Many problems in adaptive control can be divided into two parts; the first part is the control of plant dynamics, and the second is the control of plant disturbance. Very often, a single system is utilized to achieve both of these control objectives. The approach of this paper treats each problem separately. Control of plant dynamics can be achieved by preceding the plant with an adaptive controller whose transfer function is the inverse of that of the plant. Control plant disturbance can be achieved by an adaptive feedback process that minimizes plant output disturbance without altering plant dynamics. The adaptive controller is implemented using adaptive filters.

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The contraction theorem has many fields of application, including linear algebraic equations, differential and integral equations, control systems theory, optimization, etc. The paper aims at showing how contraction mapping can be applied to the computation and the training of adaptive structures with algebraic loops. These structures are used for the approximation of unknown functional relations (mappings) represented by training sets. The technique is extended to multilayer neural networks with algebraic loops. Application of a two-layer neural network to breast cancer diagnosis is described.

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