Synchronization of Two RCL Shunted JosephsonJunctions

Mohamed ZribiDepartment of Electrical EngineeringSchool of Engineering and Petroleum

Kuwait UniversityP.O. Box. 5969, Safat 13060, Kuwait

Email: mohamed.zribi@ku.edu.kw

Nabil KhachabDepartment of Electrical EngineeringSchool of Engineering and Petroleum

Kuwait UniversityP.O. Box. 5969, Safat 13060, Kuwait

Email: nabil.khachab@ku.edu.kw

Moneera BoufarsanDepartment of Electrical EngineeringSchool of Engineering and Petroleum

Kuwait UniversityP.O. Box. 5969, Safat 13060, Kuwait

Abstract—This paper deals with the synchronization of twoidentical RCL-shunted Josephson Junctions (RCLSJJs). TheRCLSJJ displays chaotic behavior for certain values of theparameters of the circuit. Two control schemes are proposedfor the synchronization of two identical RCLSJJs. The proofs ofthese controllers are based on the Lyapunov theory. The proposedcontrollers are validated through MATLAB simulation resultswhich indicate that the proposed schemes are very effective insynchronizing two RCLSJJs.

I. INTRODUCTION

The Josephson junction and its properties are of interestin many circuit and control applications as well as in thechaotic and Bifurcation behavior of these junctions. Thisinterest has increased due to the rapid advancement in themicroelectronic fabrication technology. These types of junc-tions have found applications in high frequency applications,such as satellite and space communications. They can alsobe used in microwave and devices and as voltage standards[1-2]. The electric properties of these junctions along withinductances have found their applications in superconductivedigital circuits [3]. These circuits operate with high speedsand low level power consumption. Other applications havebeen reported in chaos generation, secure communications,mixer systems, neural systems, information science, biologicalsystems and chemical reactions, Quantum computing, digitalsystems, millimeter wave oscillators [4-6]. Moreover applica-tions in high frequency sources and detectors are also reported.The dynamics of this junction under weak magnetic fieldis also finding applications. The rapid single-flux-quantum(RSFQ) circuit family is also finding applications. Thesecircuits depend on the transfer of a single magnetic fluxquantum across a Josephson junction [7]. Elementary cells inthis circuit family can generate, pass, memorize, and reproducepico-second voltage pulses with a nominally quantized areacorresponding to transfer of a single magnetic flux quantumacross a Josephson junction. Functionally, each cell can beviewed as a combination of a logic gate and an output latch(register) controlled by clock pulses, which are physicallysimilar to the signal pulses [3]. Recently, a superconductinglogic family whose operation relies on the availability of acurrent gain greater than one namely The Complementary

Josephson Junction (CJJ) logic. This logic family utilizes twotypes of non-latching devices: a conventional device and acomplementary device. This is similar to the CMOS logicfamily. The conventional device has a finite critical current,and the complementary device has zero critical current with noinput applied [8]. The area of A/D converters has also find itsapplications by using a series of arrays of shunted Josephsonjunctions to make a 14-b D/A converter. The reported circuit isa fast accurate dc reference, and it makes possible the digitalsynthesis of ac waveforms [9]. Josephson junctions can beutilized as voltage standards. This is based on an of arrayseries of pulse-biased, non-hysteretic Josephson junctions. Theoutput voltage can be rapidly and continuously programmedover a wide range by changing the pulse width. Two type ofvoltage standard systems have been reported. The first typeis a programmable Josephson voltage standard. The secondtype is the Josephson arbitrary waveform synthesizer. Thiswave form synthesizer uses perfectly quantized Josephsonpulses to generate arbitrary waveforms with low harmonicdistortion [10]. A New type of amplifier whose primarypurpose is the readout of superconducting quantum bits hasbeen realized by using a Josephson junction. It is based onthe transition of an rf-driven Josephson junction between twodistinct oscillation states near a dynamical bifurcation point[11]. A high TC Josephson junction can be operated as aharmonic mixer, if a source signal and a local oscillator signalare both applied at the junction. The study of the effects of thenonlinear inductance of huge Josephson junction arrays opennew opportunities for new, high-quality, RF circuit designs.A tunable high frequency band-pass filter using stacks ofJosephson junctions in LC resonators have been realized.The center frequency was tuned by changing the appliedmagnetic field [12]. In the process in these applications, manymodels have been used to study and characterize and representthe junction properties. These include the SNRCJ (shuntednonlinear resistive-capacitive junction. RCSJ shunted linearresistive-capacitive junction. The other model is the shuntedlinear resistive-capacitive-inductance junction model (RCLSJ).This model is used extensively and widely investigated. [1,3].The behavior of the junction under the an ac signal is ofinterest. This interest is translated in the effect of external

