Graph-Theoretic Stability Conditions for Metzler Matrices and Monotone Systems

TL;DR

This paper develops graph-theoretic stability conditions for Metzler matrices and monotone systems, linking cycle structures in the interconnection graph to system stability through input-to-state stability and small-gain theory.

Contribution

It introduces novel graph-based stability criteria for Metzler systems using input-to-state gains and extends these results to nonlinear monotone systems.

Findings

Cyclic small-gain theorem is necessary and sufficient for Metzler system stability.

New graph-theoretic conditions based on sum-interconnection gains.

Structural properties of the interconnection graph influence stability.

Abstract

This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from input-to-state stability and network small-gain theory, we first establish necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices. Specifically, we derive two sets of stability conditions based on two forms of input-to-state stability gains for Metzler systems, namely max-interconnection gains and sum-interconnection gains. Based on the max-interconnection gains, we show that the cyclic small-gain theorem becomes necessary and sufficient for the stability of Metzler systems; based on the sum-interconnection gains, we obtain novel graph-theoretic conditions for the stability of Metzler systems. All these conditions highlight the role of cycles in the interconnection graph and unveil how the structural properties of the graph affect stability. Finally, we extend our results to the nonlinear monotone system and obtain similar sufficient conditions for global asymptotic stability.

Equations

$$\Vert x{\Vert }_{[0,t]}=\underset{s\in [0,t]}{\mathrm{esssup}}|x(s)|.$$

$$\sum _{i=1}^n{x}_i\le \underset{i\in \{1,\dots ,n\}}{\max }\{{\alpha }_i{x}_i\}.$$

$$\sum _{i=1}^n{x}_i\le \sum _{i=1}^n\frac{{\alpha }_s{x}_s}{{\alpha }_i}\le {\alpha }_s{x}_s=\underset{i\in \{1,\dots ,n\}}{\max }\{{\alpha }_i{x}_i\}.$$

$$M=\left[\begin{array}{cc}{M}_{n-1}& b_{n-1}\\ c_{n-1}^{\top }& d_{n-1}\end{array}\right]$$

$$\dot{x}=f(x,u),$$

$$|x(t)|\le \max \{\beta (|x_0|,t),\gamma (\Vert u{\Vert }_{\infty })\}.$$

$${\dot{x}}_i=f_i(x_i,x_{{\mathcal{N}}_i},u_i),\text{for all }i\in \{1,\dots ,n\},$$

$$|x_i(t)|\le \underset{j\in {\mathcal{N}}_i}{\max }\{{\beta }_i(|x_i(0)|,t),{\Psi }_{ij}(\Vert x_j{\Vert }_{[0,t]}),{\Psi }_i(\Vert u_i{\Vert }_{\infty })\}.$$

$$|x_i(t)|\le {\beta }_i(|x_i(0)|,t)+\sum _{j\in {\mathcal{N}}_i}{\Gamma }_{ij}(\Vert x_j{\Vert }_{[0,t]})+{\Gamma }_i(\Vert u_i{\Vert }_{\infty }).$$

$${\Psi }_{i_2{i}_1}\circ {\Psi }_{i_3{i}_2}\circ \cdots \circ {\Psi }_{i_1{i}_k}(s)<s,$$

$$\dot{x}=Mx+u,$$

$${\dot{x}}_i=m_{ii}{x}_i+\sum _{j\in {\mathcal{N}}_i}{m}_{ij}{x}_j+u_i,\text{for all }i\in \{1,\dots ,n\}.$$

$$m_{ii}<0,\text{for all }i\in \{1,\dots ,n\};$$

$$\frac{m_{ij}}{-m_{ii}}\le {\gamma }_{ij},\text{for all }i\in \{1,\dots ,n\},j\in {\mathcal{N}}_i;$$

$$\sum _{j\in {\mathcal{N}}_i}\left(\frac{m_{ij}}{-m_{ii}}\right){\psi }_{ij}^{-1}<1,\text{for all }i\in \{1,\dots ,n\};$$

$$x_i(t)=e^{m_{ii}t}{x}_i(0)+\sum _{j\in {\mathcal{N}}_i}{m}_{ij}{\int }_0^t{e}^{m_{ii}(t-\tau )}{x}_j(\tau )d\tau +{\int }_0^t{e}^{m_{ii}(t-\tau )}{u}_i(\tau )d\tau ,$$

$$\begin{array}{rl}|x_i(t)|& \le e^{m_{ii}t}|x_i(0)|+\sum _{j\in {\mathcal{N}}_i}{m}_{ij}{\int }_0^t|e^{m_{ii}(t-\tau )}{x}_j(\tau )|d\tau +{\int }_0^t|e^{m_{ii}(t-\tau )}{u}_i(\tau )|d\tau \\ & \le e^{m_{ii}t}|x_i(0)|+\sum _{j\in {\mathcal{N}}_i}{m}_{ij}\Vert x_j{\Vert }_{[0,t]}{\int }_0^t{e}^{m_{ii}(t-\tau )}d\tau +\Vert u_i{\Vert }_{\infty }{\int }_0^t{e}^{m_{ii}(t-\tau )}d\tau \\ & \le e^{m_{ii}t}|x_i(0)|+\sum _{j\in {\mathcal{N}}_i}\frac{m_{ij}}{-m_{ii}}\Vert x_j{\Vert }_{[0,t]}+\frac{1}{-m_{ii}}\Vert u_i{\Vert }_{\infty }.\end{array}$$

$$|x_i(t)|\le \max \{{\alpha }_1{e}^{m_{ii}t}|x_i(0)|,{\alpha }_2\sum _{j\in {\mathcal{N}}_i}\frac{m_{ij}}{-m_{ii}}\Vert x_j{\Vert }_{[0,t]},{\alpha }_3\frac{1}{-m_{ii}}\Vert u_i{\Vert }_{\infty }\},$$

$$\sum _{j\in {\mathcal{N}}_i}\frac{m_{ij}}{-m_{ii}}\Vert x_j{\Vert }_{[0,t]}<\underset{j}{\max }\{{\psi }_{ij}\Vert x_j{\Vert }_{[0,t]}\}.$$

$$\sum _{j\in {\mathcal{N}}_i}\frac{m_{ij}}{-m_{ii}}\Vert x_j{\Vert }_{[0,t]}\le \frac{1}{{\alpha }_2}\underset{j}{\max }\{{\psi }_{ij}\Vert x_j{\Vert }_{[0,t]}\},$$

$${\psi }_c=\left({\psi }_{i_2{i}_1}\right)\left({\psi }_{i_3{i}_2}\right)\dots \left({\psi }_{i_1{i}_k}\right),$$

$${\gamma }_c=\left(\frac{m_{i_2{i}_1}}{-m_{i_2{i}_2}}\right)\left(\frac{m_{i_3{i}_2}}{-m_{i_3{i}_3}}\right)\dots \left(\frac{m_{i_1{i}_k}}{-m_{i_1{i}_1}}\right).$$

$$M=\left[\begin{array}{ccccc}{m}_{11}& m_{12}& 0& \cdots & 0\\ 0& m_{22}& m_{23}& \cdots & 0\\ ⋮& ⋮& \ddots & \ddots & ⋮\\ 0& 0& \cdots & m_{n-1,n-1}& m_{n-1,n}\\ m_{n1}& 0& \cdots & 0& m_{nn}\end{array}\right],$$

$$\det (-M)=\prod _{i=1}^n(-m_{ii})-m_{n1}\prod _{i=1}^{n-1}{m}_{i,i+1}.$$

$$\prod _{i=1}^n(-m_{ii})>m_{n1}\prod _{i=1}^{n-1}{m}_{i,i+1},$$

$$M=\left[\begin{array}{cc}g-f& f\\ f& -d-f\end{array}\right],$$

$$g-f<0,-d-f<0,\text{and }\frac{f^2}{(f-g)(d+f)}<1.$$

$$d>g\text{ and }f>\frac{dg}{d-g}.$$

$$\sum _{j\in {\mathcal{N}}_i}\left(\frac{m_{ij}}{-m_{ii}}\right){\psi }_{ij}^{-1}<1,$$

$${\psi }_c<1,$$

$$\sum _{j\in {\mathcal{N}}_i}\left(\frac{m_{ij}}{-m_{ii}}\right){\psi }_{ij}^{-1}<1,\text{for all }i\in \{1,\dots ,n\}.$$

Full text

{AMS}

15B48, 93D20, 93D25

1 Introduction

Problem description and motivation

Much attention in recent years has been focused on multi-agent systems, but the majority of efforts has been devoted to averaging dynamics and consensus behavior. Much less attention has been drawn to dynamical flow systems, modeled as monotone or cooperative systems [13, 22]. Notable exceptions are a collection of recent papers motivated by applications to traffic and biological systems [2, 7] as well as the long-standing interest in positive systems [12, 19]. Despite these remarkable recent works, many open questions remain.

