Stability analysis of recurrent neural networks with applications
Date
2008
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Abstract
Recurrent neural networks are an important tool in the analysis of data with temporal structure. The ability of recurrent networks to model temporal data and act as dynamic mappings makes them ideal for application to complex control problems. Because such networks are dynamic, however, application in control systems, where stability and safety are important, requires certain guarantees about the behavior of the network and its interaction with the controlled system. Both the performance of the system and its stability must be assured. Since the dynamics of controlled systems are never perfectly known, robust control requires that uncertainty in the knowledge of systems be explicitly addressed. Robust control synthesis approaches produce controllers that are stable in the presence of uncertainty. To guarantee robust stability, these controllers must often sacrifice performance on the actual physical system. The addition of adaptive recurrent neural network components to the controller can alleviate, to some extent, the loss of performance associated with robust design by allowing adaptation to observed system dynamics. The assurance of stability of the adaptive neural control system is prerequisite to the application of such techniques. Work in [49, 2] points toward the use of modern stability analysis and robust control techniques in combination with reinforcement learning algorithms to provide adaptive neural controllers with the necessary guarantees of performance and stability. The algorithms developed in these works have a high computational burden due to the cost of the online stability analysis. Conservatism in the stability analysis of the adaptive neural components has a direct impact on the cost of the proposed system. This is due to an increase in the number of stability analysis computations that must be made. The work in [79, 82] provides more efficient tools for the analysis of time-varying recurrent neural network stability than those applied in [49, 2]. Recent results in the analysis of systems with repeated nonlinearities [19, 52, 17] can reduce the conservatism of the analysis developed in [79] and give an overall improvement in the performance of the on-line stability analysis. In this document, steps toward making the application of robust adaptive neural controllers practical are described. The analysis of recurrent neural network stability in [79] is not exact and reductions in the conservatism and computational cost of the analysis are presented. An algorithm is developed allowing the application of the stability analysis results to online adaptive control systems. The algorithm modifies the recurrent neural network updates with a bias away from the boundary between provably stable parameter settings and possibly unstable settings. This bias is derived from the results of the stability analysis, and its method of computation is applicable to a broad class of adaptive control systems not restricted to recurrent neural networks. The use of this bias term reduces the number of expensive stability analysis computations that must be made and thus reduces the computational complexity of the stable adaptive system. An application of the proposed algorithm to an uncertain, nonlinear, control system is provided and points toward future work on this problem that could further the practical application of robust adaptive neural control.
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Subject
linear matrix inequalities
recurrent neural networks
reinforcement learning
robust control
electrical engineering
computer science