Generalized Dissipativity and Nonlinear Negative Imaginary Systems

Nonlinear negative imaginary systems find application in a range of engineering fields, including the control of flexible structures and air vehicles. Nevertheless, unlike their linear counterparts, the theory for nonlinear negative imaginary systems is not as well-established. In this paper, we propose a generalized k-th order dissipativity framework with respect to a supply rate which is a function of the k-th time-derivative of the system output. It is shown that positive realness and negative imaginaryness can be defined in this general framework in a unified manner. Then, necessary and sufficient conditions for first order dissipativity of nonlinear systems are obtained. These capture and are more general than the negative imaginary property. Moreover, the concept of exponentially negative imaginary systems for both linear and nonlinear systems is developed and the required conditions are obtained.