# Control of Large-Scale Dynamical Systems via Vector Lyapunov Functions

# Control of Large-Scale Dynamical Systems via Vector Lyapunov Functions

This chapter introduces the notion of a control vector Lyapunov function as a generalization of control Lyapunov functions, showing that asymptotic stabilizability of a nonlinear dynamical system is equivalent to the existence of a control vector Lyapunov function. These control vector Lyapunov functions are used to develop a universal decentralized feedback control law for a decentralized nonlinear dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. The chapter also describes the connections between the notion of vector dissipativity and optimality of the proposed decentralized feedback control law. The proposed control framework is then used to construct decentralized controllers for large-scale nonlinear dynamical systems with robustness guarantees against full modeling uncertainty.

*Keywords:*
control vector Lyapunov function, control Lyapunov function, asymptotic stabilizability, nonlinear dynamical system, feedback control law, gain margin, sector margin, vector dissipativity, optimality, decentralized controller

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