Extending small gain and passivity theory for large-scale system interconnections

In this paper, we use classical Nyquist arguments to derive stability results for large-scale interconnections of “mixed” linear, time-invariant (LTI) systems. We compare our results with P. J. Moylan and D. J. Hill, Stability criteria for large-scale systems, IEEE Transactions on Automatic Control, vol. 23, no. 2, 1978, pp. 143-149. Our results suggest that, if one relaxes the assumptions on the subsystems in an interconnection from assumptions of passivity or small gain to assumptions of “mixedness,” then the Moylan- and Hill-like conditions on the interconnection matrix become more stringent. We also explore the stability of large-scale, time-varying interconnections of strictly positive real systems. We determine a condition that guarantees the existence of a Lyapunov function for the interconnected system.