A feedforward stable approximation to uncancelable inverse dynamics in discrete time systems

In this dissertation a design methodology is developed to compensate unstable zeros in linear discrete time systems with precision tracking objectives. Unstable zeros are defined to be those zeros of a rational transfer function that are outside the unit circle. Since unstable zeros cannot be canceled inside the feedback loop, a feedforward control scheme is considered. The proposed scheme exploits the fact that the noncausal expansion of unstable inverse dynamics is convergent in the region of the complex plane encompassing the unit circle. In fact, the coefficients of the noncausal expansion are identical to the impulse response coefficients of a system with poles at locations given by the inverse of the the zeros. A stable and implementable approximation to the unstable inverse dynamics follows by truncating the series and utilizing the necessary preview information. It is shown that the error in the approximation can be made arbitrarily small if enough preview information is available. If the amount of preview information is limited, then a frequency weighted optimization criteria can be introduced to yield lower order filters. The robustness of the proposed scheme is discussed. Simulation results are provided to corroborate the theoretical findings of this work. Finally, an experimental study for end point tracking of a flexible beam was performed to validate the effectiveness of the proposed scheme.