Graphon LQR的子空间分解：在谐振子VLSNs中的应用
Subspace Decomposition for Graphon LQR: Applications to VLSNs of Harmonic Oscillators
摘要：
图子控制是在文献[1]-[3]中提出并发展起来的，用于近似解决基于图子极限的线性动力系统的超大规模网络（VLSNs）控制问题。针对一类图子线性二次调节（LQR）问题，提出了一种基于不变子空间分解的求解方法，该问题的局部动力学参数是齐次的，但图子间的耦合可能是异质的.本文中的图子耦合出现在状态、控制和代价中，这些耦合可以用不同的图子来表示。通过寻找一个共同的不变子空间，将原问题分解为一个有限维网络耦合LQR问题和一个解耦的无限维LQR问题。一个集中式的最优解，和节点协作的最优控制解决方案，其中每个代理本地计算的最优解的一部分，建立。将这些解应用于有限网络LQR问题可以通过（i）图子控制方法[3]，或（ii）将有限LQR问题表示为图子LQR问题的特殊情况。这些解决方案的复杂性涉及解决一个ND X ND维Riccati方程和一个N X N Riccati方程，其中N是每个节点代理状态的维度，d是耦合算子的非平凡公共不变子空间的维度，而直接的方法涉及解决一个N X N N维Riccati方程，其中N是网络的大小。对于图子耦合不承认确切的低秩表示的情况下，近似控制开发的基础上低秩近似。最后，一个应用程序的监管谐振子耦合在大型网络的不确定性。
英文摘要：
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.
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