State Feedback Control and Observer Based Adaptive Synchronization of Chaos in a Memristive Murali-Lakshmanan-Chua Circuit
aa r X i v : . [ n li n . AO ] A ug Pramana – J. Phys. (2016) : / s78910-011-012-3 c (cid:13) Indian Academy of Sciences
State Feedback Control and Observer Based Adaptive Synchronization ofChaos in a Memristive Murali-Lakshmanan-Chua Circuit
A. ISHAQ AHAMED and M. LAKSHMANAN Department of Physics, Jamal Mohamed College, (A ffi liated to Bharathidasan University), Tiruchirappalli-620020, India Department of Nonlinear Dynamics, School of Physics, Bharathidasan University,Tiruchirappalli-620024, India * Corresponding author. E-mail: [email protected] received 23 January 2020; revised 23 January 2020; accepted 23 January 2020
Abstract.
In this paper we report the control and synchronization of chaos in a Memristive Murali-Lakshmanan-Chua circuit. This circuit, introduced by the present authors in 2013, is basically a non-smooth system having twodiscontinuity boundaries by virtue of it having a flux controlled active memristor as its nonlinear element. Whilethe control of chaos has been e ff ected using state feedback techniques, the concept of adaptive synchronizationand observer based approaches have been used to e ff ect synchronization of chaos. Both of these techniques arebased on state space representation theory which is well known in the field of control engineering. As in our earlierworks on this circuit, we have derived the Poincar´e Discontinuity Mapping (PDM) and Zero Time DiscontinuityMapping (ZDM) corrections, both of which are essential for realizing the true dynamics of non-smooth systems.Further we have constructed the observer and controller based canonical forms of the state space representations,have set up the Luenberger observer, derived the controller gain vector to implement state feedback control andcalculated the gain matrices for switch feed back and finally performed parameter estimation for e ff ecting observerbased adaptive synchronization. Our results obtained by numerical simulation include time plots, phase portraits,estimation of the parameters and convergence of errors graphs and phase plots showing complete synchronization. Keywords.
Memristive MLC circuit, state space representations, canonical forms, Luenberger Observer, feed-back control, gain vectors and matrices, pole placement.
PACS Nos 12.60.Jv; 12.10.Dm; 98.80.Cq; 11.30.Hv1. Introduction
Chaotic systems are characterised by their high sensi-tivity to even infinitesimal changes in their initial con-ditions. As a result these systems, by their intrinsic na-ture, defy attempts at control or synchronization. Nev-ertheless many techniques have been proposed by a largegroup of researchers to control and synchronize chaoticsystems. Control of chaos refers to a process wherein ajudiciously chosen perturbation is applied to a chaoticsystem, in order to realize a desirable behaviour [1].Since the seminal contribution by Ott, Grebogi and Yorkein 1990 [2] the concept of control of chaos has beenmodified and developed by many researchers [3] andapplied to a large number of physical systems [4]. Syn-chronization of chaos, on the other hand, can be de-scribed as a process wherein two or more chaotic sys-tems (either equivalent or non-equivalent) adjust a givenproperty of their motion to a common behaviour, due to coupling or forcing. This may range from completeagreement of trajectories to locking of phases [5, 6, 7].In this paper we describe the general principles ofcontrol of chaos using state feedback mechanism andsynchronization of chaotic systems using observer basedadaptive techniques. Further using these, we report thecontrol of chaos in a single Memristive Murali-Lakshmanan-Chua (MLC) oscillator and the synchronization of chaosin a two coupled Memristive MLC oscillator system.The paper is organized as follows. In Sec. 2 we givea brief introduction of the Memristive MLC circuit, itscircuit realization, its circuit equations and their normal-ized forms and the description of the circuit as a non-smooth system. In Sec. 3 the various algorithms for thecontrol of chaos are outlined. In Sec. 4 the control ofchaos in the Memristive MLC circuit using state feedback control technique is dealt with. Similarly in Secs.5 and 6 the concept of synchronization of chaos and itsrealization are explained, while in Sec. 7 the observer
Pramana – J. Phys. (2016) :
Figure 1.
The memristive MLC circuit based adaptive synchronization of chaos in a system oftwo coupled Memristive MLC oscillator is described.Finally in Sec. 8 the results and further discussions aregiven.
2. Memristive Murali-Lakshmanan-Chua Circuit
The memristive MLC circuit was introduced by the presentauthors [8] by replacing the Chua’s diode in the clas-sical Murali-Lakshmanan-Chua circuit with an activeflux controlled memristor as its non-linear element. Theanalog model of the memristor used in this work wasdesinged by [9]. The schematic of the memristive MLCcircuit is shown in Fig. 1, while the actual analog real-ization based on the prototype model for the memristoris shown in Fig. 2.Applying Kircho ff ’s laws, the circuit equations canbe written as a set of autonomous ordinary di ff erentialequations (ODEs) for the flux φ ( t ), voltage v ( t ), current i ( t ) and the time p in the extended coordinate system as d φ dt = v , C dvdt = i − W ( φ ) v , L didt = − v − Ri + F sin( Ω p ) , d pdt = . (1)Here W ( φ ) is the memductance of the memristor and isas defined in [10], W ( φ ) = dq ( φ ) d φ = ( G a , | φ | > G a , | φ | ≤ , (2)where G a and G a are the slopes of the outer and innersegments of the characteristic curve of the memristorrespectively. We can rewrite Eqs. (1) in the normalizedform as ˙ x = x , ˙ x = x − W ( x ) x , ˙ x = − β ( x + x ) + f sin( ω x ) , ˙ x = . (3)Here dot stands for di ff erentiation with respect to thenormalized time τ (see below) and W ( x ) is the normal-ized value of the memductance of the memristor, givenas W ( x ) = dq ( x ) dx = ( a , | x | > a , | x | ≤ a = G a / G and a = G a / G are the normalizedvalues of G a and G a mentioned earlier and are nega-tive. The rescaling parameters used for the normaliza-tion are x = G φ C , x = v , x = iG , x = G pC , G = R , (5) β = CLG , ω = Ω CG = πν CG , τ = GtC , f = F β. In our earlier work on this memristive MLC circuit,see [8], we reported that the addition of the memris-tor as the nonlinear element converts the system into apiecewise-smooth continuous flow having two discon-tinuous boundaries, admitting grazing bifurcations , atype of discontinuity induced bifurcation (DIB). Thesegrazing bifurcations were identified as the cause forthe occurrence of hyperchaos, hyperchaotic beats andtransient hyperchaos in this memristive MLC system.Further we have reported discontinuity induced Hopfand Neimark-Sacker bifurcations in the same circuit, re-fer [11]. Thus the memristive MLC circuit shows richdynamics by virtue of it being a non-smooth system.Hence we give a brief description of the memristiveMLC circuit in the frame work of non-smooth bifurca-tion theory.2.1
Memristive MLC Circuit as a Non-smooth System
The memristive MLC circuit is a piecewise-smooth con-tinuous system by virtue of the discontinuous nature ofits nonlinearity, namely the memristor. An active fluxcontrolled memristor is known to switch state with re-spect to time from a more conductive ON state to a lessconductive OFF state and vice versa at some fixed val-ues of flux across it, see [11]. In the normalized coordi-nates this switching is found to occur at x = + x = −
1. These switching states of the memristor giverise to two discontinuity boundaries or switching man-ifolds, Σ , and Σ , which are symmetric about the ori-gin and are defined by the zero sets of the smooth func-tions H i ( x , µ ) = C T x , where C T = [1 , , ,
0] and x = [ x , x , x , x ], for i = ,
2. Hence H ( x , µ ) = ( x − x ∗ ), x ∗ = − H ( x , µ ) = ( x − x ∗ ), x ∗ = +
1, respec-tively. Consequently the phase space D can be dividedinto three subspaces S , S and S due to the presence ramana – J. Phys. (2016) :
Figure 2 . A Multisim Prototype Model of a memristive MLC circuit. The memristor part is shown by the dashed outline. The parametervalues of the circuit are fixed as L = mH , R = Ω , C = . nF . The frequency of the external sinusoidal forcing is fixed as ν ext = . kHz and the amplitude is fixed as F = mV pp ( peak-to-peak voltage). of the two switching manifolds. The memristive MLCcircuit can now be rewritten as a set of smooth ODEs ˙ x ( t ) = F , ( x , µ ) , H ( x , µ ) < H ( x , µ ) > , x ∈ S , F ( x , µ ) , H ( x , µ ) > H ( x , µ ) < , x ∈ S (6) where µ denotes the parameter dependence of the vec-tor fields and the scalar functions. The vector fields F i ’sare F i ( x , µ ) = x − a i x + x − β x − β x + f sin ( ω x )1 , i = , , a = a .The discontinuity boundaries Σ , and Σ , are notuniformly discontinuous. This means that the degreeof smoothness of the system in some domain D of theboundary is not the same for all points x ∈ Σ i j ∩ D .This causes the memristive MLC circuit to behave as anon-smooth system having a degree of smoothness ofeither one or two . In such a case it will behave either asa Filippov system or as a piecewise-smooth continuousflow respectively, refer Appendix A in [11].2.2
Equilibrium Points and their Stability
In the absence of the driving force, that is if f =
0, thememristive MLC circuit can be considered as a three-dimensional autonomous system with vector fields givenby F i ( x , µ ) = x − a i x + x − β x − β x , i = , , . (8)This three dimensional autonomous system has a triv-ial equilibrium point E , two admissible equilibrium -0.5-0.25 0 0.25 0.5 -2 -1 0 1 2 x x Figure 3 . Figure showing the equilibrium points E ± in the sub-spaces S and S for the parameter value above β c = . x = . x = . x = .
