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 me...[ Read more ]

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.