# Negative-Imaginary Systems

Mechanical structures with colocated force actuators and position sensors do not yield positive-real systems because the product of force and position is not equal to the power provided by the actuator. In this case, the transfer function matrix from the force actuator inputs $u_1(t),…,u_m()t$ to the position sensor outputs $y_1(t),…,y_m()t$ is of the form \begin{align} P(s) = \displaystyle\sum_{i=1}^{\infty} \frac{s}{s^2} \psi_i \end{align} where, for all $i$,$k_i>0$,$\omega_i>0$, and $\psi_i$ is an mx1 vector.