The synchronisation of two floating memristor-based oscillators and the circuit design

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The synchronisation of two ﬂoating memristor-based oscillators and the circuit design

HONGMIN DENG ^∗and QIONGHUA WANG

College of Electronics and Information Engineering, Sichuan University, No. 24, South Section 1, Yihuan Road, Chengdu 610065, People’s Republic of China

∗Corresponding author. E-mail: denghongmin@aliyun.com

MS received 23 June 2018; revised 13 February 2019; accepted 26 February 2019; published online 10 July 2019 Abstract. The synchronisation between two ﬂoating memristor-based Colpitts oscillators is studied in this paper.

Firstly, the mathematical and circuit models of Colpitts oscillator based on a floating memristor with a diode bridge structure are built. On this basis, numerical simulations on the features of both the independent memristor and the floating memristor are conducted and compared using MATLAB software. Secondly, circuit simulation is made on the synchronisation of two floating memristor-based systems by using MULTISIM software. Finally, the physical circuit on the synchronisation of the two coupling Colpitts systems based on the diode bridge memristors is implemented by using the linear error feedback scheme and improved by the capacitor coupling scheme and the adaptive nonlinear feedback control scheme, respectively. The experimental results by the oscilloscope and simulation results show that approximate synchronisation is achieved.

Keywords. Floating memristor; synchronisation; Colpitts oscillator; physical implementation.

PACS Nos 05.45.Xt; 07.50.Ek; 05.45.–a

1. Introduction

The synchronisation of chaotic systems has been an important topic especially in secure communication since Pecorra and Carrol realised the drive–response synchronisation of two systems [1]. So far, there have been different synchronisation modes and control methods, e.g., exponential synchronisation [2], cluster synchronisation [3], asymptotic synchronisation [4], lag synchronisation [5], multiple switching combination synchronisation [6], etc. As for the control approaches, there are observer-based control [4], adaptive control [5,7,8], sliding mode control [9], unidirectional or bidirectional coupling control [3,10], state feedback control [11], impulsive control [12], intermittent control [2], active control [13], etc.

On the other hand, memristor, a newly realised device known as the fourth fundamental passive element, has drawn much attention because of its great application prospects: (i) memristor is usually expected to be used as the electrical synapse of the neural network because of its ability to remember charge quantity that passed through it, which is associated with the neural network’s memory of the past history. In the neural network, it

was important to investigate the synchronisation of the neurocircuits with electrical synapses [11,14]. Simul- taneously, the synchronisation in the coupled neural networks played a vital role in the fundamental science (e.g. the self-organised behaviour in the brain) [15]. (ii) The memristor’s intrinsic feature of the pinched hysteresis loop and the memory of the internal state beneﬁtted its potential applications in secure communication by producing complex transient transition [16], where synchronisation became a key technology. So many works have been focussed on the synchronisation of memristive systems since the memristor prototype was realised in the HP laboratory [17]. As an example of investigat- ing the synchronisation of memristor-based systems, the synchronisation of the memristor-based recurrent neural network was studied in ref. [11]. Adaptive synchronisation for fractional-order memristive systems was presented in ref. [18]. Correspondingly, different control methods were applied to the memristive systems, such as adaptive feedback control [19,20], pinning control [21], impulsive control [22], intermittent control [12,23] and so on.

The differences from our proposed work and works in the already available literatures on memristor-based

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synchronisation are as follows: (i) different research tools, means and roles of memristors. To the best of our knowledge, memristor-based synchronisation was usually investigated by means of theoretical analysis and simulation in most existing studies. In addition to these means and corresponding tools, we physically imple- ment the approximate synchronisation in the proposed paper. For instance, Zhang and Liao [24] utilised the circuit simulation software to demonstrate the synchronisation of coupled Fitzhugh–Nagumo circuits. Min and Jing [7] studied the implementation of a memristor- based circuit, but did not involve the synchronisation.

