International Journal of Dynamics and Control https://doi.org/10.1007/s40435-020-00701-3 An LMI based approach to stabilize a type of nonlinear uncertain neutral-type delay systems Chong Ke 1 · Xingyong Song 2 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, popu- lation dynamics, and beam structured mechanical systems, can be described by the NDDEs. Particularly, the hyper- bolic 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], loss- less 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. B Xingyong Song songxy@tamu.edu Chong Ke cyruskirn@gmail.com 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 def- inite time derivative in the sense of Lyapunov. Researchers have tried to find different forms of LKF to ensure Lya- punov stability condition, such as the reciprocally convex approach [4,5], delay partitioning approach [6,7], construct- ing a novel LKF with triple/quadruple integral terms [8,9], etc. In addition, various inequality techniques have been applied to further reduce the upper bound of the derivative of LKF. These techniques include Jensen’s inequality [10], Bessel-Legendre inequality [11], Wirtinger-based inequal- ity [12], and auxiliary function based inequality [13], etc. Recently, an LQR (Linear Quadratic Regulator) based slid- ing mode control is proposed using an integral switching surface [14]. And a new envelope approach that ensures all the poles are inside this envelope is proposed to ensure α- stability [15]. An example of real-world application of the nonlinear neutral-type time-delay system is down-hole drilling, which is widely used for down-hole energy production especially in the oil and gas industry. It is mainly used to create a well- bore for energy extraction. Due to the slenderness of the 123