Solutions to the positive real control problem for linear systems via reduced-order observer

The problem of rendering a transfer function strictly positive real using reduced-order observer is studied in this paper. It is shown that the problem always has a solution that is parameterized by a stabilizing feedback gain K and a positive definite matrix P
K
satisfying certain Lyapunov matrix equation. In addition, the resultant positive real transfer function is explicitly obtained and has the property that its H
∞
norm can be made arbitrary small if K and P
K
are properly chosen. The results are illustrated with some examples.