# (M, [beta])-Stability of Positive Linear Systems.

1. Introduction

1.1. Notation. Let x = [[[x.sub.1] ... [x.sub.n]].sup.T] [member of] [R.sup.n] be a vector. The Holder vector p-norm is defined as [mathematical expression not reproducible].

Let A = [[a.sub.ij] [member of] [R.sup.n x n] be a square matrix. Let [sigma](A) be the spectrum of matrix A; the eigenvalues of A are denoted by [[lambda].sub.i](A) [member of] [sigma](A), i = [bar.1, n]. The norm of matrix A induced by a vector norm [parallel] [parallel] (not necessarily a Holder p-norm) is defined as [mathematical expression not reproducible]. The matrix measure of A with respect to a matrix norm [parallel] [parallel] is given by [[mu].sub.[parallel][parallel]] (A) = [lim.sub.h[??]0] (1/h) [[parallel]I + hA [parallel] - 1], where I stands for the unit matrix of order n. In the particular case of Holder p-norms p [member of] {1,2, [infinity]}, the expressions of the induced matrix norms are [mathematical expression not reproducible], and the corresponding matrix measures are [mathematical expression not reproducible], as per Fact 11.15.7 from .

1.2. Concept of (M, [beta])--Stability. The concept of (M, [beta])-stability has been developed by the monographic work in , aiming to offer a refined characterization of the short-term behavior (also called transient behavior) of the exponentially stable systems, in the sense of the dynamics properties exhibited by the free response.

In particular, the cited work provides adequate instruments for the analysis of both long -term and short-term behavior of linear systems with discrete-time (abbreviated DT) dynamics,

x (t + 1) = Ax (i), A [member of] [R.sup.n x n], x(0) = [x.sub.0], t [member of] [Z.sub.+], (1-DT)

and continuous-time (abbreviated CT) dynamics,

[??] (t) = Ax (i), A [member of] [R.sup.n x n], x (0) = [x.sub.0], t [member of] [R.sub.+]. (1-CT)

Throughout the text, we intend to develop a parallel analysis of the DT and CT cases, reason for which the equation numbering includes the extensions -DT and -CT, respectively, as above. We are going to refer concomitantly to equations numbered as (#-DT) and (#-CT) by using the formulation "(#-DT) (resp., (#-CT))". This type of parallel approach will also have a more general formulation, in the sense of "property of DT system (resp., property of CT system)".

The analysis of the long-term behavior of system (1-DT) (resp., (1-CT)) relies on the notion of growth rate, denoted by [alpha](A), which is defined as in Subsection 3.3.2 of , by

[mathematical expression not reproducible], (2-DT)

for the DT case, and, respectively,

[mathematical expression not reproducible], (2-CT)

for the CT case. In algebraic terms, the growth rate equals the spectral radius of A for system (1-DT) and, respectively, the spectral abscissa of A for system (1-CT).

The analysis of the transient (short-term) behavior of system (1-DT) (resp., (1-CT)) relies on the notion of (M, [beta])-stability which is defined as follows.

Definition 1 (Definition 5.5.1 ). Let 1 [less than or equal to] M and [alpha](A) [less than or equal to] [beta], with 0 < [beta] < 1 in the DT case, and, respectively, [beta] < 0 in the CT case. Consider an absolute vector norm [parallel] [parallel] in the state-space [R.sup.n]. System (1-DT) (resp., (1-CT)) is said to be (M, [beta])-stable relative to the norm [parallel] [parallel], if its trajectories satisfy the inequality

[mathematical expression not reproducible], (3-DT)

in the DT case, and, respectively,

[mathematical expression not reproducible], (3-CT)

in the CT case. The scalar M is called the transient bound and the scalar [beta] is called the exponential rate.

In colloquial terms, a "good" transient behavior relative to norm [parallel] [parallel] means M close to 1 and [beta] close to [alpha](A). The "ideal" transient behavior relative to norm [parallel] [parallel] can be introduced by formal specifications, namely, M = 1 and [beta] = [alpha](A) (i.e., both minimal exponential rate and minimal transient bound). Obviously, for these particular values of M and [beta], inequality (3-DT) (resp., (3-CT)) becomes inequality

[mathematical expression not reproducible], (4-DT)

in the case of system (1-DT), and, respectively,

[mathematical expression not reproducible], (4-CT)

in the case of system (1-CT).

Both growth rate and (M, [beta])-stability are rigorously related to the properties of the semigroup of operators generated by A (discrete- or continuous-time), for example, . Consider the semigroup of linear operators generated by matrix A. Then, by using the operator norm induced by the vector norm [parallel] [parallel], inequality (3-DT) can be equivalently written for operators [A.sup.t], i [member of] [Z.sub.+], in the form

[parallel] [A.sup.t] [parallel] [less than or equal to] M[[beta].sup.t], [for all]t [member of] [Z.sub.+], (5-DT)

and inequality (3-CT) can be equivalently written for operators [e.sup.At], i [member of] [R.sub.+], in the form

[parallel] [e.sup.At] [parallel] [less than or equal to] [e.sup.[beta]t], [for all]t [member of] +] [R.sub.+]. (5-CT)

If we take M = 1 ininequality (5-DT) (resp., (5-CT)), then the initial growth rate relative to the norm [parallel] [parallel] (see Definition 5.5.7 and Proposition 5.5.8 in ), denoted by [[alpha].sub.[parallel][parallel]] (A), is defined as

[mathematical expression not reproducible], (6-DT)

for the DT case, and, respectively, as

[mathematical expression not reproducible], (6-CT)

for the CT case.

