Nonlinear Analysis: Hybrid Systems 2 (2008) 993–1001 www.elsevier.com/locate/nahs
Chaos control in AFM systems using nonlinear delayed feedback via sliding mode control Mehdi Tabe Arjmand, Hoda Sadeghian, Hassan Salarieh, Aria Alasty ∗ Center of Excellence in Design, Robotics and Automation, Department of Mechanical Engineering, Sharif University of Technology, PO Box 11365-9567, Tehran, Iran Received 25 May 2007; accepted 11 October 2007
Abstract In this paper a nonlinear delayed feedback control is proposed to control chaos in an Atomic Force Microscope (AFM) system. The chaotic behavior of the system is suppressed by stabilizing one of its first-order Unstable Periodic Orbits (UPOs). At first, it is assumed that the system parameters are known, and a nonlinear delayed feedback control is designed to stabilize the UPO of the system. Then, in the presence of model parameter uncertainties, the proposed delayed feedback control law is modified via sliding mode scheme. The effectiveness of the presented methods is numerically investigated by stabilizing the unstable first-order periodic orbit of the AFM system. Simulation results show the high performance of the methods for chaos elimination in AFM systems. c 2007 Elsevier Ltd. All rights reserved.
Keywords: AFM; Chaos; Delayed feedback; Sliding mode
1. Introduction Atomic force microscopy (AFM) has been widely used for surface inspection with nanometer resolution in engineering applications and fundamental research since the time of its invention, 1986 [1]. As an imaging tool, the AFM is capable of resolving surface features at the atomic level for conducting and non-conducting samples. Currently, the AFM is used in many imaging applications ranging from biological systems to semiconductor manufacturing. The mechanism of AFM basically depends on the interaction of a microcantilever with surface forces. The tip of the microcantilever interacts with surface through a surface–tip interaction potential. One approach to measure the surface forces is to monitor the deflection of the microcantilever through a photodiode. This approach is named “contact mode”. Another approach termed “tapping mode”, is performed by vibrating the microcantilever close to its resonance frequency and monitoring the changes in its effective spring constant. In this method, the driving amplitude is set to a constant value and typical resonant frequencies are in the range from a few kilohertz to some megahertz [2]. A microcantilever in tapping mode may exhibit chaotic behavior under certain conditions. This matter has been experimentally observed in [3–5]. Theoretical studies, based on the techniques of Melnikov theory, have been ∗ Corresponding author. Tel.: +98 21 6616 5504; fax: +98 21 6600 0021.
E-mail address:
[email protected] (A. Alasty). c 2007 Elsevier Ltd. All rights reserved. 1751-570X/$ - see front matter doi:10.1016/j.nahs.2007.10.002
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performed in [1,6] to prove the existence of chaotic invariant sets and to determine the region in the space of physical parameters where chaotic motion is present. In addition, using nonlinear analysis techniques on attracting limit sets, the the presence of chaotic invariant sets is numerically verified in [2,7]. Typically, in theoretical and numerical works, the interaction between the sample and the microcantilever is modeled by a Lennard-Jones potential which consists of a short-range repulsive potential and a long-range Van der Waals attractive potential. By analyzing the dynamics of the microcantilever–sample system subjected to a harmonic force, it is concluded that chaos can exist in AFM depending on the damping value, excitation amplitude and frequency, and average tip–sample distance. The existence of chaotic behavior in AFM is highly undesirable for its performance since this type of complex irregular motion causes the AFM to give inaccurate measurements and low resolution of the achieved sample topography. Accordingly, it is always required to ensure good performance of the microscope and to eliminate the possibility of chaotic motion of the microcantilever either by changing the AFM operating conditions to a region of the parameter space where regular motion