Stable-protocol admissible synchronizability for high-order singular complex networks with switching topologies

Abstract

Stable-protocol admissible synchronization analysis and design problems for singular complex networks with both connected switching topologies and jointly connected switching topologies are investigated. Firstly, a singular observer-type synchronization protocol is introduced, where the errors of protocol states among neighboring agents are applied to make complex networks satisfy a specific separation principle which can simplify synchronization analysis and design problems. Then, by the orthonormal projection operators between the synchronization subspace and the complement synchronization subspace, stable-protocol admissible synchronization problems are transformed into admissible ones of multiple subsystems with lower dimensions, and an approach is given to determine the synchronization function. Furthermore, stable-protocol admissible synchronizability criteria for connected switching topology cases and jointly connected switching topology cases are proposed, respectively. Finally, two numerical examples are given to demonstrate the effectiveness of theoretical results.

Introduction

The synchronization of complex networks which consist of multiple distributed nodes has potential applications in various fields such as congestion control of distributed sensor networks [10], phase synchronization of multiple oscillators [24], [29], attitude synchronization of multiple spacecrafts [6], [13] and small-world dynamical networks [19], [20].

Synchronization problems for first-order complex networks were dealt with in [1], [11], [16], [17], [18], [26], where the dynamics of each agent was modeled as a first-order integrator. In [3], [15], second-order complex networks were dealt with, where each agent was modeled as a second-order integrator. It should be pointed out that the dynamics of each agent both first-order and second-order complex networks has a specific structure, so synchronization analysis and design problems can be simplified. For general high-order complex networks, since each agent has no specific structure, synchronization problems are more complex and challenging. Guan et al. [8] investigated synchronization analysis and design problems for high-order linear time-invariant (LTI) complex networks, where it was assumed that all state information is available to construct synchronization protocols. Synchronization problems for high-order LTI complex networks with dynamic output feedback synchronization protocols were addressed in [14], [25]. Cheng et al. [2] considered the impacts of communication noises and proposed mean square synchronization criteria. Ni and Cheng [23] discussed leader-following synchronization problems and considered both fixed and switching topology cases. Leader-following synchronization for complex networks with switching topologies were investigated in [27], where each agent has Lipschitz-type dynamics. Output synchronization analysis and design problems for high-order LTI complex networks were dealt with in [30]. By the event-triggered strategy, sampled-data consensus of complex networks was studied in [9], [5].

In aforementioned literatures about high-order complex networks, the dynamics of each agent was modeled as a normal system. However, each agent can only be modeled as a singular system rather than a normal one in many practical complex networks. As shown in [21], the transistor circuit in Fig. 1(a) is equivalent to the one in Fig. 1(b), which can be modeled by

$$\left\{\begin{array}{c}\left[\begin{array}{cc}\hfill C\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\dot{v}}_C(t)\hfill \\ \hfill {\dot{i}}_E(t)\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill v_C(t)\hfill \\ \hfill i_E(t)\hfill \end{array}\right]+\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]v_{\text{S}}(t)\text{,}\hfill \\ v_R(t)=\left[0\text{,}\alpha R\right]\left[\begin{array}{c}\hfill v_C(t)\hfill \\ \hfill i_E(t)\hfill \end{array}\right]\text{.}\hfill \end{array}\right.$$

It can be found that the coefficient matrix is singular, which means that there exists an impulse when the circuit is turned on at $t=0$ if $v_C(0)\ne -v_{\text{S}}(0)$ or $i_E(0)\ne 0$. Moreover, multi-agent supporting systems with each block supported by several pillars can only be modeled by singular complex networks rather than normal ones, as shown in [34], where the impacts of switching topologies on synchronization for high-order LTI singular complex networks were considered, but it was supposed that each topology in the switching set is connected and all state information are available to construct synchronization protocols. Yang and Liu [35] proposed static output feedback synchronization protocols and gave necessary and sufficient conditions for synchronization of singular complex networks under the assumption that the communication topology is fixed. For complex networks with dynamic output feedback synchronization protocols, if protocol states do not tend to zero as time tends to infinity; that is, the synchronization protocol is not stable, then the control input of each agent may not tend to zero as time tends to infinity. In this case, a continuous expenditure of energy is required even if synchronization is achieved, which is not expected in practical applications. To the best of our knowledge, stable-protocol admissible synchronization for singular complex networks with dynamic output feedback synchronization protocols and jointly connected switching topologies are still not comprehensively investigated.

