Deployment of the first-order and second-order nonlinear multi agent systems over desired open (and, as a particular case, closed) smooth curves in 2D or 3D space is considered. The considered nonlinearities are globally Lipschitz. We assume that the agents have access to the local information of the desired curve and to their positions with respect to their closest neighbors (as well as to their velocities for the second-order systems), whereas in addition a leader agent is able to measure its absolute position. We assume that a small number of leaders (distributed in the spatial domain) transmit their measurements to other agents through a communication network. We take into account the following network imperfections: variable sampling, transmission delay and quantization. We propose a static output-feedback controller and model the resulting closed-loop system as a disturbed (due to quantization) nonlinear heat equation (for the first-order systems) or damped wave equation (for the second-order systems) with delayed point state measurements, where the state is the relative position of the agents with respect to the desired curve. In order to cope with the open curve we consider Neumann boundary conditions that ensure mobility of the boundary agents. We derive linear matrix inequalities (LMIs) that guarantee the input-to-state stability (ISS) of the system. The advantage of our approach is in the simplicity of the control law and the conditions. Numerical examples illustrate the efficiency of the method.

考虑在一阶或二阶非线性多主体系统在2D或3D空间中的所需开放（以及特定情况下为封闭）平滑曲线上的部署。所考虑的非线性在全球范围内都是Lipschitz。我们假设代理可以访问所需曲线的本地信息以及它们相对于其最近邻居的位置（以及它们对于二阶系统的速度），而另外，领导者能够测量它的绝对位置。我们假设少量的领导者（分布在空间域中）通过通信网络将其测量结果传输给其他代理。我们考虑了以下网络缺陷：可变采样，传输延迟和量化。我们提出了一个静态输出反馈控制器，并将得到的闭环系统建模为具有延迟点的扰动（由于量化）非线性热方程（对于一阶系统）或阻尼波方程（对于二阶系统）状态测量 其中状态是代理相对于所需曲线的相对位置。为了应对开放曲线，我们考虑诺伊曼边界条件，确保边界试剂的流动性。我们得出线性矩阵不等式（LMI），以保证系统的输入到状态稳定性（ISS）。我们方法的优点是控制律和条件简单。数值算例说明了该方法的有效性。