Signal Decoupling with Preview in the Geometric Context: Exact Solution for Nonminimum-Phase Systems

Abstract

The problem of making the output of a discrete-time linear system totally insensitive to an exogenous input signal known with preview is tackled in the geometric approach context. A necessary and sufficient condition for exact decoupling with stability in the presence of finite preview is introduced, where the structural aspects and the stabilizability aspects are considered separately. On the assumption that structural decoupling is feasible, internal stabilizability of the minimal self-bounded controlled invariant satisfying the structural constraint $\mathcal{V}_m$ guarantees the stability of the dynamic feedforward compensator. However, if the structural decoupling is feasible, but $\mathcal{V}_m$ is not internally stabilizable, exact decoupling is nonetheless achievable with a stable feedforward compensator on the sole assumption that $\mathcal{V}_m$ has no unassignable internal eigenvalues on the unit circle, provided that the signal to be rejected is known with infinite preview. An algorithmic framework based on steering-along-zeros techniques, completely devised in the time domain, shows how to compute the convolution profile of the feedforward compensator in each case.