网络控制系统(Networked Control Systems, NCS)是通过通信网络代替传统的点对点式的连接方式构成闭环控制系统.相对于传统的控制系统, 网络控制系统有成本低、安装维护简便、灵活性高等特点^[1].无源性作为耗散性的一个特例, 将系统输入输出的乘积作为能量供给率, 体现了系统在有界输入条件下能量的衰减特性, 系统无源可以保持系统内部稳定.无源性和稳定性之间有着紧密的联系, 无源性理论可用来解决非线性系统的稳定性问题.另外无源性也为Lyapunov函数的构造提供了新的有效途径, 无源性与Lyapunov稳定性的关系可以通过将存储函数用作Lyapunov函数来建立^[2].

有关网络控制系统和无源性, 近年来学者已做了大量工作, 文献[3]考察了线性广义系统的无源性控制问题, 得到了闭环系统严格无源的充分条件; 文献[4-5]研究一类具有时延网络控制系统的无源控制问题, 推导出了闭环系统渐近稳定且满足无源性的充分条件以及无源控制器的设计方法; 文献[6]针对一类有界范数不确定性的仿射非线性系统, 研究了它的鲁棒无源性问题; 文献[7]研究分析了带有时滞的网络控制系统的鲁棒稳定性; 文献[8]则研究了离散广义系统在有界能量外部输入作用下的无源控制问题, 得到了离散广义系统容许且严格无源的充分条件并且给出了状态反馈控制器; 文献[9]给出了具有时变时滞的离散广义系统稳定的充分条件, 并且推广了离散广义系统稳定与镇定的相关结果.但据作者所知具有多时滞和非线性扰动的网络控制系统还较少研究.因此, 本文针对一类非线性时滞网络控制系统, 研究了其无源性.运用线性矩阵不等式（LMI）和Lyapunov理论, 推导出系统满足无源性的充分条件.

在本文中, 我们主要考虑如下具有非线性扰动的网络控制对象模型:

$\begin{equation} \left\{ \begin{array}{l} {{\dot x}_p}(t){\kern 1pt} {\kern 1pt} = {\kern 1pt} {\kern 1pt} {A_p}{x_p}(t) + {B_p}{u_p}(t) + {G_p}{\omega _p}(t) + {B_f}f(x(t)),\\ {y_p}(t) = {C_p}{x_p}(t) + {E_p}{\omega _p}(t), \end{array} \right. \end{equation}$

(2.1)

其中$x_{p}(t)$是控制对象状态, $u_{p}(t)$是对象输入, $y_{p}(t)$是对象输出, $\omega_{p}(t)$是外部干扰输入, $A_{p},B_{p}$和$C_{p},B_{f},G_{p}, E_p$是具有适当维数的实常数矩阵. $f(x(t))$是外部非线性扰动, 且满足:

$\begin{equation} ||f(x(t))|| \le \alpha ||x(t)||,\forall t > 0, \end{equation}$

(2.2)

其中$\alpha > 0$为正常数.由(2.2) 式可以看出存在一个正常数$\kappa > 0$使得下面不等式成立:

$\begin{equation} \kappa ({\alpha ^2}{x^T}(t)x(t) - {f^T}(x(t))f(x(t)) \ge 0. \end{equation}$

(2.3)

网络控制系统中具有时滞的控制器可以描述为:

$\begin{equation} \left\{ \begin{array}{l} {{\dot x}_c}(t) = {A_c}{x_c}(t) + {B_c}{u_c}(t) + {G_c}{\omega _c}(t),\\ {y_c}(t) = {C_c}{x_c}(t - {\tau _c}(t)) + {D_c}{u_c}(t - {\tau _c}(t)), \end{array} \right. \end{equation}$

(2.4)

其中${x_c}(t)$是控制器状态, ${u_c}(t)$是控制器输入, ${y_c}(t)$是控制器输出, ${\omega _c}(t)$是外部干扰输入, ${A_c}, {B_c}$和${C_c},{G_c}$是具有适当维数的实常数矩阵, ${\tau _c}(t)$是控制器中的时滞.

