Extended Lyapunov Stability Theorem and Its Applications in Control System with Constrained Input

The Lyapunov stability theorem has been proposed for more than 100 years, and it is still one of the most important theories in control science and other fields. In this paper, a new stability theorem (Extended Lyapunov stability theorem) is proposed and proved to be different from Lyapunov stability theorem. The Lyapunov stability theorem demands that the time derivative of Lyapunov function is negative. But according to Extended Lyapunov stability theorem, the system can still keep stable when the time derivative of Lyapunov function is positive in some horizon and even in infinite horizon . So the conditions of Extended Lyapunov stability theorem is widely relaxed compared with the Lyapunov stability theorem. Inputs of actual systems are always limited by energy, under this background, a control law is designed to make system stable according to Extended Lyapunov stability theorem. And it can be proved that no Lyapunov function can be found to make the system stable. So with the help of Extended Lyapunov stability theorem better results can be obtained than those by Lyapunov stability theorem. At last, the numerical simulation result shows that Extended Lyapunov stability theorem is greatly different from Lyapunov stability theorem, such as the time derivative of energy function can be positive in infinite horizon. So it can be concluded that Extended Lyapunov stability theorem contains Lyapunov stability theorem and also it can be used more widely in many fields.