Abstract
The main purpose of this work is to show that the Perron-Frobenius eigenstructure of a positive linear system is involved not only in the characterization of long-term behavior (for which well-known results are available) but also in the characterization of short-term or transient behavior. We address the analysis of the short-term behavior by the help of the “-stability” concept introduced in literature for general classes of dynamics. Our paper exploits this concept relative to Hölder vector -norms, , adequately weighted by scaling operators, focusing on positive linear systems. Given an asymptotically stable positive linear system, for each , we prove the existence of a scaling operator (built from the right and left Perron-Frobenius eigenvectors, with concrete expressions depending on ) that ensures the best possible values for the parameters and , corresponding to an “ideal” short-term (transient) behavior. We provide results that cover both discrete- and continuous-time dynamics. Our analysis also captures the differences between the cases where the system dynamics is defined by matrices irreducible and reducible, respectively. The theoretical developments are applied to the practical study of the short-term behavior for two positive linear systems already discussed in literature by other authors.
1. Introduction
1.1. Notation
Let be a vector. The Hölder vector -norm is defined as for , and for .
Let be a square matrix. Let be the spectrum of matrix ; the eigenvalues of are denoted by . The norm of matrix induced by a vector norm (not necessarily a Hölder -norm) is defined as . The matrix measure of with respect to a matrix norm is given by , where stands for the unit matrix of order . In the particular case of Hölder -norms , the expressions of the induced matrix norms are , , , and the corresponding matrix measures are , , , as per Fact 11.15.7 from [1].
1.2. Concept of -Stability
The concept of -stability has been developed by the monographic work in [2], 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,and continuous-time (abbreviated CT) dynamics,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 (resp., ) relies on the notion of growth rate, denoted by , which is defined as in Subsection 3.3.2 of [2], byfor the DT case, and, respectively, for the CT case. In algebraic terms, the growth rate equals the spectral radius of for system and, respectively, the spectral abscissa of for system .
The analysis of the transient (short-term) behavior of system (resp., ) relies on the notion of -stability which is defined as follows.
Definition 1 (Definition 5.5.1 [2]). Let and , with in the DT case, and, respectively, in the CT case. Consider an absolute vector norm in the state-space . System (resp., ) is said to be -stable relative to the norm , if its trajectories satisfy the inequalityin the DT case, and, respectively, in the CT case. The scalar is called the transient bound and the scalar is called the exponential rate.
In colloquial terms, a “good” transient behavior relative to norm means close to 1 and close to . The “ideal” transient behavior relative to norm can be introduced by formal specifications, namely, and (i.e., both minimal exponential rate and minimal transient bound). Obviously, for these particular values of and , inequality (resp., ) becomes inequality in the case of system , and, respectively, in the case of system .
Both growth rate and -stability are rigorously related to the properties of the semigroup of operators generated by (discrete- or continuous-time), for example, [3]. Consider the semigroup of linear operators generated by matrix . Then, by using the operator norm induced by the vector norm , inequality can be equivalently written for operators , in the formand inequality can be equivalently written for operators , in the formIf we take in inequality (resp., ), then the initial growth rate relative to the norm (see Definition and Proposition in [2]), denoted by , is defined asfor the DT case, and, respectively, as for the CT case.
By comparing with (resp., with ), one can simply notice that the inequality 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 is characterized by the equality .
Obviously, inequality (resp., ) is satisfied by the trajectories of any system (resp., ) whose matrix is diagonalizable, once the considered vector norm 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 -stability provides relevant information only when the considered norm is able to preserve the key role played by the original state-space vector.
In work [2], the “ideal” transient behavior is illustrated by a single class of linear systems, namely, those where matrix in equality (resp., ) is a normal matrix, and the norm considered in inequality (resp., ) is the 2-norm.
1.3. Paper Objective and Organization
The main objective of our paper is to expand the analysis framework of -stability and “ideal” transient behavior by focusing on the class of positive linear systems, whose dynamics are generated by matrices nonnegative in equality , and essentially nonnegative in equality . 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 -norm () considered in inequality (resp., ), 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 . In norm terms, the use of the scaled state-space variables represents a simple weighting of the standard vector -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 , the properties of scaled positive systems presented by the recent paper [9] in Propositions and and Remark .
