MEMS-Based Atomic Force Microscope: Nonlinear Dynamics Analysis and Its Control

Abstract

In this chapter, we explore a mathematical modelling that describes the nonlinear dynamic behavior of atomic force microscopy (AFM). We propose two control techniques for suppressing the chaotic motion of the system. The proposed model considers the interatomic interactions between the analyzed sample and the cantilever. These acting forces are van der Waals type, and we add a mathematical term that is a simple approximation to the viscoelasticity that possibly occurs in biological samples. We analyzed the behavior of the initial conditions of the proposed mathematical model, which showed a degree of complexity of the basins of attraction that were detected by entropy and uncertainty parameter, both detect if the basins have a fractal behavior. Numerical results showed that the nonlinear dynamic behavior has chaotic regions with the Lyapunov exponent, bifurcation diagram, and the Poincaré map. And, we propose two control techniques to suppress the chaotic movement of the AFM cantilever. First technique is the optimal linear feedback control (OLFC), which does not consider the nonlinearities of mathematical model. On the other hand, the control state dependent Riccati equation (SDRE) considers the nonlinearities of mathematical model. Both control techniques for a desired periodic orbit proved to be efficient.

1. Introduction

It is well known that many practical electromechanical devices can be modeled by a coupled equation; they can be understood in the context of simple lumped mechanical masses and electric and magnetic circuits. Electromechanical systems fall into three groups: Conventional electromechanical systems (MACRO), microelectromechanical systems (MEMS), and nanoelectromechanical systems (NEMS) [1].

Many of the “NEMS” device technologies use “MEMS” as a bridge to the nanoworld. “MEMS” will provide a bridge to enable applications of nanotechnology, as illustrated in the cantilever sensor. As an example, the atomic force microscope (AFM) used in this technique has a tip made of silicon, using a typical MEMS device process, and is of micrometer dimensions. Cantilever sensors recognize resonant frequency shifts with the addition of mass, indicating the presence or absence of specific compounds in the environment tested. First, “MEMS” technology is based on multidisciplinary foundations. Designing a commercially viable microsystem with required dynamic performance requires an in-depth understanding and accurate prediction of its dynamic characteristics. Furthermore, the macro-properties of materials may change as the size of the feature of mechanical elements is reduced, leading to difficulties in modeling their dynamic characteristics [2].

We remark in an AFM that a microscale cantilever with a sharp tip is used to scan the specimen surface, and the vibration of the cantilever is measured to identify the distance between the tip and the specimen surface. The AFM is composed of an elastic cantilever, and the achievable sensitivity and resolution of the AFM are largely dependent on the geometry of the cantilever. Currently, AFM is one of the most effective imaging techniques that is being used at the nanoscale and sub-nanoscale levels. This technique has been applied to multiple problems in the field of natural sciences and can record a range of surface properties of materials in both liquid media and air. Nowadays, AFM includes a wide variety of methods in which the probe interacts with the sample in different ways to characterize various material properties. AFM can characterize a wide array of mechanical properties (e.g., adhesion, stiffness, friction, and dissipation), electrical properties (e.g., capacitance, electrostatic forces, work function, and electrical current), magnetic properties, and optical spectroscopic properties. In addition to imaging, the AFM probe can be used to manipulate, write, or even pull-on substrates in lithography and molecular pull experiments.

A general overview [3] was written concerning nonlinear and chaotic behavior and their controls of an atomic force microscopy (AFM) vibrating problem, which has been dedicated to tapping mode operation, considered the presence of hydrodynamic damping, based on papers of the research group in Brazil. We also discussed an AFM mathematical modeling with phase-locked loops (PLLs), inspired by Ref. [4, 5].

The MEMS-based atomic force microscope (AFM) device consists of a microcantilever beam with a tip that interacts with the surface of a sample. The sample surface topography causes vibrations in the microcantilever beam; the microcantilever reflects a laser beam that is captured by a photodiode. The laser beam deflection is used to generate the topographic images of the sample. The common operation modes of AFMs are noncontact, contact, intermittent, and trolling modes [3, 6, 7] and underserved of others. The atomic force microscope (AFM) was first presented in Ref. [8].

The field of scanning probe microscopy (SPM) began in the early 1980s with the invention of the scanning tunneling microscope (STM) by Gerd Binnig and Heinrich Rohrer, which was awarded the Nobel Prize in Physics in 1986.

Many authors developed theories on this topic. We remarked that mathematical models in (AFM) are used to analyze the behavior of strongly nonlinear dynamics, to determine the presence of irregular displacements that are caused by the interaction forces between the microcantilever tip and the atoms of the sample surface. Another contributing factor to irregular displacement in the microcantilever is the viscoelastic phenomenon. The control design are applied to suppress the irregular (chaotic) displacement of microcantilever the AFM to keep the motion regular (periodic). AFM is an essential technique for the study of surfaces and their interactions with atomic resolution, showing the strong influence of fractional nonlinear dynamics.

It is well known that the nonlinear dynamics of atomic force microscopy (AFM) is an emerging topic of research and is a widely used tool for atomic-level surface analysis. In addition, nanodevices has become an important equipment in the industry for developing nanotechnology, being used to develop substances, compounds, artifacts, and nano chips. In fact, (AFM) is a force sensor. When the surface under investigation attracts or repels the tip, the cantilever bends to or from the surface.

