Stability Analysis of Systems With Generalized Frequency Variables

Abstract

Abstract-A class of large-scal, multi-agent systems with decentralized information structures can be represented by a linear system with a generalized frequency variable. In this paper, we investigate fundamental properties of such systems, stability and D-stability, exploiting the dynamical structure. Specifically, we first show that such system is stable if and only if the eigenvalues of the connectivity matrix lie in a region on the complex plane specified by the generalized frequency variable. The stability region is characterized in terms of polynomial inequalities, leading to an algebraic stability condition. We also show that the stability test can be reduced to a linear matrix inequality (LMI) feasibility problem involving generalized Lyapunov inequalities and that the LMI result can be extended for robust stability analysis of systems subject to uncertainties in the interconnection matrix. We then extend the result to D-stability analysis to meet practical requirements, and provide a unified treatment of D-stability conditions for ease of implementation. Finally, numerical examples illustrate utility of the stability conditions for the analysis of biological oscillators and for the design of cooperative stabilizers.

Generalizing the KYP Lemma to Multiple Frequency Intervals

“…The sufficient lmi condition is valid for a larger class of curves Λ(Φ, Ψ). For instance, for the analysis of systems with generalized frequency variables [8], a frequency domain inequality needs to be checked, not on the imaginary axis, but on a curve recognized as the Nyquist plot of a rational function. Such a curve can be characterized as a set of roots to a polynomial [6], and multiple segments of such a curve are represented as Λ(Φ, Ψ), with Assumptions 1 and 2 possibly violated.…”

“…Such curves appear, for instance, when considering multiple segments on a circle or straight line, which correspond to multiple frequency ranges for discrete-time and continuous-time systems, respectively. The more interesting situation that fits into our setting is the frequency domain analysis of linear time-invariant systems with a generalized frequency variable [6,8]. The transfer function of such systems is expressed by C(λI − A) −1 B + D, where the generalized frequency variable λ = p(s) varies on the Nyquist plot of p(s) in the complex plane.…”

“…This class of systems was proposed to treat a class of linear homogeneous multiagent systems, where 1/p(s) is the common proper transfer function of the agents and (A, B, C, D) captures the interaction or information exchange structure. When p(s) is a rational function, its Nyquist plot coincides with the set of roots of a polynomial equality [6,8].…”

Design of Coupled Harmonic Oscillators for Synchronization and Coordination

“…The framework of the master stability equation [18,19] is directly useful for both analysis and design of coupling functions with a specific (e.g. diffusive) structure for synchronization with orbital stability [20].…”

Range extension control system for electric vehicles by LTI modeling with generalized frequency variable

“…As an alternative, distributed control is an ideal solution to such problems. Proposed by Hara etc., a formation control methodology was investigated for multiple homogeneous dynamic agents which are locally stabilized by identical controllers [9]- [11]. One of the main contributions of this literature is the establishment of a stability criterion for a linear system with generalized frequency variables.…”