Thus, as an overall comment on the contribution brought by this paper, we notice that, besides the extension of the investigation for -stability along the lines proposed by [2], it also reveals deeper connections between the Perron-Frobenius eigenstructure 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 that defines the dynamics of system (resp., ) is irreducible (e.g., [8, Chapter 2, Definition ]). Matrix is irreducible if and only if the oriented graph associated with is strongly connected (e.g., [8, Chapter 2, Theorem ]). In the DT case matrix is nonnegative (i.e., all its entries are nonnegative). In the CT case matrix is essentially nonnegative or Metzler (i.e., all its off-diagonal entries are nonnegative).
The meanings of the Perron-Frobenius eigenstructure agree for nonnegative and essentially nonnegative, in the sense of the following properties:(i)Matrix has a simple real eigenvalue, called the Perron-Frobenius eigenvalue and denoted by , which satisfies , , in the DT case, and , , in the CT case.(ii)In both DT and CT cases, matrix has right and left positive eigenvectors , associated with , which are called the right and left Perron-Frobenius eigenvectors and satisfy and .
Given the irreducible matrix , for any , , we define the -type Perron-Frobenius scaling operator
built from the positive eigenvectors , , of , which satisfy .
Theorem 2. Let . Consider an asymptotically stable, positive linear system of form (resp., ), with matrix irreducible.
Then, system (resp., ) is -stable relative to the norm , where is the scaling operator defined by .
Proof. For any , is a vector norm, defined by the standard -norm, weighted by the scaling operator .
(a) In the DT case, for any , for an arbitrary trajectory we can writeOn the other hand, we have as resulting from the proof of the theorem presented by paper [15].
Thus, from and we get the inequalityand, eventually,proving that system is -stable, with and , in accordance with .
(b) For the CT case, for any , we can writeOn the other hand, we have Indeed, the first equality in results from Fact 11.15.7 [1]. For the second equality, let us pick an such that is nonnegative. Since matrices and have the same left and right Perron-Frobenius eigenvectors, by using part (a) of our proof, for any , we get , which yields . By taking , for , we obtain .
Thus, from and we get the inequalityNow, if holds for , let us consider the differential equation with the initial condition . According to Theorem in [3], we haveproving that system is -stable, with and , in accordance with .
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 for (e.g., [16]), we show that the system dynamics exhibit set-invariance properties for sets defined by both right and left eigenvectors. Indeed, for any , the existence of the “ideal” transient behavior relative to the norm (proven by Theorem 2) is equivalent to the invariance of the exponentially contractive sets and, respectively,with respect to system (resp., ). In particular, for an arbitrary (but fixed) time (resp., ) and an arbitrary (but fixed) constant , the Minkowski functional of the constant set defined by (resp., ) equals multiplied by a positive scalar. It is worth saying that the invariance of the sets of form (resp., ) has already been mentioned by our previous works [17, 18]. The invariance analysis for sets of form (resp., ) can be extended to positive interval systems, by using Corollary and Theorem in our paper [19], dealing with dynamics defined by interval matrices.(ii)The existence of the invariant contractive sets of form (resp., ) defined for all , allows a deeper insight into the dynamics of positive linear systems, for which the classical property presented in literature (e.g., [4]) is the invariance of the nonnegative orthant . In fact, all the nonnegative sets , , , with defined by are invariant with respect to positive system , and all the nonnegative sets , , , with defined by are invariant with respect to positive system .
Remark 4. The proof of Theorem 2 also highlights the following dynamical properties of system (resp., ) typical to the “ideal” transient behavior:(i)For any , , can serve for system (resp., ), as a Lyapunov function (in a form called “norm Lyapunov function”), which decreases with the fastest possible rate, namely, , along the trajectories of system (resp., ).(ii)Propositions and and Remark in [9] refer to the existence of norm Lyapunov functions corresponding to , but the possibility of ensuring the fastest decreasing rate (when the Perron-Frobenius eigenvectors are used) is not investigated.(iii)The semigroup of operators , (resp., , ) is contractive relative to any operator norm subordinated to a vector norm of form , . For all , the contraction rate is precisely .(iv)Form of the scaling operator uses the right and left eigenvectors of , uniquely defined by the norm equalities , . Obviously, Theorem 2 holds true if is replaced by any diagonal matrix , , 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 , is not compulsory).