In this unit chapter paper, we will mention an example of a cantilever modeled as being a single spring-mass-damper system, and a nonlinear dynamic model is developed to study the cantilever-sample interaction by using the L. J. potential including the long-range attractive forces and short-range repulsive forces. A comprehensive investigation of the nonlinear dynamics and chaos is carried out based on this model, including samples related to damping; it is possible to see that the behavior is not only viscous but viscoelastic and will be discussed, using bifurcation diagrams, phase portraits, Poincare maps, and Lyapunov exponents. An active control strategy has also been proposed by us to be effective in suppressing micro-oscillations, although these methods require significant use of control resources.

Note that active methods also require sensors capable of constantly providing accurate measurements for feedback to the controller. Here, the analysis is performed considering the effects of an optimal active linear control and time-delay control.

Through computer simulations, the efficiency and robustness to parametric errors of each control technique are verified. The results obtained were in complete agreement with earlier theories and experiments. This bending is then measured by position-sensitive photodiodes via the displacement of the laser beam reflected by the back of the cantilever.

In summary, this chapter presents a short review of AFM applications of nonlinear dynamics and control. This is organized as follows. In Section 2, we presented a state of the art to give the position of the problem in the current literature. In Section 3, we exhibit the mathematical model used. In Section 4, we discuss its nonlinear dynamic behavior. In Section 5, we develop a control design to control its chaotic behavior. Finally, we present the concluding remarks and give some acknowledgments. And list the main references used.

2. Atomic force microscopy: a state of the art

A typical (AFM) system consists of a micro-machined cantilever probe and a sharp tip mounted to a piezoelectric (PZT) actuator and a position-sensitive photodetector for receiving a laser beam reflected off the endpoint of the beam to provide cantilever deflection feedback. The fundamental principle of the operation of the AFM is the measurement of the deflections of a support at the free end on which the probe is mounted. These deflections are caused by the forces acting between the probe and the sample. The effects of a variety of forces acting between the tip sample can be analyzed during the scan, as shown in Figure 1a (an AFM control block diagram). The diagram shows a scanned sample design, where the tip and cantilever are fixed, and the sample is moved under the tip by the piezo actuator. In this mode, the controller attempts to maintain a constant level of deflection, which corresponds to a constant level of contact force. The quantity to be measured, the surface profile, comes as an unknown disturbance to the control loop. The deflection of the cantilever is detected by optical detection. Figure 1b simplify the AFM schematic.

Figure 1.

(a) An AFM control block diagram and (b) AFM schematics. Adapted to [3, 4, 5, 9, 10].

Note that depending on the tip’s interaction with the specimen surface, an AFM can work various imaging modes available, such as contact, noncontact, and intermittent-contact modes, tapping (where the tip oscillates and touches the surface occasionally), trolling mode (where the analysis tip is replaced by a nanoneedle that is inserted into aqueous media, the analysis splint is used in biological samples), and others. Some examples, undeserved of many others, have studied the cantilever of atomic force microscopy based on its nonlinear dynamics, listed next [3, 4, 5, 6, 8, 9, 11, 12, 13, 14].

The AFM microcantilever suffers from severe sensitivity degradation and noise intensification while operating in liquid; the large hydrodynamic drag between the cantilever and the surrounding liquid overwhelms the tip-sample interaction forces that are important in controlling the process. Therefore, [9] study the dynamic modeling of the manipulation process in trolling-mode AFM. The role of local and global dynamics to assess system robustness and actual safety in operating conditions is investigated, by also studying the effect of different local and global control techniques on the nonlinear behavior of a noncontact AFM. First, the nonlinear dynamical behavior of a single-mode noncontact AFM model is analyzed in terms of stability of the main periodic solutions, as well as the robustness of the attractors and the integrity of the basins [10]. The focus of the paper by Ref. [12] was on the investigation of local and global bifurcations in a continuum mechanics-based resonator model proposed for the measurement of electron spin by magnetic resonance force microscopy (MRFM).

Tapping mode AFM is one of the most potent techniques for topographic imaging of substrates. The cantilever is oscillated vertically near its resonance frequency so that the tip contacts the sample surface only briefly in each cycle of oscillation. Because of the short intermittent contact, it greatly reduces irreversible destruction of the sample surfaces, so it has been widely used for the study of soft materials, such as polymers and biological samples. When the tip is brought close to the sample surface, the vibrational characteristics of the cantilever vibration change due to the tip-sample interaction. In the imaging method, the cantilever is usually driven at the resonance frequency of the free cantilever with the driving amplitude. In Ref. [15] the authors showed how machine learning and data-driven approaches could be used to capture intermodal coupling. We employ a quasi-recurrent neural network (QRNN) for identifying mode coupling and verifying its applicability on experimental data obtained from tapping mode atomic force microscopy (AFM). The QRNN is an approach that adds convolutions to recurrence and recurrence to convolutions in the layers of the neural network to determine patterns in the system’s experimental data (AFM). For details on QRNN see Ref. [16].

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 ensured or by designing an active controller that stabilizes the system on one of its unstable periodic orbits.

