Synchronization of Chaotic Oscillators With Partial Linear Feedback Control
SSynchronization of Chaotic Oscillators With Partial Linear Feedback Control
Synchronization of Chaotic Oscillators With Partial Linear Feedback Control
K. Mistry, S. Dash, a) and S. Tallur b) Indian Institute of Technology (IIT) Bombay, Mumbai,India c) (Dated: 24 January 2019) We present a methodology for synchronization of chaotic oscillators with linear feed-back control. The proposed method is based on analyzing the chaotic oscillator as amulti-mode linear system and deriving sufficient conditions for asymptotic stability.The oscillators are synchronized in a master-slave configuration, wherein a subset ofthe state variables for implementing the feedback control, enabling applications incryptography for message encryption using the unused chaotic state variables. Con-troller stability is ensured through conventional root-locus technique for designingappropriate loop gain. We validate the methodology presented here with numericalsimulations and experimental results obtained using an operational amplifier (op-amp) based electronic chaotic oscillator circuit.PACS numbers: Valid PACS appear hereKeywords: Chaos; synchronization; partial linear feedback control; multi-mode sys-tem a) B.Tech. (Electronics and Communication Engineering) student at National Institute of Technology (NIT),Trichy, India. b) c) Electronic mail: [email protected] a r X i v : . [ n li n . C D ] J a n ynchronization of Chaotic Oscillators With Partial Linear Feedback Control I. INTRODUCTION
All second order dynamical systems exhibit one of three categories of trajectory in statespace : 1) stable (convergent) 2) unstable (divergent) and 3) limit cycle (oscillatory). Higherorder dynamical systems may exhibit another type of trajectory, namely chaotic behavior .Such systems may be emulated through simple electronic circuits , that exhibit rich non-linear dynamics while appearing deceptively deterministic from a circuit analysis perspective.Synchronization of chaotic oscillator circuits can enable several interesting applications inelectronic message encryption . Numerous methods for synchronization of chaotic sys-tems have been proposed over the decades , however all such implementations requireeither all state variables of the individual oscillators to generate the necessary locking signalto entrain the slave oscillators to the master oscillator , or a non-linear feedbacksignal employing a subset of state variables .In this work we report a methodology to design a linear feedback controller to synchronizetwo chaotic oscillators represented by third order non-linear differential equations. Theoscillators are analyzed as piecewise linear systems in different modes of operation. Usinglinear control theory and root locus method, the controller coefficients can be appropriatelydesigned to ensure stability across all modes of operation, and utilizing a partial subsetof state variables to generate the feedback signal. The unused state variables can then beemployed for message encryption by adding these to a small-amplitude message signal at thetransmitter in a communication system. The encrypted message could then be recovered atthe receiver end by synchronizing the local oscillator at the receiver end to the transmitteroscillator, and subtracting the corresponding states used in encryption. We present a prooffor the stability of this technique and provide validation with Scilab simulations of a thirdorder non-linear system and experimental measurements obtained through an operationalamplifier (op-amp) circuit implementation of the oscillators and the controller.The paper is structured as follows: section II describes the notations and section IIIintroduces the chaotic oscillator circuit used in this work. Section IV introduces some controlsystems techniques for synchronization, along with their limitations. Section V describesthe method presented in this work in detail and a methodology for designing the controller,and section VI presents numerical simulations and experimental results corroborating thismethod. 2ynchronization of Chaotic Oscillators With Partial Linear Feedback Control II. NOTATIONS