is assured [8,9] or by designing an active controller that stabilizes the system on one of its UPOs [10]. Chaos control via stabilizing a UPO has attracted a great deal of research interest in recent years. In this area, some techniques have been developed based on low energy control signals [11,12]. Also, Pyrages designed a controller based on the linear delayed feedback control [13] that has been extensively employed for stabilizing periodic solutions of nonlinear systems [14]. Other approaches like sliding mode, Fuzzy control and feedback linearization also have been used in diverse chaotic systems [15–17]. In this paper, we have developed a nonlinear delayed feedback control algorithm to stabilize the AFM microcantilever on its first-order UPO when specific parameters lead the system into a chaotic zone. This UPO is actually an unstable solution of the AFM system with a period equal to that of the exciting force on the system. Also a delayed feedback control via sliding mode scheme is proposed for chaos elimination in the presence of some uncertainties in the model parameters. The methods are used for stabilizing the first-order periodic orbits of a chaotic AFM to numerically investigate the effectiveness of the control algorithm. Simulation results show that the proposed techniques can be successfully implemented for chaos suppression in the AFM, even when the exact values of model parameters are not accessible. Moreover, the robustness of the designed controller is checked for possible noise in the measured states. 2. Problem statement The forced dynamical system describing tapping mode operation can be obtained based on the model suggested by Ashhab et al. in [1]. In their work, the microcantilever tip–sample interaction is modeled by the Lennard-Jones interaction potential. The system can be written in a non-dimensionalized form as [7]: ξ˙1 (t) = ξ2 (t) (1) ξ˙2 (t) = −δξ2 (t) − ξ1 (t) + f˜ cos(Ω t) + f IL (ξ1 (t)), where t is the non-dimensional time; ξ1 (t) and ξ2 (t) are the non-dimensional position and velocity of the microcantilever tip; f˜ and Ω are the amplitude and the frequency of the forcing term, respectively; δ is the damping factor; and f IL (ξ1 (t)) =
σ˜ 6 d 30 (α + ξ1 (t))
8
−
d (α + ξ1 (t))2
(2)
denotes the attraction/repulsion interaction force derived from Lennard-Jones interaction potential. Under certain conditions for the values of σ˜ , d, α, Ω , δ and f˜, the above described system exhibits chaotic behavior. The main goal is to stabilize the system (1) on one of its UPOs when chaotic motion emerges. Here, the unstable periodic orbit of the system with a period equal to that of the exciting force, i.e. 2π Ω , is chosen as the desired trajectory for the system to follow. To this effect, based on the assumption that both the state variables ξ1 (t) and ξ2 (t) are available, at first a nonlinear delayed feedback control is proposed to impose the chaotic system to follow a regular periodic trajectory. Then, a delayed feedback control via sliding mode scheme is proposed for the case that there exists some uncertainties in the values of parameters in the microcantilever tip–sample model, e.g. d, σ˜ and α. In this case,
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it is assumed that the uncertainties in the mentioned parameters are all bounded and the lower and upper bounds of the uncertainties for each parameter are known. Note that in practice the parameter α is not exactly known but its approximated value and its lower and upper limits can be easily obtained. 3. Chaos control of AFM In the standard tapping mode operation, the microcantilever is sinusoidally forced by a piezo actuator near its natural frequency, and periodically ‘taps’ the surface [10]. The implementation of nonlinear control techniques for suppressing chaotic dynamics in the AFM system will be achieved by superimposing a feedback control forcing signal onto the sinusoidal forcing signal both produced by the piezo. The new equations of motion including the dimensionless form of control force, u, can be written as: ξ˙1 (t) = ξ2 (t) (3) ξ˙2 (t) = −δξ2 (t) − ξ1 (t) + f˜ cos (Ω t) + f IL (ξ1 (t)) + u(t) or equivalently: ξ¨1 (t) = −δ ξ˙1 (t) − ξ1 (t) + f˜ cos (Ω t) + f IL (ξ1 (t)) + u(t).