By comparing (2-DT) with (6-DT) (resp., (2-CT) with (6-CT)), one can simply notice that the inequality [alpha](A) [less than or equal to] [[alpha].sub.[parallel][parallel]](A) holds true regardless of the considered norm. (In algebraic terms, the spectral radius is less than or equal to any matrix norm, and, respectively, the spectral abscissa is less than or equal to any matrix measure.) This fact makes completely clear the difference between the global sense of the growth rate and the local sense of the initial growth rate. Subsequently, the "ideal" transient behavior relative to norm [parallel] [parallel] is characterized by the equality [alpha](A) = [[alpha].sub.[parallel][parallel]] (A).

Obviously, inequality (4-DT) (resp., (4-CT)) is satisfied by the trajectories of any system (1-DT) (resp., (1-CT)) whose matrix A is diagonalizable, once the considered vector norm [parallel] [parallel] is defined as an absolute norm, weighted by the diagonalizing matrix. This represents a trivial example of the "ideal" transient behavior, with low practical interest, since the state variables of the diagonalized system are linear combinations of the original state variables, and the transient bounds of the former may seldom have a useful meaning for the latter. Therefore, the study of (M, [beta])-stability provides relevant information only when the considered norm [parallel][parallel] is able to preserve the key role played by the original state-space vector.

In work , the "ideal" transient behavior is illustrated by a single class of linear systems, namely, those where matrix A in equality (1-DT) (resp., (1-CT)) is a normal matrix, and the norm [parallel] [parallel] considered in inequality (4-DT) (resp., (4-CT)) is the 2-norm.

1.3. Paper Objective and Organization. The main objective of our paper is to expand the analysis framework of (M, [beta])-stability and "ideal" transient behavior by focusing on the class of positive linear systems, whose dynamics are generated by matrices A nonnegative in equality (1-DT), and essentially nonnegative in equality (1-CT). Literature includes several remarkable monographs on positive systems, among which we mention [4-6], that cover both analysis and design topics, by creating a wide perspective on the structure and behavioral particularities of various types of systems (social, economic, biological, and technical). The connections between the algebraic characterization of (essentially) nonnegative matrices and the dynamical properties of positive linear systems are deeply explored by [7, 8].

Our paper shows that positive linear system exhibits an "ideal" transient behavior relative to any vector p-norm (1 [less than or equal to] p [less than or equal to] [infinity]) considered in inequality (4-DT) (resp., (4-CT)), if the state-space variables are individually scaled by appropriate values. Concisely speaking, these scaling values are intimately related to the elements of the left and right Perron-Frobenius eigenvectors of matrix A. In norm terms, the use of the scaled state-space variables represents a simple weighting of the standard vector p-norms by positive definite diagonal matrices, fact which does not alter the meaning of the state-space vector components. Our approach includes, as particular cases corresponding to p [member of] {1,2, [infinity]}, the properties of scaled positive systems presented by the recent paper  in Propositions 1 and 2 and Remark 1.

Thus, as an overall comment on the contribution brought by this paper, we notice that, besides the extension of the investigation for (M, [beta])-stability along the lines proposed by , it also reveals deeper connections between the Perron-Frobenius Eigen structure and the dynamics of positive linear systems. Connections of this type are mentioned by many works devoted to positive linear systems, such as [4-14]. Nevertheless, the cited works do not explore the role of the Perron-Frobenius eigenstructure in the characterization of the short-term behavior of positive linear systems.

The remainder of the text is organized as follows. The main results are presented by Section 2, for positive systems defined by irreducible matrices, and by Section 3, for positive systems defined by reducible matrices. Section 4 illustrates the applicability of the theoretical results in studying the short-term dynamics of two positive linear systems previously discussed by other works. Section 5 formulates some concluding remarks on the importance of our research.

2. Results for Positive Systems Defined by Irreducible Matrices

Throughout this section, matrix A that defines the dynamics of system (1-DT) (resp., (1-CT)) is irreducible (e.g., [8, Chapter 2, Definition 1.2]). Matrix A is irreducible if and only if the oriented graph G(A) associated with A is strongly connected (e.g., [8, Chapter 2, Theorem 2.7]). In the DT case matrix A is nonnegative (i.e., all its entries are nonnegative). In the CT case matrix A is essentially nonnegative or Metzler (i.e., all its off-diagonal entries are nonnegative).

