The Kalman-Yakubovich-Popov lemma in a behavioural framework and polynomial spectral factorization

Abstract

The classical Kalman-Yakubovich-Popov lemma provides a link between dissipativity of a system in state-space form and the solution to a linear matrix inequality. In this paper we derive the KYP lemma for linear systems described by higher-order differential equations. The result is an LMI in terms of the original coefficients in which the dissipativity problem is posed. Subsequently we study the connection between dissipativity and spectral factorization of polynomial matrices. This enables us to derive a new algorithm for polynomial spectral factorization in terms of an LMI in the coefficients of the polynomial matrix.

Time domain characterization of finite frequency properties via behavioral approach

“…Based on QDFs, Willems and Trentelman [8] has proved that dissipativity of a behavior is equivalent to a certain frequency domain inequalities on the entire frequency range. This also leads an equivalent LMI characterization of the inequalities [5]. But there has not been derived both time domain and LMI characterizations of the FFDIs in the behavioral framework.…”

Energy and power optimization in a behavioural framework

“…A QDF is a quadratic function of a signal and some of its higher-order derivatives, and therefore it is particularly apt to describe expressions involving the variables of a linear differential system, see Willems and Trentelman [13]. The current paper is the sequel of our papers Van der Geest and Trentelman [3] and [4], in which we studied dissipativity of linear differential systems in terms of QDFs. The exchange of energy between a system and its environment is related to the behaviour of the external variable of the system; it does not depend on the partition of this external variable into inputs and outputs.…”