Model reduction of linear multi-agent systems by clustering with varvec{mathcal {H}_2} and varvec{mathcal {H}_infty } error bounds

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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 mathcal {H}_2 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 mathcal {H}_2 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 mathcal {H}_infty model reduction error. Thirdly, we will prove an a priori upper bound for the mathcal {H}_infty 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.