Regulation and Internal Stabilization in Linear Multivariable Systems

For the multivariable control system described by \[ x = Ax + Bu, y = Cx, z = Dx,\] constructive necessary and sufficient conditions are given for the existence of state feedback $ u = Fx$ such that (i) F ⊃ C (observability constraint), (ii) Dexp [t(A + BF)] → 0 as t → ∞ (output regulation), and (iii) any unstable modes of $A + BF$ are either uncontrollable or unobservable at y (internal stability). It is assumed that C is A-invariant, or equivalently that an observer or dynamic compensator is utilized. A common application is treated, and sensitivity is considered for a simple example.