Lyapunov-Krasovskii泛函三重积分项求导_原理

在之前转载的Lyapunov-Krasovskii泛函二重积分项求导_原理文章的基础上，这里给出Lyapunov-Krasovskii泛函三重积分项求导简单的计算过程及不等式放缩引理，主要是在文章的基础上进行一个总结。

1 Lyapunov-Krasovskii泛函三重积分项举例

这里给出研究时滞系统、网络化控制系统时常出现的Lyapunov-Krasovskii泛函三重积分项，主要参考的是以下三篇文章。

1. P. Park, W. Lee, and S. Lee, “Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems,” Journal of the Franklin Institute, vol. 352, no. 4, pp. 1378-1396, Apr. 2015.
2. J, Zhang and Y. Ma, “Event-triggered dissipative double asynchronous controller for interval type-2 fuzzy semi-Markov jump systems with state quantization and actuator failure,” ISA Transactions, vol. 138, pp. 226-242, Jul. 2023.
3. D. Zhang, Z. Ye, G. Feng and H. Li, “Intelligent event-based fuzzy dynamic positioning control of nonlinear unmanned marine vehicles under DoS attack,” IEEE Transactions on Cybernetics, vol. 52, no. 12, pp. 13486-13499, Dec. 2022.

定义时变时延$h(t)$满足$0\le h_1\le h(t)\le h_2$，$h_{12}=h_2-h_1$。给出如下几种常见的Lyapunov-Krasovskii泛函三重积分项。
$$\begin{array}{rl}{V}_1(t)& ={\int }_{-h_1}^0{\int }_{-h_1}^{\gamma }{\int }_{t+\beta }^t{\dot{x}}^T(\alpha )Z_1\dot{x}(\alpha )d\alpha d\beta d\gamma \\ V_2(t)& ={\int }_{-h_2}^{-h_1}{\int }_{\gamma }^{-h_1}{\int }_{t+\beta }^t{\dot{x}}^T(\alpha )Z_2\dot{x}(\alpha )d\alpha d\beta d\gamma \\ V_3(t)& ={\int }_{-h_2}^{-h_1}{\int }_{-h_2}^{\gamma }{\int }_{t+\beta }^t{\dot{x}}^T(\alpha )Z_3\dot{x}(\alpha )d\alpha d\beta d\gamma \end{array}\text{\hspace{1em}(1)}$$

定义时变时延$\tau (t)$满足$0\le {\tau }_1\le \tau (t)\le {\tau }_2$，${\tau }_{12}={\tau }_2-{\tau }_1$。给出如下几种常见的Lyapunov-Krasovskii泛函三重积分项。
$$\begin{array}{rl}{V}_1(t)& ={\int }_{-{\tau }_2}^{-{\tau }_1}{\int }_{\beta }^{-{\tau }_1}{\int }_{t+u}^t{e}^{2\alpha (s-u)}{\dot{x}}^T(s)R_1\dot{x}(s)dsdud\beta \\ V_2(t)& ={\int }_{-{\tau }_2}^{-{\tau }_1}{\int }_{-{\tau }_2}^{\beta }{\int }_{t+u}^t{e}^{2\alpha (s-u)}{\dot{x}}^T(s)R_2\dot{x}(s)dsdud\beta \end{array}\text{\hspace{1em}(2)}$$

定义时变时延$d(t)$满足$0\le d_m\le d(t)\le d_M$，$d_{12}=d_M-d_m$。给出Lyapunov-Krasovskii泛函三重积分项。
$$V_1(t)={\int }_{-d_M}^{-d_m}{\int }_v^{-d_m}{\int }_{t+\theta }^t{e}^{2\alpha (t-u)}{\dot{x}}^T(u)Q_1\dot{x}(u)dud\theta dv\text{\hspace{1em}(3)}$$