Remark 5. Paper [13] addresses the transient behavior of positive linear systems and considers state-space transforms defined as in by matrix . However, these transforms of type are not regarded as individual scalings of the state-space variables (as illustrated by Theorem 2). The -weighted norms are used in the -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 -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 .
3. Results for Positive Systems Defined by Reducible Matrices
Throughout this section, matrix that defines the dynamics of system (resp., ) is reducible (i.e., the oriented graph associated with is not strongly connected). Matrix is nonnegative (resp., essentially nonnegative). To deal with the reducibility of matrix , we consider two distinct cases, specified by the structure of the communication classes of matrix , for example, Section , Chapter 2 in [8].
3.1. All Communication Classes of Matrix Are Basic and Final
Theorem 2 can still be used if all the communication classes are basic and final. Indeed Theorem , Chapter 2, from [8] guarantees the existence of positive right and left eigenvectors , , associated with the multiple eigenvalue . (In other words, we are able to generalize the “ irreducible” case, where the eigenvalue is unique.) By a proof similar to the proof of the Theorem presented in [15], one can show that equality (resp., equality ) holds true for nonnegative (resp., essentially nonnegative).
3.2. Structure of Communication Classes of Matrix Is Different from Section 3.1
There exists nonnegative right and left eigenvectors associated with the eigenvalue , but at least one of them contains 0 elements, in accordance with Theorems and , Chapter 2, from [8]. Hence, the diagonal operator defined by cannot be used any longer.
However, the same type of information becomes available, if instead of we consider the slightly modified matrix:where is small enough. Matrix is irreducible, so that the -type Perron-Frobenius scaling operator is well defined in accordance with applied to as follows:where , are the Perron-Frobenius eigenvectors of , which satisfy , .
Note that, for simplicity, in one can use a matrix with , and the results of the current section preserve their validity.
Theorem 6. Let . Consider an asymptotically stable, positive linear system of form (resp., ), with matrix reducible. Let matrix be defined by .
Then, for any arbitrarily small, there exists such that, for each , system (resp., ) is -stable relative to the norm , where is the scaling operator defined by .
Proof. For any , is a vector norm, defined by the standard -norm, weighted by the scaling operator .
(a) In the DT case, for any , we first prove that This proof relies on the following steps: (i)The inequality is obvious.(ii)For any , we can write the componentwise inequalities and the monotonicity of the vector -norms implies .(iii)The matrix norm means .(iv)Equality written for and the diagonal operator ensures .(v)Since the Perron-Frobenius eigenvalue is continuous and strictly increasing with respect to (Theorem in [20])), we also have , .Thus, the proof of is completed and we can use it for an arbitrary trajectory of system . In accordance with , where , we can writeSince , we getproving that system is -stable, with and , in accordance with .
(b) In the CT case, for any , we first prove that This proof relies on the following steps: (i)The inequality is ensured by Fact in [1].(ii)For small we can write and, subsequently, that yields .(iii)The second equality in written for and the diagonal operator ensures .(iv)To complete the proof of , we use the equalities , and the same approach as in the proof of part (a).For an arbitrary trajectory of system , in accordance with where , we can write since as per Fact in [1].
By using the same proof as for implication ⇒ , as well as the equality , we getproving that system is -stable, with and , in accordance with .
Remark 7. If system (resp., ) is asymptotically stable, for the concrete use of Theorem 6, we chose such that (resp., ) and then we search for a such that , for both DT and CT case. This search requires the computation of for some and relies on the fact that is continuous and strictly increasing with respect to (as mentioned in the proof of Theorem 6). Once such is found, the diagonal operator is built in accordance with .
Remark 8. If system (resp., ) is asymptotically stable and matrix has the structure of the communication classes considered by the current subsection, Theorem 2 can be used in the following particular cases:(i)For , if the basic and final classes of coincide, then the diagonal operator can be built in accordance with , since there exists a positive left eigenvector (as per Theorem , Chapter 2, from [8]).(ii)For , if the basic and final classes of coincide, then the diagonal operator can be built in accordance with , since there exists a positive right eigenvector (as per Theorem , Chapter 2, from [8]).
4. Case Studies
Example 1. Consider the electric circuit in Figure 1 that was also used in [21], where the following state-space model is given: For and , , , , , the circuit dynamics are described byMatrix is irreducible. For the numerical values of the entries presented above, it has the Perron-Frobenius eigenvalue , with the associated right eigenvector and left eigenvector .