01 for the fixedpoint E + in the subspace S and x = . x = − . x = − . E − in the subspace S . points E ± and two boundary equilibrium points E B ± .The trivial equilibrium point is given as E = { ( x , x , x ) | x = x = x = } (9)The two admissible equilibria E ± are E ± = { ( x , x , x ) | x = x = , x ∗ = constant and not equal to ± } (10)The two boundary equilibrium points are E B ± = { ( x , x , x ) | x = x = , ˆ x = ± } (11)The multiplicity of equilibrium points arises because ofthe non-smooth nature of the nonlinear function, namely Pramana – J. Phys. (2016) : W ( x ) given in Eq. (4). To find the stability of theseequilibrium states, we construct the Jacobian matrices N i , i = , , N i = − a i − β − β , i = . (12)The characteristic equation associated with the system N i in these equilibrium states is λ + p λ + p λ = , (13)where λ ’s are the eigenvalues that characterize the equi-librium states and p i ’s are the coe ffi cients, given as p = β (1 + a i ) and p = ( β + a i ). The eigenvalues are λ = , λ , = − ( β + a i )2 ± p ( β − a i ) − β . (14)where i = , ,
3. Depending on the eigenvalues, thenature of the equilibrium states di ff er.1. When ( β − a i ) = β , the equilibrium state willbe a stable / unstable star depending on whether( β + a i ) is positive or not.2. When ( β − a i ) > β , the equilibrium state willbe a saddle.3. When ( β − a i ) < β , the equilibrium state willbe a stable / unstable focus.For the third case, the circuit admits self oscillationswith natural frequency varying in the range p(cid:2) ( β − a ) − β (cid:3) / < ω o < p(cid:2) ( β − a ) − β (cid:3) / F ( x , µ ) and F ( x , µ ) are sym-metric about the origin, that is F ( x , µ ) = F ( − x , µ ),the admissible equilibria E ± are also placed symmetricabout the origin in the subspaces S and S . These areshown in Fig. 3 for a certain choice of parametric val-ues.2.3 Sliding Bifurcations and Chaos
Let us assume the bifurcation points at the two switch-ing manifolds to be E B ± = { ( x , x , x ) | x , , x = , ˆ x = ± . } (15)Then we find from Eqs. (8) that F ( x , µ ) , F ( x , µ ) at x ∈ Σ , and F ( x , µ ) , F ( x , µ ) at x ∈ Σ , . Undersuch conditions the system is said to have a degree of -3-1.5 0 1.5 32.6 2.8 3 3.2a(i) x Time (t x 10 )-1.5-0.75 0 0.75 1.5-2.5 -1.25 0 1.25 2.5a(ii) x x Figure 4 . The chaotic dynamics of the memristive MLC oscillatorarising due to sliding bifurcations occurring in the circuit, with a(i)the time plot of the x variable and a(ii) phase portrait in the ( x − x )plane. The step size is assumed as h = (2 π/ω ), with ω = . f = . smoothness of order one , that is r = . Hence the mem-ristive MLC circuit can be considered to behave as a
Filippov system or a
Filippov flow capable of exhibit-ing sliding bifurcations .Sliding bifurcations are Discontinuity Induced Bi-furcations ( DIB’s ) arising due to the interactions be-tween the limit cycles of a Filippov system with theboundary of a sliding region. Four types of slidingbifurcations have been identified by Feigin [12] andwere subsequently analysed by di Bernado, Kowalczykand others [13, 14, 15, 16] for a general n − dimensionalsystem. These four sliding bifurcations are crossing-sliding bifurcations, grazing-sliding bifurcations, switching-sliding bifurcations and adding-sliding bifurcations.The memristive MLC circuit is found to admit threetypes of sliding bifurcations, namely crossing-sliding , grazing-sliding and switching sliding bifurcations [17].Let the parameters be chosen as a , = − . a = − . β = . f = .
20 and ω = .
65. For thesechoice of parameters, the memristive MLC circuit un-dergoes repeated sliding bifurcations at the discontinu- ramana – J. Phys. (2016) : Σ , and Σ , , giving rise to a chaotic stateas shown in Fig. 4. Here a(i) shows the time plot ofthe x variable and a(ii) shows the phase portrait in the( x − x ) plane. In the subsequent section we will showthat this chaotic behaviour exhibited by the memristiveMLC circuit can be controlled using state feedback con-trol technique.
