On Synchronization of Dynamical Systems over Directed Switching Topologies: An Algebraic
and Geometric Perspective
In this paper, we aim to investigate the synchronization problem of dynamical systems, which can be of generic linear or Lipschitz nonlinear type, communicating over directed switching network topologies. A mild connectivity assumption on the switching topologies is imposed, which allows them to be directed and jointly connected. We propose a novel analysis framework from both algebraic and geometric perspectives to justify the attractiveness of the synchronization manifold. Specifically, it is proven that the complementary space of the synchronization manifold can be spanned by certain subspaces. These subspaces can be the eigenspaces of the nonzero eigenvalues of Laplacian matrices in linear case. They can also be subspaces in which the projection of the nonlinear self-dynamics still retains the Lipschitz property. This allows to project the states of the dynamical systems into these subspaces and transform the synchronization problem under consideration equivalently into a convergence one of the projected states in each subspace. Then, assuming the joint connectivity condition on the communication topologies, we are able to work out a simple yet effective and unified convergence analysis for both types of dynamical systems. More specifically, for partial-state coupled generic linear systems, it is proven that synchronization can be reached if an extra condition, which is easy to verify in several cases, on the system dynamics is satisfied. For Lipschitz-type nonlinear systems with positive-definite inner coupling matrix, synchronization is realized if the coupling strength is strong enough to stabilize the evolution of the projected states in each subspace under certain conditions. The above claims generalize the existing results concerning both types of dynamical systems to so far the most general framework. Some illustrative examples are provided to verify our theoretical findings.
Over the last couple of years, consensus and synchronization problems have been popular subjects in systems and control [10, 8, 12], inspired by their applications in physics, social sciences, biology, and engineering [4, 5, 9]. The essence of these kinds of problems is the collective objective to reach agreement about some variables of interest [6, 7, 11, 12]. A widely used control protocol to achieve the above-mentioned goal is the linear controller using nearest neighbors’ information . In determining the collective behavior using the distributed linear controller, three different factors are fundamental, namely, the self-dynamics, the coupling configuration (e.g., type and strength of couplings), and the coupling topology [3, 10]. To date, intensive analyses on such issues as how these factors influence the collective behaviors of networked systems has been conducted and fruitful conclusions have been obtained. Synchronization over switching communication topology is one of these issues and is attracting great attentions. So far, necessary and/or sufficient connectivity conditions to achieve consensus for first-order dynamics have been very well developed, e.g., [1, 8, 22, 13, 30, 37, 36]. Unfortunately, when it comes to higher-order linear systems or nonlinear systems with complex self-dynamics and coupling configuration, there are still many open problems. We aim to further address synchronization of dynamical systems over switching topology driven by static controller in this work.
For inter-connected generic linear systems and nonlinear systems, (common/multiple) Lyapunov method is commonly adopted to perform synchronization analysis [21, 15]. The Lyapunov function is appropriately designed such that the factors as coupling topology are involved . The difficulty of applying Lyapunov method is each switched sub-system may not be a convergent one (because the Laplacian matrix has multiple zero eigenvalues). To overcome this difficulty, it is usually required that the communication graph has a well connectivity property [21, 19, 18]. Specifically, in  synchronization among partial-state coupled identical linear systems (viz., input matrix is not invertible) using dynamic controller is investigated, where the communication topology is assumed to have a well-defined average that is connected. In , the authors seek synchronization via multiple Lyapunov function approach assuming that the communication graph is frequently connected, that is, the graph is connected over at least one sub-interval for a period of time. Matrix inequalities are proposed with respect to system matrix and Laplacian matrix to guarantee practical synchronization  in the presence of input disturbance, where the communication graph remains connected all the time. Very recently, a dynamic controller is designed in  to address the bounded synchronization for uncertain linear systems which communicate over frequently connected undirected graphs.
The contraction analysis [15, 14] is another approach to dealing with synchronization problem over switching topology. This approach focuses on deriving the contraction property of synchronization error/disagreement vector. In particular, it is aimed to show that the synchronization error decreases strictly over sufficiently long time, usually with mild connectivity condition at the cost of additional constraints on system model  or stringent sufficient algebraic condition . Specifically, ref.  considers the synchronization problem among nonlinear system dynamics, which satisfies Lipschitz condition. By considering the contraction of the norm of the state deviation, the factors related to nonlinear system dynamics, switching communication graph that is jointly strongly connected, and coupling strength are implicitly involved in the sufficient condition taking algebraic form. In ref. , similar technique to that used in  is applied to derive synchronization condition with relaxed joint connectivity condition. However, from the sufficient algebraic condition provided therein, one may not figure out how the switching scheme, the coupling topology as well as the coupling strength influence the synchronization behavior, which is one of the most fundamental issues in examining the collective behavior of dynamical systems. Although ref.  succeeds in addressing this issue by working on the networks of linear systems under mild constraint on communication topology, it is, however, assumed that the input matrix is invertible.
