On uniform global asymptotic stability of nonlinear discrete-time systems with applications

This paper presents new characterizations of uniform global asymptotic stability for nonlinear and time-varying discrete-time systems. Under mild assumptions, it is shown that weak zero-state detectability is equivalent to uniform global asymptotic stability for globally uniformly stable systems. By employing the notion of reduced limiting systems, another characterization of uniform global asymptotic stability is proposed on the basis of the detectability for the reduced limiting systems associated with the original system. As a by-product, we derive a generalized, discrete-time version of the well-known Krasovskii-LaSalle theorem for general time-varying, not necessarily periodic, systems. Furthermore, we apply the obtained stability results to analyze uniform asymptotic stability of cascaded time-varying systems, and show that some technical assumptions in recent papers can be relaxed. Through a practical application, it is shown that our results play a similar role to the classic LaSalle invariance principle in guaranteeing attractivity, noting that reduced limiting systems are used instead of the original system. To validate the conceptual characterizations, we study the problem of sampled-data stabilization for the benchmark example of nonholonomic mobile robots via the exact discrete-time model rather than approximate models. This case study also reveals that in general, sampled-data systems may become non-periodic even though their original continuous-time system is periodic. A novel sampled-data stabilizer design is proposed using the new stability results and is supported via simulation results. Index Terms--Cascaded systems, discrete-time systems, mobile robots, reduced limiting systems, sampled-data controllers, uniform global asymptotic stability, weak zero-state detectability.