Abstract In this paper, transport control and synchronization are investigated be-tween two periodically driven non-identical, inertial ratchets that are able to exhibitdirected transport. One of the two ratchets is acting as a drive system, while theother one represents the response system. Based on the Lyapunov stability theorem,the essential conditions, under which the error nonlinear system is transformed intoan equivalent passive system and globally asymptotically stabilized at equilibriumpoints, are established. With these results, synchronization, not only between twonon-identical ratchets with known parameters but also between two different uncer-tain ratchets, are realized via adaptive passive controllers and parameter update al-gorithm. The direction of transporting particles can be dominated along expectedone and it is useful to control the motion of tiny particles, ratchetlike devices innanoscience.
Keywords Inertial ratchets · Transport control · Chaos synchronization · Passivecontrol
Communicated by F.L. Chernousko.
P. Lu (�) · Q. WuSchool of Automation, Beijing Institute of Technology, Beijing 100081, Chinae-mail: [email protected]
Transport phenomena of nonlinear systems, extracting usable work from unbiasednonequilibrium fluctuations, have recently been proposed for a variety of applica-tions, including the separation of molecules or small particles [1, 2], as well as fortransport in biological systems, e.g., through ion channels in cell membranes [3].These so-called ratchets (or Brownian motors) can be modeled by a Brownian par-ticle undergoing random walk on a periodic asymmetric potential and being actedupon by an external time-dependent force of zero average. An interesting feature ofthe ratchet model is the possibility to use diffusion to convert an ac driving signal(either a multiplicative [4, 5] or an additive [6] fluctuating term) into a net dc cur-rent corresponding to the unidirectional motion of the particle through the system(ratchet effect). The ratchet effect has an extensive applications in rectifiers, pumps,particle separation devices, molecular switches, and transistors (see [7, 8] and the ref-erences therein). It is also of great interest in biology, since the working principles ofmolecular motors can be conveniently explained in terms of ratchet mechanisms [9].Moreover, it is possible to demonstrate quantum ratchet effects [10–12] by using coldatoms. The phenomenon of the ratchet effect can be observed both in overdamped de-terministic systems [13, 14] and in underdamped chaotic ones [15]. This was recentlystimulated some interest in the study of deterministic underdamped ratchets [16, 17].These ratchets possess a classical chaotic dynamics that modifies significantly thetransport properties [15, 18, 19].
Savel’ev [20] investigated the transport properties of interacting particles andproved that an repelling interaction among identical particles can result in the in-version of their net current. In this paper, it is proved that the reversed current isalso existent between two non-identical particles with different initial values. In ad-dition, to achieve the desired transports directed by one of these particles, a passivecontroller [21, 22], based on Lyapunov stability theorem, is added to the responseratchet to track the drive ratchet. Furthermore, the synchronized dynamics can alsobe realized between the two non-identical systems, namely, the drive ratchet and theresponse ratchet.
The study of synchronization in the interacting or coupled ratchets is certainlyof wide concern. In particular, Vincent et al. [23] studied the synchronized dynam-ics for unidirectionally coupled deterministic ratchets. Anticipated synchronizationis also considered for unidirectionally coupled ratchets with time-delayed feedbackin paper Kostur, but only numerical results are obtained. More recently, Vincent andLaoye [24] researched the chaos synchronization and control of chaotic ratchets viaactive control in detail. Most of the researches are based on the exactly knowing ofthe ratchet parameters. However, in many practical situation, the values of these pa-rameters are unknown, and the synchronization will be destroyed with the effects ofthese uncertainties. Therefore, the design of adaptive controller for the control andsynchronization of ratchets with unknown parameters is an important issue.
To the best of our knowledge, the synchronization in the non-identical inertialratchets with unknown parameters has not been addressed up to now. Accordingly,another central work, in this paper, is to design an adaptive control law to synchro-nize the two non-identical ratchets using by passive control. The essential conditions
890 J Optim Theory Appl (2013) 157:888–899
under which the error system is transformed into an equivalent passive system andglobally asymptotically stabilized at equilibrium points are established. With theseresults, synchronization between two different uncertain ratchets is realized via adap-tive passive controllers and parameter update algorithm. Simulation results verifiedthe applicability of the proposed approach and robust against the uncertainty in sys-tem parameters.