This paper focuses on a key foundational question for linear monotone systems, i.e., positive systems modeled by Metzler matrices, and on its application to the study of nonlinear monotone systems: what are graph-theoretical conditions for the Hurwitzness of a Metzler matrix? While a graph theoretical treatment is available for a subclass of Metzler matrices known as “compartmental matrices” [27], a general treatment is lacking. This is in stark contrast with the comprehensive understanding of the graph theoretical conditions guaranteeing convergence to consensus for row-stochastic matrices in averaging systems. Related to this open question is the work in [3]. The graph-theoretic conditions are particularly useful because they allow us to analyze stability based on the structural properties of the interconnection network given the existence of perturbations or uncertainties on the parameters.

For nonlinear monotone systems, much recent progress is documented in [8, 9], where a basic fundamental connection is built between monotone systems and contractive systems. A notable gap, however, remains, in explaining the relationship between the treatment of monotone contractive systems and the stability theory of network small gain developed in [10, 15].

In summary, we aim to develop an algebraic graph theory for monotone dynamical systems, starting with the linear case of Metzler matrices and continuing with the nonlinear setting and its connections with network small-gain theorems.

Literature review

Monotone dynamical systems appear naturally in numerous applications and have many appealing properties. The mathematical theory of nonlinear monotone systems has been vastly studied in dynamical system literature [21, 13, 22]. In control community, the notion of monotonicity has been extended to systems with inputs and outputs, and properties of the interconnected monotone systems have been studied [2]. It is well known that linear monotone systems (also referred to as linear positive systems) are described by Metzler matrices. Conditions for stability of Metzler matrices have been studied extensively in the literature. Narendra and Shorten, et al. established an iterative method based on the Schur complement to check the Hurwitzness of Metzler matrices in [18, 26]. A graph-theoretic characterization for diagonal stability of matrices whose underlying digraph is a cactus graph was proposed in [3]. Briat studied the sign stability of Metzler matrices and block Metzler matrices in [5]. Blanchini et al. studied switched Metzler systems and Hurwitz convex combinations in [4]. Stability of switched Metzler systems has also been studied in [17], where the authors provided guarantees for robustness with respect to delays. In [19], scalable methods for analysis and control of large-scale linear monotone systems have been studied. The admissibility, stability, and persistence of interconnected positive heterogeneous systems have been studied in [11]. For nonlinear monotone systems, using novel connections to the contraction theory, Coogan established sufficient conditions for global stability of monotone systems [8, 9]. We refer the interested readers to [12] for a detailed study of linear positive systems and to the survey paper [24] for theoretical results and applications of interconnected monotone systems.

Small-gain theorems are arguably one of the fundamental results for stability of interconnected systems. Started with the works by Zames [28], the early classical studies on small-gain theorems mostly focused on stability analysis using linear gains [20]. Introduction of the notion of input-to-state stability (ISS) in the seminal paper [23] triggered a paradigm shift in the study of small-gain theorems. More recent works on small-gain theorems focused on the input-to-state framework and they provided results in terms of nonlinear notions of input-to-state gains [14, 10].

Contributions

In this paper, we study the graph-theoretic stability conditions for Metzler matrices. By using concepts from the small-gain theorems for interconnected systems, we obtain necessary and sufficient conditions for Hurwitzness of Metzler matrices in terms of the input-to-state gains, and we also extend our results to the nonlinear monotone systems. Our main contributions are as follows. First, we characterize two types of input-to-state stability gains for linear Metzler systems, namely max-interconnection gains and sum-interconnection gains. Second, using the max-interconnection and the sum-interconnection gains, we obtain two main theorems on graph-theoretic characterizations for Hurwitzness of Metzler matrices. Our conditions highlight the role of cycles and cycle gains and provide valuable insights for connections between the network structure and network functions. In particular, our characterizations for Hurwitzness of Metzler matrices using the max-interconnection gains coincide with the well-known cyclic small gain theorem [15, Theorem 3.1], which becomes necessary and sufficient in our case; based on the sum-interconnection gains, in addition to necessary and sufficient cycle gain conditions that depend on the cycle structure of the interconnection graph, we also show that all cycle gains being less than $1$ is a necessary condition and the sum of cycle gains being less than $1$ is a sufficient condition. Finally, we extend our stability analysis using max-interconnection and sum-interconnection gains to nonlinear monotone systems. As a result, we provide two equivalent sufficient conditions for global stability of monotone nonlinear systems.

Paper organization

We review the known stability results for Metzler matrices in Section 2. The input-to-state stability and two forms of ISS gains are introduced in Section 3, where we also characterize different ISS gains for Metzler systems. The main results on graph-theoretic conditions for Hurwitzness of Metzler matrices are presented in Section 4. We extend the conditions to nonlinear monotone systems in Section 5. A few additional concepts and proofs are included in Section 6. We conclude the paper in Section 7.

2 Review of Metzler matrices

2.1 Notation and preliminaries

Let and ≥0 be the set of real and nonnegative real numbers, respectively. For a vector $v\in^{n}$, its Euclidean norm is denoted by $|v|$. In Particular, if $v\in$, then $|v|$ is the absolute value of $v$. For a finite set $S$, $|S|$ is the cardinality. For $t\geq 0$ and a time-varying vector signal $x:[0,t]\rightarrow^{n}$, we define the norm

[TABLE]

Moreover, for $x:_{\geq 0}\mapsto^{n}$, $\|x\|_{\infty}=\mathrm{ess\,sup}_{s\geq 0}|x(s)|$. A continuous function $\alpha:_{\geq 0}\rightarrow_{\geq 0}$ is a class $\mathcal{K}$ function if it is strictly increasing and $\alpha(0)=0$; it is a class $\mathcal{K}_{\infty}$ function if it is a class $\mathcal{K}$ function and $\lim_{s\to\infty}\alpha(s)=\infty$. A continuous function $\beta:_{\geq 0}\times_{\geq 0}\rightarrow_{\geq 0}$ is a class $\mathcal{KL}$ function if $\beta(s,t)$ is a class $\mathcal{K}$ function of $s$ for fixed $t$, and a decreasing function of $t$ with $\lim_{t\to\infty}\beta(s,t)=0$ for fixed $s$.

For a matrix $A\in^{n\times n}$, its associated graph $\mathcal{G}(A)=(V,\mathcal{E},A)$ is a weighted digraph defined as follows: $V=\{1,\dots,n\}$ is the set of nodes, $\mathcal{E}=\{(j,i)\,|\,i,j\in V,a_{ij}\neq 0\}$ is the set of edges, and $A=\{a_{ij}\}$ is the weight matrix with $a_{ij}$ being the weight on edge $(j,i)\in\mathcal{E}$. For $i\in V$, the neighbor set of node $i$ is defined by $\mathcal{N}_{i}=\{j\in V\,|\,(j,i)\in\mathcal{E}\}$. A matrix $A\in^{n\times n}$ is irreducible if its associated digraph $\mathcal{G}(A)$ is strongly connected. A strongly connected component of a digraph $\mathcal{G}$ is a strongly connected subgraph such that it is not strictly contained in any other strongly connected subgraph of $\mathcal{G}$.

