This thesis investigates two fundamental properties of dynamical systems — stability and monotonicity — via analysis of feedback interconnected models with special structures.

In the first part, a versatile framework to model and study networked control systems (NCSs) is introduced. An NCS is described as a feedback interconnection of a plant and a controller communicating through a bidirectional channel modeled by cascaded two-port networks. This model is sufficiently rich to capture various properties of a real-world communication channel, such as distortion, interference, and nonlinearity. Uncertainties in the plant, controller and communication channels can be handled simultaneously in the framework. Necessary and sufficient robust stability conditions are proposed with re...[ Read more ]

This thesis investigates two fundamental properties of dynamical systems — stability and monotonicity — via analysis of feedback interconnected models with special structures.

In the first part, a versatile framework to model and study networked control systems (NCSs) is introduced. An NCS is described as a feedback interconnection of a plant and a controller communicating through a bidirectional channel modeled by cascaded two-port networks. This model is sufficiently rich to capture various properties of a real-world communication channel, such as distortion, interference, and nonlinearity. Uncertainties in the plant, controller and communication channels can be handled simultaneously in the framework. Necessary and sufficient robust stability conditions are proposed with respect to different dynamical models of systems and uncertainties, such as linear time-invariant, linear time-varying and nonlinear systems. Based on the associated robust stability margins, an elementary polynomial approach is developed and demonstrated for an optimal robust controller synthesis problem, and an approximation method is proposed for another synthesis problem with frequency-wise specifications.

In the second part, an exact characterization of monotone dynamical systems is developed with the aid of the Shapiro theorem and via analysis of the diagonally feedback-controlled model. To reduce the conservatism in the theory, the Shapiro theorem, which bridges the simplicity of eigenvalues and the irreducible cone-invariance property of matrices, is generalized by a geometric approach. Furthermore, the associated Shapiro problem and its original form — the Frisch-Kalman problem, a constrained rank minimization problem — are analyzed and approached by convex relaxations and optimizations. Conditions on the tight relaxation are developed for the proposed method.