Containment Control for Partial Differential Multi-Agent Systems via Iterative Learning Algorithm

Zhang Dan, Fu Qin,, Chen Zhenjie

 基金资助: 国家自然科学基金.  11971343

 Fund supported: the NSFC.  11971343

Abstract

In this paper, we deals with the containment control problem for a class of partial differential multi-agent systems, which are composed of the second-order parabolic equations or the second-order hyperbolic equations. Based on the framework of network topologies, the P-type iterative learning law is designed depending on the output form of the follower system, and the convergence condition of the system in the sense of iterative learning stability is obtained. By using the contraction mapping method, it is proved that the containment errors of two kinds of systems on the finite time interval converge to zero on $L^2$ space with the increasing of iterations. Finally, simulation examples demonstrate the validity of the theoretical analysis.

Keywords： Iterative learning algorithm ; Partial differential equation ; Multi-agent systems ; Containment control

Zhang Dan, Fu Qin, Chen Zhenjie. Containment Control for Partial Differential Multi-Agent Systems via Iterative Learning Algorithm. Acta Mathematica Scientia[J], 2021, 41(4): 1111-1123 doi:

2 预备知识与问题描述

$$${\left\{ \begin{array}{l} { } \frac{{{\partial ^\gamma }{Q_{i, k}}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}{Q_{i, k}}\left( {x, t} \right)}}{{\partial {x^2}}} + {u_{i, k}}\left( {x, t} \right)\\ {z_{i, k}}\left( {x, t} \right) = C{Q_{i, k}}\left( {x, t} \right) + B{u_{i, k}}\left( {x, t} \right) \end{array}, \right.{\rm{ }}i = 1, 2, \cdots , M, \left( {x, t} \right) \in \left[ {0, 1} \right] \times \left[ {0, T} \right], }$$$

$$$\left\{ \begin{array}{l} { } \frac{{{\partial ^\gamma }{Q_i}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}{Q_i}\left( {x, t} \right)}}{{\partial {x^2}}}\\ {z_i}\left( {x, t} \right) = C{Q_i}\left( {x, t} \right) \end{array}, \right.{\rm{ }}i = M + 1, M + 2, \cdots , N, \left( {x, t} \right) \in \left[ {0, 1} \right] \times \left[ {0, T} \right],$$$

(或${\left. {\frac{{\partial {Q_{i, k}}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 0}} = {r_i}\left( x \right)$), ${Q_{i, k}}\left( {1, t} \right) = {\tau _i}\left( t \right)$ (或${\left. {\frac{{\partial {Q_{i, k}}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 1}} = {v_i}\left( x \right)$), $t \in (0, T]$, $i = 1, $$2, \cdots , M , k = 0, 1, 2, \cdots . i 个领航者智能体的初、边值条件设定为 (或 {\left. {\frac{{\partial {Q_i}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 0}} = {r_i}\left( x \right) ), {Q_i}\left( {1, t} \right) = {\tau _i}\left( t \right) (或 {\left. {\frac{{\partial {Q_i}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 1}} = {v_i}\left( x \right) ), t \in (0, T] , i = M + 1,$$ M + 2, $$\cdots , N . 基于迭代学习的包容控制的目标是: 构建 {u_{i, k}}\left( {x, t} \right)$$ (i = 1, 2, \cdots , M$, $k = 0, 1, 2, \cdots$), 使得

$$$\left\{ \begin{array}{l} { } \frac{{{\partial ^\gamma }{Q_{f, k}}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}{Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}} + {u_k}\left( {x, t} \right)\\ {z_{f, k}}\left( {x, t} \right) = C{Q_{f, k}}\left( {x, t} \right) + B{u_k}\left( {x, t} \right) \end{array}, \right.\left( {x, t} \right) \in \left[ {0, 1} \right] \times \left[ {0, T} \right].$$$

