Equivalences of stability properties for discrete-time nonlinear systems

Several qualitative equivalences are demonstrated between various robust stability prop-erties for discrete-time nonlinear systems. In particu-lar, two different summation-to-summation estimates are shown to be equivalent to the standard defini-tions of Input-to-State Stability (ISS) and integral ISS (iISS), respectively. Furthermore, ISS and iISS are shown to be qualitatively equivalent, via a nonlinear change of coordinates, to linear and nonlinear l₂-gain properties, respectively. These equivalences, together with a previous result on the equivalence of 0-input global asymptotic stability and iISS provide interesting relationships between discrete-time robust stability properties that do not hold in continuous-time.