107 Chapter 4: Stability of MIMO via transfer functions

Example (Damped harmonic oscillator)

The system is

{{dX}\over{dt}}=AX+U,

where $\gamma>0$ and $\beta$ real in

A=\begin{bmatrix}0&1\\ -\gamma&-\beta\end{bmatrix}.

The characteristic equation of $A$ is

\det\begin{bmatrix}\lambda&-1\\ \gamma&\lambda+\beta\end{bmatrix}=\lambda^{2}+\lambda\beta+\gamma=0,

so eigenvalues are

\lambda_{\pm}=2^{-1}(-\beta\pm\sqrt{\beta^{2}-4\gamma}),