Robust stability of LTI systems over semialgebraic sets using sum-of-squares matrix polynomials

This paper deals with the robust stability of discrete-time linear time-invariant systems with parametric uncertainties belonging to a semialgebraic set. It is asserted that the robust stability of any system over any semialgebraic set (satisfying a mild condition) is equivalent to solvability of a semidefinite programming (SDP) problem, which can be handled using the available software tools. The particular case of a semialgebraic set associated with a polytope is then investigated, and a computationally appealing method is proposed to attain the SDP problem by means of a sampling technique, introduced recently in the literature. Furthermore, it is shown that the current result encompasses the ones presented in some of the recent works. The efficacy of the proposed method is demonstrated through some illustrative examples, and the results are compared to some of the existing methods.