  We present a new method for the automated synthesis of digital controllers
with formal safety guarantees for systems with nonlinear dynamics, noisy output
measurements, and stochastic disturbances. Our method derives digital
controllers such that the corresponding closed-loop system, modeled as a
sampled-data stochastic control system, satisfies a safety specification with
probability above a given threshold. The proposed synthesis method alternates
between two steps: generation of a candidate controller pc, and verification of
the candidate. pc is found by maximizing a Monte Carlo estimate of the safety
probability, and by using a non-validated ODE solver for simulating the system.
Such a candidate is therefore sub-optimal but can be generated very rapidly. To
rule out unstable candidate controllers, we prove and utilize Lyapunov's
indirect method for instability of sampled-data nonlinear systems. In the
subsequent verification step, we use a validated solver based on SMT
(Satisfiability Modulo Theories) to compute a numerically and statistically
valid confidence interval for the safety probability of pc. If the probability
so obtained is not above the threshold, we expand the search space for
candidates by increasing the controller degree. We evaluate our technique on
three case studies: an artificial pancreas model, a powertrain control model,
and a quadruple-tank process.
