不等式的推广以及在加性时变时滞系统中的应用

5 主要结论

在本节中, 我们将基于扩展不等式, 给出加性时变时滞系统的保守性较低的稳定性准则. 简单起见, 定义以下向量和矩阵

$\begin{array}{l} e_{j}=\left[0_{n \times(j-1) n} I_{n} 0_{n \times(9-j) n}\right],(j=1,2, \cdots, 9), \\ \tilde{e}_{j}=\left[0_{n \times(j-1) n} I_{n} 0_{n \times(8-j) n}\right],(j=1,2, \cdots, 8), \\ \xi(t)=\left[x^{T}(t), x^{T}\left(t-d_{1}(t)\right), x^{T}\left(t-d_{1}(t)-d_{2}(t)\right), x^{T}\left(t-d_{1}(t)-\bar{d}_{2}\right), x^{T}\left(t-\bar{d}_{1}-\bar{d}_{2}\right),\right. \end{array}$

$\begin{array}{l} \frac{1}{d_{1}(t)} \int_{t-d_{1}(t)}^{t} x^{T}(s) \mathrm{d} s, \frac{1}{d_{2}(t)} \int_{t-d_{1}(t)-d_{2}(t)}^{t-d_{1}(t)} x^{T}(s) \mathrm{d} s, \frac{1}{\bar{d}_{2}-d_{2}(t)} \int_{t-d_{1}(t)-\bar{d}_{2}}^{t-d_{1}(t)-d_{2}(t)} x^{T}(s) \mathrm{d} s, \\ \left.\frac{1}{\bar{d}_{1}-d_{1}(t)} \int_{t-\bar{d}_{1}-\bar{d}_{2}}^{t-d_{1}(t)-\bar{d}_{2}} x^{T}(s) \mathrm{d} s\right]^{T}, \end{array}$

$\begin{array}{l} \Gamma_{1}=\left[\begin{array}{c} x\left(t-d_{1}(t)-\bar{d}_{2}\right)-x\left(t-\bar{d}_{1}-\bar{d}_{2}\right) \\ x\left(t-d_{1}(t)-\bar{d}_{2}\right)+x\left(t-\bar{d}_{1}-\bar{d}_{2}\right)-\frac{2}{d_{1}-d_{1}(t)} \int_{t-\bar{d}_{1}-\bar{d}_{2}}^{t-d_{2}(t)-\bar{d}_{2}} x(s) \mathrm{d} s \end{array}\right], \\ \Gamma_{2}=\left[\begin{array}{c} x\left(t-d_{1}(t)-d_{2}(t)\right)-x\left(t-d_{1}(t)-\bar{d}_{2}\right) \\ x\left(t-d_{1}(t)-d_{2}(t)\right)+x\left(t-d_{1}(t)-\bar{d}_{2}\right)-\frac{2}{d_{2}-d_{2}(t)} \int_{t-d_{1}(t)-\bar{d}_{2}}^{t-d_{1}(t)-d_{2}(t)} x(s) \mathrm{d} s \end{array}\right], \\ \Gamma_{3}=\left[\begin{array}{c} x\left(t-d_{1}(t)\right)-x\left(t-d_{1}(t)-d_{2}(t)\right) \\ x\left(t-d_{1}(t)\right)+x\left(t-d_{1}(t)-d_{2}(t)\right)-\frac{2}{d_{2}(t)} \int_{t-d_{1}(t)-d_{2}(t)}^{t-d_{1}(t)} x(s) \mathrm{d} s \end{array}\right], \\ \Gamma_{4}=\left[\begin{array}{c} x(t)-x\left(t-d_{1}(t)\right) \\ x(t)+x\left(t-d_{1}(t)\right)-\frac{2}{d_{1}(t)} \int_{t-d_{1}(t)}^{t} x(s) \mathrm{d} s \end{array}\right], \\ \Gamma=\left[\begin{array}{l} \Gamma_{1} \\ \Gamma_{2} \\ \Gamma_{3} \\ \Gamma_{4} \end{array}\right]. \end{array}$

