Robust Guaranteed Cost Consensus for High-order Discrete-time Multi-agent Systems With Switching Topologies and Time Delays
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摘要: 研究存在参数不确定性的高阶离散时间多智能体系统在时延和联合连通切换通信拓扑条件下的鲁棒保性能一致性问题，给出一种线性一致性协议的设计方法.1）引入高阶离散时间不确定多智能体系统的鲁棒保性能一致性问题，定义基于智能体邻居状态误差和控制输入的保性能函数；2）通过构造合适的Lyapunov函数并利用离散时间系统稳定性理论，推导出一个使高阶离散时间不确定多智能体系统在该条件下获得保性能一致性的线性矩阵不等式（Linear matrix inequality，LMI）充分条件，并给出相应的保性能上界；3）以一致性序列的形式给出参数不确定条件下的高阶离散时间多智能体系统的一致性收敛结果；4）数值仿真验证了本文理论的正确性和有效性.Abstract: The robust guaranteed cost consensus problem for high-order discrete-time linear multi-agent systems with parameter uncertainties is studied, under the condition of jointly-connected interconnections and time-varying delays. A corresponding linear consensus protocol design is proposed. The idea of robust guaranteed cost control is introduced to the consensus problem. After that, a cost function is defined based on the state errors among neighboring agents and control inputs of all the agents. By constructing a suitable Lyapunov function and using the stability theory of discrete-time linear systems, a sufficient linear matrix inequality (LMI) condition, as well as an upper bound of the cost function, is derived to ensure the robust guaranteed cost consensus of the concerned systems. Then, convergence results are provided as final consensus values for the high-order discrete-time linear multi-agent systems with switching topologies and time-varying delays. Numerical experiment is carried out to demonstrate the correctness and effectiveness of the theoretical results.
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Key words:
- Time delays /
- switching topologies /
- uncertain multi-agent systems /
- consensus /
- guaranteed cost
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近年来, 多智能体系统的分布式协调控制问题受到了研究者的极大关注, 在卫星编队[1-2]、多无人系统[3-4]、无线传感器网络[5-6]等实际系统都有广泛应用.一致性问题作为多智能体系统分布式协调控制的一个基础问题, 一经提出就得到研究者的普遍重视[7-12], 它要求每个智能体仅利用自身和邻居的状态信息, 通过设计合适的一致性协议, 使得多智能体系统的某个感兴趣的状态变量相同或趋于一致.解决一致性问题的基本框架最早由Olfati-Saber等在文献[7]提出, 主要研究多智能体系统的一阶一致性问题. Ren等在文献[8]考虑了二阶多智能体系统的一致性问题, 给出了二阶多智能体系统获得一致性的充分条件.文献[9]考虑了邻域交互结构优化的二阶多智能体快速蜂拥控制算法. Xi等在文献[10]提出高阶连续时间多智能体系统的一致性问题, 并将高阶多智能体系统的模型描述为一般线性系统.文献[11]研究了一类控制器模态和系统模态不匹配的异步切换高阶多智能体系统的输出调节一致性问题.文献[12]研究了时延和切换拓扑条件下高阶离散时间多智能体系统的一致性问题.文献[13]讨论了有领导者条件下高阶离散时间多智能体系统的一致性问题.随着研究的深入, 研究者发现之前的研究仅考虑了系统的稳定性, 没有考虑系统的性能指标, 例如系统的能量消耗.基于此考虑, Cheng等在文献[14]中提出了保性能一致性问题.保性能一致性的优点在于它不仅考虑了多智能体系统的一致性稳定性, 还考虑了多智能体系统的能量消耗.实际的多智能体系统所携带的能量往往都是有限的, 因此研究多智能体系统的保性能一致性问题具有十分重要的工程意义.文献[15]和文献[16]研究了连续时间多智能体系统的一阶保性能一致性问题.文献[17]研究了连续时间多智能体系统的二阶保性能一致性问题.上述文献研究的是连续时间多智能体系统的保性能一致性问题, 且没有考虑系统的鲁棒性. Xu等在文献[18]中考虑了高阶离散时间多智能体系统的鲁棒保性能一致性问题.但是, 文献[18]仅仅讨论了固定拓扑和无时延条件下的高阶离散时间多智能体系统的鲁棒保性能一致性问题, 没有涉及时延和切换拓扑的情况.受此启发, 本文考虑研究时延和联合连通切换拓扑条件下的高阶离散时间多智能体系统的鲁棒保性能一致性问题.
本文符号说明如下: ${{\mathbf{1}}_N}$表示维数为$N$的元素全为1的列向量; $I$表示适当维数的单位矩阵; rank表示矩阵的秩; 表示Kronecker直积[17]; 表示对称矩阵的中的相应对称元素.
1. 图论知识
用表示一个有向图, 其中, ${\cal V}=$ $\{ {v_1}$, $\cdots, {v_N} \}$表示图的$N$个节点集; 表示图的边集, 节点序号属于一个固定的节点序列集 $=\{ 1, 2, \cdots , N\}$, 图$G$的边用表示, ${e_{ij}}$表示从${v_i}$到${v_j}$的信息流, ${v_i}$称做父节点, ${v_j}$称做子节点.用非负矩阵表示其邻接矩阵, 图$G$的边${e_{ij}}$关联的权重${a_{ij}}$非负.另外假定 $0$, $i \in {\cal I}$.如果, 则称图$G$为一个无向图.图$G$的Laplacian矩阵定义为 $=\Delta - {\cal A}$, 其中, $i=1, 2, \cdots$, 为图$G$的度矩阵. ${e_i}$是第$i$个元素为1其余元素为0的列向量.
2. 问题描述
文献[10]指出, 高阶多智能体系统一致性问题可以用一般线性群系统一致性问题描述.因此, 本文考虑一个由$N$个同构智能体组成的高阶离散时间不确定线性多智能体系统.
$ \begin{align} {{{\pmb{x}}}_i}(k + 1) = (A + \Delta A){{{\pmb{x}}}_i}(k) + (B + \Delta B){{{\pmb{u}}}_i}(k) \end{align} $
(1) 其中, $i = 1, 2, \cdots, N$, 和分别表示智能体$i$的系统状态和控制向量, $A$和$B$是适当维数的常数矩阵, 且$(A, B)$可控. $\Delta A$和是适当维数的不确定矩阵, 假定形式如下:
$ \begin{align} [\begin{array}{*{20}{c}} {\Delta A}&{\Delta B} \end{array}] = DF[\begin{array}{*{20}{c}} {{E_1}}&{{E_2}} \end{array}] \end{align} $
(2) 其中, $D$, $E_1$和$E_2$是合适维数的常数矩阵, 反映了离散时间多智能体系统(1)中的每一个智能体的不确定性结构, $F$是一个满足
$ \begin{align} {F^{\rm T}}F \le I \end{align} $
(3) 的未知矩阵, 且可以是时变的.
注1. 在实际多智能体系统中, 由于系统器件老化等原因, 系统的参数常常会发生变化, 但一般不会太大, 因此本文做出式(3)的假设.另外, $\Delta A$和$\Delta B$在实际系统中也不完全一定始终满足$\Delta A=$ $DFE_1$和$\Delta B = DFE_2$, 如果$\Delta A \ne DF{E_1}$或$\Delta B$ $\ne$ $DF{E_2}$, 高阶离散时间不确定多智能体系统(1)的鲁棒保性能问题将变得十分复杂, 本文不做讨论.
假定高阶离散时间不确定多智能体系统(1)在$k$时刻的通信拓扑为, 并用表示通信拓扑所有可能拓扑结构的集合, 具体表示为
$ {\Gamma _N} = \left\{ {{G_{\sigma (k)}} = \left( {{{\cal V}_{\sigma (k)}}, {{\cal E}_{\sigma (k)}}, {{\cal A}_{\sigma (k)}}} \right):} \right.\left. {\sigma (k) \in {{\cal I}_{{\Gamma _N}}}} \right\} $
其中, ${{\cal I}_{{\Gamma _N}}} = \{1, 2, \cdots , M\} $, 表示通信拓扑的切换信号, 即通信拓扑集中元素的索引. 为系统在$k$时刻通信拓扑${G_{\sigma (k)}}$的Laplacian矩阵.
针对高阶离散时间不确定多智能体系(1)的一致性问题, 本文采用如下线性一致性协议:
$\begin{align} {{\pmb{u}}_i}(k) = &\ K\sum\limits_{j = 1}^N {a_{ij}}(\sigma (k)) \times \nonumber\\ &\ ({{{\pmb{x}}}_j}(k - \tau (k)) - {{\pmb{x}}_i}(k - \tau (k))) \end{align} $
(4) 其中, $K$是合适维的常值反馈矩阵, $\tau (k)$是变时延, 且满足, ${\tau _{\max }}$为最大允许时延, ${{a_{ij}}(\sigma (k))}$表示当切换信号为${\sigma (k)}$时, 智能体$j$到智能体$i$的作用权重.
