Infinite dimensional nonlinear systems with state-dependent delays and state suprema: Analysis, observer design and applications
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The prime objective of this research is to investigate systems with state-dependent delays in a unified, well rounded, global and coherent framework from first principles. Such systems are inherently nonlinear and infinite dimensional in nature. The delay may either depend explicitly or implicitly on the state of the system. Our goal and contribution is four-fold. Firstly, to give an information structure i.e., to define the Cauchy problem and characterize the state space for such systems. We use the framework of Banach function spaces with the topology of uniform convergence. In systems with time-varying and/or state-dependent delays, causality plays a crucial role. Throughout the analysis and synthesis problems in this research, causality of the system is ensured. Secondly, we perform the inversion of the state-dependent delay i.e., we use the information on the delay and retrieve the state vector of the system. We use our newly established technique of Delay Injection to recover the state of the system and solve the observer design problem. Thirdly, we analyze and design controllers and observers for a special class of systems with state-dependent delays namely systems evolving with state suprema. We nail down the rich structure possessed by these systems and show that the discrete counterpart of these systems can be expressed as Multi-Mode Multi-Dimensional (M3D) systems. Fourthly, we investigate the spectrum of higher order linear time delay systems in the framework of matrix Lambert W functions and give some counter examples to show that the already existing well established literature suffers from some serious limitations. We support our theory with practical applications including estimator design for a subsonic rocket car for soft landing, nonlinear observer design for a turning process for high precision machining and temperature control of a tank with state-dependent delay; and analysis of gene expression.