Application of graph theory to stability of non linear systems

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For a given composite system~ stability is Ltsually the most important perf ormance at tr i bt.tte to be determined. If the composite system is linear many stability criteria are available. Among them are the I\lyquist criterion~ RoLtth's stability ~ etc. .. If the system is nonlinear, then SLtCh stability criteria do ~ not apply, but the di rect method of LyapLtnOV (DML) is the most general method for the determination of stability of nonlinear composite systems. Despite its elegance and generality~the usefLtlness of (DML) is severely limited when applied to systems of high dimension. For this reason it may be advantageous to view the composite system as being composed of several subsystems, which when interconnected in an appropriate fashion~ yield the original composite or interconnected system.We use graph theoretic decomposition techni qLte based on identifying the strongly connected components (SCC) of the digraph model. The basic concept of sol vi ng a composi te system thrOLtgh an analysis of its subsystem (SCC) and their interconnections is to simplify determination of its stability.This stems from the fact easier to find LyapLtnOV fLtnction for the lower order subsystem than the higher order composite system.