An "impossibility" theorem on a class of high-order discrete-time nonlinear control systems

A high-order discrete-time nonlinear system with unknown parameters and Gaussian noise is studied in this paper, and it is shown that under a simple algebraic condition concerning the nonlinearity of the system, there does not exist any feedback controller that globally stabilizes the uncertain system. This impossibility result attempts to understand the capability and limitation of feedback mechanism of a high-order discrete-time nonlinear system and continues a series of previous research on similar first-order systems. This result shows that for the high-order discrete-time nonlinear system studied here, a polynomial corresponding to the first-order principle part of the system determines the limit of the feedback mechanism.