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![Example of a multi-agent system defined on an undirected, weighted, connected graph. Vertices 6 and 7 are leaders. [9]](profile/Petar-Mlinaric/publication/309429386/figure/fig1/AS:613984143167488@1523396635010/Example-of-a-multi-agent-system-defined-on-an-undirected-weighted-connected-graph.png)
Example of a multi-agent system defined on an undirected, weighted, connected graph. Vertices 6 and 7 are leaders. [9]
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In this paper, we extend our clustering-based model order reduction method for multi-agent systems with single-integrator agents to the case where the agents have identical general linear time-invariant dynamics. The method consists of the Iterative Rational Krylov Algorithm, for finding a good reduced order model, and the QR decomposition-based cl...
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In [3] it was shown that four seemingly different algorithms for computing low-rank approximate solutions $X_j$ to the solution $X$ of large-scale continuous-time algebraic Riccati equations (CAREs) $0 = \mathcal{R}(X) := A^HX+XA+C^HC-XBB^HX $ generate the same sequence $X_j$ when used with the same parameters. The Hermitian low-rank approximations...
Citations
... In this work, we extend the clustering-based approach for linear time-invariant multiagent systems from [21,22]. There, we proposed a method combining the iterative rational Krylov algorithm (IRKA) [1] and QR decomposition-based clustering [27]. ...
... For multi-agent systems (1) with agents of order n, we have the matrices V and W as in (6). QR decomposition-based clustering can then be extended as in Algorithm 2 from [22] by clustering the block-columns of V T (or W T ). For the k-means algorithm, we can show in a similar way as in the single-integrator case that clustering the block-rows leads to minimizing an upper bound of the largest principal angle. ...
Clustering by projection has been proposed as a way to preserve network structure in linear multi-agent systems. Here, we extend this approach to a class of nonlinear network systems. Additionally, we generalize our clustering method which restores the network structure in an arbitrary reduced-order model obtained by projection. We demonstrate this method on a number of examples.
... The first one is to reduce the number of agents in the network. Typical methods 35 are based on graph clustering and generalized balanced truncation, see e.g., [16,17,18,19,20,21,22]. The other direction is to reduce the dimension of each subsystem, which is of particular interest in this paper. ...
This paper proposes a model order reduction scheme that reduces the complexity of diffusively coupled homogeneous Lur'e systems. We aim to reduce the dimension of each subsystem and meanwhile preserve the synchronization property of the overall network. Using the Laplacian spectral radius, we characterize the robust synchronization of the Lur'e network by a linear matrix inequality (LMI), whose solutions then are treated as generalized Gramians for the balanced truncation of the linear component of each Lur'e subsystem. It is verified that, with the same communication topology, the resulting reduced-order network system is still robustly synchronized, and an a priori bound on the approximation error is guaranteed to compare the behaviors of the full-order and reduced-order Lur'e subsystems.
... Model reduction techniques specifically for networked multi-agent systems with first-order agents have been proposed in [6,15,16,22]. Extensions to second-order agents have been considered in [7,14] and to more general higher-order agents in [4,17,23,25]. Some of these methods are based on clustering nodes in the network. ...
In the recent paper (Monshizadeh et al. in IEEE Trans Control Netw Syst 1(2):145–154, 2014. https://doi.org/10.1109/TCNS.2014.2311883), model reduction of leader–follower multi-agent networks by clustering was studied. For such multi-agent networks, a reduced order network is obtained by partitioning the set of nodes in the graph into disjoint sets, called clusters, and associating with each cluster a single, new, node in a reduced network graph. In Monshizadeh et al. (2014), this method was studied for the special case that the agents have single integrator dynamics. For a special class of graph partitions, called almost equitable partitions, an explicit formula was derived for the model reduction error. In the present paper, we will extend and generalize the results from Monshizadeh et al. (2014) in a number of directions. Firstly, we will establish an a priori upper bound for the model reduction error in case that the agent dynamics is an arbitrary multivariable input–state–output system. Secondly, for the single integrator case, we will derive an explicit formula for the model reduction error. Thirdly, we will prove an a priori upper bound for the model reduction error in case that the agent dynamics is a symmetric multivariable input–state–output system. Finally, we will consider the problem of obtaining a priori upper bounds if we cluster using arbitrary, possibly non almost equitable, partitions.
... Nevertheless, the approximation accuracy of these methods highly relies on the selection of node clusters, and generally finding a reduced network with the best approximation is an NPhard problem, see [15]. A combination of the Krylov subspace method with graph clustering is proposed by [21], where a reduced-order model is firstly found by the Iterative Rational Krylov Algorithm (IRKA), and then the partition of network nodes is obtained by the QR decomposition with column pivoting on the projection matrix. However, no error bound is provided for the network approximation. ...
... Unlike the clustering-based approaches in e.g. [14,5,6], the effort of state observability is also considered, and no special structure of the output matrix is required as in [23,21]. ...
... If k = N and r < n, the above problem setting can be specialized to reduce the dimension of each subsystems as in [22,29]. When k < N and r = n, then the simplification of the network structure is discussed as in [2,21], where the reduction is based on graph clustering. ...
This paper studies model order reduction of multi-agent systems consisting of identical linear passive subsystems, where the interconnection topology is characterized by an undirected weighted graph. Balanced truncation based on a pair of specifically selected generalized Gramians is implemented on the asymptotically stable part of the full-order networked model, which leads to a reduced-order system preserving the passivity of each subsystem. To restore the network structure, we then apply a coordinate transformation to convert the resulting reduced-order model to a state-space model of Laplacian dynamics. The proposed method simultaneously reduces the complexity of the network structure and individual agent dynamics. Moreover, it preserves the passivity of the subsystems and allows for the a priori computation of a bound on the approximation error. Finally, the feasibility of the method is demonstrated by an example.
Clustering by projection has been proposed as a way to preserve network structure in linear multi-agent systems. Here, we extend this approach to a class of nonlinear network systems. Additionally, we generalize our clustering method which restores the network structure in an arbitrary reduced-order model obtained by projection. We demonstrate this method on a number of examples.