定理 5.1 给定标量 ${\bar d_j} > 0$ , ${\mu _j} > 0,(j = 1,2)$ 以及满足条件 $k_1^2 \le 36$ 的常复数 ${k_1}$ , 则满足(2.2) 式的时滞系统 (2.1) 是渐近稳定的, 如果存在 $2n \times 2n$ 的矩阵 $\left[ \begin{matrix} {{P}_{1}} & {{P}_{2}} \\ * & {{P}_{3}} \\ \end{matrix} \right]>0$ , $n \times n$ 的矩阵 ${Q_j} > 0,\left({j = 1,2,3,4} \right)$ , $W > 0$ , $8n \times 8n$ 的矩阵 $M$ , 使如下线性矩阵不等式 (5.1)-(5.4) 成立,

(5.1)

(5.2)

(5.3)

(5.4)

其中

$\begin{align*} & {{\Xi }_{1}}=Sym\{e_{1}^{T}({{P}_{1}}A+{{P}_{2}}){{e}_{1}}+e_{1}^{T}{{P}_{1}}B{{e}_{3}}-e_{1}^{T}{{P}_{2}}B{{e}_{5}}\}, \\ & {{\Xi }_{2}}=e_{1}^{T}({{Q}_{1}}+{{Q}_{2}}+{{Q}_{3}}+{{Q}_{4}}){{e}_{1}}-(1-{{\mu }_{1}})e_{2}^{T}{{Q}_{1}}{{e}_{2}}\\ & \hskip.8cm -(1-{{\mu }_{1}}-{{\mu }_{2}})e_{3}^{T}{{Q}_{2}}{{e}_{3}}-(1-{{\mu }_{1}})e_{4}^{T}{{Q}_{3}}{{e}_{4}}-e_{5}^{T}{{Q}_{4}}{{e}_{5}}, \\ & {{\Xi }_{3}}=({{{\bar{d}}}_{1}}+{{{\bar{d}}}_{2}})(e_{1}^{T}{{A}^{T}}WA{{e}_{1}}+e_{3}^{T}{{B}^{T}}WB{{e}_{3}}+Sym\{e_{1}^{T}{{A}^{T}}WB{{e}_{3}}+{{E}^{T}}ME\}), \\ & {{\Phi }_{j}}=Sym\{e_{1}^{T}({{A}^{T}}{{P}_{2}}+{{P}_{3}}){{e}_{j+5}}+e_{3}^{T}{{B}^{T}}{{P}_{2}}{{e}_{j+5}}-e_{5}^{T}{{P}_{3}}{{e}_{j+5}}\},(j=1,2,3,4),\\ & {{E}_{j}}=\left[ \begin{matrix} {{{\tilde{e}}}_{2j-1}} \\ {{{\tilde{e}}}_{2j}} \\ \end{matrix} \right],(j=1,2,3,4),&\\ &\tilde{W}=\left[ \begin{matrix} W & 0 \\ 0 & \frac{432+k_{1}^{2}({{\pi }^{2}}-12)}{144}W \\ \end{matrix} \right],&\\ &E=\left[ \begin{matrix} {{e}_{4}}-{{e}_{5}} \\ {{e}_{4}}+{{e}_{5}}-2{{e}_{9}} \\ {{e}_{3}}-{{e}_{4}} \\ \begin{matrix} {{e}_{3}}+{{e}_{4}}-2{{e}_{8}} \\ {{e}_{2}}-{{e}_{3}} \\ {{e}_{2}}+{{e}_{3}}-2{{e}_{7}} \\ \begin{matrix} {{e}_{1}}-{{e}_{2}} \\ {{e}_{1}}+{{e}_{2}}-2{{e}_{6}} \\ \end{matrix} \\ \end{matrix} \\ \end{matrix} \right].& \end{align*}$

证构造以下 LKF,

$\begin{align*} V(t)={V_1}(t) + {V_2}(t) + {V_3}(t), \end{align*}$

(5.5)

