On synchronizing sampling and quantization for stabilizing the double integrator under binary sensing

We revisit the problem of constructing finite state ρ/μ approximations for the purpose of certified-by-design control synthesis. We investigate in this context the problem of picking the `initial partition´ to enable successful control design for a benchmark problem, namely that of exponentially stabilizing the double integrator by switching between two available feedback gains under binary sensing constraints. We motivate the problem through two illustrative case studies, provide an analysis of the intuition gleamed from special instances of it, propose a general algorithm for choosing the initial partition taking into account the sampling frequency and available choices of feedback gains, and demonstrate the use of our algorithm in a set of illustrative examples.