The current paper deals with singular complex networks with dynamic output feedback synchronization protocols, where both connected switching topology cases and jointly connected switching topology cases are discussed. Based on outputs and protocol states among neighboring agents, a singular observer-type synchronization protocol is applied, whose structures can guarantee the stability of the synchronization protocol and can simplify synchronization problems. Furthermore, based on the orthonormal projection between the synchronization subspace and the complement synchronization subspace, a necessary and sufficient condition for stable-protocol admissible synchronization is proposed and an explicit expression of the synchronization function which determines the synchronization behavior of all agents in a complex network is presented. Moreover, based on the generalized Riccati equation and Barbalat’s lemma, stable-protocol admissible synchronizability conditions for connected switching topology cases and jointly connected switching topology cases are given, respectively.

Compared with the existing works [34], [35] about synchronization for singular complex networks, the current paper has the following three novel features. Firstly, the current paper proposes a singular dynamic output feedback synchronization protocol with switching topologies and no rank constraint involves in synchronizability criteria. All state information was required to construct synchronization protocols in [34] and static output feedback synchronization protocols in [35] brought in a rank constraint in synchronizability criteria. Especially, it was supposed that communication topologies are fixed in [35]. Secondly, the current paper considers the stability of synchronization protocols and determines the impacts of protocol states on the synchronization function. The approaches in [34], [35] cannot directly be used to deal with the influences of protocol states on both stable-protocol admissible synchronization and the synchronization function. Thirdly, synchronizability criteria in the current paper are analytic; that is, the existence of the solution of synchronizability criteria can be guaranteed. However, synchronizability criteria in terms of the linear matrix inequality (LMI) in [34] are not analytic.

This paper is organized as follows. In Section 2, basic concepts and conclusions about graph theory and singular systems are shown, respectively, and the problem description is given. Section 3 converts stable-protocol admissible synchronization problems into admissible ones of multiple subsystems and an approach to determine the synchronization function is proposed. Stable-protocol admissible synchronizability criteria for connected switching topology cases and jointly connected switching topology cases are addressed in Sections 4 Synchronizability criteria for connected switching topology cases, 5 Synchronizability criteria for jointly connected switching topology cases, respectively. Two numerical examples are shown to demonstrate theoretical results in Section 6. Finally, concluding remarks are stated in Section 7.

The following notations are applied throughout the current paper. ${\mathbb{R}}^d$ and ${\mathbb{C}}^d$ denote the sets of all d-dimensional real and complex column vectors, respectively. $I_d$ represents the d-dimensional identity matrix. The notations $\mathbf{1}$ and ${\mathbf{1}}_{N-1}$ denote N- and $N-1$-dimensional column vectors with all components 1, respectively, ${\mathbf{1}}_{N\times N}$ represents an $N\times N$ matrix with all components 1, and $e_i(i=1\text{,}2\text{,}\dots \text{,}N)$ represent N-dimensional column vectors with the ith component 1 and 0 elsewhere. The notations $\otimes$ and $\oplus$ denote the Kronecker product and the direct sum, respectively, and 0 is applied to denote zero matrices of any size with zero vectors and zero number as special cases. For a symmetric matrix, $R>0$ and $R<0$ mean that R is positive definite and negative definite, respectively. The notation $\mathrm{diag}\left\{{\Delta }_1\text{,}{\Delta }_2\text{,}\dots \text{,}{\Delta }_N\right\}$ represents a diagonal matrix with diagonal elements ${\Delta }_1\text{,}{\Delta }_2\text{,}\dots \text{,}{\Delta }_N$.