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I know that for a system to be BIBO stable the eigenvalues need to be stable so real(lambda)<0 so get that is only valid for 4 but how can i find out the other systems which are BIBO stable?

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In order for a linear time invariant system to be BIBO all modes who are observable and controllable need to have a negative eigenvalue. A quick way to check the observability and controllability is with the Hautus lemma. Calculating the eigenvalues of $A$ should be straightforward when it is diagonal or upper/lower triangular. For the matrices that are not of that form you could use the determinant and trace of the matrix and from that deduce the sign of the real part of the eigenvalues (at least for two by two matrices). Namely the determinant and trace of a matrix are equal to the product and sum of all its eigenvalues respectively.

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  • $\begingroup$ so i need to check individually for every system if they are controllable and observable? $\endgroup$
    – vansh
    Commented Jan 19, 2019 at 23:06
  • $\begingroup$ @vansh Yes, but if $A$ is diagonal or triangular, or all eigenvalues are all stable or unstable you can spot this quite fast. $\endgroup$
    – fibonatic
    Commented Jan 20, 2019 at 1:51
  • $\begingroup$ the matrix in 1 has 1 stable ev and 1 unstable ev how can this be BIBO stable $\endgroup$
    – vansh
    Commented Jan 20, 2019 at 9:15
  • $\begingroup$ @vansh you can check observability and controllability of the unstable eigenvalues with the Hautus lemma. $\endgroup$
    – fibonatic
    Commented Jan 20, 2019 at 11:49

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