令, 定义保性能函数
$ \begin{align} {J_C} = {J_{Cx}} + {J_{Cu}} \end{align} $
(5) 其中,
$ \begin{align*}&{J_{Cx}} = \sum\limits_{k = 0}^\infty {\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}{\pmb{\delta }}_{ij}^{\rm T}(k){Q_x}{{\pmb{\delta }}_{ij}}(k)} } }\nonumber\\ & {J_{Cu}} = \sum\limits_{k = 0}^\infty {\sum\limits_{i = 1}^N {{\pmb{u}}_i^{\rm T}(k){Q_u}{{\pmb{u}}_i}(k)} } \end{align*} $
其中, ${Q_x}$和${Q_u}$为给定的对称正定加权矩阵.
定义1.对于高阶离散时间不确定多智能体系统(1), 如果存在一个序列${\pmb{c}(k)}$和一个正数$J_C^*$使得和成立, 则称不确定多智能体系统(1)达到保性能一致性, $c(k)$称做一致性序列, $J_C^*$称做保性能代价.
注2. ${J_C}$是一个与高阶离散时间不确定多智能体系统(1)参数相关的保性能函数, 其中, ${J_{Cu}}$描述了系统控制输入的能量消耗, ${J_{Cx}}$描述了一致性协议(4)造成的能量消耗, 保性能一致性的实质就是寻找一个${J_{Cu}}$和${J_{Cx}}$之间的平衡, 具体就是寻找一个合适的反馈增益矩阵$K$使得保性能函数${J_C}$存在一个上界${J_C^*}$.由此可以看出, 保性能一致性本质上是从能量优化的角度考虑了多智能体系统的一致性, 但最优解一般很难得到, 一般求出次优解即可.
3. 保性能一致性协议设计
令, 则高阶离散时间多智能体系统(1)在一致性协议(4)作用下的闭环系统为
$ \begin{align} \begin{cases} {\pmb{x}}(k + 1) = ({I_N} \otimes (A + \Delta A)){\pmb{x}}(k) - \\ \qquad\qquad\quad ({L_{\sigma (k)}} \otimes (B + \Delta B)K){\pmb{x}}(k - \tau (k))\\ {\pmb{x}}(k) =\phi (k), \quad\ \ k \in \{ - {\tau _{\max }}, - {\tau _{\max }} + 1, \cdots , 0\} \end{cases} \end{align} $
(6) 其中, .为了分析联合连通条件下高阶离散时间多智能体闭环系统(6)的一致性问题.
引理1[19].用$L$表示一个无向图$G$的Laplacian矩阵, 则零是它的一个特征值, 如果$G$连通, 则零是$L$的单一特征值, 并且$L$的其他所有特征值均为正实数.
定义2.设存在一个有限时间段, 使得切换通信拓扑${G_{\sigma (k)}}$在时间段内的联合作用通信拓扑是连通的, 则称切换通信拓扑${G_{\sigma (k)}}$在有限时间段联合连通.其中表示${G_{\sigma (k)}}$在时间段内的并集, 所对应的邻接矩阵和Laplacian矩阵${L_\kappa }$分别满足和${L_\kappa }$ $=$ .
注3.联合连通与一般意义的切换拓扑的区别是联合连通的子拓扑不一定是连通的, 而一般意义的切换拓扑的每个子拓扑都是连通的, 这样多智能体系统在联合连通条件下的一致性问题将更加复杂.
令$\lambda _{\sigma (k)}^{(i)}$, $i = 1, 2, \cdots , N$为Laplacian矩阵${L_{\sigma (k)}}$的$N$个特征值, 其中, 所对应的一个标准特征向量为, 同时其他特征值满足, 则存在一个正交矩阵
$ \begin{align*}{U_0} = \left[ {\begin{array}{*{20}{c}} {\dfrac{1}{{\sqrt N }}}&{\dfrac{{{\mathbf{1}}_{N - 1}^{\rm T}}}{{\sqrt N }}}\\[5mm] {\dfrac{{{\mathbf{1}}_{N - 1}^{\rm T}}}{{\sqrt N }}}&{{{\bar U}_0}} \end{array}} \right]\end{align*} $
满足.其中是对称的.由于通信拓扑不一定连通, 则${L_{\sigma (k)}}$的秩满足 $-$ $1$.因此, ${L_{\sigma (k)}}$有个非零特征值, 即$0 <$ .为方便起见, 本文定义:
$ \begin{align*} &{\lambda _{\min }} = \mathop {\arg}\min\limits_{{\lambda _i}} \left\{ {\lambda _{\sigma (k)}^{(N - {r_{\sigma (k)}} + 1)}|{G_{\sigma (k)}} \in {\Gamma _N}} \right\}\\[1mm] & {\lambda _{{\mathop{\rm m}\nolimits} {\rm{ax}}}} = \mathop {\arg}\max\limits_{{\lambda _i}} \left\{ {\lambda _{\sigma (k)}^{(N)}|{G_{\sigma (k)}} \in {\Gamma _N}} \right\}\end{align*} $
再令
$ \begin{align} \tilde {{\pmb{x}}}(k) =&\ \left( {U_{\sigma (k)}^{\rm{T}} \otimes {I_d}} \right){\pmb{x}}(k) = \nonumber\\ &\ {\left[\tilde {{\pmb{x}}}_c^{\rm T}(k), \tilde {{\pmb{x}}}_r^{\rm T}(k)\right]^{\rm T}} = \nonumber\\ &\ {\left[\tilde {{\pmb{x}}}_1^{\rm T}(k), \tilde {{\pmb{x}}}_2^{\rm T}(k), \cdots , \tilde {{\pmb{x}}}_N^{\rm T}(k)\right]^{\rm T}} \end{align} $
(7) 其中, $\tilde {{\pmb{x}}}_c(k) \in {{\mathbf R}^d}$, ${\tilde {{\pmb{x}}}_{r, N}^{\rm T}(k)]^{\rm T}}$.然后, 根据引理1, 闭环多智能体系统(6)可重新写为
$ \begin{align} {\tilde {{\pmb{x}}}_c}(k + 1) =&\ (A + \Delta A){\tilde {{\pmb{x}}}_c}(k) \\[1mm] {{\tilde {{\pmb{x}}}}_r}(k + 1) =&\ ( {{I_N} \otimes (A + \Delta A)} ){{\tilde {{\pmb{x}}}}_{r, i}}(k) -\notag \\ &\ ( {{{\tilde L}_{\sigma (k)}} \otimes (B + \Delta B)K} ){{\tilde {{\pmb{x}}}}_r}(k - \tau (k)){\kern 1pt} {\kern 1pt} \end{align} $
(8) 显然, 如果子系统(9) Schur稳定, 则闭环多智能体系统(6)获得一致性.需要说明的是, 这个条件与文献[10]中的高阶连续时间多智能体系统情况类似, 区别是对于高阶连续时间多智能体系统, 子系统(9)需要的是Hurwitz稳定.
另外, 保性能函数(5)可写为
$ \begin{align} {J_{Cx}} = \sum\limits_{k = 0}^\infty {\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}{\delta }_{ij}^{\rm T}(k){Q_x} {{\delta }_{ij}}(k)} } } =\\ \sum\limits_{k = 0}^\infty {{{{\pmb{x}}}^{\rm T}}(k)(2{L_{\sigma (k)}} \otimes {Q_x}){{\pmb{x}}}(k)} \nonumber\end{align} $
(10) $ \begin{align} {J_{Cu}} = &\ \sum\limits_{k = 0}^\infty {\sum\limits_{i = 1}^N {{{\pmb{u}}}_i^{\rm T}(k){Q_u}{{{\pmb{u}}}_i}(k)} }= \nonumber\\ &\ \sum\limits_{k = 0}^\infty {{{{{\pmb{x}}}}^{\rm T}}(k - \tau (k)){\Pi}{{{\pmb{x}}}}(k - \tau (k))} \end{align} $
(11) 其中, $\Pi=(L_{\sigma (k)}^2 \otimes {K^{\rm T}}{Q_u}K)$.因此, 保性能函数(5)可写为
$ \begin{align} {J_C} =&\sum\limits_{k = 0}^\infty {{{{\pmb{x}}}^{\rm T}} (k)(2{L_{\sigma (k)}} \otimes {Q_x}){{\pmb{x}}}(k)} + \nonumber\\ & \sum\limits_{k = 0}^\infty {{{{\pmb{x}}}^{\rm T}}(k - \tau (k)){\Pi}{{\pmb{x}}}(k - \tau (k))} \end{align} $
(12) 引理2 (Jensen不等式)[20].对于任意常值矩阵$M \ge 0$, $M \in {{\mathbf R}^{n \times n}}$, , 正整数${\alpha _1}$, , 且${\alpha _2} \ge {\alpha _1} \ge 1$, 如下不等式成立:
$ \begin{align} &- \left( {{\alpha _2} - {\alpha _1} + 1} \right)\sum\limits_{i = {\alpha _1}}^{{\alpha _2}} {{\psi ^{\rm{T}}}\left( i \right)M\psi \left( i \right)} \le \nonumber\\ & \qquad - {\left( {\sum\limits_{i = {\alpha _1}}^{{\alpha _2}} {\psi \left( i \right)} } \right)^{\rm{T}}}M\left( {\sum\limits_{i = {\alpha _1}}^{{\alpha _2}} {\psi \left( i \right)} } \right) \end{align} $
(13) 引理3. 考虑一个高阶离散时间不确定多智能体系统(1), 同时假定它的通信作用拓扑是联合连通的, 时间延迟为.如果存在合适维的矩阵$K$, $d$ $\times$ $d$维的矩阵${R_1}$, ${R_2}$, ${Q_1}$, ${Q_2}$满足
$ \begin{align} &{\Omega _i} = \left[ {\begin{array}{*{20}{c}} {{\Omega _{i11}}}&{{\Omega _{i12}}}&0&{{\Omega _{i14}}}\\ *&{{\Omega _{i22}}}&0&0\\ *&*&{{\Omega _{i33}}}&0\\ *&*&*&{{\Omega _{i44}}} \end{array}} \right] < 0, \quad i = 1, 2 \end{align} $
(14) $ \begin{align} &(A + \Delta A)P(A + \Delta A) - P < 0 \end{align} $
(15) 其中,
$ \begin{align*} {\Omega _{i11}} =&\ X^{\rm T}PX -P+\tau_{\max }{Q_1}+ {Q_2} +\\ &\ \tau _{\max }X^{\rm T}R_1(X-I) + \\&\ \tau_{\max }(X-I)^TR_2(X-I)- \\&\frac{{R_1}}{\tau _{\max }}+2{{\tilde \lambda }_i}{Q_x}\\ {\Omega _{i12}} =&\ {{\tilde \lambda }_{\max}}(X-I)^{\rm T}{R_1}(B + \Delta B)K +\\ &\ {{\tilde \lambda }_{\max}}(X-I)^{\rm T}{R_2}(B + \Delta B)K +\\ &\ {{\tilde \lambda }_{\max}}X^TP(B + \Delta B)K\end{align*} $
以上两式中, .
