Lyapunov-Krasovskii泛函三重积分项求导_原理
- 1 Lyapunov-Krasovskii泛函三重积分项举例
- 2 Lyapunov-Krasovskii泛函三重积分项求导
- 3 基于辅助函数的一重积分不等式
- 4 基于辅助函数的二重积分不等式
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。
- 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.
- 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.
- 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 ) h(t) h(t)满足 0 ≤ h 1 ≤ h ( t ) ≤ h 2 0\leq h_1\leq h(t)\leq h_2 0≤h1≤h(t)≤h2, h 12 = h 2 − h 1 h_{12}=h_2-h_1 h12=h2−h1。给出如下几种常见的Lyapunov-Krasovskii泛函三重积分项。
V 1 ( t ) = ∫ − h 1 0 ∫ − h 1 γ ∫ t + β t x ˙ T ( α ) Z 1 x ˙ ( α ) d α d β d γ V 2 ( t ) = ∫ − h 2 − h 1 ∫ γ − h 1 ∫ t + β t x ˙ T ( α ) Z 2 x ˙ ( α ) d α d β d γ V 3 ( t ) = ∫ − h 2 − h 1 ∫ − h 2 γ ∫ t + β t x ˙ T ( α ) Z 3 x ˙ ( α ) d α d β d γ (1) \begin{aligned}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{aligned}\tag{1} V1(t)V2(t)V3(t)=∫−h10∫−h1γ∫t+βtx˙T(α)Z1x˙(α)dαdβdγ=∫−h2−h1∫γ−h1∫t+βtx˙T(α)Z2x˙(α)dαdβdγ=∫−h2−h1∫−h2γ∫t+βtx˙T(α)Z3x˙(α)dαdβdγ(1)
定义时变时延 τ ( t ) \tau(t) τ(t)满足 0 ≤ τ 1 ≤ τ ( t ) ≤ τ 2 0\leq \tau_1\leq \tau(t)\leq \tau_2 0≤τ1≤τ(t)≤τ2, τ 12 = τ 2 − τ 1 \tau_{12}=\tau_2-\tau_1 τ12=τ2−τ1。给出如下几种常见的Lyapunov-Krasovskii泛函三重积分项。
V 1 ( t ) = ∫ − τ 2 − τ 1 ∫ β − τ 1 ∫ t + u t e 2 α ( s − u ) x ˙ T ( s ) R 1 x ˙ ( s ) d s d u d β V 2 ( t ) = ∫ − τ 2 − τ 1 ∫ − τ 2 β ∫ t + u t e 2 α ( s − u ) x ˙ T ( s ) R 2 x ˙ ( s ) d s d u d β (2) \begin{aligned}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)ds dud\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)ds dud\beta\end{aligned}\tag{2} V1(t)V2(t)=∫−τ2−τ1∫β−τ1∫t+ute2α(s−u)x˙T(s)R1x˙(s)dsdudβ=∫−τ2−τ1∫−τ2β∫t+ute2α(s−u)x˙T(s)R2x˙(s)dsdudβ(2)
定义时变时延 d ( t ) d(t) d(t)满足 0 ≤ d m ≤ d ( t ) ≤ d M 0\leq d_m\leq d(t)\leq d_M 0≤dm≤d(t)≤dM, d 12 = d M − d m d_{12}=d_M-d_m d12=dM−dm。给出Lyapunov-Krasovskii泛函三重积分项。
V 1 ( t ) = ∫ − d M − d m ∫ v − d m ∫ t + θ t e 2 α ( t − u ) x ˙ T ( u ) Q 1 x ˙ ( u ) d u d θ d v (3) 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)du d\theta dv\tag{3} V1(t)=∫−dM−dm∫v−dm∫t+θte2α(t−u)x˙T(u)Q1x˙(u)dudθdv(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。
V ( t ) = ∫ − h 1 0 ∫ t + β t x ˙ T ( α ) M 1 x ˙ ( α ) d α d β d V ( t ) d t = ∫ − h 1 0 [ x ˙ T ( t ) M 1 x ˙ ( t ) − x ˙ T ( t + β ) M 1 x ˙ ( t + β ) ] d β = ∫ − h 1 0 x ˙ T ( t ) M 1 x ˙ ( t ) d β − ∫ − h 1 0 x ˙ T ( t + β ) M 1 x ˙ ( t + β ) d β = x ˙ T ( t ) M 1 x ˙ ( t ) − ∫ t − h 1 t x ˙ T ( β ) M 1 x ˙ ( β ) d β (5) \begin{aligned}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{aligned}\tag{5} V(t)dtdV(t)=∫−h10∫t+βtx˙T(α)M1x˙(α)dαdβ=∫−h10[x˙T(t)M1x˙(t)−x˙T(t+β)M1x˙(t+β)]dβ=∫−h10x˙T(t)M1x˙(t)dβ−∫−h10x˙T(t+β)M1x˙(t+β)dβ=x˙T(t)M1x˙(t)−∫t−h1tx˙T(β)M1x˙(β)dβ(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。
