LOSSLESSNESS, FEEDBACK EQUIVALENCE, AND THE GLOBAL STABILIZATION OF DISCRETE-TIME NONLINEAR-SYSTEMS

In this paper a necessary and sufficient condition for a nonlinear system of the form [GRAPHICS] to be lossless is given, and it is shown that a lossless system can be globally asymptotically stabilized by output feedback if and only if the system is zero-state observable. Then, we investigate conditions under which a nonlinear system Sigma can be rendered lossless via smooth state feedback. In particular, we show that this is possible if and only if the system in question has relative degree (0, ..., 0) and has lossless zero dynamics. Under suitable controllability-like rank conditions, we prove that nonlinear systems having relative degree (0, ..., 0) and lossless zero dynamics can be globally stabilized by smooth state feedback. As a consequence, we obtain sufficient conditions for a class of cascaded systems to be globally stabilizable. The global stabilization problem of the nonlinear system Sigma without output is also investigated in this paper by means of feedback equivalence. The philosophy of this paper was inspired by [5]. Indeed, some of the results are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear.