Eigenvalue regions for positive systems - Semantic Scholar
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Systems & Control Letters 51 (2004) 325 – 330
www.elsevier.com/locate/sysconle
Eigenvalue regions for positive systems Luca Benvenuti∗ , Lorenzo Farina Dipartimento di Informatica e Sistemistica “A. Ruberti”, Universita degli Studi di Roma “La Sapienza”, Via Eudossiana 18, Roma 00184, Italy Received 4 November 2002; received in revised form 16 April 2003; accepted 8 September 2003
Abstract Positive systems are systems in which the state variables take only nonnegative values for all times. In this paper, we derive eigenvalue regions for discrete and continuous-time positive linear systems by resorting to the immediately available information on the values of the main diagonal entries of the system matrix. c 2003 Elsevier B.V. All rights reserved. Keywords: Positive systems; Positive matrices; Metzler matrices; Dominant eigenvalues; Eigenvalue regions
1. Introduction
for the discrete-time case, and
Positive systems are systems in which the state variables take only nonnegative values for all times. In the last decade, such systems have drawn the attention of researchers working in many diverse areas of applied mathematics and engineering (see [3] for details and bibliography). The main reasons appear to rely— on one hand—on positive systems ability of modeling a wide variety of phenomena in which positivity of relevant variables is the key issue and—on the other hand—on the fact that positivity enforces many peculiar properties, thus often simplifying the analysis and the design of their behavior. In this paper we shall focus on homogeneous positive linear systems of the form
x(t) ˙ = Ax(t)
x(k + 1) = Ax(k)
∗
(1)
Corresponding author. E-mail address: [email protected] (L. Benvenuti).
(2)
for the continuous-time case. Nonnegativity of each state vector element for all times enforces a speci@c sign pattern on the matrix A and consequently a number of properties regarding the modal decomposition (eigenvalues and eigenvectors of A) of the state trajectories do hold. Such properties, for discrete-time positive systems have been a subject of study over the last century: in fact, the @rst relevant result on the eigenvalues of positive matrices is due to Perron and is dated 1907 [7]. More importantly, this research area is still very active. In what follows, we shall study eigenvalue regions for discrete- and continuous-time positive linear systems. This problem—for the discrete-time case—has been addressed by Karpelevich [5] by providing such region only on the basis of the a priori information regarding positivity of the matrix A and its dimension n. The main result of this paper consists in providing a straightforward method for reducing the Karpelevich
c 2003 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2003.09.009
326
L. Benvenuti, L. Farina / Systems & Control Letters 51 (2004) 325 – 330
region by resorting to the immediately available information on the values of main diagonal entries of the matrix A. Moreover, we shall extend such results to the case of continuous-time positive linear systems. The paper is organized as follows: Section 2 contains some basic de@nitions used throughout the paper. In Section 3, we begin our study by considering the location of dominant eigenvalues only. To this purpose, we @rst recall some known basic results and provide tight bounds for the spectral radius of a positive matrix. In Section 4, we illustrate the Karpelevich theorem and provide the main results of the paper on eigenvalue regions for positive systems. 2. Denitions Given a matrix A; (A) denotes its spectrum and deg , with ∈ (A), is the size of the largest diagonal block containing in the Jordan canonical form of A. A matrix A is said to be positive if all its entries are nonnegative and at least one is positive (so to avoid the trivial case of an all-zero matrix). Any eigenvalue of a nonnegative matrix A such that ||= (A)=max{||: ∈ (A)} will be called a dominant radius eigenvalue of A and (A) will be called the spectral radius of A. A matrix A is said to be Metzler if all its oJ-diagonal entries are nonnegative and at least one entry is not zero (so to avoid the trivial case of an all-zero matrix), and a Metzler matrix having at least one negative main diagonal entry is said to be a proper Metzler matrix. Any eigenvalue of a Metzler matrix A such that Re() = (A) = max{Re(): ∈ (A)} will be called a dominant abscissa eigenvalue of A and (A) will be called the spectral abscissa of A. A square matrix A is said to be reducible if there exists a permutation matrix P such that PAP T is block triangular. A matrix that is not reducible is said to be irreducible. General references on positive matrices are [2,6]. 3. Dominant eigenvalues It is well known that positivity imposes a speci@c sign pattern on the system matrix A, in particular, a discrete-time positive system of form (1) is fully characterized by having a positive matrix A, while a continuous-time positive system of form (2) is fully
