TY - GEN

T1 - Further results on computing the distance to uncontrollability via LMIs

AU - Ebihara, Yoshio

AU - Hagiwara, Tomomichi

PY - 2007

Y1 - 2007

N2 - This paper concerns the computation of the distance to uncontrollability (DTUC) of a given controllable pair A ∈ Cn×n and B ∈ Cn×m. This problem can be regarded as a special case of the structured singular value computation problems and motivated from this fact, in our preceding work, we derived an semidefinite program (SDP) to compute the lower bounds of the DTUC. In the first part of this paper, we show via convex duality theory that the lower bounds can be computed by solving a very concise dual SDP. In particular, this dual SDP enables us to derive a rank condition on the dual variable under which the computed lower bound coincides with the exact DTUC. This rank condition is surely effective in practice, and we will show thorough numerical examples that we can obtain exactness certificate even for those problems where the common rank-one exactness principle fails. On the other hand, in the second part of the present paper, we consider the problem to compute the similarity transformation matrix T that maximizes the lower bound of the DTUC of (T-1AT,T-1B). Based on the SDP for the lower bounds computation, we clarify that this problem can be reduced to generalized eigenvalue problem and thus solved efficiently. In view of the correlation between the DTUC and the numerical difficulties of the associated pole placement problem, this computation of the similarity transformation matrix would lead to an effective and efficient conditioning of the pole placement problem for the pair (A, B).

AB - This paper concerns the computation of the distance to uncontrollability (DTUC) of a given controllable pair A ∈ Cn×n and B ∈ Cn×m. This problem can be regarded as a special case of the structured singular value computation problems and motivated from this fact, in our preceding work, we derived an semidefinite program (SDP) to compute the lower bounds of the DTUC. In the first part of this paper, we show via convex duality theory that the lower bounds can be computed by solving a very concise dual SDP. In particular, this dual SDP enables us to derive a rank condition on the dual variable under which the computed lower bound coincides with the exact DTUC. This rank condition is surely effective in practice, and we will show thorough numerical examples that we can obtain exactness certificate even for those problems where the common rank-one exactness principle fails. On the other hand, in the second part of the present paper, we consider the problem to compute the similarity transformation matrix T that maximizes the lower bound of the DTUC of (T-1AT,T-1B). Based on the SDP for the lower bounds computation, we clarify that this problem can be reduced to generalized eigenvalue problem and thus solved efficiently. In view of the correlation between the DTUC and the numerical difficulties of the associated pole placement problem, this computation of the similarity transformation matrix would lead to an effective and efficient conditioning of the pole placement problem for the pair (A, B).

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U2 - 10.1109/ACC.2007.4282315

DO - 10.1109/ACC.2007.4282315

M3 - Conference contribution

AN - SCOPUS:46449085516

SN - 1424409888

SN - 9781424409884

T3 - Proceedings of the American Control Conference

SP - 6157

EP - 6162

BT - Proceedings of the 2007 American Control Conference, ACC

T2 - 2007 American Control Conference, ACC

Y2 - 9 July 2007 through 13 July 2007

ER -