In a digraph $\mathcal{G}=(V,\mathcal{E})$, a simple cycle $c$ in $\mathcal{G}$ is a directed path that starts and ends at the same node and has no repetitions other than the starting and ending nodes. Two simple cycles $c_{1}$ and $c_{2}$ in $\mathcal{G}$ intersect if they share at least one common node, i.e., $c_{1}\cap c_{2}\neq\emptyset$; $c_{1}$ is a subset of $c_{2}$ if all the nodes on $c_{1}$ are also on $c_{2}$. Self loops are not considered as simple cycles in this paper.

For a matrix $A\in^{n\times n}$, the leading principal submatrices of $A$ are given by $A_{I}$, where $I=\{1,\dots,i\}$ is the set of indices for all $i\in\{1,\dots,n\}$. In particular, when $I=\{1,\dots,n\}$, we have $A_{I}=A$. A matrix $M\in^{n\times n}$ is Metzler if all its off-diagonal elements are nonnegative.

The following lemma will be useful later in the paper.

Lemma 2.1** (Bounding sum by maximum).**

Let $\{x_{1},\dots,x_{n}\}$ and $\{\alpha_{1},\dots,\alpha_{n}\}$ be a set of nonnegative and positive real numbers respectively. If $\sum_{i=1}^{n}\frac{1}{\alpha_{i}}\leq 1$, then

[TABLE]

Proof 2.2**.**

Let $s\in\{1,\dots,n\}$ satisfy $\alpha_{i}x_{i}\leq\alpha_{s}x_{s}$ for all $i\in\{1,\dots,n\}$. Then

[TABLE]

2.2 Algebraic conditions for Hurwitzness of Metzler matrices

We collect a few well-known equivalent conditions for the Hurwitzness of Metzler matrices in the following lemma.

Lemma 2.3** (Properties of Hurwitz Metzler matrices [6, Theorem 15.17] [12, Theorem 13]).**

Let $M\in^{n\times n}$ be a Metzler matrix, then the following statements are equivalent:

1. (i)

$M$* is Hurwitz;* 2. (ii)

$M$* is invertible and $-M^{-1}\geq 0$;* 3. (iii)

all leading principal minors of $-M$ are positive; 4. (iv)

there exists $\xi\in^{n}$ such that $\xi>\mathbb{0}_{n}$ and $M\xi<\mathbb{0}_{n}$; 5. (v)

there exists $\eta\in^{n}$ such that $\eta>\mathbb{0}_{n}$ and $\eta^{\top}M<\mathbb{0}_{n}$; 6. (vi)

there exists a diagonal matrix $P\succ 0$ such that $M^{\top}P+PM\prec 0$.

The inequalities in (ii), (iv) and (v) of Lemma 2.3 are componentwise. The matrix inequalities in (vi) indicate positive/negative definiteness.

Remark 2.4**.**

1. (i)

To the best of our knowledge, the equivalence of parts (i) and (iii) in Lemma 2.3 has not been fully exploited in the literature, and we build one of our main results based on this condition. 2. (ii)

If the Metzler matrices are symmetric, then the necessary and sufficient condition in Lemma 2.3(iii)* is exactly the Sylvester’s criterion for negative definiteness of general symmetric matrices.* 3. (iii)

The equivalence of parts (i) and (vi) in Lemma 2.3 implies that for Metzler matrices, the Hurwitzness and diagonal stability are equivalent.

Based on the Schur complement, Narendra et al. propose an iterative method to verify the Hurwitzness of a Metzler matrix [18]. Partition a Metzler matrix $M\in^{n\times n}$ as follows

[TABLE]

where $d_{n-1}$ is a scalar. The Schur complement of $M$ with respect to $d_{n-1}$ is given by $M[n-1]=M_{n-1}-\frac{b_{n-1}c_{n-1}^{\top}}{d_{n-1}}$. For $k\in\{1,\dots,n-1\}$, define $M[k]$ iteratively as the Schur complement of $M[k+1]$ with respect to $d_{k}$, where $M[n]=M$, then the following statement holds.

Lemma 2.5** (Necessary and sufficient condition based on the Schur complement [18]).**

*A Metzler matrix $M\in^{n\times n}$ is Hurwitz if and only if for all $k\in\{1,\dots,n\}$, all the diagonal elements of $M[k]$ are negative. *

By Lemma 2.5, we have the following necessary condition.

Corollary 2.6** (Negativity of diagonal elements [18]).**

*If a Metzler matrix $M\in^{n\times n}$ is Hurwitz, then all the diagonal elements of $M$ are negative. *

3 Review of ISS, interconnected systems and ISS gains

We review the concepts of input-to-state stability and introduce the gain functions in two different forms for interconnected input-to-state stable systems [15, 10].

3.1 Input-to-state stability

Consider the system

[TABLE]

where $x\in^{N}$ is the state, $u\in^{m}$ is the input, and $f:^{N}\times^{m}\mapsto^{N}$ is a locally Lipschitz function and satisfies $f(\mathbbold{0}_{N},\mathbbold{0}_{m})=\mathbbold{0}_{N}$. Then, we have the following definition for input-to-state stability.

Definition 3.1** (Input-to-state stability [23, Definition 2.1]).**

System (1) is input-to-state stable if there exist $\beta\in\mathcal{KL}$ and $\gamma\in\mathcal{K}$ such that for any initial state $x(0)=x_{0}$ and any measurable and locally essentially bounded input $u$, the solution $x(t)$ satisfies, for all $t\geq 0$,

[TABLE]

The class $\mathcal{K}$ function $\gamma$ in (2) is the ISS gain of the system.

Remark 3.2** (ISS Lyapunov function).**

*To verify ISS using Definition 3.1, we need to find an estimate for the trajectory of the system, which is computationally hard in general, if not impossible. However, one can show that ISS is equivalent to the existence of an ISS Lyapunov function. We refer the interested readers to [25, Theorem 1]. *

3.2 Interconnection, ISS gains, and cyclic small-gain

theorem

In this subsection, we study input-to-state stability for networked interconnected systems. Suppose the interaction between subsystems is described by a directed graph $\mathcal{G}=(V,\mathcal{E})$, where $V=\{1,\dots,n\}$ is the set of nodes and for all $i,j\in V$ and $i\neq j$, $(j,i)\in\mathcal{E}$ if $x_{j}$ is an input to subsystem $i$. We consider a network of $n$ interconnected dynamical systems with the interconnection graph $\mathcal{G}$:

[TABLE]

where $x_{i}\in^{n_{i}}$ and $x_{\mathcal{N}_{i}}=\begin{bmatrix}x_{i_{1}},\ldots,x_{i_{k_{i}}}\end{bmatrix}^{\top}\in^{n_{\mathcal{N}_{i}}}$ with $\mathcal{N}_{i}=\{i_{1},\ldots,i_{k_{i}}\}$ and $n_{\mathcal{N}_{i}}=\sum_{j=1}^{k_{i}}n_{i_{j}}$. For every $i\in V$, the function $f_{i}:^{n_{i}+n_{\mathcal{N}_{i}}+m_{i}}\rightarrow^{n_{i}}$ is locally Lipschitz satisfying $f_{i}(\mathbbold{0}_{n_{i}},\mathbbold{0}_{n_{\mathcal{N}_{i}}},\mathbbold{0}_{m_{i}})=\mathbbold{0}_{n_{i}}$. For the interconnected system (3), it is desirable to study ISS of the interconnection using the ISS of each subsystem. We first introduce componentwise ISS for network systems.

Definition 3.3** (Componentwise ISS).**

*An interconnected system (3) is componentwise ISS if every subsystem $i$ is ISS for the input $\begin{bmatrix}x_{\mathcal{N}_{i}}&u_{i}\end{bmatrix}^{\top}\in^{n_{\mathcal{N}_{i}}+m_{i}}$. *

In other words, an interconnected network system is componentwise ISS if each subsystem, separated from the whole system, is ISS. In general, componentwise ISS does not guarantee ISS of the whole interconnected system, and conditions on the interconnection structure and composition of suitable gains are required to ensure ISS of the whole system. In the following, we introduce two notions of gains.

