  This paper investigates adaptive model predictive control (MPC) for a class
of constrained linear systems with unknown model parameters. This is also posed
as the dual control problem consisting of system identification and regulation.
We first propose an online strategy for simultaneous unknown parameter
identification and uncertainty set estimation based on the recursive least
square technique. The designed strategy provides a contractive sequence of
uncertain parameter sets, and the convergence of parameter estimates is
achieved under certain conditions. Second, by integrating tube MPC with
proposed estimation routine, the developed adaptive MPC provides a less
conservative solution to handle multiplicative uncertainties. This is made
possible by constructing the polytopic tube based on the consistently updated
nominal system and uncertain parameter set. In addition, the proposed method is
extended with reduced computational complexity by sacrificing some degrees of
optimality. We theoretically show that both designed adaptive MPC algorithms
are recursively feasible, and the perturbed closed-loop system is
asymptotically stable under standard assumptions. Finally, numerical
simulations and comparison are given to illustrate the proposed method.
