  This paper deals with the problem of formulating an adaptive Model Predictive
Control strategy for constrained uncertain systems. We consider a linear
system, in presence of bounded time varying additive uncertainty. The
uncertainty is decoupled as the sum of a process noise with known bounds, and a
time varying offset that we wish to identify. The time varying offset
uncertainty is assumed unknown point-wise in time. Its domain, called the
Feasible Parameter Set, and its maximum rate of change are known to the control
designer. As new data becomes available, we refine the Feasible Parameter Set
with a Set Membership Method based approach, using the known bounds on process
noise. We consider two separate cases of robust and probabilistic constraints
on system states, with hard constraints on actuator inputs. In both cases, we
robustly satisfy the imposed constraints for all possible values of the offset
uncertainty in the Feasible Parameter Set. By imposing adequate terminal
conditions, we prove recursive feasibility and stability of the proposed
algorithms. The efficacy of the proposed robust and stochastic Adaptive MPC
algorithms is illustrated with detailed numerical examples.