The meanings of the Perron-Frobenius eigenstructure agree for A nonnegative and A essentially nonnegative, in the sense of the following properties:

(i) Matrix A has a simple real eigenvalue, called the Perron-Frobenius eigenvalue and denoted by [[lambda].sub.max](A), which satisfies [absolute value of [[lambda].sub.i](A)] [less than or equal to] [[lambda].sub.max](A), i = 1, ..., n, in the DT case, and Re{[[lambda].sub.i](A)} [less than or equal to] [[lambda].sub.max](A), i = 1, ..., n, in the CT case.

(ii) In both DT and CT cases, matrix A has right and left positive eigenvectors v = [[[v.sub.1] ... [v.sub.n]].sup.T] [much greater than] 0, w = [[[w.sub.1] ... [w.sub.n]].sup.T] [much greater than] 0 associated with [[lambda].sub.max](A), which are called the right and left Perron-Frobenius eigenvectors and satisfy Av = [[lambda].sub.max](A)v and [A.sup.T]w = [[lambda].sub.max](A)w.

Given the irreducible matrix A, for any p, 1 [less than or equal to] p [less than or equal to] [infinity], we define the p-type Perron-Frobenius scaling operator

[mathematical expression not reproducible], (7-DT//CT)

built from the positive eigenvectors v [much greater than] 0, w [much greater than] 0, of A, which satisfy [parallel] v [parallel][sub.[infinity]] = 1, [[parallel]w[parallel].sub.[infinity]] = 1.

Theorem 2. Let 1 [less than or equal to] p [less than or equal to] [infinity]. Consider an asymptotically stable, positive linear system of form (1-DT) (resp., (1-CT)), with matrix A irreducible.

Then, system (1-DT) (resp., (1-CT)) is (1,[alpha](A))-stable relative to the norm [V.sub.p](x) = [[parallel][D.sub.p]x[parallel].sub.P], where [D.sub.P] is the scaling operator defined by (7-DT//CT).

Proof. For any 1 [less than or equal to] p [less than or equal to] [infinity], [V.sub.p](x) is a vector norm, defined by the standard _p-norm, weighted by the scaling operator [D.sub.p] (7-DT//CT).

(a) In the DT case, for any 1 [less than or equal to] p [less than or equal to] [infinity], for an arbitrary trajectory we can write

[mathematical expression not reproducible]. (8-DT)

On the other hand, we have

[parallel][D.sub.p][AD.sup.-1.sub.p][parallel][sub.p] = [[lambda].sub.max] (A) = [alpha](A) (9-DT)

as resulting from the proof of the theorem presented by paper .

Thus, from (8-DT) and (9-DT) we get the inequality

[V.sub.p] (x (t + 1)) [less than or equal to] [alpha](A) [V.sub.p] (x (i)), [for all]t [member of] [Z.sub.+], (10-DT)

and, eventually,

[mathematical expression not reproducible], (11-DT)

proving that system (1-DT) is (M, [beta])-stable, with M = 1 and [beta] = [alpha](A), in accordance with (3-DT).

(b) For the CT case, for any 1 [less than or equal to] p [less than or equal to] [infinity], we can write

[mathematical expression not reproducible]. (8-CT)

On the other hand, we have

[mathematical expression not reproducible]. (9-CT)

Indeed, the first equality in (9-CT) results from Fact 11.15.7 . For the second equality, let us pick an s > 0 such that sI + A is nonnegative. Since matrices sI + A and A have the same left and right Perron-Frobenius eigenvectors, by using part (a) of our proof, for any [mathematical expression not reproducible]. By taking s = 1/h, for h [down arrow] 0, we obtain (9-CT).

Thus, from (8-CT) and (9-CT) we get the inequality

[D.sup.+.sub.t][V.sub.p] (x (t)) [less than or equal to] [alpha](A) [V.sub.p] (x (i)), [for all]t [member of] [R.sub.+]. (10-CT)

Now, if (10-CT) holds for x(i) = x(t; 0, [x.sub.0]), let us consider the differential equation [??](t) [less than or equal to] [alpha](A) y (t) with the initial condition y(0) = [V.sub.p](x(0)) = [V.sub.p]([x.sub.0]). According to Theorem 4.2.11 in , we have

[mathematical expression not reproducible], (11-CT)

proving that system (1-CT) is (M, [beta])-stable, with M = 1 and [beta] = [alpha](A), in accordance with (3-CT).

Remark 3. Theorem 2 reveals new connections between the Perron-Frobenius eigenstructure and the dynamics of asymptotically stable positive linear systems.

(i) Besides the information on the long-term behavior referring to the role of the right eigenvector in guiding (as asymptote) any trajectory x(i) for i [right arrow] [infinity] (e.g., ), we show that the system dynamics exhibit set-invariance properties for sets defined by both right and left eigenvectors. Indeed, for any p, 1 [less than or equal to] p [less than or equal to] [infinity], the existence of the "ideal" transient behavior relative to the norm [V.sub.p](x) (proven by Theorem 2) is equivalent to the invariance of the exponentially contractive sets

[mathematical expression not reproducible], (12-DT)

and, respectively,

[mathematical expression not reproducible], (12-CT)

with respect to system (1-DT) (resp., (1-CT)). In particular, for an arbitrary (but fixed) time i [member of] [Z.sub.+] (resp., i [member of] [R.sub.+]) and an arbitrary (but fixed) constant c > 0, the Minkowski functional of the constant set [X.sup.c.sub.p](t) defined by (12-DT) (resp., (12-CT)) equals [V.sub.p] (x) multiplied by a positive scalar. It is worth saying that the invariance of the sets of form (12-DT) (resp., (12-CT)) has already been mentioned by our previous works [17, 18]. The invariance analysis for sets of form (12-DT) (resp., (12-CT)) can be extended to positive interval systems, by using Corollary 3 and Theorem 4 in our paper , dealing with dynamics defined by interval matrices.

