A graphtheoretic approach to global inputtostate stability for coupled control systems
 Yu Qiao^{1}Email author,
 Yue Huang^{2} and
 Minghao Chen^{3}
https://doi.org/10.1186/s136620171129y
© The Author(s) 2017
Received: 23 October 2016
Accepted: 6 March 2017
Published: 5 May 2017
Abstract
In this paper, the inputtostate 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 inputtostate 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 inputtostate stability of this coupled Lurie system is established. Two examples are provided to demonstrate the effectiveness of the theoretical results.
Keywords
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, inputtostate 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 wellknown 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 onedimensional 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 wellknown 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 globalstability problem of equilibrium and developed a systematic approach that allows one to construct global Lyapunov functions for largescale coupled systems from building blocks of individual vertex systems. Later, this technique was appropriately developed and extended to some other coupled systems. In [6–8] several delayed coupled systems were discussed, and some sufficient conditions were obtained. Li et al. in [9–12] 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 discretetime coupled systems [13, 14], and Zhang et al. extended this technique to multidispersal coupled systems [15]. Besides, this technique is also applied to many practical applications, such as biological systems [16–18], neural networks [19, 20], and mechanical systems [20–23]. 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 ndimensional 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 semidefinite (negative semidefinite). 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 mdimensional 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]
In the remainder of this section, we shall give the model formulation and state some definitions that will be used in the main results.
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
In the proof of our main results, we need to find a global ISSLyapunov function for CCS (1). For the convenience of the proof, we now define vertex ISSLyapunov 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 ith 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 ISSLyapunov 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
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
 (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 thatfor 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 \(u^{\mathrm{T}}=uu_{T}\).$$ \bigl\vert x(t,x_{0},u) \bigr\vert \leq\beta_{1}\bigl( \vert x_{0} \vert ,t\bigr)+\beta_{2}\bigl( \Vert u_{T} \Vert ,tT\bigr)+\gamma \bigl( \bigl\Vert u^{\mathrm{T}} \bigr\Vert \bigr) $$
 (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 wellstudied and fruitful area of research.
 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
The proof of Lemma 2 can be seen in [4].
Since it is complex to construct a vertex ISSLyapunov function for CLS (7), we firstly construct a section of the vertex ISSLyapunov function in the following lemma. And then we give the entire vertex ISSLyapunov function for CLS (7) in the main theorem.
Lemma 3
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 \(\alpha _{i}(y_{i})\leqx_{i}/4k\) and \(x_{i}\leq4k\alpha_{i}(y_{i})\), and using \(\sigma_{i}'(z)\leq\pi_{i}(z)\), we can obtainConsidering the two cases \(y_{i}\leqx_{i}/4k\) and \(x_{i}\leq 4ky_{i}\), 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$$ \sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)kx_{i}y_{i}\leq\frac {1}{4} \sigma_{i}'\bigl(x_{i}^{\mathrm{T}}Q_{i}x_{i} \bigr)x_{i}^{2} +4k^{2}y_{i}^{2} \pi_{i}\bigl(16\lambda_{i}k^{2}y_{i}^{2} \bigr). $$(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)
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 ISSLyapunov 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 LyapunovKrasovskii 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
Declarations
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Sontag, E, Wang, Y: On characterizations of the inputtostate stability property. Syst. Control Lett. 24, 351359 (1995) MathSciNetView ArticleMATHGoogle Scholar
 Grüne, L: Inputtostate dynamical stability and its Lyapunov function characterization. IEEE Trans. Autom. Control 47, 14991504 (2002) MathSciNetView ArticleGoogle Scholar