Definition 3.4** (Max-interconnection ISS gains).**

Consider the interconnected system (3). The family of functions $\{\Psi_{ij}\}\in\mathcal{K}\cup\{0\}$ is a max-interconnection gain if, for every $i\in\{1,\dots,n\}$, there exists $\beta_{i}\in\mathcal{KL}$ and $\Psi_{i}\in\mathcal{K}$ such that for any initial state $x(0)=x_{0}$, and any measurable and locally essentially bounded inputs $u_{i}$, the solution $x_{i}(t)$ satisfies, for all $t\geq 0$,

[TABLE]

Definition 3.5** (Sum-interconnection ISS gains).**

Consider the interconnected system (3). The family of functions $\{\Gamma_{ij}\}\in\mathcal{K}\cup\{0\}$ is a sum-interconnection gain if, for every $i\in\{1,\dots,n\}$, there exists $\beta_{i}\in\mathcal{KL}$ and $\Gamma_{i}\in\mathcal{K}$ such that for any initial state $x(0)=x_{0}$, and any measurable and locally essentially bounded inputs $u_{i}$, the solution $x_{i}(t)$ satisfies, for all $t\geq 0$,

[TABLE]

The following lemma provides conditions on a set of max-interconnection ISS gains which guarantee ISS of the interconnected system (3).

Lemma 3.6** (Cyclic small-gain theorem [15, Theorem 3.2]).**

Consider an interconnected system (3) where each subsystem $i$ is componentwise ISS and has a family of max-interconnected gains $\{\Psi_{ij}\}$. The interconnected system (3) is ISS with $x$ as the state and $u$ as the input if, for every simple cycle $c=(i_{1},i_{2},\dots,i_{k},i_{1})$ in the interconnection graph $\mathcal{G}$ and every $s>0$,

[TABLE]

*where $\circ$ is the function composition. *

3.3 ISS gains for Metzler systems

In this subsection, we characterize the ISS gains for Metzler systems. Consider the continuous-time linear system

[TABLE]

where $M\in^{n\times n}$ is a Metzler matrix and $u\in_{\geq 0}^{n}$ is the control input. The Metzler system (5) can be viewed as a network of $n$ interconnected scalar systems, where the interconnection is characterized by the digraph $\mathcal{G}(M)$. More specifically, one can write the Metzler system (5) in the interconnection form (3) as,

[TABLE]

We characterize the sum-interconnection and max-interconnection ISS gains for the Metzler system (5) in the following lemma. Some parts of Lemma 3.7 may be known in the literature, and we hereby provide self-contained proofs.

Lemma 3.7** (ISS Metzler systems).**

The Metzler system (5) with interconnection digraph $\mathcal{G}(M)=(V,\mathcal{E},M)$

1. (i)

is componentwise ISS if and only if

[TABLE] 2. (ii)

has sum-interconnection gains $\{s\mapsto\Gamma_{ij}(s)=\gamma_{ij}s\}$, if it is componentwise ISS and the set of scalars $\{\gamma_{ij}\}$ satisfies $\gamma_{ij}=0$ for all $j\notin\mathcal{N}_{i}$ and

[TABLE] 3. (iii)

has max-interconnection gains $\{s\mapsto\Psi_{ij}(s)=\psi_{ij}s\}$, if it is componentwise ISS and the set of scalars $\{\psi_{ij}\}$ satisfies $\psi_{ij}=0$ for all $j\notin\mathcal{N}_{i}$ and

[TABLE] 4. (iv)

is ISS if and only if $M$ is Hurwitz.

Proof 3.8**.**

Regarding part (i), since the dynamics of the $i$th subsystem given by (6) is linear, it is ISS if and only if $m_{ii}<0$ [15, Theorem 1.3]. Therefore, the Metzler system (5) is componentwise ISS if and only if, for every $i\in\{1,\dots,n\}$, we have $m_{ii}<0$.

Regarding part (ii), the state trajectory $x_{i}(t)$ satisfies

[TABLE]

which implies

[TABLE]

Therefore, the Metzler system (5) has sum-interconnection ISS gains $\{s\mapsto\Gamma_{ij}(s)=\gamma_{ij}(s)\}$ if we have $\frac{m_{ij}}{-m_{ii}}\leq\gamma_{ij}$.

Regarding part (iii), by Lemma 2.1 and (9), we have

[TABLE]

where $\alpha_{1},\alpha_{2},\alpha_{3}>0$ and $\sum_{i=1}^{3}\frac{1}{\alpha_{i}}\leq 1$. If (8) holds, then by Lemma 2.1, we have

[TABLE]

Therefore, we can pick $\alpha_{2}>1$ properly such that

[TABLE]

which combined with (10) imply that $\{\psi_{ij}\}$ are max-interconnection gains.

*Regarding part (iv), this is a straightforward application of [15, Theorem 1.3]. *

4 Graph-theoretic conditions for Hurwitzness of Metzler

matrices

In this section, we first show that we only need to consider irreducible Metzler matrices. Then, we show that different ISS gains result in different graph-theoretic conditions for the stability of Metzler systems. In particular, if we use the max-interconnection ISS gains, then the cycle condition (4) in Lemma 3.6 is a necessary and sufficient condition for the stability of Metzler systems. On the other hand, if we use the sum-interconnection ISS gains, then we can obtain new necessary and sufficient graph-theoretic conditions.

4.1 Metzler matrices with reducible graphs

The following lemma allows us to restrict our attention to irreducible Metzler matrices.

Lemma 4.1** (Hurwitzness and strongly connected components).**

*For a Metzler matrix $M\in^{n\times n}$, $M$ is Hurwitz if and only if all the submatrices corresponding to the strongly connected components of $\mathcal{G}(M)$ are Hurwitz. *

Proof 4.2**.**

If $M$ is irreducible, then the statement holds true since there is only one strongly connected component in $\mathcal{G}(M)$, which is $\mathcal{G}(M)$ itself.

*If $M$ is reducible, then there exists a permutation matrix such that $M$ can be brought into block upper triangular form where each block on the diagonal corresponds to a strongly connected component of $\mathcal{G}(M)$. Therefore, $M$ is Hurwitz if and only if all the submatrices corresponding to the strongly connected components of $\mathcal{G}(M)$ are Hurwitz. *

If $\mathcal{G}(M)$ is acyclic, then we have the following corollary.

Corollary 4.3** (Necessary and sufficient condition for acyclic graphs [5, Theorem 3.4]).**

*For a Metzler matrix $M\in^{n\times n}$ whose associated digraph $\mathcal{G}(M)$ is acyclic, $M$ is Hurwitz if and only if all the diagonal elements of $M$ are negative. *

Hereafter, we focus on irreducible Metzler matrices with negative diagonal elements.

4.2 Cycle gains and the case of a simple cycle

In this subsection, we define the sum-cycle gains and max-cycle gains for Metzler matrices, and we emphasize the importance of cycles through the case of a simple cycle. Note that self loops are not considered as simple cycles in this paper.

Definition 4.4** (Cycle gains for Metzler matrices).**

Let $M\in^{n\times n}$ be an irreducible Metzler matrix with negative diagonal elements and $c=(i_{1},i_{2},\dots,i_{k},i_{1})$ be a simple cycle in $\mathcal{G}(M)$. Then

1. (i)

a max-cycle gain of $c$ is

[TABLE]

where the scalars $\{\psi_{ij}\}$ satisfy (8); and 2. (ii)

the sum-cycle gain of $c$ is

[TABLE]

Remark 4.5** (Uniqueness of cycle gains).**

*The max-interconnection gains and sum-interconnection gains as characterized in Lemma 3.7 are not unique. In Definition 4.4, the max-cycle gains as in (11) are not unique, and for every solution of (8), one can compute a set of max-cycle gains for simple cycles. However, the sum-cycle gains in (12) are uniquely defined for simple cycles in $\mathcal{G}(M)$ since we pick the natural lower bound for the sum-interconnection gains in (7). *

For an irreducible Metzler matrix $M\in^{n\times n}$ with negative diagonal elements, if the associated digraph $\mathcal{G}(M)$ is a simple cycle, i,e, $M$ has the following structure,

[TABLE]

then we have the following lemma.

