Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview

on identification error. The significant improvement in response in the latter case is evident.


7. SUMMARY

In the past decade investigators from across the scientific spectrum, including neurobiologists, psychologists, computer scientists, and engineers, have become interested in the field of neural networks. From a systems theoretic point of view, artificial neural networks represent tractable parameterized families of mappings. As such, they have found extensive application in pattern recognition problems where the decision surfaces are nonlinear. With the introduction of dynamics and feedback, the scope of such networks for the representation of systems has increased significantly. In this chapter, the use of artificial neural networks as identifiers and controllers in dynamical systems is investigated in detail.

After introducing many well-known concepts in control theory, the chapter first addressed the question of identification of nonlinear dynamical systems. Equivalent state vector and input-output representations were suggested. It was shown, using simulation studies, that from a practical stand-point, input-output models are preferable. Following this, the problem of adaptive control of nonlinear systems was explored. Once again, it was shown that, using currently available gradient approaches, neural network based nonlinear controllers result in significantly better response than linear controllers.

Gradient methods, when used for the adjustment of identifier and controller parameters, result in improved performance. They do not assure stability, however, when the parameters are adjusted online. To assure global stability, the unknown plant must be restricted to specific classes of nonlinear systems. Such systems are described by nonlinear difference equations that are linear in the unknown parameters, as well as the control inputs. Current investigations in this area seek to increase the range of applicability of such models while retaining their practical advantages.

The environments encountered by a system can be broadly classified under two categories: anticipated and unanticipated. These include external disturbances, parameter variations, sensor and actuator failures, and so forth. In Sec. 5.2.3, it was shown that the methods used for unknown parameters can be extended to the case when external disturbances are present. Hence, control in the different environments mentioned involves identification of plant models and determination of corresponding controllers. Assuming that a number of environments have been encountered before and that identification models and controllers have been determined, they can be stored for use at a later time. The controller will then be capable of responding rapidly and accurately to such (anticipated) environments by switching and tuning. For environments that are distinctly different from those that have been identified and stored (i.e., unanticipated), the con-

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