3. Control of Chaos
Control of chaos refers to purposeful manipulation ofthe chaotic behaviour of a nonlinear system to somedesired or preferred dynamical state. As chaotic be-haviour is considered undesired or harmful, a need wasfelt for suppression of chaos or at least reducing it asmuch as possible. For example, control of chaos isnecessary in avoiding fatal voltage collapses in powergrids, elimination of cardiac arrhythmias, guiding cel-lular neural networks to reach certain desirable patternformations, etc. The earliest attempts at controllingchaos were focussed on eliminating the response of achaotic system, which resulted in the destruction of thedynamics of the system itself. However it was Ott, Gre-bogi and Yorke [2] who showed that it would be ben-eficial to force the chaotic system to one of its infiniteunstable periodic orbits (UPO) which are embedded inthe chaotic attractor of the system without totally de-stroying the dynamics of the system. Following thismany workers have developed newer techniques to con-trol chaos and have applied them successfully on a va-riety of systems to realize di ff erent desired behaviours.Generally all the known methods of chaos control canbe grouped into two categories, either feedback controlmethods or non-feedback control algorithms.3.1 Feedback Controlling Algorithms
Feedback control algorithms essentially make use ofthe intrinsic properties of chaotic systems to stabilizeorbits which are already existing in the systems. TheAdaptive Control Algorithm (ACA) developed by [18]and applied by [19, 3], the Ott-Grebogi-Yorke (OGY)Algorithm developed by [2] and applied by [20, 21, 22,5, 23], the Control Engineering Approach, developedby [24, 25] are all examples of these algorithms.3.2
Non-feedback Methods
The non-feedback methods refer to the use of somesmall perturbing external force, or noise, or a constantbias potential, or a weak modulating signal to some sys-tem parameter. The parametric control of chaos wasdemonstrated by, [26, 27, 3, 28, 29, 30, 5]. The con-trol of chaos by applying a constant weak biasing volt- age was demonstrated by [5] in the case of MLC os-cillator and Du ffi ng oscillator and by addition of noisewas demonstrated in a BVP oscillator by [3]. The othercontrol algorithms are Entrainment or Open Loop Con-trol method developed and applied by [31, 32, 33, 34,35], the Oscillation Absorber Method developed by [36,37].3.3 Control of Chaos using State Feedback
As the feedback and nonfeed back methods of chaoscontrol have many drawbacks, a continuous time feed-back control using small perturbations was proposednumerically by [38]. This control scheme was provideda rigorous basis by Chen and Dong and was demon-strated successfully in time continuous systems like Du ff -ing Oscillator [39], Chua’s Circuit [25, 40] and so on.However the drawbacks of these methods are1. they can be applied only when the dynamical equa-tions for the system are known a priori
2. the internal state variables are assumed to be avail-able to construct control forces3. the controller structure, in some cases, is extremelycomplicated4. limited information may be available and the onlymeasurable quantity of the system is its output5. Further, for nonsmooth systems, these conven-tional techniques, in particular addition of a sec-ond weak periodic excitation or the addition of aconstant bias do not seem to enforce control ofchaosUnder such conditions a parallel state reconstruction bymeans of either a Kalman filter or Luenberger type ob-server must be used to implement control laws. For thispurpose, the state space representation of the systemand their transformations to either controller canonicalform or observer canonaical form are derived, refer Ap-pendix A.The state space representation refers to the mod-elling of dynamical systems in terms of state vectorsand matrices so that the analyses of such systems aremade conveniently in the time domain, using the basicknowledge of matrix algebra [41, 42]. This represen-tation is a well researched area in the field of controlengineering [43, 44, 41]. The main advantage of thisapproach is that it presents a uniform platform for rep-resenting time varying as well as time invariant systems,linear as well as piece-wise nonlinear systems. Furtherthe vector fields for all the sub-spaces of the system takeon a uniform form. Some of the methods of control thatfall in this category are adaptive control [45], observer
Pramana – J. Phys. (2016) : x o = ˜ Ax o + B T u , y = C To x o + D T u , (16)where ˜ A ∈ R n × n , B ∈ R n × r , C ∈ R n × l and D ∈ R l × r arematrices, u is a r -dimensional vector denoting the con-trol input and y is a l -dimensional vector representingthe output of the system. This system is often called as open-loop system in control theory.Being in the observer canonical form, the systemmatrix ˜ A is given as˜ A = − ˜ a · · · − ˜ a · · · ... ... ... · · · ... − ˜ a n − · · · − ˜ a n · · · , (17)where ˜ a i ’s are the coe ffi cients of the characteristic poly-nomial {| sI − ˜ A |} .If we want the states of the system to approach zerostarting from any arbitrary state, then we should have todesign a control input which would regulate the statesof the system to the desired equilibrium conditions. Toachieve this we assume a state feedback control lawu = − ˜ K x o , (18)where ˜ K is called the control gain vector and can bedesigned using pole placement technique, familiar incontrol theory.Substituting this control law, Eq. (18) in the statespace representation of the open-loop system, Eq. (16),the system now becomes a closed-loop system repre-sented as ˙ x o = ( ˜ A − B T ˜ K ) x o , y = C To x o + D T u , (19)where B T is the transpose of the vector B and the closed-loop system matrix is given as( ˜ A − B T ˜ K ) = − (˜ a − ˜ k n ) 1 0 · · · − (˜ a − ˜ k n − ) 0 1 · · · ... ... ... · · · ... − (˜ a n − − ˜ k ) 0 0 · · · − (˜ a n − ˜ k ) 0 0 · · · . (20) If the values of ˜ K are so chosen that the eigen valuesof the matrix ( ˜ A − B T ˜ K ) lie within the unit circle inthe complex plane, then the system can be controlledto a desired stable equilibrium state. The problem ofchaos control thus reduces to just determining a statefeedback control gain vector ˜ K such that the control law,Eq. (18), places the poles of the closed loop system,Eq. (19), in the desired locations. An illustration ofthis concept is shown in the block diagram in Fig. 5.A necessary and su ffi cient condition for successfulpole placement is that the nonlinear system, that is, thepair of matrices ( ˜ A , B ), must be controllable.Let the characteristic polynomial { sI − ( ˜ A − B T ˜ K ) } of the closed-loop system, Eq. (19), be given as s n + (˜ a − ˜ k n ) s n − + (˜ a − ˜ k n − ) s n − + · · · + (˜ a n − ˜ k ) = . (21)Let the characteristic equation of the desired controlstate of the system be( s − s )( s − s )( s − s ) · · · ( s − s n ) = , s n + α s n − + α s n − + · · · α n − s + α n = , (22)where s i , i = , , · · · n are the desired poles to whichthe system should be guided and α i , i = , , · · · n arethe coe ffi cients of the desired characteristic equation.By comparing Eqs. (21) and (22) we get the elementsof the transformed control gain vector ˜ K as˜ k n = α − ˜ a , ˜ k n − = α − ˜ a , ˜ k n − = α − ˜ a , · · · ˜ k = α n − ˜ a n .
4. Control of Chaos in Memristive MLC Circuit
In the earlier sections we have seen that the memris-tive MLC circuit is a piecewise-smooth dynamical sys-tem having two discontinuity boundaries causing thestate space of the system to be split up into three sub-spaces. Consequently the memristive MLC circuit isrepresented by a set of smooth ODE’s, refer Eqs. (6).Further we have seen that for the boundary equilibriumpoints given by Eqs. (15), the memristive MLC circuitbecomes a
Filippov system.Linearising the vector fields about the equilibriumpoints defined by Eqs. (15), the observer canonicalform of the state space representation of the memris-tive MLC oscillator as a SISO system, refer Eq. (A.9)in Appendix A, can be given as˙ x o ( t ) = ˜ A x o + B T u if x ∈ S , ˜ A , x o + B T u if x ∈ S , , y = C T x + D T u , (23) ramana – J. Phys. (2016) :
Controller K Nonlinear System
State FeedbackInput u Output y
Figure 5.
Block diagram illustrating the concept of state feedback control. where the system matrices ˜ A i ’s are calculated for theabove chosen parameters as˜ A ( x ) = . . . . . . . . . , (24)while˜ A , ( x ) = − . . . − . . . − . . . . (25)Further the vectors B T , C T and D T are chosen as B T = (cid:16) (cid:17) , (26) C T = (cid:16) (cid:17) , (27) D T = (cid:16) (cid:17) . (28)We assume here that no disturbance is present in thesystem, that is, the vector D T is a null vector D = S , for the above mentioned parameters are given as P c , = . .
00 0 . . . − . . − .
95 0 . . (29)Similarly the controllability matrix for the sub-space S is P c = . .
00 1 . . .
02 0 . . − . − . . (30) As the controllability matrices in all the three sub-spaceshave a full rank of 3, we find that the matrices ( ˜ A i , B )form controllable pairs. Hence the linearised parts ofthe memristive MLC circuit are controllable. To achievestate feedback control, we assume a switched state feed-back control law [50], u = − ˜ K x o if x ∈ S , − ˜ K , x o if x ∈ S , , (31)where ˜ K i ’s are the control gain vectors in the three sub-regions of the phase space and are found using the pro-cedure outlined in the previous section as˜ K = (cid:16) − . . − . (cid:17) , (32)and˜ K , = (cid:16) . . . (cid:17) . (33)The closed loop system for the memristive MLC circuitupon application of gain is˙ x o ( t ) = ( ˜ A − B T ˜ K ) x o if x ∈ S , ( ˜ A , − B T ˜ K , ) x o if x ∈ S , , y = C To x + D T u . (34)As the eigen values of the matrices ( ˜ A i − B T ˜ K i ), i = , , x = − . , x = − . , x = − .