2 Lyapunov-Krasovskii泛函三重积分项求导

对于Lyapunov-Krasovskii泛函二重积分项，参考之前的文章Lyapunov-Krasovskii泛函二重积分项求导_原理，可以得到如下的计算过程和结果。
$$\begin{array}{rl}V(t)& ={\int }_{-h_1}^0{\int }_{t+\beta }^t{\dot{x}}^T(\alpha )M_1\dot{x}(\alpha )d\alpha d\beta \\ \frac{dV(t)}{dt}& ={\int }_{-h_1}^0[{\dot{x}}^T(t)M_1\dot{x}(t)-{\dot{x}}^T(t+\beta )M_1\dot{x}(t+\beta )]d\beta \\ & ={\int }_{-h_1}^0{\dot{x}}^T(t)M_1\dot{x}(t)d\beta -{\int }_{-h_1}^0{\dot{x}}^T(t+\beta )M_1\dot{x}(t+\beta )d\beta \\ & ={\dot{x}}^T(t)M_1\dot{x}(t)-{\int }_{t-h_1}^t{\dot{x}}^T(\beta )M_1\dot{x}(\beta )d\beta \end{array}\text{\hspace{1em}(5)}$$

同样的原理，对于Lyapunov-Krasovskii泛函三重积分项，我们可以看成是二重积分项和一重积分项的嵌套，那么，我们可以按照以下的步骤进行Lyapunov-Krasovskii泛函三重积分项求导的计算。
$$\begin{array}{rl}V(t)& ={\int }_{-h_2}^{-h_1}{\int }_{\gamma }^{-h_1}{\int }_{t+\beta }^t{\dot{x}}^T(\alpha )M_2\dot{x}(\alpha )d\alpha d\beta d\gamma \\ \frac{dV(t)}{dt}& ={\int }_{-h_2}^{-h_1}{\int }_{\gamma }^{-h_1}[{\dot{x}}^T(t)M_2\dot{x}(t)-{\dot{x}}^T(t+\beta )M_2\dot{x}(t+\beta )]d\beta d\gamma \\ & ={\int }_{-h_2}^{-h_1}{\int }_{\gamma }^{-h_1}{\dot{x}}^T(t)M_2\dot{x}(t)d\beta d\gamma -{\int }_{-h_2}^{-h_1}{\int }_{t+\gamma }^{t-h_1}{\dot{x}}^T(\beta )M_2\dot{x}(\beta )d\beta d\gamma \end{array}\text{\hspace{1em}(6)}$$

到这一步，我们看到，得到的结果和Lyapunov-Krasovskii泛函二重积分项求导的结果很类似，对于第一项，由于被积函数 ${\dot{x}}^T(t)M_2\dot{x}(t)$ 不含有积分变量 $\beta$ 和 $\gamma$，因此，我们可以将此项进一步处理，得到如下结果。
$$\begin{array}{rl}{\int }_{-h_2}^{-h_1}{\int }_{\gamma }^{-h_1}{\dot{x}}^T(t)M_2\dot{x}(t)d\beta d\gamma & ={\int }_{-h_2}^{-h_1}{\int }_{\gamma }^{-h_1}{\dot{x}}^T(t)M_2\dot{x}(t)d\gamma \\ & ={\int }_{-h_2}^{-h_1}(-h_1-\gamma ){\dot{x}}^T(t)M_2\dot{x}(t)d\gamma \\ & =\frac{h_{12}^2}{2}{\dot{x}}^T(t)M_2\dot{x}(t)\end{array}\text{\hspace{1em}(7)}$$