(4)
A proper control method to eliminate chaos via stabilizing a UPO in the AFM system is nonlinear delayed feedback control. In the face of model parameter uncertainties, a modified version of this technique, named nonlinear delayed feedback control via sliding mode, may be applied to the system. These two methods are described in the following subsections. 3.1. Delayed feedback control via sliding mode An approach to control chaos used by numerous authors is the so-called delayed feedback control which was originally proposed by Pyragas. This method uses a linear combination of the error signals defined as: ei (t) = ξi (t) − ξi (t − T ),
(5)
where T is a delay time which can be equal to the period of a UPO of the system, and ξi (t) is a state variable. In [18] a nonlinear delayed feedback control has been presented which modifies the commonly used Pyragas’ method to achieve much better performance in chaos control. Here, at first a brief review of the method proposed in [18] is presented. Consider the delayed error in Eq. (5) as: e(t) = ξ1 (t) − ξ1 (t − T ),
(6)
where T = 2π Ω is the period of the exciting force in the AFM system of Eq. (4). Rewriting Eq. (4) for a T time units earlier yields: ξ¨1 (t − T ) = −δ ξ˙1 (t − T ) − ξ1 (t − T ) + f˜ cos (Ω t − Ω T ) + f IL (t − T ) + u(t − T ),
(7)
where f IL (t) is f IL (ξ1 (t)). Subtracting Eq. (7) from (5) and using the error state defined in (6) one may write, e(t) ¨ = −δ e(t) ˙ − e(t) − f IL (t − T ) − u(t − T ) + f IL (t) + u(t).
(8)
We need to stabilize the error signal and enforce it to converge asymptotically to zero. To do so, the controlling signal is proposed to be: u(t) = u(t − T ) + f IL (t − T ) − f IL (t) + δ e(t) ˙ + e(t) − [λ1 e(t) ˙ + λ2 e(t)].
(9)
Thus, Eq. (9) would be, e(t) ¨ + λ1 e(t) ˙ + λ2 e(t) = 0.
(10)
λ1 and λ2 are selected such that the above error dynamics is exponentially stable. So the system with the designed controller is asymptotically stable on its periodic solution (T = 2π Ω ).
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In practice some uncertainties may exist, especially on the α parameter, in Eq. (2). Here, we design a control to overcome possible uncertainties in the system model; hence the control algorithm can be practically applied in actual cases. For simplicity in formulations, we first define: χ , σ˜ 6 d.
(11)
The uncertainties are assumed to be on the values of d, χ and α as: d − d¯ < εd , |α − α| |χ − χ¯ | < εχ , ¯ < εα ,
(12)
where α, ¯ d¯ and χ¯ are the nominal known or approximated values of α, d and χ . Also εα , εd and εχ are the known upper bounds of the errors between the actual and nominal values of α, d and χ , respectively. Now define: ξ1 (t − T ) = ξ¯1
(13)
u(t − T ) = u. ¯ Adding and subtracting some equal terms, Eq. (8) is rewritten as: e¨ + δ e˙ + e = u − u¯ + f IL (ξ¯1 + e) − f¯IL (ξ¯1 + e) + f¯IL (ξ¯1 + e) − f IL (ξ¯1 ) + f¯IL (ξ¯1 ) − f¯IL (ξ¯1 ),
(14)
where, f¯IL (ξ¯1 ) =
χ¯ 30 α¯ + ξ¯1
8 −
d¯ α¯ + ξ¯1
2 .
(15)
One can write, f IL (ξ¯1 ) − f¯IL (ξ¯1 ) =
=
χ 30 α + ξ¯1
8 −
d α + ξ¯1
2 −
χ¯ 30 α¯ + ξ¯1
8 +
d¯ α¯ + ξ¯1
2
χ − χ¯ + χ¯ d¯ d − d¯ + d¯ χ¯ 8 − 2 − 8 + 2 . 30 α + ξ¯1 α + ξ¯1 30 α¯ + ξ¯1 α¯ + ξ¯1
Using Eq. (12), it is concluded that: f IL (ξ¯1 ) − f¯IL (ξ¯1 ) < N (α, ξ¯1 ),
(16)
(17)
where, 1
1 8 + εd 2 ¯ 30 α + ξ1 α + ξ¯1 1 1 1 1 ¯ + χ¯ − + d − . 8 2 2 30 α + ξ¯ 8 30 α¯ + ξ¯1 α + ξ¯1 α¯ + ξ¯1 1
N (α, ξ¯1 ) = εχ
(18)
Now considering the variation of α according to Eq. (12), it is obvious that f IL (ξ¯1 ) − f¯IL (ξ¯1 ) < max(N (α¯ − εα , ξ¯1 ), N (α¯ + εα , ξ¯1 )) , M(ξ¯1 ).
