THESIS
2019
xvi, 183 pages : illustrations ; 30 cm
Abstract
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...[
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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.
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