$\begin{aligned} V_{1}(t)= & {\left[\begin{array}{c} x(t) \\ \int_{t-\bar{d}_{1}-\bar{d}_{2}}^{t} x(s) \mathrm{d} s \end{array}\right]^{T}\left[\begin{array}{ll} P_{1} & P_{2} \\ * & P_{3} \end{array}\right]\left[\begin{array}{c} x \\ \int_{t-\bar{d}_{1}-\bar{d}_{2}}^{t} x(t) \mathrm{d} s \end{array}\right] }, \\ V_{2}(t)= & \int_{t-d_{1}(t)}^{t} x^{T}(s) Q_{1} x(s) \mathrm{d} s+\int_{t-d_{1}(t)-d_{2}(t)}^{t} x^{T}(s) Q_{2} x(s) \mathrm{d} s \\ & +\int_{t-d_{1}(t)-\bar{d}_{2}}^{t} x^{T}(s) Q_{3} x(s) \mathrm{d} s+\int_{t-\bar{d}_{1}-\bar{d}_{2}}^{t} x^{T}(s) Q_{4} x(s) \mathrm{d} s, \\ V_{3}(t)= & \int_{-\bar{d}_{1}-\bar{d}_{2}}^{0} \int_{t+\theta}^{t} \dot{x}^{T}(s) W \dot{x}(s) \mathrm{d} s \mathrm{~d} \theta, \end{aligned}$

对 $V(t)$ 求导有以下结果

$\begin{align*} & {{{\dot{V}}}_{1}}(t)={{\xi }^{T}}(t)\left[ {{\Xi }_{1}}+{{d}_{1}}(t){{\Phi }_{1}}+{{d}_{2}}(t){{\Phi }_{2}}+({{{\bar{d}}}_{2}}-{{d}_{2}}(t)){{\Phi }_{3}}+({{{\bar{d}}}_{1}}-{{d}_{1}}(t)){{\Phi }_{4}} \right]\xi (t), \nonumber \\ & {{{\dot{V}}}_{2}}(t)={{\xi }^{T}}(t)\left[ e_{1}^{T}({{Q}_{1}}+{{Q}_{2}}+{{Q}_{3}}+{{Q}_{4}}){{e}_{1}}-(1-{{{\dot{d}}}_{1}}(t))e_{2}^{T}{{Q}_{1}}{{e}_{2}}-(1-{{{\dot{d}}}_{1}}(t)-{{{\dot{d}}}_{2}}(t))e_{3}^{T}{{Q}_{2}}{{e}_{3}}) \right. \nonumber\\ & \left. \hskip1.2cm -(1-{{{\dot{d}}}_{1}}(t))e_{4}^{T}{{Q}_{3}}{{e}_{4}}-e_{5}^{T}{{Q}_{4}}{{e}_{5}} \right]\xi (t) \le {{\xi }^{T}}(t){{\Xi }_{2}}\xi (t), \nonumber\\ &{{{\dot{V}}}_{3}}(t)=({{{\bar{d}}}_{1}}+{{{\bar{d}}}_{2}}){{{\dot{x}}}^{T}}(t)W\dot{x}(t)-\int_{t-{{{\bar{d}}}_{1}}-{{{\bar{d}}}_{2}}}^{t}{{{{\dot{x}}}^{T}}(s)W\dot{x}(s)}{\rm d}s \nonumber\\ & \hskip.8cm =({{{\bar{d}}}_{1}}+{{{\bar{d}}}_{2}}){{{\dot{x}}}^{T}}(t)W\dot{x}(t)-\int_{t-{{{\bar{d}}}_{1}}-{{{\bar{d}}}_{2}}}^{t-{{d}_{1}}(t)-{{{\bar{d}}}_{2}}}{{{{\dot{x}}}^{T}}(s)W\dot{x}(s)}{\rm d}s-\int_{t-{{d}_{1}}(t)-{{{\bar{d}}}_{2}}}^{t-{{d}_{1}}(t)-{{d}_{2}}(t)}{{{{\dot{x}}}^{T}}(s)W\dot{x}(s)}{\rm d}s \nonumber \\ & \hskip1.2cm -\int_{t-{{d}_{1}}(t)-{{d}_{2}}(t)}^{t-{{d}_{1}}(t)}{{{{\dot{x}}}^{T}}(s)W\dot{x}(s)}{\rm d}s-\int_{t-{{d}_{1}}(t)}^{t}{{{{\dot{x}}}^{T}}(s)W\dot{x}(s)}{\rm d}s, \end{align*}$

