Qualitative equivalences of ISS and l_p-gain stability properties for discrete-time nonlinear systems

Input-to-state stability (ISS) and input–output L₂-gain are two important robust stability concepts in analyzing the interconnections of nonlinear dynamical systems. In this paper, we demonstrate several qualitative equivalences between different ISS related properties and l_p-gain properties for discrete-time systems via nonlinear changes of coordinates. Prior to that, certain summation-to-summation estimates are shown to be equivalent to the standard definitions of ISS and integral ISS (iISS) which lead to sufficient Lyapunov function conditions to verify l_p-gain properties. Combined with known results on the equivalence of 0-input global asymptotic stability and iISS, and a superposition principle, we subsequently outline interesting implicative relationships between various discrete-time robust stability properties.