This paper develops an adaptive actuator failure compensation control scheme for discrete-time and possibly nonminimum-phase systems with unknown actuator failures of characterizations that first, some unknown system inputs are stuck at some unknown fixed or varying values at unknown time instants, and second, the actuator failures as system disturbances are not matched by other inputs. Such a failure compensation control problem is solved by using a new control separation design based on a new linear quadratic (LQ) control formulation: One control component is used for disturbance rejection and actuator failure compensation, and one component is used for closed-loop stability and regulation, to achieve the desired control objective. The nominal LQ control solution uses a new cost function to derive the two-component control signal, which has improved performance over a traditional LQ control design. The adaptive actuator failure compensation control problem is solved using a stable adaptive law for parameter estimation, to guarantee the system stability and output regulation performances in the presence of plant and actuator failure uncertainties. Simulation results are presented to verify the satisfactory performance of the proposed adaptive LQ control scheme.