Equivalences of Stability Properties for Discrete-Time Nonlinear Systems

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Abstract

Several qualitative equivalences are demonstrated between various robust stability properties for discrete-time nonlinear systems. In particular, two different summation-to-summation estimates are shown to be equivalent to the standard definitions 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.

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1

School of Electrical Engineering and Computer Science, The University of Newcastle, Newcastle NSW 2308, Australia.

2

Department of Electrical & Electronic Engineering, University of Melbourne, Melbourne, Victoria 3010, Australia.

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