多速率分布式预测控制及其在热连轧活套系统中的应用
  自动化学报  2018, Vol. 44 Issue (5): 935-942   PDF    
多速率分布式预测控制及其在热连轧活套系统中的应用
张晓东, 高绍姝, 刘新平     
中国石油大学(华东)计算机与通信工程学院 青岛 266580
摘要: 多智能体协调控制系统更适合采用分布式控制方式,但是处理智能体之间的耦合影响是分布式控制的一个难点.本文针对串联结构下的多智能体系统,提出一类多速率分布式预测控制策略,异步更新多智能体的控制律,能够充分考虑智能体之间的耦合影响,提高系统的稳定性,并给出了系统稳定的充分条件.最后,将多速率分布式控制算法应用到热连轧活套系统,仿真验证了方法的有效性和可行性.
关键词: 分布式预测控制     多速率     活套系统     多智能体    
Multi-rate Distributed Predictive Control and Its Application to Looper Control System of Hot Strip Rolling
ZHANG Xiao-Dong, GAO Shao-Shu, LIU Xin-Ping     
College of Computer and Communication Engineering, China University of Petroleum, Qingdao 266580
Manuscript received : April 26, 2016, accepted: April 10, 2017.
Foundation Item: Supported by Natural Science Foundation of Shandong Provincial (ZR2014FP013), Qingdao Research Foundation for Basic Research China (14-2-4-115jch), The Economy & Technology Development of Zone of Qingdao Development of Science and Technology Plan Major Projects (2013-1-37)
Author brief: ZHANG Xiao-Dong  Lecturer at the College of Computer and Communication Engineering, China University of Petroleum. He received his Ph. D. degree in control theory and control engineering at the Beijing Institute of Technology in 2011. His research interest covers predictive control, robust control, and data driven control;
LIU Xin-Ping  Associate professor at the College of Computer and Communication Engineering, China University of Petroleum. He received his Ph. D. degree in mechanical and electronic engineering from China University of Petroleum in 2009. His research interest covers fuzzy control and predictive control
Corresponding author. GAO Shao-Shu  Lecturer at the College of Computer and Communication Engineering, China University of Petroleum, Qingdao, Shandong. She received her Ph. D. degree in optical engineering from the Beijing Institute of Technology in 2013. Her research interest covers image processing and color night. Corresponding author of this paper
Recommended by Associate Editor YANG Chun-Hua
Abstract: Distributed model predictive control strategy is more suitable for dealing with multi-agent cooperative control systems. However the coupling effects among multiple agents are still a difficult point. In the paper, a multi-rate distributed model predictive control strategy is proposed for series structural systems, in which all the controllers of agents can be updated asynchronously and the coupling effects are fully taken into account to improve stability. Furthermore, a sufficient condition for the stability is generated. Finally, the control strategy is applied into a looper control system of hot strip rolling and simulation results show its feasibility and effectiveness.
Key words: Distributed model predictive control     multi-rate     looper system     multi-agent    

随着多智能体系统的广泛应用, 分布式预测控制成为研究的热点[1-4].采用分布式控制方式能够把整个系统的优化问题分散到各个子系统中去解决, 适用于各子系统间动态独立的对象.基于预测控制原理, 学者们提出了多种分布式控制控制策略.如子系统控制律同步获取、同步实施[5-6]; 顺序获取、同步实施[7-8]以及迭代方式获取、同步实施[9]等方式.热连轧活套系统是具有串联结构的多智能体控制系统.研究人员在热连轧活套系统控制器的设计方面主要采用分散式控制策略, 针对单机架的多输入多输出活套系统设计控制器[10].如采用滑膜控制、鲁棒控制、模糊控制、预测控制等获得了较好的控制效果.由于活套高度与张力控制系统具有强耦合特性, 增加了控制器设计的难度.其中文献[11]采用多通路控制策略[12-13], 活套高度和张力系统交互更新控制律, 对子系统之间的耦合影响具有较好的抑制效果, 并建立了活套关联系统模型, 在设计单机架控制器的时候, 考虑了下游机架轧制速度的影响, 提高了活套系统的控制性能.但是, 如果采用同步更新控制策略, 下游机架轧制速度更新的同时会对上游机架的张力系统产生影响.整个活套系统的稳定性及协调性有待进一步提高.本文针对具有串联结构的多智能体控制系统, 提出一类多速率分布式预测控制策略, 并将该算法应用到热连轧活套控制系统, 提高系统的稳定性和协调性, 通过仿真实验验证了该方法的有效性和可行性.

