# Large-Scale Discrete-Time Interconnected Dynamical Systems

# Large-Scale Discrete-Time Interconnected Dynamical Systems

This chapter develops vector dissipativity notions for large-scale nonlinear discrete-time dynamical systems. In particular, it introduces a generalized definition of dissipativity for large-scale nonlinear discrete-time dynamical systems in terms of a vector dissipation inequality involving a vector supply rate, a vector storage function, and a nonnegative, semistable dissipation matrix. On the subsystem level, the proposed approach provides a discrete energy flow balance in terms of the stored subsystem energy, the supplied subsystem energy, the subsystem energy gained from all other subsystems independent of the subsystem coupling strengths, and the subsystem energy dissipated. The chapter also develops extended Kalman–Yakubovich–Popov conditions, in terms of the local subsystem dynamics and the interconnection constraints, for characterizing vector dissipativeness via vector storage functions for large-scale discrete-time dynamical systems.

*Keywords:*
vector dissipativity, discrete-time dynamical system, vector dissipation inequality, vector supply rate, vector storage function, semistable dissipation matrix, energy flow, subsystem energy, Kalman–Yakubovich–Popov conditions

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