# Lyapunov–Krasovskii stable T2FNN controller for a class of nonlinear time-delay systems

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## Abstract

In this paper, a type-2 fuzzy neural network (T2FNN) controller has been designed for a class of nonlinear time-delay systems using the feedback error learning (FEL) approach. In the FEL strategy, the T2FNN controller is in the feedforward path to overcome the nonlinearity and time delay and a classical controller is in the feedback path to guarantee the stability of the controlled system. Using the Lyapunov–Krasovskii stability theorem, the adaptation rules for training of T2FNN controller have been achieved in a way that, in the presence of the unknown disturbance and time-varying delay, the tacking error becomes zero. In the proposed stability criteria and adaptation laws, since just the training error is utilized, i.e., the mathematical model of the system or its parameters is not needed, the overall training and control algorithm is computationally simple. In the present study, the effect of delay has been considered in tuning the T2FNN parameters and, therefore, the performance of the proposed controller has been improved. The proposed strategy has been applied to systems with time-varying input delay and measurement noise and compared with indirect type-1 fuzzy sliding controller. The effectiveness of the proposed controller is shown by some simulation results.

## Keywords

Nonlinear time-delay system Type-2 fuzzy neural network controller Lyapunov–Krasovskii functional Measurement noise and stability## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of this paper.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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