Prescribed performance synchronization of neural networks with impulsive effects

Abstract

In this paper, the prescribed performance synchronization problem is addressed for a class of neural networks with impulsive effects. According to the prescribed performance control principle and the Lyapunov’s second stability theorem, a preset performance control protocol is designed. For neural networks with impulsive effects, the proposed control scheme can not only guarantee the steady-state performance of synchronization errors, but also ensure the transient performance of the synchronization process. This improves the performance of the neural networks effectively. Finally, a numerical simulation is given to illustrate the effectiveness and feasibility of the proposed control scheme.

Appendix

We assume that the transformation error $\xi_{i} \left( t \right)$ is bounded,so we can obtain

$$ |\xi_{i} \left( t \right)| = \frac{1}{2}|\ln \frac{{\rho_{i} + \frac{{e_{i} \left( t \right)}}{{\eta_{i} \left( t \right)}}}}{{\rho_{i} - \frac{{e_{i} \left( t \right)}}{{\eta_{i} \left( t \right)}}}}| < A $$

$$ \to - 2A < \ln \frac{{\rho_{i} + \frac{{e_{i} \left( t \right)}}{{\eta_{i} \left( t \right)}}}}{{\rho_{i} - \frac{{e_{i} \left( t \right)}}{{\eta_{i} \left( t \right)}}}} < 2A $$

$$ \to e^{ - 2A} < \frac{{\rho_{i} + \frac{{e_{i} \left( t \right)}}{{\eta_{i} \left( t \right)}}}}{{\rho_{i} - \frac{{e_{i} \left( t \right)}}{{\eta_{i} \left( t \right)}}}} < e^{2A} $$

Then, we consider the right-hand side inequality and simplify it, we get

$$ e_{i} \left( t \right) < \frac{{e^{2A} - 1}}{{e^{2A} + 1}}\rho_{i} \eta_{i} \left( t \right) < \rho_{i} \eta_{i} \left( t \right) $$

In the same way, we also have

$$ e_{i} \left( t \right) > \frac{{e^{ - 2A} - 1}}{{e^{ - 2A} + 1}}\rho_{i} \eta_{i} \left( t \right) > - \rho_{i} \eta_{i} \left( t \right) $$

To sum up, we can conclude that as long as $\xi_{i} \left( t \right)$ is bounded, $e_{i} \left( t \right)$ can satisfy the prescribed performance (4).