Input-to-State Stability of Distributed Parameter Systems

Titelangaben

Mironchenko, Andrii:
Input-to-State Stability of Distributed Parameter Systems.
Passau , 2023 . - 285 S.
( Habilitation, 2022 , Universität Passau, Fakultät für Informatik und Mathematik)
DOI: https://doi.org/10.48550/arXiv.2302.00535

Volltext

Link zum Volltext (externe URL):

Abstract

Nonlinear distributed parameter systems with both distributed and boundary inputs are used to model a broad range of phenomena, including chemical reactors, fluid and gas dynamics, traffic networks, multi-body systems (e.g., robotic arms, flexible elements), adaptive optics, fluid-structure interactions (e.g., dynamics of aircraft wings), etc. For many classes of such systems, it is known that small disturbances coming from actuator and observation errors, hidden dynamics, and external disturbances can dramatically reduce the performance, alter the stability, or even destabilize the control system. Counteracting to these challenges requires the development of methods for the design of robust controllers and observers for nonlinear distributed parameter systems that ensure the reliability and efficiency of closed-loop systems. This objective has to be achieved despite the fact that usually infinite-dimensional systems have to be controlled using finitely many (and usually very few) actuators and sensors that can be accessed possibly only at some discrete moments of time and which can frequently be placed merely at the boundary of the spatial domain.

A systematic framework that solves the counterparts of these problems for finite-dimensional nonlinear systems was developed around the notion of input-to-state stability (ISS), introduced by E. Sontag at the dawn of the 1990s. ISS combines two different types of stability behavior: stability in the sense of Lyapunov and input-output stability. The unified treatment of external and internal stability has made ISS a central tool in robust stability analysis. ISS plays a vital role in many fields of nonlinear control, including robust stabilization of nonlinear systems, stabilization via controllers with saturation, design of robust nonlinear observers, nonlinear detectability, ISS feedback redesign, stability of nonlinear networked control systems, supervisory adaptive control, and others.

In this work, we develop an input-to-state stability theory for a broad class of infinite-dimensional systems encompassing partial differential equations (PDEs), time-delay systems, ordinary differential equations (ODEs), ensembles, as well as interconnections of heterogeneous systems consisting of an arbitrary number of finite- and infinite-dimensional components, with both in-domain and boundary couplings. On the one hand, this provides a solid basis to treat the questions in robust control and observation of infinite-dimensional systems summarized above. On the other hand, the interest in such a general theory is driven by the desire to create an overarching framework that provides a unifying view of ISS theories for various classes of systems.