1) is shown inFig. 6a(i) while the periodic attractor in the ( x − x )phase plane in the asymptotic limit is shown in Fig. Pramana – J. Phys. (2016) : -3-1.5 0 1.5 32.5 3.0 3.5 4.0 a(i) x Time (t x 10 ) -3-1.5 0 1.5 32.5 3.0 3.5 4.0 a(ii) x Time (t x 10 ) -0.5-0.25 0 0.25 0.5-1.2 -1 -0.8 -0.6 -0.4 b(i) x x -0.5-0.25 0 0.25 0.5 0.4 0.6 0.8 1 1.2 b(ii) x x Figure 6 . The periodic oscillations of the memristive MLC oscillator after the application of the state feedback control shown by a(i) & a(ii)the time plots and b(i) & b(ii) phase portraits in the ( x − x ) plane. A change in the initial conditions form ( x = − . , x = − . , x = − . x = − . , x = − . , x = − .
2) results in the symmetric interchange of the time plots and attractors about the origin. The step size isassumed as h = (2 π/ω ), with ω = .
65 and f = . x = − . , x = − . , x = − . x and theperiodic attractor in ( x − x ) phase space as are shownin the corresponding Figs. 6a(ii) and 6b(ii).It is pertinent to state here that from Figs. 3 and6 the memristive MLC system may possess multista-bility. This is because we see that in these two cases,a mere change in the initial conditions forces the sys-tem to exhibit di ff erent dynamics. If the system were topossess multistability, then we strongly believe that bytweaking the control gain vectors K and K , it can bedirected to take on any of the desired multistable states.
5. Synchronization of Chaos
The feasibility of synchronization of chaotic systemsand the conditions to be satisfied for the same werefirst demonstrated by [54] by introducing the concept of
Drive-Response systems. Here a chaotic system is con-sidered as the drive system and a part of or subsystemof this drive system is considered as the response . Un-der the right conditions ( the conditional Lyapunov ex-ponents (CLEs) of the error dynamics being negative),the signals of the response part will converge to those ofthe drive system as time elapses. Ever since this groundbreaking work, many researchers have proposed syn-chronization of chaos in di ff erent systems based on the-oretical analysis and even experimental realizations. Forexample, this methodology has been successfully ap-plied to synchronize chaos in Lorenz systems [54, 55,56], R¨ossler systems [54], the hysteretic circuits [57],Chua’s circuits [58], driven Chua’s circuits [59], Chua’sand MLC circuits [60, 61], ADVP oscillators [62, 63],phase locked loops (PLL) [64, 65], etc.Further the possibility of applying this approach forsecure communication has been demonstrated. The ideaof chaotic masking and modulation and chaotic switch-ing for secure communication of information signals ramana – J. Phys. (2016) : ffi culties of the drive-response concept.
6. Observer Based Adaptive Synchronization ofChaos
Let us consider the state space representation of a sin-gle input single output (SISO) nonlinear system [42],defined as in Eq. (A.1) in Appendix A,˙ x = ˜ Ax + B T u , y = C T x + D T u , (35)where ˜ A ∈ R n × n , B ∈ R n × r , C ∈ R n × l and D ∈ R l × r are matrices, u is a r -dimensional vector denoting thecontrol input and y is a l -dimensional vector represent-ing the output of the system. The control input can begiven as u = d + θ T f ( x , y ) , (36)where d ∈ R is a bounded disturbance, θ ∈ R p is the con-stant parameter vector and f ( x , y ) is a p -dimensionalvector di ff erential function.When all the state variables of this system are un-available for measurement, then according to controltheory, the states of the system may be estimated by de-signing a parametric model of the original system. Thisparametric model is called an observer and is consid-ered as the response system. The concept of observerdesign is a well established branch of control engineer-ing and is widely used in the state feedback control ofdynamical systems [43, 44, 41]. In this method, oncethe drive system and its related observer are chosen,then under certain conditions, local or global synchro-nization between the drive and observer system is guar-anteed [87].Let us assume that the output y ( t ) is the only vari-able that can be measured for the system Eq. (35). Thenan observer based on the available signal can be derivedto estimate the state variables. This observer is knownin the literature as the Luenberger Observer [42] and isgiven as ˙ˆ x = ˜ A ˆ x + L T ( y − ˆ y ) + B T ˆ u , ˆ y = C T ˆ x + D T ˆ u , (37)where ˆ x denotes the dynamic estimate of the state vari-able x , L ∈ R n is a n -dimensional vector called as the observer gain vector . It is essential that Eq. (37) is inobserver canonical form, refer Eq. (A.10) in AppendixA. The control law can be derived asˆ u = ˆ d + ˆ θ T f ( x , y ) , (38)where ˆ d and ˆ θ are the estimates of the disturbances andthe parameters of the system and are updated according Pramana – J. Phys. (2016) : d = ( y − ˆ y ) , ˙ˆ θ = f ( x , y )( y − ˆ y ) . (39)The Luenberger observer Eqs. (37) has a feedback termthat depends on the output observation error ˜ y = y − ˆ y .Then the state observation error ˜ x = x − ˆ x satisfies theequation˙˜ x = ( ˜ A − L T C ) ˜ x + B T h ( d − ˆ d ) + ( θ T − ˆ θ T ) f ( x , y ) i , ˜ x (0) = x − ˆ x , (40)where we assume X = ( ˜ A − L T C ) as the augmented sys-tem matrix. The implementation of this observer basedadaptive synchronization of nonlinear systems is illus-trated in Fig. 7.6.1 Conditions for stability:
According to control theory, the system represented bythe Eq. (37) is stable in the sense of Lyapunov [42],refer section A.1 of Appendix A, if any of the followingconditions are satisfied:1. All eigen values of the augmented matrix X = ( ˜ A − L T C ), have negative real parts.2. For every positive definite matrix Q , (that is Q = Q T > X T P + PX = − Q , (41)has a unique solution P that is also positive defi-nite.3. For any given matrix C , with the pair ( C , X ) beingobservable, the equation X T P + PX = − C T C , (42)has a unique solution P , that is also positive defi-nite.If ( C T , ˜ A ) is an observable pair, then we can choose thevalues of the gain vector L such that the matrix ( ˜ A − L T C ) is stable. In fact, the eigen values of the matrix( ˜ A − L T C ), and therefore the rate of convergence of ˜ x ( t )to zero can be arbitrarily chosen by designing the vector L appropriately [41].The observer based response system given by Eq.(37) and associated with the control law given by Eq.(36) and the adaptive algorithm given by Eq. (39) will now globally and asymptotically synchronize with thedrive system given by Eq. (35), that is k ˜ x ( t ) k = k x ( t ) − ˆ x ( t ) k→ t → ∞ , for all initial conditions.Thus we find that the adaptive synchronization schemeis based on the following:1. the linear part of the system is observable, that isthe pair ( C T , ˜ A ) is observable,2. design of an suitable observer based on an adap-tive law and3. formulation of a suitable control law.
7. Observer Based Adaptive synchronization of Chaosin Coupled Memristive MLC Oscillators
In this section we report the synchronization of chaosvia an observer based design, with appropriate controllaw and adaptive algorithm in a system of two coupledmemristive MLC circuits. As in the case of control ofchaos, we assume that under appropriate choice of theboundary equilibrium points, the memristive MLC cir-cuit becomes a
Filippov system. Further we assume thesame parameter values as were fixed for e ff ecting con-trol in a single memristive MLC circuit, namely a , = − . a = − .
02 and β = . f = .
20 and ω = .
65. Also we assume the observer canonical form ofthe state space representation of the memristive MLCcircuit as given in Eq. (23), namely˙ x o ( t ) = ˜ A x o + B T u if x ∈ S , ˜ A , x o + B T u if x ∈ S , , y = C T x + D T u , (43)where the system matrices ˜ A i ’s and the vectors B T , C T and D T are the same as are given in Eqs. (24 - 28).Then the observability matrices for the sub-spaces S , are P o , = .
00 0 .
