THESIS
2020

xiv, 113 pages : illustrations ; 30 cm

**Abstract**
Magnitude (gain) analysis and phase analysis are two pillars of the classical
frequency-domain analysis of single-input-single-output (SISO) systems. For multi-input
multi-output (MIMO) linear time-invariant (LTI) systems, gain-based analysis
and control theory has obtained huge success. While the small gain theorem is well-known
in the field of robust control. Phase-based analysis has obtained much less
attention. How to define phase response of a MIMO system has been an unsettled
issue. The main obstacle in having the phase concept for MIMO systems lies in the
lack of mathematical definition of matrix phases.

In this thesis, we define the phases of a special class of complex matrices, called
sectorial matrices. We study various properties of such phases and find that they
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Magnitude (gain) analysis and phase analysis are two pillars of the classical
frequency-domain analysis of single-input-single-output (SISO) systems. For multi-input
multi-output (MIMO) linear time-invariant (LTI) systems, gain-based analysis
and control theory has obtained huge success. While the small gain theorem is well-known
in the field of robust control. Phase-based analysis has obtained much less
attention. How to define phase response of a MIMO system has been an unsettled
issue. The main obstacle in having the phase concept for MIMO systems lies in the
lack of mathematical definition of matrix phases.

In this thesis, we define the phases of a special class of complex matrices, called
sectorial matrices. We study various properties of such phases and find that they
meet our expectations. In particular, a majorization relation between the phases of
the eigenvalues of AB and the phases of A and B is established, which naturally leads
to a matrix small phase theorem. With the concept of matrix phases, we define the
phase response of a MIMO LTI stable system, assumed to be a sectorial matrix at
each frequency. This phase concept generalizes the notions of positive realness and negative imaginariness. The magnitude (singular value) response and the new phase
response can be plotted shoulder-to-shoulder to form a complete MIMO Bode plot.
We also define the half-sectorial systems and provide a time-domain interpretation. As
a starting point in an endeavour to develop a comprehensive phase theory for MIMO
systems, we establish a small phase theorem for feedback stability, which complements
the well-known small gain theorem.

As an application, we study the synchronization of heterogeneous agents interacting
over a dynamical network, where the edge dynamics can model nonuniform
communication environment between the agents, or a uniform controller shared by all
the agents. Novel synchronization conditions are obtained from a phasic perspective
by exploiting the small phase theorem. These conditions have lower conservatism
compared to gain-based conditions and they generalize positive real and negative
imaginary type conditions. They scale well with the size of the network and reveal
the trade-off between the phases of node dynamics and edge dynamics.

We also formulate and analyze the synchronizability problem, where heterogeneous
agents share a uniform controller. Sufficient conditions under which these agents
can achieve synchronization via a uniform controller have been established, and the
controller is given. These results have nice scalability and make it possible to control
the synchronization of large-scaled networks efficiently.

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