# A graph-theoretic approach to global input-to-state stability for coupled control systems

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## Abstract

In this paper, the input-to-state stability for coupled control systems is investigated. A systematic method of constructing a global Lyapunov function for the coupled control systems is provided by combining graph theory and the Lyapunov method. Consequently, some novel global input-to-state stability principles are given. As an application to this result, a coupled Lurie system is also discussed. By constructing an appropriate Lyapunov function, a sufficient condition ensuring input-to-state stability of this coupled Lurie system is established. Two examples are provided to demonstrate the effectiveness of the theoretical results.

## Keywords

input-to-state stability coupled control system Lyapunov function## 1 Introduction

In recent years, coupled control systems (CCSs) have received considerable attention for their interesting characteristics from the mathematical point of view. The main interest has been focused on the investigation of the global dynamics of the systems, with a special emphasis on the study of stability. Meanwhile, input-to-state stability (ISS) for control systems has been extensively studied due to a wide range of applications in physics, biology, social science, neural networks, engineering fields, and artificial complex dynamical systems. For example, Sontag and Wang [1] showed the importance of the well-known Lyapunov sufficient condition for ISS and provided additional characterizations of the ISS property, including one in terms of nonlinear stability margins. Grüne [2] presented a new variant of the ISS property which is based on a one-dimensional dynamical system, showed the relation to the original ISS formulation, and described the characterizations by means of suitable Lyapunov functions. In [3], Angeli presented a framework for understanding such questions fully compatible with the well-known ISS approach and discussed applications of the newly introduced stability notions. In [4], Arcak and Teel analyzed ISS for the feedback interconnection of a linear block and a nonlinear element.

As far as we know, there are a lot of papers dealing with the ISS of individual control systems but few papers dealing with the ISS of CCSs. In general, the study of ISS for CCSs is complex, because it is very difficult to straightly construct an appropriate Lyapunov function for CCSs. However, in [5], Li and Shuai studied the global-stability problem of equilibrium and developed a systematic approach that allows one to construct global Lyapunov functions for large-scale coupled systems from building blocks of individual vertex systems. Later, this technique was appropriately developed and extended to some other coupled systems. In [6, 7, 8] several delayed coupled systems were discussed, and some sufficient conditions were obtained. Li et al. in [9, 10, 11, 12] investigated the stochastic stability of coupled systems with both white noise and color noise. Moreover, by using this technique, Su et al. derived sufficient conditions ensuring global stability of discrete-time coupled systems [13, 14], and Zhang et al. extended this technique to multi-dispersal coupled systems [15]. Besides, this technique is also applied to many practical applications, such as biological systems [16, 17, 18], neural networks [19, 20], and mechanical systems [20, 21, 22, 23]. Hence, the graph theory is a great method in the study of coupled systems.

Motivated by the above discussions, in this paper, we investigate the ISS of CCSs. A systematic method of constructing a global Lyapunov function for the CCSs is provided by combining graph theory and the Lyapunov method. Consequently, some novel global stability principles are given. As an application to this result, a coupled Lurie system is also discussed. By constructing an appropriate Lyapunov function, a sufficient condition ensuring the ISS of this coupled Lurie system is established. Finally, two examples and their numerical simulations are provided to demonstrate the effectiveness and correctness of the theoretical results.

The rest of the paper is organized as follows. In Section 2, some preliminaries and the problem description are presented. In Section 3, the main theorems and their rigorous proofs are described. Finally, in Section 4, an application to a coupled Lurie system is given, and the respective simulations are also given to demonstrate the effectiveness of our results.