(19)
Similarly it can be obtained that f IL (ξ¯1 + e) − f¯I L (ξ¯1 + e) < max(N (α¯ − εα , ξ¯1 + e), N (α¯ + εα , ξ¯1 + e)) = M(ξ¯1 + e).
(20)
Therefore defining, η(εα , εd , εχ , ξ¯1 , e) , f IL (ξ¯1 + e) + f¯IL (ξ¯1 + e) − f IL (ξ¯1 ) − f¯IL (ξ¯1 )
(21)
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and using the triangular inequality, it is concluded that η(εα , εd , εχ , ξ¯1 , e) ≤ M(ξ¯1 + e) + M(ξ¯1 ).
997
(22)
So one can rewrite Eq. (14) as: e¨ + δ e˙ + e = u − u¯ + f¯IL (ξ¯1 + e) − f¯IL (ξ¯1 ) + η(εα , εd , εχ , ξ¯1 , e),
(23)
where η has an uncertainty with the bound introduced in Eq. (22). Now let u − u¯ = v − f¯IL (ξ¯1 + e) + f¯IL (ξ¯1 ),
(24)
where v is a part of controlling signal which will be obtained during this section. Substituting Eq. (24) into Eq. (23) yields: e¨ + δ e˙ + e = v + η(εα , εd , εχ , ξ¯1 , e).
(25)
To design a control law which asymptotically stabilizes the origin of the above error dynamics, we use the sliding mode method by defining a sliding surface as, S = e˙ + λe,
(26)
where λ is a positive constant. A Lyapunov function is defined as: 1 2 S . 2 The derivative of V along the error trajectories is: V =
(27)
V˙ = S S˙ = S(e¨ + λe) ˙ = S(−δ e˙ − e + v + η + λe). ˙
(28)
Set v as: v = δ e˙ + e − λe˙ − K (t)sign(S),
(29)
where K (t) must satisfy the condition of K (t) ≥ M(ξ¯1 + e) + M(ξ¯1 ) + θ
(30)
and θ is a positive constant. Substituting Eqs. (29) and (30) into Eq. (28) results in: V˙ = S(η − K (t)sgn(S)) < −θ |S| .
(31)
It is concluded that the sliding surface S = 0 is attracted by the error trajectories in a finite time, hence the tracking error asymptotically converges to zero. Thus considering Eq. (29) the control law can be summarized as: u(t) = u(t − T ) − f¯IL (ξ¯1 + e) + f¯IL (ξ¯1 ) + δ e˙ + e − λe˙ − K (t)sign(S).
(32)
3.2. Control implementation remarks Although the controlled system on the sliding surface describes a stable system, the value of corresponding u in Eq. (32) may vary in a wide range due to large value of nonlinear attraction/repulsion force. As the control signal cannot exceed the limited force range of the piezo, we need a bounded control signal when the proposed control method is experimentally implemented. Suppose that the control signal is limited to be in the range of [−u, ˆ u]. ˆ Therefore Eq. (32) is rewritten as: u 0 (t) = u(t − T ) + fˆIL (t − T ) − fˆIL (t) + δ e(t) ˙ + e(t) − [λ1 e(t) ˙ + λ2 e(t)] u(t) = u 0 (t) |u 0 (t)| < uˆ u(t) = uˆ · sign(u 0 (t)) |u 0 (t)| > u. ˆ
(33)
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Fig. 1. Chaotic attractor of the AFM (a) in the phase space, (b) in the Poincare section (10 000 points).
Moreover, when the error signal reaches a desired magnitude, i.e. the UPO of the system is being stabilized, the saturation bounds can be decreased because when the UPO of the uncontrolled system is achieved, the control should be equal to zero. For example one may use an exponential function for this purpose: u(t) ˆ = as ebs (t−tˆ) .
