Proportional–Integral State Estimator for Quaternion-Valued Neural Networks With Time-Varying Delays

This brief investigates the problem of state estimation of quaternion-valued neural networks (QVNNs) with time-varying delays. First, by extending the Jensen inequality to quaternion domain, an extended Jensen inequality with quaternion term is derived. Second, a class of proportional–integral state estimator (PISE) with exponential decay term is proposed. Then, by constructing a suitable Lyapunov–Krasovskii functional (LKF), some sufficient conditions are obtained to ensure the existence of the designed PISE and the gain matrices of the designed PISE can be directly computed. Simulations are given to illustrate the advantage of the proposed method.