Systems and methods provided herein relate to a pump system. The pump system includes a pump disposed within a well, an actuator operable to move a rod including a surface end coupled to the actuator and a downhole end coupled to the pump, and a controller. The controller is configured to identify a first impulse response and a second impulse response associated with the pump system based on a first model of the pump system. The controller is further configured to generate a second model of the pump system based on the first impulse response and the second impulse response. The controller is further configured to operate the pump system based on the second model.