A novel inequality property of the second-order modes of linear continuous-time state-space systems

This paper reveals a novel inequality property of the second-order modes of linear continuous-time state-space systems. It is shown that, in the family of state-space systems having identical denominator polynomials and magnitude responses, the minimum phase system has the smallest second-order modes while the maximum phase system has the largest second-order modes. It is also shown that the second-order modes of any other system in such family are larger than those of the minimum phase system and smaller than those of the maximum phase system. These results lead to the derivation of the inequality property of the theoretical error bound of the balanced model reduction; in the above-mentioned family of state-space systems, the minimum phase system yields the smallest error bound and the maximum phase system causes the largest one. This means that the minimum phase system is easiest to approximate and the maximum phase system is hardest to approximate through the balanced model reduction. The proof of the inequality property of the second-order modes is achieved by the description of the Gramians of power complementary systems with the help of the bounded-real Riccati equation.