**6.2. Feedback control for a model with cubic spring**

56 Nonlinearity, Bifurcation and Chaos – Theory and Applications

The system (18) can be represented as follows:

3 3

deviations as:

where

1 2

*e e*

*ae x ax*

2 2 2

*pe x px <sup>u</sup>*

 

3 3 11 1

Representing the system (24) in the form:

() ()

0

frequency equal to ( ), then:

3

*ay ay a y*

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

*x x* 

Replacing (19) in (18) and defining the deviation from the desired trajectory as:

*e*

2 2 1 11 1 22 88

with *<sup>u</sup> uF u* , and the feedback control: *u Ke* . The system (23) can be represented in

0 1 <sup>0</sup> () () <sup>1</sup>

88 33 11 1 11 1

Defining the desired trajectory as the periodic orbit, with amplitude less than (a) and

Considering the parameters values: 1; r 0.1 ; b 0.05 ; c 0.35 ; d 4/27 ; e 0.0001 ;

2 2 2

11 1 2 2

0

*<sup>g</sup> x gx u e e b r* 

3 3

*gx gx d d*

*e e pe x px ae x ax ae x ax*

 

 

*ddee e re be c e x cx*

1 1 2 2

*ce x x*

g 0.2 ; p 0.005 e a 1.6 the matrices A and B, are given by:

*e e*

2 8 <sup>3</sup> cos

*ae x ax ae x ax*

11 1 11 1

*e Ae g x g x Bu* () () (24)

(25)

*ae x ax*

11 1

*x t* 1.3sin( ) (26)

(22)

(21)

(23)

Considering the following parameters: 0.14668 , 0.17602 *b* , 2.6364 , 2.5 *z* , 1*a* 1 , <sup>2</sup>*a* 14.5 , and the control *U* in (16) results:

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= \mathbf{x}\_2\\ \dot{\mathbf{x}}\_2 &= -a a\_1 \mathbf{x}\_1 - \beta a\_2 \mathbf{x}\_1^3 - \frac{b}{\left(\mathbf{z} + \mathbf{x}\_1\right)^2} + \delta \sin b + \mathcal{U} \end{aligned} \tag{28}$$

where: *U uu* , *u* is the feedback control, *u* is the feedforward control, given by :

$$
\tilde{\mu} = \dot{\tilde{\mathbf{x}}}\_2 + aa\_1 \tilde{\mathbf{x}}\_1 + aa\_2 \tilde{\mathbf{x}}\_1^3 + \frac{b}{\left(\mathbf{z} + \tilde{\mathbf{x}}\_1\right)^2} - \delta \sin b \tag{29}
$$

Replacing (29) in (28) and defining the deviation from the desired trajectory as:

$$y = (\mathbf{x} - \tilde{\mathbf{x}}) \tag{30}$$

where *x* is the desired orbit, and rewriting the system in deviations, results:

$$\begin{aligned} \dot{y}\_1 &= y\_2\\ \dot{y}\_2 &= -aa\_1y\_1 - aa\_2\left(y\_1 + \tilde{\mathbf{x}}\_1\right)^3 + aa\_2\tilde{\mathbf{x}}\_1^{-3} - \frac{b}{\left(z + y\_1 + \tilde{\mathbf{x}}\_1\right)^2} + \frac{b}{\left(z + \tilde{\mathbf{x}}\_1\right)^2} + u \end{aligned} \tag{31}$$

Considering the system (31) in the following way:

$$
\dot{y} = Ay + g(\mathbf{x}) - g(\tilde{\mathbf{x}}) + Bu\tag{32}
$$

On an Overview of Nonlinear and Chaotic Behavior and

Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 59

**Figure 13.** (a): Tip displacement without control (b): Tip Displacement with control

**Figure 14.** Microcantilever oscillatory behavior. Source: (Bueno et al., 2011).

In the dynamic mode the microcantilever is deliberately vibrated. The Amplitude Modulated AFM and the Frequency Modulated AFM are the most important techniques. In both AM-AFM and FM-AFM the amplitude and frequency of the microcantilever are kept constant by two control loops. The AGC (Automatic Gain Control) and the ADC (Automatic Distance Control). The AGC controls the amplitude of oscillation and the ADC controls the frequency by adjusting the distance between tip and sample. The oscillatory behavior of the

(a) (b)

In the FM-AFM the control signal of the AGC loop is used to generate the dissipation images and the ADC control signal is used to generate the topographic images. The FM-AFM improved image resolution and for surface studies in vacuum is the preferred AFM technique (Morita et. al., 2009; Bhushan, 2004). From Figure 15 it can be seen that the PLL generates the feedback signal for both control loops, therefore the PLL performance is vital to the FM-AFM. The PLL is a closed loop control system that synchronizes a local oscillator to a sinusoidal input. The PLLs are composed of a phase detector (usually a multiplier circuit), of a low-pass filter and of a VCO (Bueno et. al., 2010; Bueno et al., 2011), as it can be seen in Figure 16, and additionally, shows the PM and AM outputs used in the AFM system.

microcantilever is illustrated in Figure 14. The FM-AFM block is shown in Figure 15.

$$\begin{aligned} \text{where: } y &= \begin{bmatrix} y\_1 \\ y\_2 \end{bmatrix}; A = \begin{bmatrix} 0 & 1 \\ -aa\_1 & 0 \end{bmatrix}; B = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \text{ and} \\\ \text{g(x)} &= \begin{bmatrix} 0 & 1 \\ -aa\_2\left(y\_1 + \tilde{x}\_1\right)^3 + aa\_2\tilde{x}\_1^3 - \frac{b}{\left(z + y\_1 + \tilde{x}\_1\right)^2} + \frac{b}{\left(z + \tilde{x}\_1\right)^2} \\\ \vdots \end{bmatrix}. \end{aligned} $$

solving the following equation:

$$
\mu = -R^{-1}B^T P \,\text{y} \tag{33}
$$

where *P* is a symmetric matrix, solution of the reduced Riccati equation:

$$PA + A^T P - PBR^{-1}B^T P + Q = 0\tag{34}$$

Defining the desired trajectory as:

$$
\tilde{\mathfrak{X}} = \mathfrak{Z}\cos(t) \tag{35}
$$

The matrices 0 1 0.14668 0 *<sup>A</sup>* , <sup>0</sup> 1 *<sup>B</sup>* , and defining Q and R as 250 0 0 20 *<sup>Q</sup>* , *R* 0.1 , results after using the matlab(R) to obtain *u*:

$$u = -49.8535y\_1 - 17.3120y\_2\tag{36}$$

In Figure 13 it can observed the tip displacement with and without control.