In the paper by Ref. [15], the authors investigate the mechanism of atomic force microscopy in tapping mode (AFM-TM) under the Casimir and van der Waals (VdW) force; 0–1 test was implemented to analyze the dynamics of the system, allowing the identification of the chaotic and periodic regimes of the AFM system. The dynamic results of the conventional derivative and fractional models reveal the need for the application of control techniques, such as Optimum Linear Feedback Control (OLFC), state-dependent Riccati equations (SDRE) by using feedback control, and the time-delayed feedback control. The results of the control techniques are efficient with and without the fractional-order derivative.

Ref. [16] also investigated the nonlinear dynamic model of the atomic force microscopy model (AFM) with the influence of a viscoelastic term. For the analysis of the system, we used the classic tooling of nonlinear dynamics (bifurcation diagram, 0–1 test, Poincaré maps, and the maximum Lyapunov exponent), however, the results showed the chaotic and periodic regions of the fractional system. In Ref. [17], the nonlinear dynamics and control of atomic force microscopy (AFM) in fractional order are also investigated observing the existence of chaotic behavior for some regions in the parameter space. To bring the system from a chaotic state to a periodic one, the nonlinear saturation control (NLSC) and time-delayed feedback control (TDFC) techniques for fractional order systems are applied with and without accounting for fractional order. In Ref. [18] for (AFM) fractional-order case, the results showed the influence of derivative order on the dynamics of the AFM system. Due to the fractional order, some phenomena come up, which were confirmed through detailed numerical investigations by 0–1 test. The time-delayed feedback control technique was efficient in controlling the chaotic motion of the AFM in fractional order. Furthermore, the robustness of the proposed time-delayed feedback control was tested by a sensitivity analysis of parametric uncertainties. Recently [19] considering (AFM) that the system is operating in intermittent mode, the damping dynamics of the squeeze film damping can be represented by fractional calculus through numerical simulation and dynamic analysis to prove chaotic regimes. To suppress chaotic behavior, the authors used and analyzed two control strategies, the SDRE (Riccati equation dependent states) and OLFC (Linear Control for Optimum Feedback) controls.

3. Mathematical modeling

The mathematical model analyzed is based on the atomic force microscopy system considering a viscoelastic term that is assigned to the medium in which the tip performs the analysis. For this, the proposed model considers the interatomic forces between the probe and the surface of the samples. These interatomic forces are of the van der Waals type that arises from the Lennard-Jones potential. Eqs. (1) and (2) describe the Lennard Jones potential (U_LJ) and the van der Waals force (F_WD), respectively [20, 21].

$$U_{\mathit{LJ}}=\frac{A_{1}R}{1260z^7}-\frac{A_{2}R}{1260z}$$E1

$$F_{\mathit{WD}}=\frac{-\partial U_{\mathit{LJ}}}{\partial z}=\frac{A_{1}R}{180z^8}-\frac{A_{2}R}{6z^2}$$E2

Where R is the radius of the tip, z is the distance of the tip, $A_1={\pi }^2{\rho }_1{\rho }_2{c}_1$, and $A_2={\pi }^2{\rho }_1{\rho }_2{c}_2$ are called Hamarke constants (where ${\rho }_1$ and ${\rho }_2$ are the number densities of the two interacting kinds of particles, and c₁ and c₂ are the London coefficient). Therefore, the mathematical model analyzed is the one considered by Ref. [20] in which the deflection is determined by the following:

$$w\left(x,t\right)=u\left(x,t\right)+w\left(x\right)+y\left(t\right)$$E3

where $w\left(x,t\right)$ is microcantilever beam deflection, $u\left(x,t\right)$ is a relative deflection of displacement of the actuator, described as $y\left(t\right)=Ysin\left(\mathit{\Omega t}\right)$. We consider the term viscoelastic to the AFM system considering $z$ [15, 17]:

$$F_{\mathit{CS}}=\frac{-{\mu }_{\mathit{eff}}{b}^{3}L}{{\left[z-w\left(L\right)-u\left(L,t\right)-\mathit{\text{Ysin}}\left(\mathit{\Omega t}\right)\right]}^3}\dot{u}$$E4

Where ${\mu }_{\mathit{eff}}$ is the coefficient of effective viscosity, and $b$ and $L$ are the width and length of microcantilever, respectively. Considering the vibrations on the intermittent configuration, described by:

$$\begin{array}{ll}\mathit{\rho A}\ddot{u}\left(x,t\right)+\mathit{EI}\left(u^{\mathrm{″″}}\left(x,t\right)+w^{″″}\left(x\right)\right)=& (\frac{-A_{1}R}{180{\left(Z-w\left(L\right)-u\left(L,t\right)-Ysin\left(\mathit{\Omega t}\right)\right)}^8}\\ & +\frac{A_{2}R}{6{\left(Z-w\left(L\right)-u\left(L,t\right)-Ysin\left(\mathit{\Omega t}\right)\right)}^2}\\ & -\frac{{\mu }_{\mathit{eff}}{b}^{3}L\dot{u}\left(x,t\right)}{6{\left(Z-w\left(L\right)-u\left(L,t\right)-Ysin\left(\mathit{\Omega t}\right)\right)}^3})\times \delta \left(x-L\right)\\ & +\mathit{\rho A}{\Omega }^2\mathit{\text{Ysin}}\left(\mathit{\Omega t}\right)\end{array}$$E5

Eq. (5) is nonlinear and nonautonomous, and its discretization can be achieved through a dynamic projection on the linear modes of the system. According to Ref. [19], an approximation of the solution of Eq. (5) is by using the linear modes and frequencies of the microcantilever around its electrostatic equilibrium that are different from those of a microcantilever located far from the surface. Therefore, calculations of linear modes and microcantilever frequencies over nonlinear electrostatic equilibrium are rigorously calculated using Galerkin’s method. The Galerkin’s method is used to analyze problems of beams subjected to moving loads with time-varying velocities.

