Computing the Distance to Uncontrollability via LMIs: Lower and Upper Bounds Computation and Exactness Verification

In this paper, we consider the problem to compute the distance to uncontrollability of a given controllable pair A isin C ^{n times n} and B isin C^{n times m}. It is known that this problem is equivalent to computing the minimum of the smallest singular value of [ A - zI B ] over z isin C. With this fact, Gu proposed an algorithm that correctly estimates the distance at a computation cost O(n⁶). On the other hand, in the community of control theory, remarkable advances have been made on the techniques to deal with parametrized linear matrix inequalities (LMIs) as well as the analysis of positive polynomials. This motivates us to explore an alternative LMI-based algorithm and shed more insight on the problem to estimate the distance to uncontrollability. In fact, this paper shows that we can establish an effective method to compute a lower bound of the distance by simply applying the existing techniques to solve parametrized LMIs. To obtain an upper bound, on the other hand, we analyze in detail the solutions resulting from the LMI optimization carried out to compute the lower bound. This enables us to estimate the location of the local, but potentially global optimizer z* isin C in a reasonable fashion. We thus provide a novel technique to obtain an upper bound of the distance by evaluating the smallest singular value on the estimated optimizer z*. It turns out that the lower and upper bounds are very close in all tested numerical examples. We finally derive an algebraic condition under which the exactness of the suggested computation method of the lower bound can be ensured, based on the convex duality theory. Furthermore, we show that the suggested computation method of the upper bound is closely related to the obtained algebraic condition for the exactness verification