Improved Zhang neural network model and its solution of time-varying generalized linear matrix equations

A new variant of the Zhang neural network for solving online time-varying linear inequalities

“…It is worth pointing out that, when the initial error E(0) > 0 ∈ R m×n , the proposed NVZNN design formula (2.3) reduces toĖ(t) = −gF(E(t)), which is the so-called ZNN design formula for solving online time-varying matrix/vector/scalar equations (Zhang et al 2002;Zhang & Ge 2005;Zhang & Li 2009;Li & Zhang 2010;Guo et al 2011). In this situation, different choices of activation function arrays may lead to different convergence performances of the error function E(t).…”

“…Even if the linear activation functions are used, the nonlinear phenomena may appear in the hardware implementations. Thus, based on the previous work (Zhang et al 2002;Zhang & Ge 2005;Zhang & Li 2009;Li & Zhang 2010;Guo et al 2011), the (nonlinear) activation function array F(·) : R m×n → R m×n is introduced and used for the construction of the NVZNN design formula. The generalized NVZNN design formula is then proposed aṡ…”

“…the so-called ZNN design formula) is based on a matrix/vector-valued lower-unbounded error function (instead of the conventional scalar-valued lowerbounded energy function that is usually involved in the gradient-descent method and Hopfield neural networks), which aims at making every element of the error function globally and exponentially converge to zero. The resultant ZNN models depicted in implicit dynamics exploit methodologically the time-derivative information of time-varying coefficient(s), and thus solve the time-varying problems effectively and efficiently (Zhang et al 2002;Zhang & Ge 2005;Zhang & Li 2009;Li & Zhang 2010;Guo et al 2011). Note that the abovementioned ZNN design formula is presented for solving time-varying equations, but it cannot be used directly to solve time-varying LIs by defining the corresponding simple matrix/vector-valued error functions.…”

“…Because of the parallel distributed nature and the suitability for hardware realization, neural networks have been proposed, developed and studied for scientific research and engineering applications (Ramasamy et al 1995;Cichocki & Bargiela 1997;Labonte 1997;Xia et al 1999;Lin et al 2000;Liao et al 2002;Zhang et al 2002;Novo et al 2005;Zhang & Ge 2005;Chetwynd et al 2006;Yu et al 2007;Liu 2009;Zhang & Li 2009;Li & Zhang 2010;Guo et al 2011;Kaslik & Sivasundaram 2011), drawing extensive interest and investigation by researchers. Recently, owing to in-depth research on neural networks, the artificial neural-dynamic approach based on recurrent neural networks has been viewed as a powerful alternative to online solution of LMIs as well as LMVIs.…”

“…Since March 2001, a special class of recurrent neural networks termed the Zhang neural network (ZNN) has been proposed by Zhang and co-workers for solving time-varying problems (Zhang et al 2002;Zhang & Ge 2005;Zhang & Li 2009;Li & Zhang 2010;Guo et al 2011), such as time-varying Sylvester equation solving and time-varying matrix inversion. The design of such a ZNN (i.e.…”