Lyapunov stability analysis in the training of type 2 Neuro-Fuzzy Identifier with a swarm-based hybrid intelligent algorithm

Training stability of a model in an identification process has been one of the primitive requirements in recent control researches. This paper aims at analyzing the training stability of the interval type 2 adaptive Neuro-Fuzzy inference system (IT2ANFIS) with a swarm-based hybrid algorithm. The antecedent and the consequent parts of the model are trained by particle swarm optimization (PSO) and Kalman filter (KF) algorithms, respectively (PSO+KF). The Lyapunov stability theorem with a newly found Lyapunov function is employed to assess the stability conditions. These conditions led to adaptive stabilizing boundaries in the adjustable parameters of the algorithms (APAs), such as the covariance matrix in KF, inertia factor, and maximum gain in PSO. The selection of APAs within these boundaries guaranteed the stability of the training process. The analytical approach of this study resulted in finding new and broader stabilizing boundaries for the APAs. Implementation of the theorem to the training and predicting the future values of the Mackey-Glass chaotic time series and a stochastic non‐linear system revealed the superiority of the theorem in terms of their root mean square errors (RMSEs), simulation times, and their entrapment in the local minimums.