1 多速率分布式预测控制策略

通常情况下, 基于预测控制原理的分布式控制采用同步更新控制律的方式.但是, 对于相互耦合影响的多智能体系统, 在控制律同步更新的同时也会对其他子系统产生扰动影响, 这种扰动影响难以消除, 尤其是对信息有向传输的串联结构的多智能体系统, 会随着信息的逐级传输影响整个系统的稳定性, 降低智能体之间的协调性能.基于此, 本文提出多速率分布式预测控制策略, 如图 1所示.设有$M$个智能体, 基于预测控制原理, 实施多速率异步控制策略.即:在一个输出采样周期$T$内, 顺序获得控制序列${\pmb u}_{i}(k+j)$后开始新的循环.其中$ i=\{1, \cdots, M\}, \quad j=\{0, \cdots, N_{u}-1\} $.针对第$i$个子系统, 在一个周期$T$内, 生成${\pmb u}_{i}(k)$, ${\pmb u}_{i}(k+1)$, $\cdots, {\pmb u}_{i}(k+N_{u}-1)$$N_{u}$个控制律, 控制律更新周期为$T/M$.其中$Nu=M$为控制时域, $L$是最大预测时域.令$\Delta {\pmb u}_{i}(k+l)=0$, 其中$l=\{N_{u}, \cdots, L\}$, 即在$N_{u} $步后控制律不变化.控制信息和输出信息从第$i$个智能体向第$i+1$个智能体传递.

图 1 分布式预测控制方框图 Figure 1 The diagram of distributed predictive control strategy

考虑如下离散时不变系统模型[14]

$ {\pmb x}_{i\_(k+1)} = {A_i}{\pmb x}_{i\_k} + {B_i}{\pmb u}_{i\_k}\\ $ (1)
$ {\pmb y}_{i\_k} = {C_i}{\pmb x}_{i\_(k)}+D_i{\pmb u}_{i\_k} $ (2)

其中, ${x}_{i}$, ${y}_{i}$${ u}_{i}$分别是第$i$个系统的状态、输入和输出向量, $A_{i}$, $B_i$, $C_i$$D_i$是相应维数的矩阵.基于多速率控制策略, 系统输出${\pmb y}_{i\_k}=\{{ y}_{i\_k0}, \cdots, {y}_{i\_k}, { y}_{i\_k+M}, \cdots\}$的采样周期设为$T$, 控制输入信号${\pmb u}_{i\_k}=\{{u}_{i\_k0}, \cdots, {y}_{i\_k}, { u}_{i\_k+1}, \cdots\}$的采样周期为$T/M$.

我们定义如下表示形式

$ \begin{align} S_{k|M}=[s_k, s_{k+1}, \cdots, s_{k+M+1}] \end{align} $ (3)

基于提升技术, 控制信号可以表示为

$ \begin{align} {\pmb u}_{i\_k}=\{{u}_{i\_k_0|M}, \cdots, {u}_{i\_k|M}, { u}_{i\_k+M|M}, \cdots\} \end{align} $ (4)

如果$M=2$, 则控制信号表示为

$ \begin{equation} {\pmb u}_{i\_k}\!=\!\left\{ \left[\!\!{\begin{array}{*{20}c} {{u}_{i\_k_0}}\\ {{u}_{i\_k_0+1}}\\ \end{array}} \!\!\right], \cdots, \left[\!\! {\begin{array}{*{20}c} {{u}_{i\_k}}\\ {{u}_{i\_k+1}}\\ \end{array}}\!\! \right], \left[\!\!{\begin{array}{*{20}c} {{u}_{i\_k+2}}\\ {{u}_{i\_k+3}}\\ \end{array}}\!\! \right], \cdots\!\!\right\} \end{equation} $ (5)

根据式(1)和(2), 多速率系统模型可以表示为

$ {\pmb x}_{i\_{k+M}} = {A_i}{\pmb x}_{i\_k} + \bar{B}_i{{\pmb u}_{i\_k|M}} $ (6)
$ {\pmb y}_{i\_k} = {C_i}{{\pmb x}_{i\_k}}+D_{i\_M}{\pmb u}_{i\_k|M} $ (7)