However, we consider under near-resonant forcing, and in the absence of additional internal resonances, only one mode of the microcantilever is assumed to participate in the response.

$$U\left(x,t\right)={\varphi }_1\left(x\right)q_1\left(t\right)$$E6

where ${\varphi }_1\left(x\right)$ is the first approximate eigenfunction about the chosen equilibrium. Substitution of (6) into (5), multiplication of (5) by ${\varphi }_1\left(x\right)$, subsequent integration over the domain, and the introduction of a modal damping consistent with the Q factors listed in Table 1 yields the single-degree-of-freedom model:

Description	Symbol	Silicio-Silicio
Length	L	449 $\mathrm{nm}$
Width	b	46$\mathrm{\mu m}$
Thickness	h	$1.7\mathrm{\mu m}$
Tip radius	R	150 $\mathrm{nm}$
Material density	$\rho$	2330 $\mathrm{kg}{\mathrm{m}}^3$
Static stiffness	k	0.11$\mathrm{N}{\mathrm{m}}^{-1}$
Elastic modulus	E	176$\mathrm{GPa}$
1st resonance	$f_1$	11.804 $\mathrm{kHz}$
Q fator (air)	Q	100
Hamaker constant (rep.)	$A_1$	$1.3596\times {10}^{-70}\mathrm{J}{\mathrm{m}}^6$
Hamaker constant (att.)	$A_2$	$1.865\times {10}^{-19}\mathrm{J}$
Coefficient of effective viscosity	${\mu }_{\mathit{eff}}$	$\approx 4.2101\times {10}^{-8}\mathrm{Ns}{\mathrm{m}}^{-2}$

Table 1.

Parameters used for numerical analysis.

$$\begin{array}{ll}\ddot{\overline{\eta }}=& -d_1\dot{\overline{\eta }}-\overline{\eta }+B_1+\frac{C_1}{{\left(1-\overline{\eta }-\overline{y}sin\left(\mathit{\Omega \tau }\right)\right)}^8}+\frac{C_2}{{\left(1-\overline{\eta }-\overline{y}sin\left(\mathit{\Omega \tau }\right)\right)}^2}\\ & -\frac{p\dot{\overline{\eta }}}{{\left(1-\overline{\eta }-\overline{y}sin\left(\mathit{\Omega \tau }\right)\right)}^3}+\overline{y}{\overline{\Omega }}^2{E}_{1}sin\left(\mathit{\Omega \tau }\right)\end{array}$$E7

where : $\overline{\eta }=\frac{x_1\left(\tau \right)}{\eta }$, $x_1={\varphi }_1\left(L\right)q_1\left(\tau \right)$, $\eta =z-\mathit{wL}$, $x_1={\varphi }_1\left(L\right)q_1\left(\tau \right)$, $\eta =z-w\left(L\right)$, $\tau ={\omega }_{1}t$, $\overline{\Omega }=\frac{\Omega }{{\omega }_1}$$d_1=\frac{C_1}{{\omega }_1\mathit{\rho A}{\int }_0^L{\varphi }_1^2\mathit{dx}}$, $B_1=\left(1-\overline{\alpha }\right){\Gamma }_1$, $C_1=\frac{A_{1}R}{180k{\left(\eta \right)}^9}{\Gamma }_1$,$C_2=\frac{A_{2}R}{6k{\left(\eta \right)}^3}{\Gamma }_1$, ${\omega }_1^2=\frac{\mathit{EI}{\int }_0^L{\varphi }_1{\varphi }_1^{⁗}\mathit{dx}}{\mathit{\rho A}{\int }_0^L{\varphi }_1^2\mathit{dx}}$, $\overline{\alpha }=\frac{z}{\eta }$, ${\Gamma }_1=\frac{{\mathit{k\varphi }}_1^2\left(L\right)}{{\omega }_1^2\mathit{\rho A}{\int }_0^L{\varphi }_1^2\mathit{dx}}$, $k=\frac{3\mathit{EI}}{L^3}$, $\zeta =\frac{Y}{\eta }$, $E_1=\frac{{\varphi }_1\left(L\right){\int }_0^L{\varphi }_1\mathit{dx}}{{\int }_0^L{\varphi }_1^2\mathit{dx}}$, and $p={\mu }_{\mathit{eff}}{b}^{2}l$.

Thus, considering, $\overline{\eta }\equiv x_1$ and $\dot{\overline{\eta }}\equiv x_2$ we can rewrite Eq. (7) in the following system of differential equations:

$$\frac{d{x}_1}{\mathit{dt}}=x_2\frac{d{x}_2}{\mathit{dt}}=-d_1{x}_2-x_1+B_1+\frac{C_1}{{\left(1-x_1-\zeta sin\left(\mathit{\Omega \tau }\right)\right)}^8}+\frac{C_2}{{\left(1-x_1-\zeta sin\left(\mathit{\Omega \tau }\right)\right)}^2}$$E8

$$\frac{-p{x}_2}{{\left(1-x_1-\zeta sin\left(\mathit{\Omega \tau }\right)\right)}^3}+\zeta {\overline{\Omega }}^2{E}_{1}sin\left(\mathit{\Omega \tau }\right)$$

Figure 2 shows the schemes simplify of deflection behavior of nano cantilever of AFM″ for completeness.