978-1-4577-2209-7/11/$26.00 ©2011 IEEE

sinusoidal forces on the dynamics of the junction. Thesedynamics that can be controlled the chaos in these junctionsand turning these chaotic oscillations to periodic ones .

The paper is organized as follows. The dynamic model ofthe RCL-shunted Josephson Junction is presented in section2. Section 3 presents the design of two Lyapunov based con-trollers to synchronize two identical RCLSJJs. The simulationsresults are presented and discussed in section 5. Finally, someconcluding remarks are given in section 5.

II. THE MODEL OF THE RCL-SHUNTED JOSEPHSON

JUNCTION

The normalized dynamic model of the RCL-shunted Joseph-son Junction (RCLSJJ) can be described by the following setof ordinary differential equations:

𝜃 = 𝑣

�̇� =1

𝛽𝐶(𝐼 − 𝑔(𝑣)𝑣 − 𝑠𝑖𝑛(𝜃)− 𝑖𝑠)

�̇�𝑠 =1

𝛽𝐿(𝑣 − 𝑖𝑠) (1)

The states of the system 𝜃, 𝑣 and 𝑖𝑠 are defined as follows.𝜃 : the phase difference of the superconductor pair.𝑣 : the normalized junction voltage,𝑖𝑠 : the normalized shunt current,𝐼 : the normalized external bias current,𝛽𝐶 : the capacitive parameter,𝛽𝐿 : the inductive parameter,𝑔(𝑣) : the nonlinear damping function defined such that,

𝑔(𝑣) =

{0.366 if ∣𝑣∣ > 2.90.061 if ∣𝑣∣ ≤ 2.9

(2)

Let 𝑥 = 𝜃, 𝑦 = 𝑣 and 𝑧 = 𝑖𝑠. The model of RCLSJJ canbe written as:

�̇� = 𝑦

�̇� =1

𝛽𝐶(𝐼 − 𝑔(𝑦)𝑦 − 𝑠𝑖𝑛(𝑥)− 𝑧)

�̇� =1

𝛽𝐿(𝑦 − 𝑧) (3)

The RCLSJJ displays chaotic behavior for certain values ofthe parameters of the circuit. Therefore, this paper deals withthe synchronization of two RCLSJJs. Hence, we define themaster RCLSJJ chaotic system as follows:

�̇�𝑚 = 𝑦𝑚

�̇�𝑚 =1

𝛽𝐶(𝐼𝑚 − 𝑔(𝑦𝑚)𝑦𝑚 − 𝑠𝑖𝑛(𝑥𝑚)− 𝑧𝑚)

�̇�𝑚 =1

𝛽𝐿(𝑦𝑚 − 𝑧𝑚) (4)

where 𝑥𝑚, 𝑦𝑚 and 𝑧𝑚 are the states of the master system.

The slave RCLSJJ chaotic system can be written as follows:

�̇�𝑠 = 𝑦𝑠

�̇�𝑠 =1

𝛽𝐶(𝐼𝑠 − 𝑔(𝑦𝑠)𝑦𝑠 − 𝑠𝑖𝑛(𝑥𝑠)− 𝑧𝑠) + 𝑢

�̇�𝑠 =1

𝛽𝐿(𝑦𝑠 − 𝑧𝑠) (5)

Note that the second ordinary differential equation (ODE)of the slave system contains the forcing term 𝑢. This termrepresents the controller of the system. This controller will bedesigned such that the master system and the slave system aresynchronized after starting from different initial conditions.Also, note that from an implementation point view, it is betterto minimize the number of inputs. Therefore, we assumed thatthe slave system given by equation (5) has a single input.

