Adaptive controller design and disturbance attenuation for SISO linear systems with zero relative degree under noisy output measurements

In this paper, we present robust adaptive controller design for SISO linear systems with zero relative degree under noisy output measurements. We formulate the robust adaptive control problem as a nonlinear H^∞ optimal control problem under imperfect state measurements, and then solve it using game theory. By using the a priori knowledge of the parameter vector, we apply a soft projection algorithm, which guarantees the robustness property of the closed-loop system without any persistency of excitation assumption on the reference signal. Due to our formulation in state space, we allow the true system to be uncontrollable, as long as the uncontrollable part is stable in the sense of Lyapunov, and the uncontrollable modes on the jw-axis are uncontrollable from the exogenous disturbance input. This assumption allows the adaptive controller to asymptotically cancel out, at the output, the effect of exogenous sinusoidal disturbance, inputs with unknown magnitude, phase, and frequency. These strong robustness properties are illustrated by a numerical example.