${Q_l}\left( {x, t} \right) = {\left[ {{Q_{M + 1}}\left( {x, t} \right), \cdots , {Q_N}\left( {x, t} \right)} \right]^{\rm{T}}}$, ${z_l}\left( {x, t} \right) = {\left[ {{z_{M + 1}}\left( {x, t} \right), \cdots , {z_N}\left( {x, t} \right)} \right]^{\rm{T}}}$, 则系统(2.2)可以写成如下的紧凑形式

$$$\left\{ \begin{array}{l} { } \frac{{{\partial ^\gamma }{Q_l}\left( {x, t} \right)}}{{\partial {t^\gamma }}} = \frac{{{\partial ^2}{Q_l}\left( {x, t} \right)}}{{\partial {x^2}}}\\ {z_l}\left( {x, t} \right) = C{Q_l}\left( {x, t} \right) \end{array}, \right.\left( {x, t} \right) \in \left[ {0, 1} \right] \times \left[ {0, T} \right].$$$

3 主要结果

$$${u_{k + 1}}\left( {x, t} \right) = {u_k}\left( {x, t} \right) - \Gamma {\delta _k}\left( {x, t} \right),$$$

$$$\rho = \left| {1 - B\Gamma } \right| < 1.$$$

$\begin{eqnarray} && \int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right\}} {\rm{d}}x {}\\ & = &\int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}}} \right\}} {\rm{d}}x - \Gamma \int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}{\delta _k}\left( {x, t} \right)} \right\}} {\rm{d}}x. \end{eqnarray}$

$$$\int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right\}} {\rm{d}}x = \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2,$$$

$$$- \Gamma \int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}{\delta _k}\left( {x, t} \right)} \right\}} {\rm{d}}x \le \frac{{\left| \Gamma \right|}}{2}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2 + \frac{{\left| \Gamma \right|}}{2}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2.$$$

$\begin{eqnarray} &&\int_0^1 {\left\{ {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}}} \right\}} {\rm{d}}x{}\\ & = & \left. {{{\left( {\Delta {Q_{f, k}}\left( {x, t} \right)} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 0}^{x = 1} - \int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right\}} {\rm{d}}x{}\\ & = & - \int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right\}} {\rm{d}}x = - \left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial x}}} \right\|_{{L^2}}^2 \le 0. \end{eqnarray}$

$\begin{eqnarray} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}& \le &\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {\frac{{1 - {{\rm{e}}^{ - 2\lambda t}}}}{{2\lambda }}} \right\}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}{}\\ & = & \left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}. \end{eqnarray}$

$\begin{eqnarray} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - \lambda t}}{{\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|}_{{L^2}}}} \right\} \le \sqrt {\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - \lambda t}}{{\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|}_{{L^2}}}} \right\}. \end{eqnarray}$

3.2 双曲型多智能体系统的包容控制($\gamma = 2$)

(3.5)式两端左乘${\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)^{\rm{T}}}$并对$x$$0 \to 1$积分, 能够得到

$\begin{eqnarray} &&\int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {t^2}}}} \right\}} {\rm{d}}x {}\\ & = &\int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}}} \right\}} {\rm{d}}x - \Gamma \int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}{\delta _k}\left( {x, t} \right)} \right\}} {\rm{d}}x. {\qquad} \end{eqnarray}$

$$$\int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {t^2}}}} \right\}} {\rm{d}}x = \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2,$$$

$$$- \Gamma \int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}} \left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}{\delta _k}\left( {x, t} \right)} \right\}} {\rm{d}}x \le \frac{{\left| \Gamma \right|}}{2}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2 + \frac{{\left| \Gamma \right|}}{2}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2.$$$

$\begin{eqnarray} &&\int_0^1 {\left\{ {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial {x^2}}}} \right\}} {\rm{d}}x{}\\ & = & \left. {{{\left( {\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t}}} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right|_{x = 0}^{x = 1} - \int_0^1 {\left\{ {{{\left( {\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial t\partial x}}} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right\}} {\rm{d}}x{}\\ & = &- \int_0^1 {\left\{ {{{\left( {\frac{{{\partial ^2}\Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x\partial t}}} \right)}^{\rm{T}}}\frac{{\partial \Delta {Q_{f, k}}\left( {x, t} \right)}}{{\partial x}}} \right\}} {\rm{d}}x = - \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial x}}} \right\|_{{L^2}}^2. \end{eqnarray}$