Padmanaban et al [10] investigated the engineering synchronisation of chaotic oscillators without memristors. In the newest published work [25], the field programmable gate array (FPGA) toolbox in Simulink software was used to realise the synchronisation of the fractional-order memristor-based discrete chaotic systems. As for the roles, two memristors were connected antiparallel as the coupling elements for the synchronisation of two systems in ref. [26]. In our paper, each memristor is set as a part of its corresponding oscillating system and this is a real circuit construction of the synchronisation, (ii) different strategies of error feedback and coupling: in most of the existing studies on synchronisation, two systems were usually classified as the master (drive) and the slave (response) systems. The error feedback control was usually applied to the slave system. In our paper, there is no master–slave difference between the two systems, and the bidirectional coupling is used to control the two systems via simple error feedback of a single-state variable from each system, (iii) as different research objects: contrary to most of the literatures which were concentrated on grounding memristor- based synchronisation, this paper takes the more flexible floating memristors as the research objects.

2. Circuit models of two coupled memristor-based Colpitts oscillators

Figure 1 describes the model of two coupled oscillating systems based on ﬂoating memristors.

In ﬁgure 1, the modules ‘linear error feedback’ and

‘controller’ are two parts closely related to each other:

the former is the error producing circuit, where the output is the error E1 and the latter is the linear control circuit, where the output is the control variable U, as shown in ﬁgure 2. It is also described by eq. (1), where mis the coupling strength factor:

U =m E1 = −2m

Ve₁ −Ve₂

= −2m

vC₂−vC_2S

. (1) For simplicity, letm = −1 here, and soU = −E1.

Figure 1. Schematic diagram of two coupled Colpitts oscillating systems based on ﬂoating memristors.

Figure 2. Error-producing circuit and controlling circuit corresponding to the linear error feedback module and the controller module, respectively.

3. Mathematical models of memristor and ﬂoating memristor-based Colpitts oscillators

In this paper, the ﬁrst Colpitts system of two bidirectional coupled oscillators is described as

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ dvC1

dt = 1

C₁iL− 1

C₁ic+ 1 R_f₁C₁

U −vC₁−vC₂

, dvC₂

dt = − 1 C2

vC₂ −VE E

/RE E + 1 C2

iL− vC₂

R3C2

+ 1 Rf₁C2

U −vC1−vC2

, diL

dt = −1

LvC₁− 1

LvC₂+ 1

L(VCC−vM)− 1 LiLR.

(2) The second Colpitts system is described as

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ dvC_1S

dt = 1

C_1SiL S− 1

C_1SicS+ 1 R_f₂C_1S

×

U−vC1S−vC2S

, dvC_2S

dt = − 1 C2S

vC_2S−VE E S

/RE E S+ 1 C2S

iL S

− vC_2S

R₁C_2S + 1 R_f₂C_2S

U −vC_1S −vC_2S

, di_{L S}

dt = − 1

LSvC_1S− 1

LSvC_2S + 1

LS(VCC S−vM S)

− 1

L_SiL SRS,

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where ic≈ie=IS S

e^v^be^/^V^{T T} −1

=IS S

e^−v^C²^/V^{T T} −1 (4)

icS ≈ieS =IS S S

e^v^beS/^{VT T S} −1

=IS S S

e^−v^C^2S^/^V^{T T S}−1 . (5)

IS SandIS S Sare the reverse saturation currents, andVT T

andVT T S are the thermal voltages of the transistorsT andTS, respectively.

Different emulated circuits have been proposed as alternatives of the memristors because they are not mature enough to be made on a product line. For instance, a memristor-based chaotic circuit was analysed theoretically and implemented physically by two AD633JN multipliers plus three AD711 operational ampliﬁers [27]. In another scheme, a memristor was implemented based on the second-generation current conveyors (CCII+devices) [28]. In ref. [29], an analogue model of a memristor using a light-dependent resistor (LDR) was presented and realised physically.

Next, a diode bridge circuit was utilised to emulate the memristor, and applied to a Shinriki’s circuit and a Col- pitts circuit, respectively [30,31]. In our previous work [32], a memristor-based Colpitts oscillator was studied where the emulated memristor required a large number of elements, such as many pairs of MOSFETS, operational ampliﬁers, analogue multiplier and many other discrete components. Therefore, it was neither conve- nient nor reliable for building the multimemristor node circuit, which was recognised as an important application of memristors in a multisynaptic neural network.

After a comprehensive comparison, we adopt the diode bridge approach which needs a simple circuit conﬁgu- ration and is easy to realise.