The rest of this paper is organized as follows: Section 2 presents a brief descriptionof the ratchets. Section 3 gives some basic characters of the passive system, while inSect. 4, we give we give the passive control law which guarantees the synchronizationof the drive and response non-identical inertial ratchets with known and unknownparameters, respectively. Simulation results are also studied in this section to verifyour approaches. Section 5 shows our conclusions of this paper.
2 The Model
Consider the motion of a particle driven by a periodic time-dependent external forcein a spatially periodic potential with an asymmetric profile. Here, the dynamics of theparticle is deterministic due to the absence of the noise. The differential equation ofmotion, which is also investigated in friction modeling areas [25], in dimensionlessvariables, is written as [16, 24, 26]
x + bx + dV (x)
dx= a cos(ωt), (1)
where b is the friction coefficient. V (x) is the asymmetric ratchet periodic potential,and a and ω are the amplitude and frequency of the driver, respectively. The dimen-sionless potential is given by
V (x) = C − 1
4π2δ
[sin 2π(x − x0) + 1
4sin 4π(x − x0)
], (2)
where the constants C and x0 are introduced in order to have the potential minimumV (0) = 0 (see Fig. 1) in x = 0. Accordingly, the constants
C = − 1
4π2δ
[sin 2πx0 + 1
4sin 4πx0
].
It is obvious that the period of the ratchet potential V (x) with regard to x is TV = 1which is also shown in Fig. 1. In this paper, δ � 1.600, and x0 = −0.19; see also thereferences [16, 24, 26].
The nonlinear dynamics (1) can be embedded into a corresponding two-dimension-al state space dynamics, which is
x1 = x2,
x2 = −bx2 − dV (x1)
dx1+ a cos(ωt).
(3)
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Fig. 1 The ratchet periodicpotential with parametersx0 = −0.19, δ � 1.600
3 Passivity Control
Consider the following differential equation:
x = f (x) + g(x)u,
y = h(x),(4)
where x ∈ Rn is the state variable, f (x) and g(x) are two smooth vector fields, h is
smooth mapping, and u ∈ Rm, y ∈ R
m are the input and output, respectively. With-out any loss of generality, it is supposed that the vector function f has at least oneequilibrium point.
Definition 3.1 [21] The system (4) is passive if the following two conditions aresatisfied.
(1) f (x) and g(x) exist and are smooth vector fields, and h(x) is also a smoothmapping.
(2) For any t ≥ 0, there is a real value β that satisfies the inequality
∫ t
0uT (τ)y(τ ) dτ ≥ β. (5)
The physical meaning of passive system is that the energy of the nonlinear systemcan be increased only through the supply from the external source. The followinglemma can be found in the literature [22].
Lemma 3.1 Suppose the nonlinear system (4) is passive with storage function V (x)
which is positive definite. Let ϕ be any smooth function, the control law u = −ϕ(y)
asymptotically stabilizes the equilibrium point of the nonlinear system (4).
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4 Transport Control and Synchronization
Let the ratchet system below
x1 = x2,
x2 = −bx2 − dV (x1)
dx1+ a cos(ω1t),
(6)
be the drive system, and the response system is
y1 = y2,
y2 = −by2 − dV (y1)
dy1+ a cos(ω2t) + u,
(7)
where u is the control function to be designed.Subtracting the system (6) from the system (7), the error dynamical system is
obtained between the drive system and the response one as follows:
e1 = e2,
e2 = −be2 − f (e1, y1) + a(cos(ω2t) − cos(ω1t)
) + u,(8)
where
e1 = y1 − x1,
e2 = y2 − x2,
f (e1, y1) = dV (y1)
dy1− dV (x1)
dx1.