In this paper, we propose a novel analysis framework, which is totally different from most existing works, from an unified algebraic and geometric perspective to revisit the synchronization problems for both generic linear systems and Lipschitz-type nonlinear systems over switching directed topologies. It is interesting to observe that the complementary space of the synchronization manifold can be spanned by certain subspaces. These subspaces can be the eigenspaces of the nonzero eigenvalues of Laplacian matrices induced from communication topologies in linear case. They can also be subspaces in which the projection of the nonlinear self-dynamics still retains the Lipschitz property. This allows us to project the states of the systems onto these subspaces. Subsequently, to guarantee synchronization, it suffices to show that the states of the systems vanish along each of these subspaces by employing techniques developed from matrix analysis and stability theory. To the best of our knowledge, no approach has been developed so far that can simultaneously deal with the synchronization analysis of both generic linear and Lipschitz-type nonlinear systems.
We are able to, with the above analysis framework, tackle the difficulty confronted by the approach in [27, 28] and  that the associated Laplacian matrices may have eigenvalue zero with algebraic multiplicity larger than one and cannot apply Lyapunov function method to the cases with directed and disconnected topologies. Moreover, different from  and  where the contraction analysis is performed directly with respect to state deviation, we analyze the contraction in the subspaces in which the projection of nonlinear system dynamics still retains Lipschitz property. Hence, the contraction property, which is guaranteed in [15, 18] by an algebraic condition, follows easily with strong couplings.
With these observations, we are able to work out the following contributions under a mild joint connectivity condition.
It is proven that synchronization for linear partial-state coupled systems can be achieved if an algebraic condition with regard to system dynamics and Laplacian matrix is satisfied. This generalizes the results in  and  from undirected communication topologies to the directed case. As a byproduct of the observation that eigenvalue zero of the graph Laplacian matrix is semisimple, we show that if the communication graph switches slowly, then synchronization can be achieved for a class of marginally stable and positive linear systems under the joint connectivity condition. A lower bound of the dwell time is also explicitly specified to guarantee the synchronization.
For Lipschitz-type nonlinear systems, it is found that with sufficiently strong couplings to ensure the decay of the projected states onto the subspaces in which the Lipschitz property of nonlinear system still holds, the synchronization can be guaranteed. This reveals that the desynchronization coming from self-dynamics should be dominated by the synchronization contributed by the jointly connected communication graph provided a certain geometric property of the subspaces holds. Although sufficient conditions that are required to synchronize the Lipschitz-type nonlinear systems have also been developed, e.g., in refs.  and , no such information as how the self-dynamics, the coupling strength and/or coupling topologies influence the synchronization behavior have been revealed intuitively and explicitly.
The remainder of the paper is arranged as follows. In Section II, we introduce the relevant graph notions and formulate the problem. Some technical lemmas are provided in Section III. The evolution analysis of the projection state in a fixed interval is provided in Section IV, followed by Section V where we provide the main results on synchronization by invoking the joint connectivity condition. Some technical analyses of the main results are presented in Section VI. The paper is concluded at last in Section VII
Notations: Let denote the Euclidean norm of a finite dimensional vector . Denote by the identity matrix (if the subscript is dropped, denotes the identity matrix of compatible dimension) and by the zero matrix in . Let denote the diagonal matrix with being the -th diagonal element. Let and denote the kernel and range space of a square matrix , respectively.
Ii Graph Theory and Problem Formulation
Ii-a Graph and Matrix Theory Notions
The interaction topology of a collection of systems is represented by the directed graph of order with a finite nonempty set of nodes a set of edges and a weighted adjacency matrix , where is the weight, also called coupling strength in this work, of the directed edge satisfying if is an edge of and otherwise. Moreover, we assume for all The Laplacian matrix of is defined as , where . An important fact of is that is a right eigenvector associated with eigenvalue . A directed path is a sequence of edges in a directed graph of the form A digraph has a directed spanning tree if there exists at least one node, called the root, having a directed path to every other node.
For simplicity, let denote the set of all possible interaction graphs, each associated with the Laplacian for . Herein with being an integer. Consider an infinite sequence of nonempty, bounded, and contiguous time intervals with and where . In each interval , there is a sequence of non-overlapping subintervals , , , with , satisfying , for an integer and a positive constant which is also coined as the dwell time in the literature. The digraph remains unchanged during each subinterval and switches at . In particular, define a right continuous switching signal and the dynamically changing digraph is denoted by (with Laplacian matrix ).
