Gramian-Based Model Reduction of Directed Networks\thanksreffootnoteinfo
This paper investigates a model reduction problem for linear directed network systems, in which the interconnections among the vertices are described by general weakly connected digraphs. First, the definitions of pseudo controllability and observability Gramians are proposed for semistable systems, and their solutions are characterized by Lyapunov-like equations. Then, we introduce a concept of vertex clusterability to guarantee the boundedness of the approximation error and use the newly proposed Gramians to facilitate the evaluation of the dissimilarity of each pair of vertices. An clustering algorithm is thereto provided to generate an appropriate graph clustering, whose characteristic matrix is employed as the projections in the Petrov-Galerkin reduction framework. The obtained reduced-order system preserves the weakly connected directed network structure, and the approximation error is computed by the pseudo Gramians. Finally, the efficiency of the proposed approach is illustrated by numerical examples.
Cheng]Xiaodong Cheng, Cheng]Jacquelien M.A. Scherpen
Willems Center for Systems and Control, Faculty of Science and Engineering, University
Nijenborgh 4, 9747 AG Groningen, The Netherlands.
Key words: Model reduction; Balanced truncation; Passivity; Laplacian matrix; Network topology.
A network system captures the behaviors of a collection of dynamical subsystems. In recent decades, the study of such systems gradually becomes a popular topic of interdisciplinary research, which appears in e.g., social and ecological interactions, chemical reactions and physical networks, see e.g. [Newman2010NetworksIntroduction, Banasiak2015EvolutionaryEquations, Higham2008ChemicalReactions, boccaletti2006complex, Gunawardena2009nature, Schaft2017PhysicalNetwork] for an overview. An important property of network systems is consensus, which occurs when certain agreements are reached via exchanging the information among the vertices [ren2005survey]. Formation control of mobile vehicles, coordination of distributed sensors, or balancing in chemical kinetics can be viewed as different applications of network consensus[jadbabaie2003coordination, Scutari2008SensorNetworks, Rao2015balancing].
The model reduction of consensus network systems is motivated by the challenges of their large scale and high complexity that cause limitations for both theoretical analysis and experimental investigations. It is worth addressing a structure preserving model reduction problem, which aims to derive a lower-dimensional model that can approximate the behavior of the original network with an acceptable accuracy and without being too expensive to evaluate. Furthermore, in the reduction process, it is desirable to preserve a network structure, since such a structure determines the consensus property of a network and is essential for the further applications of reduced-order models to e.g., distributed controller design or sensor allocation.
A variety of techniques are available in the literature for reducing the dimension of a linear state-space system. Classic approaches include balanced truncation, Hankel norm approximation, and Krylov subspace methods, see e.g., [moore1981principal, antoulas2005approximation, astolfi2010model]. They provide systematic procedures to generate reduced-order models that approximate the input-output characteristics of the original large-scale systems. Nevertheless, a direct application of these conventional methods may not maintain a network architecture in the reduced-order model, as they do not impose any structure for the reduced state space. Consequently, the states of network vertices are mixed and thus lose a network interpretation. In [XiaodongBT2017], a structure preserving method is developed using the generalized balanced truncation for undirected networks. Even though it yields a reduced-order model that can be realized as a network system by a proper coordinate transformation, the topology information from the original network eclipses during the process and the obtained network is fully connected.
Recently, graph clustering (or graph partition) has shown a great potential in the structure preserving model reduction of network systems. By assimilating the vertices in each cluster into a single vertex, the essential information of the original topology can be retained. It has to be emphasized that the idea of grouping vertices is relevant to the problem of community or cluster detection in static networks, see e.g., [Aggarwal2013Clustering, Schaeffer2007SurveyClustering]. For dynamical networks that exhibit consensus properties, the clustering process has to take into account the evolution of vertex states driven by external excitation and disturbance signals.
For dynamical systems on undirected networks, the methods developed by [Schaft2014, Monshizadeh2014, Petar2015CDC, Ishizaki2014] formulate the model reduction problem in the Petrov-Galerkin framework, and the projections are generated from selected graph clusterings. However, many applications are restricted to directed networks, e.g., chemical reaction networks [Higham2008ChemicalReactions] or metabolic processes [Banasiak2015EvolutionaryEquations], where the mass/energy exchanges among different species are usually directional.