V ( t ) = ∫ − h 2 − h 1 ∫ γ − h 1 ∫ t + β t x ˙ T ( α ) M 2 x ˙ ( α ) d α d β d γ d V ( t ) d t = ∫ − h 2 − h 1 ∫ γ − h 1 [ x ˙ T ( t ) M 2 x ˙ ( t ) − x ˙ T ( t + β ) M 2 x ˙ ( t + β ) ] d β d γ = ∫ − h 2 − h 1 ∫ γ − h 1 x ˙ T ( t ) M 2 x ˙ ( t ) d β d γ − ∫ − h 2 − h 1 ∫ t + γ t − h 1 x ˙ T ( β ) M 2 x ˙ ( β ) d β d γ (6) \begin{aligned}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{aligned}\tag{6} V(t)dtdV(t)=∫−h2−h1∫γ−h1∫t+βtx˙T(α)M2x˙(α)dαdβdγ=∫−h2−h1∫γ−h1[x˙T(t)M2x˙(t)−x˙T(t+β)M2x˙(t+β)]dβdγ=∫−h2−h1∫γ−h1x˙T(t)M2x˙(t)dβdγ−∫−h2−h1∫t+γt−h1x˙T(β)M2x˙(β)dβdγ(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 x ˙ T ( t ) M 2 x ˙ ( t ) \dot{x}^T(t)M_2\dot{x}(t) x˙T(t)M2x˙(t) 不含有积分变量 β \beta β 和 γ \gamma γ,因此,我们可以将此项进一步处理,得到如下结果。
∫ − h 2 − h 1 ∫ γ − h 1 x ˙ T ( t ) M 2 x ˙ ( t ) d β d γ = ∫ − h 2 − h 1 ∫ γ − h 1 x ˙ T ( t ) M 2 x ˙ ( t ) d γ = ∫ − h 2 − h 1 ( − h 1 − γ ) x ˙ T ( t ) M 2 x ˙ ( t ) d γ = h 12 2 2 x ˙ T ( t ) M 2 x ˙ ( t ) (7) \begin{aligned}\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^2_{12}}{2}\dot{x}^T(t)M_2\dot{x}(t)\end{aligned}\tag{7} ∫−h2−h1∫γ−h1x˙T(t)M2x˙(t)dβdγ=∫−h2−h1∫γ−h1x˙T(t)M2x˙(t)dγ=∫−h2−h1(−h1−γ)x˙T(t)M2x˙(t)dγ=2h122x˙T(t)M2x˙(t)(7)
因此,结合公式 (6) 和 (7),可得到如下结果。
d V ( t ) d t = ∫ − h 2 − h 1 ∫ γ − h 1 [ x ˙ T ( t ) M 2 x ˙ ( t ) − x ˙ T ( t + β ) M 2 x ˙ ( t + β ) ] d β d γ = ∫ − h 2 − h 1 ∫ γ − h 1 x ˙ T ( t ) M 2 x ˙ ( t ) d β d γ − ∫ − h 2 − h 1 ∫ t + γ t − h 1 x ˙ T ( β ) M 2 x ˙ ( β ) d β d γ = h 12 2 2 x ˙ T ( t ) M 2 x ˙ ( t ) − ∫ − h 2 − h 1 ∫ t + γ t − h 1 x ˙ T ( β ) M 2 x ˙ ( β ) d β d γ = h 12 2 2 x ˙ T ( t ) M 2 x ˙ ( t ) − ∫ − h ( t ) − h 1 ∫ t + γ t − h 1 x ˙ T ( β ) M 2 x ˙ ( β ) d β d γ − ∫ − h 2 − h ( t ) ∫ t + γ t − h ( t ) x ˙ T ( β ) M 2 x ˙ ( β ) d β d γ − ( h 2 − h ( t ) ) ∫ t − h ( t ) t − h 1 x ˙ T ( β ) M 2 x ˙ ( β ) d β (8) \begin{aligned}\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^2_{12}}{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^2_{12}}{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{aligned}\tag{8} dtdV(t)=∫−h2−h1∫γ−h1[x˙T(t)M2x˙(t)−x˙T(t+β)M2x˙(t+β)]dβdγ=∫−h2−h1∫γ−h1x˙T(t)M2x˙(t)dβdγ−∫−h2−h1∫t+γt−h1x˙T(β)M2x˙(β)dβdγ=2h122x˙T(t)M2x˙(t)−∫−h2−h1∫t+γt−h1x˙T(β)M2x˙(β)dβdγ=2h122x˙T(t)M2x˙(t)−∫−h(t)−h1∫t+γt−h1x˙T(β)M2x˙(β)dβdγ−∫−h2−h(t)∫t+γt−h(t)x˙T(β)M2x˙(β)dβdγ−(h2−h(t))∫t−h(t)t−h1x˙T(β)M2x˙(β)dβ(8)
注意到,最后结果对项 ∫ − h 2 − h 1 ∫ t + γ t − h 1 x ˙ T ( β ) M 2 x ˙ ( β ) d β d γ \int_{-h_2}^{-h_1}\int_{t+\gamma}^{t-h_1}\dot{x}^T(\beta)M_2\dot{x}(\beta)d\beta d\gamma ∫−h2−h1∫t+γt−h1x˙T(β)M2x˙(β)dβdγ 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 e 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 e 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
4 基于辅助函数的二重积分不等式
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.