Definition 1 (Union of Graphs).
The union of a collection of graphs , each of order , is a graph with node set given by and edge set given by . The union graph across any time interval is defined by .
Definition 2 (Generalized Eigenvector ).
If is an matrix, a generalized eigenvector of corresponding to the eigenvalue is a nonzero vector satisfying for some positive integer . Equivalently, it is a nonzero element of the nullspace of . Specifically, if , then the generalized eigenvector becomes the eigenvector.
Ii-B System Model and Problem of Interest
Consider the following partial-state coupled linear systems
while the inter-connected nonlinear systems are described by
for , where denotes the state of the -th agent, is the feedback matrix to be designed, is a continuous function, is a positive diagonal matrix, and is the coupling strength.
There exists a positive constant such that the union graph across any time interval with length contains a directed spanning tree, i.e., contains a directed spanning tree for any .
In view of Assumption 1, assume throughout this paper, without loss of generality, that the union of communication graphs over contains a directed spanning tree, that is, contains a directed spanning tree. Besides, our results to be established also base on the following technical assumptions.
The matrix pair is stabilizable, i.e., there exists a compatible matrix such that is Hurwitz.
The nonlinear function satisfies Lipschitz condition with Lipschitz constant being , i.e., , .
Assumption 1 imposes a joint connectivity assumption on switching communication graph, which is milder than those considered in existing works for synchronization of linear or Lipschitz-type nonlinear systems [18, 28]. It is worth pointing out that Assumption 1 is not the weakest connectivity condition. Weaker constraints on connectivity include infinite joint-connectivity  and extensible joint-connectivity . Assumption 2 is a necessary condition for consensusability of linear systems via state-feedback controller , while Assumption 3 is satisfied by many well-known systems, such as Lorenz systems .
Iii Technical Lemmas
In this section, we shall present several useful results. They lay the foundation of the analysis framework to be developed and enlighten the proof of our main results. Please refer to the Appendix for the proofs of all the lemmas proposed in this paper.
Given any Laplacian matrix of a non-negatively weighted graph, its zero eigenvalue is semi-simple, that is, the algebraic multiplicity of the zero eigenvalue equals to the geometric multiplicity.
Given a collection of non-negatively weighted graphs , each of order , if contains a directed graph, then 1) and 2) , where is the Laplacian matrix of for .
Consider two Laplacian matrices and which are respectively defined as
It is easy to know that is spanned by the vectors , , and . While is spanned by and . Moreover is spanned by , and is spanned by and . It is easily verified that and that .
Given any linear time-invariant system with , the following two propositions hold.
) Provided there exists a direct sum such that with being of dimension and being invariant with respect to the linear mapping . Construct the transformation matrix
where is chosen as the basis vector of the invariant subspace . Then, describes the evolution of 222 denotes the projection of into . That describes the evolution of means . with defined by and being the sub-matrix of
) For any decomposition , where , which is of dimension , and ,
) if for each , then there exists a positive constant such that
) if then for each and some positive constant depending on , the decomposition of , and the state, one has
where is a upper bound of ;
) given a different decompositions , where , which is of dimension , and , if , then
where denotes the space of a direct sum of a subset of that contains . The subscripts of and indicate the space with where the corresponding variable evolves.
It is worth pointing out that the conclusion drawn in ) of Lemma 3 can also be applied to an autonomous nonlinear system. The following remark specifies how to obtain .
in Proposition ) of 2) can be obtained as follows: Let , then can be chosen in such a way that . Here, is a diagonal matrix having 1’s from the ()-th to the ()-th entry. and denote the maximum and the minimum singular value of a matrix, respectively. It is worth pointing out that relies on the decomposition of and the initial choice of the upper bound to avoid the case , which is illustrated in the following example.
See Fig. 3 for an example where and are assumed to be the projection states onto two eigenvectors of . Let be the projection state onto another pair of basis vectors. Suppose with . The increase of the norm of projection state onto is described by with for . Then, the evolution of can be given by if and are nonnegative scalars. Here, is bounded, uniformly with respect to the initial value of . However, if and , is not uniformly bounded. Moreover, it is possible that in this case. This is why we choose an upper bound in ii) of proposition 2).
Given subspaces , there exists a direct sum of such that and for some .
Example 3 (A Demonstration of Lemma 4).
Consider the following two Laplacian matrices:
is spanned by and . While is spanned by and . It is easy to verify that . One then obtains , , and .