A pioneering approach dealing with semistable directed networks is proposed in [ishizaki2015clustereddirected]. The graph clustering is formed based on a notion of cluster reducibility, characterized by the uncontrollability of local states. Merging the vertices in the reducible clusters then yields a reduced-order model that preserves the structural information of a directed network. In this method, the studied network essentially has a strongly connected topology, since a positive Frobenius eigenvector of the system matrix is needed for the projection. An alternative approach in [XiaodongCDC2017Digraph] focuses on the behavior of individual vertices described by transfer functions and pairwise dissimilarities evaluated by function norms. The vertices behaving similarly are sequentially assimilated to a single vertex. This approach is preferable for a consensus network and applicable to a strongly connected topology. In broader applications of dynamical networks, for instance, biochemical systems, sensor coordination, gene regulation, weakly connected spatial structures commonly appear in the networks, see e.g., [Gunawardena2009nature, Mirzaev2013LaplacianDynamics, Scutari2008SensorNetworks, Ahsendorf2014GeneRegulation].
This motivates us to consider weakly connected directed networks. As undirected networks and strongly connected networks are only subcategories of weakly connected ones, the systems studied in this paper describe more general scenarios, and the proposed method can be also applied to the former two cases. It is worth noting that a model reduction problem of weakly connected directed networks has been absent from the literature so far. The major difficulty for such networks is an appropriate clustering selection scheme. The approximation accuracy heavily relies on the resulting graph clustering, whereas finding an optimal clustered network is roughly an NP-hard problem even for static networks [Aggarwal2013Clustering, SurveyClustering]. More importantly, in [XiaodongCDC2017Digraph, ishizaki2015clustereddirected], projections are generated using the positive Frobenius eigenvectors of the system matrix. However, such vectors may not exist in the weakly connected case. Furthermore, a weakly connected network may not reach a global consensus as strongly connected ones do. Instead, a local consensus is achievable among the vertices that are able to influence each other. Consequently, the clustering for a weakly connected graph has to be prudently selected to avoid an unbounded approximation error.
To tackle the above difficulties, this paper introduces a definition of vertex clusterability for weakly connected networks and shows that the boundedness of the approximation error is guaranteed if and only if clusterable vertices are aggregated. Thereby, the concept of dissimilarity is defined only for clusterable vertices. In contrast to [XiaodongECC2016, XiaodongCDC2017Digraph], the input and output dissimilarities are considered based on the responses of the vertex states to the external inputs and the measurement of the state discrepancy from the output channels, respectively. Thus, the pairwise dissimilarities are evaluated by combining the input and output efforts. Then, according to the vertex clusterability and dissimilarity, a graph cut algorithm is designed to partition the underlying network into a desired number of clusters. Then, a clustering-based projection is employed to reduce the dimension of the original network system, where the projection matrix is generated from the left kernel space of the system matrix. The proposed method yields a reduced-order model that preserves not only the structure and connectedness of directed network but several fundamental properties, including consensus, semistability, and asymptotic behaviors of the vertex.
Another contribution of this paper is to summarize the notion of controllability Gramians in [XiaodongCDC2016Gramian, ishizaki2015clustereddirected] and extend the results to propose a pair of pseudo controllability and observability Gramians for general semistable systems. The new Gramians can be viewed as the generalization of standard Gramians for asymptotically stable systems. Moreover, the pseudo Gramians are characterized by a set of Lyapunov equations, and their ranks are strongly related to the controllability and the observability of a semistable system. Using the pseudo Gramians, the -norm of a semistable system can be easily evaluated. Therefore, this paper employs them to facilitate the computation of input and output dissimilarities and thus provides a crucial step in the clustering-based model reduction.
The rest of this paper is organized as follows: In Section LABEL:sec:Gramian, we introduce the definition of pseudo Gramians for semistable systems, and some important properties of the new Gramians are the discussed. Section LABEL:sec:networksystem presents the model of directed networks, and the Petrov-Galerkin reduction framework is established based on graph clustering. Then, in Section LABEL:sec:Reduction we define the vertex clusterability and dissimilarity, and propose a scheme for model reduction of directed networks. The proposed method is illustrated through an example in Section LABEL:sec:example, and finally, concluding remarks are made in Section LABEL:sec:conclusion.
Notation: Denote as the set of real numbers and as set of real nonnegative numbers. is a vector space of dimension. Let be a subspace of , then denotes the orthogonal complement of in . The cardinality of a set is denoted by , and represents the dimension of space The identity matrix of size is given as , and denotes a -entries vector of all ones. The subscript is omitted when no confusion arises. is the -th column vector of , and . The trace, rank, image and nullspace of are denoted by , , , and , respectively. Besides, we use the following abbreviations and symbols throughout this paper.
|-norm of a system|
|SCC||strongly connected component|
|LSCC||leading strongly connected component|
|, ,||sets of weakly connected digraphs, quasi strongly connected digraphs, and strongly connected digraphs|