控制对象和控制器之间的闭环系统中的通信时滞可以建模为:

$\begin{equation} \left\{ \begin{array}{l} {u_c}(t) = {y_p}(t - {\tau _{sc}}(t)),\\ {u_p}(t) = {y_c}(t - {\tau _{ca}}(t)), \end{array} \right. \end{equation}$

(2.5)

其中${\tau _{sc}}(t)$是传感器到控制器的网络时滞, ${\tau _{ca}}(t)$是控制器到执行器的网络时滞.

类似于文献[7]的方法, 我们可以得到如下形式的网络控制系统:

$\begin{equation} \left\{ \begin{array}{l} \dot x(t) = {A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}(t)) + {{\tilde B}_f}f(x(t)) + G\omega (t),} \\ y(t) = Cx(t) + E\omega (t),\\ x(t) = \phi (t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} t \in [ - \bar \tau ,0], \end{array} \right. \end{equation}$

(2.6)

其中

$\begin{eqnarray*}&& x(t) = {\left[ {\begin{array}{*{20}{c}} {x_p^T(t)}&{x_c^T(t)} \end{array}} \right]^T},\omega (t) = {\left[ {\begin{array}{*{20}{c}} {\omega _p^T(t)}&{\omega _c^T(t)} \end{array}} \right]^T},\\ && {A_0} = \left[ \begin{array}{l} {A_p}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {A_c}\end{array} \right], {A_1} = \left[ \begin{array}{l} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0\\ {B_c}{C_p}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \end{array} \right], {A_2} = \left[ \begin{array}{l} {B_p}{D_c}{C_p}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \end{array} \right], {A_3} = \left[ \begin{array}{l} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {B_p}{C_c}\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \end{array} \right],\\ && C = \left[ \begin{array}{l} {C_p}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \end{array} \right], G = \left[ \begin{array}{l} {G_p}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {G_c} \end{array} \right], {{\tilde B}_f} = \left[ {\begin{array}{*{20}{c}} {{B_f}}&0\\ 0&0 \end{array}} \right], E = \left[ \begin{array}{l} {E_p}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \end{array} \right]. \end{eqnarray*}$

网络控制系统中的时变时滞满足

$\left\{ \begin{array}{l} 0 \le {\tau _1}(t) = {\tau _{sc}}(t) \le {{\bar \tau }_1},\\ 0 \le {\tau _2}(t) = {\tau _{sc}}(t) + {\tau _{ca}}(t) + {\tau _c}(t) \le {{\bar \tau }_2},\\ 0 \le {\tau _3}(t) = {\tau _{ca}}(t) + {\tau _c}(t) \le {{\bar \tau }_3}. \end{array} \right.$

令$\bar \tau = \max \{ {\bar \tau _1},{\bar \tau _2},{\bar \tau _3}\} $.

为了后续讨论, 我们首先给出如下定义及引理.

定义2.1 ^[10] 对于闭环系统(2.6), 如果对于任意的$T > 0$, 不等式

$\begin{equation} \int_0^T {{\omega ^T}(t)} y(t)dt \ge 0,\forall T > 0, \end{equation}$

(2.7)

对于任意的外部干扰输入$\omega (t) \in {R^p}$都成立, 则称系统(2.6) 无源.特别地, 若存在$\eta > 0$使得

$\begin{equation} \int_0^T {[{\omega ^T}(t)y(t) - \eta {\omega ^T}(t)\omega (t)} ]dt \ge 0,\forall T > 0. \end{equation}$

(2.8)

对所有零初始条件$\omega (t) \in {R^p}$的轨迹以及任意的外部干扰输入都成立, 则称系统(2.6) 是严格无源的.

引理2.2 ^[11] 对给定的对称矩阵$S = \left[ {\begin{array}{*{20}{c}} {{S_{11}}}&{{S_{12}}}\\ {{S_{21}}}&{{S_{22}}} \end{array}} \right]$, 则以下三个命题等价:

(1) $S < 0,$

(2) ${S_{22}} - {S_{21}}S_{11}^{ - 1}{S_{12}} < 0,$

(3) ${S_{11}} - {S_{12}}S_{22}^{ - 1}{S_{12}} < 0.$

在本文中, 我们主要针对如下两种形式的时滞进行无源性分析:

情形1 时滞是常数${\tau _1},{\tau _2},{\tau _3}$;

情形2 时滞${\tau _i}(t)$是时变时滞, 且可导, 并满足:

$\begin{equation} 0 \le {\tau _i}(t) \le {\bar \tau _i},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \le {\dot \tau _i}(t) \le {\mu _i} < 1. \end{equation}$

(2.9)

其中是${\tau _i},{\bar \tau _i},{\mu _i}, i = 1,2,3$非负常数, 令$\bar \tau = \max \{ {\bar \tau _1}, {\bar \tau _2},{\bar \tau _3}\}$.

