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 /spl Sigma/, given by x(k+1)=f(x(k))+g(x(k))u(k), y(k)=h(x(k))+J(x(k))u(k), 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 /spl 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 /spl lcub/0,...,0/spl rcub/ and has lossless zero dynamics. Under suitable controllability-like rank conditions, we prove that nonlinear systems having relative degree /spl lcub/0,...,0/spl rcub/ 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 /spl Sigma/ without output is also investigated in this paper by means of feedback equivalence. 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.<
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