THESIS
2022
1 online resource (xv, 97 pages) : illustrations (some color)
Abstract
The gain and phase information of a single-input-single-output linear time-invariant (LTI)
is valuable in the frequency domain analysis in classic control theory. The related research
results are well-established and often come out hand in hand. The extension of gain and
phase analysis to multi-input-multi-output (MIMO) LTI systems is not trivial. In contrast
to the well-developed gain analysis based on the small gain theorem, the research on the
phase analysis does not share the same status. This study explores the phases of discrete-time
MIMO LTI systems and heterogeneous networks.
We first introduce the phase response of a class of discrete-time (DT) LTI multivariable
systems by exploiting a notion of matrix phases based on the numerical range. Positive
frequency joint sectorial syst...[
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The gain and phase information of a single-input-single-output linear time-invariant (LTI)
is valuable in the frequency domain analysis in classic control theory. The related research
results are well-established and often come out hand in hand. The extension of gain and
phase analysis to multi-input-multi-output (MIMO) LTI systems is not trivial. In contrast
to the well-developed gain analysis based on the small gain theorem, the research on the
phase analysis does not share the same status. This study explores the phases of discrete-time
MIMO LTI systems and heterogeneous networks.
We first introduce the phase response of a class of discrete-time (DT) LTI multivariable
systems by exploiting a notion of matrix phases based on the numerical range. Positive
frequency joint sectorial systems are also defined, which generalize the positive real and
negative imaginary systems. The interpretation of system phases is given in terms of the
input and output signals of the system. A sectored real lemma is obtained to characterize
the phase information from a state-space realization. Motivated by finding a phasic
counterpart to the small gain theorem, we develop a small phase theorem for the internal
stability of a closed-loop system. The phase properties are also investigated for parallel
and feedback interconnected systems.
The output synchronization in large-scale discrete-time networks with dynamic edges is
examined by utilizing the developed phase tool. Both agent dynamics and edge dynamics
are assumed to be heterogeneous. For the synchronization analysis problem, conditions are
obtained from the phase perspectives. The trade-off between phases of nodes and edges are
demonstrated, which can be treated as the application of phase lead-lag compensation idea
to the heterogeneous network. The synchronization synthesis problem is also formulated
and investigated. We provide a sufficient condition, answering the solvability question
that under what condition there exists a uniform controller such that a group of agents
will reach synchronization. If the condition is satisfied, a design procedure is given,
which produces a low gain synchronizing controller. Numerical examples are given to
demonstrate the effectiveness of phase tool.
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