We define the errors between the states of the master andthe slave systems as follows:

𝑒1 = 𝑥𝑠 − 𝑥𝑚

𝑒2 = 𝑦𝑠 − 𝑦𝑚

𝑒3 = 𝑧𝑠 − 𝑧𝑚 (6)

The error vector 𝑒(𝑡) is such that:

𝑒(𝑡) =

⎛⎝ 𝑒1(𝑡)

𝑒2(𝑡)𝑒3(𝑡)

⎞⎠ (7)

Using equations (4) - (6), the error dynamics can be writtenas follows:

�̇�1 = 𝑒2 (8)

�̇�2 = 𝑓 + 𝑢 (9)

�̇�3 =1

𝛽𝐿(𝑒2 − 𝑒3) (10)

𝑓 =1

𝛽𝐶[−𝑒3 + 𝐼𝑠 − 𝐼𝑚 − 𝑔(𝑦𝑠)𝑦𝑠 + 𝑔(𝑦𝑚)𝑦𝑚

−𝑠𝑖𝑛(𝑥𝑠) + 𝑠𝑖𝑛(𝑥𝑚)]

The objective of this paper is to design a controller to forcethe errors to asymptotically converge to zero as 𝑡tends toinfinity. We will use Lyapunov based controllers to force theerrors to converge to zero as 𝑡 tends to infinity.

III. DESIGN OF THE CONTROLLERS

A. Design of the First Controller

Let 𝛼 be a positive scalar.Proposition 1: The control law:

𝑢 = −𝑓 − 𝑒1 − 𝛼𝑒2 − 1

𝛽𝐿𝑒3 (11)

with

𝑓 =1

𝛽𝐶[−𝑒3 + 𝐼𝑠 − 𝐼𝑚 − 𝑔(𝑦𝑠)𝑦𝑠 + 𝑔(𝑦𝑚)𝑦𝑚

−𝑠𝑖𝑛(𝑥𝑠) + 𝑠𝑖𝑛(𝑥𝑚)]

when applied to the error system (8)-(10) guarantees theconvergence of the errors 𝑒1, 𝑒2 and 𝑒3 to zero as 𝑡 tendsto infinity.Proof:

Let the Lyapunov function candidate 𝑉1 be such that:

𝑉1 =1

2𝑒21 +

1

2𝑒22 +

1

2𝑒23 (12)

Using the dynamic model of the errors in (8)-(10) and thecontrol law given by (11), the derivative of 𝑉1 with respect totime is such:

�̇�1 = 𝑒1�̇�1 + 𝑒2�̇�2 + 𝑒3�̇�3

= 𝑒1𝑒2 + 𝑒2(𝑓 + 𝑢) +1

𝛽𝐿(𝑒2 − 𝑒3)𝑒3

= 𝑒1𝑒2 + 𝑒2(−𝑒1 − 𝛼𝑒2 − 1

𝛽𝐿𝑒3) +

1

𝛽𝐿(𝑒2 − 𝑒3)𝑒3

= −𝛼𝑒22 −1

𝛽𝐿𝑒23 (13)

Clearly, 𝑉1 is a continuously differentiable positive definitefunction defined over 𝑅3 such that �̇�1 is negative semi-definitein 𝑅3.

Define the compact set Ω1 such that:

Ω1 = {𝑒(𝑡) ∈ 𝑅3∣�̇�1 = 0}. (14)

Solving the equation �̇�1 = 0 and using the fact that 𝛼 and𝛽𝐿 are positive, we get:

�̇�1 = 0 ⇒ −𝛼𝑒22 −1

𝛽𝐿𝑒23 = 0

⇒ 𝑒2(𝑡) = 0 and 𝑒3(𝑡) = 0 (15)

Note that,

𝑒2(𝑡) ≡ 0 ⇒ �̇�2(𝑡) = 0 (16)

𝑒3(𝑡) ≡ 0 ⇒ �̇�3(𝑡) = 0. (17)

Using (16), (9), (11) and (15), yield:

�̇�2(𝑡) = 0 ⇒ 𝑓 + 𝑢 = 0 ⇒ 𝑒1(𝑡) = 0. (18)

Therefore, the only solution of the equation �̇�1 = 0 is𝑒1(𝑡) = 0, 𝑒2(𝑡) = 0 and 𝑒3(𝑡) = 0. This means thatno solution can stay identically in Ω1, other than the trivialsolution (0, 0, 0)𝑇 . Hence, using LaSalle’s invariance theorem,it can be concluded that the origin is asymptotically stable. Inaddition, the Lyapunov function 𝑉1 is such that:

lim∣∣𝑒∣∣→0

𝑉1 = ∞ (19)

Thus 𝑉1 is radially unbounded and hence the origin is globallyasymptotically stable.

