Robust stability and control for systems with outputs

• Applicants: Professor Dr. Sergey Dashkovskiy; Professor Dr. Andrii Mironchenko
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 552243517

This project aims at developing the input-to-output stability (IOS) theory for infinite-dimensional systems. It is an essential concept in mathematical systems theory and engineering applications such as the design of observers and stabilizing feedbacks with applications to chemical reactors, traffic networks, multi-body systems, etc. For many classes of control systems with outputs, it is known that unmodeled dynamics, quantization errors, external perturbations, etc., can dramatically reduce the performance or even destabilize the control system. Overcoming these challenges requires methods for the verification of output stability and robustness, robust estimation of the full state, and for stability analysis of interconnected systems to ensure the reliability and the efficiency of closed-loop systems. If the full state is known (i.e., when output equals the state), these problems have been successfully studied within the infinite-dimensional input-to-state stability theory. However, this theory is not applicable to practical problems where the full state is not available, and merely the output dynamics should be analyzed and/or used for control purposes. To address these applications, we establish in this project the overarching IOS theory for infinite-dimensional systems with outputs. IOS combines the internal stability of the output dynamics with its robustness with respect to external disturbances of various kinds, which makes this concept appealing from the mathematical point of view. Our results will comprise equivalent characterizations of IOS in terms of weaker stability properties, Lyapunov criteria for IOS, as well as general small-gain conditions for interconnections of infinite-dimensional IOS systems. This will unify the finite-dimensional IOS theory and infinite-dimensional input-to-state stability theory, intensively developed during the last decade utilizing the tools from functional analysis, PDE theory, dynamical systems, and control theory. Our framework will serve as a foundation for nonlinear observer design, robust adaptive control, and robust tracking of nonlinear systems. The infinite-dimensional input-output-to-state stability theory that we initiate in this proposal (and which is fully open even in the special class of time-delay systems) will serve as an ultimate tool for developing the nonlinear detectability theory and the theory of strict dissipativity. In turn, the latter concept is the cornerstone of the mathematical foundations of model predictive control. We will demonstrate the efficiency of our methods by applying them to robust observer design for reaction-diffusion systems as well as for analysis of teleoperation systems with time delays. Overall, we see this project as an initial step in a wide-reaching research program for developing robust control and observation methods for distributed parameter systems with outputs.

• DFG Programme: Research Grants