This note presents a robust adaptive controller design for a special class of linear system, consisting of two sequentially interconnected SISO linear subsystems, S-1 and S-2, under noisy output measurements and with additional feedback. We formulate the robust adaptive control problem as a nonlinear H-infinity-optimal control problem under imperfect state measurements, and then solve it using game theory. A cost-to-come function formulation is utilized in the analysis in order to derive identifiers for S-1 and S-2, and integrator backstepping methodology is applied recursively to obtain the control law, which guarantees the boundedness of closed-loop signals, and achieves asymptotic tracking, under some assumptions. The closed-loop system exhibits a guaranteed disturbance attenuation level with respect to the exogenous disturbance inputs, where the ultimate attenuation lower bound for the achievable performance level is equal to the noise intensity in the measurement channel of S-1.