定理3.1 对于给定的常数${\tau _1}, {\tau _2},{\tau _3},\eta $, 若存在对称的正定矩阵使$P,{Q_i}$得有如下线性矩阵不等式（LMI）成立:

$\begin{equation} \left[ {\begin{array}{*{20}{c}} \Sigma &{P{A_1}}&{P{A_2}}&{P{A_3}}&{P{{\tilde B}_f}}&{PG - {C^T}}\\ *&{ - {Q_1}}&0&0&0&0\\ *&*&{ - {Q_2}}&0&0&0\\ *&*&*&{ - {Q_3}}&0&0\\ *&*&*&*&{ - \kappa I}&0\\ *&*&*&*&*&{2\eta I - E - {E^T}} \end{array}} \right] < 0, \end{equation}$

(3.1)

其中$\Sigma = P{A^T}_0 + P{A_0} + \kappa {\alpha ^2} + \sum\limits_{i = 1}^3 {{Q_i}} $, 则系统(2.6) 是严格无源的.

证 构造如下的Lyapunov-Krasovskii泛函:

$\begin{equation} V(t) = {\textstyle{1 \over 2}}[x(t)Px(t) + \sum\limits_{i = 1}^3 {\int_{t - {\tau _i}}^t {{x^T}(s){Q_i}x(s)} } ds]. \end{equation}$

(3.2)

对Lyapunov-Krasovskii泛函沿(2.6) 式求导得

$\begin{eqnarray*} \dot V(t) &=& {\textstyle{1 \over 2}}[{{\dot x}^T}(t)Px(t) + {x^T}(t)P\dot x(t) + \sum\limits_{i = 1}^3 {{x^T}(t){Q_i}x(t) - } \sum\limits_{i = 1}^3 {{x^T}(t - {\tau _i}){Q_i}x(t - {\tau _i})} \\ & =& {[{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}) + {{\tilde B}_f}f(x(t))} + G\omega (t)]^T}Px(t)\\&& + {x^T}(t)P[{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}) + {{\tilde B}_f}f(x(t))} + G\omega (t)]\\ && + \sum\limits_{i = 1}^3 {{x^T}(t){Q_i}x(t) - } \sum\limits_{i = 1}^3 {{x^T}(t - {\tau _i}){Q_i}x(t - {\tau _i})} ]. \end{eqnarray*}$

由(2.3) 式可得

$\begin{eqnarray*} &&\dot V(t) - {\omega ^T}(t)y(t) + \eta {\omega ^T}(t)\omega (t)\\ &\le &{\textstyle{1 \over 2}}\{ {[{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}) + {{\tilde B}_f}f(x(t))} + G\omega (t)]^T}Px(t) \\&& + {x^T}(t)P[{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}) + {{\tilde B}_f}f(x(t))} + G\omega (t)]\\&& + \sum\limits_{i = 1}^3 {{x^T}(t){Q_i}x(t) - } \sum\limits_{i = 1}^3 {{x^T}(t - {\tau _i}){Q_i}x(t - {\tau _i})} \\&&- 2{\omega ^T}(t)Cx(t) - 2{\omega ^T}(t)E\omega (t) + 2\eta {\omega ^T}(t)\omega (t)\} + {\textstyle{1 \over 2}}[\kappa ({\alpha ^2}{x^T}(t)x(t) - {f^T}(x(t))f(x(t))]\end{eqnarray*}$

$\begin{eqnarray*} & = &{\textstyle{1 \over 2}}{\xi ^T}(t)\left[ {\begin{array}{*{20}{c}} \Sigma &{P{A_1}}&{P{A_2}}&{P{A_3}}&{P{{\tilde B}_f}}&{PG - {C^T}}\\ *&{ - {Q_1}}&0&0&0&0\\ *&*&{ - {Q_2}}&0&0&0\\ *&*&*&{ - {Q_3}}&0&0\\ *&*&*&*&{ - \kappa I}&0\\ *&*&*&*&*&{2\eta I - E - {E^T}} \end{array}} \right]\xi (t), \end{eqnarray*}$