Figure 2.

Scheme of the AFM system. (a) System in initial start and (b) intermittent.

4. Nonlinear dynamic behavior outline

The basin of attraction is the set of all starting points (initial values) that converge to the attractor. A qualitative change in the behavior (i.e., attractor) of a dynamic system is associated with a change in the control parameter. It is a qualitative leap that progresses to more complex dynamics [22, 23, 24, 25, 26].

Therefore, we analyzed the behavior of the basin of attraction considering the parameters p ∈ [0.0.08] and η ∈ [0.0.1]. And thus determine an initial condition to perform the scan of other parameters of nonlinear dynamics, such as the maximum Lyapunov exponent (MLE) and the bifurcation diagram.

The MLE describes the divergence rate of the trajectories described by Eq. (8). The MLE has some characteristics: (i) with positive values for the Lyapunov exponent, they indicate that the trajectories of the system are divergent, this corresponds to the system having a chaotic behavior, (ii) if the MLE is negative, there is a contraction of the phase space, corresponding to a periodically stable system. For our calculations of the MLE exponent, we use the algorithm proposed by Ref. [27] with the variational calculus and the Jacobian matrix of Eq. (8).

The bifurcation diagram, in this work, is calculated considering the maximum points of the time series resulting from the integration of the system of Eq. (8) with the variation of the control parameter p. With this, we can observe the periods formed with this parametric variation. And so, to establish possible values of intervals that have multiple periods and thus diagnosing a chaotic behavior of the system, together with the MLE. And thus, obtaining the phase maps that describe the behavior of the trajectory for a given value of the parameter p.

For the dynamics calculations, we considered the parameters and properties of the cantilever described in Table 1 for the silicon-silicon system, as seen in Ref. [28].

Table 2 are the dimensionless parameters used with the values from Table 1 and dimensionless Eq. (8).

Parameters	Values
$d_1$	0.01
B1	−0.148967
$C_1$	$-4.59118\times {10}^{-5}$
$C_2$	$0.149013$
$\Omega$	1.0
$E_1$	1.57367

Table 2.

Parameters used for numerical analysis.

4.1 Basins of attraction analysis: entropy basins and uncertainty coefficient

Suppose that we have a dynamic system with N_A attractors for a choice of parameters in a certain region Ω of the phase space. We discretize Ω through a finite number of boxes so that we cover Ω with a grid of linear size ε. Now we build an application C: Ω → N that relates each initial condition to its attractor (which will have an associated color). Each box contains, in principle, infinite trajectories, each of which leads to an attractor labeled from 1 to N_A [22, 23].

We consider the colors in the box to be randomly distributed according to some proportions. We can assign a probability to each color j within a box i, as p_i,j is evaluated by calculating statistics about the trajectories inside the box. Considering that the trajectories inside a box are statistically independent, we can define the Gibbs entropy of each box i is begin:

$$S_i=\sum _{j=1}^{m_i}{p}_{\mathit{ij}}\mathit{log}\left(\frac{1}{p_{\mathit{ij}}}\right)$$E9

Where m_i ∈ [1, N_A] is the number of colors (attractors) in box i in p_i,j is the probability that each color j is determined by the number of trajectories leading to that color divided by the total number of trajectories in the box. We choose nonoverlapping boxes covering Ω so that the entropy of the entire grid is calculated by adding the entropy associated with each of the N boxes given by:

$$S_i=\sum _{i=1}^N{S}_i$$E10

$$S=\sum _{i=1}^N\sum _{j=1}^{m_i}{p}_{\mathit{ij}}\mathit{log}\left(\frac{1}{p_{\mathit{ij}}}\right)$$E11

Therefore, we can consider the entropy of the basin of attraction (S_b) as follows:

$$S_b=\frac{S}{N}$$E12

An interpretation of this quantity is associated with the degree of basin uncertainty, ranging from 0 (a single attractor) to log(N_A) (completely random basins with equiprobable N_A attractors). This latter higher value is rarely realized in practice, even for extremely chaotic systems. In some cases, we may only be interested in the uncertainty of boundaries between basins of attraction. We often want to know if the boundary is fractal. For this purpose, we can restrict the calculation of the basin entropy to the boxes that fall within the boundaries of the basin of attraction. We can calculate the entropy only for the N_b boxes that contain more than one attractor (color),

$$S_{\mathit{bb}}=\frac{S}{N_b}$$E13

where S is defined by Eq. (11). We refer to this S_bb number as the basin entropy quantifies the uncertainty regarding the boundaries only. The nature of this S_bb quantity is different from the entropy of the Sb basin, since Sb is sensitive to the size of the basins, so it can distinguish between different basins with smooth boundaries, S_bb provides a sufficient condition to easily assess that some boundaries are fractals [22, 23, 24].

Another way to quantify this uncertainty in the initial conditions for its final state is through the uncertainty coefficient. The uncertainty coefficient is related to the sensitivity of the final state of the trajectories in the phase space. An exponent close to 1 means that the basin has smooth contours, while an exponent close to 0 represents fully fractalized basins, also called sieve basins [26].

