In classical control theory, gain and phase are two fundamental and parallel concepts for single-input single-output linear time-invariant (LTI) systems, and they are equal partners in serving control system analysis and synthesis. For more general systems, such as multi-input multi-output (MIMO) LTI systems and nonlinear systems, the gain-based techniques, e.g., L₂-gain control theory, have been flourishing, while the phase counterparts are somewhat inadequate and ambiguous. This thesis is dedicated to exploring two phasic notions in nonlinear systems and to using the notions in stability analysis of feedback systems.

First, we propose a definition of phase for a class of stable nonlinear systems, called semi-sectorial systems, from an input-output perspective. The definition involves...[ Read more ]

In classical control theory, gain and phase are two fundamental and parallel concepts for single-input single-output linear time-invariant (LTI) systems, and they are equal partners in serving control system analysis and synthesis. For more general systems, such as multi-input multi-output (MIMO) LTI systems and nonlinear systems, the gain-based techniques, e.g., L₂-gain control theory, have been flourishing, while the phase counterparts are somewhat inadequate and ambiguous. This thesis is dedicated to exploring two phasic notions in nonlinear systems and to using the notions in stability analysis of feedback systems.

First, we propose a definition of phase for a class of stable nonlinear systems, called semi-sectorial systems, from an input-output perspective. The definition involves the Hilbert transform as a key instrument to complexify real-valued signals since the notion of phase arises most naturally in the complex domain. The proposed nonlinear system phase, serving as a counterpart of L₂-gain, quantifies the passivity and is highly related to the dissipativity. It also possesses a nice physical interpretation which quantifies the tradeoff between the real energy and reactive energy. A nonlinear small phase theorem is then established for feedback stability analysis of semi-sectorial systems. Additionally, its generalized version is proposed via the use of multipliers. The nonlinear small phase theorems generalize a version of the classical passivity theorem and a recent small phase theorem for MIMO LTI systems.

Second, we introduce an angle notion, called the singular angle, for stable nonlinear systems from an input-output perspective. The proposed system singular angle, based on the angle between L₂-signals, describes an upper bound for the “rotating effect” from the system input to output signals. It is thus different from the aforementioned nonlinear system phase which adopts the complexification of real-valued signals using the Hilbert transform. It can quantify the passivity and serve as another angular counterpart to the system L₂-gain. It also provides an alternative to the nonlinear system phase. A nonlinear small angle theorem, which involves a comparison of the loop system angle with π, is established for feedback stability analysis. When dealing with MIMO LTI systems, we further come up with the frequency-wise and H_∞ singular angle notions based on the matrix singular angle, and develop the corresponding LTI small angle theorems.