B. Nagy

Minimal positive realizations of transfer functions with nonnegative multiple poles

This note concerns a particular case of the minimality problem in positive system theory. A standard result in linear system theory states that any nth-order rational transfer function of a discrete time-invariant linear single-input-single-output (SISO) system admits a realization of order n. In some applications, however, one is restricted to realizations with nonnegative entries (i.e., a positive system), and it is known that this restriction may force the order N of realizations to be strictly larger than n. A general solution to the minimality problem (i.e., determining the smallest possible value of N) is not known. In this note, we consider the case of transfer functions with nonnegative multiple poles, and give sufficient conditions for the existence of positive realizations of order N=n. With the help of our results we also give an improvement of an existing result in positive system theory.

A lowerbound on the dimension of positive realizations

A basic phenomenon in positive system theory is that the dimension N of an arbitrary positive realization of a given transfer function H(z) may be strictly larger than the dimension n of its minimal realizations. The aim of this brief is to provide a nontrivial lowerbound on the value of N under the assumption that there exists a time instant k/sub 0/ at which the (always nonnegative) impulse response of H(z) is 0 but the impulse response becomes strictly positive for all k>k/sub 0/. Transfer functions with this property may be regarded as extremal cases in positive system theory.

Order Bound for the Realization of a Combination of Positive Filters

In a problem on the realization of digital filters, initiated by Gersho and Gopinath, we extend and complete a remarkable result of Benvenuti, Farina and Anderson on decomposing the transfer function t(z) of an arbitrary linear, asymptotically stable, discrete, time-invariant single-input-single-output system as a difference t(z)=t1(z)-t2(z) of two positive, asymptotically stable linear systems. We give an easy-to-compute algorithm to handle the general problem, in particular, also the case of transfer functions t(z) with multiple poles, which was left open in a previous paper. One of the appearing positive, asymptotically stable systems is always one-dimensional, while the other has dimension depending on the order and, in the case of nonreal poles, also on the location of the poles of t(z). The appearing dimension is seen to be minimal in some cases and it can always be calculated before carrying out the realization

Nonnegative realizations of matrix transfer functions

Abstract Various aspects of the nonnegative, finite-dimensional realizability of time-invariant discrete linear systems are considered. A new proof of the basic result of the nonnegative realizability of a primitive (scalar-valued) transfer function with nonnegative impulse response function is given. An algorithm for establishing whether a scalar-valued transfer function with nonnegative impulse response has a nonnegative realization is presented. The main result characterizes the nonnegative realizability of a scalar-valued transfer function with the help of primitive transfer functions, and is extended to the general case of matrix-valued transfer functions. Then conditions for the existence of some special nonnegative realizations of transfer functions are presented, e.g., where the middle (main) matrix is irreducible, strictly positive or primitive.

Spectral Properties of Operator Polynomials with Nonnegative Coefficients

We study properties of the polynomial Q(λ) = λmI − S(λ) where S(λ) = λlAl + ...+ λA1 + A0 and 1 ≤ m < l. The coefficients Al,...,A0 are in the positive cone of an ordered Banach algebra or are positive operators on a complex Banach lattice E and I is the identity. We study the properties of the spectral radius of S(λ) if λ is a nonnegative real number, and its connection with the existence of spectral divisors with nonnegative coefficients in the considered sense. We prove factorization results for nonnegative elements in an ordered decomposing Banach algebra with closed normal algebra cone and in the Wiener algebra. Earlier results on monic (nonnegative) operator polynomials are applied to the operator polynomial class studied here.

Algorithm for positive realization of transfer functions

The aim of this brief is to present a finite-step algorithm for the positive realization of a rational transfer function H(z). In comparison with previously described algorithms we emphasize that we do not make an a priori assumption on (but, instead, include a finite step procedure for checking) the nonnegativity of the impulse-response sequence of H(z). For primitive transfer functions a new method for reducing the pole order of the dominant pole is also proposed.

On nonnegative realizations of rational matrix functions and nonnegative input-output systems

We consider realizations (A, B, C) of rational l × κ matrix functions W, i.e. W(λ) = C(λ — A)-1 B if λ ∉ σ(A). Generalizing the scalar case we prove that W has a realization with nonnegative matrices iff for each of its minimal realizations (A 0, B 0, C 0) there exists an A 0-invariant polyhedral proper cone P 0 with B 0ℝκ + ⊂ P 0 and C 0 P 0 ⊂ ℝl +. For such functions W the McMillan degree is less than or equal to the size of nonnegative-minimal realizations, and the latter is less than or equal to the minimum of the number of extreme rays of polyhedral cones P 0 with the properties mentioned above. We give an example for which both these inequalities are strict.

Spectral properties of rational matrix functions with nonnegative realizations

Abstract If ( A,B,C ) is an (entrywise) nonnegative realization of a rational matrix function W (i.e. W ( λ ) = C ( λ − A ) −1 B for λ ∉ σ ( A )) vanishing at infinity, then r ( W ) := inf{ r ⩾ 0: W has no poles λ with r λ |} is a pole of W and r ( A ) := spectral radius of A is an eigenvalue of A . We prove that, if the realization is minimal-nonnegative, then 1. 1. r ( W ) = r ( A ), 2. 2. order of the pole r ( W ) of W = order of the pole r ( A ) of (· − A ) −1 . We characterize the order of these poles in the spirit of Rothblums index theorem, namely as the length of the longest chains of singular vertices in the reduced graph of A with a suitable new access relation, which incorporates B and C into the familiar access relation of A .

Finitely spectral operators

In the theory of spectral (and prespectral) operators in a Banach space or in a locally convex topological vector space the countable additivity (in some topology) of a resolution of the identity of the operator is a standing assumption. One might wonder why. Even if one cannot completely agree with the opinion of Diestel and Uhl ([6, p. 32]) stating that “countable additivity [of a set function] is often more of a hindrance than a help”, it might be interesting to study which portions of the theory of (pre)spectral operators and in which form extend to the more general situation described below.

On Boolean Algebras of Projections and Prespectral Operators

Complete and σ-complete Boolean algebras of projections in a complex Banach space were studied first by Bade [1] (see also [3; XVII.3]). The purpose of this paper is to find the appropriate extensions of several of his results to the more general case of G-complete and G-σ-complete Boolean algebras of projections, where G is a total linear manifold in the dual of the underlying Banach space. We shall prove e.g. that a Boolean algebra of projections is G-σ-complete if and only if it coincides with the range of a spectral measure of class G (Theorem 2), and we shall give a sufficient condition ensuring that the uniformly closed operator algebra generated by a G-σ-complete Boolean algebra B of projections coincides with the first commutant of B (Theorem 3). The new techniques will include the application of certain weak topologies and some of the duality theory of paired linear spaces as well as an idea due to Palmer [5].