Zhang X, Liu X. Consensus of linear multi-agent systems with exogenous disturbance generated from heterogeneous exosystems[J]. International Journal of Systems Science, 2017, 48(15): 3147-3159.

文章目录

1. Introduction
2. Preliminaries and problem formulation
- 2.1. Graph theory
- 2.2. Input-to-state stability and some useful lemmas
- 2.3. Problem formulation
3. Consensus analysis with exogenous disturbances generated by linear heterogeneous exosystems
4. Consensus analysis with exogenous disturbances generated by non-linear heterogeneous exosystems
5. Simulation examples

1. Introduction

2. Preliminaries and problem formulation

2.1. Graph theory

2.2. Input-to-state stability and some useful lemmas

2.3. Problem formulation

领航者为
x˙0=Ax0(5)\dot{x}_0 = A x_0 \tag{5}x˙0=Ax0(5)

跟随者为
x˙i=Axi+B(ui+di)(6)\begin{aligned} \dot{x}_i = A x_i + B (u_i + d_i) \end{aligned} \tag{6}x˙i=Axi+B(ui+di)(6)

3. Consensus analysis with exogenous disturbances generated by linear heterogeneous exosystems

干扰信号 did_idi 由以下动力学产生
ξ˙i=A0iξidi=C0iξi(8)\begin{aligned} \dot{\xi}_i = A_{0i} \xi_i \\ d_i = C_{0i} \xi_i \end{aligned} \tag{8}ξ˙i=A0iξidi=C0iξi(8)

观测器为
z˙i=(A0i−KiBC0i)(zi+Kixi)−Ki(Axi+Bui)ξ^i=zi+Kixid^i=C0iξ^i(9)\begin{aligned} \dot{z}_i &= (A_{0i}-K_iBC_{0i}) (z_i + K_i x_i) - K_i(Ax_i + Bu_i) \\ \hat{\xi}_i &= z_i + K_i x_i \\ \hat{d}_i &= C_{0i} \hat{\xi}_i \end{aligned} \tag{9}z˙iξ^id^i=(A0i−KiBC0i)(zi+Kixi)−Ki(Axi+Bui)=zi+Kixi=C0iξ^i(9)

控制器为
ui=cF[∑j=1Naij(xj−xi)+gi(x0−xi)]−C0iξ^i(15)\begin{aligned} u_i &= cF \left[ \sum_{j=1}^N a_{ij} (x_j - x_i) + g_i(x_0 - x_i) \right] - C_{0i} \hat{\xi}_i \end{aligned} \tag{15}ui=cF[j=1∑Naij(xj−xi)+gi(x0−xi)]−C0iξ^i(15)

4. Consensus analysis with exogenous disturbances generated by non-linear heterogeneous exosystems

干扰信号 did_idi 由以上动力学产生
ξ˙i=A0iξi+ϕi(ξi)di=C0iξi(35)\begin{aligned} \dot{\xi}_i &= A_{0i} \xi_i \red{+ \phi_i(\xi_i)} \\ d_i &= C_{0i} \xi_i \end{aligned} \tag{35}ξ˙idi=A0iξi+ϕi(ξi)=C0iξi(35)

观测器为
z˙i=(A0i−KiBC0i)(zi+Kixi)+ϕi(zi+Kixi)−Ki(Axi+Bui)+Qζi−1ζieiζ˙i=eiTeiξ^i=zi+Kixid^i=C0iξ^i(37)\begin{aligned} \dot{z}_i &= (A_{0i}-K_iBC_{0i}) (z_i + K_i x_i) \red{+\phi_i(z_i + K_i x_i)} - K_i(Ax_i + Bu_i) + \red{Q^{-1}_{\zeta_i} \zeta_i e_i} \\ \red{\dot{\zeta}_i} &\red{= e_i^{\text{T}} e_i} \\ \hat{\xi}_i &= z_i + K_i x_i \\ \hat{d}_i &= C_{0i} \hat{\xi}_i \end{aligned} \tag{37}z˙iζ˙iξ^id^i=(A0i−KiBC0i)(zi+Kixi)+ϕi(zi+Kixi)−Ki(Axi+Bui)+Qζi−1ζiei=eiTei=zi+Kixi=C0iξ^i(37)

论文中的观测器（37），写出来的结果无法运行，因为维度都不匹配。经过我的测试，感觉应该是少了个负号，我给补充上去了，结果就变好了。

ei=ξi−ξ^i(38)\begin{aligned} {e}_i &= \xi_i - \hat{\xi}_i \end{aligned} \tag{38}ei=ξi−ξ^i(38)

5. Simulation examples

Example 5.1

对应程序为 Main_Ex51.m
效果如下：

Example 5.2

对应程序为 Main_Ex52.m
效果如下：

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