Stability and robustness issues for linear and nonlinear time delay systems

• Applicant: Professorin Irina Aleksandrova, Ph.D.
• Subject Area: Mathematics ; Automation, Mechatronics, Control Systems, Intelligent Technical Systems, Robotics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 552846308

Time delay equations have found various applications in engineering, robotics, medicine, biology and population dynamics. In this project, the exponential, asymptotic and input-to-state stability (ISS) as well as robustness issues are studied for linear and nonlinear time delay systems of both retarded and neutral types. The long term goal is to advance the Lyapunov-Krasovskii approach. For a class of linear systems, a special attention is paid to the delay Lyapunov matrix concept which naturally extends the solution of the classical Lyapunov matrix equation to the case of time delay systems. The focus is on developing new effective stability criteria expressed exclusively through the delay Lyapunov matrix and, most importantly, verifiable in a finite number of operations. This topic has received outstanding attention in recent years. It is thus of particular importance to address the computational aspects of stability criteria, applying various approximation schemes for Lyapunov-Krasovskii functionals. The resulting stability criteria can be expressed, for instance, via the positive definiteness of a block matrix consisting of discrete Lyapunov matrix values. New computational algorithms for the delay Lyapunov matrix in difficult cases will also be developed, and transferring the framework to a class of linear time-varying systems with delay will be discussed. For a class of nonlinear systems, the focus is on proving new asymptotic stability conditions at the junction of Lyapunov-Krasovskii and Razumikhin frameworks as well as on the practical aspects of using the Lyapunov-Krasovskii functionals. A recent open problem of existence of the ISS Lyapunov-Krasovskii functional with point-wise dissipation will be addressed for a class of nonlinear time delay systems in close connection with the delay Lyapunov matrix framework. Expected outcomes are new constructive stability conditions for both linear and nonlinear time delay systems, new computational algorithms for the delay Lyapunov matrix for linear systems with non-commensurate delays, as well as new ISS conditions and applications in the field of robustness analysis.

• DFG Programme: WBP Position