It can be concluded that the control law (11) when appliedto the error system (8)-(10) guarantees the convergence of theerrors 𝑒1, 𝑒2 and 𝑒3 to zero as 𝑡 tends to infinity.

It should be noted that since the errors 𝑒1, 𝑒2 and 𝑒3converge to zero as 𝑡 tends to infinity, then we are guaranteedthat 𝑥𝑠, 𝑦𝑠 and 𝑧𝑠 converge to 𝑥𝑚, 𝑦𝑚 and 𝑧𝑚 respectively as𝑡 tends to infinity. Therefore, the states of the master and theslave chaotic systems are synchronized.

B. Design of the Second Controller

Let 𝛼 and 𝑊 be positive scalars. Also, define the 𝑠𝑔𝑛function such that:

𝑠𝑔𝑛(𝑆) =

⎧⎨⎩

1 if 𝑆 > 00 if 𝑆 = 0−1 if 𝑆 < 0

Proposition 2: The control law:

𝑢 = −𝑓 − 𝑒1 − 𝛼𝑒2 − 1

𝛽𝐿𝑒3 −𝑊𝑠𝑔𝑛(𝑒2) (20)

with

𝑓 =1

𝛽𝐶[−𝑒3 + 𝐼𝑠 − 𝐼𝑚 − 𝑔(𝑦𝑠)𝑦𝑠 + 𝑔(𝑦𝑚)𝑦𝑚

−𝑠𝑖𝑛(𝑥𝑠) + 𝑠𝑖𝑛(𝑥𝑚)]

when applied to the error system (8)-(10) guarantees theconvergence of the errors 𝑒1, 𝑒2 and 𝑒3 to zero as 𝑡 tendsto infinity.Proof:

Let the Lyapunov function candidate 𝑉2 be such that:

𝑉2 =1

2𝑒21 +

1

2𝑒22 +

1

2𝑒23 (21)

Using the dynamic model of the errors in (8)-(10) and thecontrol law given by (20), the derivative of 𝑉2 with respect totime is such:

�̇�2 = 𝑒1�̇�1 + 𝑒2�̇�2 + 𝑒3�̇�3

= 𝑒1𝑒2 + 𝑒2(𝑓 + 𝑢) +1

𝛽𝐿(𝑒2 − 𝑒3)𝑒3

= −𝛼𝑒22 −1

𝛽𝐿𝑒23 −𝑊𝑒2𝑠𝑔𝑛(𝑒2)

= −𝛼𝑒22 −1

𝛽𝐿𝑒23 −𝑊 ∣𝑒2∣ (22)

Clearly, 𝑉2 is a continuously differentiable positive definitefunction defined over 𝑅3 such that �̇�2 is negative semi-definitein 𝑅3.

Define the compact set Ω2 such that:

Ω2 = {𝑒(𝑡) ∈ 𝑅3∣�̇�2 = 0}. (23)

Solving the equation �̇�2 = 0 and using the fact that 𝛼, 𝛽𝐿

and 𝑊 are positive, we get:

�̇�2 = 0 ⇒ 𝑒2(𝑡) = 0 and 𝑒3(𝑡) = 0 (24)

Noting that 𝑒2(𝑡) ≡ 0 implies that �̇�2(𝑡) = 0 and using (24),(9) and (20), yield:

�̇�2(𝑡) = 0 ⇒ 𝑓 + 𝑢 = 0 ⇒ ⇒ 𝑒1(𝑡) = 0. (25)

Therefore, the only solution of the equation �̇�2 = 0 is𝑒1(𝑡) = 0, 𝑒2(𝑡) = 0 and 𝑒3(𝑡) = 0. This means thatno solution can stay identically in Ω2, other than the trivialsolution (0, 0, 0)𝑇 . Hence, using LaSalle’s invariance theorem,it can be concluded that the origin is asymptotically stable. Inaddition, the Lyapunov function 𝑉2 is is radially unboundedand hence the origin is globally asymptotically stable.

