Passivity Analysis for Discrete-Time Periodically Controlled Nonlinear Systems

In this paper, we generalize the classical definition of passivity to discrete-time periodically controlled nonlinear systems that are open loop nonpassive and closed-loop feedback passive. In the classical passivity theory, a system is said to be passive if there exists a storage function such that the increase in storage function is bounded by the energy supplied to it at every time step. This definition may be overly restrictive. Therefore, we consider a periodically controlled system to be passive as long as the increase in storage function at the end of a period as compared to the beginning of the period is bounded by the accumulated energy supplied to it within this period. Because the input/output structure switches periodically, we need to exploit the zero dynamics of the closed-loop system and relate it to the passivity of the periodically controlled system. We prove that: 1) There exists a maximum allowable transmission ratio (MATR) between the time steps at which the system evolves open and closed-loop within a period such that the generalized passivity of system zero dynamics is guaranteed; and 2) If the system zero dynamics is passive, the original periodically controlled nonlinear system is also locally passive.