## 1 Introduction
Therefore, it is still very challenging to consider the leader-following consensus of multi-agent systems under followers input quantization. In fact, many practical systems that can be transformed to second-order multi-agent systems, such as synchronous motion of GWCF. In this paper, the followers' input are quantized by the newly proposed adaptive dynamic quantizer, which solve the problem leader-following consensus with limited communication bandwidth. The static quantizer in [30] is a special case of the newly proposed dynamic quantizer, the dynamic gain parameter is 1 and applied to the SISO system [31]. At present, the synchronous motion of the second-order system has not been resolved very well. For example, the synchronized rise of GWCF adopts switches for one-to-one control of each node. During the lifting process, there are four observers around. When the GWCF in that direction rises and falls, slowly or quickly, the switches of corresponding nodes are controlled by the indoor personnel through the walkie-talkie. It makes the whole rises process very inefficient and the workload of the staff is huge. To effectively solve this problem and be more suitable for practical working conditions, we adopt the distributed control law. This law cannot only make current node state within a reasonable range according to the state of the around nodes, but also require a small number of calculations by a single node. To apply the distributed control law, we first build the mathematical model by analyzing the force on the practical object, and then calculate it, such as the vertical force of each hanging point in the climbing frame, the horizontal force of a single mobile robot and so on, under the condition that the friction force does not changed by the time. A mathematical model of a second-order integrator systems is established for the longitudinal force of the single hanging point in the GWCF. The second-order integrator system of all hanging points constitutes a network structure, which is regarded as a second-order multi-agent systems. Supposing one of the hanging point with constant speed is the leader, the other hanging points are the followers. The pull of the motor at a single suspension point is the input to a single second-order integrator system. Under the action of a novel adaptive dynamic quantizer, the input of a single hanging point can be adjusted intermittently in the initial stage of the motion process by means of distributed control law, and the synchronous motion can be achieved in a short time. In this paper, we first introduce self-proposed adaptive dynamic quantizer which has dynamic quantization interval, limited quantization level and smaller quantization error compared with the logarithmic quantizer, uniform quantizer and dynamic uniform quantizer. On this basis, the follower input is quantized by a dynamic quantizer in a second-order integrator multi-agent systems, and the leader-following consensus problem with limited communication bandwidth is solved by a distributed control law. Finally, the distributed control law is applied to practical systems, such as synchronous motion of GWCF. According to the simulation example, the better results are achieved. The main contributions of this paper include three aspects. (1) First, a novel adaptive dynamic quantizer is proposed. Compared with the previous quantizer, this quantizer can effectively reduce quantization error, has dynamic quantization interval and limited quantization level. (2) Secondly, most of researcher solve the multi-agent system consensus problem after the input is quantized, using leaderless method and the quantizer adopts the static quantizer. In this paper, using second-order systems and having a leader, the problem of leader-following consensus of multi-agent systems is realized by using the adaptive dynamic quantizer proposed by ourselves. The second-order system and the self-proposed adaptive dynamic quantizer are more realistic.