A Simple Stability Conjecture and Proofs to Lower Dimensional Phase Space’s Examples for Discrete-Time Systems: Applications to PID control Theory

This paper is concerned with an important conjecture of stability of discrete systems and its exact analysis to lower dimensional case. This is simple, but important to control theory, for example, PID control theory. We prove this by four popular methods and finally we state what is an difficuly in higher dimension case. This problem is very classical and interesting. We state a slightly real conjecture on control theory and solve it under several necessary assumptions in this paper.