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The paper describes the concepts and background theory for the analysis of a neural-like network for learning and replication of periodic signals containing a finite number of distinct frequency components. The approach is based on the combination of ideas from dynamic neural networks and systems and control theory where concepts of dynamics, adaptive control and tracking of specified time signals are fundamental. The proposed procedure is a two stage process consisting of a learning phase when the network is driven by the required signal followed by a replication phase where the network operates in an autonomous feedback mode whilst continuing to generate the required signal to a desired acccuracy for a specified time. The analysis draws on currently available control theory and, in particular, on concepts from model reference adaptive control.

The paper describes the concepts and background theory of the analysis of a neural-like network for the learning and replication of periodic signals containing a finite number of distinct frequency components. The approach is based on a two stage process consisting of a learning phase when the network is driven by the required signal followed by a replication phase where the network operates in an autonomous feedback mode whilst continuing to generate the required signal to a desired accuracy for a specified time. The analysis focusses on stability properties of a model reference adaptive control based learning scheme via the averaging method. The averaging analysis provides fast adaptive algorithms with proven convergence properties.

We present the concept of a universal adaptive tracking controller for classes of linear systems. For the class of scalar minimum phase systems of relative degree one, adaptive tracking is shown for arbitrary finite dimensional reference signals. The controller requires no identificaiton of the system parameters. Robustness properties are explored.

We consider universal adaptive stabilization and tracking controllers for classes of linear systems. Under the technical assumption of linear scaling invariance necessary and sufficient conditions for adaptive stabilization are derived. For scalar systems sufficient conditions for adaptive tracking of finite dimensional reference signals are explored.

A multiparameter, polynomial feedback strategy is introduced to solve the universal adapative tracking problem for a class of multivariable minimum phase system and reference signals generated by a known linear time-invariant differential equation. For 2-input, 2-output, minimum phase systems (A,B,C) with det(CB)0, a different polynomial tracking controller is given which does not invoke a spectrum unmixing set.

Stability and Robustness Properties of Universal Adaptive Controllers for First Order Linear Systems
(1987)

The question: What is an adaptive controller? is as old as the word adaptive control itself. In this paper we will adopt a pragmatic viewpoint which identifies adaptive controllers with nonlinear feedback controllers, designed for classes (families) of linear systems. In contrast to classical linear feedback controllers which are designed for individual systems, these non-linear controllers are required to achieve a specific design objective (such as e.g. stability, tracking or decoupling) for a whole prescribed family of linear systems.

To a network N(q) with determinant D(s;q) depending on a parameter vector q Î Rr via identification of some of its vertices, a network N^ (q) is assigned. The paper deals with procedures to find N^ (q), such that its determinant D^ (s;q) admits a factorization in the determinants of appropriate subnetworks, and with the estimation of the deviation of the zeros of D^ from the zeros of D. To solve the estimation problem state space methods are applied.