Perfect decoupling in nonminimum-phase multivariable systems: a complete geometric framework

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 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, namely Vm, guarantees stability of the dynamic feedforward compensator. However, if structural decoupling is feasible but Vm is not internally stabilizable, exact decoupling is nonetheless achievable with a stable feedforward compensator, on the sole assumption that Vm 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.