其中, $\bar{B}_i=[A_i^{M-1}B_i, \cdots, A_iB_i, B_i]$.令$D_{i\_M}=[0, \cdots, 0]$, 基于提升技术上述方程可以改写为

$ \begin{align} {\pmb y}_{i\_k+M|L}= &{M_i}{{\pmb y}_{i\_k|L}}+{N_i}{\pmb u}_{i\_k|(L+1)M}= \notag\\ &{M_i^L}{{\pmb y}_{i\_k-(L-1)M|L}}+\notag\\ &\left[M_i^{L-1}N_i, M_i^{L-2}N_i, \cdots, N_i\right]\times\notag\\ &\left[ {\begin{array}{*{20}c} {{\pmb u}_{i\_k-(L-1)M|(L+1)M}} \\ {{\pmb u}_{i\_k-(L-2)M|(L+1)M}} \\ \vdots\\ {{\pmb u}_{i\_k|(L+1)M}} \\ \end{array}} \right]=\notag\\ &{M_i^L}{{\pmb y}_{i\_k-(L-1)M|L}}+F_i{\pmb u}_{i\_k|(L+1)M}+\notag\\ &P_i{\pmb u}_{i\_k-(L-1)M|(L-1)M} \end{align} $ (8)

其中

$ \begin{align*} &M_i=O_{i\_l}A_i^M(O_{i\_L}^{\rm T}O_{i\_L}^{-1}O_{i\_L}^{\rm T})\\ &O_{i\_L}=\left[ {\begin{array}{*{20}c} {C_i} \\ {C_iA_i^M} \\ \vdots\\ {C_iA_i^{M(L-1)}} \\ \end{array}} \right] \end{align*} $
$ \begin{align*} &N_i=\left[{O_{i\_L}\bar{B}_i\quad H_{i\_L}}\right]-M_i\left[H_{i\_L}\quad 0\right]\\ &H_{i\_L}=\left[ \begin{array}{*{20}c} D_{i\_M} & \cdots & 0& 0\\ C_i\bar{B}_i & \cdots & 0& 0\\ C_iA_i^M\bar{B}_i & \cdots & 0& 0\\ \vdots & \ddots & \vdots & \vdots\\ C_iA_i^{(L-2)M}\bar{B}_i & \cdots & C_i\bar{B}_i& 0\\ \end{array} \right] \end{align*} $

式(8)中

$ \begin{align*} &P_i{\pmb u}_{i\_k-(L-1)M|(L-1)M}+F_i{\pmb u}_{i\_k|(L+1)M} =\\ &\qquad\left[p_1\quad p_2\quad \cdots\quad p_{L-1}\right] \left[ \begin{array}{*{20}c} {\pmb u}_{i\_k-(L-1)M|M}\\ {\pmb u}_{i\_k-(L-2)M|M}\\ \vdots\\ {\pmb u}_{i\_k-M|M}\\ \end{array} \right]+\\ &\qquad\left[ \begin{array}{*{20}c}f_1\quad f_2\quad\cdots\quad f_{L-1}\end{array} \right]\times\\ &\qquad \left[ \begin{array}{*{20}c} {\pmb u}_{i\_k|M}\\ {\pmb u}_{i\_k+1|M}\\ \vdots\\ {\pmb u}_{i\_k+LM|M}\\ \end{array} \right]\end{align*} $

由矩阵

$ \begin{equation*} \left[\begin{array}{*{20}ccc} N_i\\ M_iN_i\\ \vdots\\ M_i^{L-1}N_i\\ \end{array}\right] =\left[\begin{array}{*{20}c} H_{11} & H_{12} &\cdots &H_{1, L+1}\\ H_{21} & H_{22} &\cdots &H_{2, L+1}\\ \vdots& \vdots &\ddots &\vdots\\ H_{L1} & H_{L2} &\cdots & H_{L, L+1}\\ \end{array}\right] \end{equation*} $

得到参数

$ \begin{align*} &f_1=H_{11}+H_{22}+\cdots+H_{L, L}\\ &f_2=H_{12}+H_{23}+\cdots+H_{L, L+1}\\ &\qquad\qquad \vdots\\ &f_{L+1}=H_{1, L+1}\\ &p_1=H_{L1}\\ &p_2=H_{L1}+H_{L2}\\ &\qquad\qquad\vdots\\ &p_{L-1}=H_{21}+H_{32}+\cdots+H_{L, L-1} \end{align*} $

基于本文提出的多速率分布式控制策略, 在$k$时刻每一个智能体的控制序列分为两部分.如图 1所示, 一部分为$k-1$时刻求解的控制律, 可以视为已知量, 另一部分为当前$k$时刻求解的控制律.

