Claudia Califano - Nonlinear Time-Delay Systems: A Geometric Approach

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This book is devoted to nonlinear time-delay control systems. Although they include the class of linear time-delay systems, the specific mathematical tools valid (only) for this subclass of systems will not be developed or used herein.

Thus, in this introductory chapter, a sketch is given of what can be found elsewhere (Richard 2003) and which will not be considered herein.

For the subclass of linear time-delay systems in continuous time, the use of the Laplace transform yields quasi-polynomials in the Laplace variable and in Gu et al. (2003), Michiels et al. (2007), Niculescu (2001). Among those systems, one may distinguish betweem the so-called retarded systems (Fridman 2014) described by differential equations where the highest differentiation order of the output, or the state, is not delayed, and the so-called neutral systems (Fridman 2001) whose equations involve delayed values of the highest differentiation order of the output, or the state.

Thus, the Laplace transform still enables an inputoutput analysis of linear time-delay systems (Olgac and Sipahi 2002, Sipahi et al. 2011). This approach can hardly be extended to more general nonlinear time-delay systems which are the main focus of this book. Finite dimension approximations may be appealing, but have a limited interest due to stability issues (Insperger 2015).

Discrete-time linear systems with unknown delays are under interest in Shi et al. (1999). Discrete-time linear systems with varying delays are under interest with an ad hoc predictor design in Mazenc (2008).

Stability analysis and stabilization of linear time-delay systems require general tools derived from the Lyapunov theory as in Fridman (2001), Kharitonov and Zhabko (2003).

Stability of linear systems with switching delays is tackled in Mazenc (2021) using trajectory-based methods and the so-called sup-delay inequalities.

Some of the most significant historic results obtained for the general class of nonlinear time-delay systems are about the analysis of their stability, thanks to a generalization of the Lyapunov theory, the so-called Krasovskii-type approach (Gu et al. 2003). This approach can hardly be circumvented even in the case of linear time-delay systems (Fridman 2001).