Finite, fixed and prescribed-time stability and stabilization of nonlinear negative imaginary systems?

One of the most challenging problems in the negative imaginary (NI) systems framework is to achieve stability, particularly rated stability for the interconnected systems as it uses the positive-feedback connection, which may have a destabilizing effect. In this paper, rated convergence properties are introduced in the framework of nonlinear negative imaginary (NNI) systems theory. New NI definitions for a general nonlinear system are provided using dissipative notions and after that the relation between these NI definitions and rated stability is investigated. Further, several positive-feedback interconnections are discussed to establish finite-time, fixed-time and prescribed-time NI properties and their respective stabilities about origin under certain assumptions. Moreover, a methodology of finite-time, fixed-time and prescribed-time NI based control that renders a NNI system, finitetime, fixed-time and prescribed-time stable about origin inspite of the positive-feedback is developed. Finally, encouraging academic and real-world examples are reported to test the validity of the theoretical results.

Automated summary

This paper introduces the application of rated convergence properties in the framework of nonlinear negative imaginary systems theory. New definitions of negative imaginary for general nonlinear systems are provided, and the relationship between these definitions and rated stability is investigated. Additionally, positive-feedback interconnections are discussed and a methodology of control for maintaining stability under positive feedback is developed. Theoretical results are validated through academic and real-world examples.