It can be concluded that the control law (20) when appliedto the error system (8)-(10) guarantees the convergence of theerrors 𝑒1, 𝑒2 and 𝑒3 to zero as 𝑡 tends to infinity.

It should be noted that since the errors 𝑒1, 𝑒2 and 𝑒3converge to zero as 𝑡 tends to infinity, then we are guaranteedthat 𝑥𝑠, 𝑦𝑠 and 𝑧𝑠 converge to 𝑥𝑚, 𝑦𝑚 and 𝑧𝑚 respectively as𝑡 tends to infinity. Therefore, the states of the master and theslave chaotic systems are synchronized.

IV. SIMULATION RESULTS

The performances of the RCL-shunted Josephson Junctionis simulated using the MATLAB software. The parameters ofthe circuit are such that: 𝐵𝑐 = .707 and 𝐵𝑙 = 2.6. The RCLSJJdisplays chaotic behavior for certain values of the parametersof the circuit. If 𝐼 = 0.8, The system trajectory decays to theorigin (see Fig. 1); if 𝐼 = 1.01, the system’s trajectory displaysperiodic behavior (see Fig. 2). If 𝐼 = 1.12, the system’strajectory displays limit cycle (see Fig. 3). If 1.2 ≤ 𝐼 ≤ 1.25the system’s trajectory displays chaotic behavior (see Fig. 4-5)

Then, the proposed controllers given by equations (11) and(20) are used to synchronize the two identical RCL-shuntedJosephson Junctions whose equations are given by (4) and (5).The initial conditions of the master system (4) are taken to be𝑥𝑚(0) = 0, 𝑦𝑚(0) = 0 and 𝑧𝑚(0) = 0. On the other hand,the initial conditions of the slave system (5) are taken to be𝑥𝑠(0) = 2, 𝑦𝑠(0) = 2 and 𝑧𝑠(0) = 3. The parameter of thecontroller is chosen such that 𝛼 = 10.

The simulation results when controller (11) is used are pre-sented in Figures 6-8. The controller is applied at 𝑡 = 20 𝑠𝑒𝑐.The plots of the currents versus time are shown in Figures 6.The plots of the voltages versus time are shown in Figures 7.The plots of the errors versus time are shown in Figures 8. Itis clear that the simulation results show that the master andthe slave circuits are synchronized and hence the controller(11) works well.

Then, the second controller given by equation (20) is appliedto the master and slave systems given by (4) and (5). Theparameters of the controller is chosen such that 𝛼 = 10and 𝑊 = 10.The controller is applied at 𝑡 = 20 𝑠𝑒𝑐. Thesimulation results when controller (20) is used are presented inFigures 9-11. The plots of the currents versus time are shownin Figures 9. The plots of the voltages versus time are shownin Figures 10. The plots of the errors versus time are shownin Figures 11. It is clear that the simulation results show thatthe master and the slave circuits are synchronized and hencethe controller (20) works well.

V. CONCLUSION

Two Lyapunov based controllers are used to synchronizetwo identical RCL-shunted Josephson Junctions. It is shownthat both controllers guarantee the asymptotic convergenceof the errors between the master and slave circuits to zero.Simulation results are presented to show the effectiveness ofthe proposed controllers.

REFERENCES

[1] Feng, Y. L. and Shen, K., ”Controlling Chaos in RLC Shunted JosephsonJunction by delayed Linear Feedback,” Chinese Physics, vol. 17, no. 1,pp. 111-116, 2008.

[2] Benz, S. P. and Hamilton, C. A., ”A pulse-driven programmable Joseph-son voltage standard,” Applied physics letters, vol. 68, no. 22, pp. 3171-3173, 1996.

[3] Likharev, K. K. Semenov, V. K., ”RSFQ logic/memory family: a newJosephson-junction technology for sub-terahertz-clock-frequency digitalsystems,” IEEE Transactions on Applied Superconductivity, vol. 1, no.1. pp. 3-28, 1991.

[4] Guo, R., Vincent, U. E. and Idowu, B. A., ”Synchronization of Choasin RLC-Shunted Josephson Junction using a simple adaptive controller,”Physica Scripta, vol. 79, no 3, pp. 1-6, 2009.