## 2 Preliminaries and model formulation

Throughout the paper, unless otherwise specified, the following notations will be used. As we usually use, \(\mathbb{R}^{n}\) denotes the *n*-dimensional Euclidean space. Notations \(\mathbb{R}^{1}_{+}=[0,+\infty)\), \(\mathbb {Z}^{+}=\{1,2,\ldots\}\), \(\mathbb{L}=\{1,2,\ldots, l\}\), \(n=\sum_{i=1}^{l} n_{i}\), and \(m=\sum_{i=1}^{l} m_{i}\) for \(n_{i}, m_{i}\in \mathbb{Z}^{+}\) are used. For any \(x\in\mathbb{R}^{n}\), \(x^{\mathrm {T}}\) is its transpose and \(|x|\) is its Euclidean norm. Let \(\mathbb {R}^{n\times n}\) denote the set of \(n\times n\) real matrix space. For a matrix *P*, \(P\geq0\) (≤0) means that *P* is positive semi-definite (negative semi-definite). The symbol \(\psi_{1}\circ\psi_{2}\) stands for the composition of two functions \(\psi_{1}\) and \(\psi_{2}\). The gradient function of a function *f* is indicated by ▽*f*. In an *m*-dimensional space, the symbol \(L_{\infty}^{m}\) indicates the set of all the functions which are endowed with essential supremum norm \(\|u\| =\sup\{|u(t)| \mid t\geq0\}\leq\infty\).

We recall some knowledge of graph theory that will be used in the rest of the paper. Define a weighted digraph \(\mathcal{G}=\{V,E,A\}\), in which set \(V=\{v_{1},v_{2},\ldots,v_{l}\}\) denotes *l* vertices of the graph, element \(e_{ij}\) of *E* denotes the arc leading from initial vertex *j* to terminal vertex *i*, and the element \(a_{ij}\) of a weighted adjacency matrix *A* denotes the weight of arc \(e_{ij}\). We denote \(a_{ij}>0\) if and only if there exists an arc from vertex *i* to vertex *j* in \(\mathcal{G}\), otherwise \(a_{ij}=0\), and we denote \(a_{ii}=0\) for all \(i\in\mathbb{L}\). Denote the digraph with weight matrix *A* as \((\mathcal{G},A)\). If a graph \(\mathcal{S}\) has the same vertex as \(\mathcal{G}\), we call it a subgraph of \(\mathcal{G}\). The weight \(W(\mathcal{S})\) of a subgraph \(\mathcal{S}\) is the product of the weights on all its arcs. If a connected subgraph has no cycle, it is a tree. We call \(v_{i}\) the root of the tree if vertex *i* of the tree is not a terminal vertex of any arcs and each of the remaining vertices is a terminal vertex of one arc. A subgraph *Q* is unicyclic when it is a disjoint union of rooted trees whose roots form a directed cycle. The Laplacian matrix of \(\mathcal{G}\) is defined as \(L=(b_{ij})_{l\times l}\), where \(b_{ij}=-a_{ij}\) for \(i\neq j\) and \(b_{ij}= \sum_{k\neq i}a_{ik}\) for \(i=j\).

The following lemma will be used in the proof of our main results.

### Lemma 1

[5]

*Assume*\(l\geq2\).

*Then the following identity holds*:

*Here*\(F_{ij} (x_{i},x_{j})\)

*are arbitrary functions for any*\(1\leq i, j\leq l\), \(a_{ij}\)

*are elements of matrix*

*A*, \(\mathbb{Q}\)

*is the set of all spanning unicyclic graphs of*\((\mathcal{G},A)\), \(W(\mathcal{Q})\)

*is the weight of*\(\mathcal {Q}\),

*and*\(C_{\mathcal{Q}}\)

*denotes the directed cycle of*\(\mathcal {Q}\).

*And*\(c_{i}\)

*denotes the cofactor of the*

*ith diagonal element of*

*L*,

*in particular*,

*if*\((\mathcal{G},A)\)

*is strongly connected*,

*then*\(c_{i}>0\)

*for*\(1\leq i \leq l\).

In the remainder of this section, we shall give the model formulation and state some definitions that will be used in the main results.

