An LMI based approach to stabilize a type of nonlinear uncertain neutral-type delay systems
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An LMI based approach to stabilize a type of nonlinear uncertain neutral-type delay systems Chong Ke1
· Xingyong Song2
Received: 13 December 2019 / Revised: 14 September 2020 / Accepted: 27 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper proposes a linear matrix inequality (LMI) based approach for stability analysis of a type of nonlinear uncertain neutral-type delay systems. The delays in states, state derivatives and inputs are known and constant. By constructing a Lyapunov–Krasovskii functional (LKF), a stability criterion in the form of LMI is derived. In this paper, we treat these nonlinear terms to be norm-bounded and construct a Lyapunov matrix inequality considering system states, delayed states, delayed state derivatives, and delayed inputs. The stability criterion of this nonlinear neutral-type delay system is then obtained. To this end, a numerical example is given to demonstrate the feasibility of the proposed approach. The developed method for stability analysis can potentially be applied to critical real-world applications such as the down-hole drilling system. Keywords Lyapunov–Krasovskii functional · Delay-dependent stability · Linear matrix inequality · Neutral-type delay
1 Introduction Over the past few decades, stability analysis and stabilization of neutral delay differential equations (NDDEs) have drawn much attention both in theory and in practice. Many of the practical systems, such as chemical reaction plants, population dynamics, and beam structured mechanical systems, can be described by the NDDEs. Particularly, the hyperbolic partial differential equations (PDEs) can be converted to neutral-type time-delay systems [1], for applications such as lossless transmission line in electrical networks [2], lossless propagation models [3], etc. Compared with the retarded delay differential equations (RDDEs), the existence of the delayed derivative terms in the NDDEs can often deteriorate the system performance and become a common source of instability. Conventional approaches to determine and analyze the stability of the NDDEs can be divided into two categories.
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Xingyong Song [email protected] Chong Ke [email protected]
1
HDD R&D, Western Digital Corporation, 5601 Great Oaks Pkwy, San Jose, CA 95119, USA
2
College of Engineering, Texas A&M University, 510 Ross St, College Station, TX 77843, USA
The first is the spectrum approach, which is based on the eigenvalue analysis by solving the characteristics equation. Note that the presence of exponential type transcendental terms inside the characteristics equation makes the stability analysis difficult for the NDDEs. The other method, named as the Lyapunov–Krasovskii functional approach, is to find a positive definite Lyapunov-like function with a negative definite time derivative in the sense of Lyapunov. Researchers have tried to find different forms of LKF to ensure Lyapunov stability condition, such as the reciprocally convex approach [4,5], delay partitioning approach [6,7], constructing
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