(ii) The existence of the invariant contractive sets of form (12-DT) (resp., (12-CT)) defined for all p, 1 [less than or equal to] p [less than or equal to] [infinity], allows a deeper insight into the dynamics of positive linear systems, for which the classical property presented in literature (e.g., ) is the invariance of the nonnegative orthant [R.sup.n.sub.+]. In fact, all the nonnegative sets [[bar.X].sup.c.sub.p](t) = [X.sup.c.sub.p](t) [intersection] [R.sup.n.sub.+], t [member of] [Z.sub.+], c > 0, with [X.sup.c.sub.p](t) defined by (12-DT) are invariant with respect to positive system (1-DT), and all the nonnegative sets [[bar.X].sup.c.sub.p](t) = [X.sup.c.sub.p](t) [intersection] [R.sup.n.sub.+], t [member of] [R.sub.+], c > 0, with [X.sup.c.sub.p] (t) defined by (12-CT) are invariant with respect to positive system (1-CT).

Remark 4. The proof of Theorem 2 also highlights the following dynamical properties of system (1-DT) (resp., (1-CT)) typical to the "ideal" transient behavior:

(i) For any p, 1 [less than or equal to] p [less than or equal to] [infinity], [V.sub.p](x) can serve for system (1-DT) (resp., (1-CT)), as a Lyapunov function (in a form called "norm Lyapunov function"), which decreases with the fastest possible rate, namely, [alpha](A), along the trajectories of system (1-DT) (resp., (1-CT)).

(ii) Propositions 1 and 2 and Remark 1 in  refer to the existence of norm Lyapunov functions corresponding to p [member of] {1,2, [infinity]}, but the possibility of ensuring the fastest decreasing rate [alpha](A) (when the Perron-Frobenius eigenvectors are used) is not investigated.

(iii) The semigroup of operators [A.sup.t], t [member of] [Z.sub.+] (resp., [e.sup.At], t [member of] [R.sub.+]) is contractive relative to any operator norm subordinated to a vector norm of form [V.sub.p] (x), 1 [less than or equal to] p [less than or equal to] [infinity]. For all 1 [less than or equal to] p [less than or equal to] [infinity], the contraction rate is precisely [alpha](A).

(iv) Form (7-DT//CT) of the scaling operator [D.sub.p] uses the right and left eigenvectors of A, uniquely defined by the norm equalities [parallel]v[parallel][sub.[infinity]] = 1, [parallel] w [parallel][sub.[infinity]] = 1. Obviously, Theorem 2 holds true if D, is replaced by any diagonal matrix k[D.sub.p], k > 0, fact showing that the essential information is offered by the directions of the right and left Perron-Frobenius eigenvectors (i.e., the fulfillment of the conditions [parallel]v[parallel][sub.[infinity]] = 1, [parallel] w [parallel][sub.[infinity]] = 1 is not compulsory).

Remark 5. Paper  addresses the transient behavior of positive linear systems and considers state-space transforms defined as in (7-DT//CT) by matrix [D.sub.p]. However, these transforms of type (7-DT//CT) are not regarded as individual scalings of the state-space variables (as illustrated by Theorem 2). The [D.sub.p]-weighted norms are used in the (M, [beta])-stability analysis in the sense of eccentricity with respect to the nonweighted norms, and, therefore, the approach misses the sharp interpretation of "ideal" transient behavior for the scaled variables. This is because the eccentricity caused by the [D.sub.p]-scaling can be produced by various nondiagonal weighting matrices, where the practical meaning of the state-space variables differs drastically from the original form to the weighted form. Moreover, all the results presented by the cited paper are limited to the particular cases p [member of] {1,2, [infinity]}.

3. Results for Positive Systems Defined by Reducible Matrices

Throughout this section, matrix A that defines the dynamics of system (1-DT) (resp., (1-CT)) is reducible (i.e., the oriented graph G(A) associated with A is not strongly connected). Matrix A is nonnegative (resp., essentially nonnegative). To deal with the reducibility of matrix A, we consider two distinct cases, specified by the structure of the communication classes of matrix A, for example, Section 3, Chapter 2 in .

3.1. All Communication Classes of Matrix A Are Basic and Final. Theorem 2 can still be used if all the communication classes are basic and final. Indeed Theorem 3.14, Chapter 2, from  guarantees the existence of positive right and left eigenvectors v [much greater than] 0, w [much greater than] 0, associated with the multiple eigenvalue [[lambda].sub.max](A). (In other words, we are able to generalize the "A irreducible" case, where the eigenvalue [[lambda].sub.max](A) is unique.) By a proof similar to the proof of the Theorem presented in , one can show that equality (9-DT) (resp., equality (9-CT)) holds true for A nonnegative (resp., essentially nonnegative).