Lemma 4.6** (Necessary and sufficient condition for simple cycles).**

Let $M\in^{n\times n}$ be an irreducible Metzler matrix with negative diagonal elements whose associated digraph $\mathcal{G}(M)$ is a simple cycle $c=(1,n,\dots,2,1)$. Then the following statements are equivalent:

1. (i)

$M$* is Hurwitz;* 2. (ii)

$\gamma_{c}<1$; 3. (iii)

there exists a solution to (8) such that $\psi_{c}<1$.

Proof 4.7**.**

Regarding the equivalence between (i) and (ii): by Lemma 2.3(iii), $M$ is Hurwitz if and only if all the leading principal minors of $-M$ are positive. If $i<n$ and $I=\{1,\dots,i\}$, then the leading principal submatrices $(-M)_{I}$ of $-M$ are upper triangular with positive diagonal elements and thus $\det((-M)_{I})>0$. When $I=\{1,\dots,n\}$, we have

[TABLE]

Then, $\det(-M)>0$ if and only if

[TABLE]

which is equivalent to $\gamma_{c}<1$.

*Regarding the equivalence between (ii) and (iii): notice that if we pick $\psi_{ij}=\frac{m_{ij}}{-m_{ii}}+\epsilon$ for sufficiently small $\epsilon>0$, then (8) is satisfied and $\psi_{c}<1$ is equivalent to $\gamma_{c}<1$. *

It is worth mentioning that the necessary and sufficient condition in Lemma 4.6 is a special case of a more general result in [3, Proposition 2] regarding diagonal stability.

Example 4.8** (A two by two Metzler matrix describing a flow system [6, Exercise 9.8]).**

We apply Lemma 4.6 to a simple two by two case where the Metzler matrix describes a symmetric flow system $\dot{x}=Mx$. Suppose the Metzler matrix $M$ has the following form

[TABLE]

where $f>0$ is the flow rate between two nodes, $g>0$ is the growth rate at node $1$ and $d>0$ is the decay rate at node $2$. By Lemma 4.6, the flow system $\dot{x}=Mx$ is asymptotically stable if and only if

[TABLE]

Equivalently, we have

[TABLE]

*This condition has a clear physical interpretation that in order for the two-node flow system $\dot{x}=Mx$ to be asymptotically stable, i.e., the flow does not accumulate in the system, the decay rate at one node must be larger than the growth rate at the other node and the flow rate between the nodes should be sufficiently large. *

Lemma 4.6 states that a Metzler matrix whose associated digraph is a simple cycle is Hurwitz if and only if the cycle gain is less than $1$. It turns out that, for irreducible Metzler matrices with general digraphs, the gains of the simple cycles play a central role in determining the Hurwitzness. Moreover, cycle gains in different forms (sum or max) lead to different graph-theoretic conditions.

4.3 Max-cycle gains and Hurwitz Metzler matrices

In this subsection, we use the max-cycle gains of the Metzler system (5) to characterize the Hurwitzness of a Metzler matrix, and the cyclic small gain theorem in Lemma 3.6 becomes necessary and sufficient in this case.

Theorem 4.9** (Max-interconnection characterization).**

Let $M\in^{n\times n}$ be an irreducible Metzler matrix with negative diagonal elements, $\mathcal{G}(M)=(V,\mathcal{E},M)$ be the associated digraph, and $\Phi$ be the set of simple cycles of $\mathcal{G}(M)$. Then the following conditions are equivalent:

1. (i)

$M$* is Hurwitz;* 2. (ii)

for every $i\in V$ and $j\in\mathcal{N}_{i}$, there exists $\psi_{ij}>0$ such that

[TABLE]

Proof 4.10**.**

(ii) $\implies$ (i):* Since the diagonal entries of $M$ are negative, the Metzler system (5) is componentwise ISS by Lemma 3.7(i). By Lemma 3.7(iii), there exist max-interconnection gains $\{\psi_{ij}\}$ such that*

[TABLE]

Thus, the sufficient condition in Lemma 3.6 is equivalent to the existence of $\psi_{ij}>0$, for $i\in V$ and $j\in\mathcal{N}_{i}$ such that

[TABLE]

Therefore, by Lemma 3.6, the Metzler system (5) is ISS and asymptotically stable, which implies that $M$ is Hurwitz.

(i) $\implies$ (ii):* Suppose that $M$ is Hurwitz, then by Lemma 2.3(iv) there exists $\xi>\mathbb{0}_{n}$ such that $M\xi<\mathbb{0}_{n}$. Therefore, $\textup{diag}(\xi^{-1})M\textup{diag}(\xi)$ is a Metzler matrix with negative row sums, which implies*

[TABLE]

Note that, for every $(i_{1},\ldots,i_{k},i_{1})\in\Phi$, we have

[TABLE]

Thus, we have

[TABLE]

By a straightforward continuity argument, one can show that, for every $i\in V$ and $j\in\mathcal{N}_{i}$, there exists $\psi_{ij}>0$ such that

[TABLE]

*This completes the proof. *

By Theorem 4.9, we can prove the following corollary.

Corollary 4.11** (Diagonal Stability and Hurwitzness of Metzler matrices).**

Let $M\in^{n\times n}$ be an irreducible Metzler matrix with negative diagonal elements, $\mathcal{G}(M)=(V,\mathcal{E},M)$ be the associated digraph, and $\Phi$ be the set of simple cycles of $\mathcal{G}(M)$. Assume that any two simple cycles of $\mathcal{G}(M)$ have at most one vertex in common, i.e., $\mathcal{G}(M)$ is cactus. Then the following conditions are equivalent:

1. (i)

$M$* is Hurwitz;* 2. (ii)

for every $c\in\Phi$ and every node $i\in c$, there exists positive constant $\theta^{c}_{i}>0$ such that

[TABLE]

where $\gamma_{c}$ is defined in equation (12).

Proof 4.12**.**

*We postpone the proof to Appendix B. *

Remark 4.13**.**

1. (i)

The condition in Corollary 4.11(ii)* for Metzler matrices is the same as conditions (11) and (12) in [3, Theorem 1] for the diagonal stability of arbitrary matrices with cactus graphs. Therefore, in the context of Metzler matrices, Theorem 4.9 is a generalization of [3, Theorem 1] to arbitrary topologies.* 2. (ii)

One can compute the positive constants $\psi_{ij}$ in Theorem 4.9(ii)* by solving the following feasibility problem*

[TABLE]

Then, for $i\in V$ and $j\in\mathcal{N}_{i}$, we can compute $\psi_{ij}$ as

[TABLE]

where $0<\delta<1$ is given by

[TABLE]

The problem (LABEL:eq:lp) is not a linear programming due to the strict inequalities. However, one can easily transform it to the following linear programming.

[TABLE]

In order to check conditions (13) and (14), we need to compute the max-interconnection ISS gains using the method in Remark 4.13(ii). This computation is essentially equivalent to the well-known condition in Lemma 2.3(iv).

4.4 Sum-cycle gains and Hurwitz Metzler

matrices

In this subsection, we use sum-cycle gains to characterize the Hurwitzness of Metzler matrices. We first introduce the concept of disjoint cycle sets.

Definition 4.14** (Disjoint cycle sets).**

Let $M\in^{n\times n}$ be a Metzler matrix with the associated digraph $\mathcal{G}(M)$ and $\Phi=\{c_{1},\dots,c_{r}\}$ be the set of simple cycles in $\mathcal{G}(M)$, the disjoint cycle sets $K_{\ell}^{M}$ for $\ell\in\{1,\dots,r\}$ are defined by

[TABLE]

Intuitively, the disjoint cycle sets $K_{\ell}^{M}$ are sets where each element is a set of $\ell$ cycles that are mutually disjoint. We collect the graph-theoretic interpretations for the disjoint cycle sets in Section 6.1. With the disjoint cycle sets, we are ready to define the notion of total cycle gain of a Metzler matrix and its leading principal submatrices.