A phase portrait with a fractal boundary can cause uncertainty in the final state of the dynamical system for a given initial condition. To determine the uncertainty coefficient, one must probe the basin of attraction with balls of size ε at random. If there are at least two initial conditions that lead to different attractors, a ball is marked “uncertain.” In this way, we can denominate the fraction of “uncertain balls” (f_ε) for the total number of attempts in the basin. In analogy to the fractal dimension, there is a scaling law between, f_ε ∼ ε^α. The number that characterizes this scale is called the uncertainty exponent α [26]. For our analysis, we considered the set of differential equations that describe the interactions of the atomic force microscopy system and the parameters described in Table 1.

For this we consider p ∈ [0,0.08] and η∈[0,0.1], and for the numerical analysis of $S_b$, $S_{\mathit{bb}}$, and α we use an interval of initial conditions $x_1^0\times x_2^0=\left[-\mathrm{0.9,0.9}\right]\times \left[-\mathrm{0.9,0.9}\right]$ with approximately 250000 initial conditions and making a 200 x 200 grid considering ε=0.002 for each attraction basin formed during the numerical analysis. Figure 3a shows the behavior of $S_b$, Figure 3b shows behavior of $S_{\mathit{bb}},$ and Figure 3c shows behavior of $\alpha$ for set parameter $p\in \left[\mathrm{0,0.08}\right]$ e $\eta \in \left[\mathrm{0,0.1}\right]$.

Figure 3.

(a) shows the behavior of $S_b$, (b) shows behavior of $S_{\mathit{bb}},$ and (c) shows behavior of $\alpha$ for set parameter $p\in \left[\mathrm{0,0.08}\right]$ e $\eta \in \left[\mathrm{0,0.1}\right]$.

We can see in Figure 3a and b that the red regions show the maximum values for S_b and S_bb, that is, the regions where the basins of attraction have more attractors and their edges are fractalized. However, for the uncertainty coefficient in Figure 3c the uncertainty coefficient is close to 1 showing the smooth basins. According to Ref. [20] the difference between S_b and S_bb and the uncertainty coefficient of the attraction basins is that when S_b and S_bb are minimum the uncertainty coefficient is maximum, or when S_b and S_bb are maximum the uncertainty coefficient is minimum. This corroborates the analysis of the behavior of the initial conditions with the parametric variation of p and η. Table 3 shows the parameters p and η that provide the maximum values for S_b and S_bb and the minimum value of α that produces the basins of attraction.

	$p$	$\eta$
$S_b$	0.0012	0.0045
$S_{\mathit{bb}}$	0.0012	0.0045
$\alpha$	0.0764	0.0121

Table 3.

Summarizes the parameters p and η that provide the maximum. values for S_b and S_bb and the minimum value of α that produce the basins of attraction.

Figure 4a–c show the behavior of the attraction basins considering the maximum values of$S_b\text{and}{S}_{\mathit{bb}}$ and for the minimum value of α.

Figure 4.

Basins of attraction. (a) p = 0.0012 and $\eta$ = 0.0045, (b) p = 0.0012 and $\eta$ = 0.0045, and (c) p = 0.0764 and $\eta$ = 0.0121.

Figure 5a–c represent attractor points referring to the basins of attraction of Figure 5a–c.

Figure 5.

Attractive orbits that form the basins of attraction of the figure. (a) p = 0.0012 and $\eta$ = 0.0045, (b) p = 0.0012 and $\eta$ = 0.0045, and (c) p = 0.0764 and $\eta$ = 0.0121.

4.2 Numerical dynamics analysis

Taking into account the analysis of the basins of attraction for the interval of p∈[0,0.08] and η∈[0.0.25], we adopted a larger interval for η, for analysis of the maximum Lyapunov exponent (MLE) and the initial condition [0.1, 0.0], because depending on the values of p and η the initial condition can participate in different attractors, as we saw in Figures 4a and b and 5a and b.

Using the Jacobian matrix for the variational calculus and the sweep of p∈ [0,0.08] and η∈ [0,0.25], we have the behavior of the MLE. Figure 6 shows the space of MLE parameters in which the region of white to black shows the regions in Eq. (7) has a periodic behavior. However, for the region between yellow and green, it shows the chaotic behavior with the parameter sweep p×η.

Figure 6.

Behavior of the maximum Lyapunov exponent (MLE) for the parameters$p\in \left[\mathrm{0,0.08}\right]$ and $\eta \in \left[\mathrm{0,0.25}\right]$.

Figure 7a shows the behavior of the bifurcation diagram for the following parameters, so we can observe the periodic windows p∈ [0, 0.0777] (black region), the intervals are confirmed by the maximum Lyapunov exponent. However, for the interval p∈ [0.0777, 0.08] there is a chaotic window (red region).

Figure 7.

(a) Bifurcation diagram for displacement x₁ in black region shows the periodic behavior, red region shows the chaotic behavior, and (b) Lyapunov exponent.

Therefore, considering the previous nonlinear dynamic results, we obtain the phase portrait and the Poincare map for p =0.0791 and $\eta$ = 0.2043 (Figure 8). Figure 9a shows the phase portrait (Gray Line) and Poincare map (Black Dots). Figure 9b and c show the time series of displacement x₁ and velocity x_2.

Figure 8.