3.2. Structure of Communication Classes of Matrix A Is Different from Section 3.1. There exists nonnegative right and left eigenvectors associated with the eigenvalue [[lambda].sub.max](A), but at least one of them contains 0 elements, in accordance with Theorems 3.10 and 3.14, Chapter 2, from . Hence, the diagonal operator [D.sub.p] defined by (7-DT//CT) cannot be used any longer.

However, the same type of information becomes available, if instead of A we consider the slightly modified matrix:

[mathematical expression not reproducible], (13-DT//CT)

where [delta] is small enough. Matrix A([delta]) is irreducible, so that the _p-type Perron-Frobenius scaling operator [D.sub.p]([delta]) is well defined in accordance with (7-DT//CT) applied to A([delta]) as follows:

[mathematical expression not reproducible], (14-DT//CT)

where v([delta]) [much greater than] 0, w([delta]) [much greater than] 0 are the Perron-Frobenius eigenvectors of A([delta]), which satisfy [[parallel]v([delta])[parallel].sub.[infinity]] = 1, [[parallel]w([delta])[parallel].sub.[infinity]] = 1.

Note that, for simplicity, in (13-DT//CT) one can use a matrix U = [[u.sub.ij]] with [u.sub.ij] = 1, i, j = 1, ..., n, and the results of the current section preserve their validity.

Theorem 6. Let 1 [less than or equal to] p [less than or equal to] [infinity]. Consider an asymptotically stable, positive linear system of form (1-DT) (resp., (1-CT)), with matrix A reducible. Let matrix [D.sub.p]([delta]) be defined by (14-DT//CT).

Then, for any [epsilon] > 0 arbitrarily small, there exists [delta]([epsilon]) > 0 such that, for each [delta] [member of] (0,[delta]([epsilon])], system (1-DT) (resp., (1-CT)) is (1, [alpha](A) + [epsilon])-stable relative to the norm [V.sup.[delta].sub.p](x) = [[parallel][D.sub.p]([delta]) x [parallel].sub.p], where [D.sub.p]([delta]) is the scaling operator defined by (14-DT//CT).

Proof. For any 1 [less than or equal to] p [less than or equal to] [infinity], [V.sup.[delta].sub.p](x) is a vector norm, defined by the standard p-norm, weighted by the scaling operator [D.sub.p]([delta]) (14-DT//CT).

(a) In the DT case, for any 1 [less than or equal to] p [less than or equal to] [infinity], we first prove that

[mathematical expression not reproducible]. (15-DT)

This proof relies on the following steps:

(i) The inequality [[lambda].sub.max](A) [less than or equal to] [parallel][D.sub.p]([delta]) A [([D.sub.p]([delta])).sup.-1][parallel][sub.p] is obvious.

(ii) For any y [member of] [R.sup.n], we can write the componentwise inequalities [mathematical expression not reproducible] and the monotonicity of the vector p-norms implies [mathematical expression not reproducible].

(iii) The matrix norm means [mathematical expression not reproducible].

(iv) Equality (9-DT) written for A([delta]) and the diagonal operator [D.sub.p]([delta]) ensures [parallel] [D.sub.p],([delta])A([delta])([D.sub.p][([delta])).sup.-1] [parallel][sub.p] = [[lambda].sub.max](A([delta])).

(v) Since the Perron-Frobenius eigenvalue [[lambda].sub.max](A([delta])) = [[lambda].sub.max](A + [delta]U) is continuous and strictly increasing with respect to [delta] > 0 (Theorem 8.1.18 in )), we also have [mathematical expression not reproducible].

Thus, the proof of (15-DT) is completed and we can use it for an arbitrary trajectory of system (1-DT). In accordance with (8-DT), where [V.sup.[delta].sub.p](x) = [parallel][D.sub.p]([delta])x[parallel][sub.p], we can write

[mathematical expression not reproducible]. (16-DT)

Since [[lambda].sub.max](A) = [alpha](A), we get

[mathematical expression not reproducible], (17-DT)

proving that system (1-DT) is (M, [beta])-stable, with M = 1 and [beta] = [alpha](A) + [epsilon], in accordance with (3-DT).

(b) In the CT case, for any 1 [less than or equal to] p [less than or equal to] [infinity], we first prove that

[mathematical expression not reproducible]. (15-CT)

This proof relies on the following steps:

(i) The inequality [mathematical expression not reproducible] is ensured by Fact 11.15.7 in .

(ii) For small [mathematical expression not reproducible].

(iii) The second equality in (9-CT) written for A([delta]) and the diagonal operator [mathematical expression not reproducible].

(iv) To complete the proof of (15-CT), we use the equalities [mathematical expression not reproducible] and the same approach as in the proof of part (a).