00 0 . .
00 1 .
00 0 . .
00 0 .
55 1 . . (44)Similarly the observability matrix for the sub-space S is P o = .
00 0 .
00 0 . .
00 1 .
00 0 . .
00 1 .
02 1 . . (45)As these observability matrices in all the three sub-spaceshave a full rank of 3, we find that the matrices ( C T , A i ) ramana – J. Phys. (2016) :
Input u q < q < . u x < x < _ S + x x < _ x = ~ Controller State Observer Nonlinear SystemParameter Estimation
Figure 7.
Block diagrammatic representation of the observer based adaptive synchronization of nonlinear systems. form an observable pair. Hence the linearised parts ofthe memristive MLC circuit are observable. Under thiscondition the Luenberger observer for the memristiveMLC circuit can be derived as˙ˆ x ( t ) = ˜ A ˆ x + L T ( y − ˆ y ) + B T ˆ u if ˆ x ∈ S , ˜ A , ˆ x + L T , ( y − ˆ y ) + B T ˆ u if ˆ x ∈ S , , ˆ y = C T ˆ x + D T ˆ u , (46)where the control law isˆ u = ˆ θ T f ( x , y ) , (47)with the vector f ( x , y ) given as f ( x , y ) = y , f ( x , y ) = | y + | − | y − | , (48)is the di ff erential function and ˆ θ are the estimates of theparameters of the system and are updated according tothe adaptive algorithm˙ˆ θ = − ( y − ˆ y ) f ( x , y ) , ˙ˆ θ = − ( y − ˆ y ) f ( x , y ) . (49)The state error ˜ x = ˙ x − ˙ˆ x dynamics is represented by˙˜ x = ( ˜ A − L T C ) ˜ x + B T ( θ − ˆ θ T ) f ( x , y ) , ( ˜ A , − L T , C ) ˜ x + B T ( θ − ˆ θ T ) f ( x , y ) . (50)The augmented matrices for the system can be definedas X i = ( ˜ A i − L Ti C ) for i = , , . (51)For the choice of parameters of the system mentionedabove, the observer gain vectors L i for each of the sub-spaces S i are chosen so as to have the augmented matri-ces X i to be exponentially stable. For the sub-spaces S , , the gain vectors are chosen as L , = (cid:16) . . − . (cid:17) T . (52)Due to this choice of the observer gain vectors L i , theaugmented matrices X , in these sub-spaces S , willhave poles at { . , − . ± i . } . Similarly,for the sub-space S , the gain vector is chosen as L = (cid:16) . . . (cid:17) T . (53)This will cause the augmented matrix X to have poles at {− . , . ± i . } .The Lyapunov equation for stability, Eq. (41), may bewritten separately for the three sub-spaces as, X Ti P i + P i X i = − Q for i = , , , (54)where we assume the matrix Q to be a 3 - dimensionalunit matrix. The solutions of the above Lyapunov equa-tion for stability for the sub-spaces S , are positive def-inite matrices P , given as P , = . − . − . − . . − . − . − . . . (55)The matrix P for the sub-space S is given as P = − . − . − . − . − . − . − . − . − . × , (56)We find that the matrix P for the sub-space S is not asolution of the Lyapunov equation, Eq. (54). Thereforethe trajectories in this sub-space should be, as per Lya-punov theory, unstable. Hence the augmented matrix X in region S is also unstable. However the combined Pramana – J. Phys. (2016) : ff ect of the dynamics in the outer two sub-spaces S , represented by the augmented matrices X , and the pos-itive definite matrices P , will impress upon the systemas a whole to become asymptotically stable and exhibita bounded behaviour asymptotically. Further as the con-ditions for the Lyapunov asymptotic stability, Eq. (54)are satisfied by the system as a whole, we find that un-der the action of the control law, Eq. (47) and the adap-tive algorithm, Eq. (49), the estimated values of the un-known parameters of the observer system ˆ a i ’s convergefinally to the true values of the parameters a i ’s as timeprogresses. These are shown in Figs. 8, where we findin Fig. 8(a) the value of the parameter ˆ a converges toits true value of − .
02, while in Fig. 8(b) the value ofthe parameter ˆ a converges to its true value of − . k ˜ x ( t ) k = k x ( t ) − ˆ x ( t ) k→ t → ∞ . The convergence of the error dynamics ˜ x to zero is shownin Fig. 9. Here the convergence of the errors ˜ x , ˜ x and ˜ x are shown in plots (a), (b) and (c) of Fig. 9respectively. These convergences of the parameters totheir true values and that of the error dynamics to zero,cause the observer system dynamics to converge to theoriginal system dynamics as time elapses. This meansthat the response system dynamics evolves as time pro-ceeds to that of the drive system dynamics. Hence ifthe drive system is in a chaotic state, then the responsesystem should also exhibit identical chaotic state. Thisis shown in Fig. 10.Had the drive system been in a periodic state, thenone would expect the response system also to take onasymptotically the periodic state by virtue of the adap-tive synchronization. As both the drive and the responsesystems exhibit identical behaviour, they are said to bein complete synchronization (CS) with each other. Thisis shown by the diagonal lines for the variables in the( x − x ′ ), ( x − x ′ ) and ( x − x ′ ) phase planes in plots(a), (b) and (c) respectively in Fig. 11.For e ff ecting this, it is essential that the gain vectors L i for all the sub-spaces S i ’s are properly chosen. Dueto the di ff erences in the gain vectors in the three sub-regions of the phase space, this observer based adap-tive synchronization is also referred to in literature as switched state feedback method of adaptive synchro-nization [50].
8. Conclusion
In this work, we have studied the control of chaos inan individual memristive MLC circuit as well as the -1.5-1-0.5 0 0.5 0 1 2 3 4 5 (b) a = -0.55 a Time (t x 10 ) -1.5-1-0.5 0 0.5 0 1 2 3 4 5 (a) a = -1.02 a Time (t x 10 ) Figure 8 . The estimation of (a) the parameter a and (b) the param-eter a of response system of the two coupled Memristive MLC Cir-cuit in the synchronized state using the adaptive observer scheme.It is to be noted that the asymptotic values of a = − .
02 and a = − .