For the vector -norms defined by , this example is able to offer nice illustrations of the differences between the concepts of -stability and -stability.
Thus, the short-term behavior fulfills the condition of -stability relative to the vector norm , meaning thatwhereas the condition of -stability is fulfilled relative to the vector norm , , meaning thatSimilarly, the short-term behavior fulfills the condition of -stability relative to the vector norm , meaning thatwhereas the condition of -stability is fulfilled relative to the vector norm , , meaning thatBoth cases discussed above emphasize the role played by the scaling operator defined by for , when we are interested in describing the short-term behavior with the best decreasing rate (i.e., ).
Note that the decreasing rate can also be obtained by using and the scaling operator , meaning that In other words, the condition of -stability is also fulfilled relative to the vector norm , but we cannot talk about -stability relative to the vector norm , since .
Example 2. Consider the three-compartment mammillary CPB (cardiopulmonary bypass) model used in pharmacokinetics and discussed in Subsection of [6]. The model has form with where , , , , are the values used in the simulation example at page 122 in [6]. 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 -norms relative to which the considered system has an “ideal” transient behavior; that is, it is -stable.
Matrix is irreducible. For the numerical values of the entries presented above, it has the Perron-Frobenius eigenvalue , with the associated right eigenvector and left eigenvector . Thus, for any , Theorem 2 ensures the -stability, relative to vector norm , where is the scaling operator defined by .
Our example provides graphical plots that support the intuitive understanding of the theoretical result stated by Theorem 2, for the frequently used vector -norms. Thus, for each , we simulate the system evolution for two initial conditions; namely, (studied at page 122 in [6]) and with , 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 in [6], i.e., solid line for , dashdot line for , dashed line for ),(ii)the left-hand side of inequality in Definition 1 plotted in red,(iii)the right-hand side of inequality in Definition 1 plotted in blue.These figures offer a nice graphical illustration for the fulfillment of the -stability condition, expressed by Definition 1 and tested by Theorem 2. For all plots, the simulation horizon was selected , as corresponding to a characterization of the short-term dynamics and also permitting direct comparisons to the simulation results presented in Figure in [6].
The results corresponding to the use of Theorem 2 with are presented in Figure 2. Figure 2(a) considers the dynamics started from the initial condition , and Figure 2(b) refers to dynamics started from the initial condition , for which we have the equality , where , . In both figures, the blue line is identical and depicts the exponentially decreasing function meaning the right-hand side of inequality in Definition 1. The red line depicts the function in Figure 2(a) and the function in Figure 2(b), meaning the left-hand side of inequality in Definition 1 for the two discussed cases. Figures 2(a) and 2(b) graphically illustrate the inequality and, respectively, the inequality , for , 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 are presented in Figure 3. Figure 3(a) considers the dynamics started from the initial condition , and Figure 3(b) refers to dynamics started from the initial condition , for which we have the equality , where , . In both figures, the blue line is identical and depicts the exponentially decreasing function meaning the right-hand side of inequality in Definition 1. The red line depicts the function in Figure 3(a), and the function in Figure 3(b), meaning the left-hand side of inequality in Definition 1 for the two discussed cases. Figures 3(a) and 3(b) graphically illustrate the inequality , and, respectively, the inequality , for .
The results corresponding to the use of Theorem 2 with are presented in Figure 4. Figure 4(a) considers the dynamics started from the initial condition , and Figure 4(b) refers to dynamics started from the initial condition , for which we have the equality , where , . In both figures, the blue line is identical and depicts the exponentially decreasing function meaning the right-hand side of inequality in Definition 1. The red line depicts the function in Figure 4(a), and the function in Figure 4(b), meaning the left-hand side of inequality in Definition 1 for the two discussed cases. Figures 4(a) and 4(b) graphically illustrate the inequality , and, respectively, the inequality , for .
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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 -stability relative to scaled vector -norms, . 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., ), as well as the fastest exponential rate (i.e., as close to as we want). The concrete expression of the scaling operator depends on , in the sense that for 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 , namely, the fulfillment of the equality .
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 -stability and -stability associated with the operation of an electrical circuit. The second one constructs the scaling operators corresponding to for a mammillary compartmental system and proves their role in ensuring the “ideal” short-term behavior.
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.