We shall explain in Sections IV and V how to employ Lemmas 1 to 4 to conduct the analysis. Note that under joint connectivity condition, Lemmas 2 and 4 tell us that the space complementary to the synchronization manifold can be written into the direct sum of subspaces (see an illustration in Fig. 3). These subspaces can be the eigenspaces of the nonzero eigenvalues of Laplacian matrices in linear case. They can also be subspaces in which the projection of the nonlinear self-dynamics still retains the Lipschitz property. Although the underlying topology might be disconnected at any time, in the subspaces mentioned above, the convergence property can be guaranteed. Then, one can invoke Lemma 4 to extend the above convergence property to the complementary space of the synchronization manifold.
The matrix pair is observable if and only if is observable for any compatible matrix .
Finally, we introduce a useful result on how to construct the eigenvectors of zero eigenvalues of a Laplacian matrix. The following notations and definitions are needed: Let be the set containing and all other nodes such that there exists a directed path from to . A set of nodes in a graph will be called a reach if it is a maximal reachable set; in other words, is a reach if for some and there is no such that (properly). For each reach of a graph, define the exclusive part of to be the set . Likewise, define the common part of to be .
Lemma 6 (cf. ).
Suppose , where is a nonnegative diagonal matrix and is stochastic. Suppose has reaches, denoted by , where we denote the exclusive and common parts of each by and , respectively. Then the nullspace of has a basis in whose elements satisfy: 1) for ; 2) for ; 3) for ; and 4) .
It is obvious that is actually a Laplacian matrix of some non-negatively weighted digraph . To better understand Lemma 6, we write , the Laplacian matrix of the -th reach , into the Frobenius normal form :
where corresponds to the subgraph of that is strongly connected for Moreover, , the node set of , belongs to the exclusive part of for . Otherwise, any node in can be reached by the node outside, which contradicts the fact that is a reach. We call the nodes in non-reachable nodes.
Iv Evolution Of Systems Over Intervals Without Connectivity Requirement
In this section, with the help of the established results in Section III, we will illustrate how the linear or Lipschitz-type nonlinear system evolves no matter how the communication graph is structured. To this aim, we first find the subspaces which can be the eigenspaces of the nonzero eigenvalues of Laplacian matrices in linear case and can also be subspaces in which the projection of the nonlinear self-dynamics still retains the Lipschitz property. Then, the analysis is performed with respect to these subspaces
The following analysis is performed in regard to the evolution of synchronization error. We start this section by the derivation of error system dynamics. For brevity, hereafter, we assume that is a periodic signal, which implies that and . However, all the results obtained in this paper can be easily extended to the case that is not periodic.
Iv-a Error System Dynamics and State Space Decomposition of Linear System
Now, consider the commonly used synchronization error
where It is obvious that if , then for .
Now consider the time interval , which contains subintervals . Moreover, the union of communication graph over contains a directed spanning tree. For the subinterval , , the compact form of the error system dynamics is
where is the submatrix of with satisfying and being an unspecified row vector. To perform our analysis, the following fact is important.
If the corresponding nonnegatively weighted graph of the Laplacian matrix contains a directed spanning tree, then
where denotes the space spanned by the generalized eigenvectors associated with eigenvalue 0.
Iv-B Error System Dynamics and State Space Decomposition of Nonlinear System
We discard the synchronization error used in the linear case due to the difficulty to maintain the Lipschitz property of the projection of nonlinear system dynamics onto or . Instead, in order to capture the property of the evolution of system state in a certain subspace of , we construct a series of suitable error variables that evolve in subspaces where the projection of the nonlinear self-dynamics still retains the Lipschitz property. These error variables turn out to serve as the synchronization error in the sense that if all of them equal zero, then synchronization is realized. The convergence analysis can then be performed with respect to each error variable.
We first write into the Frobenius normal form (5):
Note that fix , there exists at least one such that . in (5) corresponds to a subgraph of that is strongly connected for and , the node set in , belongs to the exclusive part of the -th reach for .
With the concept provided in Lemma 6, we consider the following error variable
In (6), is the right eigenvector of associated with eigenvalue zero corresponding to the reach (see Lemma 6). Moreover, is defined such that where , denotes the number of nodes in , and satisfies and . Hence, .
There are two important properties of . First, for Second, . This is true because and . The latter can be known from the fact that
The following lemma shows that if for any under joint connectivity condition, then for
Consider a collection of graphs , each of order , whose union contains a directed spanning tree. Then,
The compact form of the error system dynamics is given by
where and . The third equality is obtained by the observation that
and . It is easy to verify that .
We will show the structure of using a simple yet illustrative example. Consider a given communication graph in Fig. 4. There are two reaches and . Moreover, . Hence, , , , and . One can then obtain that
Iv-C Evolution Property of the Projection of System State
The following analysis is performed with respect to without loss of generality. Recall that