$ \begin{align*} {\Omega _{i22}} =&\ {{\tilde \lambda }}_{i}^2K^{\rm T}(B + \Delta B)^{\rm T}P(B + \Delta B)K +\\ &\ {\tau _{\max }}{{\tilde \lambda }}_{i}^2K^{\rm T}(B + \Delta B)^{\rm T}{R_1}(B + \Delta B)K +\\ &\ {\tau _{\max }}{{\tilde \lambda }}_{i}^2K^{\rm T}(B + \Delta B)^{\rm T}{R_2}(B + \Delta B)K -\\ &\ \frac{{R_2}}{\tau _{\max }}\\ {\Omega _{i33}} =&-Q_1+{{\tilde \lambda }}_{i}^2K^{\rm T}Q_uK\\ {{\tilde \lambda }}_{1}= \lambda&_{\min}, ~~{{\tilde \lambda }}_{2}=\lambda_{\max} \\{\Omega _{i14}} =&\ \frac{{R_1}}{\tau _{\max }}\\ {\Omega _{i44}} = &-Q_2-\frac{{R_1}}{\tau _{\max }} \end{align*} $
则高阶离散时间不确定多智能体系统(1)能在一致性协议(4)作用下获得保性能一致性, 且它的保性能函数上界$J_C^*$满足
$ \begin{align} J_C^* =&\ {{{\pmb{x}}}^{\rm{T}}}(0)\left( {\Theta \otimes P} \right){{\pmb{x}}}(0) + \nonumber\\ &\ {\tau _{\max }}{{{\pmb{x}}}^{\rm{T}}}(0)\left( {\Theta \otimes ({Q_1} + {Q_2})} \right){{\pmb{x}}}(0) \end{align} $
(16) 其中,
$ \Theta = \left[ {\begin{array}{*{20}{c}} {\dfrac{{{\mathbf 1}_{N - 1}^{\rm{T}}{{\mathbf 1}_{N - 1}}}}{N}}&{\dfrac{{{\mathbf 1}_{N - 1}^{\rm{T}}{{\bar U}^{\rm{T}}}}}{{\sqrt N }}}\\ {\dfrac{{\bar U{{\mathbf 1}_{N - 1}}}}{{\sqrt N }}}&{\bar U{{\bar U}^{\rm{T}}}} \end{array}} \right] $
证明.考虑Lyapunov函数候选
$ \begin{align} V(k) = \sum\limits_{l = 1}^6 {{V_l}(k)} \end{align} $
(17) 其中,
$ \begin{align} & {V_1}\left( k \right) = \tilde {{\pmb{x}}}_r^{\rm{T}}(k)({I_{N - 1}} \otimes P){\tilde {{\pmb{x}}}_r}(k) \end{align} $
(18) $ \begin{align} & {V_2}\left( k \right) = \sum\limits_{s = k - \tau (k)}^{k - 1} {\tilde {{\pmb{x}}}_r^{\rm{T}}(s)({I_{N - 1}} \otimes {Q_1}{\rm{)}}{{\tilde {{\pmb{x}}}}_r}(s)} \end{align} $
(19) $ \begin{align} & {V_3}\left( k \right) = \sum\limits_{s = k - {\tau _{\max }}}^{k - 1} {\tilde {{\pmb{x}}}_r^{\rm{T}}(s)({I_{N - 1}} \otimes {Q_2}{\rm{)}}{{\tilde {{\pmb{x}}}}_r}(s)} , \end{align} $
(20) $ \begin{align} & {V_4}\left( k \right) = \sum\limits_{s = - {\tau _{\max }} + 1}^{ - 1} {\sum\limits_{l = k + s}^{k - 1} {\tilde {{\pmb{x}}}_r^{\rm{T}}(l)({I_{N - 1}} \otimes {Q_1}{\rm{)}}{{\tilde {{\pmb{x}}}}_r}(l)} } \end{align} $
(21) $ \begin{align} & {V_5}\left( k \right) = \sum\limits_{s = - {\tau _{\max }}}^{ - 1} {\sum\limits_{l = k + s}^{k - 1} {\tilde {\pmb\eta} _r^{\rm{T}}(l)({I_{N - 1}} \otimes {R_1}{\rm{)}}{{\tilde {\pmb\eta} }_r}(l)} } \end{align} $
(22) $ \begin{align} & {V_6}\left( k \right) = \sum\limits_{s = - \tau (k)}^{ - 1} {\sum\limits_{l = k + s}^{k - 1} {\tilde {\pmb\eta} _r^{\rm{T}}(l)({I_{N - 1}} \otimes {R_2}{\rm{)}}{{\tilde {\pmb\eta} }_r}(l)} } \end{align} $
(23) 其中, , ${\tilde {\pmb\eta} _{r, i}}(l)$ $=$ ${\tilde {{\pmb{x}}}_{r, i}}(l + 1) - {\tilde {{\pmb{x}}}_{r, i}}(l)$.因此, $V(k)$关于时间的差分为
$ \begin{align} \Delta V(k) =&\ V(k + 1) - V(k) =\sum\limits_{i = 1}^6 {\Delta {V_i}(k)} = \nonumber\\ &\ \sum\limits_{i = 1}^6 {\left( {{V_i}(k + 1) - {V_i}(k)} \right)} \end{align} $
(24) 由${\tilde L_{\sigma (k)}}$性质可知, 存在一个正交矩阵使得
$ \begin{align} &\tilde U_{\sigma (k)}^{\rm{T}}{\tilde L_{\sigma (k)}}{\tilde U_{\sigma (k)}} =\notag \\ &\qquad {\rm diag}\left\{ {\underbrace {0, \cdots , 0}_{N - {r_{\sigma (k)}} - 1}, \lambda _{\sigma (k)}^{(N - {r_{\sigma (k)}} + 1)}, \cdots , \lambda _{\sigma (k)}^{(N)}} \right\} \end{align} $
(25) 进而, 令
$ \begin{align} &{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _r}(k)\! =\! \left( {\tilde U_{\sigma (k)}^{\rm{T}} \otimes {I_d}} \right){\tilde {{\pmb{x}}}_r}(k) \!=\! {\left[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _2^{\rm T}(k), \cdots , \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _N^{\rm T}(k)\right]^{\rm T}} \end{align} $
(26) $ \begin{align} &{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } _r}(k)\! =\! \left( {\tilde U_{\sigma (k)}^{\rm{T}} \otimes {I_d}} \right){\tilde {\pmb\eta} _r}(k)\!=\! {\left[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } _2^{\rm T}(k), \cdots , \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } _N^{\rm T}(k)\right]^{\rm T}} \end{align} $
(27) 将式(25)和(26)代入式(24)可得, $\Delta {V_1}(k)$沿着式(6)的差分为
$ \begin{align} &\Delta {V_1}(k) = {V_1}(k + 1) - {V_1}(k) = \nonumber\\ &\quad \sum\limits_{i = 2}^N \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _{r, i}^{\rm{T}}(k)\left( {(A + \Delta A)P{{(A + \Delta A)}^{\rm{T}}} - P} \right)\times\nonumber\\ &\quad {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_{r, i}}(k) + \sum\limits_{i = N -{r_{\sigma (k)}} + 1}^N \Big( \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _{r, i}^{\rm{T}}(k)\Big( (A + \Delta A - \nonumber\\ &\quad \lambda _{\sigma (k)}^{(i)}(B + \Delta B)K)^{\rm{T}} P(A + \Delta A - \nonumber\\ &\quad \lambda _{\sigma (k)}^{(i)}(B + \Delta B)K) - P \Big){{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_{r, i}}(k)\Big) + \nonumber\\ &\quad 2\sum\limits_{i = N - {r_{\sigma (k)}} + 1}^N \Big( \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _{r, i}^{\rm{T}}(k)(A + \Delta A -\nonumber\\ &\quad \lambda _{\sigma (k)}^{(i)}(B + \Delta B)K)^{\rm{T}} P(B + \Delta B) \times\nonumber\\ &\quad K\Big( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle \frown$}} \over {{\pmb{x}}}} }_{r, i}}(k) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_{r, i}}(k - \tau (k)){\kern 1pt} } \Big) \Big) + \nonumber\\ &\quad \sum\limits_{i = N - {r_{\sigma (k)}} + 1}^N \Big( {{{\Big( {\lambda _{\sigma (k)}^{(i)}} \Big)}^2}\sigma (k)} \times \nonumber\\&\quad \Big( {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_{r, i}}(k) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_{r, i}}(k - \tau (k)){\kern 1pt} \Big)^{\rm{T}}\times\nonumber\end{align} $
$ \begin{align}&{K^{\rm{T}}}{(B + \Delta B)^{\rm{T}}} P(B + \Delta B) \times\nonumber\\& K\Big( {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_{r, i}}(k) - {{\tilde {{\pmb{x}}}}_{r, i}}(k - \tau (k)){\kern 1pt} \Big) \Big) \end{align} $
(28) $\Delta {V_2}(k)$沿着式(6)的差分为
$ \begin{align} \Delta {V_2}&(k) = {V_2}(k + 1) - {V_2}(k)\nonumber\le \\& \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _r^{\rm{T}}(k)({I_{N - 1}} \otimes {Q_1}{\rm{)}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k) - \nonumber\\& \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _r^{\rm{T}}(k - \tau (k)){({I_{N - 1}} \otimes {Q_1}{\rm{)}}_1}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k - \tau (k)) + \nonumber\\& \sum\limits_{s = k + 1 - {\tau _{\max }}}^{k - 1} {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _r^{\rm{T}}(s)({I_{N - 1}} \otimes {Q_1}{\rm{)}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(s)} \end{align} $
(29) $\Delta {V_3}(k)$沿着式(6)的差分为
