An alternative Kalman-Yakubovich-Popov lemma and some extensions

نویسندگان

Matthew R. Graham

Maurício C. de Oliveira

Raymond A. de Callafon

چکیده

This paper introduces an alternative formulation of the Kalman–Yakubovich–Popov (KYP) Lemma, relating an infinite dimensional Frequency Domain Inequality (FDI) to a pair of finite dimensional Linear Matrix Inequalities (LMI). It is shown that this new formulation encompasses previous generalizations of the KYP Lemma which hold in the case the coefficient matrix of the FDI does not depend on frequency. In addition, it allows the coefficient matrix of the frequency domain inequality to vary affinely with the frequency parameter. One application of this results is illustrated in an example of computing upper bounds to the structured singular value with frequency-dependent scalings. © 2009 Elsevier Ltd. All rights reserved. 0. Notation The scalar j = √ −1. We denote by Cm×n (Rm×n) the space of rectangular complex (real) matrices of dimension m × n, and by HC n the space of Cn×n Hermitianmatrices. For a matrix X ∈ Cm×n: X , X∗, X⊥ are, respectively, the complex-conjugate, the complexconjugate transpose, and a basis for the null space of X , i.e., a full column rank matrix such that XX⊥ = 0 and [ X X⊥ ] has also full column rank. He{X} is short-hand notation for X + X∗. We use X ≻ 0 (X 0) to denote that X ∈ HC is positive (semi)definite. X ⊗ Y is the Kronecker product of X and Y .

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

H infinity Analysis Revisited

—This paper proposes a direct, and simple approach to the H∞ norm calculation in more general settings. In contrast to the method based on the Kalman–Yakubovich–Popov lemma, our approach does not require a controllability assumption, and returns a sinusoidal input that achieves the H∞ norm of the system including its frequency. In addition, using a semidefinite programming duality, we present a...

متن کامل

MIT EECS 6 . 241 ( FALL 2006 ) LECTURE NOTES BY A . MEGRETSKI 18 Kalman - Yakubovich - Popov Lemma

Kalman-Yakubovich-Popov (KYP) Lemma (also frequently called “positive real lemma”) is a major result of the modern linear system theory. It is a collection of statements related to existence and properties of quadratic storage functions for LTI state space models and quadratic supply rates. The KYP Lemma is used in the derivation of H2 and H-Infinity optimal controllers, Hankel optimal reduced ...

متن کامل

On the Kalman-Yakubovich-Popov lemma and the multidimensional models

This paper focuses on Kalman–Yakubovich–Popov lemma for multidimensional systems described by Roesser model that possibly includes both continuous and discrete dynamics. It is shown that, similarly to the standard 1-D case, this lemma can be studied through the lens of S-procedure. Furthermore, by virtue of this lemma, we will examine robust stability, bounded and positive realness of multidime...

متن کامل

A Krein Space Interpretation of the Kalman - Yakubovich - PopovLemma 1

In this note we give a Krein space interpretation of the celebrated Kalman-Yakubovich-Popov (KYP) Lemma by introducing state-space models driven by inputs that lie in an indeenite-metric space. Such state-space models can be considered as generalizations of standard stochastic state-space models driven by stationary stochastic processes (that lie in a deenite, or so-called Hilbert, space). In t...

متن کامل