characterized by having a Metzler matrix A. Hence, we will concentrate on the eigenvalue regions of those kind of matrices only (see [3] for details). We begin with studying the location of dominant eigenvalues of positive matrices for which the celebrated Perron–Frobenius theorem (see [6]) holds. In what follows we will disregard the case of a positive matrix with all zero eigenvalues (nilpotent matrix), since, in this case, the eigenvalue region is trivial: it reduces to the origin of the complex plane. The Perron–Frobenius theorem, which holds for irreducible positive matrices, can be used to gain insight also into the case of a generic positive matrix (which is not nilpotent). In fact, a reducible matrix can be reduced to a block triangular matrix with irreducible diagonal blocks. Since the spectrum of such a matrix is the union of the spectra of each main diagonal block, the following theorem is immediate. Theorem 1. The dominant radius eigenvalues of a positive square matrix A of dimension n are all the roots of k − (A)k = 0 for some (possibly more than one) values of k = 1; : : : ; n. In particular, if A is not nilpotent, then one of the dominant radius eigenvalues is positive real (the Frobenius eigenvalue F = (A)) and deg F ¿ deg for any dominant radius eigenvalue . The previous theorem provides a limitation on the location of dominant radius eigenvalues for a generic positive matrix. This limitation does not take into account any information about reducibility of the matrix A. In the case one knows that the matrix A is reducible and also the dimension of the irreducible main diagonal blocks, then the dominant radius eigenvalues region can be dramatically reduced. Fig. 1 illustrates the above discussion for a matrix of dimension 4 in the case of no information about its reducibility and in case the matrix is known to be reducible with two irreducible blocks of dimension 2 on the main diagonal. The region described in Theorem 1 relies on the knowledge of the exact value of the spectral radius (A) and a simple iterative procedure to compute (A) can be established. Nevertheless, several easy-to-compute upper and lower bounds for (A) can be found in [6] and hereafter we present the following theorem which is an extension of a known result [6].
L. Benvenuti, L. Farina / Systems & Control Letters 51 (2004) 325 – 330
327
Fig. 1. Dominant radius eigenvalue possible location (bullets) for the case of a generic positive matrix of dimension 4 (left) and for the case of a reducible positive matrix with 2 diagonal blocks of dimension 2 (right).
Theorem 2. Let A be a positive square matrix of dimension n. Denote by r(i) and c(i) the sets of indices of the rows and columns not having all the entries equal to zero of a matrix A(i) . Let A(0) = A and recursively de
Systems & Control Letters 51 (2004) 325 – 330
www.elsevier.com/locate/sysconle
Eigenvalue regions for positive systems Luca Benvenuti∗ , Lorenzo Farina Dipartimento di Informatica e Sistemistica “A. Ruberti”, Universita degli Studi di Roma “La Sapienza”, Via Eudossiana 18, Roma 00184, Italy Received 4 November 2002; received in revised form 16 April 2003; accepted 8 September 2003
Abstract Positive systems are systems in which the state variables take only nonnegative values for all times. In this paper, we derive eigenvalue regions for discrete and continuous-time positive linear systems by resorting to the immediately available information on the values of the main diagonal entries of the system matrix. c 2003 Elsevier B.V. All rights reserved. Keywords: Positive systems; Positive matrices; Metzler matrices; Dominant eigenvalues; Eigenvalue regions
1. Introduction
for the discrete-time case, and
Positive systems are systems in which the state variables take only nonnegative values for all times. In the last decade, such systems have drawn the attention of researchers working in many diverse areas of applied mathematics and engineering (see [3] for details and bibliography). The main reasons appear to rely— on one hand—on positive systems ability of modeling a wide variety of phenomena in which positivity of relevant variables is the key issue and—on the other hand—on the fact that positivity enforces many peculiar properties, thus often simplifying the analysis and the design of their behavior. In this paper we shall focus on homogeneous positive linear systems of the form
x(t) ˙ = Ax(t)
x(k + 1) = Ax(k)
∗
(1)
Corresponding author. E-mail address: [email protected] (L. Benvenuti).
