Synthesis and Analysis of Nonlinear Control Systems Based on Transformations and Factorizations

Many real-world systems are inherently nonlinear by nature, hence it is of great benefit to develop control techniques for general nonlinear systems. In this thesis, we will investigate two specific questions in the nonlinear context: synthesis and analysis of nonlinear control systems. For the task of synthesis, we shall mainly consider systems which can be modeled by physical laws. For this task, two different frameworks have been developed in the last decade to describe such systems. One class is the class of feedback linearizable (differentially flat) systems and the other is that of port-controlled Hamiltonian systems. Feedback linearizable systems, which in fact encompass many mechanical and electromechanical systems, are systems which can be transformed into linear controllable ones via a set of a feedback and a coordinate transformation. Consequently such systems can be controlled after transforming them into linear ones. Port-controlled Hamiltonian systems were originally introduced to describe passive physical systems, which are generalization of classical mechanical systems. For port-controlled Hamiltonian systems, the generalized canonical transformation which is an extension of the canonical transformation in classical mechanics is introduced so that it plays a role similar to the linearizing transformation in feedback linearization. Consequently, port-controlled Hamiltonian systems can be controlled after transforming it into an appropriate form, similar to feedback linearizable systems. The strategies which utilize the intrinsic properties of the physical systems and their applications are investigated in the synthesis part of this thesis. For the analysis task, general nonlinear systems will be considered. The basic strategy is to extend analysis tools in conventional linear systems theory to nonlinear systems. In contrast to the synthesis part, factorizations and related techniques are used because it is difficult to find any useful similarity among general nonlinear systems; in other words, it is hard to determine a common property which all nonlinear systems possess, such as linearizability or passivity. To be precise, nonlinear versions of coprime factorizations, Hankel operators and related tools, namely adjoints and derivatives, which all play important roles in linear control systems theory are investigated. Coprime factorization allows us to derive a parametrization of all stabilizing plant and controller pairs, which finds many applications, such as closed-loop identification of nonlinear systems. Of course, the same parametrization can be utilized for synthesis of nonlinear control systems also. Adjoints can be used to analyze nonlinear operators gain and this technique can be applied to compute controllability and observability functions. The duality between controllability and observability of nonlinear control systems is also clarified. That is, it is proved that a nonlinear control system is controllable if and only if its adjoint is observable and vice versa. Derivatives reveal the linearized property of nonlinear operators and clarify the