The dissipativity analysis of the large-scale dynamical system with the supply rate <i>S</i> ( <i>u,ẏ</i> )

This paper investigates dissipative theory and feedback regulation within the dissipation framework, employing the vector supply rate S(u,y˙) for both linear and nonlinear large-scale dynamical systems (LSDS). A criterion, both necessary and sufficient, is established for the dissipativity of this class of systems. Furthermore, conditions that characterize the vector negative imaginary (NI) property, vector input strictly negative imaginary (ISNI) property, and vector output strictly negative imaginary (OSNI) property in nonlinear LSDS are derived. Under specific assumptions, the relationship between the vector NI property and the L2-gain condition is explored. Additionally, the connection between asymptotic stability and vector dissipativity in the context of the vector supply rate S(u,y˙) is analysed. Several examples are provided to illustrate the validity and practical significance of the findings.