**4. Chaos in mathematical model with cubic spring**

According to Ashhab (1999) the chaotic behavior of AFM depends on the damping of the excitation and on the distance between the tip and the sample, suggesting that a feedback control of the states can be used to eliminate the possibility of chaotic behavior. According to that, considering the system (7) in nondimensional form:

$$\begin{aligned} \dot{\boldsymbol{x}}\_1 &= \boldsymbol{x}\_2\\ \dot{\boldsymbol{x}}\_2 &= -a\boldsymbol{a}\_1 \boldsymbol{x}\_1 - a\boldsymbol{a}\_2 \boldsymbol{x}\_1^3 - \frac{b}{\left(\boldsymbol{z} + \boldsymbol{x}\_1\right)^2} + \delta \sin \boldsymbol{\tau} \end{aligned} \tag{16}$$

with the parameters: 0.14668 ; 0.17602 *b* , 2.6364 e 2.5 *z* , 1*a* 1 , 2*a* 14.5 , numerically simulation results can be seen Figure 9. Additionally, The FFT and the Lyapunov exponents ( <sup>1</sup> 0.336 , 2 0.336 ) are shown in Figure 10.

On an Overview of Nonlinear and Chaotic Behavior and

Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 55

**Figure 11.** (a): Phase diagram (b): Lyapunov exponents

A laser beam focus on the top of the microcantilever and the reflection is detected by a photodiode. The light is converted into an electrical signal, and stored in the computer as a reference. An oscillation of the microcantilever deflects the laser beam on the photodiode, allowing the system to compute the microcantilever motion. The error signals are then forwarded, and the piezoelectric scanner moves vertically to scan the sample, as shown in

(a) (b)

*ax ax a x*

*ay ay a y*

(18)

1 1 1

2 8 <sup>3</sup> cos

  (19)

 

*b sen* (20)

cos *<sup>u</sup>*

Figure 3. The control techniques are diverse: PID or PD, sliding mode, LQR, or other.

2 2 11 2 8 3 2 3

and defining a periodic orbit as a function of *x t* ( ) . The desired regime is given by:

*d e <sup>p</sup> x rx bx cx g x x F*

*d e <sup>p</sup> <sup>y</sup> ry by cy <sup>g</sup> y u*

Since *u* control the system in the desired trajectory, and *x t* ( ) is a solution of (19), without

01 1 2 <sup>2</sup> *y a a b sen a* cos( ) ( ) cos(2 ) (2 ) ...

**6.1. Feedback control for the model with hydrodynamic damping** 

Considering the model (17) with the inclusion of control: *uF* .

3

3

**6. Scanner position control** 

1 2

the term control *uF* , then 0 *u* , resulting:

The feedforward control *u* is given by:

*x x*

**Figure 9.** Phase diagram

**Figure 10.** (a): FFT, (b): Lyapunov exponent
