## 1 Introduction  
Because of cooperative control widespread potential applications in various fields such as synchronized rise of guide way climbing frame (GWCF), formation control of mobile robots, distributed coordination of multiple dynamic systems (also known as multi-agent systems [1; 2; 3; 4; 5; 6]), has achieved rapid development in recent decades. Consensus is one of the most popular topics in this area, which has received significant attention from numerous researchers [7; 8]. These approaches are often classified as leaderless consensus [9; 10; 11] and leader-following consensus solutions (see [12; 13; 14] and the references therein). Moreover, many of the early works were established for systems with first-order dynamics, whereas more results have been reported in recent years, such as [15] for systems with second or higher-order dynamics. However, all the above results assume that the communication among the agents under consideration is not limited [16; 17], and that real-time changes are entered in the context of continuous time. This assumption does not hold in practical systems, and mechanical equipment cannot be adjusted in real time under continuous-time conditions. Therefore, the consensus problem under the limitation of communication bandwidth [18; 19] has become a new hot-spot issue in the field of control. In recent years, most consensus problems have been solved in the background of dynamic discrete time [20; 21]. However, the practical system is rare in the context of dynamic discrete time. It is more in the context of continuous time, so it is very meaningful to solve the consensus problem of bandwidth limitation in a continuous-time background. Therefore, this paper uses the quantization approach to solve the bandwidth limitation or finite capacity problem in continuous-time dynamics. Quantization can be seen as a map from continuous signals to discrete finite sets [22]. Quantizer digitizes incoming information and usually gives piecewise constant output and input. Therefore, only finite bits of channel capacity are required for the transmission of quantized data among agents [23; 24]. To our best knowledge, the current research on the consensus of multi-agent systems under input quantization is not very extensive. Static quantizer include logarithmic quantizer and uniform quantizer, which usually contains a fixed quantization interval and an infinite quantization level. Compared with newly proposed adaptive dynamic quantizer, it is inefficient and unrealistic. When the input of system is large, the use of logarithmic quantizer will lead to large quantization error, which makes the system instability. When the input of system is small, the use of uniform quantizer also will lead to relatively large quantization error. To overcome the shortcomings of two static quantizers, which are static and have large errors under specific conditions, the newly proposed adaptive dynamic quantizer combines the logarithmic quantizer and the uniform quantizer by introducing dynamic gain parameters, and gives the boundary conditions. The error of the newly proposed adaptive dynamic quantizer is less than that of the dynamic uniform quantizer [25] when the input of system is small, and which is equal to that of the dynamic uniform quantizer when the input of system is large. Generally speaking, the quantization error of this quantizer has dynamic quantization interval, limited quantization level and small quantization error compared with the logarithmic quantizer [26], uniform quantizer [27] and dynamic uniform quantizer. To our best knowledge, dynamic uniform quantizer has been applied to leaderless multi-agent systems. For systems with information quantization, input is quantized by a quantizer which results in an inevitable quantization error. Because when an actual value is sent from an agent to its neighbor, the value will be truncated and only a quantified version will be received by the neighbor, which may cause undesirable oscillating behavior in the process of leader-following consensus [28; 29].