Distributed Neural Fault-Tolerant Supervisory Control Against Backlash-Like Hysteresis

Abstract

Neural networks (NNs) and fuzzy logic systems (FLSs) (Wang 1993, 1994; Shen et al. 2019; Su et al. 2000), as two universal function approximations, are utilized widely in the control design of nonlinear systems. For the approximation theories of NNs and FLSs, it is under the necessary condition that they are able to approximate a continuous unknown function as precisely as possible. The necessary condition is that all the variables in the function must always be within a compact set. In fact, the boundedness of approximation error is guaranteed by the condition. Now, let’s take function f(y) as an example to illustrate clearly it, where y can be a scalar or vector. It can be approximated by NNs or FLSs. Without losing the generality, its estimate is denoted by $\hat{f}(y)$. Obviously, approximation error can be described as $\varepsilon (y)=f(y)-\hat{f}(y)$. There exists a compact set $\varOmega _y$ is a compact set related to y. For the function f(y), only if y or each element of y always belongs to the set $\varOmega _y$, it can be accurately approximated by neural networks or fuzzy logic systems, which implies that $\varepsilon (y)$ is bounded on the set.