使用引理 4.1, 对上述积分项进行估计, 结果如下

$\begin{aligned} & -\int_{t-\bar{d}_{1}-\bar{d}_{2}}^{t-d_{1}(t)-\bar{d}_{2}} \dot{x}^{T}(s) W \dot{x}(s) \mathrm{d} s-\int_{t-d_{1}(t)-\bar{d}_{2}}^{t-d_{1}(t)-d_{2}(t)} \dot{x}^{T}(s) W \dot{x}(s) \mathrm{d} s \\ & -\int_{t-d_{1}(t)-d_{2}(t)}^{t-d_{1}(t)} \dot{x}^{T}(s) W \dot{x}(s) \mathrm{d} s-\int_{t-d_{1}(t)}^{t} \dot{x}^{T}(s) W \dot{x}(s) \mathrm{d} s \\ \leq & -\frac{1}{\bar{d}_{1}-d_{1}(t)} \Gamma_{1}^{T} \tilde{W} \Gamma_{1}-\frac{1}{\bar{d}_{2}-d_{2}(t)} \Gamma_{2}^{T} \tilde{W} \Gamma_{2}-\frac{1}{d_{2}(t)} \Gamma_{3}^{T} \tilde{W} \Gamma_{3}-\frac{1}{d_{1}(t)} \Gamma_{4}^{T} \tilde{W} \Gamma_{4} \\ = & {\left[\begin{array}{l} \Gamma_{1} \\ \Gamma_{2} \\ \Gamma_{3} \\ \Gamma_{4} \end{array}\right]\left[\begin{array}{cccc} T-\frac{1}{\bar{d}_{1}-d_{1}(t)} \tilde{W} & 0 & 0 \\ 0 & -\frac{1}{\bar{d}_{2}-d_{2}(t)} \tilde{W} & 0 & 0 \\ 0 & 0 & -\frac{1}{d_{2}(t)} \tilde{W} & 0 \\ 0 & 0 & 0 & -\frac{1}{d_{1}(t)} \tilde{W} \end{array}\right]\left[\begin{array}{l} \Gamma_{1} \\ \Gamma_{2} \\ \Gamma_{3} \\ \Gamma_{4} \end{array}\right] }, \end{aligned}$

(5.6)

接着, 运用引理 2.4 对上述式子进行处理

$\begin{aligned} & {\left[\begin{array}{l} \Gamma_{1} \\ \Gamma_{2} \\ \Gamma_{3} \\ \Gamma_{4} \end{array}\right]^{T}\left[\begin{array}{cccc} -\frac{1}{\bar{d}_{1}-d_{1}(t)} \tilde{W} & 0 & 0 & 0 \\ 0 & -\frac{1}{\bar{d}_{2}-d_{2}(t)} \tilde{W} & 0 & 0 \\ 0 & 0 & -\frac{1}{d_{2}(t)} \tilde{W} & 0 \\ 0 & 0 & 0 & -\frac{1}{d_{1}(t)} \tilde{W} \end{array}\right]\left[\begin{array}{l} \Gamma_{1} \\ \Gamma_{2} \\ \Gamma_{3} \\ \Gamma_{4} \end{array}\right] } \\ \leq & \Gamma^{T}\left[M+M^{T}+M^{T}\left[\begin{array}{cccc} \left(\bar{d}_{1}-d_{1}(t)\right) \tilde{W}^{-1} & 0 & 0 & 0 \\ 0 & \left(\bar{d}_{2}-d_{2}(t)\right) \tilde{W}^{-1} & 0 & 0 \\ 0 & 0 & d_{2}(t) \tilde{W}^{-1} & 0 \\ 0 & 0 & 0 & d_{1}(t) \tilde{W}^{-1} \end{array}\right] M\right] \Gamma, \end{aligned}$

可以获得

$\begin{aligned} \dot{V}_{3}(t) \leq & \left(\bar{d}_{1}+\bar{d}_{2}\right) \dot{x}^{T}(t) W \dot{x}(t) \\ & +\Gamma^{T}\left[M+M^{T}+M^{T}\left[\begin{array}{cccc} \left(\bar{d}_{1}-d_{1}(t)\right) \tilde{W}^{-1} & 0 & 0 & 0 \\ 0 & \left(\bar{d}_{2}-d_{2}(t)\right) \tilde{W}^{-1} & 0 & 0 \\ 0 & 0 & d_{2}(t) \tilde{W}^{-1} & 0 \\ 0 & 0 & 0 & d_{1}(t) \tilde{W}^{-1} \end{array}\right] M\right] \Gamma \end{aligned}$