因此，结合公式 (6) 和 (7)，可得到如下结果。
$$\begin{array}{rl}\frac{dV(t)}{dt}& ={\int }_{-h_2}^{-h_1}{\int }_{\gamma }^{-h_1}[{\dot{x}}^T(t)M_2\dot{x}(t)-{\dot{x}}^T(t+\beta )M_2\dot{x}(t+\beta )]d\beta d\gamma \\ & ={\int }_{-h_2}^{-h_1}{\int }_{\gamma }^{-h_1}{\dot{x}}^T(t)M_2\dot{x}(t)d\beta d\gamma -{\int }_{-h_2}^{-h_1}{\int }_{t+\gamma }^{t-h_1}{\dot{x}}^T(\beta )M_2\dot{x}(\beta )d\beta d\gamma \\ & =\frac{h_{12}^2}{2}{\dot{x}}^T(t)M_2\dot{x}(t)-{\int }_{-h_2}^{-h_1}{\int }_{t+\gamma }^{t-h_1}{\dot{x}}^T(\beta )M_2\dot{x}(\beta )d\beta d\gamma \\ & =\frac{h_{12}^2}{2}{\dot{x}}^T(t)M_2\dot{x}(t)-{\int }_{-h(t)}^{-h_1}{\int }_{t+\gamma }^{t-h_1}{\dot{x}}^T(\beta )M_2\dot{x}(\beta )d\beta d\gamma \\ & -{\int }_{-h_2}^{-h(t)}{\int }_{t+\gamma }^{t-h(t)}{\dot{x}}^T(\beta )M_2\dot{x}(\beta )d\beta d\gamma -(h_2-h(t)){\int }_{t-h(t)}^{t-h_1}{\dot{x}}^T(\beta )M_2\dot{x}(\beta )d\beta \end{array}\text{\hspace{1em}(8)}$$

注意到，最后结果对项 ${\int }_{-h_2}^{-h_1}{\int }_{t+\gamma }^{t-h_1}{\dot{x}}^T(\beta )M_2\dot{x}(\beta )d\beta d\gamma$ 拆分成了三项，这是为了让更多的时滞信息得到利用，降低设计的保守性。对于结果中的一重积分项和二重积分项的处理，即通过一些不等式关系来放缩，将会在下面两节来介绍。

对于带有 $e$ 指数的Lyapunov-Krasovskii泛函三重积分项，可以看出是常规的Lyapunov-Krasovskii泛函三重积分和 $e$ 指数乘在一起构成的，直接运用乘积求导法则即可得到求导结果，这里不再给出详细的推导过程。

3 基于辅助函数的一重积分不等式

参考文章 “Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems”，可以得到比常规的 Jensen 不等式更紧的放缩结果，引理如下：

引理1. 对于可积函数 { x ( s ) ∣ s ∈ [ a , b ] } \{x(s)|s\in[a,b]\} {x(s)∣s∈[a,b]} 和 任意的正定矩阵 $R>0$，下列不等式成立：
$$\begin{array}{rl}{\int }_a^b{x}^T(s)Rx(s)ds& \ge \frac{1}{b-a}({\int }_a^{b}x(s)ds{)}^{T}R({\int }_a^{b}x(s)ds)+\frac{3}{b-a}{\Omega }_1^{T}R{\Omega }_1\\ {\int }_a^b{x}^T(s)Rx(s)ds& \ge \frac{1}{b-a}({\int }_a^{b}x(s)ds{)}^{T}R({\int }_a^{b}x(s)ds)+\frac{5}{b-a}{\Omega }_2^{T}R{\Omega }_2\\ {\int }_a^b{x}^T(s)Rx(s)ds& \ge \frac{1}{b-a}({\int }_a^{b}x(s)ds{)}^{T}R({\int }_a^{b}x(s)ds)+\frac{3}{b-a}{\Omega }_1^{T}R{\Omega }_1+\frac{5}{b-a}{\Omega }_2^{T}R{\Omega }_2\end{array}$$

其中
$$\begin{array}{rl}{\Omega }_1& ={\int }_a^{b}x(s)ds-\frac{2}{(b-a)}{\int }_a^b{\int }_u^{b}x(s)dsdu\\ {\Omega }_2& ={\int }_a^{b}x(s)ds-\frac{6}{(b-a)}{\int }_a^b{\int }_u^{b}x(s)dsdu+\frac{12}{(b-a{)}^2}{\int }_a^b{\int }_u^b{\int }_{\beta }^{b}x(s)dsdud\beta \end{array}$$