Definition 4.15** (Total cycle gain).**

*Let $M\in^{n\times n}$ be an irreducible Metzler matrix with negative diagonal elements. For $i=\{1,\dots,n\}$ and $I=\{1,\dots,i\}$, the leading principal submatrix $M_{I}$ has the associated digraph $\mathcal{G}(M_{I})$, set of simple cycles $\Phi_{M_{I}}=\{c_{1},\dots,c_{r_{M_{I}}}\}$ and disjoint cycle sets $K_{\ell}^{M_{I}}$, $\ell\in\{1,\dots,r_{M_{I}}\}$, then the total cycle gain of $M_{I}$ is defined by *

[TABLE]

Example 4.16** (Disjoint cycle sets and total cycle gain).**

We illustrate the definitions of the disjoint cycle sets and the total cycle gain in this example. Let $M\in^{6\times 6}$ be an irreducible Metzler matrix with negative diagonal elements as follows

[TABLE]

The associated weighted digraph $\mathcal{G}(M)$ is shown in Fig. 1.

There are five cycles in $\mathcal{G}(M)$, i.e., $c_{1}=(1,2,1)$, $c_{2}=(2,3,2)$, $c_{3}=(4,5,4)$, $c_{4}=(6,1,6)$, $c_{5}=(1,2,3,4,5,6,1)$, and the disjoint cycle sets of $M$ are:

[TABLE]

According to (17), the total cycle gains of the leading principal submatrices are given by:

[TABLE]

With the above definitions, we now present a useful lemma.

Lemma 4.17** (Determinant and total cycle gain).**

Let $M\in^{n\times n}$ be an irreducible Metzler matrix with negative diagonal elements and let $\gamma_{M_{I}}$ be the total cycle gain of $M_{I}$ for $i\in\{1,\dots,n\}$ and $I=\{1,\dots,i\}$. Then

[TABLE]

Proof 4.18**.**

*We postpone the proof to Appendix A. *

We are now ready to write the leading principal minor condition in Lemma 2.3(iii) in the graph-theoretic language.

Theorem 4.19** (Sum-interconnection characterization).**

Let $M\in^{n\times n}$ be an irreducible Metzler matrix with negative diagonal elements, $\mathcal{G}(M)=(V,\mathcal{E},M)$ be the associated digraph, and $\Phi$ be the set of simple cycles of $\mathcal{G}(M)$. Then the following statements hold:

1. (i)

(necessary condition) if $M$ is Hurwitz then

[TABLE] 2. (ii)

(sufficient condition) if

[TABLE]

then $M$ is Hurwitz; 3. (iii)

(necessary and sufficient condition) $M$ is Hurwitz if and only if, for all $i\in\{1,\dots,n\}$

[TABLE]

Proof 4.20**.**

Regarding part (i), we postpone the proof to Section 6.2, where an expansion algorithm for $\mathcal{G}(M)$ is given so that all the simple cycles can be identified by the leading principal submatrices and a simple proof is constructed.

Regarding part (ii), we prove the result by showing that Theorem 4.19(iii) holds. For all $i\in\{1,\dots,n\}$ and $I=\{1,\dots,i\}$, the leading submatrix $M_{I}$ only involves a subset of $\Phi$. If $\Phi_{M_{I}}$ is empty, then $\gamma_{M_{I}}=0<1$. Otherwise, from (17), we know that $\gamma_{M_{I}}$ has the following form:

[TABLE]

Since for all $c\in\Phi$, we have $\gamma_{c}>0$ and $\sum_{c\in\Phi}\gamma_{c}<1$ by assumption, then we have that $\gamma_{c}<1$ for all $c\in\Phi$ and $\sum_{\{c_{i_{1}}\}\in K_{1}^{M_{I}}}\gamma_{c_{i_{1}}}<1$. Note that by the definition of $K_{\ell}^{M_{I}}$, for any $\{c_{i_{1}},\dots,c_{i_{\ell}}\}\in K_{\ell}^{M_{I}}$, we must have that all the subsets of $\{c_{i_{1}},\dots,c_{i_{\ell}}\}$ with $\ell-1$ elements are contained in $K_{\ell-1}^{M_{I}}$. Thus, we have that, for all $k\geq 1$,

[TABLE]

Hence, we have for all $i\in\{1,\dots,n\}$ and $I=\{1,\dots,i\}$, $\gamma_{M_{I}}<1$, and by Theorem 4.19(iii), $M$ is Hurwitz.

Regarding part (iii), by Lemma 4.17, we have that for $i\in\{1,\dots,n\}$ and $I=\{1,\dots,i\}$,

[TABLE]

By Lemma 2.3(iii), $M$ is Hurwitz if and only if for all $i\in\{1,\dots,n\}$ and $I=\{1,\dots,i\}$, $\det((-M)_{I})>0$ , i.e.,

[TABLE]

*which is equivalent to $\gamma_{M_{I}}<1$. *

Remark 4.21** (Necessary and sufficient condition in special graphs).**

*The sufficient condition for Hurwitzness in Theorem 4.19(ii) becomes necessary and sufficient when any two cycles share at least one common node in the digraph associated with the Metzler matrix. *

We give two simple examples illustrating that the condition in Theorem 4.19(i) is not sufficient and the condition in Theorem 4.19(ii) is not necessary.

Example 4.22** (Insufficiency of condition (i) in Theorem 4.19).**

Consider an irreducible Metzler matrix $M\in^{3\times 3}$ as follows

[TABLE]

The associated weighted digraph $\mathcal{G}(M)$ is shown in Fig. 2. There are two cycles in $\mathcal{G}(M)$, i.e., $c_{1}=(1,2,1)$ and $c_{2}=(2,3,2)$, and the cycle gains are $\gamma_{c_{1}}=\gamma_{c_{2}}=\frac{1}{2}$. The cycle gains satisfy the condition in Theorem 4.19(i), but $M$ is not Hurwitz since it has a zero eigenvalue.

Example 4.23** (Lack of necessity of condition (ii) in Theorem 4.19).**

Consider an irreducible Metzler matrix $M\in^{4\times 4}$ as follows

[TABLE]

The associated weighted digraph $\mathcal{G}(M)$ is shown in Fig. 3. There are three cycles in $\mathcal{G}(M)$, i.e., $c_{1}=(1,2,1)$, $c_{2}=(2,3,2)$ and $c_{3}=(3,4,3)$, and the cycle gains are $\gamma_{c_{1}}=\frac{3}{5}$, $\gamma_{c_{2}}=\frac{1}{5}$ and $\gamma_{c_{3}}=\frac{1}{5}$. The cycle gains do not satisfy the sufficient condition in Theorem 4.19(ii), but one can check that $M$ is Hurwitz.

We give the Hurwitzness conditions for Example 4.16.

Example 4.24** (continues=exam:concepts).**

By Theorem 4.19(iii) and (19), the necessary and sufficient conditions for $M$ to be Hurwitz are given by

[TABLE]

which are equivalent to

[TABLE]

It is not obvious whether the necessary conditions in Theorem 4.19(i) hold in this example. We show that (21)-(23) imply those necessary conditions in the following. From (21), since the cycle gains are positive, we know that $\gamma_{c_{1}}<1$ and $\gamma_{c_{2}}<1$. We can rewrite (22) as follows

[TABLE]

which along with (21) imply that $\gamma_{c_{3}}<1$. By using (19), we can rearrange (23) as follows

[TABLE]

which is equivalent to

[TABLE]

*Since all the terms on the left hand side of (24) are positive, and on the right hand side we have $\gamma_{c_{2}}<1$ and $\gamma_{c_{3}}<1$, thus we must have that $\gamma_{c_{4}}<1$. At the same time, since the term on the right hand side of (24) is less than $1$, we must have that $\gamma_{c_{5}}<1$. *

5 Graph-theoretic conditions for stability of nonlinear

monotone systems

In this section, we extend our stability results to monotone nonlinear systems. We consider a network of $n$ interconnected dynamical systems with the interconnection graph $\mathcal{G}$:

[TABLE]

where $x_{i}\in$ and $x_{\mathcal{N}_{i}}=\begin{bmatrix}x_{i_{1}},\ldots,x_{i_{k_{i}}}\end{bmatrix}^{\top}\in^{|{\mathcal{N}_{i}}|}$ with $\mathcal{N}_{i}=\{i_{1},\ldots,i_{k}\}$. For every $i\in\{1,\ldots,n\}$, the function $f_{i}:^{|\mathcal{N}_{i}|+1}\rightarrow$ is continuously differentiable. We assume that the interconnected system (25) is monotone, i.e., for every $x\in^{n}_{\geq 0}$, the Jacobian matrix $J(x)$ is Metzler. Moreover, we assume that $f(\mathbbold{0}_{n})=\mathbbold{0}_{n}$. We show that our characterizations of stability for linear Metzler systems can be generalized to sufficient conditions for global stability of nonlinear monotone systems. In particular, we prove two global results for asymptotic stability of monotone interconnected networks based on the max-interconnection gains and the sum-interconnection gains.

