Crossover frequency limitations in MIMO nonminimum phase feedback systems

This note investigates limitations and design tradeoffs of the closed-loop sensitivity/performance of linear-time-invariant nonminimum-phase uncertain multiple-input-multiple-output plants, with l inputs and m outputs, where m ≤ l. It is shown that if rows i₁,..., i_k of the plant transfer function form a k × l nonminimum phase transfer matrix, and if the design is such that the sensitivity gain of k - 1 rows among the rows i₁,..., i_k of the closed-loop transfer function is low, then by necessity the sensitivity gain of the remaining row is high. This sensitivity constraint is quantified with the help of the crossover frequency restriction of a specially constructed single-input-single-output transfer function that includes the right half plane zeros and poles of the k × l transfer matrix.