其中${\xi ^T}(t) = [ {\begin{array}{*{20}{c}} {{x^T}(t)}&{{x^T}(t - {\tau _1}(t))}&{{x^T}(t - {\tau _2}(t))}&{{x^T}(t - {\tau _3}(t))}&{{f^T}(x(t))}&{{\omega ^T}(t)} \end{array}} ].$则由(3.1) 式可得$\dot V(t) - {y^T}(t)\omega + \eta {y^T}(t)y(t) < 0$成立, 由定义2.1可得系统(2.6) 是严格无源的.

定理3.2 对于给定的常数$\bar \tau ,{\mu _1},{\mu _2},{\mu _3},\eta $, 若存在对称的正定矩阵$P,{Z_i},{Q_i}$使得有如下线性矩阵不等式（LMI）成立

$\begin{equation} \left[ {\begin{array}{*{20}{c}} {{\Psi _{11}}}&{{\Psi _{12}}}\\ {{\Psi ^T}_{12}}&{{\Psi _{22}}} \end{array}} \right] < 0, \end{equation}$

(3.3)

其中

$\begin{eqnarray*}&& {\Psi _{11}} = P{A_0} + A_0^TP + \kappa {\alpha ^2} + \sum\limits_{i = 1}^3 {({Q_i} + \bar \tau A_0^T{Z_i}{A_0})} ,\\ && {\Psi _{12}} = [\begin{array}{*{20}{c}} {{J_1}}&{{J_2}}&{{J_3}}&{{J_4}}&{{J_5}} \end{array}],\\ && {\Psi _{22}} = \left[ {\begin{array}{*{20}{c}} {{W_1}}&{\sum\limits_{i = 1}^3 {A_1^T{Z_i}{A_2}} {\kern 1pt} }&{\sum\limits_{i = 1}^3 {A_1^T{Z_i}{A_3}} }&{\sum\limits_{i = 1}^3 {A_1^T{Z_i}{{\tilde B}_f}} }&{\sum\limits_{i = 1}^3 {A_1^T{Z_i}G} }\\ *&{{W_2}{\kern 1pt} }&{\sum\limits_{i = 1}^3 {A_2^T{Z_i}{A_3}} }&{\sum\limits_{i = 1}^3 {A_2^T{Z_i}{{\tilde B}_f}} }&{\sum\limits_{i = 1}^3 {A_2^T{Z_i}G} }\\ *&*&{{W_{3{\kern 1pt} }}}&{\sum\limits_{i = 1}^3 {A_3^T{Z_i}{{\tilde B}_f}} }&{\sum\limits_{i = 1}^3 {A_3^T{Z_i}G} }\\ *&*&*&{{W_4}}&{\sum\limits_{i = 1}^3 {\tilde B_f^T{Z_i}G} }\\ *&*&*&*&{{W_5}} \end{array}} \right],\\ && \begin{array}{l} {J_1} = P{A_1} + \sum\limits_{i = 1}^3 {\bar \tau A_0^T{Z_i}{A_1}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } ,{J_2} = P{A_1} + \sum\limits_{i = 1}^3 {\bar \tau A_0^T{Z_i}{A_2}{\kern 1pt} {\kern 1pt} {\kern 1pt} ,{J_3}{\kern 1pt} = P{A_1} + \sum\limits_{i = 1}^3 {\bar \tau A_0^T{Z_i}{A_3}{\kern 1pt} {\kern 1pt} } ,{\kern 1pt} {\kern 1pt} } \\ {J_4} = P{{\tilde B}_f} + \sum\limits_{i = 1}^3 {\bar \tau A_0^T{Z_i}{{\tilde B}_f}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } {\kern 1pt} ,{J_5} = PG + \sum\limits_{i = 1}^3 {\bar \tau A_0^T{Z_i}G - {C^T}}, \end{array}\\ && \begin{array}{l} {W_1} = \sum\limits_{i = 1}^3 {A_1^T{Z_i}{A_1} - {Q_1}(1 - {\mu _1})} , {W_2} = \sum\limits_{i = 1}^3 {A_2^T{Z_i}{A_2} - {Q_2}(1 - {\mu _2})} {\kern 1pt},\\ {\kern 1pt} {\kern 1pt} {W_3} = \sum\limits_{i = 1}^3 {A_3^T{Z_i}{A_3} - {Q_3}(1 - {\mu _3})} {\kern 1pt} ,{W_4} = \sum\limits_{i = 1}^3 {\tilde B_f^T{Z_i}{{\tilde B}_f} - \kappa I{\kern 1pt} } ,\\ {W_5} = \sum\limits_{i = 1}^3 {{G^T}{Z_i}G} + 2\eta I - E - {E^T}. \end{array}\end{eqnarray*}$