$\begin{aligned} = & \xi^{T}(t)\left[\Xi_{3}+d_{1}(t) E^{T} M^{T} E_{4}^{T} \tilde{W}^{-1} E_{4} M E+d_{2}(t) E^{T} M^{T} E_{3}^{T} \tilde{W}^{-1} E_{3} M E\right. \\ & \left.+\left(\bar{d}_{2}-d_{2}(t)\right) E^{T} M^{T} E_{2}^{T} \tilde{W}^{-1} E_{2} M E+\left(\bar{d}_{1}-d_{1}(t)\right) E^{T} M^{T} E_{1}^{T} \tilde{W}^{-1} E_{1} M E\right] \xi(t), \end{aligned}$

综上所述, 有

$\begin{aligned} \dot{V}(t) \leq & \xi^{T}(t)\left[\Xi_{1}+\Xi_{2}+\Xi_{3}+d_{1}(t)\left(\Phi_{1}+E^{T} M^{T} E_{4}^{T} \tilde{W}^{-1} E_{4} M E\right)\right. \\ & +d_{2}(t)\left(\Phi_{2}+E^{T} M^{T} E_{3}^{T} \tilde{W}^{-1} E_{3} M E\right)+\left(\bar{d}_{2}-d_{2}(t)\right)\left(\Phi_{3}+E^{T} M^{T} E_{2}^{T} \tilde{W}^{-1} E_{2} M E\right) \\ & \left.+\left(\bar{d}_{1}-d_{1}(t)\right)\left(\Phi_{4}+E^{T} M^{T} E_{1}^{T} \tilde{W}^{-1} E_{1} M E\right)\right] \xi(t), \end{aligned}$

基于凸组合技术, 下式成立则保证了 $\dot V(t) < 0$ 成立

$\begin{array}{l} \Xi_{1}+\Xi_{2}+\Xi_{3}+\bar{d}_{1}\left(\Phi_{1}+E^{T} M^{T} E_{4}^{T} \tilde{W}^{-1} E_{4} M E\right)+\bar{d}_{2}\left(\Phi_{2}+E^{T} M^{T} E_{3}^{T} \tilde{W}^{-1} E_{3} M E\right)<0, \\ \Xi_{1}+\Xi_{2}+\Xi_{3}+\bar{d}_{1}\left(\Phi_{1}+E^{T} M^{T} E_{4}^{T} \tilde{W}^{-1} E_{4} M E\right)+\bar{d}_{2}\left(\Phi_{3}+E^{T} M^{T} E_{2}^{T} \tilde{W}^{-1} E_{2} M E\right)<0, \\ \Xi_{1}+\Xi_{2}+\Xi_{3}+\bar{d}_{1}\left(\Phi_{4}+E^{T} M^{T} E_{1}^{T} \tilde{W}^{-1} E_{1} M E\right)+\bar{d}_{2}\left(\Phi_{2}+E^{T} M^{T} E_{3}^{T} \tilde{W}^{-1} E_{3} M E\right)<0, \\ \Xi_{1}+\Xi_{2}+\Xi_{3}+\bar{d}_{1}\left(\Phi_{4}+E^{T} M^{T} E_{1}^{T} \tilde{W}^{-1} E_{1} M E\right)+\bar{d}_{2}\left(\Phi_{3}+E^{T} M^{T} E_{2}^{T} \tilde{W}^{-1} E_{2} M E\right)<0, \end{array}$

依据引理 2.5, 如果式 (5.1)-(5.4) 成立则成立, 结论得证.

注 5.1 在上述推导过程中, 我们使用了引理 4.1 对积分项进行了估计. 此外, 本文中的引理 2.1, 引理 2.2, 引理 4.2 和引理 4.3 等均可只改变 $\tilde W$ 项实现对积分项的估计. 因此, 本文所提出的扩展不等式并未增加计算量.

