On Finite-time Stable Theory And Adaptive Controller Design

Automatic control plays an important role in many realistic problems in industry and daily life. The goal of automatic control is to regulate the system controller to achieve desired performance. Finite time stable problem and adaptive control problem are two of the important problems in study on control theory. The studies on these two problems have received increasing attention in recent years. And a lot of literature, including researches on finite time stable for impulsive systems and adaptive control for systems with quantized input, are published.In this paper, we study the problem of designing adaptive controller and finite time controller for nonlinear uncertain systems. Firstly, we develop an adaptive controller for a class of nonlinear uncertain systems with time varying by using new basis functions based on Fourier expansion. Then we develop two sufficient and necessary conditions for finite time stable of discrete time systems. And we proof that they are equivalent. The new conditions help us interpret the finite time stability property in an energetic function way. In addition, we develop a sufficient condition for finite time stable of continuous time systems. And we proof that it is equivalent to the condition used in literature and analyze the relations of all these four conditions. Then basing on the conditions developed here, we design adaptive finite time stability controllers for nonlinear uncertain systems with both quantized input and general input. By analyzing several Lyapunov functions, we show the effectiveness of the controllers presented. Finally, we show some illustrative numerical examples.