则系统(2.6) 对于任意的时变时滞都是严格无源的.

证 构造如下的Lyapunov-Krasovskii泛函

$\begin{equation} \begin{array}{l} V(t) = {\textstyle{1 \over 2}}[{V_1}(t) + {V_2}(t) + {V_3}(t)],\\ \end{array} \end{equation}$

(3.4)

其中

$\begin{eqnarray*}&& {V_1}(t) = {x^T}(t)Px(t), {V_2}(t) = \sum\limits_{i = 1}^3 {\int_{ - {\tau _i}(t)}^0 {\int_{t + \beta }^t {{{\dot x}^T}(\alpha ){Z_i}\dot x(\alpha )d\alpha d\beta } } },\\ && {V_3}(t) = \sum\limits_{i = 1}^3 {\int_{t - {\tau _i}(t)}^t {{x^T}(\alpha ){Q_i}x(\alpha )d\alpha .} }\end{eqnarray*}$

对Lyapunov-Krasovskii泛函沿(2.6) 式求导可得

$\begin{eqnarray*} {{\dot V}_1}(t) &=& {{\dot x}^T}(t)Px(t) + {x^T}(t)P\dot x(t) = [{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}(t)) + {{\tilde B}_f}f(x(t))} \\ && + G\omega (t){]^T}Px(t) + {x^T}(t)P[{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}(t)) + {{\tilde B}_f}f(x(t))} + G\omega (t)], \\ {{\dot V}_2}(t) &\le& \sum\limits_{i = 1}^3 {\bar \tau {{[{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}(t)) + {{\tilde B}_f}f(x(t))} + G\omega (t)]^T}}} {Z_i}[{A_0}x(t) \\&& + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}(t)) + {{\tilde B}_f}f(x(t))} + G\omega (t)]], \\ {{\dot V}_3}(t) &=& \sum\limits_{i = 1}^3 {{x^T}(t){Q_i}x(t) - } \sum\limits_{i = 1}^3 {{x^T}(t - {\tau _i}(t)){Q_i}x(t - {\tau _i}(t))} (1 - {{\dot \tau }_i}(t))\\ & \le& \sum\limits_{i = 1}^3 {{x^T}(t){Q_i}x(t) - } \sum\limits_{i = 1}^3 {{x^T}(t - {\tau _i}(t)){Q_i}x(t - {\tau _i}(t))} (1 - {\mu _i}), \end{eqnarray*}$

则由(2.3) 式可得

$\begin{eqnarray*} && \dot V(t) - {\omega ^T}(t)y(t) + \eta {\omega ^T}(t)\omega (t) \le {\textstyle{1 \over 2}}\{ [{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i} x(t - {\tau _i}(t)) + {{\tilde B}_f}f(x(t))} \\ && + G\omega (t){]^T}Px(t) + {x^T}(t)P[{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}(t)) + {{\tilde B}_f}f(x(t))} + G\omega (t)] \\&& + \sum\limits_{i = 1}^3 {\bar \tau {{[{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}(t)) + {{\tilde B}_f}f(x(t))} + G\omega (t)]^T}}} {Z_i}[{A_0}x(t) + \sum\limits_{i = 1}^3 {{A_i}x(t - {\tau _i}(t))} \\ && + {{\tilde B}_f}f(x(t)) + G\omega (t)] + \sum\limits_{i = 1}^3 {{x^T}(t){Q_i}x(t) - } \sum\limits_{i = 1}^3 {{x^T}(t - {\tau _i}(t)){Q_i}x(t - {\tau _i}(t))} (1 - {\mu _i})\\ && - 2{\omega ^T}(t)Cx(t) - 2{\omega ^T}(t)E\omega (t) + 2\eta {\omega ^T}(t)\omega (t)\} + {\textstyle{1 \over 2}}[\kappa ({\alpha ^2}{x^T}(t)x(t) - {f^T}(x(t))f(x(t))]\\ & \le& {\textstyle{1 \over 2}}{\xi ^T}(t)\left[ {\begin{array}{*{20}{c}} {{\Psi _{11}}}&{{\Psi _{12}}}\\ {{\Psi ^T}_{12}}&{{\Psi _{22}}} \end{array}} \right]\xi (t), \end{eqnarray*}$

