PID state estimator for Lagrangian systems

The PID (proportional-integral-derivative) state estimator to make an output feedback PID controller is suggested for Lagrangian systems. When an inverse optimal PID controller (full state feedback) is utilised with a PID state estimator, an output feedback PID control system recovers the extended disturbance input-to-state stability like a full state feedback PID control system, if only one condition for an estimator gain parameter is satisfied. The condition is derived from the stability proof and it depends only on the proportional gain value among the gains in the PID controller. Also, the performance of the suggested PID state estimator can be predicted by the proportional relation between the size of estimation error and the square of the estimator gain parameter.