(34)
Here as and bs are two arbitrary constants, and tˆ is the time when the error signal is sufficiently close to zero. Decreasing the saturation bound guaranties that the stabilized periodic solution is a UPO of the uncontrolled system. It must be noted that in recursive control techniques such as the method proposed in this paper and the one presented in [18], the stabilized periodic solution may not be the UPO of the uncontrolled system. The UPO of the uncontrolled system is obtained when u(t) converges to zero, so by decreasing the saturation limit this condition is satisfied, nevertheless setting any saturation bound may have undesired effects on the stabilization property of the controller. The saturation bound is chosen arbitrarily since it depends on the real-world condition not the controller algorithm. There are some other methods which guaranty that the stabilized periodic orbit is the UPO of the uncontrolled system. One of them is to apply the controller when the delayed error becomes smaller than a sufficiently small desired value. This condition will be satisfied because chaotic systems have the recurrence or ergodic property. However all of these techniques may result in decreasing the controller performance. 4. Simulation results The above described control scheme is now used to control the states of a chaotic AFM system. The aforementioned AFM system in Eq. (1) exhibits chaotic behavior with the following parameters [2]: 4 . (35) 27 For these parameters, the phase plane diagram of the system chaotic attractor is shown in Fig. 1(a). Also this attractor, in the Poincare section of t = nT (T = 2π Ω ), is illustrated in Fig. 1(b). Now the objective is to eliminate the chaotic behavior by stabilizing the system on its one cycle UPO, i.e. with the period of T = 2π Ω , using the delayed feedback control scheme described in Section 3. The simulation consists of two parts. First in the absence of uncertainties, the described controller in [18] is implemented. Then, by considering some uncertainties in the model parameters, the proposed delayed feedback via sliding mode is applied to the system. In both the methods the controller signal is equal to zero for the first T time units. For time greater than T , the controller acts in a constant saturation limit until the stability of the periodic orbit is assured. This matter can be checked by observing the position error, e1 (t). The constant saturation bound in the simulation is set to 0.1. When the position error becomes less than a desired magnitude, that we have set to 0.1, the exponential saturation in Eq. (34) with as = 0.1 and bs = −0.1 would be the saturation limit of the control signal. This guaranties that the stabilized periodic orbit is a UPO of the uncontrolled AFM. δ = 0.04,
σ˜ = 0.3,
α = 0.8,
f˜ = 2.0,
Ω = 1,
d=
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Fig. 2. Simulation results for the nonlinear delayed feedback control (a) ξ1 time series (b) u time series (c) Stabilized UPO.
The controller in delayed nonlinear feedback control follows the pattern defined in Eq. (9), with λ1 = 3 and λ2 = 2. Fig. 2 shows the results for this case. In the presence of uncertainties in the system, the sliding mode delayed feedback is used to control the chaotic motion of the system. Regarding Eq. (9), the following uncertainties are considered in the simulation: εα = 0.1α¯
εd = 0.1d¯
εχ = 0.1χ¯ ,
(36)
4 where α¯ = 0.3, d¯ = 27 and χ¯ = 0.000108. Also, in Eq. (36) λ and θ are set to 1. Fig. 3 represents the results of the sliding mode version of the controller. Also, in this case, the robustness of the designed controller against the output noise is checked for 1% noise added to the measured values of states. The results are shown in Fig. 4. As can be seen from the simulation results, the stability of unstable periodic orbits is completely achieved in less than 100 non-dimensional time units, which is in the order of 0.01 s in a real AFM system. The amplitude of the control signal is sufficiently small and converges to zero in a finite small duration of time, and the system is strongly robust to disturbances and measurement noise. To show the ability of the controller to stabilize periodic solutions in the absence of saturation, an extra simulation is done by setting the bound of saturation to infinity. The results are shown in Fig. 5. It is observed that the rate of convergence towards the periodic orbit is higher than the previous examples. Besides the control action does not vanish, because the stabilized orbit is nearly different from the main UPO of the uncontrolled system.
5. Conclusion In this paper, a new method is developed to eliminate chaos in AFM system based on stabilizing a UPO of the system in the presence of model uncertainties. The nonlinear delayed feedback control is developed and used to control
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Fig. 3. Simulation results for the sliding mode control (a) ξ1 time series (b) u time series (c) Stabilized UPO.
Fig. 4. Simulation results for the nonlinear delayed feedback control via sliding mode with a 1% noise in feedback states (a) ξ1 time series (b) u time series.