The diode bridge model of the memristor is shown in figure 3 [30,31]. In our paper, it is applied as a floating memristor to each of the Colpitts oscillators in figure 1.

Hence, the memristor here should be described by the diode bridge model as in eqs (6) and (7):

M = vM

2IS Dexp(−kv0)sinh(kvM), (6) dv0

dt = − v0

R0C0 +2IS Dexp(−kv0)cosh(kvM) C0

−2IS D

C₀ , (7)

wherek = 1/2nVTD,n = 1.9, VTD = 26 mV, IS D = 26 nA, n is the emission coefﬁcient, I_SD is the reverse saturation current of the diode and V_TD is the thermal voltage of the diode.

Figure 3. Emulated memristor model with a diode bridge conﬁguration.

4. Numerical simulation, circuit simulation and physical implementation

In this section, three schemes including a linear error feedback control scheme, a capacitor coupling scheme and an adaptive and nonlinear feedback control scheme are applied to the memristor-based Colpitts systems and it is seen that synchronisation can be achieved by each of the three schemes.

4.1 Numerical simulation for memristive emulator and the system

Notably, both ref. [31] and this paper discuss about applying the memristor configured by the diode bridge circuit to the Colpitts oscillator, but they differ in three aspects: (i) the way that the emulated memristor is added to the Colpitts oscillator – in ref. [31], the the memristor is grounded, but the memristor in our paper is connected to the oscillator in a more flexible and universal floating form, (ii) as for the purposes of circuit implementation, the former is aimed to realise the chaotic oscillator and the latter is dedicated to the synchronisation of two memristor-based Colpitts oscillators for extending to the multinode network in the future. So the control part of two systems is also presented in our paper, (iii) in our paper, the circuit is confirmed by the physical implementation of approximate synchronisation between two memristor-based systems, in addition to the theoretical analysis, the numerical simulation in MATLAB and the circuit simulation in MULTISIM.

The features of the diode bridge memristor with an independent sinusoidal input are shown in ﬁgure 4a–4d.

The attractors of the Colpitts circuit and the features of the floating memristor are shown in more detail in figure 5, where figures 5a–5j are the attractors of the Colpitts circuit, and figures 5a–5jare theiM–vM features of the floating memristor with the corresponding parameters in figures 5a–5j. The bifurcation diagram for iM Svs.C0Sof system (2) is shown in figure 6. It exhibits chaotic behaviour withC0S =300 nF in this paper.

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Figure 4. Features of the diode bridge memristor: (a)iM−vM characteristic of the pinched hysteresis loop in the diode bridge memristor with the sine input signalvM =sin(120πt), (b) time evolution waveformv0in the diode bridge memristor circuit (see figure 3) with the sine input signalvM =sin(120πt), (c)iM–vM characteristics of the floating bridge memristor in the Colpitts oscillator and (d) time evolution outputv0in the diode bridge circuit which is applied to the Colpitts oscillator as the floating memristor.

4.2 Synchronisation of two Colpitts oscillators by MULTISIM circuit simulation

In this subsection, the features of the memristor are further investigated by MULTISIM circuit simulation, with an independent sine wave input and with a dynamic and ﬂoating input added into the Colpitts circuit (only one type of input is added each time). TheiM–vMcharacter- istics of the memristor are shown in ﬁgures 7a and 7b.

Next, approximate synchronisation of the two ﬂoating memristor-based Colpitts oscillators through bidirectional coupling is achieved and shown in ﬁgure 8.

4.3 Theoretical analysis and physical experiments on the synchronisation of two ﬂoating memristor-based Colpitts oscillators

The oscillating frequency of the Colpitts oscillator can be designed in a wide range based on eq. (8). In this paper,C1 =C2 =100 nF andLtakes different values under different conditions:

f = 1

2π√

L(C1C2/(C1+C2)). (8) In the following analysis, we assume that the parameters are identical between eqs (2) and (3) for simplicity,C1= C1S, C2 =C2S,L = LS,R3 = R1,R = RS,Rf₁ = R_f₂ = R_f.