First, transport control and synchronization problem is considered for the two ratchetswith known parameters.
Theorem 4.1 The error dynamical system (8) will be equivalent to a passive systemand asymptotically stabilized at equilibrium points through the controller as follows:
u = ke2 − e1 + f (e1, y1) − a[cos(ω2t) − cos(ω1t)
] + ν, (9)
where k < b is a real constant which can adjust the performance of adaptation algo-rithm, and ν is an external signal which is connected to the reference input.
Proof Choose the following Lyapunov function:
V (e1, e2) = 1
2e2
1 + 1
2e2
2.
Its time derivative along the solutions of system (8) is
V = e1e1 + e2e2
= e1e2 + e2[−be2 − f (e1, y1) + a
[cos(ω2t) − cos(ω1t)
] + u].
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Therefore, from the control input (9), we have
V = (k − b)e22 + νe2.
Since k < b, the following inequality is established:
V ≤ νe2. (10)
If e2 is taken as the output of error dynamical system (8), namely, ye(t) = e2(t), theinequality (10) is rewritten as
V ≤ νye. (11)
Assuming the initial condition of Lyapunov function V (e1, e2) is V (e10, e20) andtaking integration over both sides of (11), we obtain
V (e1, e2) − V (e10, e20) ≤∫ t
0ν(τ)ye(τ ) dτ.
For V (e1, e2) ≥ 0, the following inequality can be obtained:
∫ t
0ν(τ)ye(τ ) dτ ≥ β, (12)
where β = −V (e10, e20). The inequality (12) satisfied Definition 3.1, therefore, theerror nonlinear system (8) is equivalent to a passive system. From Lemma 3.1, thecontroller u asymptotically stabilizes the equilibrium point of the error system (8). �
Suppose that the equilibrium point of error system (8) is (e1m, e2m). Let e1 = 0,
e2 = 0, and we have e2m = 0, e1m = ν. If the external signal is elected as ν = 0,namely, e1m = 0, the following formulations can be guaranteed:
x1 = y1, x2 = y2,
which means that the response ratchet system (7) will synchronize the drive ratchetsystem (6) with controller (9).
In order to investigate the efficiency of Theorem 4.1, simulation results are shownin Figs. 2, 3 and 4 with parameters a = 0.156, b = 0.1,ω1 = 0.67,ω2 = 0.65 andinitial values
First, the controller u is switched off. It is exhibited from Fig. 2 that the chaotic at-tractor (solid) of drive system generates a positive current and the periodic attractor(dashed) of response system generates a negative one which is studied in [27]. Theparameters of the two ratchets presented in [27] are identical, however, here the pa-rameter ω and initial values are different. This result implies that we can reverse thedirection of the current by choosing the appropriate attractor. In addition, it is shownthat these attractors are mixed in a complex way. To achieve the desired transports
894 J Optim Theory Appl (2013) 157:888–899
Fig. 2 Trajectories for the driveand response ratchet withoutcontroller: (a) dynamics ofx1, y1, (b) the error e1 betweenx1 and y1
directed by these attractors, passive controller u is applied to the response ratchet invirtue of Theorem 4.1.
Comparatively, the controllers u is switched on at t = 100, and the transport prop-erties are denoted in Figs. 3 and 4 with k = −0.1 and k = −5, respectively. It isshown that the directed transports in the negative direction has been controlled totrack the direction of the positive one. It is also depicted from Figs. 3 and 4 thatdifferent transitional time is consumed with different k. In addition, a great deal ofsimulation results show that the smaller the control parameter k is, the shorter timerequired for transition and the synchronization is to be fast realized between the tworatchets. Accordingly, the efficient k can be used to adjust the performance of controlalgorithm. However, the parameter k cannot be decreased infinitely in view of theregulation in practical dynamical systems.