Theorem 5.1** (Max-interconnection stability).**

Consider an interconnected nonlinear system (25) evolving on the positive orthant ${}^{n}_{\geq 0}$ with the interconnection graph $\mathcal{G}=(V,\mathcal{E})$. Assume that $f(\mathbbold{0}_{n})=\mathbbold{0}_{n}$, and for every $x\in^{n}_{\geq 0}$, the matrix $J(x)$ is Metzler with negative diagonal entries. Moreover, assume there exists a family of positive numbers $\{\psi_{ij}\}$ for $i\in V$ and $j\in\mathcal{N}_{i}$ such that:

1. (i)

for every $i\in\{1,\ldots,n\}$,

[TABLE] 2. (ii)

for every $c=(i_{1},\ldots,i_{k},i_{1})\in\Phi$,

[TABLE]

*Then $\mathbbold{0}_{n}$ is globally asymptotically stable for system (25). *

Proof 5.2**.**

Given $c>0$, we define the set $B(c)$ and the real number $\delta(c)$ as follows:

[TABLE]

Since $B(c)$ is a compact set and (26) holds, we have that $\delta(c)>0$. Let $\beta:_{\geq 0}\times_{\geq 0}\mapsto$ be a class $\mathcal{KL}$ function given by $\beta(s,t)=se^{-\delta(s)t}$, where $\delta(s)>0$ is a nonincreasing function with respect to $s$. Consider the control system

[TABLE]

where $u\in^{n}_{\geq 0}$. We first show that, for every $t\geq 0$ and every $i\in\{1,\ldots,n\}$,

[TABLE]

Suppose that the statement (28) is not true. Therefore, there exist $i\in\{1,\ldots,n\}$, $t^{*}\geq 0$, and $\epsilon>0$ such that

[TABLE]

and for every $t\in(t^{*},t^{*}+\epsilon)$,

[TABLE]

Since ${}^{n}_{\geq 0}$ is convex, by the Mean Value Theorem [1, Proposition 2.4.7], there exists $\mathbbold{0}_{n}\leq\xi_{x}\leq x$ such that

[TABLE]

By (29) and (30), we have that, for every $j$ such that $(j,i)\in\mathcal{E}$ and every $t\in[t^{*},t^{*}+\epsilon)$, we have $\|x_{j}\|_{[0,t]}\leq\psi^{-1}_{ij}x_{i}(t)$. Therefore, by (31), we have

[TABLE]

We consider two cases in the following.

1. (i)

$x_{i}(t^{*})=\beta(x_{i}(0),t^{*})$: In this case, for small enough $\epsilon$ and, for every $t\in[t^{*},t^{*}+\epsilon)$, we have $\xi_{x(t)}\in B(x_{i}(0))$. Thus, by (32), we have

[TABLE]

which implies that $x_{i}(t)\leq e^{-\delta(x_{i}(0))(t-t^{*})}x_{i}(t^{*})$. Thus, along with (29), we have, for every $t\in[t^{*},t^{*}+\epsilon)$,

[TABLE]

which is contradictory to (30). 2. (ii)

$x_{i}(t^{*})>\beta(x_{i}(0),t^{*})$: In this case, we have $x_{i}(t^{*})>x_{i}(0)e^{-\delta(x_{i}(0))t^{*}}$ and therefore

[TABLE]

By (32), we have $\dot{x}_{i}(t)\leq 0$ for every $t\in[t^{*},t^{*}+\epsilon)$. Since $\|x_{j}\|_{[0,t]}$ is nondecreasing with respect to $t$, for every $t\in[t^{*},t^{*}+\epsilon)$,

[TABLE]

which is contradictory to (30).

*In both cases, we have a contradiction. Therefore, for every $t\geq 0$ and every $i\in\{1,\dots,n\}$, $x_{i}(t)$ satisfies (28). Moreover, Theorem 5.1(ii) ensures that $\{\psi_{ij}\}_{(i,j)\in\mathcal{E}}$ satisfies $\psi_{c}<1$, for every $c\in\Phi$. Therefore, by cyclic small-gain theorem 3.6, the control system (27) is ISS, which implies that $\mathbbold{0}_{n}$ is globally asymptotically stable for nonlinear dynamical system (25). *

Theorem 5.3** (Sum-interconnection stability).**

Consider an interconnected nonlinear system (25) evolving on the positive orthant ${}^{n}_{\geq 0}$ with the interconnection graph $\mathcal{G}=(V,\mathcal{E})$. Assume that $f(\mathbbold{0}_{n})=\mathbbold{0}_{n}$, and for every $x\in^{n}_{\geq 0}$, the matrix $J(x)$ is Metzler with negative diagonal entries. Moreover, assume there exists a family of positive numbers $\{\gamma_{ij}\}$ for $i\in V$ and $j\in\mathcal{N}_{i}$ such that:

1. (i)

for every $i\in\{1,\ldots,n\}$,

[TABLE] 2. (ii)

for every $i\in\{1,\ldots,n\}$ and $I=\{1,\dots,i\}$,

[TABLE]

where the Metzler matrix $M$ is defined as, for $i^{\prime},j^{\prime}\in V$

[TABLE]

*Then $\mathbbold{0}_{n}$ is globally asymptotically stable for system (25). *

Proof 5.4**.**

By (ii) and Theorem 4.19(iii), $M$ is Hurwitz. Thus, by Theorem 4.9, there exists a family of positive numbers $\{\psi_{ij}\}_{(i,j)\in\mathcal{E}}$ such that, for every $i\in\{1,2,\ldots,n\}$,

[TABLE]

and $\psi_{c}<1$ for every $c\in\Phi$. This implies that, for every $x\in^{n}_{\geq 0}$, we have

[TABLE]

Therefore, for the family of positive numbers $\{\psi_{ij}\}_{(i,j)\in\mathcal{E}}$,

[TABLE]

*and $\psi_{c}<1$ for every $c\in\Phi$. Therefore, by Theorem 5.1, $\mathbbold{0}_{n}$ is globally asymptotically stable for the dynamical system (25). *

6 Additional Concepts and proofs

6.1 Cycle graphs, complementary cycle graphs and disjoint cycle sets

Let $M\in^{n\times n}$ be an irreducible Metzler matrix with negative diagonal elements and $\Phi=\{c_{1},\ldots,c_{r}\}$ be the set of simple cycles in $\mathcal{G}(M)$. Then the associated cycle graph of $\mathcal{G}(M)$ is the graph $\mathcal{G}_{\Phi}(M)=(V_{\Phi},\mathcal{E}_{\Phi})$ with the node set $V_{\Phi}=\{1,\ldots,r\}$ and the edge set $\mathcal{E}_{\Phi}$ given by

[TABLE]

We define the complementary cycle graph of $\mathcal{G}(M)$ by $\mathcal{G}^{\rm c}_{\Phi}(M)=(V_{\Phi},\mathcal{E}^{\rm c}_{\Phi})$. Note that while the graph $\mathcal{G}(M)$ is a weighted digraph, the graphs $\mathcal{G}_{\Phi}(M)$ and $\mathcal{G}^{\rm c}_{\Phi}(M)$ are unweighted undirected graphs. Moreover, since $M$ is irreducible, the cycle graph $\mathcal{G}_{\Phi}(M)$ is always connected. The disjoint cycle set $K_{\ell}^{M}$ is a set in which each element is a nonempty set of $\ell\geq 1$ cycles in $\Phi$ that form a complete graph in $\mathcal{G}^{\rm c}_{\Phi}(M)$.