(a) Phase Portrait (gray lines) and Poincare Map (black dots), (b) time series of displacement x₁ and (b) velocity x_2..

Figure 9.

(a) Portrait phase (black line) and orbit (red line), (b) time series of displacement x_1,, and (c) and time series of velocity x₂.

5. Control design

The control techniques are Optimal linear feedback control (OLFC) and state-dependent Riccati equation (SDRE).

The OLFC has control feedback as its main application for controlling nonlinear systems, it has this name because it is characteristic of the controller to have the control signal as a function of the difference between real values and the values expected by the state variables. The application of this technique in nonlinear systems is due to the simplicity of implementation. In other words, the OLFC control does not consider the nonlinearities of the system of equations for the suppression of the chaos that occurred in the microcantilever of the AFM system. However, the SDRE controller considers the nonlinearities of the system, and its state matrix is not fixed. The SDRE control technique guarantees an asymptotically stable solution over the origin [7, 29, 30, 31, 32, 33].

In both cases, there is a need to propose an orbit for the system to be controlled. In this chapter, we opted for simplicity of calculations to consider an orbit described by Eq. (14):

$$\left\{\begin{array}{c}{x}_1=0.5\mathit{sin}\left(\mathit{\omega t}\right)\\ x_2=0.5\mathit{cos}\left(\mathit{\omega t}\right)\end{array}\right.$$E14

We can consider the system Eq. (8) in the matrix form given by:

$$\dot{x}=\mathit{Ax}+g\left(x\right)+U$$E15

Where Ax is error dependent state matrix, g(x) is a nonlinear matrix, and U is control signal [34, 35].

Both control techniques use two controllers called feedback (u_f) and feedforward (ũ), so the control signal of the nonlinear system is defined U=ũ+u𝑓. While u_f has the characteristic of correcting the difference between the real values and the stipulated values, taking the system to the stipulated orbit, ũ has the purpose of keeping the system in the desired orbit [24, 25]. The SDRE controller design, like the OLFC, follows some steps to obtain the optimal solution to the dynamic control problem [10, 31]:

• Define the state space of the model and parameterize the model in the form of state-dependent coefficients.

• Measures the state of the system x(t), that is, to define x (0) = x₀, and to choose the coefficients of the matrices Q and R.

• Solve the Riccati equation for the state x(t);

• Calculate the input signal of the state feedback control equation.

• Check controllability

• Assume the output value of the system as a new initial value and update the state of the system x (t). Recalculate the Riccati equation and repeat the process until the defined stopping criterion is reached.

5.1 Optimal linear feedback control OLFC

The optimal linear feedback control is used in nonlinear systems due to its simplicity in its implementation since the control uses fixed k gains. The value of K is obtained by solving the Riccati equation [34, 35, 36].

The nonlinear system of Eq. (7) can be written in matrix form using the general equation of nonlinear systems 14.

Then Eq. (16) shows the system rewritten in matrix form and as a function of errors, knowing that $=\left(X-\stackrel{\sim }{X}\right)$ , then it is possible to obtain the state matrix A error-dependent and nonlinearity matrix $g\left(e,\stackrel{\sim }{X}\right)$,

which will not be used in the control and, therefore, is indicated in the equation.

$$\dot{e}=\left(\begin{array}{cc}0& 1\\ -1& D1\end{array}\right).\left(\begin{array}{c}{e}_1\\ e_2\end{array}\right)+g\left(e,\stackrel{\sim }{X}\right)+U$$E16

The gain is described by$k=R^{-1}{B}^{T}P$ where $P$ is the Riccati matrix, $R$=$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$, $B$=$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),$ and $Q$=$1000\ast \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$[23, 24, 33].

The matrices $P,R,B,$ and $Q$ guarantee the stability of the solution of the Riccati equation. Therefore, the gain found by the system is defined by

$$k=\left(\begin{array}{cc}1000& 5\times {10}^{-6}\\ 5\times {10}^{-6}& 1000\end{array}\right)$$E17

Figure 9a shows the portrait phase of the uncontrolled system in black and controlled in red, and (b) and (c) the time series of the system is shown.

Figure 10 shows the control signal u_𝑓 in the transient regime, that is, at the moment when the control is taking the system to the desired orbit in the plot of ũ is the steady-state control signal.

Figure 10.

u_f in the transient regime and ũ is the steady-state control signal.

In Figures 11 and 12, the controller errors in transient and steady state are presented for each time series of the system.

Figure 11.

$e_1^{u_f}$represents the feedback control and $e_2^{u_f}$ represents the feedback control.

Figure 12.

$e_1^{\stackrel{\sim }{u}}$ represents the error of x₁ in steady state for the feedforward control. $e_2^{\stackrel{\sim }{u}}$ represents the trajectory error of x₂ in the control feedforward.

