  A posteriori error estimates based on residuals can be used for reliable
error control of numerical methods. Here, we consider them in the context of
ordinary differential equations and Runge-Kutta methods. In particular, we take
the approach of Dedner & Giesselmann (2016) and investigate it when used to
select the time step size. We focus on step size control stability when
combined with explicit Runge-Kutta methods and demonstrate that a standard I
controller is unstable while more advanced PI and PID controllers can be
designed to be stable. We compare the stability properties of residual-based
estimators and classical error estimators based on an embedded Runge-Kutta
method both analytically and in numerical experiments.
