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An adaptive interpolating control (AIC) algorithm is proposed for constrained linear systems with parametric uncertainty and additive disturbance. This adaptive algorithm consists of an iterative set membership identification algorithm, which updates the uncertain parameter set at each time step, and an interpolating controller, which robustly stabilizes the uncertain system with state and input constraints. We prove that the AIC algorithm is recursively feasible and guarantees robust constraint satisfaction and robust asymptotic stability of the closed-loop system in the presence of uncertainties. Moreover, we detail two possible extensions of the AIC algorithm: (a) persistent excitation conditions can be embedded into the AIC algorithm to accelerate the convergence of system parameters and (b) the combination of the AIC algorithm and aggressive learning is able to enlarge the size of the feasible region with every iteration by exploiting information from previous iterations. We illustrate the effectiveness of the proposed algorithms through comparisons with adaptive model predictive control and one example of mobile carrier robot.

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