The principle of contraction mapping in nonlinear and adaptive control systems

The principle of contraction mapping is well established in mathematics. It has been used in control theory and its application to control systems considered as recursive processes has been investigated. The principle is valid for nonlinear systems and can be used to establish stability in nonlinear control. Also, because the existence of a solution can be obtained by a recursive algorithm, the principle can be used in adaptive or learning systems to generate a desired output function recursively. In this paper, after reviewing the background and recent results on the application of contraction mapping to control systems, the principle is applied to aircraft terrain following. Next, a scheme is presented for a learning model, involving the control of a system with undetermined parameters. In conclusion, the connection between a discrete iterative process and a continuous operation is discussed.