  Stably inverting a dynamic system model is the foundation of numerous servo
designs. Existing inversion techniques have provided accurate model
approximations that are often highly effective in feedforward controls.
However, when the inverse is implemented in a feedback system, additional
considerations are needed for assuring causality, closed-loop stability, and
robustness. In pursuit of bridging the gap between the best model matching and
a robust feedback performance under closed-loop constraints, this paper
provides a modern review of frequency-domain model inversion techniques and a
new treatment of unstable zeros. We provide first a pole-zero-map-based
intuitive inverse tuning for motion control systems. Then for general
nonminimum-phase and unstable systems, we propose an optimal inversion
algorithm that can attain model accuracy at the frequency regions of interest
and meanwhile constrain noise amplification elsewhere to guarantee system
robustness. The design goals are achieved by a multi-objective H infinity
formulation and all-pass factorization that consider model matching, causality
of transfer functions, frequency-domain gain constraints, and factorization of
unstable system modes in a unified scheme. The proposed algorithm is validated
on motion control systems and complex high-order systems.