 Angeli, D: A Lyapunov approach to incremental stability properties. IEEE Trans. Autom. Control 47, 410421 (2002) MathSciNetView ArticleGoogle Scholar
 Arcak, M, Teel, A: Inputtostate stability for a class of Lurie systems. Automatica 38, 19451949 (2002) MathSciNetView ArticleMATHGoogle Scholar
 Li, MY, Shuai, Z: Globalstability problem for coupled systems of differential equations on networks. J. Differ. Equ. 248, 120 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Jin, T, Li, W, Feng, J: Outer synchronization of stochastic complex networks with timevarying delay. Adv. Differ. Equ. 2015, 359 (2015) MathSciNetView ArticleGoogle Scholar
 Wang, R, Li, W, Li, X: The almost sure stability of coupled system of stochastic delay differential equations on networks. Adv. Differ. Equ. 2015, 133 (2015) MathSciNetView ArticleGoogle Scholar
 Li, W, Zhang, X, Zhang, C: A new method for exponential stability of coupled reactiondiffusion systems with mixed delays: combining Razumikhin method with graph theory. J. Franklin Inst. Eng. Appl. Math. 352, 11691191 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Wang, G, Li, W, Feng, J: Stability analysis of stochastic coupled systems on networks without strong connectedness via hierarchical approach. J. Franklin Inst. Eng. Appl. Math. 354, 11381159 (2017) MathSciNetView ArticleMATHGoogle Scholar
 Liu, Y, Li, W, Feng, J: Graphtheoretical method to the existence of stationary distribution of stochastic coupled systems. J. Dyn. Differ. Equ. (2017). doi:10.1007/s108840169566y Google Scholar
 Zhang, C, Li, W, Wang, K: Graph theorybased approach for stability analysis of stochastic coupled systems with Lévy noise on networks. IEEE Trans. Neural Netw. Learn. Syst. 26, 16891709 (2015) Google Scholar
 Li, W, Su, H, Wang, K: Global stability analysis for stochastic coupled systems on networks. Automatica 47, 215220 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Su, H, Li, W, Wang, K: Global stability analysis of discretetime coupled systems on networks and its applications. Chaos 22, 033135 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Su, H, Wang, P, Ding, X: Stability analysis for discretetime coupled systems with multidiffusion by graphtheoretic approach and its application. Discrete Contin. Dyn. Syst., Ser. B 21, 253269 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, C, Li, W, Wang, K: Graphtheoretic approach to stability of multigroup models with dispersal. Discrete Contin. Dyn. Syst., Ser. B 20, 259280 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Liu, M, Bai, C: Analysis of a stochastic tritrophic foodchain model with harvesting. J. Math. Biol. 73, 597625 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Liu, M, Fan, M: Permanence of stochastic LotkaVolterra systems. J. Nonlinear Sci. (2016). doi:10.1007/s0033201693372 Google Scholar
 Liu, M, Fan, M: Stability in distribution of a threespecies stochastic cascade predatorprey system with time delays. IMA J. Appl. Math. (2016). doi:10.1093/imamat/hxw057 Google Scholar
 Li, W, Pang, L, Su, H, Wang, K: Global stability for discrete CohenGrossberg neural networks with finite and infinite delays. Appl. Math. Lett. 25, 22462251 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, C, Li, W, Su, H, Wang, K: A graphtheoretic approach to boundedness of stochastic CohenGrossberg neural networks with Markovian switching. Appl. Math. Comput. 219, 91659173 (2013) MathSciNetMATHGoogle Scholar
 Li, W, Su, H, Wei, D, Wang, K: Global stability of coupled nonlinear systems with Markovian switching. Commun. Nonlinear Sci. Numer. Simul. 17, 26092616 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, C, Li, W, Wang, K: A graphtheoretic approach to stability of neutral stochastic coupled oscillators network with timevarying delayed coupling. Appl. Math. Comput. 37, 11791190 (2014) MathSciNetMATHGoogle Scholar
 Zhang, X, Li, W, Wang, K: The existence and global exponential stability of periodic solution for a neutral coupled system on networks with delays. Appl. Math. Comput. 264, 208217 (2015) MathSciNetGoogle Scholar
 Su, H, Qu, Y, Gao, S, Song, H, Wang, K: A model of feedback control system on network and its stability analysis. Commun. Nonlinear Sci. Numer. Simul. 18, 18221831 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Shatyrko, A, Diblik, J, Khusainov, D, Ruzickova, M: Stabilization of Lur’etype nonlinear control systems by LyapunovKrasovskii functionals. Adv. Differ. Equ. 2012, 229 (2012) MathSciNetView ArticleGoogle Scholar