Kalman-Yakubovich-Popov Lemma

Abstract

The Kalman–Yakubovich–Popov Lemma (also called the Yakubovich–Kalman–Popov Lemma) is considered to be one of the cornerstones of Control and Systems Theory due to its applications in absolute stability, hyperstability, dissipativity, passivity, optimal control, adaptive control, stochastic control, and filtering. Despite its broad field of applications, the lemma has been motivated by a very specific problem which is called the absolute stability Lur’e problem [1, 2], and Lur’e’s work in [3] is often quoted as +being the first time the so-called KYP Lemma equations have been introduced. The first results on the Kalman–Yakubovich–Popov Lemma are due to Yakubovich [4, 5] . The proof of Kalman [6] was based on factorization of polynomials, which were very popular among electrical engineers. They later became the starting point for new developments. Using general factorization of matrix polynomials, Popov [7, 8] obtained the lemma in the multivariable case. In the following years, the lemma was further extended to the infinite-dimensional case (Yakubovich [9], Brusin [10], Likhtarnikov and Yakubovich [11]) and discrete-time case (Szego and Kalman [12]).