Learning the Uncertainty Sets for Control Dynamics via Set Membership: A
  Non-Asymptotic Analysis
Yingying Li Jing Yu Lauren Conger Adam Wierman
Set-membership estimation is commonly used in adaptive/learning-based control
algorithms that require robustness over the model uncertainty sets, e.g.,
online robustly stabilizing control and robust adaptive model predictive
control. Despite having broad applications, non-asymptotic estimation error
bounds in the stochastic setting are limited. This paper provides such a
non-asymptotic bound on the diameter of the uncertainty sets generated by set
membership estimation on linear dynamical systems under bounded, i.i.d.
disturbances. Further, this result is applied to robust adaptive model
predictive control with uncertainty sets updated by set membership. We
numerically demonstrate the performance of the robust adaptive controller,
which rapidly approaches the performance of the offline optimal model
predictive controller, in comparison with the control design based on least
square estimation's confidence regions.