[5] Yang, T. and Chua, L.O., ”Secure communication via chaotic parametermodulation,” IEEE Transactions on Circuits and Systems-I, vol. 44, no.5, pp. 469-472, 1997.

[6] Finger, L. and Tavsanoglu, V., ”Mapping of one-dimensional Josephsonjunction arrays onto cellular neural networks and their dynamics,”IEEE Transactions on Circuits and Systems I: Fundamental Theory andApplications, vol. 44, no. 5 pp. 438-445, 1997.

[7] Mukhanov, O.A., Rylov, S.V., Gaidarenko, D.V., Dubash, N.B. andBorzenets, V.V., ”Josephson output interfaces for RSFQ circuits,” IEEETransactions on Applied Superconductivity, vol. 7, no. 2, pp. 2826-2831,1997.

[8] Terzioglu, E., Beasley, M.R. and Edward L. Ginzton., ”ComplementaryJosephson junction devices and circuits: a possible new approach tosuperconducting electronics,” IEEE Transactions on Applied Supercon-ductivity, vol. 8, no. 2. pp. 48-53, 1998.

[9] Hamilton, C.A., Burroughs, C.J. and Kautz, R.L., ”Josephson D/Aconverter with fundamental accuracy,” IEEE Transactions on Instrumen-tation and Measurement, vol. 44, no. 2, pp. 223-225, 1995.

[10] Benz, S.P., Dresselhaus, P.D. and Burroughs, C. J., ”Nanotechnologyfor next generation Josephson voltage standards,” IEEE Transactions onInstrumentation and Measurement, vol. 50, no. 6, pp. 1513-1518, 2001.

[11] Benz, S.P., Hamilton, C.A. Burroughs, C.J. Harvey, T.E., Christian, L.A.and Przybysz, J.X., ”Pulse-driven Josephson digital/analog converter[voltage standard],” IEEE Transactions on Applied Superconductivity,vol. 8, no. 2, pp. 42-47, 1998.

[12] Siddiqi, I, Vijay, R, Pierre, F., Wilson, C. M., Metcalfe,M., Rigetti,C., Frunzio, L. and Devoret, M. H. ”RF-Driven Josephson BifurcationAmplifier for Quantum Measurement,” Physical Review Letters, vol. 93,no. 20, pp. 207002-1-4, 2004.

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

The normalized junction voltage

The

norm

aliz

ed s

hunt

cur

rent

Fig. 1. Plots 𝑖𝑠 versus 𝑣 when I=0.8

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

The normalized junction voltage

The

norm

aliz

ed s

hunt

cur

rent

Fig. 2. Plots 𝑖𝑠 versus 𝑣 when I=1.01

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

The normalized junction voltage

The

norm

aliz

ed s

hunt

cur

rent

Fig. 3. Plots 𝑖𝑠 versus 𝑣 when I=1.12

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

The normalized junction voltage

The

norm

aliz

ed s

hunt

cur

rent

Fig. 4. Plots 𝑖𝑠 versus 𝑣 when I=1.2

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

The normalized junction voltage

The

norm

aliz

ed s

hunt

cur

rent

Fig. 5. Plots 𝑖𝑠 versus 𝑣 when I=1.25

0 10 20 30 40 50 60 70 80−0.5

0

0.5

1

1.5

2

2.5

3

time

curre

nts

im

is

Fig. 6. Plots of currents versus time

0 10 20 30 40 50 60 70 80−3

−2

−1

0

1

2

3

4

time

volta

ges

vm

vs

Fig. 7. Plots of voltages versus time

0 10 20 30 40 50 60 70 80−6

−5

−4

−3

−2

−1

0

1

2

3

time

erro

rs

e1

e2

e3

Fig. 8. Plots of errors versus time

0 10 20 30 40 50 60 70 80−0.5

0

0.5

1

1.5

2

2.5

3

time

curre

nts

im

is

Fig. 9. Plots of currents versus time

0 10 20 30 40 50 60 70 80−3

−2

−1

0

1

2

3

4

time

volta

ges

vm

vs

Fig. 10. Plots of voltages versus time

0 10 20 30 40 50 60 70 80−6

−5

−4

−3

−2

−1

0

1

2

3

time

erro

rs

e1

e2

e3

Fig. 11. Plots of errors versus time