$ \begin{align} \Delta {V_3}&(k) = {V_3}(k + 1) - {V_3}(k) = \nonumber\\& \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _r^{\rm{T}}(k)({I_{N - 1}} \otimes {Q_2}{\rm{)}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k) - \nonumber\\& \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _r^{\rm{T}}(k - {\tau _{\max }})({I_{N - 1}} \otimes {Q_2}{\rm{)}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k - {\tau _{\max }}) \end{align} $
(30) $\Delta {V_4}(k)$沿着式(6)的差分为
$ \begin{align} \Delta {V_4}&(k) = {V_4}(k + 1) - {V_4}(k) = \nonumber\\&\ ({\tau _{\max }} - 1)\tilde {{\pmb{x}}}_{r, i}^{\rm{T}}(k){Q_1}{{\tilde {{\pmb{x}}}}_{r, i}}(k) - \nonumber\\&\ \sum\limits_{l = k - {\tau _{\max }} + 1}^{k - 1} {\tilde {{\pmb{x}}}_{r, i}^{\rm{T}}(l){Q_1}{{\tilde {{\pmb{x}}}}_{r, i}}(l)} \end{align} $
(31) $\Delta {V_5}(k)$沿着式(6)的差分为
$ \begin{align} \Delta {V_5}&(k) = {V_5}(k + 1) - {V_5}(k) = \nonumber\\&\ {\tau _{\max }}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } _r^{\rm{T}}(k)({I_{N - 1}} \otimes {R_1}{\rm{)}}{{\tilde {\pmb\eta} }_r}(k) - \nonumber\\&\ \sum\limits_{l = k - {\tau _{\max }}}^{k - 1} {\tilde {\pmb\eta} _r^{\rm{T}}(l)({I_{N - 1}} \otimes {R_1}{\rm{)}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } }_r}(l)} \end{align} $
(32) 另外, 等价于
$ \begin{align} {\tau _{\max }}&\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } _r^{\rm{T}}(k)({I_{N - 1}} \otimes {R_1}{\rm{)}} {{\tilde {\pmb\eta} }_r}(k) = \nonumber\\ & {\tau _{\max }}{\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k + 1) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k)} \right)^{\rm{T}}}({I_{N - 1}} \otimes {R_1}{\rm{)}} \times\nonumber\\ & \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k + 1) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k)} \right) = {\tau _{\max }}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _r^{\rm{T}}(k) \times\nonumber\\ & {(A + \Delta A - \lambda _{\sigma (k)}^{(i)}(B + \Delta B)K - I)^{\rm{T}}}\times\nonumber\\& ({I_{N - 1}} \otimes {R_1}{\rm{)}}(A + \Delta A - \lambda _{\sigma (k)}^{(i)}(B + \Delta B)K -\nonumber\\& I){{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k) + 2{\tau _{\max }}\lambda _{\sigma (k)}^{(i)}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _r^{\rm{T}}(k)(A + \Delta A -\nonumber\\& {\lambda _i}(\sigma (k))(B + \Delta B)K - I{)^{\rm{T}}}({I_{N - 1}} \otimes {R_1}{\rm{)}} \times\nonumber\\ &(B +\Delta B)K \Big( {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k - \tau (k)){\kern 1pt} \Big) + \nonumber\\ & {\tau _{\max }}{\left( {\lambda _{\sigma (k)}^{(i)}} \right)^2}{\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k - \tau (k)){\kern 1pt} } \right)^{\rm{T}}}{K^{\rm{T}}} \times\nonumber\end{align} $
$ \begin{align}& {(B + \Delta B)^{\rm{T}}}({I_{N - 1}} \otimes {R_1}{\rm{)}} \times\nonumber\\ & (B + \Delta B)K\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_r}(k - \tau (k)){\kern 1pt} } \right) \end{align} $
(33) 同时, 根据引理2, 可得
$ \begin{align} & - \sum\limits_{l = k - {\tau _{\max }}}^{k - 1} {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } _r^{\rm{T}}(l)({I_{N - 1}} \otimes {R_1}{\rm{)}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle \frown$}} \over {\pmb\eta} } }_r}(l)} \le \nonumber\\ &\qquad - {\left( {\sum\limits_{l = k - {\tau _{\max }}}^{k - 1} {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } _r^{\rm{T}}(l)} } \right)^{\rm{T}}}\dfrac{{({I_{N - 1}} \otimes {R_1}{\rm{)}}}}{{{\tau _{\max }}}}~\times\nonumber\\&\qquad\left( {\sum\limits_{l = k - {\tau _{\max }}}^{k - 1} {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } }_r}(l)} } \right) = \nonumber\\ &\qquad - {\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb x}} }_r}(k) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb x}} }_r}(k - {\tau _{\max }})} \right)^{\rm{T}}} \times\nonumber\\ &\qquad \frac{{({I_{N - 1}} \otimes {R_1}{\rm{)}}}}{{{\tau _{\max }}}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb x}} }_r}(k) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb x}} }_r}(k - {\tau _{\max }})} \right) \end{align} $
(34) $\Delta {V_6}(k)$沿着式(6)的差分为
$ \begin{align} \Delta {V_6}&(k) = {V_6}(k + 1) - {V_6}(k) = \nonumber\\ & \tau (k)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } _r^{\rm{T}}(k)({I_{N - 1}} \otimes {R_2}{\rm{)}}{{\tilde {\pmb\eta} }_r}(k) - \nonumber\\ & \sum\limits_{l = k - \tau (k)}^{k - 1} {\tilde {\pmb\eta} _r^{\rm{T}}(l)({I_{N - 1}} \otimes {R_2}{\rm{)}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } }_r}(l)}\le \nonumber\\ & {\tau _{\max }}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } _r^{\rm{T}}(k)({I_{N - 1}} \otimes {R_2}{\rm{)}}{{\tilde {\pmb\eta} }_r}(k) - \nonumber\\ & \sum\limits_{l = k - \tau (k)}^{k - 1} {\tilde {\pmb\eta} _r^{\rm{T}}(l)({I_{N - 1}} \otimes {R_2}{\rm{)}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\eta} } }_r}(l)} \end{align} $
(35) 进一步, 做出如下定义
$ \begin{align} \Delta J(k) = \Delta V(k) + {\bar J_C} \end{align} $
(36) 其中,
$ \begin{align*} {{\bar J}_C} = &\ \sum\limits_{i = N - {r_{\sigma (k)}}}^N {2\lambda _{\sigma (k)}^{(i)}\tilde {{\pmb{x}}}_{r, i}^{\rm{T}}(k){Q_x}{{\tilde {{\pmb{x}}}}_{r, i}}(k)} + \\ &\ \sum\limits_{i = N - {r_{\sigma (k)}} + 1}^N {{{\left( {\lambda _{\sigma (k)}^{(i)}} \right)}^2}\tilde {{\pmb{x}}}_{r, i}^ {\rm{T}}(k - \tau (k))} \times\\ &\ {({K^{\rm{T}}}{Q_u}K){{\tilde {{\pmb{x}}}}_{r, i}}(k - \tau (k))} \end{align*} $
令
$ \begin{align*} {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {\pmb\xi} } }_i}(k) =&\Bigg[ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _{r, i}^{\rm{T}}(k), {{\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_{r, i}}(k) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_{r, i}}(k - \tau (k)){\kern 1pt} } \right)}^{\rm{T}}}} , \\ &\qquad\qquad{ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _{r, i}^{\rm{T}}(k - \tau (k)), \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} _{r, i}^{\rm{T}}(k - {\tau _{\max }})} \Bigg]^{\rm{T}}} \end{align*} $
注意到如果$\Delta J(k) \le 0$, 有$\Delta V(k) \le 0$, 接着可以得到
$\begin{align} \Delta&J(k) \le \sum\limits_{i = 2}^N \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle \frown$}} \over {{\pmb{x}}}} _{r, i}^{\rm{T}}(k)~\times\nonumber\\ &\left( {(A + \Delta A)P{{(A + \Delta A)}^{\rm{T}}} - P} \right){{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over {{\pmb{x}}}} }_{r, i}}(k)~ + \nonumber\\ & \sum\limits_{i = 2}^N {\tilde {\pmb\xi} _i^{\rm{T}}(k){\Omega _i}{{\tilde {\pmb\xi} }_i}(k)} \end{align} $
(37) 再由线性系统理论知识和凸集优化理论可以得到, 如果矩阵不等式(14)和(15)成立, 则闭环系统(6)渐近稳定.
