## 1 Introduction
NCSs, coupled with novel analytical techniques and controller design schemes, have been used to solve problems in dynamic control systems and networks [24; 25; 26]. The combination of networked control and iterative learning control (NILC) has received a great deal of attention, owing to its advantages including simple installation, decreased system wiring, simple system maintenance and diagnosis, among others [27]. However, NILC's potential has been curtailed by the physical characteristics of wired or wireless network communication devices, compounded by several factors that may degrade the tracking performance of ILC systems. For example, the quantized error, temporal oscillation of the network, time delays, and packet dropout can all contribute to such deterioration. Numerous studies have been conducted to overcome the limitations of NILC. The common method used to deal with the problem of communication delays is to replace the delayed data with the most recently captured data when the delay is smaller than the sampling step length [28; 29]. The solution to the packet dropout issue is to replace the dropped data with the latest captured data [30; 31]. In order to alleviate the transmission burden, the quantizer converts the continual signals into the piecewise constant signals with a finite set of values in control system. For instance, quantization as a form of data compression that reduces the resolution of the signal by rounding its values to the nearest allowable value. However, quantization may result in chaotic vibrations [32; 33]. The stabilities of ILC for linear and nonlinear with data quantization have been studied by [PERSON] and [PERSON] et al. [34]. The convergence conditions related to the quantized density are investigated by using a sector-bound approach for the logarithmic quantizer and the contraction mapping approach of the ILC, but the tracking error of the system with the data quantized signal still converges to a small bound. Later on, [PERSON] and [PERSON] improved such theoretical results by proposing an ILC law using the quantized error information for linear and nonlinear systems; the tracking error of this scheme is zero-error convergence [35]. Two new multi-lagged input-based quantized iterative learning control methods are proposed using the output quantization and the error quantization in the paper [36]. Later, by using the lifting representation, some schemes for the convergence conditions are given for the different data quantization in papers [37; 38; 39], but when the convergence conditions are guaranteed, the tracking error of ILC with system output quantized signal and control input quantized signal converges to a bound. Collectively, these studies underscore the significance of employing appropriate algorithms in digitally controlled systems with quantization. In the existing analyses of convergence for the discrete-time systems with data quantization, if the static logarithmic quantizer is adopted such that the quantized density is determined, the convergence condition is provided to guide the selection of learning gain. However, the learning gain for the iterative learning control scheme with data quantization is chosen using the \"trial and error method\" according to experience in the convergence conditions. The learning mechanism is passive and blind, and the tracking error of the system with data quantization cannot converge to zero. Some cases have shown that the iterative learning control law exhibits asymptotic convergence, and the convergence rate is slow. Consequently, the selection of a suitable learning gain for accurate tracking characteristics is an important issue in the ILC system with data quantization. We are motivated by the preceding discussions and inspired by the concept of the iteration time-variable derivative learning-gain vector from the minimization problem in our prior work [23].