Then, assume that the parameters a, b are unknown constants. Adaptive passivecontroller will be designed in order to realize the synchronization between the drive
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Fig. 3 Trajectories for the driveand response ratchet incompletely synchronization withcontroller u and k = −0.1:(a) dynamics of x1, y1,(b) the error e1 between x1and y1
ratchet (6) and the response ratchet (7). Therefore, the following adaptive passivecontroller is established based on Lyapunov theory.
Theorem 4.2 The error dynamical system (8) will be equivalent to a passive systemand asymptotically stabilized at equilibrium points through the controller as follows:
u = k1e2 − e1 + f (e1, y1) − a(cos(ω2t) − cos(ω1t)
) + be2 + ν,
˙a = e2(cos(ω2t) − cos(ω1t)
),
˙b = −k2e22,
(13)
where k1 is a real negative constant which can adjust the performance of adaptationalgorithm, and ν is an external signal which is connected to the reference input. a, b
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Fig. 4 Trajectories for the driveand response ratchet incompletely synchronization withcontroller u and k = −5:(a) dynamics of x1, y1,(b) the error e1 between x1and y1
are the estimation values of undeterministic parameters a, b. ˙a and ˙b are adaptionalgorithms, and the efficient k2 is an arbitrary scalar.
Proof Choose the following Lyapunov function:
V (e1, e2) = 1
2e2
1 + 1
2e2
2 + 1
2(a − a)2 + 1
2k2(b − b)2.
Its time derivative along the solutions of system (8) is
V = e1e1 + e2e2
= e1e2 + e2[−be2 − f (e1, y1) + a
[cos(ω2t) − cos(ω1t)
] + u]
+ (a − a) ˙a + 1
k2(b − b) ˙b.
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Therefore, from the control input (13), we have
V = k1e22 + νe2.
Since k1 < 0, the following inequality is established:
V ≤ νe2. (14)
If e2 is taken as the output of error dynamical system (8), namely, ye(t) = e2(t), theinequality (14) is rewritten as
V ≤ νye. (15)
Assuming the initial condition of Lyapunov function V (e1, e2) is V (e10, e20) andtaking integration over both sides of (15), we obtain
V (e1, e2) − V (e10, e20) ≤∫ t
0ν(τ)ye(τ ) dτ.
For V (e1, e2) ≥ 0, the following inequality can be obtained:∫ t
0ν(τ)ye(τ ) dτ ≥ β, (16)
where β = −V (e10, e20). The inequality (16) satisfied Definition 3.1, therefore, theerror nonlinear system (8) is equivalent to a passive system. From Lemma 3.1,the controller (13) asymptotically stabilizes the equilibrium point of the error sys-tem (8). �
The efficient k1 plays the same role as k of Theorem 4.1 in the adaptive con-trol algorithm. Similarly, the external input v of adaptive controller u is selected aszero, and the synchronization will be realized between the drive ratchet (6) and theresponse one (7) with unknown parameters a and b.
Utilizing Theorem 4.2, the adaptive controllers (13) is switched on at t = 0, andthe transport properties are illustrated in Fig. 5 with the same initial state values asbefore and the initial estimated parameters
a0 = 0.12, b0 = 0.21.
Simulation results show that the adaptive passive controller (13) is effective and pos-sesses robust performance against the uncertainty in the parameters of ratchets.
5 Conclusions
In this paper, we theoretically studied the synchronization of two non-identical iner-tial ratchets by using passive control theory. The sufficient conditions, under whichthe stable synchronization can be obtained between not only the two non-identicaltransporting ratchets with known parameters but also between the two different un-certain transporting ratchet, have been established. Moreover, with these controllers,
898 J Optim Theory Appl (2013) 157:888–899
Fig. 5 Trajectories for the driveand response ratchet incompletely synchronization:(a) dynamics of x1, y1,(b) the error e1
the direction of particle transport can be dominant along the expected one. It is use-ful for chemists and biologists to note that it may well suggest an efficient controlmethod upon which the motion of tiny particles, ratchetlike devices in nanoscience,may be changed and optimized.
Acknowledgements This work is supported by the National Science Foundation of China under Grants60904003, 60874011.
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