ABSTRACT
The concept of vector Lyapunov functions (V LF ) was coincidentally introduced by R. Bellman [13] in the linear systems setting and by V. M. Matrosov [236] in the general nonlinear systems framework. Matrosov continued to develop the VLF concept to large-scale nonlinear systems [237], [238]. It became the basic mathematical tool for studying stability properties of complex (interconnected and large-scale) dynamic systems [131], [237], [243], [298]. The V LF was the mathematical mean to e¤ectively construct a scalar Lyapunov function for the complex dynamic systems and to reduce their stability test to simple algebraic conditions imposed on constant matrices the dimension of which was reduced to the number of subsystems of a high dimensional overall system. The analysis of the application of a scalar Lyapunov function for control
synthesis meets the mathematical problem of how to separate the control from the Lyapunov function gradient and how to accommodate it to the tracking task. In order to overcome this drawback of the scalar Lyapunov function approach it was proposed in [149], [162], [173]-[175] to use the VLF in its real vector form without any need for the scalar Lyapunov function application to the whole system in order to ensure its tracking. We will present it in its simpli…ed form adequate to the need of the tracking control synthesis in the framework of the linear systems.