For an arbitrary trajectory of system (1-CT), in accordance with (8-CT) where [V.sup.[delta].sub.p](x) = [[parallel][D.sub.p]([delta])x[parallel].sub.p], we can write

[mathematical expression not reproducible], (16-CT)

since [mathematical expression not reproducible] as per Fact 11.15.7 in .

By using the same proof as for implication (10-CT) [??] (11-CT), as well as the equality [[lambda].sub.max](A) = [alpha](A), we get

[mathematical expression not reproducible], (17-CT)

proving that system (1-CT) is (M, [beta])-stable, with M = 1 and [beta] = [alpha](A) + [epsilon], in accordance with (3-CT).

Remark 7. If system (1-DT) (resp., (1-CT)) is asymptotically stable, for the concrete use of Theorem 6, we chose e > 0 such that [[lambda].sub.max](A) + [epsilon] < 1 (resp., [[lambda].sub.max](A) + [epsilon] < 0) and then we search for a [[delta].sup.*] > 0 such that [[lambda].sub.max](A) < [[lambda].sub.max] (A([[delta].sup.*])) < [[lambda].sub.max](A) + [epsilon], for both DT and CT case. This search requires the computation of [[lambda].sub.max](A([delta])) for some [delta] > 0 and relies on the fact that [[lambda].sub.max](A([delta])) is continuous and strictly increasing with respect to [delta] > 0 (as mentioned in the proof of Theorem 6). Once such [[delta].sup.*] > 0 is found, the diagonal operator [D.sub.p]([[delta].sup.*]) is built in accordance with (14-DT//CT).

Remark 8. If system (1-DT) (resp., (1-CT)) is asymptotically stable and matrix A has the structure of the communication classes considered by the current subsection, Theorem 2 can be used in the following particular cases:

(i) For p = 1, if the basic and final classes of [A.sup.T] coincide, then the diagonal operator [D.sub.1] can be built in accordance with (7-DT//CT), since there exists a positive left eigenvector w [much greater than] 0 (as per Theorem 3.10, Chapter 2, from ).

(ii) For p = [infinity], if the basic and final classes of A coincide, then the diagonal operator [D.sub.[infinity]] can be built in accordance with (7-DT//CT), since there exists a positive right eigenvector v [much greater than] 0 (as per Theorem 3.10, Chapter 2, from ).

4. Case Studies

Example 1. Consider the electric circuit in Figure 1 that was also used in , where the following state-space model is given:

[mathematical expression not reproducible]. (18-CT)

For [mathematical expression not reproducible], the circuit dynamics are described by

[mathematical expression not reproducible]. (19-CT)

Matrix A is irreducible. For the numerical values of the entries presented above, it has the Perron-Frobenius eigenvalue [alpha](A) = -9.3695, with the associated right eigenvector v = [[1.0000 0.8365].sup.T] and left eigenvector w = [[1.0000 0.2788].sup.T].

For the vector p-norms defined by p [member of] {2, [infinity]}, this example is able to offer nice illustrations of the differences between the concepts of (1,[alpha](A))-stability and (1,[[alpha].sub.[parallel][parallel]](A))-stability.

Thus, the short-term behavior fulfills the condition of (1,-7.1583)-stability relative to the vector norm [[parallel] x [parallel].sub.2], meaning that

[mathematical expression not reproducible], (20-CT)

whereas the condition of (1, -9.3695)-stability is fulfilled relative to the vector norm [mathematical expression not reproducible], meaning that

[mathematical expression not reproducible]. (21-CT)

Similarly, the short-term behavior fulfills the condition of (1,-7.6923)-stability relative to the vector norm [[parallel] x [parallel].sub.[infinity]], meaning that

[mathematical expression not reproducible], (22-CT)

whereas the condition of (1,-9.3695)-stability is fulfilled relative to the vector norm [V.sub.[infinity]](x) = [[parallel][D.sub.[infinity]] x [parallel].sub.[infinity]], [D.sub.[infinity]] = diag{1/[v.sub.1], 1/[v.sub.2]} = diag{1,1.1955}, meaning that

[mathematical expression not reproducible]. (23-CT)

Both cases discussed above emphasize the role played by the scaling operator [D.sub.P] defined by (7-DT//CT) for p [member of] {2, [infinity]}, when we are interested in describing the short-term behavior with the best decreasing rate (i.e., [alpha](A) = -9.3695).

Note that the decreasing rate [alpha](A) = -9.3695 can also be obtained by using [[parallel] x [parallel].sub.1] and the scaling operator [D.sub.1] = diag{[[w.sub.1], [w.sub.2]]} = diag{[1.0000,0.2788]}, meaning that

[mathematical expression not reproducible]. (24-CT)

In other words, the condition of (1, [alpha](A))-stability is also fulfilled relative to the vector norm [V.sub.1](x) = [[parallel][D.sub.1] x [parallel].sub.1], but we cannot talk about [mathematical expression not reproducible] (A))-stability relative to the vector norm [mathematical expression not reproducible].