This section introduces the notations we use to describe the system mathematically. Wefocus on a chaotic oscillator represented by a third order non-linear differential equation: d xdt = c d xdt + b dxdt + f ( x ) . (1)where, f ( x ) is piecewise linear function that captures the non-linearity in the system. Wechoose the following form of f ( x ): f ( x ) = ax + u x < u x ≥ a, b, c, u and u are all real constants. Defining state variables x = x , x = dxdt and x = d xdt , state space realization of equation 1 is expressed as follows:˙ x = x ˙ x = x ˙ x = f ( x ) + bx + cx = g ( x , x , x ) (3)The state vector for this state space model is expressed as X = (cid:104) x x x (cid:105) T . Forsynchronization of oscillators, we introduce a control signal to dictate the dynamics of theslave oscillator. The control signal is modeled as signal u ( t ), and the combined model isexpressed below: d xdt = c d xdt + b dxdt + f ( x ) + u ( t ) . (4)˙ x = x ˙ x = x ˙ x = g ( x , x , x ) + u ( t ) (5)When two oscillators are synchronized, the trajectory in state-space is identical for bothoscillators. We consider a master-slave locking scheme for two oscillators and denote thestate space variables of the master oscillator as x i and those of the slave oscillator as y i , i = 1 , ,
3. The slave oscillator dynamics are also controlled through the controller output u ( t ). The state space representation of both oscillators are then written as below:3ynchronization of Chaotic Oscillators With Partial Linear Feedback Control˙ x = x ˙ x = x ˙ x = g ( x , x , x ) , ˙ y = y ˙ y = y ˙ y = g ( y , y , y ) + u ( t ) . (6)The system is easier to analyze in terms of the error states: e i = y i − x i , i = 1 , , e , e and e converge to zero.From equations 3 and 6, we obtain:˙ e = e ˙ e = e ˙ e = f ( y ) − f ( x ) + be + ce + u ( t ) (7)The error states can be expressed as a vector E = (cid:104) e e e (cid:105) T . III. CIRCUIT IMPLEMENTATION OF THE CHAOTIC OSCILLATOR
For experimental validation of the technique, we implement the system differential equa-tion (1) using an analog circuit containing resistors, capacitors, operational amplifiers (op-amps) and diodes as shown in Figure 1. The chaotic behavior of such circuits has beenextensively studied and documented by Kiers et al. . The difference in this circuit is the im-plementation of the “f-block” as shown in Figure 2, which implements a modified precisionrectifier circuit. The characteristic differential equation of this circuit is expressed as: d xdt = − R v C d xdt − R C dxdt + 1 R C f ( x ) (8)The f-block circuit in Figure 2 implements the following function: f ( x ) = − R R x + 0 . x < − . x ≥ x ≥
0, the function f ( x ) has a non-zero value due to the forwardbias voltage drop across diode D
1. This modification does away with the requirement of anexternal bias voltage that is necessary in the implementation reported by Kiers et al. .4ynchronization of Chaotic Oscillators With Partial Linear Feedback Control FIG. 1. Circuit diagram of the chaotic signal generator (oscillator), based on an architectureproposed by Kiers et al. . The “f-block” is a non-linear circuit shown in Figure 2.FIG. 2. (a) Circuit diagram for the “f-block” in Figure 1, that implements the equation for f ( x ) asin equation (9). (b) Experimentally measured transfer function of the non-linear f-block, verified byapplying sinusoidal signal to the f-block circuit and observing output vs input graph on oscilloscope(configured to display in XY mode). IV. CONTROL SYSTEMS TECHNIQUES FOR SYNCHRONIZATION
Since the system under consideration is governed by a non-linear transfer function, sev-eral non-linear control techniques can be used to control the dynamics and achievesynchronization of the two oscillators. Consider feedback linearization technique appliedto this system, wherein we design u ( t ) such that the overall system becomes linear in na-ture. Observing equation (7) we can select u ( t ) = − f ( y ) + f ( x ) + v ( t ). The state spacerepresentation of the error states can then be rewritten as follows:˙ e = e ˙ e = e ˙ e = be + ce + v ( t ) (10)˙ E = b c E + v ( t ) (11)As evident from equations (10) and (11), the system is transformed to a linear system,with state space equation of form ˙ E = AE + Bv ( t ) as shown in equation (11). The signal v ( t )is chosen as a linear combination of the error states, i.e. v ( t ) = KE , such that the overallstate matrix A + BK is Hurwitz. This is a necessary and sufficient condition for stability ofthe controller, as will be explained in detail in section V B. In this technique, the controllerimplementation u ( t ) depends on the non-linearity in the system f ( x ). Even though one maydiscretely implement a controller by externally implementing the non-linearity, the techniqueis susceptible to drifts in the system that may change the nature of f ( x ), and implementingsuch a robust controller may not be feasible practically.Another method to design the controller is by approximating the non-linearity in thesystem transfer function as a smooth (continuous and differentiable) response e.g. as ahigher order polynomial. However this approximation is effective only in the vicinity ofequilibrium point(s) of the system (in this case origin) i.e. the errors are low for small signalamplitudes . This scheme is not robust as large signals at any of the circuit nodes at start-up (initial conditions) will lead to large diverging errors and the controller may not achievesynchronization. 6ynchronization of Chaotic Oscillators With Partial Linear Feedback Control V. ANALYSIS AS MULTI-MODE LINEAR SYSTEM