从条件(37)和$\Delta J(k) \le 0$可以得到${\bar J_C}$ .再利用和, 对${\bar J_C} \le - \Delta V(k)$两边进行从$k = 0$到求和, 利用线性系统稳定性理论可以得到.根据定义可知, 矩阵不等式(14)和(15)可确保高阶离散时间不确定多智能体系统(1)获得保性能一致性, 并且系统的保性能函数(5)满足.另外, 通过式(7)可以得到.又因为$ [ 0, {I_{N - 1}} ]$ $\times$ $\otimes$ , 所以有
$ \begin{align}& {V_1}\left( k \right) = {{{\pmb{x}}}^{\rm{T}}}(k)(\Theta \otimes P){{\pmb{x}}}(k) \end{align} $
(38) $ \begin{align} & {V_2}\left( k \right) = \sum\limits_{s = k - \tau (k)}^{k - 1} {{{{\pmb{x}}}^{\rm{T}}}(s)(\Theta \otimes {Q_1}){{\pmb{x}}}(s)} \end{align} $
(39) $ \begin{align} & {V_3}\left( k \right) = \sum\limits_{s = k - {\tau _{\max }}}^{k - 1} {{{{\pmb{x}}}^{\rm{T}}}(s)(\Theta \otimes {Q_2}){{\pmb{x}}}(s)} \end{align} $
(40) $ \begin{align} & {V_4}\left( k \right) = \sum\limits_{s = - {\tau _{\max }} + 1}^{ - 1} {\sum\limits_{l = k + s}^{k - 1} {{{{\pmb{x}}}^{\rm{T}}}(l)(\Theta \otimes {Q_1}){{\pmb{x}}}(l)} } \end{align} $
(41) $ \begin{align} & {V_5}\left( k \right) = \sum\limits_{s = - {\tau _{\max }}}^{ - 1} {\sum\limits_{l = k + s}^{k - 1} {{{\pmb\eta} ^{\rm{T}}}(l)(\Theta \otimes {R_1}){\pmb\eta} (l)} } \end{align} $
(42) $ \begin{align} & {V_6}\left( k \right) = \sum\limits_{s = - \tau (k)}^{ - 1} {\sum\limits_{l = k + s}^{k - 1} {{{\pmb\eta} ^{\rm{T}}}(l)(\Theta \otimes {R_1}){\pmb\eta} (l)} } \end{align} $
(43) 其中,
$ \begin{align*} &{{\pmb\eta} ^{\rm{T}}}(k) = {[{\pmb\eta} _1^{\rm T}(l), {\pmb\eta} _2^{\rm T}(l), \cdots , {\pmb\eta} _N^{\rm T}(l)]^{\rm T}}\\[1mm] &{{\pmb\eta} _i}(l) = {{\pmb x}_i}(l + 1) - {{\pmb x}_i}(l)\end{align*} $
由于$U{U^{\rm{T}}}={I_N}$, 所以有和.这样可以得到.又因为$x(k) = x(0)$, $k$ $\in$ , 可以得到 $=$ .从而有
$ \begin{align} {J_C} \le&{{{\pmb{x}}}^{\rm{T}}}(0)\left( {\Theta \otimes P} \right){{\pmb{x}}}(0) + \nonumber\\ & {\tau _{\max }}{{{\pmb{x}}}^{\rm{T}}}(0)\left( {\Theta \otimes ({Q_1} + {Q_2})} \right){{\pmb{x}}}(0) \end{align} $
(44) 根据定义1, 综上可得, 式(16)是这种情况下保性能函数(5)的一个上界.
注4. 引理3利用状态空间分解法, 将不确定多智能体系统(1)的鲁棒保性能一致性问题转换为$N$ $-$ $1$个子系统(9)的鲁棒保性能控制问题, 并通过构造合适的Lyapunov函数, 寻找到了一个高阶离散时间不确定多智能体系统(1)在时延和切换通信拓扑条件下获得鲁棒保性能一致性的充分条件, 并给出了保性能函数(5)的一个与不确定多智能体系统(1)的初始状态$ {\pmb x}({\mathbf 0})$和最大通信时延相关的上界.可行矩阵$P$, $R_1$, $R_2$, $Q_1$, $Q_2$和$K$的求解与系统(1)的通信拓扑有关, 对保性能函数(5)的上界具有间接的影响.但易发现矩阵不等式(14)和(15)含有非线性不确定项, 因此很难利用引理3直接求解出反馈增益矩阵$K$.
引理4[21].给定合适维的矩阵$Y$, $D$和$E$, 且$Y$是对称矩阵, 则
$ \begin{align} Y + DFE + {E^{\rm T}}{F^{\rm T}}{D^{\rm T}} < 0 \end{align} $
(45) 对于所有满足的矩阵$F$成立的充要条件是存在$\varepsilon > 0$满足
$ \begin{align} Y + \varepsilon D{D^{\rm T}} + {\varepsilon ^{ - 1}}{E^{\rm T}}E < 0 \end{align} $
(46) 引理5[22].线性矩阵不等式
$ \left( {\begin{array}{*{20}{c}} {Q(x)}&{S(x)}\\ {S^{\rm T}{{(x)}}}&{R(x)} \end{array}} \right) > 0 $
其中, $Q(x) = Q^{\rm T}{(x)}$, $R(x) = R^{\rm T}{(x)}$和$S(x)$与$x$相关, 它等价于
1) $Q(x) > 0 $, $R(x) - S^{\rm T}{(x)}Q^{ - 1}{(x)}S(x) > 0$;
2) $R(x) > 0 $, $Q(x) - S(x)R^{ - 1}{(x)}S^{\rm T}{(x)} > 0$.