Example 2. Consider the three-compartment mammillary CPB (cardiopulmonary bypass) model used in pharmacokinetics and discussed in Subsection 4.3 of . The model has form (1-CT) with

[mathematical expression not reproducible], (25-CT)

where [k.sub.11] = 0.001, [k.sub.21] = 0.2, [k.sub.12] = 0.2, [k.sub.31] = 0.01, [k.sub.13] = 0.02 are the values used in the simulation example at page 122 in . The mentioned example rules out partial monotonicity with respect to any compartment. Despite the lack of this property, our Theorem 2 is able to prove the existence of scaled p-norms relative to which the considered system has an "ideal" transient behavior; that is, it is (1, [alpha](A))-stable.

Matrix A is irreducible. For the numerical values of the entries presented above, it has the Perron-Frobenius eigenvalue [alpha](A) = -0.0004, with the associated right eigenvector v = [[0.9980 1.0000 0.5091].sup.T] and left eigenvector w = [[0.9801 0.9821 1.0000].sup.T]. Thus, for any 1 [less than or equal to] p [less than or equal to] [infinity], Theorem 2 ensures the (1, [alpha](A))-stability, relative to vector norm [V.sub.p](x) = [[parallel][D.sub.p]x[parallel].sub.p], where [D.sub.p] is the scaling operator defined by (7-DT//CT).

Our example provides graphical plots that support the intuitive understanding of the theoretical result stated by Theorem 2, for the frequently used vector p-norms. Thus, for each p [member of] {1,2, [infinity]}, we simulate the system evolution for two initial conditions; namely, [x.sub.0] = [[0.5 1 1].sup.T] (studied at page 122 in ) and [mathematical expression not reproducible], and the simulation results are given in two distinct figures. Each figure displays the time dependence of the following functions:

(i) the three state variables plotted in black (by using line-types similar to Figure 4.2 in , i.e., solid line for [x.sub.1](i), dashdot line for [x.sub.2](i), dashed line for [x.sub.3] (i)),

(ii) the left-hand side of inequality (3-CT) in Definition 1 plotted in red,

(iii) the right-hand side of inequality (3-CT) in Definition 1 plotted in blue.

These figures offer a nice graphical illustration for the fulfillment of the (1, [alpha](A))-stability condition, expressed by Definition 1 and tested by Theorem 2. For all plots, the simulation horizon was selected t [member of] [0, 300], as corresponding to a characterization of the short-term dynamics and also permitting direct comparisons to the simulation results presented in Figure 4.2 in .

The results corresponding to the use of Theorem 2 with p = 1 are presented in Figure 2. Figure 2(a) considers the dynamics started from the initial condition [x.sub.0] = [[0.5 1 1].sup.T], and Figure 2(b) refers to dynamics started from the initial condition [[??].sub.0](1) = [[0.5 0.5 1.4910].sup.T], for which we have the equality [V.sub.1]([[??].sub.0(1)]) = [V.sub.1]([x.sub.0]) = 2.4721, where [V.sub.1](x) = [parallel][D.sub.1] x[parallel][sub.1], [D.sub.1] = diag{0.9801,0.9821,1.0000}. In both figures, the blue line is identical and depicts the exponentially decreasing function [V.sub.1]([[??].sub.0](1)])[e.sup.[alpha](A)t] = [V.sub.1] ([x.sub.0])[e.sup.[alpha](A)t] meaning the right-hand side of inequality (3-CT) in Definition 1. The red line depicts the function [V.sub.1](x(t;0, [x.sub.0])) in Figure 2(a) and the function [V.sub.1] (x(t; 0, [[??].sub.0](1)])) in Figure 2(b), meaning the left-hand side of inequality (3-CT) in Definition 1 for the two discussed cases. Figures 2(a) and 2(b) graphically illustrate the inequality [V.sub.1](x(t;0,[x.sub.0])) [less than or equal to] [V.sub.1]([x.sub.0])[e.sup.[alpha](A)t] and, respectively, the inequality [V.sub.1](x(t;0, [[??].sub.0(1)])) [less than or equal to] [V.sub.1]([[??].sub.0(1)]) [e.sup.[alpha](A)t], for t [member of] [0,300], which, for this numerical example, are satisfied as equalities (and, consequently, the plots coincide for the left-hand sides (in red) and right-hand sides (in blue)).

The results corresponding to the use of Theorem 2 with p = 2 are presented in Figure 3. Figure 3(a) considers the dynamics started from the initial condition [x.sub.0] = [[0.5 1 1].sup.T], and Figure 3(b) refers to dynamics started from the initial condition [[??].sub.0(2)] = [[0.5 0.5 1.1726].sup.T], for which we have the equality [V.sub.2]([[??].sub.0(2)]) = [V.sub.2]([x.sub.0]) = 1.7865, where [V.sub.2](x) = [parallel][D.sub.2]x[parallel][sub.2], [D.sub.2] = diag{0.9910,0.9910,1.4015}. In both figures, the blue line is identical and depicts the exponentially decreasing function [V.sub.2]([[??].sub.0(2)])[e.sup.[alpha](A)t] = [V.sub.2] ([x.sub.0])[e.sup.[alpha](A)t] meaning the right-hand side of inequality (3-CT) in Definition 1. The red line depicts the function [V.sub.2](x(t; 0, [x.sub.0])) in Figure 3(a), and the function [V.sub.2](x(t; 0, [[??].sub.0(2)])) in Figure 3(b), meaning the left-hand side of inequality (3-CT) in Definition 1 for the two discussed cases. Figures 3(a) and 3(b) graphically illustrate the inequality [V.sub.2](x(t;0,[x.sub.0])) [less than or equal to] [V.sub.2]([x.sub.0])[e.sup.[alpha](A)t], and, respectively, the inequality [mathematical expression not reproducible].