The dynamics of a chaotic systems may also be viewed as a trajectory switching acrossvarious modes, and studied as a Linear Complimentarity System (LCS) . The system underconsideration can be expressed in LCS form as follows:˙ X ( t ) = C X ( t ) + C w (cid:48) ( t ) + C u (cid:48) ( t ) (12) y (cid:48) ( t ) = C X ( t ) + C w (cid:48) ( t ) + C u (cid:48) ( t ) (13)where C i ( i = 1 , , . . . ,
6) are matrices of appropriate size, ˙ X = (cid:104) ˙ x ˙ x ˙ x (cid:105) T , X = (cid:104) x x x (cid:105) T and u (cid:48) ( t ) ≥ , y (cid:48) ( t ) ≥ , u (cid:48) ( t ) T y (cid:48) ( t ) = 0. For the oscillator circuit, this translatesto: ˙ X ( t ) = − R C − R v C X ( t ) + − . R C + u (cid:48) ( t ) (14) y (cid:48) ( t ) = (cid:104) R C R R (cid:105) X ( t ) − . R C + u (cid:48) ( t ) (15)The system input is denoted as w (cid:48) ( t ) and the switching vectors in the system, i.e. u (cid:48) ( t )and y (cid:48) ( t ), evolve such that one of them will be zero and other will be non-negative at everyinstant in time . If u (cid:48) ( t ) = 0, we obtain state space equation with constraint x ≥ y (cid:48) ( t ) = 0 we obtain another state space equation with constraint x <
0. While one mayuse stability theories for LCS to design a suitable controller, a more intuitive approach isto analyze the system as a multi-mode linear system and study stability of each mode usingstandard linear control theory. This technique forms the heart of the work presented here,and is described in detail below: A. Multi-mode representation of the control system
The piecewise linear function f ( x ) appears in equation (7), and hence the system showsfour modes of operation, depending on the signs of x and y : MODE-I ( x ≥ y ≥ ): E = b c E + u ( t ) MODE-II ( x < y ≥ ): ˙ E = b c E + u − ax − u + u ( t ) MODE-III ( x < y < ): ˙ E = a b c E + u ( t ) MODE-IV ( x ≥ y < ): ˙ E = b c E + ay + u − u + u ( t )The equation in mode IV can be rewritten as below, by writing ay = ae + x :˙ E = a b c E + ax + u − u + u ( t ) B. Conditions for stability of controller
For any linear autonomous system ˙ X = AX , the matrix A is called state matrix of thesystem, and its eigenvalues are the poles of the system transfer function. The eigenvaluesof matrix A are the roots of its characteristic polynomial, ∆ A ( s ) = det ( sI − A ). Matrix A is called a Hurwitz matrix if all roots of ∆ A ( s ) lie in the left half of the complex plane, i.e.all roots have strictly negative real part. Consequently a linear system is asymptoticallystable at origin if it has a Hurwitz state matrix . For a linear system ˙ X = AX + Bu , ifmatrix A is Hurwitz then system is BIBO (bounded input bounded output) stable, i.e. if8ynchronization of Chaotic Oscillators With Partial Linear Feedback Controlthe values of the input to the system are bounded, the output of the system also necessarilyhas bounded range of values . A bounded signal in this context refers to a signal that hasfinite magnitude at every instance in time.In each of these modes, the coupled oscillators are described by a linear system of equa-tions. To stabilize such a system, u ( t ) may also be a designed as a linear feedback controller.Let us denote u ( t ) = KE , where K = (cid:104) α β γ (cid:105) . Hence u ( t ) = αe + βe + γe . The statematrix for modes I and II is rewritten as A = α b (cid:48) c (cid:48) , where b (cid:48) = b + β and c (cid:48) = c + γ .The state matrix for modes III and IV is rewritten as: A = a (cid:48) b (cid:48) c (cid:48) , where a (cid:48) = a + α .The controller coefficients α , β and γ can be tuned to ensure that all eigenvalues of A and A lie in the left half of complex plane, and consequently the system is asymptoticallystable at origin for modes I and III. For modes II and IV, the state matrix is Hurwitz, andhence the system is BIBO stable. The system equation in these modes also contains an inputterm proportional to state x . Since x is a state variable of the master chaotic oscillator(implemented as an op-amp based electronic circuit), its magnitude is bounded. Thus theerror state variables e , e and e are also bounded in modes II and IV.As the trajectory of the error state variable system evolves in time in state space, itswitches from one mode to another. Notice that if the system trajectory enters mode IIor mode IV, the following conditions are always true: i) the magnitude of the error statetrajectory