On Stability of Nonlinear Observers Based on Neural Networks

In this paper, the stability problem of neural network based observers/identifiers for nonlinear systems is revisited when nonlinear-in-parameter neural networks (NLPNN) are employed. The proposed approach is based on decomposing the neural network into two subsystems. The first subsystem (Subsystem 1) consists of the estimation error and output-layer weight error and the second subsystem (Subsystem 2) consists of the hidden-layer weight error. The key to this decomposition is that the hidden-layer weights appear in Subsystem 1, only as an argument of a sigmoidal function and its derivative which are both known to be bounded. This allows us to regard the Subsystem 1 as a linear-in-parameter neural network (LPNN) whose stability proof is more straightforward. Having shown the stability of the first subsystem, the stability of the second subsystem is also shown subsequently without the requirement of having the limiting assumptions of previous work. The bound on estimation error can be made arbitrarily small by proper selection of design parameters. The estimation scheme is then employed to estimate the state of flexible joint manipulators.