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Fig. 5. Simulation results for the nonlinear delayed feedback control via sliding mode with unsaturated control action (a) ξ1 time series (b) u time series.
the chaotic AFM system when it is assumed that the exact values of model parameters are available. The modified version, the nonlinear delayed feedback via sliding mode, is applied to the system by considering some uncertainties in the parameters. Also the robustness of the system is checked for possible measurement noise in the feedback states. Both the techniques are successfully implemented to stabilize the period-1 UPO of the AFM system. Simulation results confirm the validity of the designed controllers. In both the methods, the system is properly stabilized on a periodic trajectory in a reasonable small time and control action. References [1] M. Ashhab, M.V. Salapaka, M. Dahleh, I. Mezic, Dynamical analysis and control of micro-cantilevers, Automatica 35 (1999) 1663–1670. [2] M. Basso, L. Giarrk, M. Dahleh, I. Mezic, Numerical analysis of complex dynamics in atomic force microscopes, Proc. IEEE Int. Conf. Control Appl. (1998). [3] N.A. Burnham, A.J. Kulik, G. Gremaud, G.A.D. Briggs, Nanosubharmonics: The dynamics of small nonlinear contacts, Phys. Rev. Lett. 74 (1995) 5092–5059. [4] F. Jamitzky, M. Stark, W. Bunk, W.M. Heckl, R.W. Stark, Chaos in dynamic atomic force microscopy, Nanotechnology 17 (2006) 213–220. [5] Daniel S. Burgess, Chaos affects atomic force microscopes, Photon. Spectra 40 (4) (2006) 101–102. [6] M. Ashhab, M.V. Salapaka, M. Dahleh, I. Mezic, Melnikov-based dynamical analysis of microcantilevers in scanning probe microscopy, Nonlinear Dynam. 20 (3) (1999) 197–220. [7] M. Basso, L. Giarrk, M. Dahleh, I. Mezic, Complex dynamics in a harmonically excited Lennard-Jones Oscillator: Micro-cantilever-sample interaction in scanning probe microscopes, J. Dyn. Syst. Meas. Control 122 (2000) 240–245. [8] M. Ashhab, M. Salapaka, Rail Dahleh, I. Mezic, Control of Chaos in Atomic Force Microscopes, ACC, Albuquerque, New Mexico, 1997. [9] M. Basso, G. Bagni, Controller synthesis for stabilizing oscillations in tapping-mode atomic force microscopes, IEEE Inter. Symp. Comput. Aided Control Sys. Design (2004) 372–377. [10] M. Stonier, Dynamical analysis and control of nonlinear instabilities in nano-devices, Thesis for BE, The University of Queensland, 2002. [11] E. Ott, C. Grebogi, J.A. York, Controlling chaos, Phys. Rev. Lett. 64 (1990) 1196–1199. [12] T. Shinbort, E. Ott, C. Grebogi, J. Yorke, Using chaos to direct trajectories to targets, Phys. Rev. Lett. 65 (1990) 3215–3218. [13] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A. 170 (1992) 421–428. [14] G. Chen, X. Yu, On time delayed feedback control of chaotic systems, IEEE Trans. Circuits Syst. 1 46 (1999) 767–772. [15] H. Layeghi, M.T. Arjmand, H. Salarieh, A. Alasty, Stabilizing periodic orbits of chaotic systems using fuzzy adaptive sliding mode control, Chaos Solitons Fractals (2006), doi:10.1016/j.chaos.2006.10.021. [16] M. Bonakdar, M. Samadi, H. Salarieh, A. Alasty, Stabilizing unstable periodic orbits of chaotic systems using fuzzy control of Poincare map, Chaos Solitons Fractals (2006), doi:10.1016/j.chaos.2006.06.081. [17] H. Salarieh, M. Shahrokhi, Indirect adaptive control of discrete chaotic systems, Chaos Solitons Fractals 34 (2007) 1188–1201. [18] J. Zhu, Y.-P. Tiang, Stabilizing periodic solutions of nonlinear systems and applications in chaos control, IEEE Trans. Circuits Syst. II 52 (12) (2005) 870–874.