remains bounded due to BIBO stability of the system, and ii) the trajectory canevolve to another mode as the magnitude and sign of the state x independently changeswith time. In modes I and III, the trajectory of the error state variable system asymp-totically converges to origin. Designing A and A matrices to be Hurwitz thus stabilizesthe controller, and ensures that state vector E will converge to origin, i.e. the two chaoticoscillators will synchronize. It is worth noting that the individual stability of each modeis a sufficient, but not a necessary condition for ensuring synchronization. If the rate ofincrement in distance of the state trajectory point from origin (divergence) in the unstablemodes is lower than the rate of decrement in distance of state trajectory point from origin(convergence) in a stable mode, the overall state trajectory of the multi-mode system will9ynchronization of Chaotic Oscillators With Partial Linear Feedback Control Controller K System G ( s )uUnity Gain FeedbackInput + e Output − FIG. 3. Generalized representation of a control loop for root locus analysis. The root locustechnique is used to design a stable linear controller for synchronization of the two oscillators. eventually converge towards origin.
C. Design of controller using root-locus approach
The controller u ( t ) is constructed as a linear combination of all error states e i , i = 1 , , u ( t ) can simply be a scaled version of any one of the states. Consider u ( t ) = αe , and valuesof β and γ will be zero. The characteristic polynomial of matrix A is thus:∆ A ( s ) = s − cs − bs − α (16)To analyze how the roots of this polynomial vary with value of α , we use root locusanalysis. The root locus plot of any system graphically illustrates the trajectory of variationof the roots of the system characteristic equation in the complex plane, when some parameterof the system is varied . Consider a system with transfer function G ( s ) controlled usingnegative unity gain feedback and proportional controller with gain K as shown in Figure3. The closed loop transfer function is given by T ( s ) = KG ( s )1+ KG ( s ) and the characteristicpolynomial ∆( s ) of this closed loop system is the denominator in T ( s ). The root locus ofthis system is a plot of the roots of ∆( s ) in the complex plane as K is varied from 0 to ∞ .Now consider a system with open loop transfer function G A ( s ) as given in equation (17)and proportional controller gain K = − α . G A ( s ) = 1 s − cs − bs (17)10ynchronization of Chaotic Oscillators With Partial Linear Feedback ControlThe characteristic polynomial of this closed loop system is ∆ A ( s ) as expressed in equation(16). We can choose suitable value of α by examining the root locus of G A ( s ) such that allroots of ∆ A ( s ) lie in the left-half of the complex plane (i.e. the real part of the roots are allnegative), thus ensuring that matrix A is Hurwitz. Following a similar procedure with asuitably designed G ( s ), we can find suitable values of α such that matrix A is also Hurwitz.If no such values of α can be identified, we can instead try u ( t ) = βe or u ( t ) = γe andrepeat the same root locus exercise. If one error state alone proves insufficient to generatea stable controller, one can then explore using a linear combination of multiple states forthis exercise, depending on how many states are available for controller design based on theapplication. VI. SIMULATION AND EXPERIMENTAL RESULTS
The oscillator circuit and the f-block shown in Figure 1 and Figure 2(a) respectively areimplemented using variable resistors R v and R . The fixed resistance values are R = 47 k Ω, R = 10 k Ω and the variable resistors are tuned to operate the oscillator in the chaotic regime.All capacitors are implemented as ceramic capacitors with capacitance C = 0 . nF , and theop-amps are implemented using IC TL071 low-noise JFET-input general-purpose operationalamplifier ICs from Texas Instruments. The diodes in Figure 2(a) are implemented using1N4148 silicon diodes. The dynamics of the system are simulated by solving the differentialequation numerically in Scilab. In our simulation we modify equation (8) by scaling timeas t = ( RC ) ∗ T , to obtain the modified differential equation expressed in equation (18).Comparing equations (18) and (9) with equations (1) and (2) respectively, we obtain b = − u = 0 . u = − . d xdT = − RR v d xdT − dxdt + f ( x ) . (18)This non-linear differential equation can be simulated with different values of a and c to find the appropriate set of values to operate the oscillator in chaotic regime. Figure4(a) and Figure 4(b) show the simulated and experimentally measured phase portrait of theoscillator using R v = 71 . k Ω ( c = − .
