Model reduction of linear multi-agent systems by clustering with $$\varvec{\mathcal {H}_2}$$ H 2 and $$\varvec{\mathcal {H}_\infty }$$ H ∞ error bounds

Abstract

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 H2 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 H2 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 H∞
model reduction error. Thirdly, we will prove an a priori upper bound for the H∞
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