  In this paper, the controllability and observability of linear multi-agent
systems over matrix-weighted signed networks are analyzed. Firstly, the
definition of equitable partition of matrix-weighted signed multi-agent system
is given, and the upper bound of controllable subspace and a necessary
condition of controllability are obtained by combining the restriction
conditions of the coefficient matrix and matrix weight for the case of fixed
and switching topologies, respectively.The influence of different selection
methods of coefficient matrices on the results is discussed. Secondly, for the
case of heterogeneous systems, the upper bound of controllable subspace and the
necessary condition of controllability are obtained when the dynamics of
individuals in the same cell are the same. Thirdly, sufficient conditions for
controllable and uncontrollable union graphs are obtained by taking advantage
of the concept of switched systems and equitable partitions, respectively.
Finally, necessary condition of observability is obtained in terms of the dual
system and the constraints of the coefficient matrix, and the relationship
between the observability and the controllability of the matrix-weighted signed
multi-agent systems is discussed.
