Brief paperStability of neutral type delay systems: A joint Lyapunov–Krasovskii and Razumikhin approach☆
Introduction
This paper is devoted to an extension of the recent work (Medvedeva & Zhabko, 2015), where a new approach to the stability analysis of time-delay systems which is based on a combination of the Lyapunov–Krasovskii and Razumikhin approaches was presented (see also Alexandrova & Zhabko, 2018a), to the case of linear neutral type time-delay systems.
There is a great variety of techniques to analyze the stability of the systems of neutral type. One category of approaches contains the spectral methods which are based on different variations of the D-subdivision technique, see, for instance, Gu, Niculescu, and Chen (2005) and Michiels and Niculescu (2007) for constructing of the stability crossing curves at the space of delays or some other parameters, and Sipahi and Olgac (2006) for the so-called CTCR (cluster treatment of the characteristic roots) paradigm. The stabilizability problems are addressed in Michiels and Vyhlídal (2005), and Mikhailov criterion is extended to the neutral case in Vyhlídal and Zítek (2009), both for the systems with multiple delays. Another category of techniques is based on the construction of the various types of the Lyapunov functionals. The negative definiteness of their derivatives provides then sufficient conditions for stability expressed in the form of the linear matrix inequalities, see Fridman (2014), Han, Yu, and Gu (2004) and Niculescu (2001), to name a few.
On the contrary, our approach relies on the Lyapunov functionals with a given time-derivative. The first work relating to construction of such functionals for the systems of retarded type is due to Repin (1966). Basing on this early work, the authors in Huang (1989) and Kharitonov and Zhabko (2003) have presented explicit expressions for the functionals which are currently used in the topic. The first of them has the derivative along the solutions of a system, here is the “current” state of a time-delay system and is a positive definite matrix. This functional was shown to admit only a local cubic lower bound (Huang, 1989), whereas derivative of the second one depends on the whole state , thus providing the global quadratic bound for the functional (Kharitonov & Zhabko, 2003). Because of that, this last functional has been successfully applied in the robustness analysis (Kharitonov & Zhabko, 2003), in the construction of the exponential estimates for solutions (Kharitonov & Hinrichsen, 2004), and in other interesting applications (Jarlebring et al., 2011, Kharitonov, 2013, Ochoa et al., 2013). However, the first functional, being considered to be not suitable for solving the practical problems, was recently shown to admit a quadratic lower bound on the special Razumikhin-type set of functions (Razumikhin, 1956), see Medvedeva and Zhabko (2015), what allowed us to give a constructive procedure for the stability analysis using this functional (Medvedeva & Zhabko, 2013).
A first extension of the Lyapunov-Krasovskii approach to the case of linear neutral type delay systems goes back to the work (Castelan & Infante, 1979). A concept of the functional with a given derivative was later generalized to the neutral systems in Rodriguez, Kharitonov, Dion, and Dugard (2004). At the same work, an analogue of the functional introduced in Kharitonov and Zhabko (2003) which admits a quadratic lower bound was constructed, and the robustness issues were analyzed with the help of this functional. Further useful developments and modifications of the approach can be found in the papers (Gomez et al., 2016, Gomez et al., 2017a, Gomez et al., 2017b, Kharitonov, 2005, Velàzquez-Velàzquez and V.L. Kharitonov, 2009).
In the present paper, we prove that the system of neutral type is exponentially stable, if and only if the corresponding difference system is exponentially stable, and the functional with a given time-derivative admits a quadratic lower bound on the special reduced set of functions satisfying a Razumikhin-type inequality and a similar inequality on the derivative. In this, an analogue of the functional from Huang (1989) for the neutral case rather than the one from Kharitonov and Zhabko (2003) is used. As a consequence of this result, we suggest a constructive sufficient stability condition and test it on the example of a scalar equation with one delay. An important advantage of this condition is that it provides the stability region which converges to the exact stability region at the increase of a number of discretization segments, and allows to find the delay margins, what is demonstrated by the examples. It is worthy of mention that the ideas presented in this paper had also proven to be fruitful in the robustness analysis, see Alexandrova (2018). The results of the paper in their preliminary form can be found in Alexandrova and Zhabko (2016). Notice that the work (Gomez, Egorov, & Mondiè, 2018) allowed us to streamline the proofs and to specify the value of constant in the set .
