Pseudo-causal tracking control of a nonminimum phase system

A novel method is described to implement noncausal feedfor-ward compensators causally, i.e. without requiring any future value of the reference input trajectory. A hardware-in-the loop test facility developed for continuously variable transmissions is utilized in this paper. The test facility includes two induction motors to emulate engine and vehicle characteristics. Software models of an engine and vehicle, running in real-time, provide reference torque and speed signals for the motors, which are connected to a transmission that is the hardware in the loop. Speed control of the output motor that emulates the vehicle dynamics is used to demonstrate an application of the proposed technique. A feedforward compensator, based on transfer function inversion, is used to compensate for the nonminimum phase motor and drive system dynamics. The vehicle model cannot be run ahead of time to provide the future values required by the noncausal inversion technique because it requires the current torque at the output of the transmission. Therefore, the feedforward controller has to be applied causally. A frequency domain estimation technique and a multi-frequency test signal are utilized to estimate, within the frequency range of interest, a low relative order transfer function of the closed loop system incorporating a manually added delay in the feedback loop. A noncausal feedforward controller is designed for the delayed output of the system based on the identified transfer function. It has been shown experimentally that this compensator offers excellent tracking performance of the motor when subjected a multi-frequency speed demand signal.