55 are exactly equal to those of the drive system whichwere known apriori. ramana – J. Phys. (2016) : -0.50-0.250.00 0.25 0.50 0 1 2 3 4 5(c) e = ( x - x ^ ) Time (t x 10 )-0.50-0.250.00 0.25 0.50 0 1 2 3 4 5(b) e = ( x - x ^ ) Time (t x 10 )-0.50-0.250.00 0.25 0.50 0 1 2 3 4 5(a) e = ( x - x ^ ) Time (t x 10 ) Figure 9 . The convergences of the errors in the variables, e = x − x ′ , e = x − x ′ and e = x − x ′ of the two coupled MemristiveMLC Circuit in synchronized state under adaptive observer schemeare shown in (a), (b) and (c) respectively. synchronisation behaviour in a system of two coupledmemristive MLC circuits using state feedback control and observer based adaptive control techniques respec-tively. To realize these objectives, we have consideredthe memristive MLC circuit as a Filippov system, anon-smooth system having the order of discontinuity one and have derived the discontinuity mapping correc-tions such as (ZDM and PDM). Further we have de-rived the canonical state space representations for mem-ristive MLC circuit. Also the stability theory of Lya- -1.5-0.75 0 0.75 1.5-2.5 -1.25 0 1.25 2.5(a) x x -1.5-0.75 0 0.75 1.5-2.5 -1.25 0 1.25 2.5(b) x ^ x^ Figure 10 . The phase portraits (a) in the ( x − x ) plane and (b) inthe ( ˆ x − ˆ x ) plane showing identical chaos respectively. punov and pole-placement methods, concepts which arevery much familiar in control theory, were applied.We wish to state here that we have derived analyt-ical conditions for e ff ecting control and adaptive syn-chronization using state feedback and implemented theresults using numerical simulations. The fact that theresults of simulations agree with the predictions of theanalytical conditions point to the validity of our deriva-tions.From a di ff erent point of view, it has been shown bymany researchers, that in general any two coupled sys-tems, be they smooth or discontinuous, can be directedtowards amplitude death or oscillation death, irrespec-tive of their being in periodic, chaotic, hyper-chaotic ortime-delay systems, by the application of proper feed-back coupling, for example see [90]. The same canbe applied to the two coupled system under study, bycalculating proper observer gain vectors and choosingproper initial conditions and parametric values. How-ever we have not proceeded along these lines becauseit falls beyond the realm of this present work. We hopeto pursue this possibility in future studies.The phenomenon of control of chaos may be fur-ther studied to understand and e ff ectively prevent the Pramana – J. Phys. (2016) : -3.0-1.50.001.53.0 -3 -1.5 0 1.5 3(c) x ^ x -2.0-1.00.001.02.0 -2 -1 0 1 2(b) x ^ x -2.0-1.00.001.02.0 -2 -1 0 1 2(a) x ^ x Figure 11 . The complete synchronization of the two coupled Mem-ristive MLC circuit under the adaptive observer scheme, shown in(a) the ( x − x ′ ) plane, (b) the ( x − x ′ ) plane and (c) the ( x − x ′ )plane. incidence of nonlinear catastrophic phenomena such as blackouts in transmission lines and power grids, cardiacarrythmias, etc. The synchronisation of chaos whichwe have demonstrated using observer based adaptivescheme in memristive MLC circuits can be used to ef-fect digital modulation schemes for secure communica-tion. For example, the modulation characteristics of thememristor can be used to implement Amplitude ShiftKeying ASK, a key technique in Digital Signal Process-ing and transmission of Digitized Information. Also the switching characteristics of the memristor can beutilised to implement digital protocols for secure trans-mission of data. Appendix A. Space Representations of DynamicalSystems
The state space representation refers to the modelling ofdynamical systems in terms of state vectors and matri-ces so that the analyses of such systems are made conve-niently in the time domain, using the basic knowledgeof matrix algebra. The main advantage of this approachis that it presents a uniform platform for representingtime varying as well as time invariant systems, linear aswell as piece-wise nonlinear systems. The theoreticaldetails presented here are essentially from the availableliterature on control systems [41, 42].The generic state space representation of a n th - or-der dynamical system is given as˙ x = Ax + Bu , y = C T x + Du , (A.1)where x is an n -dimensional vector representing the statevariables, A ∈ R n × n , B ∈ R n × r , C ∈ R n × l and D ∈ R l × r are matrices, u is a r -dimensional vector denoting thecontrol input and y is a l -dimensional vector represent-ing the output of the system.The first of Eq. (A.1) is referred to as the state equa-tion while the second is referred to as the output equa-tion . The solution of the state equation is given by x ( t ) = e A ( t − t ) x ( t ) + Z tt e A ( t − τ ) Bu ( τ ) d τ, (A.2)where e At ≡ Φ ( t ) is the state transition matrix and x ( t )is the initial state of the system.1. Open-Loop System:
If the output of the systemis neither fedback to the input nor is it used tomodulate the behaviour of the system, then thesystem is called as a open-loop system .2.
Closed-Loop System:
If the output of the sys-tem is used to modulate the system and manip-ulate the control action on the system throughsome suitable feed-back mechanism, then the sys-tem is known as closed-loop system .3.
Exponential Stability:
An equilibrium state x e is said to be exponentially stable, if there existsa constant α >
0, and for every small constant ramana – J. Phys. (2016) : ǫ > | x − x e | < δ ( ǫ ) such that | x ( t ; t , x ) − x e | ≤ ǫ e − α ( t − t ) . (A.3)Here α is called as the rate of convergence.4. Asymptotic Stability:
An equilibrium state x e issaid to be asymptotically stable in the sense ofLyapunov [42] if any of the following conditionsare satisfied:(a) All eigen values of the matrix A have nega-tive real parts.(b) For every positive definite matrix Q , (thatis Q = Q T > A T P + PA = − Q , (A.4)has a unique solution P that is also positivedefinite.(c) For any given matrix C , with the pair ( C T , A )being observable, the equation A T P + PA = − C T C , (A.5)has a unique solution P , that is also positivedefinite.5. Observability:
It refers to the determination ofthe state of a system by observing or measuringits output. Mathematically it is determined byfinding the rank of the observability matrixP o = C T C T AC T A ... C T A n − . (A.6)The observability matrix is of dimension n × nl . Ifthis observability matrix has a full rank, equal to n , then the dynamical system or the pair ( C T , A )is said to be observable. However if P o is a n × n square matrix, then the system is observable if P o is non-singular.6. Detectability:
A dynamical system may not becompletely observable. However if the unobserv-able parts of the system become asymptoticallystable under the action of some control law , thenthe system is called as detectable [41]. 7.
Controllability:
It refers to the transferring of asystem from any given initial x ( t ) to any givendesired final state x ( t f ) over a finite interval oftime ( t f − t ). Mathematically it is determined bythe rank of the controllability matrixP c = BABA B ... A n − B . The controllability matrix is of dimension n × nr .If this controllability matrix has a full rank, equalto n , then the dynamical system or the pair ( A , B )is said to be controllable. However if P c is a n × n square matrix, then the system is controllable if P c is non-singular.8. Stabilizability:
A dynamical system may not becompletely controllable. However if the uncon-trollable parts of the system become asymptoti-cally stable under the action of some control law ,then the system is called as stabilizable [41].
Forms of State Space Representations
For any given dynamical system, there are essentiallyan infinite number of possible state space models thatgive identical input / output dynamics. However it is of-ten desirable to have certain standardized state spacemodel structures called as the canonical forms or canon-ical state space representations. Using similarity trans-formations it is possible to convert the state space modelfrom one canonical form to another [41]. Two of themost important canonical forms in control theory arethe observer canonical form and the controller canoni-cal form . Observer Canonical Form
Let us consider the coordinate transformation x o = W x ,where W = T P o is a transformation matrix, P o the ob-servability matrix and the matrix T is constructed usingthe coe ffi cients of the characteristic polynomial of thestate matrix A . T = · · · a · · · a ˜ a · · · ... ... ... · · · ... ˜ a n − ˜ a n − · · · · · · . (A.7) Pramana – J. Phys. (2016) : {| sI − A |} itself is given as p ( s ) = s n + ˜ a s ( n − + ˜ a s ( n − + ............... ˜ a ( n − s + ˜ a n . (A.8)Using the inverse coordinate transformation x = W − x o ,Eq. (A.1) can be transformed to the observer canonicalform as ˙ x o = ˜ A o x o + B T u , y = C To x o + D T u , (A.9)where the state matrix A o is obtained by the similaritytransformation A o = W AW − and is given as˜ A o = − ˜ a · · · − ˜ a · · · ... ... ... · · · ... − ˜ a n − · · · − ˜ a n · · · , (A.10)and C To = C T W − . (A.11) Controller Canonical Form
Let us in this case, consider an alternate coordinate trans-formation x c = M x , where M = T P c is a transforma-tion matrix, P c the controllability matrix and the matrix T is constructed using the coe ffi cients of the character-istic polynomial of the state matrix A and is as given inEq. (A.7).Using the inverse coordinate transformation x = M − x c , Eq. (A.1) can now be transformed alternativelyinto the controller canonical form as˙ x c = ˜ A c x c + B T u , y = C Tc x c + D T u , (A.12)where the state matrix A c is obtained by the similaritytransformation A c = MAM − and is given as˜ A c = · · ·
00 0 1 · · · ... ... ... · · · ... · · · − ˜ a − ˜ a − ˜ a − ˜ a n − − ˜ a n , (A.13)and C Tc = C T M − . (A.14) Acknowledgement
This work has been supported by a DST-SERB Distin-guished Fellowship to M.L.
References [1] J. M Gonzal´ez-Miranda.
