An Equivalent Condition for Stability Analysis of LTI Systems with Bounded Time-invariant Delay

Abolpour, Roozbeh; Khayatian, Alireza; Dehghani, Maryam; Rokhsari, Alireza

doi:10.1016/j.amc.2022.127585

Cited by 2 publications

(3 citation statements)

References 37 publications

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“…Assume τ^ is a positive number such that d(s,τ^) has some roots on the imaginary axis, for instance jω^. In fact, d(jω^,τ^)=0 which successively implies dm(jω^,z^)=0, where z^=real(e−jtrueω^)+j×imag(e−jtrueω^) due to (4) (Abolpour et al, 2023). This fact means z^ is a critical number of the delay system that is assuredly included in {zi}i=1M based on the statement of the theorem.…”

Section: Resultsmentioning

confidence: 99%

“…The second source is the mathematical approximations that are exploited to derive the LMI conditions. This fact motivates some researchers to bypass the LKF idea and exploit the quasi-characteristic quasi-polynomial of the delay system (Kharitonov et al, 2005; Ergenc et al, 2007; Boussaada et al, 2020; Abolpour et al, 2023). These approaches are known as numerical methods due to their dependency on some information which is numerically obtained.…”

Section: Introductionmentioning

confidence: 99%

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Finding the exact value of the maximum allowable upper bound of the delay parameters in the multiple-delay LTI systems

Abolpour

Dehghani

2023

Journal of Vibration and Control

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This paper deals with the problem of determining the exact value of the maximum allowable upper bound (MAUB) of delay of an LTI system in the presence of the delay parameters. The delay parameter is supposed to be time-invariant, and this paper proposes a systematic method to exactly solve the mentioned problem. For this purpose, a modified system matrix is defined based on the delay system’s model that involves an extra complex variable on the boundary of the unit circle. It will be shown that the characteristic polynomial of the modified system can equivalently assess the system’s stability. The characteristic polynomial of the modified system is a quadratic function of the imaginary part of the extra complex variable which enables us to rewrite the polynomial based on only the real part of the extra variable. This property is exploited to develop an explicit formula to find the MAUB based on the critical numbers (complex numbers that force the modified characteristic polynomial to have pure imaginary roots) and their related pure imaginary roots. The method is tested for some sample delay systems and the results demonstrate a tighter bound for the MAUB of the systems.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Finding the exact value of the maximum allowable upper bound of the delay parameters in the multiple-delay LTI systems

Abolpour

Dehghani

2023

Journal of Vibration and Control

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“…The Evans root locus method is not applicable to multi-input-multioutput (MIMO) systems and cannot be applied to time-varying models. It can be time-consuming as a graphical method, especially for complicated higher-order systems and it does not account for time delays (Abolpour et al, 2023;Cogan et al, 2009;Fatoorehchi and Djilali, 2023;Kavitha and Ramakalyan, 2014;Shinskey, 1996). It is worthwhile to mention that Yang et al (2022) have generalized the Routh-Hurwitz technique to fractional-order linear systems, and their method is called the fractional-order Routh-Hurwitz stability criterion.…”

Section: Introductionmentioning

confidence: 99%

A novel technique for stability analysis of linear time-invariant and nonlinear continuous-time control systems based on Markov parameters and Hankel matrices

Fatoorehchi,

Ehrhardt

2024

Transactions of the Institute of Measurement and Control

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In this paper, we present an innovative approach for assessing the stability of linear time-invariant (LTI) systems. The key innovations in our methodology include the calculation of Markov parameters, the extension to nonlinear continuous-time systems, and significantly improved computational efficiency. Our novel approach leverages Markov parameters derived from the characteristic polynomial of the system. Unlike traditional methods, we employ the Leverrier-Takeno algorithm to efficiently determine the coefficients of the characteristic polynomial for state-space representation. The systematic calculation of Markov parameters using a specialized Maclaurin series expansion enhances our method’s precision. One of our major achievements is the rigorous verification of stability through Hankel matrices, ensuring positive definiteness using the Cholesky decomposition algorithm. In addition, we introduce a second stability criterion based on iterated polynomial long divisions, although we acknowledge its limitations in handling high-order systems due to computational inefficiency. Our pioneering contributions extend beyond LTI systems. We adapt our innovative techniques to analyze the local stability of nonlinear continuous-time systems by introducing the concept of Jacobian matrices and employing the indirect Lyapunov method. This expansion broadens the applicability of our approach to a wider range of real-world systems. Notably, our approach exhibits remarkable computational efficiency. According to CPU-time analysis, our first stability criterion outperforms the classical Routh–Hurwitz method by more than 12 times for dynamic systems with orders ranging from 2 to 180. This significant reduction in computation time positions our method as a valuable option for real-time stability analysis applications.

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An Equivalent Condition for Stability Analysis of LTI Systems with Bounded Time-invariant Delay

Cited by 2 publications

References 37 publications

Finding the exact value of the maximum allowable upper bound of the delay parameters in the multiple-delay LTI systems

Finding the exact value of the maximum allowable upper bound of the delay parameters in the multiple-delay LTI systems

A novel technique for stability analysis of linear time-invariant and nonlinear continuous-time control systems based on Markov parameters and Hankel matrices

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