其中${\xi ^T}(t) = [ {\begin{array}{*{20}{c}} {{x^T}(t)}&{{x^T}(t - {\tau _1}(t))}&{{x^T}(t - {\tau _2}(t))}&{{x^T}(t - {\tau _3}(t))}&{{f^T}(x(t))}&{{\omega ^T}(t)} \end{array}} ].$由(3.3) 式可得$\dot V(t) - {y^T}(t)\omega + \eta {y^T}(t)y(t) < 0$成立, 因此系统(2.6) 对于任意的时变时滞都是严格无源的.

若取$\eta = 0$, 我们可不加证明的得到如下推论:

推论3.3 对于给定的常数$\bar \tau ,{\mu _1},{\mu _2},{\mu _3}$, 如果存在对称的正定矩阵$P,{Z_i},{Q_i}$使得有如下线性矩阵不等式（LMI）成立

$\begin{equation} \left[ {\begin{array}{*{20}{c}} {{\Theta _{11}}}&{{\Theta _{12}}}\\ {{\Theta ^T}_{12}}&{{\Theta _{22}}} \end{array}} \right] < 0, \end{equation}$

(3.5)

其中${\Theta _{11}} = P{A_0} + A_0^TP + \kappa {\alpha ^2} + \sum\limits_{i = 1}^3 {({Q_i} + \bar \tau A_0^T{Z_i}{A_0})} ,{\Theta _{12}} = {\Psi _{12}}, {\Theta _{22}} = {\Psi _{22}}.$则系统(2.6) 式对于任意的时变时滞都无源.

考虑系统(2.6), 给定系统各参数矩阵如下:

$\begin{eqnarray*} & & {A_p} = \left[ {\begin{array}{*{20}{c}} { -5} & 1\\ 2 & { -6} \end{array}} \right], {B_p} = \left[ {\begin{array}{*{20}{c}} 1 & { -2}\\ 2 & 1 \end{array}} \right], {G_p} = \left[ {\begin{array}{*{20}{c}} 2 & 6\\ 3 & { -1} \end{array}} \right], {B_f} = \left[ {\begin{array}{*{20}{c}} { -2} & {1.8}\\ 0 & 1 \end{array}} \right], \\ & & {C_p} = \left[ {\begin{array}{*{20}{c}} 3 & 1\\ 0 & 2 \end{array}} \right], {E_p} = \left[ {\begin{array}{*{20}{c}} { -2} & 0\\ 0 & { -1} \end{array}} \right], {A_c} = \left[ {\begin{array}{*{20}{c}} { -4} & { -1}\\ 2 & { -5} \end{array}} \right], {B_c} = \left[ {\begin{array}{*{20}{c}} {0.2} & { -0.1}\\ {0.1} & {0.3} \end{array}} \right], \\ & & {G_c} = \left[ {\begin{array}{*{20}{c}} 1 & 5\\ 3 & { -2} \end{array}} \right], {C_c} = \left[ {\begin{array}{*{20}{c}} { -1.72} & {0.76}\\ { -0.16} & {1.08} \end{array}} \right], {D_c} = \left[ {\begin{array}{*{20}{c}} 3 & 2\\ 0 & 1 \end{array}} \right], f = \left[ {\begin{array}{*{20}{c}} {{e^{ -0.1t}}\sin 0.5{x_1}{x_2}}\\ {\cos (0.5({x_1} + {x_2}))} \end{array}} \right], \\ & & {\tau _1}(t) = 0.5\sin t \le 0.5 = {{\bar \tau }_1}, {{\dot \tau }_1} \le 0.5 = {\mu _1}, {\tau _2}(t) = 0.2\cos t + 0.3 \le 0.5 = {{\bar \tau }_2}, {{\dot \tau }_2} \le 0.2 = {\mu _2}, \\ & & {\tau _3}(t) = 0.4\sin t + 0.2 \le 0.6 = {{\bar \tau }_3} {{\dot \tau }_3} \le 0.4 = {\mu _3}, \bar \tau = 0.6.\end{eqnarray*}$

