**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.

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 intermittentcontact 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 [3–6, 8, 9, 11–14].

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).

### *MEMS-Based Atomic Force Microscope: Nonlinear Dynamics Analysis and Its Control DOI: http://dx.doi.org/10.5772/intechopen.108880*

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 timedelayed 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.
