Diagonal Stability on Cactus Graphs and Application to Network Stability Analysis
Murat Arcak · IEEE Transactions on Automatic Control · 2011
A square matrixEis said to be diagonally stable if there exists a diagonal matrixD>; 0 satisfyingDE+ETDEcombines this passivity information with the interconnection structure. This paper presents a necessary and sufficient condition for diagonal stability when the digraph describing the network conforms to a “cactus” structure, which means that a pair of distinct simple circuits in the graph have at most one common vertex. In the special case of a single circuit, this diagonal stability test recovers the “secant criterion” that was recently derived for cyclic networks. The paper then incorporates the new diagonal stability test in network stability analysis where the diagonal entries of the matrixDserve as weights in a Lyapunov function constructed from storage functions that verify passivity properties of the components. Finally, the paper illustrates this stability test on examples motivated by gene networks and population dynamics.