*l*vertices (\(l\geq2\)) and \(A=(a_{ij})_{l\times l}\). A coupled control system can be constructed on \((\mathcal{G},A)\) by assigning each vertex its own dynamics and then coupling these vertex dynamics based on directed arcs in \((\mathcal{G},A)\). The details are as follows. Assume that the

*i*th vertex dynamic is described by the control system

*i*, \(f_{i}:\mathbb{R}^{n_{i}}\times\mathbb{R}^{m_{i}}\rightarrow\mathbb {R}^{n_{i}}\) is continuously differentiable and satisfies \(f_{i}(0,0)=0\), function \(u_{i}:\mathbb{R}^{1}_{+}\rightarrow\mathbb {R}^{m_{i}}\) denotes the input of vertex

*i*and it is measurable and locally essentially bounded. Assume that \(a_{ij}\geq0\) represents the effect factor from vertex

*j*to vertex

*i*and \(a_{ij} =0\) iff there exists no arc from

*j*to

*i*. Then we use function \(P_{ij}\) to describe the effect that subsystem

*j*has on

*i*and \(P_{ij}:\mathbb {R}^{n_{i}}\times\mathbb{R}^{n_{j}}\times\mathbb {R}^{m_{j}}\rightarrow\mathbb{R}^{n_{i}}\) is continuously differentiable and satisfies \(P_{ij}(0,0,0)=0\). For example, in a digraph with six vertices, we show the interaction in vertex

*j*and vertex

*i*(see Figure 1).

To be more precise, we recall some definitions on the ISS of CCS (1). We refer to [1, 2] for definitions as follows.

### Definition 1

A function \(\gamma:\mathbb{R}^{1}_{+}\rightarrow\mathbb{R}^{1}_{+}\) is a \(\mathcal{K}\)-function if it is continuous, strictly increasing, and \(\gamma(0)=0\). If a \(\mathcal{K}\)-function satisfies \(\gamma (s)\rightarrow\infty\) as \(s\rightarrow\infty\), we call it \(\mathcal{K}_{\infty}\)-function. A function \(\beta:\mathbb {R}^{1}_{+}\times\mathbb{R}^{1}_{+}\rightarrow\mathbb{R}^{1}_{+}\) is a \(\mathcal{K}\mathcal{\phi}\)-function if the function \(\beta(\cdot,t)\) is a \(\mathcal{K}\)-function for each fixed \(t\geq0\), and for each fixed \(s\geq0\), \(\beta(s,t)\) is decreasing to zero as \(t\rightarrow \infty\).

### Definition 2

*γ*such that for each input \(u\in L_{\infty}^{m}\) and \(x_{0}\in\mathbb{R}^{n}\), it holds that

In the proof of our main results, we need to find a global ISS-Lyapunov function for CCS (1). For the convenience of the proof, we now define vertex ISS-Lyapunov functions for CCS (1).

### Definition 3

- Q1.There exist positive constants \(\alpha_{i}\), \(\delta_{i}\), \(p\geq2\), such that$$ \alpha_{i}|x_{i}|^{p}\leq V_{i}(x_{i}) \leq\delta_{i}|x_{i}|^{p},\quad x_{i} \in\mathbb{R}^{n_{i}}. $$
- Q2.There exist constants \(\xi_{i},d_{ij}\geq0\), functions \(F_{ij} (x_{i},x_{j})\), and \(\mathcal{K}\)-function \(\chi_{i}\) such that for any \(x_{i}\in\mathbb{R}^{n_{i}}\) and \(\mu_{i}\in\mathbb {R}^{m_{i}}\) satisfying \(\sum^{l}_{i=1}c_{i}\delta_{i}|x_{i}|^{p}\geq \sum^{l}_{i=1}c_{i}\delta_{i}|\chi_{i}(|\mu_{i}|)|^{p}\), where \(D=(d_{ij})_{l\times l}\) and \(c_{i}\) is the cofactor of the
*i*th diagonal element of Laplacian matrix of \((\mathcal{G},D)\). Then we havein which$$ \dot{V}_{i}\bigl(x_{i}(t)\bigr)\leq-\xi_{i} \bigl\vert x_{i}(t) \bigr\vert ^{p}+\sum ^{l}_{j=1}d_{ij}F_{ij} \bigl(x_{i}(t),x_{j}(t)\bigr), $$$$\dot{V}_{i}\bigl(x_{i}(t)\bigr)=\bigtriangledown V_{i}\bigl(x_{i}(t)\bigr) \Biggl[f_{i} \bigl(x_{i}(t),u_{i}\bigr)+\sum ^{l}_{j=1} a_{ij}P_{ij} \bigl(x_{i}(t),x_{j}(t),u_{j}\bigr) \Biggr]. $$ - Q3.Along each directed cycle \(C_{\mathcal{Q}}\) of weighted digraph \((\mathcal{G},D)\), there is$$ \sum_{(s,r)\in E(C_{\mathcal{Q}})} F_{rs}(x_{r},x_{s}) \leq0. $$

