TY - GEN

T1 - Kalman—Yakubovich—Popov Lemma And Hilbert's 17th Problem

AU - Gusev, Sergei V.

PY - 2015

Y1 - 2015

N2 - Kalman-Yakubovich-Popov (KYP) lemma is the cornerstone of control theory. It was used in thousands of papers in many areas of automatic control. The new versions and generalizations of KYP lemma emerge in literature every year. The original formulation of KYP lemma claims the equivalence of three statements: 1) fulfillment of so-called frequency-domain inequality, 2) solvability of the KYP linear matrix inequality, and 3) solvability of the Lur'e equation. The equivalence of first two statements was proved by V.A.Yakubovich and is further called Yakubovich statement. The paper investigates whether the KYP lemma holds when the field of real numbers is replaced by some other ordered field. The necessary and suficient condition is found for Yakubovich statement to hold in ordered fields. It is shown that Yakubovich statement can hold in such fields when Lur'e equation (and corresponding Riccati equation) has no solution. Based on the statement of Hilbert's 17th problem it is shown that if the matrices in the for

AB - Kalman-Yakubovich-Popov (KYP) lemma is the cornerstone of control theory. It was used in thousands of papers in many areas of automatic control. The new versions and generalizations of KYP lemma emerge in literature every year. The original formulation of KYP lemma claims the equivalence of three statements: 1) fulfillment of so-called frequency-domain inequality, 2) solvability of the KYP linear matrix inequality, and 3) solvability of the Lur'e equation. The equivalence of first two statements was proved by V.A.Yakubovich and is further called Yakubovich statement. The paper investigates whether the KYP lemma holds when the field of real numbers is replaced by some other ordered field. The necessary and suficient condition is found for Yakubovich statement to hold in ordered fields. It is shown that Yakubovich statement can hold in such fields when Lur'e equation (and corresponding Riccati equation) has no solution. Based on the statement of Hilbert's 17th problem it is shown that if the matrices in the for

KW - KYP lemma

KW - frequency domain inequality

KW - LMI

KW - Lur'e equation

KW - SOS

KW - Hilbert's 17th problem

U2 - doi:10.1016/j.ifacol.2015.09.190

DO - doi:10.1016/j.ifacol.2015.09.190

M3 - Conference contribution

SP - 238

EP - 241

BT - IFAC-PapersOnLine

ER -