Robust Stability And Input-to-State Stability Singularly Peturbed Systems

In various fields of engineering and science, there are numerous real systems which contain two obviously different dynamic modes:fast state modes and slow state modes. Singularly perturbed systems provide a natural description of these systems. On the other hand, there are many real systems in which the states are changed abruptly at certain moments during the processes. This is the familiar impulsive phenomenon. Besides impulsive effect,a real system is usually affected by perturbation. So, it is needed to consider impulsive (perturbation) effects on the stability of singularly perturbed systems. Otherwise, competitive neural networks include the aspects of long-term and short-term memory. The dynamic of competitive neural networks is characterized by an equation of neural activity as a fast state and an equation of synaptic efficiency as a slow state. The emphasis of this thesis is focused on robust stability of singularly perturbed systems under nonlinear perturbation and ISS of competitive neural networks.The main work is as follows: 1. The robust stability problem for singularly perturbed impulsive systems under nonlinear perturbation is investigated. First, the uncertainties are assumed to be limited by their uppernorm bound. And then, by applying the vector Lyapunov function method and a two-time scale comparison principle,a sufficient condition that ensures robust exponential stability for sufficiently small singular perturbation parameter is derived. Moreover, the stability bound of the singular perturbation parameter can be obtained by solving a set of matrix inequalities. Finally, two numerical examples are provided which substantiates the usefulness and feasibility of the proposed method.2. The problem for input-to-state stability of competitive neural networks is studied. By applying the vector Lyapunov function method and a two-time scale comparison principle, a sufficient condition that ensures inpt-to-state stability for sufficiently small fast time-scale parameter is derived. By carefully estimating the solutions of comparison system, sufficient conditions for the input-to-state stability for all small enough ε>0are derived in terms of linear matrix inequalities (LMIs). Numerical simulation shows that the proposed method can significantly improve the stability upper bound of the fast time-scale parameter.