为验证引理 2.6 的优越性, 本文同样借鉴了文献 [11] 中的 LKF 构造方法和积分项估计技术, 利用基于三阶矩阵的扩展互凸矩阵不等式 (引理 2.6) 代替文献 [11] 中的引理 2 来对凸项进行估计, 从而得出满足式 (2.2) 的时滞系统 (2.1) 的稳定性准则, 该准则如下所示

定理 5.2 给定标量 ${\bar d_j} > 0$ , ${\mu _j} > 0,(j = 1,2)$ , 则满足 (2.2) 式的时滞系统 (2.1) 是渐近稳定的, 如果存在 $2n \times 2n$ 的矩阵 $\left[\begin{matrix} {{P}_{1}} & {{P}_{2}} \\ * & {{P}_{3}} \\ \end{matrix} \right]>0$ , $n \times n$ 的矩阵 ${Q_j} > 0,\left({j = 1,2,3,4} \right)$ , $W > 0$ , $8n \times 8n$ 的矩阵 $M,N$ , 使如下线性矩阵不等式 (5.7)-(5.10) 成立

$\begin{align*} & \left[ \begin{matrix} {{\Xi }_{1}}+{{\Xi }_{2}}+{{\Xi }_{3}}+{{{\bar{d}}}_{1}}{{\Phi }_{1}}+{{{\bar{d}}}_{2}}{{\Phi }_{2}} & {{E}^{T}}{{M}^{T}}E_{4}^{T} & {{E}^{T}}{{N}^{T}}E_{4}^{T} & {{E}^{T}}{{M}^{T}}E_{3}^{T} & {{E}^{T}}{{N}^{T}}E_{3}^{T} \\ * & -\frac{2}{{{{\bar{d}}}_{1}}}\tilde{W} & 0 & 0 & 0 \\ * & * & -\frac{2}{{{{\bar{d}}}_{1}}}\tilde{W} & 0 & 0 \\ * & * & * & -\frac{2}{{{{\bar{d}}}_{2}}}\tilde{W} & 0 \\ * & * & * & * & -\frac{2}{{{{\bar{d}}}_{2}}}\tilde{W} \\ \end{matrix} \right]<0, \end{align*}$

(5.7)

$\begin{align*} &\left[ \begin{matrix} {{\Xi }_{1}}+{{\Xi }_{2}}+{{\Xi }_{3}}+{{{\bar{d}}}_{1}}{{\Phi }_{1}}+{{{\bar{d}}}_{2}}{{\Phi }_{3}} & {{E}^{T}}{{M}^{T}}E_{4}^{T} & {{E}^{T}}{{N}^{T}}E_{4}^{T} & {{E}^{T}}{{M}^{T}}E_{2}^{T} & {{E}^{T}}{{N}^{T}}E_{2}^{T} \\ * & -\frac{2}{{{{\bar{d}}}_{1}}}\tilde{W} & 0 & 0 & 0 \\ * & * & -\frac{2}{{{{\bar{d}}}_{1}}}\tilde{W} & 0 & 0 \\ * & * & * & -\frac{2}{{{{\bar{d}}}_{2}}}\tilde{W} & 0 \\ * & * & * & * & -\frac{2}{{{{\bar{d}}}_{2}}}\tilde{W} \\ \end{matrix} \right]<0, \end{align*}$

(5.8)

$\begin{align*} & \left[ \begin{matrix} {{\Xi }_{1}}+{{\Xi }_{2}}+{{\Xi }_{3}}+{{{\bar{d}}}_{1}}{{\Phi }_{4}}+{{{\bar{d}}}_{2}}{{\Phi }_{2}} & {{E}^{T}}{{M}^{T}}E_{1}^{T} & {{E}^{T}}{{N}^{T}}E_{1}^{T} & {{E}^{T}}{{M}^{T}}E_{3}^{T} & {{E}^{T}}{{N}^{T}}E_{3}^{T} \\ * & -\frac{2}{{{{\bar{d}}}_{1}}}\tilde{W} & 0 & 0 & 0 \\ * & * & -\frac{2}{{{{\bar{d}}}_{1}}}\tilde{W} & 0 & 0 \\ * & * & * & -\frac{2}{{{{\bar{d}}}_{2}}}\tilde{W} & 0 \\ * & * & * & * & -\frac{2}{{{{\bar{d}}}_{2}}}\tilde{W} \\ \end{matrix} \right]<0, \end{align*}$

