Exponentially dissipative nonlinear dynamical systems: A nonlinear extension of strict positive realness
A feedback interconnection consists of a pair of control systems with the output of one system being used as control of the other and vice-versa. When both systems are linear, the positivity and small gain theorems are important tools to ascertain the stability of a feedback interconnection. This paper presents an extension of the notion of dissipative dynamical systems allowing for the construction of a nonlinear analogue to strict positive realness and strict bounded realness. The notion of exponential dissipativity is introduced using appropriate exponentially weighted storage functions and supply rates. Special attention is given to systems with quadratic supply rates. It is shown that for these systems exponential dissipativeness can be characterized in terms of the vector fields governing the system and the output function. Stability of feedback interconnections is analyzed using exponential dissipativeness instead of passivity. Also, using the notion of exponentially dissipative systems with appropriate storage function and supply rate, it is possible to construct Lyapunov functions for interconnected systems. These results are used to construct globally stabilizing globally optimal static feedback controllers and globally stabilizing locally optimal dynamic feedback controllers.
scientific article; zbMATH DE number 2166707