## 3 Main results

In this section, the ISS of CCS (1) will be investigated. The approaches used in the proof of the main results are motivated by [1, 5].

### Theorem 1

*If CCS* (1) *admits a vertex ISS*-*Lyapunov function set* \(\{ V_{i}(x_{i}), i\in\mathbb{L}\}\), *and digraph* \((\mathcal{G},D)\) *is strongly connected*, *then the solution of CCS* (1) *is ISS*.

### Proof

*i*th diagonal element of Laplacian matrix of \((\mathcal{G},D)\).

*t*in \((t_{0},\tau)\). This contradicts the minimality of

*τ*, and hence \(x(t)\in S\) for all \(t\geq t_{0}\).

From (5) and (6), we can obtain \(|x(t)|\leq\beta (|x_{0}|,t)+\gamma(\|u\|)\) for all \(t\geq0\), that is, CCS (1) is ISS. □

In [1], the ISS for individual nonlinear control system was investigated by Sontag and Wang. Some classes of stability, like robust stability and weak robust stability for control systems, were investigated and some sufficient conditions were established to guarantee these stabilities. Motivated by [1], we have the following results.

### Theorem 2

*Let the conditions in Theorem*1

*hold*.

*Then*:

- (1)
*CCS*(1)*is robustly stable*. - (2)
*There exist*\(\mathcal{K}\mathcal{\phi}\)-*functions*\(\beta _{1}\), \(\beta_{2}\)*and a*\(\mathcal{K}\)-*function**γ**such that*,*for any*\(x_{0}\in\mathbb{R}^{n}\)*and any input*\(u\in L_{\infty}^{m}\),*it holds that*$$ \bigl\vert x(t,x_{0},u) \bigr\vert \leq\beta_{1}\bigl( \vert x_{0} \vert ,t\bigr)+\beta_{2}\bigl( \Vert u_{T} \Vert ,t-T\bigr)+\gamma \bigl( \bigl\Vert u^{\mathrm{T}} \bigr\Vert \bigr) $$*for any*\(0\leq T\leq t\),*where*\(u_{T}\)*denotes the input for CCSs*(1)*when*\(t=T\)*and*\(u^{\mathrm{T}}\)*is defined by*\(u^{\mathrm{T}}=u-u_{T}\). - (3)
*For each*\(\varepsilon>0\),*there exists*\(\delta>0\)*such that*\(|x(t,x_{0},u)|\leq\varepsilon\)*for all inputs*\(u\in L_{\infty}^{m}\)*and initial states*\(x_{0}\)*with*\(|x_{0}|\leq\delta\)*and*\(\|u\|\leq\delta\). - (4)
*There exists a*\(\mathcal{K}\)-*function**γ**such that*,*for any*\(r,\varepsilon>0\),*there is*\(T>0\)*so that for every input*\(u\in L_{\infty}^{m}\),*it holds that*\(|x(t,x_{0},u)|\leq\varepsilon+\gamma(\| u\|)\),*whenever*\(|x_{0}|\leq r\)*and*\(t\geq T\). - (5)
*CCS*(1)*is weakly robustly stable*.

