  We propose self-triggered control schemes for nonlinear systems with
quantized state measurements. Our focus lies on scenarios where both the
controller and the self-triggering mechanism receive only the quantized state
measurement at each sampling time. We assume that the ideal closed-loop system
without quantization or self-triggered sampling is contracting. Moreover, a
growth rate of the open-loop system is assumed to be known. We present two
control strategies that yield the closed-loop stability without Zeno behavior.
The first strategy is implemented under logarithmic quantization and imposes no
time-triggering condition other than setting an upper bound on inter-sampling
times. The second one is a joint design of zooming quantization and periodic
self-triggered sampling, where the adjustable zoom parameter for quantization
changes based on inter-sampling times and is also used for the threshold of
self-triggered sampling. In both strategies, we employ a trajectory-based
approach for stability analysis, where contraction theory plays a key role.