The results corresponding to the use of Theorem 2 with p = [infinity] are presented in Figure 4. Figure 4(a) considers the dynamics started from the initial condition [x.sub.0] = [[0.5 1 1].sup.T], and Figure 4(b) refers to dynamics started from the initial condition [[??].sub.0([infinity])] = [[1.5 1 1].sup.T], for which we have the equality [mathematical expression not reproducible]. In both figures, the blue line is identical and depicts the exponentially decreasing function [V.sub.[infinity]]([[??].sub.0([infinity])])[e.sup.[alpha](A)t] = [V.sub.[infinity]]([x.sub.0]) [e.sup.[alpha](A)t] meaning the right-hand side of inequality (3-CT) in Definition 1. The red line depicts the function [V.sub.[infinity]](x(t; 0, [x.sub.0])) in Figure 4(a), and the function V[infinity](x(t;0, [[??].sub.0([infinity])])) in Figure 4(b), meaning the left-hand side of inequality (3-CT) in Definition 1 for the two discussed cases. Figures 4(a) and 4(b) graphically illustrate the inequality [V.sub.[infinity]](x(t;0, [x.sub.0])) [less than or equal to] [V.sub.[infinity]]([x.sub.0])[e.sup.[alpha](A)t], and, respectively, the inequality [mathematical expression not reproducible], for t [member of] [0,300].

5. Conclusions

The paper proves the existence of important connections between the Perron-Frobenius eigenstructure of a positive linear system and the short-term evolution of its state-space trajectories. These connections are explored by the help of the concept of (M, [beta])-stability relative to scaled vector p-norms, 1 [less than or equal to] p [less than or equal to] [infinity]. If the time evolution of the trajectories is monitored by such a norm, then there exists a scaling operator built from the right and left Perron-Frobenius eigenvectors, which ensures an "ideal" transient behavior, meaning the least transient bound (i.e., M = 1), as well as the fastest exponential rate (i.e., [beta] as close to [alpha](A) as we want). The concrete expression of the scaling operator depends on 1 [less than or equal to] p [less than or equal to] [infinity], in the sense that for [p.sub.1] [not equal to] [p.sub.2] the contributions of the Perron-Frobenius eigenvectors are different to the construction of the two scaling operators.

Our results cover both discrete- and continuous-time dynamics of positive linear systems. The analysis is organized so as to capture the differences between the cases where the system dynamics is defined by matrices irreducible and reducible, respectively. For the case of irreducible matrices we show that the fastest exponential rate in the "ideal" transient behavior means a unique value for [beta], namely, the fulfillment of the equality [beta] = [alpha](A).

We use our theoretical developments in two numerical case studies, both already discussed in literature by previous works. The first case study illustrates the differences between the concepts of (1, [alpha](A))-stability and (1, [[alpha].sub.[parallel][parallel]](A))-stability associated with the operation of an electrical circuit. The second one constructs the scaling operators corresponding to p [member of] {1,2, [infinity]} for a mammillary compartmental system and proves their role in ensuring the "ideal" short-term behavior.

http://dx.doi.org/10.1155/2016/9605464

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors acknowledge the financial support of UEFISCDI Romania under Grant PN-II-ID-PCE-2011-3-1038.

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Octavian Pastravanu and Mihaela-Hanako Matcovschi

Department of Automatic Control and Applied Informatics, Technical University "GheorgheAsachi"ofIasi, 700050 Iasi, Romania

Correspondence should be addressed to Mihaela-Hanako Matcovschi; mhanako@ac.tuiasi.ro

Received 5 August 2015; Revised 11 January 2016; Accepted 14 January 2016

Caption: Figure 1: Electrical circuit used in Example 1.

Caption: Figure 2: Results provided by Theorem 2 for Example 2 with p = 1 and initial condition (a) [x.sub.0] = [[0.5 1 1].sup.T] and (b) [[??].sub.0(1)] = [[0.5 0.5 1.4910].sup.T].

Caption: Figure 3: Results provided by Theorem 2 for Example 2 with p = 2 and initial condition (a) [x.sub.0] = [[0.5 1 1].sup.T] and (b) [[??].sub.0(2)] = [[0.5 0.5 1.1726].sup.T]

Caption: Figure 4: Results provided by Theorem 2 for Example 2 with p = [infinity] and initial condition (a) [x.sub.0] = [[0.5 1 1].sup.T] and (b) [[??].sub.0([infinity])] = [[1.5 1 1].sup.T].