TY - GEN

T1 - An elementary proof for the exactness of (D,G) scaling

AU - Ebihara, Yoshio

PY - 2009/11/23

Y1 - 2009/11/23

N2 - The goal of this paper is to provide an elementary proof for the exactness of the (D,G) scaling applied to the uncertainty structure with one repeated real scalar block and one full complex matrix block. The (D,G) scaling has vast application area around control theory, optimization and signal processing. This is because, by applying the (D,G) scaling, we can convert inequality conditions depending on an uncertain parameter to linear matrix inequalities (LMIs) in an exact fashion. However, its exactness proof is tough, and this stems from the fact that the proof requires an involved matrix formula in addition to the standard Lagrange duality theory. To streamline the proof, in the present paper, we clarify that the involved matrix formula is closely related to a norm preserving dilation under structural constraints. By providing an elementary proof for the norm preserving dilation, it follows that basic results such as Schur complement and congruence transformation in conjunction with the Lagrange duality theory are enough to complete a self-contained exactness proof.

AB - The goal of this paper is to provide an elementary proof for the exactness of the (D,G) scaling applied to the uncertainty structure with one repeated real scalar block and one full complex matrix block. The (D,G) scaling has vast application area around control theory, optimization and signal processing. This is because, by applying the (D,G) scaling, we can convert inequality conditions depending on an uncertain parameter to linear matrix inequalities (LMIs) in an exact fashion. However, its exactness proof is tough, and this stems from the fact that the proof requires an involved matrix formula in addition to the standard Lagrange duality theory. To streamline the proof, in the present paper, we clarify that the involved matrix formula is closely related to a norm preserving dilation under structural constraints. By providing an elementary proof for the norm preserving dilation, it follows that basic results such as Schur complement and congruence transformation in conjunction with the Lagrange duality theory are enough to complete a self-contained exactness proof.

UR - http://www.scopus.com/inward/record.url?scp=70449633032&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70449633032&partnerID=8YFLogxK

U2 - 10.1109/ACC.2009.5159874

DO - 10.1109/ACC.2009.5159874

M3 - Conference contribution

AN - SCOPUS:70449633032

SN - 9781424445240

T3 - Proceedings of the American Control Conference

SP - 2433

EP - 2438

BT - 2009 American Control Conference, ACC 2009

T2 - 2009 American Control Conference, ACC 2009

Y2 - 10 June 2009 through 12 June 2009

ER -