The optimal LQ regulator with cheap control for not strictly proper systems

The LQ optimal regulator problem with cheap control is investigated for plants of the form x=Ax+Bu, y=Cx+Du, where D ?? 0 makes the plant not strictly proper. It is found that perfect regulation of the output with closed-loop stability is achieved if and only if the plant is stabilizable-detectable, is minimum phase and there exist independent inputs equal to or greater in number to the number of independent outputs.