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

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 formulation of Yakubovich statement depend rationally on parameters, then there exists solution of KYP inequality which is also rational function of these parameters. The generalized formulation of Yakubovich statement and Hilbert's 17th problem for abstract ordered fields is presented. It is shown that generalized versions of Yakubovich statement and the statement of Hilbert's 17th problem are equivalent.