identifying overshoots in given state space system with step inputs
First obtain the Transition Matrix:
$$\Phi(s) =(sI-A)^{-1}$$

The transfer function is then:

$$H(s)=C\Phi (s)B+D $$

Where the A, B, C, D have their usual state-space interpretations (your 'L' is often referred to as 'C'; and 'D' is often zero).

However, it's possible to derive relative stability information from the transition matrix without going as far as the TF.