Abstract
This paper presents a novel frequency-domain approach to reveal the exact range of the imaginary spectra and the stability of linear time-invariant systems with two delays. First, an exact relation, i.e., the Rekasius substitution, is used to replace the exponential term caused by the delays in order to transform the transcendental characteristic equation to a quasi-polynomial. Second, this quasi-polynomial is uniquely tackled by our proposed Dixon resultant and discriminant theory, leading to the elimination of delay-related elements and the revelation of the exact range of the frequency spectra of the original system of interest. Then, by sweeping the frequency over this obtained range, the stability switching curves are declared exhaustively. Last, we deploy the cluster treatment of characteristic roots (CTCR) paradigm to reveal the exact and complete stability map. The proposed methodologies are tested and verified by a numerical method called Quasi-Polynomial mapping-based Root finder (QPmR) over an example case.
Highlights
In this paper, the frequency spectra and the stability of a general class of linear-time invariant systems with two delays are analyzed from a new perspective: dx dt = Ax (t) + B1x (t − τ1) + B2x (t − τ2) (1)where x ∈ Rn is the state vector; A, B1, B2 are constant, known matrices in Rn×n; and τ1, τ2 are rationally independent delays
Based on the obtained stability maps, various control concepts have been developed with the aim to increase the delay robustness of systems against large delays, for example, the sign inverting [10], [11] and delay scheduling controls [12], [13]
The Dixon resultant and discriminant are used successively to eliminate delay-related elements in the characteristic eqution to obtain the exact range of the frequency spectra of the original system (1)
Summary
The Dixon resultant and discriminant are used successively to eliminate delay-related elements in the characteristic eqution to obtain the exact range of the frequency spectra of the original system (1). By sweeping such frequency range, the complete set of stability switching curves are determined.
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