$\begin{eqnarray} &&\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2} \right\}{}\\ &\le& \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2 + {{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial x}}} \right\|_{{L^2}}^2} \right\}{}\\ & \le& \left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {\frac{{1 - {{\rm{e}}^{ - 2\lambda t}}}}{{2\lambda }}} \right\}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}{}\\ & = &\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\}. \end{eqnarray}$

$\lambda > \frac{1}{2}$, 则

$\begin{eqnarray} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} & \le& \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {\frac{{1 - {{\rm{e}}^{ - \left( {2\lambda - 1} \right)t}}}}{{2\lambda - 1}}} \right\}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2} \right\}{}\\ & = & \frac{{1 - {{\rm{e}}^{ - \left( {2\lambda - 1} \right)T}}}}{{2\lambda - 1}}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\frac{{\partial \Delta {Q_{f, k}}\left( { \cdot , t} \right)}}{{\partial t}}} \right\|_{{L^2}}^2} \right\}. \end{eqnarray}$

$$$\sup \limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\} \le \left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - \left( {2\lambda - 1} \right)T}}}}{{2\lambda - 1}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - 2\lambda t}}\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|_{{L^2}}^2} \right\},$$$

$$$\mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - \lambda t}}{{\left\| {\Delta {Q_{f, k}}\left( { \cdot , t} \right)} \right\|}_{{L^2}}}} \right\} \le \sqrt {\left| \Gamma \right|{{\rm{e}}^{\left| \Gamma \right|T}}\frac{{1 - {{\rm{e}}^{ - \left( {2\lambda - 1} \right)T}}}}{{2\lambda - 1}}\frac{{1 - {{\rm{e}}^{ - 2\lambda T}}}}{{2\lambda }}} \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left\{ {{{\rm{e}}^{ - \lambda t}}{{\left\| {{\delta _k}\left( { \cdot , t} \right)} \right\|}_{{L^2}}}} \right\}.$$$

4 仿真算例

2个领航者的动态模型为

图 1

Fig.1

$\Gamma = 0.5$, 则$\rho = \left| {1 - B\Gamma } \right| = 0.5 < 1$. 设置$k$次迭代时跟随者的初、边值如下: ${Q_{i, k}}\left( {x, 0} \right) = 0.01 \times i \times x \times \left( {1 - x} \right)\left( {\gamma = 1, 2} \right)$, ${\left. {\frac{{\partial {Q_{i, k}}\left( {x, t} \right)}}{{\partial t}}} \right|_{t = 0}} = 0\left( {\gamma = 2} \right)$, $x \in \left[ {0, 1} \right]$; ${Q_{i, k}}\left( {0, t} \right) = 0.01 \times i \times t$, ${Q_{i, k}}\left( {1, t} \right) = 0.01 \times 2 \times i \times t$, $t \in \left( {0, 1} \right]$, $i = 1, 2, 3$; 设置领航者的初、边值如下: ${Q_i}\left( {x, 0} \right) = 0.01 \times i \times x \times \left( {1 - x} \right) \left( {\gamma = 1, 2} \right)$, ${\left. {\frac{{\partial {Q_i}\left( {x, t} \right)}} {{\partial t}}} \right|_{t = 0}} = 0\left( {\gamma = 2} \right)$, $x \in \left[ {0, 1} \right]$; ${Q_i}\left( {0, t} \right) = 0.01 \times i \times t$, ${Q_i}\left( {1, t} \right) = 0.01 \times 2 \times i \times t$, $t \in \left( {0, 1} \right]$, $i = 4, 5$. 并取初始控制为: ${u_{1, 0}}\left( {x, t} \right) = {u_{2, 0}}\left( {x, t} \right) = {u_{3, 0}}\left( {x, t} \right) \equiv 1$. 在迭代学习律(3.1)的作用下, 两类系统的包容误差曲面与${\left\| {{\delta _k}} \right\|_{{L^2}, s}}$随着$k$变化时的仿真结果分别见图 2图 9.

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