## 4 An application to a coupled Lurie system

Now in order to illustrate the result of Theorem 1, let us apply this result to a coupled Lurie system (CLS). The absolute stability problem, formulated by Lurie and coworkers in the 1940s, has been a well-studied and fruitful area of research.

*i*th vertex system, \(B_{i}\in\mathbb{R}^{n_{i}\times m_{i}}\) is the feedback and input effect matrix, \(u_{i}\) denotes the input of vertex

*i*, and \(\alpha_{i}(\cdot): \mathbb{R}^{m_{i}}\rightarrow\mathbb {R}^{m_{i}}\) is a feedback function. Let \(D_{j}\in\mathbb {R}^{n_{i}\times n_{j}}\) describe the effect that vertex system

*j*has on

*i*. Thus a CLS is obtained as follows:

- A1:If \((A_{i},K_{i})\) is detectable and there exists matrix \(P_{i}=P_{i}^{\mathrm{T}}\geq0\) satisfying$$ A_{i}^{\mathrm{T}}P_{i}+P_{i}A_{i}+l \bigl(P_{i}^{\mathrm {T}}P_{i}+D_{i}^{\mathrm{T}}D_{i} \bigr)\leq0,\qquad K_{i}^{\mathrm{T}}=P_{i}B_{i}. $$(8)
- A2:If \(\varphi_{i}\) is a \(\mathcal{K}_{\infty}\)-function, and for all \(y_{i}\in\mathbb{R}^{m_{i}}\),$$ |y_{i}|\varphi_{i}\bigl( \vert y_{i} \vert \bigr)\leq y_{i}^{\mathrm{T}} \alpha_{i}(y_{i}). $$(9)
- A3:When \(|y_{i}|\geq\mu_{i}\), where \(\mu_{i}>0\)$$ \bigl\vert \alpha_{i}(y_{i}) \bigr\vert \leq y_{i}^{\mathrm{T}}\alpha_{i}(y_{i}). $$(10)

### Lemma 2

*For CLS*(7),

*there exist a constant*\(\theta_{i}>0\)

*and a*\(\mathcal{K}_{\infty}\)-

*function*\(\eta_{i}(\cdot)\)

*satisfying*

*when*\(|y_{i}|\geq\mu_{i}\),

*when*\(|y_{i}|\leq\mu_{i}\).

The proof of Lemma 2 can be seen in [4].

Since it is complex to construct a vertex ISS-Lyapunov function for CLS (7), we firstly construct a section of the vertex ISS-Lyapunov function in the following lemma. And then we give the entire vertex ISS-Lyapunov function for CLS (7) in the main theorem.

### Lemma 3

*Suppose that*(7)

*satisfies assumptions*A1-A3.

*Definite a function*

*where the constant*\(\varepsilon_{i}>0\)

*and the*\(\mathcal{K}\)-

*function*\(\pi_{i}(\cdot)\)

*is to be specified*.

*Let*\(S_{i}(x_{i})=\sigma _{i}(x_{i}^{\mathrm{T}}Q_{i}x_{i})\),

*in which matrix*\(Q_{i}^{\mathrm {T}}=Q_{i}>0\)

*satisfying*

*then there exists a*\(\mathcal{K}\)-

*function*\(\gamma_{i}(\cdot)\)

*satisfying*

*where*\(F_{ij}(x_{i},x_{j})=x_{j}^{\mathrm{T}}D_{j}^{\mathrm {T}}D_{j}x_{j}-x_{i}^{\mathrm{T}}D_{i}^{\mathrm{T}}D_{i}x_{i}\).

