Linear Output Regulation for Unknown Stable Systems With Uncertain Minimum-Phase Actuators

Consider a linear stable system—described by the transfer function $P(s)$ of unknown parameters, unknown order, and unknown relative degree—under a linear exosystem generating biased multisinusoidal references and/or disturbances with at most $q$ different known frequencies $\omega _{i}$, $i=1,\ldots,q$. It has been recently established that a linear regulator with minimal order $(2q+1)$ exists under the knowledge of the positive or negative signs of i) $P(0)$ and ii) either $\Re (P(j\omega _{i}))$ or $\Im (P(j\omega _{i}))$, for any $i=1,\ldots,q$ [$j$ is the imaginary unit]. This technical note explores the case in which the measurable input $u$ to the aforementioned system is provided by an unknown linear actuator. It is actually shown that the regulator design can be naturally extended to such a scenario, provided that the actuator process is minimum-phase, of known relative degree $\rho \geq 1$, and with known sign of the high-frequency gain.