引理2. 对于可积函数 { x ( s ) ∣ s ∈ [ a , b ] } \{x(s)|s\in[a,b]\} {x(s)∣s∈[a,b]} 和 任意的正定矩阵 $R>0$，下列不等式成立：
$$\begin{array}{rl}{\int }_a^b{\dot{x}}^T(s)R\dot{x}(s)ds& \ge \frac{1}{b-a}{\Omega }_3^{T}R{\Omega }_3+\frac{3}{b-a}{\Omega }_4^{T}R{\Omega }_4\\ {\int }_a^b{\dot{x}}^T(s)R\dot{x}(s)ds& \ge \frac{1}{b-a}{\Omega }_3^{T}R{\Omega }_3+\frac{3}{b-a}{\Omega }_4^{T}R{\Omega }_4+\frac{5}{b-a}{\Omega }_5^{T}R{\Omega }_5\end{array}$$

其中
$$\begin{array}{rl}{\Omega }_3& =x(b)-x(a)\\ {\Omega }_4& =x(b)-x(a)-\frac{2}{(b-a)}{\int }_a^{b}x(s)ds\\ {\Omega }_5& =x(b)-x(a)+\frac{6}{(b-a)}{\int }_a^{b}x(s)ds-\frac{12}{(b-a{)}^2}{\int }_a^b{\int }_u^{b}x(s)dsdu\end{array}$$

4 基于辅助函数的二重积分不等式

参考文章 “Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems”，给出基于辅助函数的二重积分不等式放缩结果，引理如下：

引理2. 对于可积函数 { x ( s ) ∣ s ∈ [ a , b ] } \{x(s)|s\in[a,b]\} {x(s)∣s∈[a,b]} 和 任意的正定矩阵 $R>0$，下列不等式成立：
$$\begin{array}{rl}{\int }_a^b{\int }_u^b{\dot{x}}^T(s)R\dot{x}(s)dsdu& \ge 2{\Omega }_6^{T}R{\Omega }_6+4{\Omega }_7^{T}R{\Omega }_7\\ {\int }_a^b{\int }_u^b{\dot{x}}^T(s)R\dot{x}(s)dsdu& \ge 2{\Omega }_8^{T}R{\Omega }_8+4{\Omega }_9^{T}R{\Omega }_9\end{array}$$

其中
$$\begin{array}{rl}{\Omega }_6& =x(b)-\frac{1}{(b-a)}{\int }_a^{b}x(s)ds\\ {\Omega }_7& =x(b)+\frac{2}{(b-a)}{\int }_a^{b}x(s)ds-\frac{6}{(b-a{)}^2}{\int }_a^b{\int }_u^{b}x(s)dsdu\\ {\Omega }_8& =x(a)-\frac{1}{(b-a)}{\int }_a^{b}x(s)ds\\ {\Omega }_9& =x(a)-\frac{4}{(b-a)}{\int }_a^{b}x(s)ds+\frac{6}{(b-a{)}^2}{\int }_a^b{\int }_u^{b}x(s)dsdu\end{array}$$

注：3、4节中得到的结果比常规的Jenson不等式更紧，可以在一定程度上利用更多的系统时滞信息，降低设计结果的保守性。这里只是简单列举出了放缩结果，若对放缩结果有所疑问，在文章“Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems”中有非常详细的推导和证明过程，这里不再详细介绍。

参考文献：

Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems.
Event-triggered dissipative double asynchronous controller for interval type-2 fuzzy semi-Markov jump systems with state quantization and actuator failure.
Intelligent event-based fuzzy dynamic positioning control of nonlinear unmanned marine vehicles under DoS attack.