66) and R = 58 k Ω ( a = − . XY mode.11ynchronization of Chaotic Oscillators With Partial Linear Feedback ControlTwo such circuits are constructed and the steps illustrated in section V C are imple-mented to design a linear controller u ( t ) = βe to synchronize the two chaotic circuits. Thecharacteristic polynomial of matrix A is given by:∆ A ( s ) = s + 0 . s + (1 − β ) s (19)To identify a suitable value of β to ensure controller stability, we simulate the root locusof the control loop shown in Figure 5. Figure 6 shows the root locus plot of the system shownin Figure 5, simulated using RootLocs , a freely distributed root locus plotting software.The roots always lie in the left-half of the complex plane for all values of K , and thus thesystem with state matrix A will be asymptotically stable at origin for K ∈ [0 , + ∞ ), i.e. β ∈ ( −∞ , A to be Hurwitz. The characteristicpolynomial of matrix A is expressed as:∆ A ( s ) = s + 0 . s + (1 − β ) s + 5 . A ( s ) as we tune β , we study the root locus ofclosed loop system shown in Figure 7. Figure 8 shows root locus plot of the system shownin Figure 7. Asymptotic stability of this system requires K ∈ [9 , + ∞ ), i.e. β ∈ ( −∞ , − A and A are Hurwitz, i.e.when β ∈ ( −∞ , − β in this range allows us to design the controller as a signalproportional to e = y − x , where y is the signal from the slave oscillator and x is thesignal from the master oscillator. The error signal e is thereby generated using an unitgain op-amp differential amplifier with inputs y and x , and is connected to the input ofthe slave oscillator circuit wherein it is scaled by gain β = − RR i . Figure 9 shows the circuitdiagram in its entirety.Root-locus analysis suggests that − R/R i = β ≤ − e converging to zero when the controller is turned on at time t = 0, for gain β = −
