Li-Function Activated Zhang Neural Network for Online Solution of Time-Varying Linear Matrix Inequality

Abstract

In the previous work, a typical recurrent neural network termed Zhang neural network (ZNN) has been developed for various time-varying problems solving. Based on the previous work, by exploiting a special activation function (i.e., Li activation function), the resultant ZNN model is presented and investigated in this paper for online solution of time-varying linear matrix inequality (TVLMI). For such a Li-function activated ZNN (LFAZNN) model, theoretical results are provided to show its excellent computational performance on solving the TVLMI. That is, the presented LFAZNN model has the property of finite-time convergence. Comparative simulation results with two illustrative examples further substantiate the efficacy of the presented LFAZNN model for TVLMI solving.

Appendix: Procedure of ZNN Design

In this appendix, the general procedure of the ZNN design [13, 15,16,17,18,19] for time-varying problems solving is presented.

Specifically, as to a time-varying problem $F(t)=0\in R^{m\times n}$ to be solved, the following matrix/vector-valued error function $E(t)\in R^{m\times n}$ is firstly defined:

$$\begin{aligned} E(t):=F(t). \end{aligned}$$

Next, the time derivative ${\dot{E}}(t)$ of E(t) is selected such that E(t) can globally and exponentially converge to zero. More specifically, ${\dot{E}}(t)$ can be selected via the ZNN design formula [13, 15,16,17,18,19] as follows:

$$\begin{aligned} {\dot{E}}(t)=-\gamma {\mathcal {F}}(E(t)), \end{aligned}$$

(11)

where $\gamma >0\in R$ denotes the parameter that is used to scale the convergence rate of the solution, and ${\mathcal {F}}(\cdot ): R^{m\times n}\rightarrow R^{m\times n}$ denotes the monotonically-increasing odd activation-function array. Finally, by expanding the ZNN design formula (11), the differential equation of a ZNN model is thus established for solving a specifical time-varying problem.