Model reduction of linear multi-agent systems by clustering with $$\varvec{\mathcal {H}_2}$$ H 2 and $$\varvec{\mathcal {H}_\infty }$$ H ∞ error bounds
Hidde-Jan Jongsma, Petar Mlinarić, Sara Grundel, Peter Benner, Harry L. Trentelman · Mathematics of Control Signals and Systems · 2018
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.