System characterization of positive real conditions

Necessary and sufficient conditions for positive realness in terms of state space matrices are presented under the assumption of complete controllability and complete observability of square systems with independent inputs. As an alternative to the positive real lemma and to the s-domain inequalities, these conditions provide a recursive algorithm for testing positive realness which result in a set of simple algebraic conditions. By relating the positive real property to the associated variational problem, a unified derivation of necessary and sufficient conditions for optimality of both singular and nonsingular problems is derived

On positive realness of descriptor systems

In this brief, the positive realness of descriptor systems is studied. For the continuous-time case, two positive real lemmas are given, based on a generalized algebraic Riccati equation and inequality respectively. For the discrete-time case, the positive real lemma is given in terms of a generalized algebraic Riccati inequality.published_or_final_versio

The Positive Real Lemma and Construction of All Realizations of Generalized Positive Rational Functions

We here extend the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. All state space realizations are partitioned into subsets, each is identified with a set of matrices satisfying the same Lyapunov inclusion. Thus, each subset forms a convex invertible cone, cic in short, and is in fact is replica of all realizations of positive functions of the same dimensions. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, this approach enables us to characterize systems which can be brought, through static output feedback, to be generalized positive

Frequency-Selective Vandermonde Decomposition of Toeplitz Matrices with Applications

The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices, which dates back to the early twentieth century, forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vandermonde decomposition. The existence and uniqueness of the decomposition are studied under explicit conditions on the Toeplitz matrix. The new result is connected by duality to the positive real lemma for trigonometric polynomials nonnegative on the same frequency interval. Its applications in the theory of moments and line spectral estimation are illustrated. In particular, it provides a solution to the truncated trigonometric $K$-moment problem. It is used to derive a primal semidefinite program formulation of the frequency-selective atomic norm in which the frequencies are known {\em a priori} to lie in certain frequency bands. Numerical examples are also provided.Comment: 23 pages, accepted by Signal Processin

A theory of passive linear systems with no assumptions

This is the author's accepted versionFinal version available from Elsevier via the DOI in this recordWe present two linked theorems on passivity: the passive behavior theorem, parts 1 and 2. Part 1 provides necessary and sufficient conditions for a general linear system, described by a set of high order differential equations, to be passive. Part 2 extends the positive-real lemma to include uncontrollable and unobservable state-space systems.This research was conducted in part during a Fellowship supported by the Cambridge Philosophical Society , http://www.cambridgephilosophicalsociety.org

Stability of Control Systems with Multiple Sector-Bounded Nonlinearities for Inputs Having Bounded Magnitude and Bounded Slope

This paper considers the input-output stability of a control system that is composed of a linear time-invariant multivariable system interconnecting with multiple decoupled time-invariant memoryless nonlinearities. The objectives of the paper are twofold. First and foremost, we prove (under certain assumptions) that if the multivariable Popov criterion is satisfied, then the system outputs and the nonlinearity inputs are bounded for any exogeneous input having bounded magnitude and bounded slope, and for all the nonlinearities lying in given sector bounds. As a consequence of using the convolution algebra, the obtained result is valid for rational and nonrational transfer functions. Second, for the case in which the transfer functions associated with the Popov criterion are rational functions, we develop a useful inequality for stabilizing the system by numerical methods. This is achieved by means of the positive real lemma and known results on linear matrix inequalities. To illustrate the usefulness of the inequality, a numerical example is provided

Positive and generalized positive real lemma for slice hyperholomorphic functions

In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part in the half space of quaternions with positive real part, as well as the case of (generalized) Schur functions in the open unit ball