利用Matlab中的LMI工具箱, 由推论3.3计算可得线性矩阵不等式有可行解

$\begin{eqnarray*} && P = \left( \begin{array}{llll} {\rm{3.9187}}&{\rm{- 0.9029}}&{\rm{ 0.0612}}&{\rm{- 0.1985}}\\ {\rm{- 0.9029}}&{\rm{0.8686}}&{\rm{0.1674}}&{\rm{0.7652}}\\ {\rm{0.0612}}&{\rm{ 0.1674}}&{\rm{3.5418}}&{\rm{- 0.5896}}\\ {\rm{- 0.1985}}&{\rm{ 0.7652}}&{\rm{- 0.5896}}&{\rm{3.9627}} \end{array} \right),\\ && {Q_1} = \left( \begin{array}{llll} {\rm{3.9067}}&{\rm{ - 1.0305}}&{\rm{0.0442 }}&{\rm{-0.0634}}\\ {\rm{-1.0305}}&{\rm{ 2.3763 }}&{\rm{0.0431 }}&{\rm{0.4046}}\\ {\rm{0.0442}}&{\rm{ 0.0431}}&{\rm{3.1217 }}&{\rm{-0.9596}}\\ {\rm{- 0.0634}}&{\rm{ 0.4046}}&{\rm{- 0.9596 }}&{\rm{0.2152}} \end{array} \right),\\ && {Q_2} = \left( \begin{array}{llll} {\rm{7.3990}}&{\rm{- 1.2263}}&{\rm{0.1175}}&{\rm{- 0.1227}}\\ {\rm{-1.2263}}&{\rm{ 4.9048}}&{\rm{0.1222}}&{\rm{ 0.5783}}\\ {\rm{0.1175}}&{\rm{ 0.1222}}&{\rm{5.6348}}&{\rm{- 1.5247}}\\ {\rm{-0.1227}}&{\rm{ 0.5783}}&{\rm{- 1.5247}}&{\rm{ 7.3294}} \end{array} \right),\\ && {Q_3} = \left( \begin{array}{llll} {\rm{4.6911}}&{\rm{- 1.1061}}&{\rm{ 0.0580}}&{\rm{- 0.0760}}\\ {\rm{- 1.1061}}&{\rm{ 2.9425}}&{\rm{ 0.0581}}&{\rm{ 0.4505}}\\ {\rm{0.0580}}&{\rm{0.0581}}&{\rm{ 3.7126}}&{\rm{ - 1.0949}}\\ {\rm{ - 0.0760}}&{\rm{ 0.4505}}&{\rm{ - 1.0949}}&{\rm{ 4.9524}} \end{array} \right). \end{eqnarray*}$

图 2-图 3分别给出了网络控制系统中网络控制对象与时滞控制器的状态轨迹曲线, 从图中我们可以看到系统是稳定的.图 4给出了网络控制对象的输出轨迹, 而从图 5中我们可以看到网络控制系统的性能指标$\displaystyle\int_0^T {{\omega ^T}(t)} y(t)dt \ge 0,\forall T > 0$满足, 即系统是无源的.

本文对一类非线性时滞网络控制系统的无源性问题进行了分析, 通过构造Lyapunov-Krasovskii泛函, 在考虑两种不同时滞的情况下, 以线性矩阵不等式(LMI)的形式分别给出了系统满足无源性的时滞无关与时滞相关充分条件, 最后通过仿真算例验证了结论的正确性和方法的有效性.基于本文结论, 进一步可以讨论网络控制系统其它更多的性质, 并为网络控制系统设计相应的无源控制器等.