Comments on 'Less conservative conditions for robust LQR-state-derivative controller design: an LMI approach' and new sufficient LMI conditions for invertibility of a convex combination of matrices

This note is concerned with conditions on a set of non-singular matrices A(i) epsilon R-nxn, i = 1, 2,..., r, so that any convex combination of these matrices is also non-singular. The first part of the note points out that Theorem 2.3 in a previous paper [Beteto et al. (2021). Less conservative conditions for robust LQR-state derivative controller design: An LMI approach. International Journal of Systems Science] provides only necessary conditions, which are not sufficient in the general case. In the second part, some stability results based on Linear Matrix Inequalities (LMIs) for a class of fractional order systems are used to establish new sufficient conditions. Numerical examples are presented for illustration. The results suggest that the new LMI conditions may be less conservative compared to a test proposed in the literature on P-matrices, and also to a positive-definiteness test based on matrix cross-products.

Automated summary

This note discusses conditions for a set of non-singular matrices, such that any convex combination of these matrices is also non-singular. It points out that the conditions provided in a previous paper are only necessary conditions and may not be sufficient in general. New sufficient conditions are established based on stability results using Linear Matrix Inequalities (LMIs) for a class of fractional order systems. Numerical examples suggest that the new LMI conditions may be less conservative compared to existing tests.