论文标题
在具有空间指数衰减结构的网络LQR的最佳控制上
On the Optimal Control of Network LQR with Spatially-Exponential Decaying Structure
论文作者
论文摘要
本文研究网络矩阵的网络LQR问题是网络节点之间的空间指数衰减（SED）。主要目的是研究最佳控制器是否也享有SED结构，这是确保对网络分散控制的最佳性能的吸引人的特性。我们首先研究开放环的稳定系统，并表明最佳的LQR状态反馈增益$ k $在此设置中为`quasi'-sed，即$ \ | [k] _ {ij {ij} \ | \ sim o \ left（e^{ - \ frac {c} {\ mathrm {poly} \ ln（n）}}} \ mathrm {dist}（i，j）\ right）$。衰减率$ c $取决于系统矩阵的衰减率和规范和开环指数稳定性常数。然后，在SED稳定性假设下，结果进一步推广到不稳定的系统。在$ k $上的“准SED结果）的基础上，我们对$κ$截断的本地控制器的性能进行了上限，这表明分布式控制器可以实现SED系统的近乎最佳性能。我们通过研究另一种类型的控制器，干扰响应控制的结构来开发这些结果，该控制器已在最近的在线控制文献中进行了研究和使用。因此，作为一个结果，我们还证明了最佳干扰响应控制的“准SED”属性，该响应控制是根据自己的优点的贡献。
This paper studies network LQR problems with system matrices being spatially-exponential decaying (SED) between nodes in the network. The major objective is to study whether the optimal controller also enjoys a SED structure, which is an appealing property for ensuring the optimality of decentralized control over the network. We start with studying the open-loop asymptotically stable system and show that the optimal LQR state feedback gain $K$ is `quasi'-SED in this setting, i.e. $\|[K]_{ij}\|\sim O\left(e^{-\frac{c}{\mathrm{poly}\ln(N)}}\mathrm{dist}(i,j)\right)$. The decaying rate $c$ depends on the decaying rate and norms of system matrices and the open-loop exponential stability constants. Then the result is further generalized to unstable systems under a SED stabilizability assumption. Building upon the `quasi'-SED result on $K$, we give an upper-bound on the performance of $κ$-truncated local controllers, suggesting that distributed controllers can achieve near-optimal performance for SED systems. We develop these results via studying the structure of another type of controller, disturbance response control, which has been studied and used in recent online control literature; thus as a side result, we also prove the `quasi'-SED property of the optimal disturbance response control, which serves as a contribution on its own merit.
