Optimal Wireless Control Over State-dependent Fading Channels for Heterogeneous Industrial Internet of Things Systems
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摘要: 随着工业4.0的发展, 移动智能体系统 (Mobile agent system, MAS) 与多回路无线控制系统 (Wireless control system, WCS) 被部署到工厂中, 构成异构工业物联网系统, 协作执行智能制造任务. 在协作过程中, MAS与WCS紧密耦合, 导致状态相关衰落, 两者性能相互制约. 为解决这一问题, 研究异构工业物联网系统的最优控制问题, 满足WCS控制性能约束与MAS安全生产约束的同时, 最小化系统平均通信成本. 首先, 利用有限域系统描述MAS在不同阴影衰落程度工作区间的转移, 刻画MAS与WCS耦合下的状态相关衰落信道模型. 基于此, 利用矩阵半张量积理论, 通过构建受限跟随者状态转移图, 建立最优控制问题可行性图判据, 给出关于受限集合镇定的充分必要条件. 其次, 基于加权跟随者状态转移图的最小平均环理论, 建立领航-跟随MAS最优控制序列的构造算法, 并证明其最优性. 最后, 通过仿真验证算法的有效性.Abstract: With the development of Industry 4.0, the mobile agent system (MAS) and wireless control system (WCS) are deployed to collaboratively perform smart manufacturing tasks, forming a heterogeneous Industrial Internet of Things system. In this process, the WCS is coupled with the MAS, leading to state-dependent fading and restricting each other in performance. To address this issue, this paper focuses on the optimal control of the heterogeneous system to ensure the control performance of WCS and the safety production of MAS, while minimizing the average communication cost. Firstly, the transition of MAS between cells with different levels of shadow effects is modeled by the finite-field system, and the state-dependent fading channel is characterized under MAS and WCS coupling. Using semi-tensor product of matrices, a graphical criterion is presented for the feasibility by constructing the constrained follower state transition graph, and criteria in terms of constrained set stabilization are established. Secondly, by minimum-mean cycles for weighted follower state transition graph, an algorithm is proposed to construct optimal control sequences for the leader-follower MAS, and the optimality is proved. Finally, an illustrative example is provided to demonstrate effectiveness of the algorithm.
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图 1 多回路WCS与MAS在工作区协作执行生产任务构成的异构IIoT系统
Fig. 1 A heterogeneous IIoT system where a multiloop WCS and an MAS coordinate with each other to jointly perform overall tasks in a workspace
图 4 控制策略$\mu_m$和控制序列${{\boldsymbol{u}}}^\ast$下跟随者智能体状态轨迹的对比结果
Fig. 4 Comparison results of followers' state trajectories under control policy $\mu_m$and input sequence ${{\boldsymbol{u}}}^\ast$
图 5 控制序列${{\boldsymbol{u}}}^\ast$下WCS状态经验平均值的演化
Fig. 5 The evolution of the empirical averages of the states for WCS under control sequence ${{\boldsymbol{u}}}^\ast$
图 6 状态反馈控制律$\pi_i$, $i=1,\;2,\;3,\;4$和控制序列${{\boldsymbol{u}}}^\ast$下平均成本的对比结果
Fig. 6 Comparison results of average costs under state feedback laws $\pi_i$, $i=1,\;2,\;3,\;4$and input sequence ${{\boldsymbol{u}}}^\ast$
表 1 主要符号说明
Table 1 Notations
符号 含义 $ {\bf{N}} $ 自然数集合 $ \mathcal{D}_n $ 逻辑域$ \{0,\;1,\;\cdots,\;n-1\} $ $ \mathcal{D}_n^m $ 笛卡尔乘积$ \underbrace{\mathcal{D}_n\times\cdots\times\mathcal{D}_n}_m $ $ I_n $ $ n $维单位阵 $ \delta_n^i $ 单位阵$ I_n $的第$ i $列 $ [A]_{:,\;j} $ 矩阵$ A $的第$ j $列 $ [A]_{i,\;j} $ 矩阵$ A $的$ (i,\;j) $元 $ \delta_m[i_1\;i_2\;\cdots\;i_n] $ 逻辑矩阵$ A $, $ [A]_{:,\;j}=\delta_m^{i_j} $ $ {\mathcal{L}}^{m\times n} $ $ m\times n $逻辑矩阵集合 $ {{\bf{R}}}^{m\times n} $ $ m\times n $实矩阵集合 $ Col(A) $ 矩阵$ A\in{{\bf{R}}}^{m\times n} $所有列构成的集合 $ \ltimes $ 矩阵的半张量积 $ \otimes $ 克罗内克积 $ \mathcal{A}\setminus\mathcal{B} $ 集合$ \{x\in\mathcal{A}: x\notin\mathcal{B}\} $ $ |A| $ 集合$ A $的基数 
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表 2 状态相关衰落信道参数
Table 2 State-dependent fading channel parameters
$ (i,\;b,\;a) $ $ (1,\;1,\;0) $ $ (1,\;1,\;1) $ $ (1,\;2,\;2) $ $ (1,\;2,\;3) $ $ (2,\;1,\;0) $ $ (2,\;1,\;1) $ $ (2,\;2,\;2) $ $ (2,\;2,\;3) $ $ \mathcal{Z}_1 $ 0.60 0.20 0.60 0.10 0.63 0.20 0.63 0.50 0.44 0.20 0.44 0.10 0.46 0.00 0.46 0.70 $ \mathcal{Z}_2 $ 0.17 0.30 0.17 0.10 0.18 0.20 0.18 0.40 0.33 0.20 0.33 0.10 0.35 0.50 0.35 0.20 $ \mathcal{Z}_3 $ 0.58 0.10 0.58 0.10 0.62 0.30 0.62 0.50 0.42 0.10 0.42 0.10 0.45 0.10 0.45 0.70 $ \mathcal{Z}_4 $ 0.50 0.00 0.50 0.20 0.53 0.10 0.53 0.70 0.60 0.20 0.60 0.10 0.64 0.10 0.64 0.60 粗体数字表示$\bar{\gamma}_i(a,\;z)$, 其他数字表示$\bar{\eta}_i(b,\;z)$, $\mathcal{Z}_i$, $i=1,\;\cdots,\;4$表示MAS状态集合
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