(5.9)

$\begin{align*} & \left[ \begin{matrix} {{\Xi }_{1}}+{{\Xi }_{2}}+{{\Xi }_{3}}+{{{\bar{d}}}_{1}}{{\Phi }_{4}}+{{{\bar{d}}}_{2}}{{\Phi }_{3}} & {{E}^{T}}{{M}^{T}}E_{1}^{T} & {{E}^{T}}{{N}^{T}}E_{1}^{T} & {{E}^{T}}{{M}^{T}}E_{2}^{T} & {{E}^{T}}{{N}^{T}}E_{2}^{T} \\ * & -\frac{2}{{{{\bar{d}}}_{1}}}\tilde{W} & 0 & 0 & 0 \\ * & * & -\frac{2}{{{{\bar{d}}}_{1}}}\tilde{W} & 0 & 0 \\ * & * & * & -\frac{2}{{{{\bar{d}}}_{2}}}\tilde{W} & 0 \\ * & * & * & * & -\frac{2}{{{{\bar{d}}}_{2}}}\tilde{W} \\ \end{matrix} \right]<0. \end{align*}$

(5.10)

证本定理的证明类似于定理 5.1, 此处只给出不同于定理 5.1 的部分, 即本定理利用引理 2.6 对式 (5.6) 进行估计, 估计如下

$\begin{align*} & {{\left[ \begin{matrix} {{\Gamma }_{1}} \\ {{\Gamma }_{2}} \\ {{\Gamma }_{3}} \\ {{\Gamma }_{4}} \\ \end{matrix} \right]}^{T}}\left[ \begin{matrix} -\frac{1}{{{{\bar{d}}}_{1}}-{{d}_{1}}(t)}\tilde{W} & 0 & 0 & 0 \\ 0 & -\frac{1}{{{{\bar{d}}}_{2}}-{{d}_{2}}(t)}\tilde{W} & 0 & 0 \\ 0 & 0 & -\frac{1}{{{d}_{2}}(t)}\tilde{W} & 0 \\ 0 & 0 & 0 & -\frac{1}{{{d}_{1}}(t)}\tilde{W} \\ \end{matrix} \right]\left[ \begin{matrix} {{\Gamma }_{1}} \\ {{\Gamma }_{2}} \\ {{\Gamma }_{3}} \\ {{\Gamma }_{4}} \\ \end{matrix} \right] \\ \le\ & {{\Gamma }^{T}}\left[ \frac{1}{2}{{M}^{T}}\left[ \begin{matrix} ({{{\bar{d}}}_{1}}-{{d}_{1}}(t)){{{\tilde{W}}}^{-1}} & 0 & 0 & 0 \\ 0 & ({{{\bar{d}}}_{2}}-{{d}_{2}}(t)){{{\tilde{W}}}^{-1}} & 0 & 0 \\ 0 & 0 & {{d}_{2}}(t){{{\tilde{W}}}^{-1}} & 0 \\ 0 & 0 & 0 & {{d}_{1}}(t){{{\tilde{W}}}^{-1}} \\ \end{matrix} \right]M \right. \\ & +\frac{1}{2}{{N}^{T}}\left[ \begin{matrix} ({{{\bar{d}}}_{1}}-{{d}_{1}}(t)){{{\tilde{W}}}^{-1}} & 0 & 0 & 0 \\ 0 & ({{{\bar{d}}}_{2}}-{{d}_{2}}(t)){{{\tilde{W}}}^{-1}} & 0 & 0 \\ 0 & 0 & {{d}_{2}}(t){{{\tilde{W}}}^{-1}} & 0 \\ 0 & 0 & 0 & {{d}_{1}}(t){{{\tilde{W}}}^{-1}} \\ \end{matrix} \right]N\\ &\left.+\frac{M+N+{{M}^{T}}+{{N}^{T}}}{2} \right]\Gamma, \end{align*}$

其余步骤与定理 5.1 相同, 从而证明完成.

