In recent years the Tapping Mode-Atomic Force Microscope (TM-AFM) has become one of the most important tools for imaging on the nanometer scale. In comparison with other contemporary technologies, the AFMs have been able to obtain atomic resolution both in high vacuum and liquid environments thus affirming their supremacy. The AFM can be perceived as a combination of a mechanical profilometer, where mechanical springs are used to sense the forces, and a Scanning tunneling microscope, where piezo-electric transducers are used for scanning. The AFM is widely used to generate a topographical image of the sample surface and also to study certain characteristics of the sample. The latter is aided by measuring the forces between the sample surface and the tip of the probe.

The non-linear, rapidly changing and hysteretic behavior of the tip-sample forces makes their accurate estimation extremely difficult in dynamic AFM. Moreover, the cantilever probe responds to an average of the different forces acting on the probe tip. Since several permutations of different forces can give the same periodic average, the accurate estimation of each of these forces has been evidently impossible. However, some probes exist for which the motion of the cantilever consists of super harmonic components of the tip-sample forces which provide more information about the tip-sample forces. Nevertheless the construction of these cantilevers is challenging and time consuming. Knowledge of the dynamic properties of the cantilever facilitates one to study its behaviour to a particular input. Since different forces act at different tip sample distances, a more mathematical approach towards tip-sample force estimation which includes the dynamic characteristics of the cantilever is necessary.

The accurate knowledge of the cantilever dynamics is extremely important for precise estimation of tip-sample forces, deduction of mechanical properties, controller synthesis etc. Therefore in this research, the techniques to identify the state space matrices is explored. One, rather old but an immensely useful identification method are the black box identification techniques. These techniques can be used to accurately estimate the fully parameterized state space matrices of the system. The main difficulty arises in estimating these parameters specially in the absence of one of the inputs (tip-sample interactions).

In this thesis research, an algorithm is developed to identify the fully parameterized state space matrices. A secondary cantilever of very high fundamental resonance frequency is used as a force sensor. This force sensor set up along with the periodic property of the tip-sample forces during TM-AFM is used to reconstruct the tip-sample interaction forces. Using the estimated tip-sample forces, a transfer function between the cantilever deflection and the estimated force is identified using a curve fitting technique. The curve fitting technique uses iterative least squares to reduce the two norm between the experimental frequency response and the frequency response estimated using the identified transfer function.