A Novel Proportional-Integral-Parameter Zeroing Neural Network and Its Application to the Quaternion-Valued Time-Varying Linear Matrix Inequality

Zeroing neural network (ZNN) with variable convergence parameter has been a research hotspot recently, and it can be divided into two categories: the varying-parameter ZNN (VP-ZNN) model, and the fuzzy-parameter ZNN (FP-ZNN) model. Both of the two models have their own advantages and disadvantages. VP-ZNN models are usually efficient but not intelligent, while FP-ZNN models are usually intelligent but not efficient. Inspired by the classic proportional–integral–derivative (PID) control technology, we proposed a novel proportionalintegral-parameter ZNN (PIP-ZNN) model, which is both efficient and intelligent, to solve the quaternion-valued time-varying linear matrix inequalities (QVTV-LMI) problem. Integrated with adaptive convergence parameters (ACP), the PIP-ZNN model dynamically adjusts its convergence in response to error changes and then achieves optimized performance. Compared with FPZNN models which are based on fuzzy logic systems, the PIDbased PIP-ZNN models are simpler and more efficient. With the incorporation of a robust activation function (RAF), the PIPZNN model demonstrates fixed-time stability and robustness in the presence of both attenuated and constant disturbances. Theoretical analyses in this paper establish the fixed-time stability and robustness of the PIP-ZNN model, including an estimation of the upper bound of the settling-time function. Numerical experiments here validate these advanced features further, emphasizing the efficacy and excellent performance of the proposed PIP-ZNN model.