定义如下矩阵

$ \begin{align*} &S:= \left[ \begin{array}{*{20}ccccccccccccc} I_r & 0_r & \cdots & 0_r\\ I_r & I_r & \cdots & 0_r\\ \vdots & \vdots& \ddots & \vdots\\ I_r & I_r & \cdots & I_r\\ \end{array} \right]\in {\bf R}^{(LM\times r)\times(LM\times r)} \\ &C:= \left[I_r \quad I_r \quad \cdots \quad I_r\right]^{\rm T}\in {\bf R}^{(LM\times r)\times r} \end{align*} $

其中, $r$为控制输入的维数.

控制律${\pmb u}_{k|LM}$可以表示为

$ \begin{align} {\pmb u}_{k|LM}=S\Delta {\pmb u}_{k|LM}+C{\pmb u}_{k-1} \end{align} $ (9)

因此输入输出方程(8)可以写为

$ \begin{equation} \begin{split} {\pmb y}_{i\_k+M|L}= &M_i^L{\pmb y}_{i\_k-(L-1)M|L}+\bar{N}_i\Delta {\pmb u}_{i\_k|(L+1)M}+\\ &F_iC_i{\pmb u}_{i\_k-1}+P_i{\pmb u}_{i\_k-(L-1)M|(L-1)M} \end{split} \end{equation} $ (10)

其中, $\bar N_i=F_iS $.通过优化如下目标函数求解

$ \begin{equation} \begin{split} J_i(k)= &({\pmb y}_{i\_k+M|L}-r_i)^{\rm T}({\pmb y}_{i\_k+M|L}-r_i)+\\ &\lambda_i\Delta {\pmb u}_{i\_k|LM}^{\rm T}\Delta {\pmb u}_{i\_k|LM} \end{split} \end{equation} $ (11)

其中, $r_i$为参考输出, $\lambda_i>0 $为给定的常数.

$ \begin{equation} \begin{split} \Delta {u}_{i\_k|M}= &{u}_{i\_k}-{u}_{i\_k-1}=\\ &[\underbrace{0 \quad \cdots \quad 0}_{i-1} \quad \underbrace {\Delta { u}_{i} \quad \cdots \quad \Delta { u}_{M-i}}_{M-i+1}]^{\rm T} \end{split} \end{equation} $ (12)

$ \begin{equation} \frac{{\partial J_i(k)}}{{\partial \Delta {u}_i}}=0 \end{equation} $ (13)

求得$k$时刻控制律

$ \begin{equation} \begin{split} \Delta {\pmb u}_{i\_k|(L+1)M}= &(\bar N_i^{\rm T}\bar N_i+\lambda_iI)^{-1} \bar N_i^{\rm T}\times\\ &(r_i-M_i^{L}{\pmb y}_{i\_k-(L-1)M|L}-\\ &P_i{\pmb u}_{i\_k-(L-1)M|(L-1)M}-F_iC_i{\pmb u}_{i\_k-1}) \end{split} \end{equation} $ (14)

基于信息有向传输的串联结构的多智能体系统, 由于采用多速率异步更新控制策略, 能够获得相关联子系统的输入和输出信息对其产生的耦合影响, 从而进行有效补偿, 提高控制性能.

2 稳定性分析

定理1.  在不考虑约束的情况下, 如果第$i$个子系统的特征多项式

$ \begin{equation} T_i(q^{-1})=1+a_1q^{-1}+a_2q^{-2}+\cdots+a_{L-1}q^{-(L-1)} \end{equation} $ (15)

零点均在单位圆内, 其中$\left[ a_1\quad a_2 \quad \cdots\quad a_{L-1}\right]=\Omega_iM_i^L$.则基于优化性能指标(11)获得的控制律(12)能够使闭环系统(8)稳定.

证明.  将式(12)代入到式(8)中, 可得到如下方程.

