  This paper proposes an Adaptive Stochastic Model Predictive Control (MPC)
strategy for stable linear time-invariant systems in the presence of bounded
disturbances. We consider multi-input, multi-output systems that can be
expressed by a Finite Impulse Response (FIR) model. The parameters of the FIR
model corresponding to each output are unknown but assumed sparse. We estimate
these parameters using the Recursive Least Squares algorithm. The estimates are
then improved using set-based bounds obtained by solving the Basis Pursuit
Denoising [1] problem. Our approach is able to handle hard input constraints
and probabilistic output constraints. Using tools from distributionally robust
optimization, we reformulate the probabilistic output constraints as tractable
convex second-order cone constraints, which enables us to pose our MPC design
task as a convex optimization problem. The efficacy of the developed algorithm
is highlighted with a thorough numerical example, where we demonstrate
performance gain over the counterpart algorithm of [2], which does not utilize
the sparsity information of the system impulse response parameters during
control design.
