On the phases of a complex matrix

Wang, Dan; Wei, Chen; Khong, Sei Zhen; Li, Qiu

doi:10.1016/j.laa.2020.01.035

Cited by 38 publications

(38 citation statements)

References 27 publications

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“…Recently, we discovered a more suitable definition of matrix phases based on numerical range [22]. The numerical range, also called field of values, of a matrix C ∈ C n×n is defined as W (C) = {x * Cx : x ∈ C n with x = 1}, which, as a subset of C, is compact and convex, and contains the spectrum of C [25].…”

Section: Phases Of a Complex Matrixmentioning

confidence: 99%

“…where 0 ≤ β − α < 2π, is a cone. The following lemma can be proved by modifying slightly the proof of its version for sectorial matrices in [22].…”

Section: Phases Of a Complex Matrixmentioning

confidence: 99%

“…Nevertheless, it just appears that the understandings from the two communities had not got a chance to fully cross-pollinate. Very recently, we initiated to adopt the canonical angles introduced in [19] as the phases of a sectorial complex matrix whose numerical range does not contain the origin [22]. We studied various properties of matrix phases, some of which are reviewed and extended later.…”

Section: Introductionmentioning

confidence: 99%

“…By continuity, lim ↓0 φ(B ) = φ(B U ). It follows from Theorem 6.12 in[22] that λ(A U B ) can take values in[φ(A U ) + φ(B ), φ(A U ) + φ(B )] ⊂ (γ(A ) + γ(B ) − π, γ(A ) + γ(B ) + π)and ∠λ(A U B ) ≺ φ(A U ) + φ(B ). Taking limits in both sides, we get ∠λ(A U B U ) ≺ φ(A U ) + φ(B U ) that shows items 2) and 3).It is easy to see that Lemma 3 follows from Theorem 9 immediately.The sufficiency in Lemma 4 also follows from Theorem 9 easily.…”

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confidence: 99%

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Phase Analysis of MIMO LTI Systems

Wei

Wang

Khong

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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In this paper, we introduce a definition of phase response for a class of multi-input multi-output (MIMO) linear time-invariant (LTI) systems, the frequency responses of which are cramped at all frequencies. This phase concept generalizes the notions of positive realness and negative imaginariness. We also define the half-cramped 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. In addition, we derive a sectored real lemma for phase-bounded systems as a natural counterpart of the bounded real lemma.

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Section: Phases Of a Complex Matrixmentioning

confidence: 99%

“…where 0 ≤ β − α < 2π, is a cone. The following lemma can be proved by modifying slightly the proof of its version for sectorial matrices in [22].…”

Section: Phases Of a Complex Matrixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Phase Analysis of MIMO LTI Systems

Wei

Wang

Khong

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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“…We refer readers to [4], [5] for detailed literature related to developments of phases of systems. Recently, [4] has proposed a definition of system phase for sectorial MIMO LTI systems, based on the canonical angles of matrices [6], [7]. According to [4], for a sectorial LTI system with n inputs and n outputs, n frequency-dependent phases can be defined, as the counterpart to the n magnitudes defined via its singular values.…”

Section: Introductionmentioning

confidence: 99%

When Small Gain Meets Small Phase

Zhao¹,

Chen²,

Li³

2022

Preprint

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In this paper, we investigate the feedback stability of multiple-input multiple-output linear time-invariant systems with combined gain and phase information. To begin with, we explore the stability condition for a class of so-called easily controllable systems, which have small phase at low frequency ranges and low gain at high frequency. Next, we extend the stability condition via frequency-wise gain and phase combination, based on which a mixed small gain and phase condition with necessity, called a small vase theorem, is then obtained. Furthermore, the fusion of gain and phase information is investigated by a geometric approach based on the Davis-Wielandt shell. Finally, for the purpose of efficient computation and controller synthesis, we present a bounded & sectored real lemma, which gives state-space characterization of combined gain and phase properties based on a triple of linear matrix inequalities.

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Small Phase Theorem

Chen

Wang

2020

Encyclopedia of Systems and Control

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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.

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On the phases of a complex matrix

Cited by 38 publications

References 27 publications

Phase Analysis of MIMO LTI Systems

Phase Analysis of MIMO LTI Systems

When Small Gain Meets Small Phase

Small Phase Theorem

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