$ \begin{equation} \begin{split} {\pmb y}_{i\_k+M|L}= &-(G_i-I)M_i^L{\pmb y}_{i\_k-(L-1)M|L}+\\ &G_i{\pmb r}_i+(I-G_i)P_i{\pmb u}_{i\_k-(L-1)M|(L-1)M}+\\ &(I-G_i)F_iC_i{\pmb u}_{i\_k-1} \end{split} \end{equation} $ (16)

其中, $G_i=\bar{N}_i(\bar{N}_i^{\rm T}\bar{N}_i+\lambda_iI)^{-1}\bar{N}_i^{\rm T}$, 因此上式可以写为

$ \begin{equation} \begin{split} {\pmb y}_{i\_k+M|L}= &-\Omega_iM_i^L{\pmb y}_{i\_k-(L-1)M|L}+\\ &G_ir_i-\Omega_iP_i{\pmb u}_{i\_k-(L-1)M|(L-1)M}-\\ &\Omega_iF_iC_i{\pmb u}_{i\_k-1} \end{split} \end{equation} $ (17)

其中, $\Omega_i=[G_i-I](1, :)$.引入后移算子$q^{-1}$, 上述方程等式右侧可写为

$ \begin{equation} \begin{split} {\pmb y}_{i\_k+M|L}&+\Omega_iM_i^L{\pmb y}_{i\_k-(L-1)M|L}=\\ &(1+a_1q^{-1}+a_2q^{-2}+\cdots+\\&a_{L-1}q^{-(L-1)}){\pmb y}_{i\_k+M} \end{split} \end{equation} $ (18)

因此, 如果特征多项式(15)的所有零点均在单位圆内, 则闭环系统稳定.

3 热连轧活套系统模型

热连轧系统一般是由6~7台机架组成的, 活套安装在机架中间, 活套系统示意图如图 2所示.活套系统通过轧制速度$V_i$调整张力$\tau_i$, 通过电机电流$i_i$调整活套角度$\theta_i$, 使得活套系统保持恒定的微张力, 提高产品质量.从图 2中能够看出第$i$机架活套张力同时受到第$i$$i+1$机架轧制速度的影响.

图 2 活套系统示意图 Figure 2 Configuration of looper systems

基于热连轧上、下游机架的关联特性, 考虑下游轧机轧制速度对上游机架张力控制的影响, 建立热连轧活套系统的关联模型[13].

活套高度系统模型

$ \begin{equation} \begin{split} &\left[\!\! \begin{array}{l} {\Delta {{\dot \theta }_i}(t, s)} \\ {\Delta {{\dot \omega }_i}(t, s)} \\ {\Delta {{\dot i}_i}(t, s)} \\ \end{array} \right] =\\ &\qquad\left[ \begin{array}{ccc} 0 & {\dfrac{1}{{{G_{i\_R}}}}} & 0 \\ { - \dfrac{1}{{{G_{i\_R}}}}\dfrac{{\partial {M_{i\_T}}}} {{\partial {\theta _i}}}\dfrac{1}{{{J_i}}}\dfrac{{180}}{\pi }} & 0 & {\dfrac{{{C_{i\_m}}}}{{{J_i}}}\dfrac{{180}}{\pi }} \\ 0 & 0 & { - \dfrac{1}{{{T_i}}}} \\ \end{array} \right]\times \\ &\qquad\left[ \begin{array}{l} \Delta\theta\\ \Delta\omega\\ \Delta i\\ \end{array} \right]+ \left[ \begin{array}{ccc} 0\\ 0\\ \dfrac{1}{T_{i\_i}} \end{array} \right]\times {\pmb u}_i+ \left[ \begin{array}{ccc} 0\\ 1\\ 0 \end{array} \right]\times\\ &\qquad\left(- \dfrac{1}{G_{i\_R}} \dfrac{\partial M_{i\_T}}{\partial\tau_i} \frac{1}{J_i}\dfrac{180}{\pi} \Delta\tau_i \right) \end{split} \end{equation} $ (19)