5.2 State-dependent Riccati equation control

The state-dependent Riccati equation (SDRE) control, unlike the OLFC, does not exclude the dependence of the error-dependent state matrix, so the controller gains change with each iteration. The SDRE methodology used to find error-dependent states used matrix like that is used for the OLFC control. It can be written from the following nonlinear matrix of errors:

$$\begin{array}{l}\dot{e}=\left(\begin{array}{cc}0& 1\\ \frac{2c2}{{\left(1-x_1-{\eta }_1\mathit{sin}\left(\mathit{\omega t}\right)\right)}^3}-\frac{3x_{2}p}{{\left(1-x_1-{\eta }_1\mathit{sin}\left(\mathit{\omega t}\right)\right)}^4}+\frac{8c1}{{\left(1-x_1-{\eta }_1\mathit{sin}\left(\mathit{\omega t}\right)\right)}^9}& \frac{p}{{\left(1-x_1-{\eta }_1\mathit{sin}\left(\mathit{\omega t}\right)\right)}^3}-d_1\end{array}\right).\\ \left(\begin{array}{c}{e}_1\\ e_2\end{array}\right)+g\left(\stackrel{\sim }{X}\right)+U\end{array}$$E18

where $g\left(\stackrel{\sim }{X}\right)$ is the matrix that does not depend on errors. The gain k is obtained $k=R^{-1}{B}^{T}P$ where $P$ is the Riccati matrix. The control u found from the solution of the following equation:

$$u=-R^{-1}{B}^T\mathit{Pe}$$E19

Being a symmetric matrix and obtained from the algebraic Riccati equations [7, 34, 36]:

$$A^{T}P+\mathit{PA}-\mathit{PB}{R}^{-1}{B}^{T}P+Q=0$$E20

The controller gain $k$ is defined $k=R^{-1}{B}^{T}P$ where $P$ is the Riccati matrix, $R$=$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$, $B$=$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$, and $Q$=$1000\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ [23, 24, 33].

The matrices $P,R,B,$ and $Q$ guarantee the stability of the solution of the Riccati equation. Therefore, the gain found by the system is defined in Figure 13, which shows the result of the controlled system using the SDRE technique, in Figure 13a the portrait phase without control and with control is presented, and their respective time series in Figure 13b and c.

Figure 13.

(a) Portrait phase (black line) and orbit (red line), (b) time series of displacement x_1, and (c) time series of velocity x_2.

Figure 14 represents the control signal u_f in the transient regime, that is, now when the control is taking the system to the desired orbit. The ũ plots show the steady-state control signal.

Figure 14.

SDRE signal control.

Figures 15 and 16 show the controller errors in transient and steady state for each time series of the system.

Figure 15.

$e_1^{u_f}$represents the error of x₁ for the feedback control and $e_2^{u_f}$ represents the path of x₂ for the feedback control.

Figure 16.

$e_1^{\stackrel{\sim }{u}}$ represents the error of Eq. (7) in steady state for feedforward control. $e_2^{\stackrel{\sim }{u}}$ represents the steady-state trajectory error for the feedforward control.

Figure 17 shows the behavior of gain k since it is calculated at each interaction, that is, the controller gain is variable.

Figure 17.

SDRE behavior of gain k.

Figure 18 shows the comparison of the control signal U=ũ+u_f for the two techniques applied in this text. It is possible to notice that the OLFC control, even excluding the nonlinearities of the system, has a control signal very close to the control signal of the SDRE.

Figure 18.

Comparison of the control signal.

6. Conclusions

In this chapter, we describe the applications of atomic force microscopy. We also analyzed the nonlinear dynamic behavior of a mathematical model considering the viscoelasticity term and the microcantilever deflection. In this way, we establish the behavior of the initial conditions and which ones have greater entropy and more fractality. These analyzes corroborate to determine the set of initial conditions for our dynamic analyses, as observed in Figure 3 and Table 3.

After analyzing the behavior of the initial conditions, the dynamic behavior of the dimensionless parameter was analyzed, which considers viscoelasticity and, therefore, the regions in which the system presents a chaotic behavior. This behavior was obtained using the maximum Lyapunov exponent, and for a given set of parameters, it was observed by the bifurcation diagram and the Poincaré map. In this way, the ranges for the parameters p and $\eta$ were established where a possible chaotic behavior occurs, as we see in Figures 5 and 7.

The results obtained by the analysis of the basins of attraction showed a strong influence between the parameters p and eta in the initial conditions. As we observed in the calculation of entropy and uncertainty coefficient for the grid of initial conditions $x_1^0\times x_2^0=\left[-\mathrm{0.9,0.9}\right]\times \left[-\mathrm{0.9,0.9}\right]$, have regions of high fractality and receive it from a new attractor, as shown in Figures 4a–c and 5a–c. Considering the initial condition $\left[\mathrm{0.1,0.0}\right]$ the MLE has regions of positive value, that is, there is a chaotic behavior, as shown in Figure 6.

These analyses corroborate to determine the p and η parameters for the application of two control techniques and suppress the chaotic behavior. This suppression allows us to have a better understanding of the microcantilever when reading biological samples that can generate chaotic movements. These chaotic movements can be detected as noises in the structure of the AFM device.

In general, it is possible to notice that the two control methods presented low errors as shown in Figure 16. For this system there was no difference in convergence to the intended orbit; however, it is possible to notice that the OLFC control has a simple implementation methodology in relation to the SDRE, as it excludes nonlinearities, facilitating the application of the control method and enabling a possible practical implementation using embedded systems. Figure 17 showed this comparison between the control techniques.

Due to the low computational cost, the OLFC control technique proves to be a viable alternative for embedded systems of the AFM type. Works such as [34] make a comparison of computational costs.

Acknowledgments

The authors acknowledge the CAPES and CNPq, both Brazilian research funding agencies.

Conflict of interest

The authors declare no competing interests.