The synchronisation error e = [e1,e2,e3]T is described by eq. (9) according to eqs (2) and (3):

⎛

⎝e₁ e2

e3

⎞

⎠=λ1

⎛

⎝vC1

vC₂

iL

⎞

⎠−λ2

⎛

⎝vC_1S

vC_2S

iL S

⎞

⎠, (9)

where λ1, λ2 are the constants, and λ1 = λ2 = 1 is selected for simplicity.

Deﬁne a Lyapunov function:

F(e)= 1

2e^Te. (10)

The derivative of function F(e) is shown as in eq. (11):

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Figure 5. Attractors of the circuit and the corresponding features of the ﬂoating memristor in the Colpitts oscillator under different parameter conditions. (a), (a) R0 =10,C0 = 300 nF, L =235 μH; (b), (b) R0 =1000 ,C0 = 300 nF, L =235μH; (c), (c) R0 =100,C0 =300 nF,L =235μH; (d), (d)R0 =1000,C0 =940 nF,L =235 μH; (e), (e) R0 =100,C0 =940 nF, L =235μH; (f), (f)R0=10,C0=940 nF, L =235μH; (g), (g)R0=10 ,C0=300 nF,L =500μH; (h), (h)R0=100,C0 =300 nF,L =500μH; (i), (i)R0 =100,C0 =300 nF,L = 400μH and (j), (j)R0=100,C0=300 nF,L=80μH. All situations are with the same other circuit parameters, such as R =RS =33,C1=C2=C1S=C2S=100 nF,RE E =RE E S =390, the types of transistorsT,TSare 2N2222, the diodesD1–D4,D1S–D4Sare 1N4148.

Figure 6. Bifurcation diagram ofiM Svs. the capacitorC0S

in system (2).

Figure 7. iM–vM characteristics of the memristor emulator based on the diode bridge circuit through MULTISIM simulation. (a)iM–vM characteristics of the memristor with the independent sine inputvM =√

2 sin(120πt)and (b)iM–vM

characteristics of the ﬂoating memristor applied in the Col- pitts oscillator.X scale: 1 V/Div,Y scale: 2 mA/Div in (a);

Xscale: 500 mV/Div,Y scale: 30 mA/Div in (b).

dF(e) dt =e1

de₁ dt +e2

de₂ dt +e3

de₃ dt

=e1

1

C₁e3− ic

C₁ +ics

C₁ + 1

R_fC₁(−e1−e2)

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Figure 8. Phase traces of the Colpitts oscillator with the ﬂoating memristor. Parameters:L =0.314 mH,LS =0.157 mH, C1=C2=100 nF,C1S=C2S=100 nF,R0=10,R0S=100,C0=C0S=300 nF. (a) and (b) (including (c)) belong to the two individual Colpitts oscillators without coupling. (d) and (e) are the attractors of the two Colpitts oscillators when approximate synchronisation has been achieved and (f) shows the time evolution waveforms with approximate synchronisation betweenvcb(black) andvcbS (red) wherevcbandvcbS are not exactly equal but approximately equal.X scale: 1 V/Div,Y scale: 500 mV/Div in (a), (b), (d) and (e),Xscale: 20μs/Div,Y scale: 1 V/Div, in (c) and (f).

+e2

1

R_fC₂(−e1−e2)

+e2

− 1 RE EC2

e2+ 1 C2

e3− 1 R3C2

e2

+e3

−e1

L −e2

L − Re3

L −vM

L +vM S

L

= − 1 RfC1

e²₁− 1

RfC2 + 1

R3C2 + 1 RE EC2

×e²₂− R Le²₃ +

ics

C₁ − ic

C₁

e1+vM S

L −vM

L e3

+ 1

C1 − 1 L

e1e3

+ 1

C₂−1 L

e2e3−

1

R_fC₁+ 1 R_fC₂

e1e2.

(11) It can satisfyF˙(e)≤ −k1e²₁−k2e₂²−k3e²₃ <0, where k1,k2andk3are positive real numbers, and they can be represented as

k₁= 1

RfC1, k₂= 1

RfC2 + 1

R3C2 + 1 RE EC2, k3= R

L.

Therefore, limt→∞ e(t) → 0 can be achieved. The error system is approximately stable while the error is

Figure 9. Real circuitry in terms of the schematic diagram in ﬁgure 1.

converged into zero. Then, the two systems can be syn- chronised with each other.