定理1.考虑高阶离散时间不确定多智能体系统(1)的通信时延为, 同时假定它的通信作用拓扑是联合连通的.一致性协议(4)能够解决不确定多智能体系统(1)的鲁棒保性能一致性问题的一个充分条件是存在正数, ${\varepsilon _2} > 0$, , 合适维的反馈矩阵$K$, $d \times d$维的正定矩阵$P$, $S$, $R_1$, $R_2$, $M_1$, $M_2$, $Q_1$和$Q_2$是如下优化问题的一个解
$ \begin{align} &\min\ {{\rm tr}\left( {PS + {R_1}{M_1} + {R_2}{M_2}} \right)} \notag\\ & {\rm s.t.}\quad (48)\sim(52)\end{align} $
(47) $ \begin{align}\left[ {\begin{array}{*{20}{c}} { a_1}\!&\!0\!&\!0\!&\!{ a_4}\!&\!{ a_7}\!&\!0\!&\!0\!&\!0\!&\!0\!&\!0 &0\\ \ast\!&\!{ a_2}\!&\!0\!&\!{ a_5}\!&\!{ a_7}\!&\!0\!&\!0\!&\!0\!&\!0 &0&0\\ *\!&\!*\!&\! a_3\!&\! a_5\!&\! a_7\!&\!0\!&\!0\!&\!0\!&\!0\!&\!0\!&\!0\\ *\!&\!*\!&\!*\!&\! a_6\!&\!0\!&\!0\!&\! a_{10}\!&\! a_{12}\!&\! a_{12}\!&\! a_{12}\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\! a_8\!&\!0\!&\!0\!&\! a_{13}\!&\! a_{13}\!&\! a_{13}\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\! a_9\!&\!0\!&\!0\!&\!0\!&\!0\!&\! a_{17}\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\! a_{11}\!&\!0\!&\!0\!&\!0\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\! a_{14}\!&\!0\!&\!0\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\! a_{15}\!&\!0\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\! a_{16}\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\! a_{17} \end{array}} \right]\!\! \le\! 0 \end{align} $
(48) 其中,
$ \begin{align*} & a_1=- S + {\varepsilon _1}D{D^{\rm{T}}} \\[1mm]& a_2={ - \frac{1}{{{\tau _{\max }}}}{M_1} + {\varepsilon _2}D{D^{\rm{T}}}} \\[1mm]& a_3={ - \frac{1}{{{\tau _{\max }}}}{M_2} + {\varepsilon _3}D{D^{\rm{T}}}} \\[1mm]& a_4={A - {{\tilde \lambda }_i}BK} \\[1mm]& a_5={A - {{\tilde \lambda }_i}BK - I} \\[1mm]& a_6={\tau _{\max }}{Q_1} + {Q_2} - \frac{{{R_1}}}{{{\tau _{\max }}}} + 2{\lambda _i}{Q_x} \\[1mm]& a_7={{{\tilde \lambda }_i}BK} \\[1mm]& a_8={ - \frac{{{R_2}}}{{{\tau _{\max }}}}} \\[1mm]& a_9={ - {Q_1}} \\[1mm]& a_{10}={\frac{{{R_1}}}{{{\tau _{\max }}}}} \\[1mm]& a_{11}={ - {Q_2} - \frac{{{R_1}}}{{{\tau _{\max }}}}} \\[1mm]& a_{12}={{{\left( {{E_1} - {{\tilde \lambda }_i}{E_2}K} \right)}^{\rm{T}}}} \\[1mm]& a_{13}={{{\left( {{{\tilde \lambda }_i}{E_2}K} \right)}^{\rm{T}}}} \\[1mm]& a_{14}={ - {\varepsilon _1}I} \\[1mm]& a_{15}={ - {\varepsilon _2}I} \\[1mm]& a_{16}={ - {\varepsilon _3}I} \\[1mm]& a_{17}={{K^{\rm{T}}}} \\& a_{18}={ - \frac{1}{{\tilde \lambda _i^2}}Q_u^{ - 1}} \end{align*} $
$ \begin{align} &\left[ {\begin{array}{*{20}{c}} { - S}&A&0\\ {{A^{\rm{T}}}}&{ - P}&{E_1^{\rm{T}}}\\ 0&{{E_1}}&{ - {\varepsilon _1}I} \end{array}} \right] <0 \end{align} $
(49) $ \begin{align} &\left[ {\begin{array}{*{20}{c}} P&I\\ I&S \end{array}} \right] > 0, \ \ P > 0, \ \ S > 0 \end{align} $
(50) $ \begin{align} &\left[ {\begin{array}{*{20}{c}} {{R_1}}&I\\ I&{{M_1}} \end{array}} \right] > 0, \ \ R_1 > 0, \ \ M_1 > 0 \end{align} $
(51) $ \begin{align} &\left[ {\begin{array}{*{20}{c}} {{R_2}}&I\\ I&{{M_2}} \end{array}} \right] > 0, \ \ R_2 > 0, \ \ M_2 > 0 \end{align} $
(52) 其中, ${\tilde \lambda _1} = {\lambda _{\min }}$, .在这种情况下, 如果$K$, $P$, $S$, $R_1$, $R_2$, $M_1$, $M_2$, $Q_1$和$Q_2$是优化问题(47) ~ (52)的一个可行解, 则式(16)仍然是高阶离散时间不确定多智能体系统(1)获得一致性的一个保性能上界.
证明. 矩阵不等式(14)可写为
$ \begin{align*}b_1Pb_2 + \tau _{\max }b_3{R_1}b_4+\tau _{\max }b_5 {R_2}b_4+ b_6 \le 0\end{align*} $
其中,
$ \begin{align*} & b_1= \left[ \begin{array}{*{20}{c}} {{\Phi _i^{\rm{T}}}}\\ \lambda _{\sigma (k)}^{(i)}(B + \Delta B)K)^{\rm T}\\ 0\\ 0 \end{array}\right]\\& b_2=\left[ \begin{array}{*{20}{c}} {\Phi _i}&{{\lambda _i}(\sigma (k))(B + \Delta B)K}&0&0 \end{array}\right] \\& b_3=\left[ \begin{array}{*{20}{c}} {{({\Phi _i} - I)^{\rm{T}}}}\\ \lambda _{\sigma (k)}^{(i)}(B + \Delta B)K)^{\rm T}\\ 0\\ 0 \end{array} \right] \\& b_4= \left[ \begin{array}{*{20}{c}} {{\Phi _i} - I}&{\lambda _{\sigma (k)}^{(i)}(B + \Delta B)K}&0&0 \end{array} \right] \\& b_5=\left[ \begin{array}{*{20}{c}} {({\Phi _i} - I)}^{\rm{T}}\\ \lambda _{\sigma (k)}^{(i)}(\bar B)K)^{\rm{T}}\\ 0\\ 0 \end{array} \right] \\& b_6=\left[ \begin{array}{*{20}{c}} c_1&0&0&c_4\\ *&c_2&0&0\\ *&*&c_3&0\\ *&*&*&c_5 \end{array} \right] \end{align*} $
$ \begin{align*}& c_1={ - P + {\tau _{\max }}{Q_1} + {Q_2} - \frac{{{R_1}}}{{{\tau _{\max }}}} + 2\lambda _{\sigma (k)}^{(i)}{Q_x}} \\& c_2={ - \frac{{{R_2}}}{{{\tau _{\max }}}}} \\& c_3={ - {Q_1} + {{\left( {\lambda _{\sigma (k)}^{(i)}} \right)}^2} {K^{\rm{T}}}{Q_u}K} \\& c_4={\frac{{{R_1}}}{{{\tau _{\max }}}}} \\& c_5={ - {Q_2} - \frac{{{R_1}}}{{{\tau _{\max }}}}} \end{align*} $
其中, .根据引理5, 上述矩阵不等式等价于
$ \begin{align} \begin{bmatrix} d_1&0&0&d_4 &d_7&0&0\\ *&d_2&0&d_5&d_7&0&0\\ *&*&d_3&d_5&d_7&0&0\\ *&*&*&d_6&0&0&d_{10}\\ *&*&*&*&d_8&0&0\\ *&*&*&*&*&d_9&0\\ *&*&*&*&*&*&d_{11} \end{bmatrix}\le 0. \end{align} $
(53) 其中,
$ \begin{align*}& d_1={ - {P^{ - 1}}} \\& d_2={ - \frac{1}{{{\tau _{\max }}}}R_1^{ - 1}} \\& d_3={ - \frac{1}{{{\tau _{\max }}}}R_2^{ - 1}} \\& d_4={A + \Delta A - \lambda _{\sigma (k)}^{(i)}(B + \Delta B)K} \\& d_5={A + \Delta A - \lambda _{\sigma (k)}^{(i)}(B + \Delta B)K - I} \\& d_6={ - P + {\tau _{\max }}{Q_1} + {Q_2} - \frac{{{R_1}}} {{{\tau _{\max }}}} + 2\lambda _{\sigma (k)}^{(i)}{Q_x}} \\& d_7={\lambda _{\sigma (k)}^{(i)}(B + \Delta B)K} \\& d_8={ - \frac{{{R_2}}}{{{\tau _{\max }}}}} \\& d_9={ - {Q_1} + {{\left( {\lambda _{\sigma (k)}^{(i)}} \right)}^2}{K^{\rm{T}}}{Q_u}K} \\& d_{10}={\frac{{{R_1}}}{{{\tau _{\max }}}}} \\&d_{11}={ - {Q_2} - \frac{{{R_1}}}{{{\tau _{\max }}}}} \end{align*} $
将$\Delta A = DF{E_1}$和代入矩阵不等式(53), 并利用引理4, 矩阵不等式(53)对于矩阵$F$所有可能成立, 等价于存在正数, ${\varepsilon _2} > 0$和, 满足
