Currently used imaging methods in Atomic Force Microscopy (AFM) including the use of a Lock-In Amplifier or a Phase-Locked Loop, are suboptimal. In this report, the image resolution in AFM is improved by detecting the tip-sample interaction using complete measurements of the input of the cantilever and its measured deflection. Two methods are studied while assuming that the tip-sample interaction is sparse, namely a model-based approach and a data-driven approach. Real-life experiments have shown that the model-based approach improves the image resolution with a factor of 7.5 to 0.555 nm compared to the conventional imaging method, according to a metric using Fourier Ring Correlation in which a reference image is unnecessary. The data-driven approach can be used in the model-based approach to further improve the resolution. In addition to improved resolutions, a Linear Time-Invariant model of the mechanically driven AFM-cantilever immersed in liquid – from piezo input to cantilever deflection – has been obtained through subspace identification with a Variance Accounted For of 79.2%. Recommendations for future research include applying the latter model in detecting the tip-sample interaction, improving the data-driven approach, reducing the computational effort of the model-based approach and implementing algorithms for detecting the tip-sample interaction online.

Stringent control demands from the high-tech mechatronics industry have warranted the need to explore potentially advantageous non-linear controllers. Variable Fractional Order (VFO) calculus provides one such avenue to build non-linear PID-like controllers. VFO calculus is the generalization of integer order differentiation and integration, where, in addition to the possibility of orders being real or even complex, the orders can vary as a function of a variable like time, temperature, etc. However, in this nascent field of VFO control, the focus has mainly been on controller tuning by time domain optimization of the performance for certain specific trajectories or cost functions. On the other hand, Frequency domain tools allow for analysis and tuning of controllers for performance over a wide range of exogenous inputs. For smooth adoption into industry, it is important to develop a frequency domain framework for working with VFO control. Describing function (DF) analysis is a method to obtain an approximate Frequency Response Function (FRF)-like function for non-linear systems. In this thesis, DF analysis is used for developing VFO PID controllers in the frequency domain from an industry compatibility point of view and the closed loop performance of these controllers in controlling a precision positioning stage is examined.

For large-scale system with tens of thousands of states and outputs the computation in the conventional Kalman filter becomes time-consuming such that Kalman filtering in large-scale real-time application is practically infeasible. A possible mathematical framework to lift the curse of dimensionality is to lift the problem in higher dimensions with the use of tensors and then decompose it. The tensor-train decomposition is chosen due to its computational advantages for systems with low tensor-train rank. Within this thesis two main limitations of the existing tensor Kalman filter are solved. First, a method is developed based on tensor-train rank truncation of the covariances to increase the computational speed for more general systems. Second, a MIMO tensor Kalman filter is developed for a specific class of systems. The power of the developed methods is shown on the example of adaptive optics which fits into the framework. A comparison with state-of-the-art large-scale estimation algorithms shows the computational advantage of the tensor Kalman filter at the cost of approximation errors.

An interconnected system is composed of multiple well-defined self-contained subsystems that interact among them and that together create collective behaviours. We can find many examples of interconnected systems in real life. Ranging from biological systems, such as the growth and interaction of populations in diverse and spatially distributed environments, to electric grids connecting power-generating sources, buildings and infrastructures in a country. When studying interconnected systems, a fundamental and natural question is how the properties and characteristics of the individual subsystems and the way they are connected relate to the collective behaviour of the complete system. That is the driving question of the present dissertation. Given that interconnected systems can be found in a wide variety of contexts, their representation and specific research interests can be equally varied. Because of this, it is impossible to answer the aforementioned question uniquely for all interconnected systems, and specific cases must be considered. In this dissertation, we consider two types of interconnected systems: a general class of uncertain multiple-input-multipleoutput (MIMO) systems, and agent-based opinion formation models. The investigation of uncertain MIMO interconnected systems is focused on providing topology-independent conditions for robust stability. The primary motivation for this approach is that, in real systems, it is costly or even impossible to have complete and accurate information on the network topology and subsystem parameters and dynamics. However, it is of critical interest to guarantee the system’s stability. Therefore we need stability conditions that require only partial information about the network and the subsystems to ensure the system’s stability. By studying these systems both in the time and frequency domain, we are able to provide conditions thatmeet these requirements. As for agent-based opinion formation models, we assume that each individual (or agent) in a population has an opinion about a statement. By exchanging opinions among themselves, the agents update their own internal opinion, resulting in a collective dynamic of opinion evolution. When studying these systems, the interests shifts from stability conditions, to a characterisation of the relation between the agents’ individual traits and qualitative properties of the opinion distribution in the population. Several techniques and approaches to analyse opinion formation models are proposed and applied to multiple models, one of which is new to this dissertation. The collective study of the previously mentioned interconnected systems requires the use of multiple and diverse analysis techniques and approaches, from analytical methods based on the Nyquist criterion, Bauer-Fike theorem, and Lyapunov functions to qualitative and numerical analysis techniques like histograms and binomial proportion confidence intervals. It is our hope that some of the presented results, methods, or ideas may advance the knowledge frontier in this scientific field, sparkle new research directions, and either directly or indirectly prove some value to society. @en