The real circuitry in terms of the schematic diagram in figure 1 is made in figure 9. The physical experiment on the synchronisation of two floating memristor-based Colpitts oscillators was carried out as shown in figure 10, where figures 10a and 10b are the phase traces of the two memristor-based Colpitts oscillators without coupling, figure 10c is the oscilloscope trace in theXYmode ofvcb ∼ vcbS without coupling, and figure 10d is the oscilloscope trace in theXY mode ofvcb ∼ vcbS after approximate synchronisation, respectively.

Although the linear feedback control-based synchronisation technique depicted in §4.2–4.3 is very simple and of low cost, it has some intrinsic drawbacks such as

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Figure 10. Experimental results on theXY mode of the oscilloscope for the approximate synchronisation of two ﬂoating memristor-based Colpitts oscillators.L=0.314 mH,R0=10,LS=0.157 mH, R0S=100,Rf1=10,Rf2=10. (a) Phase trajectory of memristor-based Colpitts oscillator 1, (b) phase trajectory of memristor-based Colpitts oscillator 2, (c) the Lissajous ﬁgure ofvcb−vcbSwithout coupling and (d)vcb−vcbSafter approximate synchronisation (in the real circuit, the two transistors were chosen to be 2N2219A).

heavy dependence on the accuracy of the system model and parameters, not so good generality and so on. Actu- ally, in this physical experiment, another critical factor for the synchronisation is the coupling between the out- puts of two diode bridge circuits. Through repeated experiments, we further propose a more simple synchronisation scheme, which is called capacitor coupling synchronisation in this paper. In this scheme, a capacitor (1 μF) is added between the negative terminals ofC0 andC0S, and all the linear error feedback parts can be omitted. The synchronisation result is shown in ﬁgure 11.

In a sense, the control parameter selection is an open loop and adjusted by the trial-and-error method in synchronisation through both the aforementioned control schemes, namely, the linear feedback control scheme and the capacitor coupling scheme. So, it is difﬁcult to ﬁnd appropriate controller parameters. Therefore, the

Figure 11. Approximate synchronisation of two ﬂoating memristor-based Colpitts oscillators by using the capacitor coupling method.

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Figure 12. Time evolution of the state errors with the adaptive and nonlinear feedback control scheme;L =0.235 mH, LS =0.157 mH,C1=C2=100 nF,C1S=C2S=100 nF, R0=R0S=1000,C0=C0S=300 nF andRf =80.

third approach called the adaptive and nonlinear feedback control scheme is proposed, where the feedback strength ma is automatically adjusted by the adaptive control law in eq. (12) and the results shown in ﬁgure 12 conﬁrm the effectiveness of this scheme:

U =ma·(x2−x6), dm_a

dt = −p(x1−x5)²−p(x2−x6)²−p(x3−x7)², (12) where p is a constant, and the state variables satisfy x1 = vC₁ = Vce,x2 = vC₂ = Veb,x3 = iL, x5 = vC_1S =V_ceS,x₆ =vC_2S =V_ebS,x₃ =i_{L S}.

5. Conclusion and future work

In this paper, the approximate synchronisation of two ﬂoating memristor-based Colpitts oscillators is studied.

The mathematical and circuit models of the floating memristor-based Colpitts oscillator are built. Also, the real-time characteristics of the floating memristor applied to the Colpitts oscillating circuit are analysed and simulated and compared with the characteristics of the independent bridge memristor. Numerical simulations, circuit simulations and physical circuit experiments simultaneously verified that approximate synchronisation could be achieved through a single-state error feedback control. In addition, an improved scheme called the capacitor coupling scheme is proposed after considering the linear error feedback scheme’s own drawbacks. At last, an adaptive and nonlinear feedback

control method is applied to the memristor-based Col- pitts oscillators and synchronisation is obtained because of its very attractive advantages of the control parameter selection, especially for some systems with unknown parameters or with noises. An extension of multinode synchronisation based on memristors will be our next research subject to better ﬁt the synapses of neural networks.

Acknowledgement

The work was ﬁnancially supported by a grant from the National Natural Science Foundation of China under Grant No. 61174025.

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