### Proof

- When \(|y_{i}|\geq\mu_{i}\), choosing \(\varepsilon_{i}=\theta_{i}/c_{i}\) and using (11), we have$$ \dot{S_{i}}(x_{i})\leq-\sigma_{i}' \bigl(x_{i}^{\mathrm {T}}Q_{i}x_{i} \bigr)|x_{i}|^{2}+y_{i}^{\mathrm{T}} \alpha_{i}(y_{i})+\theta _{i}|u_{i}|+ \sum_{j=1}^{l}F_{ij}(x_{i},x_{j}). $$(17)
- When \(|y_{i}|\leq\mu_{i}\), we denote by \(\lambda_{i}\) the maximum eigenvalue of \(Q_{i}\). Considering the two cases \(|\alpha _{i}(y_{i})|\leq|x_{i}|/4k\) and \(|x_{i}|\leq4k|\alpha_{i}(y_{i})|\), and using \(\sigma_{i}'(z)\leq\pi_{i}(z)\), we can obtainConsidering the two cases \(|y_{i}|\leq|x_{i}|/4k\) and \(|x_{i}|\leq 4k|y_{i}|\), we can denote$$ \sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)k \vert x_{i} \vert \bigl\vert \alpha_{i}(y_{i}) \bigr\vert \leq \frac{1}{4}\sigma_{i}' \bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i}\bigr) \vert x_{i} \vert ^{2} +4k^{2} \bigl\vert \alpha_{i}(y_{i}) \bigr\vert ^{2} \pi_{i}\bigl(16\lambda_{i}k^{2} \bigl\vert \alpha _{i}(y_{i}) \bigr\vert ^{2}\bigr). $$(18)We choose$$ \sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)k|x_{i}||y_{i}|\leq\frac {1}{4} \sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)|x_{i}|^{2} +4k^{2}|y_{i}|^{2} \pi_{i}\bigl(16\lambda_{i}k^{2}|y_{i}|^{2} \bigr). $$(19)then substituting (18) and (19) into (16) yields$$\pi_{i}(z)=\frac{1}{4k^{2}}\eta_{i} \biggl(\sqrt{ \frac{z}{16\lambda _{i}k^{2}}} \biggr), $$$$\begin{aligned} \begin{aligned}[b] \dot{S_{i}}(x_{i})\leq{}&-\frac{1}{2} \sigma_{i}'\bigl(x_{i}^{\mathrm {T}}Q_{i}x_{i} \bigr) \vert x_{i} \vert ^{2}+\eta_{i}\bigl( \vert y_{i} \vert \bigr) \vert y_{i} \vert ^{2} +\eta_{i}\bigl( \vert \alpha_{i}y_{i} \vert \bigr) \vert \alpha_{i}y_{i} \vert ^{2} \\ &{}+\theta_{i}u_{i}+\sum _{j=1}^{l}F_{ij}(x_{i},x_{j}). \end{aligned} \end{aligned}$$(20)

*z*, we can find a \(\mathcal{K}_{\infty}\)-function \(\gamma_{i}(\cdot)\) such that

### Theorem 3

*If CLS* (7) *satisfies assumptions* A1, A2 *and* A3, *then it is ISS*.

### Proof

So we conclude that \(L_{i}(x_{i})\) is a vertex ISS-Lyapunov function, then from Theorem 1, CLS (7) is ISS. □

### Remark 1

- (i)
This paper considers a coupled Lurie system, which is more complicated.

- (ii)
This paper uses graph theory combining with the Lyapunov method to derive the ISS of the considered system. This technique does not need us solving any linear matrix inequality. Literature [25] proposed Lyapunov-Krasovskii functionals which contain an exponential multiplier to solve the stabilization of an indirect control system.

- (iii)
In [25], time delay was considered, which is our further work.

Finally, two examples with their numerical simulations are provided to illustrate our results.

### Example 1

### Example 2

## Notes

### Acknowledgements

We would like to thank the editors and the anonymous reviewers for carefully reading the original manuscript and for the constructive comments and suggestions to improve the presentation of this paper.

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