Eigenvalues of Linear Time Varying systems

[sodemus]

The usual eigenvalues of a LTV system does not say much about the stability but my intuition tells me there should be some kind of extension that applies to LTV systems as well. Like including some kind of inner derivative of the eigenvalues or something, I don't know...

I guess in some way part of my question is something like, what invalidates the 'frozen' eigenvalues as a stability analysis tool for LTV systems? What is overlooked?

I can understand that my question can appear a bit fuzzy so please try ask follow-ups!

 

[dwlink]

Hard to say for a non-linear LTV in general, but if you consider the simple example of v^2 drag [Vdot = -k/2*V*abs(V)], then you can write the state matrix for a given velocity as...

dVdot = -k/2(abs(V0)+V0*sign(V0))*dV

For V0 > 0 dVdot = -k*V0*dV
For V0 < 0 dVdot = -k*abs(V0)*dV

For this system its very simple to see that for any V0, the eigenvalue is always going to be a negative value with magnitude -k*V0. The pole of this system is neutrally stable when V0 = 0, but it otherwise always located in the stable left half plane.

For an LTV, we simply replace V0 with V, and the same relationship holds. You need to investigate the dynamic behavior of the states as well as their interactions with one another if you want to ensure and LTV system is stable.