(2)
for the continuous-time case. Nonnegativity of each state vector element for all times enforces a speci@c sign pattern on the matrix A and consequently a number of properties regarding the modal decomposition (eigenvalues and eigenvectors of A) of the state trajectories do hold. Such properties, for discrete-time positive systems have been a subject of study over the last century: in fact, the @rst relevant result on the eigenvalues of positive matrices is due to Perron and is dated 1907 [7]. More importantly, this research area is still very active. In what follows, we shall study eigenvalue regions for discrete- and continuous-time positive linear systems. This problem—for the discrete-time case—has been addressed by Karpelevich [5] by providing such region only on the basis of the a priori information regarding positivity of the matrix A and its dimension n. The main result of this paper consists in providing a straightforward method for reducing the Karpelevich
c 2003 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2003.09.009
326
L. Benvenuti, L. Farina / Systems & Control Letters 51 (2004) 325 – 330
region by resorting to the immediately available information on the values of main diagonal entries of the matrix A. Moreover, we shall extend such results to the case of continuous-time positive linear systems. The paper is organized as follows: Section 2 contains some basic de@nitions used throughout the paper. In Section 3, we begin our study by considering the location of dominant eigenvalues only. To this purpose, we @rst recall some known basic results and provide tight bounds for the spectral radius of a positive matrix. In Section 4, we illustrate the Karpelevich theorem and provide the main results of the paper on eigenvalue regions for positive systems. 2. Denitions Given a matrix A; (A) denotes its spectrum and deg , with ∈ (A), is the size of the largest diagonal block containing in the Jordan canonical form of A. A matrix A is said to be positive if all its entries are nonnegative and at least one is positive (so to avoid the trivial case of an all-zero matrix). Any eigenvalue of a nonnegative matrix A such that ||= (A)=max{||: ∈ (A)} will be called a dominant radius eigenvalue of A and (A) will be called the spectral radius of A. A matrix A is said to be Metzler if all its oJ-diagonal entries are nonnegative and at least one entry is not zero (so to avoid the trivial case of an all-zero matrix), and a Metzler matrix having at least one negative main diagonal entry is said to be a proper Metzler matrix. Any eigenvalue of a Metzler matrix A such that Re() = (A) = max{Re(): ∈ (A)} will be called a dominant abscissa eigenvalue of A and (A) will be called the spectral abscissa of A. A square matrix A is said to be reducible if there exists a permutation matrix P such that PAP T is block triangular. A matrix that is not reducible is said to be irreducible. General references on positive matrices are [2,6]. 3. Dominant eigenvalues It is well known that positivity imposes a speci@c sign pattern on the system matrix A, in particular, a discrete-time positive system of form (1) is fully characterized by having a positive matrix A, while a continuous-time positive system of form (2) is fully
characterized by having a Metzler matrix A. Hence, we will concentrate on the eigenvalue regions of those kind of matrices only (see [3] for details). We begin with studying the location of dominant eigenvalues of positive matrices for which the celebrated Perron–Frobenius theorem (see [6]) holds. In what follows we will disregard the case of a positive matrix with all zero eigenvalues (nilpotent matrix), since, in this case, the eigenvalue region is trivial: it reduces to the origin of the complex plane. The Perron–Frobenius theorem, which holds for irreducible positive matrices, can be used to gain insight also into the case of a generic positive matrix (which is not nilpotent). In fact, a reducible matrix can be reduced to a block triangular matrix with irreducible diagonal blocks. Since the spectrum of such a matrix is the union of the spectra of each main diagonal block, the following theorem is immediate. Theorem 1. The dominant radius eigenvalues of a positive square matrix A of dimension n are all the roots of k − (A)k = 0 for some (possibly more than one) values of k = 1; : : : ; n. In particular, if A is not nilpotent, then one of the dominant radius eigenvalues is positive real (the Frobenius eigenvalue F = (A)) and deg F ¿ deg for any dominant radius eigenvalue . The previous theorem provides a limitation on the location of dominant radius eigenvalues for a generic positive matrix. This limitation does not take into account any information about reducibility of the matrix A. In the case one knows that the matrix A is reducible and also the dimension of the irreducible main diagonal blocks, then the dominant radius eigenvalues region can be dramatically reduced. Fig. 1 illustrates the above discussion for a matrix of dimension 4 in the case of no information about its reducibility and in case the matrix is known to be reducible with two irreducible blocks of dimension 2 on the main diagonal. The region described in Theorem 1 relies on the knowledge of the exact value of the spectral radius (A) and a simple iterative procedure to compute (A) can be established. Nevertheless, several easy-to-compute upper and lower bounds for (A) can be found in [6] and hereafter we present the following theorem which is an extension of a known result [6].
L. Benvenuti, L. Farina / Systems & Control Letters 51 (2004) 325 – 330
327
Fig. 1. Dominant radius eigenvalue possible location (bullets) for the case of a generic positive matrix of dimension 4 (left) and for the case of a reducible positive matrix with 2 diagonal blocks of dimension 2 (right).
Theorem 2. Let A be a positive square matrix of dimension n. Denote by r(i) and c(i) the sets of indices of the rows and columns not having all the entries equal to zero of a matrix A(i) . Let A(0) = A and recursively de