The paper is organized as follows. Preliminary results on the linear time-delay systems of neutral type are given in Section 2. New criterions for the exponential stability and instability are formulated and proven in Sections 3 Stability theorem, 4 Instability theorem respectively. In Section 5, we suggest a constructive procedure for the stability analysis which is based on the proven criterion, and give some illustrative examples. The concluding comments end the paper.
Notation: In this paper, the Euclidean norm for vectors and the corresponding induced norm for matrices, both denoted by , are used; stands for the space of times continuously differentiable -valued functions defined on with the uniform norm is the identity matrix; denotes the zero matrix or vector; is the zero function: , stands for the real part of the complex value denotes the imaginary unit, is the smallest eigenvalue of a matrix notation , where , , means that is an integer between and , inclusively.
Section snippets
Preliminaries
Consider a linear time-delay system of neutral type where , , are the constant matrices, and are nonnegative constant delays. Denote , and suppose that the initial function . Let be a solution of the system with initial condition , , and be a segment of solution corresponding to the values , . The solution is continuous for and continuously differentiable for ,
Main result
Define , and introduce the set Notice that is a compact set. Besides, as it will be shown below, contains the segments of solutions corresponding to the eigenvalues with positive real parts, if the system is unstable. This fact provides a basis for the exponential stability criterion expressed in terms of the positive definiteness of functional on the set
Theorem 2 Let Assumption 1 hold. Given a positive definite matrix
Instability theorem
The instability criterion is a direct consequence of Theorem 2, if the functional satisfying (2) is known to exist. However, to the best of our knowledge, there is no well-developed theory of functionals with prescribed derivative in the case , where an assumption of exponential stability is dropped, except for some particular issues discussed in Gomez et al. (2017b) and Velàzquez-Velàzquez and V.L. Kharitonov (2009) for the systems with multiple commensurate delays. For this reason, we
Methodology for stability analysis. Scalar case
In this section, we present a finite constructive procedure to the stability analysis which is based on Theorem 2 and the fact that the set is compact. For the illustration, consider a scalar equation where , , are the constant coefficients, and is a single delay. The functional satisfying (2) now has the form (see Kharitonov, 2013)
Conclusion
In this paper, a new constructive criterion for the exponential stability of linear neutral type time-delay systems is presented. This criterion reveals the fact that the functional whose time-derivative along the solutions of system (1) is equal to the negative definite quadratic form (2) admits a quadratic lower bound on the special Razumikhin-type set of functions , if the system is exponentially stable. It is worthy of mention that this result can be extended to some classes of
Acknowledgments
The authors would like to thank the anonymous reviewers for the interesting comments and suggestions that helped us to improve the quality of the paper.
Irina V. Alexandrova graduated in 2011 and received her Ph.D. degree in 2015, both from St. Petersburg State University, St. Petersburg, Russia. She is currently an Associate Professor of the Department of Computational Methods in Continuum Mechanics at the Faculty of Applied Mathematics and Control Processes of this university. Her research interests include time delay systems, stability and robust stability analysis.
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Irina V. Alexandrova graduated in 2011 and received her Ph.D. degree in 2015, both from St. Petersburg State University, St. Petersburg, Russia. She is currently an Associate Professor of the Department of Computational Methods in Continuum Mechanics at the Faculty of Applied Mathematics and Control Processes of this university. Her research interests include time delay systems, stability and robust stability analysis.
Alexey P. Zhabko received the M.A. Degree in 1973, the Candidate of Science Degree in Automatic Control in 1981, and the Doctor of Science Degree in Automatic Control in 1992, all from the Leningrad State University, Russia. He is currently the Head of the Control Theory Department at the Faculty of Applied Mathematics and Control Processes, St. Petersburg State University, St. Petersburg, Russia. His research interests include control theory, time delay systems, stability, identification and robust control.
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The research was supported by St. Petersburg State University, Russia (projects 9.37.157.2014 and 9.42.1045.2016). The material in this paper was partially presented at the 20th International Conference on System Theory, Control and Computing (ICSTCC 2016), October 13–15, 2016, Sinaia, Romania. This paper was recommended for publication in revised form by Associate Editor Akira Kojima under the direction of Editor Ian R. Petersen.