10. In our experiment we observe that the two chaotic systems synchronize when R i ≤ k Ω, i.e. β = − R/R i ≤ − .
4. Figure 10(b) shows experimentally measured resultobtained on an oscilloscope when R i = 5 k Ω. The two signals captured on the oscilloscope12ynchronization of Chaotic Oscillators With Partial Linear Feedback Controlare the error signal e ( t ) (top) which converges to a small value when the controller is turnedon using a Texas Instruments CD4066B electronic switch (bottom signal in Figure 10(b) isthe switch control signal). The simulated time constant for the decay in error signal e ,computed by fitting an exponential function to the envelope of the signal in Figure 10(a) is τ sim = 5 × RC = 23 . µs . The experimentally measured time constant for the decay in errorsignal e ( t ) upon turning on the controller is τ expt = 300 µs . Figures 10 and 11 show thesignals x and y in unsynchronized and synchronized states as observed on the oscilloscope. VII. CONCLUSION
While synchronization of chaotic oscillator circuits has been demonstrated through sev-eral methods largely in the previous three decades, we present a method that utilizes alinear controller implemented using only one state signal from each oscillator circuit. Thissimultaneously makes the controller implementation extremely simple in an electronic cir-cuit, and also enables cryptography applications wherein the unused state signals can beused for message encryption . We also present a method to design a robust controller toachieve synchronization by analyzing the non-linear chaotic system as a multi-linear modesystem and present a design methodology for the linear controller using root locus techniquefor ensuring stability. The analysis in this work and the method presented was developedspecifically for the non-linearity in the oscillator circuit chosen for analysis in this work, andour future work will focus on developing a generalized design methodology and necessary andsufficient conditions for stability of any arbitrary multi-linear mode system, and exploringextending this result to a network of oscillators. REFERENCES H. K. Khalil, “Nonlinear Systems,” 3rd ed. Prentice-Hall, (2002). J. C. Sprott, Physics Letters A 266, 16 (2000). K. Kiers, T. Klein, J. Kolb, S. Price, and J. C. Sprott, International Journal of Bifurcationand Chaos 14, 2867 (2004). H. P. W. Gottlieb, American Journal of Physics 64, 525 (1996).13ynchronization of Chaotic Oscillators With Partial Linear Feedback Control S. J. Linz, American Journal of Physics 65, 523 (1997). J. C. Sprott, American Journal of Physics 65, 537 (1997). K. Kiers, and D. Schmidt, American Journal of Physics 72, 503 (2004). G. Chen, and T. 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Wang, Physics Letters A 320, 271 (2004). H. N. Agiza, and M. T. Yassen, Physics Letters A 278, 191 (2001). M. Ho, Y. Hung, and C. Chou, Physics Letters A 298, 43 (2002). W. P. M. H. Heemels, “Linear Complementarity Systems: A Study in Hybrid Dynamics,”Ph.D. thesis, Technische Universiteit Eindhoven (1999). M. Gopal, “Control systems: principles and design,” 4th ed. McGraw Hill Education(2002). RootLocs download link: http : FIG. 4. (a) Simulated phase portrait of the oscillator described in equation (18), obtained usingnumerical simulation in Scilab. (b) Experimentally measured phase portrait of the oscillator onan oscilloscope configured to display in XY mode. The Y-axis displays signal at the node x andX-axis displays signal at node x . K = 1 − β G ( s ) = s +0 . s Unity Gain Feedback+ − FIG. 5. Block diagram of control loop for root locus analysis of characteristic polynomial of matrix A . The trajectory of the roots of the loop transfer function are analyzed in complex plane as thegain parameter K is varied from 0 to + ∞ .FIG. 6. Root locus for controller shown in Figure 5. The roots lie in the left half of the complexplane for all values of gain K ≥ K = 1 − β G ( s ) = ss +0 . s +5 . Unity Gain Feedback+ − FIG. 7. Block diagram of control loop for root locus analysis of characteristic polynomial of matrix A . The trajectory of the roots of the loop transfer function are analyzed in complex plane as thegain parameter K is varied from 0 to + ∞ .FIG. 8. Root locus for controller shown in Figure 7. The roots lie in the left half of the complexplane for values of gain K ≥ FIG. 9. Circuit diagram showing the (a) master chaotic circuit, (b) linear controller circuit, and(c) slave chaotic circuit with feedback controller: u ( t ) = − RR i e ( t ) FIG. 10. Response time of controller: (a) Numerical simulation (in Scilab) shows the error state e converging to zero after connecting the control signal u ( t ) = − e ( t ) at time T = 0. (b)Experimental result observed on an oscilloscope, wherein the error state e converges to zero(upper trace) when the control signal is turned on using an electrical switch gated by the voltagestep signal shown in the bottom trace. FIG. 11. Signals x (master oscillator, Y-axis) and y (slave oscillator, X-axis) observed on anoscilloscope configured to display in XY mode. (a) In the unsynchronized state, the two signalsare not correlated to each other. (b) When the two oscillators are synchronized, the two signalstrack each other and are equal in magnitude. FIG. 12. Signals x (master oscillator, upper trace), y (slave oscillator, bottom trace) observedon an oscilloscope configured to display in time domain. The difference between the two signals iscomputed and displayed on the oscilloscope (middle trace). (a) When the two oscillators are notsynchronized, the difference is non-zero. (b) The difference between x and y is very small, andthe two traces look identical when the oscillators are synchronized.is very small, andthe two traces look identical when the oscillators are synchronized.