) for the following values of the parameters: a =, b = 5, and (a) n = 5, (b) n = 8, (c) n =. delay devices, like delay lines and LCL T-type filters. Thanks to the modularity of the approach, the time delay can be also tuned, allowing to carry out an analysis of the circuit behavior with respect to different time delays. REFERENCES Atherton, D. (98). Nonlinear control engineering. Van Nostrand Reinhold Co. Ltd., Wokingham, UK. Daryanani, G. (976). Principles of active network synthesis and design. John Wiley & sons, Singapore. Fortuna, L., Frasca, M., and Xibilia, M. (9). Chua s Circuit Implementations: Yesterday, Today, Tomorrow. World Scientific, Singapore. Gelb, A. and Velde, W.E.V. (968). Multiple input describing functions and nonlinear system design. McGraw Hill Book Co., New York. Genesio, R. and Tesi, A. (99). Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica, 8, Ghosh, D., Chowdhury, A., and Saha, P. (8). Multiple delay ro ssler system bifurcation and chaos control. Chaos, Solitons and Fractals, 5, Ikeda, K. and Matsumoto, K. (987). High dimensional chaotic behavior in systems with time delayed feedback. Physica D, 9, 5. Mackey, M. and Glass, L. (977). Oscillation and chaos in physiological control system. Science, 97, Niculescu, S.I., Verriest, E., Dugard, L., and Dion, J.M. (998). Stability and robust stability of time-delay systems: A guided tour. In L. Dugard and E. Verriest (eds.), Stability and Control of Time-delay Systems, volume I, chapter, 7. Lecture Notes in Control and Information Sciences 8, Springer, Berlin. Pyragas, K. and Tamas evic ius, A. (99). Experimental control of chaos by delayed self-controlling feedback. Phys. Lett. A, 8, 99. Slotine, J. and Li, W. (99). Applied nonlinear control. Prentice Hall International Editions, London, UK. Xia, Y., Fu, M., and Shi, P. (9). Analysis and Synthesis of Dynamical Systems with Time-Delays. Lecture Notes in Control and Information Sciences, Springer, Berlin. Zhang, X., Cui, Z., and Zhu, Y. (7). A simple chaotic delay differential equation. Phys. Lett. A, 66, Zhang, X., Cui, Z., and Zhu, Y. (9). Synchronization and circuit experiment simulation of chaotic time-delay systems. Proc. of 9 Pacific-Asia Conference on Circuits,Communications and System. 57