  We consider a general class of nonlinear, constrained, discrete-time systems
whose dynamics are parametrized by a set of gains. We define the semiglobal,
practical, asymptotic stability (SPAS) of compact sets for this class of
systems, and we provide a Lyapunov characterization of such sets. A set A that
is SPAS with respect to a given system need not be an attractor for that
system. Relative to existing characterizations of similar qualitative
behaviors, our SPAS theorem does not require the existence of an asymptotically
stable attractor associated to a nominal counterpart of the given dynamics.
