Computing the distance to uncontrollability via LMIs: Lower bound computation with exactness verification
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概要
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In this paper, we consider the problem to compute the distance to uncontrollability(DTUC) of a given controllable pair A ∈ Cn×n and B ∈ Cn×m. It is knownthat this problem is equivalent to computing the minimum of the smallest singularvalue of [ A − zI B ] over z ∈ C. With this fact, Gu et al. proposed an algorithmthat correctly estimates the DTUC at a computation cost O(n4). From theviewpoints of linear control system theory, on the other hand, this problem can beregarded as a special case of the structured singular value computation problemsand thus it is expected that we can establish an alternative LMI-based algorithm.In fact, this paper first shows that we can compute a lower bound of the DTUC bysimply applying the existing techniques to solve robust LMIs. Moreover, we showvia convex duality theory that this lower bound can be characterized by a veryconcise dual SDP. In particular, this dual SDP enables us to derive a conditionon the dual variable under which the computed lower bound surely coincides withthe exact DTUC. On the other hand, in the second part of the paper, we considerthe problem to compute the similarity transformation matrix T that maximizes thelower bound of the DTUC of (T−1AT, T−1B). We clarify that this problem can bereduced to a generalized eigenvalue problem and thus solved efficiently. In view ofthe correlation between the DTUC and the numerical difficulties of the associatedpole placement problem, this computation of the transformation matrix would leadto an effective and efficient conditioning of the pole placement problem for the pair(A,B).
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