$ \begin{align} &e+\varepsilon_1f_1+\varepsilon_1^{-1}g_1g_2+ \varepsilon_2f_2+\varepsilon_2^{-1}g_1g_2 +\nonumber\end{align} $
$ \begin{align}&\qquad\varepsilon_3f_3+\varepsilon_3^{-1}g_1g_2\le 0 \end{align} $
(54) 其中,
$ \begin{align*}& e= \begin{bmatrix} e_1&0&0&e_4 &e_7&0&0\\ *&e_2&0&e_5&e_7&0&0\\ *&*&e_3&e_5&e_7&0&0\\ *&*&*&e_6&0&0&e_{10}\\ *&*&*&*&e_8&0&0\\ *&*&*&*&*&e_9&0\\ *&*&*&*&*&*&e_{11} \end{bmatrix} \\& e_1={ - {P^{ - 1}}} \\& e_2={ - \frac{1}{{{\tau _{\max }}}}R_1^{ - 1}} \\& e_3={ - \frac{1}{{{\tau _{\max }}}}R_2^{ - 1}} \\& e_4={A - \lambda _{BK}^{i, \sigma (k)}} \\& e_5={A - \lambda _{BK}^{i, \sigma (k)} - I} \\& e_6={ - P + {\tau _{\max }}{Q_1} + {Q_2} - \frac{{{R_1}}}{{{\tau _{\max }}}} + 2{\lambda _i}(\sigma (k)){Q_x}} \\ & e_7={\lambda _{BK}^{i, \sigma (k)}} \\& e_8={ - \frac{{{R_2}}}{{{\tau _{\max }}}}} \\& e_9={ - {Q_1} + {{\left( {\lambda _{\sigma (k)}^{(i)}} \right)}^2}{K^{\rm{T}}}{Q_u}K} \\& e_{10}={\frac{{{R_1}}}{{{\tau _{\max }}}}} \\& e_{11}={ - {Q_2} - \frac{{{R_1}}}{{{\tau _{\max }}}}} \\& f_1=\left[ {\begin{array}{*{20}{c}} D\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} D&0&0&0&0&0&0 \end{array}} \right] \\& g_1=\left[ {\begin{array}{*{20}{c}} 0\\ 0\\ 0\\ {{{\left( {{E_1} - \lambda _{{E_2}K}^{i, \sigma (k)}} \right)}^{\rm{T}}}}\\ {{{\left( {\lambda _{{E_2}K}^{i, \sigma (k)}} \right)}^{\rm{T}}}}\\ 0\\ 0 \end{array}} \right] \end{align*} $
$ \begin{align*}& g_2=\left[ {\begin{array}{*{20}{c}} 0&0&0&{{E_1} - \lambda _{{E_2}K}^{i, \sigma (k)}}&{\lambda _{{E_2}K}^{i, \sigma (k)}}&0&0 \end{array}} \right] \\& f_2=\left[ {\begin{array}{*{20}{c}} 0\\ D\\ 0\\ 0\\ 0\\ 0\\ 0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&D&0&0&0&0&0 \end{array}} \right] \\& f_3=\left[ {\begin{array}{*{20}{c}} 0\\ 0\\ D\\ 0\\ 0\\ 0\\ 0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&0&D&0&0&0&0 \end{array}} \right] \end{align*} $
其中, , .再两次使用引理5, 可得
$ \begin{align} & \left[ \begin{array}{ccccccccccccccccccccccc} h_1\!&\!0\!&\!0\!&\!h_4\!&\!h_7\!&\!0\!&\!0\!&\!0\!&\!0\!&\!0\!&\!0\\ *\!&\!h_2\!&\!0\!&\!h_5\!&\!h_7\!&\!0\!&\!0\!&\!0\!&\!0\!&\!0\!&\!0\\ *\!&\!*\!&\!h_3\!&\!h_5\!&\!h_7\!&\!0\!&\!0\!&\!0\!&\!0\!&\!0\!&\!0\\ *\!&\!*\!&\!*\!&\!h_6\!&\!0\!&\!0\!&\!h_{10}\!&\!h_{12}\!&\!h_{12}\!&\!h_{12}\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!h_8\!&\!0\!&\!0\!&\!h_{13}\!&\!h_{13}\!&\!h_{13}\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!h_9\!&\!0\!&\!0\!&\!0\!&\!0\!&\!h_{17}\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!h_{11} \!&\!0\!&\!0\!&\!0\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!h_{14} \!&\!0\!&\!0\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!h_{15} \!&\!0\!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!h_{16} \!&\!0\\ *\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!*\!&\!h_{18} \end{array}\right]\nonumber\\&\qquad \le 0. \end{align} $
(55) 其中,
$ \begin{align*}& h_1={ - {P^{ - 1}} + {\varepsilon _1}D{D^{\rm{T}}}} \\& h_2={A - \lambda _{BK}^{i, \sigma (k)}} \\& h_3={ - \frac{1}{{{\tau _{\max }}}}R_2^{ - 1} + {\varepsilon _3}D{D^{\rm{T}}}} \\& h_4={ - \frac{1}{{{\tau _{\max }}}}R_1^{ - 1} + {\varepsilon _2}D{D^{\rm{T}}}} \\& h_5={A - \lambda _{BK}^{i, \sigma (k)} - I} \\& h_6={ - P + {\tau _{\max }}{Q_1} + {Q_2} - \frac{{{R_1}}} {{{\tau _{\max }}}} + 2{\lambda _i}{Q_x}} \end{align*} $
$\begin{align*}& h_7={\lambda _{BK}^{i, \sigma (k)}} \\& h_8={ - \frac{{{R_2}}}{{{\tau _{\max }}}}} \\& h_9={ - {Q_1}} \\& h_{10}={\frac{{{R_1}}}{{{\tau _{\max }}}}} \\& h_{11}={ - {Q_2} - \frac{{{R_1}}}{{{\tau _{\max }}}}} \\& h_{12}={{{\left( {{E_1} - \lambda _{{E_2}K}^{i, \sigma (k)}} \right)}^{\rm{T}}}} \\& h_{13}={{{\left( {\lambda _{{E_2}K}^{i, \sigma (k)}} \right)}^{\rm{T}}}} \\& h_{14}={ - {\varepsilon _1}I} \\& h_{15}={ - {\varepsilon _2}I} \\& h_{16}={ - {\varepsilon _3}I} \\& h_{17}={{K^{\rm{T}}}} \\& h_{18}={ - \frac{1}{{\lambda _i^2}}Q_u^{ - 1}} \end{align*} $
另外, 再对矩阵不等式(15)做同样处理.令$S$ $={P^{ - 1}}$, 和, 再利用锥补线性化方法[23]和凸集优化理论, 非线性矩阵不等式(54)的求解问题等价于非线性最小化问题(46) ~ (51).
注5.通过定理1, 非线性线性矩阵不等式(14)和(15)转化为优化问题(46) ~ (51), 它可以利用算法1进行求解, 从而得到反馈增益矩阵, 从而使得多智能体系统(1)在一致性协议(4)下获得鲁棒保性能一致性.值得说明的是, 因为$F$是一个未知矩阵, 无法直接利用与之相关的不等式条件获得一致性协议的反馈增益矩阵.这种情况下, 引理4很好地解决了这个问题.但需注意的是, 解$K$, $P$, $S$, $R_1$, $R_2$, $M_1$, $M_2$, $Q_1$, $Q_2$与不确定矩阵$D$和$F$密切相关.因为优化问题(46) ~ (51)的线性矩阵不等式是堆成的, 所以可以利用MATLAB工具箱求解, 从而验证其有效性.
算法1.优化问题(46) ~ (51)的求解
步骤1.寻找LMI (46) ~ (51)的一个可行解${P_0}$, ${S_0}$, ${R_{1, 0}}$, ${R_{2, 0}}$, ${M_{1, 0}}$, ${M_{2, 0}}$, 如果不存在, 就退出.令$k = 0$.
步骤2.令${P_k} = {P_0}$, ${S_k} = {S_0}$, , ${R_{2, k}}$ $=$ ${R_{2, 0}}$, ${M_{1, k}} = {M_{1, 0}}$, , 并寻找${P_{k+1}}$, ${S_{k+1}}$, ${R_{1, {k+1}}}$, ${R_{2, {k+1}}}$, ${M_{1, {k+1}}}$, ${M_{2, {k+1}}}$满足LMI (46) ~ (51), 并使得tr最小.
步骤3.如果tr, 则退出.否则, 令$k = k + 1$, 返回步骤2.
定理2.当闭环高阶离散时间不确定多智能体系统(6)获得保性能一致性时, 其一致性序列满足
$ \begin{align} {\small \begin{cases} {\mathop {\lim }\limits_{{k_i} \to \infty } \left( {{{\pmb{c}}}({{\tilde k}_i}) - {{\mathbf{1}}_N} \otimes \left( {\frac{1}{N}\sum\limits_{i = 1}^N {{{\pmb{x}}}({{\tilde k}_i})} } \right)} \right) = 0} \\{\mathop {\lim }\limits_{k \to \infty } \left( {{{\pmb{c}}}(k) - {{\mathbf{1}}_N} \otimes \left( {\frac{1}{N}{{(A + \Delta A)}^{k - \tilde k}}\sum\limits_{i = 1}^N {{{{\pmb{x}}}_i}({{\tilde k}_i})} } \right)} \right) = 0} \end{cases}} \end{align} $
(56) 证明. 令, ${{{\pmb{x}}}_r}(k)$ $=$ .根据式(7), ${{\pmb{x}}}(k)$能够被唯一分解为.如前所述, 如果闭环不确定多智能体系统(6)获得保性能一致性, 则子系统(9)必须Schur稳定, 它暗示着子系统(9)由状态${{{\pmb{x}}}_r}(0)$引起的响应需满足.因此, 一致性序列${{\pmb{c}}}(k)$完全由${{{\pmb{x}}}_c}(k)$决定.下面分两种情况讨论一致性序列${{\pmb{c}}}(k)$.