活套张力关联系统模型

$ \begin{align} &\left[ \begin{array}{*{20}{ccccccccccccc}} {\Delta {{\dot \tau }_i}(t, s)} \\ {\Delta {{\dot V}_i}(t, s)} \\ \end{array}\right] = \notag\\ & \qquad\left[{\begin{array}{*{20}{cccccccccccccc}} {\dfrac{{{E_i}}}{{{l_i}}}\left( - \dfrac{{\partial {\beta _{i + 1}}}} {{\partial {\tau _i}}}{V_{i + 1}} - \dfrac{{\partial {f_i}}}{{\partial {\tau _i}}}{V_i}\right)} & \\ 0 & \\ \end{array}} \right.\notag\\ &\qquad\qquad \qquad\qquad \qquad\left.{\begin{array}{*{20}{cccccccccccccc}} & { - \dfrac{{{E_i}}}{{{l_i}}}(1 + {f_i})} \\ & { - \dfrac{1}{{{T_{i\_V}}}}} \\ \end{array}} \right]\times \notag\\ &\qquad \left[\begin{array}{*{20}{ccccccccc}} {\Delta {\tau _i}(t, s)} \\ {\Delta {V_i}} \\ \end{array}\right] + \left[\begin{array}{*{20}{cccccc}} 0 & {\dfrac{{{E_i}}}{{{l_i}}}(1 - {\beta _{i + 1}})} \\ 0 & 0 \\ \end{array}\right]\times\notag\\ &\qquad\left[\begin{array}{*{20}{cccc}} {\Delta {\tau _{i + 1}}(t, s)} \\ {\Delta {V_{i + 1}}} \\ \end{array}\right] + \left[\begin{array}{*{20}{cccccc}} 0 \\ {\dfrac{1}{{{T_{i\_V}}}}} \\ \end{array} \right]{{\pmb u}_i} +\notag\\ &\qquad\left[\begin{array}{*{20}{ccccccccccccc}} 1 \\ 0 \\ \end{array}\right] \dfrac{1}{{{G_{i\_R}}}} \dfrac{{{E_i}}}{{{l_i}}} \dfrac{{{\rm d}{l_i}}}{{{\rm d}{\theta _i}}}\Delta {\omega _i} \end{align} $ (20)

其中, $\omega_i$为活套角速度, $f_i$$\beta_i$分别为第$i$机架的后滑系数和第$i+1$机架的前滑系数.其他参数说明可以参考文献[13]的描述.基于异步更新控制策略, 在求解第$i+1$台机架轧制速度后保持不变, 顺序求解第$i$机架的轧制速度时, 能够充分考虑第$i+1$机架的轧制速度变化对当前活套张力系统的影响, 从而提高系统的稳定性和控制性能.

4 仿真实例

根据某热连轧厂轧机活套实际参数[15].

$ `\begin{align*} &E_i = 1.4\times{10^5} {\rm Mpa}, ~l_i = 5.5 {\rm m}, ~1/{T_{i\_v}} = 0.091\\ &{T_{i\_i}} = 0.0182, \quad{G_{i\_R}} = 11.638, \quad{\rm{ }}{f_i} = 0.0687\\ &{\beta _{i+1}} = 0.1595, \quad{C_{i\_m}} = 8.2404 {\rm Nm/A}\\ &{v_{i+1}} = 8.4034 {\rm m/s}, \quad J_i = 74.1342 {\rm N{m^2}}\\ & - \frac{{\partial \beta_i }}{{\partial \tau_i }} = 0.08304, \quad\frac{{\partial f_i}}{{\partial \tau_i }} = - 0.1075\\ &\frac{{{\rm d}loop_i}}{{{\rm d}\theta_i }} = 1.2312, \quad\frac{1}{{{G_{i\_R}}}}\frac{{\partial {M_{i\_T}}}}{{\partial \theta_i }} = 6.7652\\ &\frac{1}{{{G_{i\_R}}}}\frac{{\partial {M_{i\_T}}}}{{\partial \tau_i }} = 209.6705, \quad{v_i} = 6.6445 {\rm m/s} \end{align*} $

