A Novel Frequency-Domain Approach for the Exact Range of Imaginary Spectra and the Stability Analysis of LTI Systems With Two Delays

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. INDEX TERMS Time-delay system, Dixon resultant, stability, frequency domain.

Impact of load sharing schemes on the stability delay margins computed by Rekasius substitution method in load frequency control system with electric vehicles aggregator

“…The future studies will focus on the computation of robust stability regions for two-area LFC-EVs systems in the presence of uncertainties in system parameters and the delay margin computation of multiarea LFC-EVs with multiple incommensurate time delay using Rekasius substitution together with Dixon resultant and discriminant theory. 18 , i = 1,2…”

“…Direct method: Removal of exponential terms in the quasi-characteristic polynomial 15 ; 3. Rekasius substitution method [16][17][18] ; 4. Delay space rescaling approach 19 ; 5.…”

The effect of demand response control on stability delay margins of load frequency control systems with communication time-delays

“…As future work, the impact of DR control loop on stability regions in the PI controller parameters space will be investigated for robust PI controller design. Furthermore, incommensurate timedelays for multiarea realistic LFC-DR systems will be considered to investigate the impact of the DR control loop on the stability delay margin using Rekasius substitution together with Dixon resultant and discriminant theory [39,49] and time-domain indirect methods [30][31][32].…”

Stabilization of Two Electricity Generators

“…ere are several stabilization results such as the exponential stabilization [14,17,18,20], the asymptotic stabilization [5,6,16,22,23], the uniformly ultimately boundedness [9][10][11], the feedback stabilization [15,24], the stochastic stabilization [12,21], or the mean square stabilization [13].…”

“…ere are many kinds of nonlinearities which impede to reach the stabilization of electricity generators, and some examples of these nonlinearities are the arbitrary switching [9][10][11][12][13], the time-delays [14][15][16][17][18], the impulse perturbations [19,20], or the unknown nonlinearities [21][22][23][24]. e major issue is that in most of the cases, the mentioned nonlinearities are unknown.…”