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I am solving a system of equations derived from power system applications. It consists of index-1 differential and algebraic equations in the form: $$\dot{x}=f(x,y) \\ 0=g(x,y)$$

To get the eigenvalues of the system I eliminate to get the reduced jacobian matrix $$\Delta \dot{x}=(f_x-f_y*g_y^{-1}*g_x)\Delta x$$

From the eigenvalue analysis I get the eigenvalues that were expected by theory. But, I also get a pair of positive eigenvalues. $0.3\pm3.5i$. The participation factors point to a PI controller and by tuning the Kp gain I can eliminate them (move to the left plane).

My question is: Although I have these positive eigenvalues, they don't show up in time domain simulation (solving the system using trapezoidal method) over discretized time horizon. I make pulse disturbances and the system appears to be stable while I would expect to observe instability because of the positive eigenvalues.

Could this be explained or should I assume that there is something wrong with the simulation software or the eigenvalue analysis?

Thanks in advance!

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