活套张力系统的控制输入采样时间为4 ms, 离散化后得到活套系统的参数为

$ \begin{align*} &A_{dH}= \left[ \begin{array}{*{20}cccccccc} 0.9987 & 0.04922 & 0.001317 \\ -0.05229& 0.9987 & 0.04903 \\ 0 & 0 & 0.5773\\ \end{array} \right]\\[2mm] &B_{dH}= \left[ \begin{array}{*{20}ccccc} 0.0002518 \\ 0.0147 \\ 0.4227 \\ \end{array} \right], \quad d_{dH}= \left[ \begin{array}{*{20}cccc} -0.0399 \\ -1.621\\ 0 \\ \end{array} \right] \\[2mm] &A_{dT}= \left[ \begin{array}{*{20}cccc} 0.015 & -58.53 \\ 0 & 0.8949 \\ \end{array} \right], \quad B_{dT}= \left[ \begin{array}{*{20}ccccccc} -5.257\\ 0.1051\\ \end{array} \right]\\[2mm] & A_{j\_dT}= \left[ \begin{array}{*{20}ccccccccc} 0 & 50.7 \\ 0 & 0 \\ \end{array} \right], \quad d_{dT}= \left[ \begin{array}{*{20}ccc} 0.11\\ 0 \\ \end{array} \right] \\[2mm] & {\pmb x}_{T, 0}= \left[ \begin{array}{*{20}cccccc} 0.1 \quad 0.2 \end{array} \right], \quad C_{dT}= \left[ 1\quad0 \right] \\ &C_{dH}= \left[ 1 \quad 0 \quad 0 \right], \quad {\pmb x}_{H, 0}= \left[ \begin{array}{*{20}ccc} 0.1 \quad 0.2 \quad 0 \end{array} \right] \end{align*} $

其中, $A_{dH}$, $B_{dH}$, $d_{dH}$, $C_{dH}$${\pmb x}_{H, 0}$为活套高度系统参数以及初始条件, $A_{dT}$, $B_{dT}$, $A_{j\_dT}$, $d_{dT}$, $C_{dT}$${\pmb x}_{T, 0}$为活套张力系统参数及初始条件.令$L=2$$\lambda=1$, 控制时域和预测时域相同.假设热连轧活套系统都具有相同的控制模型, 选取第3, 4, 5, 6活套张力控制系统基进行仿真实验.

1) 在初始条件的作用下, 基于串联结构的活套系统, 下游机架轧制速度变化对上游机架张力系统的影响是逐步增强的.与多速率同步更新控制策略相比较(基于优化性能指标(11)求解控制律), 仿真结果如图 3 (a)(b)所示.采用异步更新控制方式能够预知关联子系统的扰动进行补偿, 控制律变化幅度相对较小, 从而提高活套张力系统的稳定性.而采用同步更新的控制方式, 第$i+1$机架控制律$\Delta {\pmb u}_{i+1}(k)$会直接作用在第$i$机架$ \Delta {\pmb u}_i(k)$上, 使得第$i$机架当前的控制律变为$\Delta \bar{u}_i(k)=\Delta {\pmb u}_{i}(k)+(1-\beta_{i+1})\Delta {\pmb u}_{i+1}(k)$从而影响系统的稳定性和控制性能.

图 3 控制器同步更新与异步更新比较 Figure 3 The comparison between simultaneous control and asynchronous control

2) 对比同步更新控制器方式(单机架活套系统的控制律更新周期与输出周期都为$T/M$), 基于式(1)、(2)和优化性能指标(11) (式(11)中令$M=1$), 求解控制律.仿真结果如图仿真结果如图 4 (a)~(c)所示.从仿真结果可以看出, 多速率异步更新控制方式与快采样同步更新控制方式的具有相近的控制性能, 但能够减小计算量, 减轻计算负担.

图 4 快采样更新与异步更新比较 Figure 4 The comparison between fast-sample control and asynchronous control

当第5个机架轧制速度产生一个幅值为1的脉冲扰动时, 其活套张力系统响应输出如图 5 (a)~(c)所示.采用异步更新的多速率控制策略能够补偿外部扰动对相邻子系统的耦合影响, 提高扰动抑制性能.

图 5 扰动情况下快采样更新与异步更新比较 Figure 5 The comparison between fast-sample control and asynchronous control with disturbance

3) 对比同步更新控制器方式(单机架活套系统的控制律更新周期与输出周期都为$T$), 基于式(1)、(2)和优化性能指标(11) (式(11)中令$M=1$), 与多速率异步控制方式相比较, 仿真结果如图 6 (a)~(d)所示.从仿真结果可以看出采用多速率异步更新方式能够明显地改善系统的控制性能.

图 6 慢采样更新与异步更新比较 Figure 6 The comparison between slow-sample control and asynchronous control with disturbance
5 结论

针对串联结构的多智能体系统, 提出一类多速率分布式控制策略.利用信息的有向传递, 顺序更新子系统的控制律, 能够有效地补偿智能体之间的耦合影响, 提高智能体之间的协调性和稳定性, 减轻计算负担, 并给出了系统稳定性的充分条件.通过热连轧活套张力系统的仿真实验表明, 该控制策略具有较好的控制效果.

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