Synchronisation and Control of Chaos:AnIntroduction for Scientists and Engineers . Imperial CollegePress, London, 2004.[2] E Ott, C Grebogi, and James A Yorke. Controlling chaos.
PRL , 64(11):1196–1199, 1990.[3] S Rajasekar and M Lakshmanan. Algorithms for controllingchaotic motion: Application for BVP oscillator.
Physica D. ,67:282–300, 1993.[4] S Boccaleti, C Grebogi, Y. C Lai, H Mancini, and D Maza.The control of chaos: Theory and applications.
Physics Re-ports , 329:103–197, 2000.[5] M Lakshmanan and K Murali.
Chaos in Nonlinear Oscilla-tors: Controlling and Synchronization . World Scientific, Sin-gapore, 1996.[6] M Lakshmanan and S Rajasekar.
Nonlinear Dynamics: In-tegrability, Chaos and Patterns . Springer-Verlag, NewDelhi,2003.[7] S Boccaleti, J Kurths, G Osipov, D. L Valladares, and Zhou C.S. Synchronisation of chaotic systems.
Physics Reports , 366:1–101, 2002.[8] Ahamed. A Ishaq and M Lakshmanan. Nonsmooth bifurca-tions, transient hyperchaos and hyperchaotic beats in a mem-ristive Murali-Lakshmanan-Chua circuit.
Int. J. Bifurcationand Chaos , 23(6):1350098 (28 pages), 2013.[9] A Ishaq Ahamed, K Srinivasan, K Murali, and M Lakshmanan.Observation of chaotic beats in a driven memristive Chua’s cir-cuit.
Int. J. Bifurcation and Chaos. , 21(3):737–757, 2011.[10] M Itoh and L. O Chua. Memristor oscillators.
Int. J. Bifurca-tion and Chaos. , 18(11):3183–3206, 2008.[11] A Ishaq Ahamed and M Lakshmanan. Discontinuity inducedhopf and neimark-sacker bifurcations in a memristive mlc cir-cuit.
Int. J. Bifurcation and Chaos. , 27(6):1730021–1730043,2017.[12] M.I Feigin.
Forced Oscillations in Systems with Discontinu-ous Nonlinearities . Nauka, Moscow, 1994.[13] M di Bernado, K. H Johansson, and F Vasca. Self-oscillationsand sliding in relay feedback systems: Symmetry and bifurca-tions.
Int. J. Bifurcation and Chaos. , 11(4):1121–1140, 2001.[14] P Kowalczyk and M di Bernado. On a novel class of bifurca-tions in hybrid dynamical systems - the case of relay feedbacksystems. In
Proc. of Hybrid Systems Computation and Con-trol , pages 361–374. Springer-Verlag, 2001.[15] M di Bernado, P Kowalczyk, and A Nordmark. Bifurcationsof dynamical systems with sliding:Derivation of normal-formmappings.
Physica D , 170:175–205, 2002.[16] M di Bernado, P Kowalczyk, and A Nordmark. Sliding bi-furcations: A novel mechanism for sudden onset of chaos infriction oscillators.
Int. J. Bifurcation and Chaos. , 13:2935–2948, 2003.[17] A Ishaq Ahamed.
Nonsmooth Bifurcations in Certain Piece-wise Continuous Nonlinear Circuits . PhD thesis, Bharathi-dasan University, Tiruchirappalli, India, 2016. ramana – J. Phys. (2016) : [18] B. A Huberman and E Lumer. Dynamics of adaptive systems.
IEEE Trans. Cir. Syst. CAS , 37(4):547–550, 1990.[19] S Sinha, R Ramaswamy, and J Subba Rao. Adaptive controlin nonlinear dynamics.
Physica D. , 43:118–128, 1990.[20] Y.C Lai, M Ding, and C Grebogi. Controlling Hamiltonianchaos.
Phys. Rev. E , 47(1):86, 1993.[21] T Tel. Controlling transient chaos.
Journal of Physics A:Mathematical and General , 24:L1359, 1991.[22] Y.C Lai, T Tel, and C Grebogi. Stabilizing chaotic-scatteringtrajectories using control.
Phys. Rev. E , 48(2):709, 1993.[23] J Singer, Y-Z Wang, and Haim H Bau. Controlling a chaoticsystem,.
PRL , 66(9):1123, 1991.[24] G Chen and X Dong. On feedback control of chaotic nonlineardynamic systems.
Int. J. Bifurcation and Chaos. , 2(2):407–411, 1992.[25] G Chen. Controlling Chua’s global unfolding circuit family.
IEEE Trans. Cir. Syst. CAS , 40(11):829–832, 1993.[26] R Lima and M Pettini. Suppression of chaos by resonant para-metric perturbations.
Phys. Rev. A , 41(2):726–733, 1990.[27] Y Liu and J. R. R Leite. Control of Lorenz chaos.
Phys. LettA. , 185:35–37, 1994.[28] K Wisenfeld and B McNamara. Small-signal amplification inbifurcating dynamical systems.
Phys. Rev. A , 33(1):629, 1986.[29] P Bryant and K Wisenfeld. Suppression of period-doublingand nonlinear parametric e ff ects in periodically perturbed sys-tems. Phys. Rev. A , 33(4):2525, 1986.[30] Y Braiman and I Goldhirsch. Taming chaotic dynamics withweak periodic perturbations.
Phys.Rev. Lett. , 66(20):2545–2548, 1991.[31] E. A Jackson and A Hubler. Periodic entrainment of chaoticLogistic Map dynamics.
Physica D. , 44:407–420, 1990.[32] E. A Jackson. The entrainment and migration controls ofmultiple-attractor systems.
Phys. Lett. A , 151:478–484, 1990.[33] E. A Jackson and A Kodogeorgiou. Entrainment and migra-tion controls of two-dimensional maps.
Physica D. , 54:253–265, 1991.[34] E. A Jackson and A Kodogeorgiou. On the control of complexdynamic systems.
Physica D. , 50:341–366, 1991.[35] E. A Jackson. Controls of dynamic flows with attractors.
Phys.Rev. A , 44(8):4839, 1991.[36] T Kapitaniak. Controlling chaotic oscillators without feed-back.
Chaos, Solitons and Fractals , 2(5):519–530, 1992.[37] T Kapitaniak, L. J Kocarev, and L.O Chua. Controlling chaoswithout feedback and control signals.
Int. J. Bifurcation andChaos. , 3(2):459, 1993.[38] K Pyragas. Continuous control of chaos by self-controllingfeedback.
Phys. Lett. A , 170:421–428, 1992.[39] G Chen and X Dong. On feedback control of chaotic continuous-time systems.
IEEE Trans. Cir. Syst. CAS , 40(9):591–601,1993.[40] C Hwang, J Hsheh, and R Lin. A linear continuous feedbackcontrol of Chua’s circuit.
CSF , 8(9):1507–1515, 1997.[41] T Kailath.
Linear Systems . Prentice Hall, Englewood Cli ff s,New Jersey, 1980.[42] P. A Ioannou and J Sun. Robust Adaptive Control . PrenticeHall, Englewood Cli ff s, New Jersey, 1996.[43] C. T Chen. Intorduction to Linear Systems Theory . Holt, Rine-hart and Winston Inc., New York, 1970.[44] C. A Desoer and M Vidyasagar.
Feedback Systems: Input-Output Properties . Academic Press Inc., New York, 1975. [45] M. T Yassen. Adaptive control and synchronization of a modi-fied Chua’s circuit system.
Appl. Math. Compute , 135(1):113–128, 2003.[46] T-L Liao. Observer based approach for controlling chaoticsystems.
Phys. Rev. E , 57(2):1604–1610, 1998.[47] H-T Yau. Design of adaptive sliding mode controller for chaossynchronization with uncertainties.
CSF , 22(2):341–347, 2004.[48] J Sun and Y Zhang. Impulsive control of Rssler systems.
Phys.Lett. A , 306:306–312, 2003.[49] M. T Yassen. Controlling, synchronization and tracking chaoticLiu system using active backstepping design.
