A speed-sensorless state feedback controller for induction machines (IMs) with LC filter is proposed. The estimation of speed and remaining states is based on a speed-adaptive observer, requiring only the measurement of the filter input currents. The motor currents are controlled by a state-feedback controller with proportional and integral control action to achieve fast and asymptotic set point tracking. Observer \emph{and} controller gains are calculated offline using linear quadratic regulator (LQR) theory and updated online (gain-scheduling), in order to guarantee stability and improve control performance in the whole operation range. The proposed control scheme is validated by simulation and experimental results including several zero-crossings of the mechanical speed. It is shown that the overall control system performs well under various load- and speed conditions; while its tuning remains easy making it attractive also for industrial application.