Small Phase Theorem

Abstract

The small gain theorem is one of the most important results in the theory of robust control. It lays the foundation for the traditional gain-based analysis and synthesis, especially within the H 1 control paradigm. This entry is concerned with the small phase theorem, which can be regarded as a fitting counterpart to the small gain theorem. With the aid of a suitable definition of complex matrix phases, small phase results for matrices and linear time-invariant (LTI) systems are described. Together, they pave the way for a comprehensive theory of phases of complex systems.

The Singular Angle of Nonlinear Systems

“…The second is to introduce complex elements to nonlinear systems on the grounds that phases are naturally defined for complex numbers. To this end, in our recent works [12] and [13], we have proposed the notion of the nonlinear system phase through complexifying real-valued signals using the Hilbert transform. Concretely, for a causal stable system P , we first define the angular numerical range of P to be…”

“…Nevertheless, we think that the passivity is only qualitatively phase-related [12]. Recently, we developed a phase definition [12], [13] for a class of stable nonlinear systems from an input-output perspective. The key idea behind this definition is to complexify real-valued signals by using the analytic signal and Hilbert transform [14], since the notion of phase (or angle) arises naturally in a complex domain.…”