1) 当$k = {\tilde k_i}$时, 因为, 所以.再根据, 可得, 即
$ \begin{array}{l} {{{\pmb{x}}}_c}({{\tilde k}_i}) = {{\bar {{\pmb{u}}}}_1} \otimes (({{\pmb{e}}}_1^{\rm T} \otimes {I_d}) \times (U_{\sigma (k)}^{\rm T} \otimes {I_d}){{\pmb{x}}}({{\tilde k}_i})) = \\ {{\bar {{\pmb{u}}}}_1} \otimes (\frac{1}{{\sqrt N }}{\mathbf 1}_N^{\rm T} \otimes {I_d}){{\pmb{x}}}({{\tilde k}_i})) = {\mathbf 1}_N^{\rm T} \otimes (\frac{1}{N}\sum\limits_{i = 1}^N {{{{\pmb{x}}}_i}({{\tilde k}_i})} ) \end{array} $
2) 当时, 根据式(8)可得
$ \begin{array}{l} {{{\pmb{x}}}_c}(k) = {(A + \Delta A)^{k - \tilde k}}\left( {{\mathbf{1}}_N^{\rm T} \otimes (\frac{1}{N}\sum\limits_{i = 1}^N {{{{\pmb{x}}}_i}({{\tilde k}_i})} )} \right) = \\ {{\mathbf{1}}_N} \otimes \left( {\frac{1}{N}{{(A + \Delta A)}^{k - \tilde k}}\sum\limits_{i = 1}^N {{{{\pmb{x}}}_i}({{\tilde k}_i})} } \right) \end{array} $
又因为${{\pmb{c}}}(k)$满足 , 综上可得式(55).
注6.从定理2可以看出, 一致性序列${{\pmb{c}}}(k)$仅与系统矩阵$A+{\Delta}A$、通信拓扑和初始状态${{\pmb{x}}}(0)$相关, 与时间延迟无关.这个条件与文献[12]中的不含不确定性的高阶离散时间多智能体系统情况类似, 区别在于定理2描述了不确定性对高阶离散时间不确定多智能体系统的一致性序列的影响.另外值得说明的是, 文献[24]利用内模原理解决了异构连续线性多智能体系统的鲁棒一致性控制问题.本文与它的区别在于, 文献[24]主要从输出一致性的角度考虑多智能体系统的一致性问题, 而本文主要从保性能的角度考虑多智能体的一致性问题.同时, 二者考虑的不确定性也有所不同, 但还是可以考虑在后续工作将两者结合起来研究.此外, 本文工作与控制的区别在于, 控制目的在于抑制外部噪声, 本文考虑的是节省能量.
4. 仿真算例
假定多智能体系统(1)的具体形式如下：
$A = \left[ {\begin{array}{*{20}{c}} {0.65}&{0.65}&{0.22}\\ { - 0.65}&{0.65}&{0.23}\\ 0&{0.03}&{1.02} \end{array}} \right], ~ B = \left[ {\begin{array}{*{20}{c}} {0.2}\\ { - 0.4}\\ 1 \end{array}} \right]$
$ D = \left[ {\begin{array}{*{20}{c}} {0.1}&0&0\\ 0&{0.2}&0\\ 0&0&{0.3} \end{array}} \right], ~~{E_2} = \left[ {\begin{array}{*{20}{c}} {0.2}\\ \begin{array}{l} 0.1\\ 0.3 \end{array} \end{array}} \right] $
$ {E_1} = \left[ {\begin{array}{*{20}{c}} {0.1}&{0.3}&0\\ {0.2}&{0.4}&0\\ 0&0&1 \end{array}} \right], ~~F = \left[ {\begin{array}{*{20}{c}} {{r_1}}&0&0\\ 0&{{r_2}}&0\\ 0&0&{{r_3}} \end{array}} \right] $
其中, ${r_1}$, ${r_2}$和${r_3}$是不确定参数, 它们满足, $- 1 \le {r_2} \le 1$和.则高阶离散时间不确定多智能体系统(1)可以重新写为
$ \begin{align} {{{\pmb{x}}}_i}(k + 1) =&\ (A + DF{E_1}){{{\pmb{x}}}_i}(k)~ +\notag\\ &\ (B + DF{E_2}){{{\pmb{u}}}_i}(k) \end{align} $
(57) 定义保性能代价加权矩阵如下:
${Q_x} = \left[ {\begin{array}{*{20}{c}} {0.6}&0&0\\ 0&{0.6}&0\\ 0&0&{0.7} \end{array}} \right]$, ${Q_u} = 1$
设高阶离散时间多智能体系统(1)包含8个智能体, 其通信拓扑是联合连通的, 具体如图 1所示, 图 1 (a)~(c)为其3种通信可能, 图 1 (d)为通信拓扑图 1 (a)~(c)的合并图.
假定多智能体系统(1)的初始状态为, , , , , , , .同时, 令${r_1} = 0.15$, ${r_2} = 0.25$, ${r_3} = 0.15$, .利用定理1可得能够使多智能体系统(1)获得保性能一致性.仿真结果如图 2~6所示.图 2~4给出了多在一致协仪(4)作用下, 智能体系统(1)在参数变化前后的状态轨迹变化曲线.图 2~4中, $\circ$表示定理2计算出来的多智能体系统(1)的通信拓扑不发生变化的初始一致性序列, $*$表示定理2计算出来的多智能体系统(1)的通信拓扑发生变化时的实际一致性序列, 其他实线表示多智能体系统(1)状态的变化轨迹.图 5给出了保性能指标${J_C}$和$J_C^*$的变化轨迹.图 6给出了多智能体系统(1)的通信拓扑切换信号的时序图.
图 2 状态${{\pmb{x}}}_{i1}$在参数${r_j} (j = 1, 2, 3)$变化前后的状态轨迹Fig. 2 Comparison of state trajectories of ${{\pmb{x}}}_{i1}$ before and after parameters ${r_j}~(j = 1, 2, 3)$ change
图 3 状态${{\pmb{x}}}_{i2}$在参数${r_j}~(j = 1, 2, 3)$变化前后的状态轨迹Fig. 3 Comparison of state trajectories of ${{\pmb{x}}}_{i2}$ before and after parameters ${r_j}~(j = 1, 2, 3)$ change
图 4 状态${{\pmb{x}}}_{i3}$在参数${r_j}~(j = 1, 2, 3)$变化前后的状态轨迹Fig. 4 Comparison of state trajectories of ${{\pmb{x}}}_{i3}$ before and after parameters ${r_j}~(j = 1, 2, 3)$ change
图 6 多智能体系统(1)的通信拓扑切换信号$\sigma (k)$Fig. 6 The switching signal $\sigma (k)$ of multi-agent system (1)从图 2~6的仿真结果可以看出, 不确定多智能体系统(1)在一致性协议(4)下获得鲁棒保性能一致性.多智能体系统(1)包含/不包含参数不确定性时, 系统都收敛于式(55), 与定理2计算出的一致性序列${{\pmb{c}}}(k)$相同.区别在于当多智能体系统(1)不包含不确定性时, 式(55)中的$A+{\Delta}A$需被$A$取代.多智能体系统(1)的一致性收敛序列与不确定参数和系统通信拓扑有关, 尤其是图 4, 可以看出系统一致性序列${{\pmb{c}}}(k)$会在系统通信拓扑切换时发生突变.图 5表明, 在实验过程中恒成立, 满足定义1的鲁棒保性能一致性要求.实验充分说明了高阶离散时间不确定多智能体系统(1)在定理1求得的增益矩阵$K$下, 受一致性协议(4)作用, 能够获得鲁棒保性能一致, 从而验证了定理1和定理2的正确性和有效性.
5. 结束语
在时延和切换通信拓扑条件下, 本文对高阶离散时间不确定多智能体系统的鲁棒保性能问题进行了研究, 给出了高阶离散时间不确定多智能体系统获得鲁棒保性能一致性的一个充分条件, 并提供了其保性能函数的上界和一致性序列.数值仿真验证了本文所提理论的正确性.另外, 本文所提方法较为复杂, 后续工作将考虑如何使其简化.
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图 2 状态${{\pmb{x}}}_{i1}$在参数${r_j} (j = 1, 2, 3)$变化前后的状态轨迹
Fig. 2 Comparison of state trajectories of ${{\pmb{x}}}_{i1}$ before and after parameters ${r_j}~(j = 1, 2, 3)$ change
图 3 状态${{\pmb{x}}}_{i2}$在参数${r_j}~(j = 1, 2, 3)$变化前后的状态轨迹
Fig. 3 Comparison of state trajectories of ${{\pmb{x}}}_{i2}$ before and after parameters ${r_j}~(j = 1, 2, 3)$ change
图 4 状态${{\pmb{x}}}_{i3}$在参数${r_j}~(j = 1, 2, 3)$变化前后的状态轨迹
Fig. 4 Comparison of state trajectories of ${{\pmb{x}}}_{i3}$ before and after parameters ${r_j}~(j = 1, 2, 3)$ change
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