Phys. Lett. A ,360:582–587, 2007.[50] J Zhang, H Zhang, and G Zhang. Controlling chaos in amemristor-based Chua’s circuit. In
Proceedings of the Interna-tional Conference on Communications, Circuits and Systems,2009.ICCCAS 2009. , Milpitas, CA, July 2009. IEEE.[51] H H C. Iu, D S. Yu, A L. Fitch, V. Sreeram, and H. Chen.Controlling chaos in a memristor based circuit using a twin-Tnotch filter.
IEEE CAS.-I , 58(6):1337–1344, 2011.[52] Y Song, Yi Shen, and Yi Chang. Chaos control of a memristor-based Chua’s oscillator via backstepping method. In
Proceed-ings of the International Conference on Information Scienceand Technology,2011. , Nanjing,Jiangsu,China, March 2011.IEEE.[53] E Sontag.
Mathematical Control Theory: Deterministic FiniteDimensional Systems . Springer-Verlag, Berlin, 1998.[54] L. M Pecora and T. L Caroll. Synchronization in chaotic sys-tems.
PRL , 64(8):821–825, 1990.[55] L. M Pecora and T. L Caroll. Synchronizing nonautonomouschaotic circuits.
IEEE Trans. Cir. Syst. CAS , 40(10):646–650,1993.[56] R He and P. G Vaidya. Driving systems with chaotic signals.
Phys. Rev. A , 46(12):7387, 1992.[57] L. M Pecora and T. L Caroll. Synchronizing chaotic circuits.
IEEE Trans. Cir. Syst. CAS , 38(4):453–456, 1991.[58] Leon O Chua, L Kocarev, K Eckert, and M Itoh. Experimentalchaos synchronisation in Chua’s circuit.
Int. J. Bifurcation andChaos. , 2(3):705, 1992.[59] Lj Kocarev, K. S Halle, K Eckert, Leon O Chua, and U Par-litz. Experimental demonstration of secure communicationvia chaotic synchronisation.
Int. J. Bifurcation and Chaos. ,2(3):709, 1992.[60] K Murali, M Lakshmanan, and L.O Chua. Controlling andsynchronisarion of chaos in simplest dissipative non-autonomouscircuit.
Int. J. Bifurcation and Chaos. , 5(2):563, 1995.[61] K Murali and Lakshmanan. Synchronization through com-pound chaotic signal in Chua’s circuit and Murali-Lakshmanan-Chua circuit.
Int. J. Bifurcation and Chaos. , 7(2):415–421,1997.[62] K Murali and M Lakshmanan. Transmission of signals bysynchronisation in a chaotic Van der Pol-Du ffi ng oscillator. Phys. Rev. E , 48(3):R1624(R), 1993.[63] M Lakshmanan and K Murali. Harnessing chaos:synchronisationand secure signal transmission.
Current Science , 67(12):989–995, 1994.[64] T Endo and L. O Chua. Synchronising chaos from electronicPhase-Locked Loops.
Int. J. Bifurcation and Chaos. , 1(3):701,1991.[65] M De Sousa Veria, A. J Lichtenberg, and M. A Liberman.Nonlinear dynamics of self-synchronising systems.
Int. J. Bi-furcation and Chaos. , 1(3):691, 1991.
Pramana – J. Phys. (2016) : [66] K. M Cuomo and A. V Oppenheim. Circuit implementationof synchronised chaos with applications to communications.
PRL , 71(8):65, 1993.[67] K. M Cuomo, A. V Oppenheim, and S. H Strogatz. Synchro-nization of Lorenz-based chaotic circuits with applications tocommunications.
IEEE Trans. Cir. Syst. CAS , 40(1):626–632,1993.[68] K. M Cuomo. Synthesising self-synchronising chaotic sys-tems.
Int. J. Bifurcation and Chaos. , 3(5):1327, 1993.[69] K. M Cuomo. Synthesising self-synchronising chaotic arrays.
Int. J. Bifurcation and Chaos. , 4(3):727, 1994.[70] U Parlitz, Leon O Chua, Lj Kocarev, K. S Halle, and A Shang.Transmission of digital signals by chaotic synchronisation.
Int.J. Bifurcation and Chaos. , 2(4):973, 1992.[71] J. H Peng, E. J Ding, and W Yang. Synchronizing hyperchaoswith a scalar transmitted signal.
PRL , 76(6):904–907, 1996.[72] R Mainieri and J Rehacek. Projective synchronization in three-dimensional chaotic systems.
PRL , 82(15):3042–3045, 1999.[73] C. Y Chee and D Xu. Control of the formation of projectivesynchronisation in lower-dimensional discrete-time systems.
Phys. Lett. A , 318:112–118, 2003.[74] D Xu. Control of projective synchronization in chaotic sys-tems.
Phys. Rev. E , 63(1):027201, 2001.[75] D Xu and C. Y Chee. Controlling the ultimate state of projec-tive synchronization in chaotic systems of arbitrary dimension.
Phys. Rev. E , 66(4):046218, 2002.[76] G Grassi and D. A Miller. Arbitrary observer scaling of allchaotic drive system states via a scalar synchronizing signal.
CSF , 39(3):1246–1252, 2009.[77] G Grassi and D. A Miller. Projective synchronization viaa linear observer:Application to time-delay,continuous- timeand discrete-time systems.
Int. J. Bifurcation and Chaos. ,17(4):1337, 2007.[78] M Hu, Z Xu, R Zhang, and A Hu. Adaptive full state hybridprojective synchronization of chaotic systems with the sameand di ff erent order. Phys. Lett. A , 365:315–327, 2007.[79] J Lu and Q Zhang. Full state hybrid lag projective synchro-nization in chaotic (hyperchaotic) systems.
Phys. Lett. A , 372:1416–1421, 2008.[80] M Hu, Z Xu, R Zhang, and A Hu. Parameters identificationand adaptive full state hybrid projective synchronization ofchaotic (hyper-chaotic) systems.
Phys. Lett. A , 361:231–237,2007.[81] K Thamilmaran and D. V Senthilkumar. Dynamics of twocoupled canonical Chua’s circuits. In
Proceedings of 2nd Na-tional Conference on Nonlinear Systems and Dynamics (NC-NSD) , pages 45–48, Aligarh, 2005.[82] P. S Swathi, S Sabarathinam, K Suresh, and K Thamilmaran.Chaos synchronisation and transmission of imformation in cou-pled SC-CNN based canonical Chua’s circuit.
Nonlinear Dyn ,78:1033–1047, 2014.[83] R Jothimurugan, K Thamilmaran, S Rajasekar, and M. A. FSanju´an. Experimental evidence for vibrational resonance andenhanced signal transmission in Chua’s circuit.
Int. J. Bifur-cation and Chaos. , 23(11):1350189 (12pages), 2013.[84] C. W Wu, T Yang, and L. O Chua. On adaptive synchro-nization and control of nonlinear dynamical systems.
Int. J.Bifurcation and Chaos. , 6(3):455–472, 1996.[85] M diBernado. An adaptive approach to control and synchro-nization of continuous-time chaotic systems.
Int. J. Bifurca-tion and Chaos. , 6(3):557–568, 1996. [86] T. L Liao. Adaptive synchronization of two Lorenz systems.
CSF , 9(9):1555–1561, 1998.[87] O Morgul and E Solak. Observer based synchronisation ofchaotic systems.
Phys. Rev. E , 54(5):4803–4811, 1996.[88] O Morgul and E Solak. On synchronization of chaotic sys-tems by using state observers.
Int. J. Bifurcation and Chaos. ,7(6):1307–1322, 1997.[89] T-L Liao and Shin-Hwa Tsai. Adaptive synchronization ofchaotic systems and its application to secure communications.
CSF , 11(9):1387–1396, 2000.[90] V Resmi, G Ambika, and RE Amritkar. General mechanismfor amplitude death in coupled systems.