Example 6.1** (Cycle graphs, complementary cycle graphs and $K_{\ell}^{M}$).**

We illustrate the a few definitions using the Metzler matrix in Example 4.16, whose associated weighted digraph $\mathcal{G}(M)$ is shown in Fig. 1.

The cycle graph $\mathcal{G}_{\Phi}(M)$ is given in Fig. 4(a) and the complementary cycle graph $\mathcal{G}^{\rm c}_{\Phi}(M)$ is given in Fig. 4(b). From Fig. 4(b), one can check that the disjoint cycle sets are clearly given by (18).

6.2 Graph expansion and proof of Theorem 4.19(i)

In this subsection, we reverse the Schur complement process and propose a graph expansion algorithm for the associated graph of a Metzler matrix. The purpose of the expansion is to separate cycles so that no cycle is strictly contained in any other cycle. Once we complete this construction, a simple proof of Theorem 4.19(i) follows.

For a Metzler matrix $M\in^{n\times n}$ associated with $\mathcal{G}(M)=(V,\mathcal{E},M)$, we construct the expansion digraph $\mathcal{G}_{\textup{exp}}(M)=(V_{\textup{exp}},\mathcal{E}_{\textup{exp}},M_{\textup{exp}})$ and the expanded Metzler matrix $M_{\textup{exp}}$ using Algorithm 1.

In words, for a Metzler matrix $M\in^{n\times n}$, Algorithm 1 inserts a node on each directed edge in $\mathcal{G}(M)$ and assigns proper weights to the added nodes and edges.

Lemma 6.2**.**

*For a Metzler matrix $M\in^{n\times n}$ and its expansion $M_{\textup{exp}}$, $M$ is Hurwitz if and only if $M_{\textup{exp}}$ is Hurwitz. *

Proof 6.3**.**

*The Metzler matrix $M$ can be recovered from $M_{\textup{exp}}$ by removing all the added nodes using the Schur complement, and the diagonal elements of the remaining nodes do not change during the elimination. Therefore, by Lemma 2.5, $M$ is Hurwitz if and only if $M_{\textup{exp}}$ is Hurwitz. *

Now we are ready to give a proof to Theorem 4.19(i).

Proof 6.4** (Proof of Theorem 4.19(i)).**

*By construction, any cycle in $\mathcal{G}_{\textup{exp}}(M)$ can show up as a leading principal submatrix after a permutation on $M_{\textup{exp}}$. Since $M$ is Hurwitz, $M_{\textup{exp}}$ is also Hurwitz and by Lemma 2.3(iii), the determinant of the negative leading principal submatrix must be positive, i.e., the cycle gain must be less than $1$. *

7 conclusion

In this paper, we obtained and characterized the graph-theoretic necessary and sufficient conditions for the Hurwitzness of Metzler matrices. By establishing connections with the well-known input-to-state stability theory and small-gain theorems, we were able to derive stability conditions for linear Metzler systems based on two different forms of ISS gains. These conditions give insights on how the cycles and cycle structures in the associated digraph of the Metzler matrices play a role in determining system stability. We also extended our results to the case of nonlinear monotone systems and obtained sufficient conditions for stability.

Appendix A Proof of Lemma 4.17

In order to prove Lemma 4.17, we need a few results regarding the graph-theoretic interpretations of determinants. For a weighted digraph $\mathcal{G}=(V,\mathcal{E},W)$, a factor $F=\{c_{1},\dots,c_{r}\}$ of $\mathcal{G}$ satisfies

1. (i)

each $c_{i}\in F$ is either a self loop or a simple cycle; 2. (ii)

$c_{i}\cap c_{j}=\emptyset$, for all $i\neq j$; 3. (iii)

$\cup_{i=1}^{r}c_{i}=V$.

Note that the set of factors may be empty and in this case the determinant of the matrix corresponding to the digraph is [math].

For a matrix $A\in^{n\times n}$, the determinant of $A$ can be computed based on the factors of $\mathcal{G}(A)$. For a simple cycle or a self loop $c$ in $\mathcal{G}(A)$, we define $A(c)$ to be the product of the edge weights along the cycle or the self loop. Then, we have the following lemma.

Lemma A.1** (Graph-theoretic interpretation of determinants [16, Theorem 1]).**

Let $A\in^{n\times n}$ be a matrix with digraph $\mathcal{G}(A)=(V,\mathcal{E},A)$. Suppose $\mathcal{G}(A)$ has factors $F_{k}=\{c_{k_{1}},c_{k_{2}},\dots,c_{k_{r_{k}}}\}$, $k=1,\dots,q$, then

[TABLE]

In the case of irreducible Metzler matrices with negative diagonal elements, we can rewrite (33) in terms of the cycle gains. Let $M\in^{n\times n}$ be an irreducible Metzler matrix with negative diagonal elements and $\Phi=\{c_{1},\dots,c_{r}\}$ be the set of simple cycles of $\mathcal{G}(M)$, then a cycle factor $F^{c}=\{c_{1},\dots,c_{t}\}$ of $\mathcal{G}(M)$ satisfies

1. (i)

$F^{c}\subset\Phi$ and $F^{c}\neq\emptyset$; 2. (ii)

$c_{i}\cap c_{j}=\emptyset$, for all $c_{i},c_{j}\in F^{c}$ and $i\neq j$.

Suppose $\mathcal{G}(M)$ has cycle factors $F^{c}_{k}=\{c_{k_{1}},c_{k_{2}},\dots,c_{k_{t_{k}}}\}$, $k=1,\dots,q$, then each cycle factor $F^{c}_{k}$ can be expanded to a factor of $\mathcal{G}(M)$ by adding the self loops at the nodes that are not on any simple cycles in $F^{c}_{k}$ and by doing this, all the factors except the one that consists of purely self loops can be recovered. Since the diagonal elements of $M$ are negative, we can factor out $\prod_{i=1}^{n}(-m_{ii})$ in the general formula (33) and rewrite the equation for $M$ as follows,

[TABLE]

By definition, the disjoint cycle sets are related to the cycle factors as $K_{\ell}^{M}=\{F_{k}^{c}\,|\,t_{k}=\ell\}$, thus we can group the cycle factors with the same cardinality in (34) and obtain (20) for $I=\{1,\dots,n\}$. For $i=\{1,\dots,n-1\}$ and $I=\{1,\dots,i\}$, the same procedure works for the leading principal submatrices $M_{I}$ and (20) follows except for the case when $\Phi_{M_{I}}$ is empty. If $\Phi_{M_{I}}$ is empty, i.e., $\mathcal{G}(M_{I})$ is acyclic, then the determinant $\det(M_{I})$ is equal to the product of the diagonal elements. By (17), we have $\gamma_{M_{I}}=0$ in this case and thus (20) holds.

Appendix B Proof of Corollary 4.11

(i) $\implies$ (ii): Since $M$ is Hurwitz, by Theorem 4.9, for every $(j,i)\in\mathcal{E}$, there exists $\psi_{ij}>0$ such that

[TABLE]

Let $c\in\Phi$ and assume that $c=(1,\ldots,k,1)$. Then, for every $k^{\prime}\in\{1,\ldots,k\}$, we define

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First note that (36) can be written as

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Since $\mathcal{G}(M)$ is connected and cactus, no two simple cycles share an edge. Therefore, one can write (35) as follows:

[TABLE]

By a straightforward continuity argument, one can show that, for every $c\in\Phi$ and $i\in c$, there exists $\theta^{c}_{i}>0$ such that

[TABLE]

(ii) $\implies$ (i): Now suppose that, for every $c\in\Phi$ and every $i\in c$, there exists $\theta^{c}_{i}>0$ which satisfies (15). Let $c=(1,\ldots,k,1)$, and for every $k^{\prime}\in\{1,\dots,k-1\}$

[TABLE]

and

[TABLE]

By a continuity argument, (15) can be written as (35) and (36). Thus, by Theorem 4.9, the matrix $M$ is Hurwitz.

Acknowledgments

The third author wishes to thank Dr. John W. Simpson-Porco for early inspiring discussions. The authors would like to thank Kevin D. Smith and Dr. Guosong Yang for numerous insightful comments and discussions on this topic.