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In 1966 Parks [22], in a paper of great historical significance in adaptive control, conclusively demonstrated using a specific example that gradient methods for adjusting the parameters in an adaptive system can result in instability. At the same time he also showed that the system could be made globally stable using a design procedure based on Lyapunov’s method. This clear demonstration that gradient-based methods could become unstable tolled the death knell of such systems, and witnessed a shift in the next decade to design based on stability methods described earlier.

The first is their ability to approximate nonlinear maps. The second is the fact that for such networks, different methods of adjusting their parameters have been developed and are generally known. 3. 2 Recurrent Networks In contrast to the feedforward networks considered thus far, which are static maps from one finite-dimensional vector space to another, recurrent networks are dynamic maps that map input time signals into output time signals. This is accomplished by providing them with memory and introducing time delays and feedback.

1 Error Models for Nonlinear Systems ............................ 2 Gradient-Based Methods and Stability ........................ 3 Adjustment of Parameters: Feedforward and Recurrent Networks ................................................................ 1 Back Propagation through Time.................................... 2 Dynamic Back Propagation............................................ 3 Interconnection of LTI Systems and Neural Networks .....................................................

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Lecture notes on maximum principles by Sweers G.


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