6 数值实例

在本节中, 应用一个经常用到的数值例子来验证所提出的扩展不等式的优越性. 考虑具有以下参数的系统 (2.1)

$A = \left[ {\begin{array}{*{20}{c}} { - 2}&0\\ 0&{ - 0.9} \end{array}} \right]\begin{array}{*{20}{c}},&{B = } \end{array}\left[ {\begin{array}{*{20}{c}} { - 1}&0\\ { - 1}&{ - 1} \end{array}} \right],$

假设时滞导数上界 ${\mu _1}$ 和 ${\mu _2}$ 分别为 0.1 和 0.8. 本研究旨在找到时滞 ${d_1}(t)$ 的上界 ${\bar d_1}$ 或时滞 ${d_2}(t)$ 的上界 ${\bar d_2}$ . 为了确保结果的可靠性, 将本文提出的准则与文献 [9,11,15] 中的稳定性准则进行比较. 在表6.1 和表6.2 中, 列出了相应的对比结果, 其中, i 为虚数. 结果表明: 1) 使用本文提出的定理 5.1 可以在不增加计算负担的情况下, 获得比文献 [11,15] 更大的时滞上界; 2) 当 ${\bar d_2=0.4,0.5}$ 时, 本文提出的定理 5.1 获得了比文献 [9] 更大的时滞 ${d_1}(t)$ 的上界 ${\bar d_1}$ ; 3) 由于时滞上界的获取不仅取决于积分不等式的估计精度, 还取决于 LKF 的构造, 因而, 本文提出的定理 5.1 没有获得比文献 [9] 更大的时滞 ${d_2}(t)$ 的上界 ${\bar d_2}$ ; 4) 鉴于 2), 3) 中 ${d_1}(t)$ 和 ${d_2}(t)$ 的取值上界并非在同一情况下取到, 侧面反映了时滞 ${d_1}(t)$ 和 ${d_2}(t)$ 是存在差异的, 故对加性时变时滞系统稳定性的研究是必要的.

表6.1 ${\bar d_1}$ 给定下的 ${d_2}(t)$ 的上界 ${\bar d_2}$

表6.2 ${\bar d_2}$ 给定下的 ${d_1}(t)$ 的上界 ${\bar d_1}$

注 6.1 关于 ${k_1}$ 的复数取值, 此处我们仅仅在纯虚数下进行了模拟仿真, 具体原因如下 1) 当 ${k_1}$ 取纯虚数值时, 仿真中的操作更为简便, 所涉及的运算主要是实数操作; 2) 仿真所应用的软件为 MATLAB, 其并不支持复数域运算.

为了验证引理 2.6 的优越性并增强实验结果的可信度, 采用了与文献 [11] 中相同的时滞系统进行仿真实验. 我们对定理 5.2 的结论与文献 [11] 中给出的加性时变时滞系统的稳定性准则进行稳定性分析比较, 并将实验结果汇总在表6.3 和表6.4 中进行展示. 通过对比表格数据, 可以得出结论: 定理 5.2 可以获得比文献 [11] 更大的时滞上界, 从而提高了系统的稳定性.

表6.3 ${\bar d_1}$ 给定下的 ${d_2}(t)$ 的上界 ${\bar d_2}$

表6.4 ${\bar d_2}$ 给定下的 ${d_1}(t)$ 的上界 ${\bar d_1}$

注 6.2 本节所采用的仿真实验平台为 MATLAB, 我们利用其内置的线性矩阵不等式 (LMI) 工具箱来求解线性矩阵不等式.

7 结语

本文首先从单调性和增加正项两个角度对 Wirtinger 不等式进行了论证. 接着, 提出了扩展不等式, 其中包括一个扩展 Wirtinger 不等式和一个基于三阶矩阵的扩展互凸不等式. 然后, 分别利用这两个扩展不等式, 给出了加性时变时滞系统的稳定性准则. 最后, 通过数值算例说明了扩展不等式的优越性. 在未来的研究工作中, 将试图从更多角度深入探讨以改进积分不等式来提高时滞系统的稳定性. 具体而言, 可从巧妙选择正项, 优化构造函数, 引入辅助不等式, 引入含参函数以及充分利用单调性等角度思考.

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