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On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form

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2 Author(s)
Shorten, R.N. ; Hamilton Inst., Nat. Univ. of Ireland, Maynooth, Ireland ; Narendra, K.S.

In this note, the problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems whose system matrices A1 and A2 are in companion form is considered. It is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product A1A2 does not have an eigenvalue that is real and negative. Examples are presented to illustrate the result.

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Automatic Control, IEEE Transactions on  (Volume:48 ,  Issue: 4 )