Liu Jason J R, Kwok Ka-Wai, Cui Yukang, Shen Jun, Lam James
IEEE Trans Neural Netw Learn Syst. 2022 Sep;33(9):4575-4583. doi: 10.1109/TNNLS.2021.3058184. Epub 2022 Aug 31.
This article addresses the distributed consensus problem for identical continuous-time positive linear systems with state-feedback control. Existing works of such a problem mainly focus on the case where the networked communication topologies are of either undirected and incomplete graphs or strongly connected directed graphs. On the other hand, in this work, the communication topologies of the networked system are described by directed graphs each containing a spanning tree, which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Specifically, the problem involves complex eigenvalues, the Hurwitzness of complex matrices, and positivity constraints, which make analysis difficult in the Laplacian matrix. First, a necessary and sufficient condition for the consensus analysis of directed networked systems with positivity constraints is given, by using positive systems theory and graph theory. Unlike the general Riccati design methods that involve solving an algebraic Riccati equation (ARE), a condition represented by an algebraic Riccati inequality (ARI) is obtained for the existence of a solution. Subsequently, an equivalent condition, which corresponds to the consensus design condition, is derived, and a semidefinite programming algorithm is developed. It is shown that, when a protocol is solved by the algorithm for the networked system on a specific communication graph, there exists a set of graphs such that the positive consensus problem can be solved as well.
本文研究了具有状态反馈控制的相同连续时间正线性系统的分布式一致性问题。关于此类问题的现有工作主要集中在网络通信拓扑为无向且不完全图或强连通有向图的情况。另一方面，在本工作中，网络系统的通信拓扑由每个都包含一棵生成树的有向图来描述，由于拉普拉斯矩阵的特征值与控制器增益之间的相互作用，这是一种更一般且新颖的情形。具体而言，该问题涉及复特征值、复矩阵的赫尔维茨性以及正性约束，这使得在拉普拉斯矩阵中进行分析变得困难。首先，利用正系统理论和图论，给出了具有正性约束的有向网络系统一致性分析的充要条件。与涉及求解代数黎卡提方程（ARE）的一般黎卡提设计方法不同，得到了一个由代数黎卡提不等式（ARI）表示的解存在条件。随后，推导出了与一致性设计条件相对应的等价条件，并开发了一种半定规划算法。结果表明，当通过该算法在特定通信图上求解网络系统的协议时，存在一组图使得正一致性问题也能得到解决。
