Passivity Margin and Related Bounds for Linear Systems

The concept of passivity margin is introduced and analyzed through the Lyapunov equation of the Kalman-Yacubovich-Popov lemma. Furthermore, using an extensive mathematical analysis, bounds for this margin are obtained for both the cases of SISO and MIMO systems. In the first case it is proven that an upper bound of the passivity margin is the minimum of either i) the absolute value of the real part of the maximum stable pole of the system or ii) the difference between the sum of the real parts of all the poles minus the sum of the real parts of all the zeros. In the second case it is proven that bound (ii) is in turn bounded by a logarithmic integral. However, this integral results in an entropy-like expression for SISO systems. Finally, some remarkable comments and an illustrative example are included