论文标题
Graphon LQR的子空间分解：应用于谐波振荡器VLSN的应用
Subspace Decomposition for Graphon LQR: Applications to VLSNs of Harmonic Oscillators
论文作者
论文摘要
在[1] - [3]中提出并开发了Graphon Control，以求助于基于Graphon限制的线性动力学系统的非常大规模网络（VLSN）的控制问题。本文提供了一种基于不变子空间分解的解决方案方法，用于一类Graphon线性二次调节（LQR）问题，其中局部动力学共享均匀参数，但是Graphon耦合可能是耦合剂之间的异质性。本文中的图形耦合出现在状态，控制和成本中，这些耦合可以由不同的图形表示。通过探索耦合的常见不变子空间，原始问题被分解为有限维度的网络耦合LQR问题和脱钩的无限尺寸LQR问题。建立了一个集中式最佳解决方案和节点协作最佳控制解决方案，其中每个代理在本地计算其最佳解决方案的一部分。这些解决方案在有限网络LQR问题上的应用可能是通过（i）Graphon Control方法[3]或（ii）将有限LQR问题表示为Graphon LQR问题的特殊情况。 The complexity of these solutions involves solving one nd X nd dimensional Riccati equation and one n X n Riccati equation, where n is the dimension of each nodal agent state and d is the dimension of the nontrivial common invariant subspace of the coupling operators, whereas a direct approach involves solving an nN X nN dimensional Riccati equation, where N is the size of the network.对于Graphon耦合不接受确切的低级别表示的情况，基于低级别近似值开发了近似控制。最后，证明了对与具有不确定性的大型网络相连的谐波振荡器调节的应用。
Graphon control has been proposed and developed in [1]-[3] to approximately solve control problems for very large-scale networks (VLSNs) of linear dynamical systems based on graphon limits. This paper provides a solution method based on invariant subspace decompositions for a class of graphon linear quadratic regulation (LQR) problems where the local dynamics share homogeneous parameters but the graphon couplings may be heterogeneous among the coupled agents. Graphon couplings in this paper appear in states, controls and costs, and these couplings may be represented by different graphons. By exploring a common invariant subspace of the couplings, the original problem is decomposed into a network coupled LQR problem of finite dimension and a decoupled infinite dimensional LQR problem. A centralized optimal solution, and a nodal collaborative optimal control solution where each agent computes its part of the optimal solution locally, are established. The application of these solutions to finite network LQR problems may be via (i) the graphon control methodology [3], or (ii) the representation of finite LQR problems as special cases of graphon LQR problems. The complexity of these solutions involves solving one nd X nd dimensional Riccati equation and one n X n Riccati equation, where n is the dimension of each nodal agent state and d is the dimension of the nontrivial common invariant subspace of the coupling operators, whereas a direct approach involves solving an nN X nN dimensional Riccati equation, where N is the size of the network. For situations where the graphon couplings do not admit exact low-rank representations, approximate control is developed based on low-rank approximations. Finally, an application to the regulation of harmonic oscillators coupled over large networks with uncertainties is demonstrated.
