An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems

By Nejat Olgaç, Rifat Sipahi. IEEE Transactions on Automatic Control, 2002

638 citations recorded by the source corpus.

A general class of linear time invariant systems with time delay is studied. Recently, they attracted considerable interest in the systems and control community. The complexity arises due to the exponential type transcendental terms in their characteristic equation. The transcendentality brings infinitely many characteristic roots, which are cumbersome to elaborate as evident from the literature. A number of methodologies have been suggested with limited ability to assess the stability in the parametric domain of time delay. This study offers an exact, structured and robust methodology to bring a closure to the question at hand. Ultimately we present a unique explicit analytical expression in terms of the system parameters which not only reveals the stability regions (pockets) in the domain of time delay, but it also declares the number of unstable characteristic roots at any given pocket. The method starts with the determination of all possible purely imaginary (resonant) characteristic roots for any positive time delay. To achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation. It is proven that the number of such resonant roots for a given dynamics is finite. Each one of these roots is created by infinitely many time delays, which are periodically distributed. An interesting property is also claimed next, that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency. These two unique findings facilitate a simple and practical stability method, which is the highlight of the work.

Subfields: Computer Networks and Communications, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/1b418ceed5b8e16366c4c9ba32e046bb54674038

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Networks and Communications

1. Yong He
2. Hongbing Zeng
3. Min Wu
4. Jinhua She
5. A. Galip Ulsoy
6. Chika O. Nwankpa
7. Sun Yi
8. Saffet Ayasun

In Control and Systems Engineering

1. Yong He
2. Min Wu
3. Hongbing Zeng
4. Jinhua She
5. Saffet Ayasun
6. A. Galip Ulsoy
7. Chika O. Nwankpa
8. Sun Yi

Citations in the corpus

• Eigenvalue Assignment via the Lambert W Function for Control of Time-delay Systems (2010, methodology): “A stability analysis called the “Direct Method”, in which a simplifying substitution is used for the transcendental terms in the characteristic equation (Olgac and Sipahi, 2002), was applied for active vibration suppression by Sipahi and Olgac (2003).”
• An Exact Method for Computing Delay Margin for Stability of Load Frequency Control Systems With Constant Communication Delays (2016, background): “The sign of root sensitivity is defined as root tendency [10], [12]”
• Free-Matrix-Based Integral Inequality for Stability Analysis of Systems With Time-Varying Delay (2015, methodology): “Note that many methods for a constant-delay system, for example, the one in [7] that exploited the characteristic equation of the system, cannot be applied to a time-varying-delay system.”
• Multiplicity-induced-dominancy in parametric second-order delay differential equations: Analysis and application in control design (2020, neutral): “…is the frequency domain approach since in the Laplace domain, where a number of effective methods have been proposed, the stability analysis amounts to studying the distribution of the characteristic quasipolynomial function’s roots, see for instance [3, 4, 18, 19, 21, 24, 35, 43, 54, 55, 59, 60].”
• Delay-Dependent Stability for Load Frequency Control With Constant and Time-Varying Delays (2012, neutral): “One is the direct methods, such as tracing critical eigenvalue [14] and cluster treatment of characteristic roots [15]–[17].”