Event-triggering Sampling Based Leader-following Consensus in Second-order Multi-agent Systems

文章目录

I. Introduction
II. Preliminaries
- A. Notations
- B. Algebraic graph theory
- C. Model description and problem formulation
III. Main results
- A. Fixed Topology
- B. Switching communication topologies
IV. Illustrative example

I. Introduction

II. Preliminaries

A. Notations

B. Algebraic graph theory

C. Model description and problem formulation

跟随者模型为
x˙i(t)=vi(t)v˙i(t)=ui(t)(1)\begin{aligned} \dot{x}_i(t) &= v_i(t) \\ \dot{v}_i(t) &= u_i(t) \\ \end{aligned}\tag{1}x˙i(t)v˙i(t)=vi(t)=ui(t)(1)

领航者模型为
x˙0(t)=v0(t)(2)\begin{aligned} \dot{x}_0(t) &= v_0(t) \\ \end{aligned}\tag{2}x˙0(t)=v0(t)(2)

III. Main results

分布式控制器为
ui(t)=∑j∈Niaij(xj(t)−xi(t)+vj(t)−vi(t))+bi(x0(t)−xi(t)+v0(t)−vi(t))(3)\begin{aligned} u_i(t) &= \sum_{j \in N_i} a_{ij} (x_j(t) - x_i(t) + v_j(t) - v_i(t)) \\ &+ b_i (x_0(t) - x_i(t) + v_0(t) - v_i(t)) \end{aligned}\tag{3}ui(t)=j∈Ni∑aij(xj(t)−xi(t)+vj(t)−vi(t))+bi(x0(t)−xi(t)+v0(t)−vi(t))(3)

ui(t)=∑j∈Niaij(xj(tki)−xi(tki)+vj(tki)−vi(tki))+bi(x0(tki)−xi(tki)+v0(tki)−vi(tki))(3)\begin{aligned} u_i(t) &= \sum_{j \in N_i} a_{ij} (x_j(\red{t_k^i}) - x_i(\red{t_k^i}) + v_j(\red{t_k^i}) - v_i(\red{t_k^i})) \\ &+ b_i (x_0(\red{t_k^i}) - x_i(\red{t_k^i}) + v_0(\red{t_k^i}) - v_i(\red{t_k^i})) \end{aligned}\tag{3}ui(t)=j∈Ni∑aij(xj(tki)−xi(tki)+vj(tki)−vi(tki))+bi(x0(tki)−xi(tki)+v0(tki)−vi(tki))(3)

定义测量误差 exi(t)e_{xi}(t)exi(t)，evi(t)e_{vi}(t)evi(t)，exij(t)e_{xij}(t)exij(t)，evij(t)e_{vij}(t)evij(t) 分别为

exi(t)=xi(tki)−xi(t)evi(t)=vi(tki)−vi(t)exij(t)={xj(tki)−xj(t),j∈Ni0,otherwiseevij(t)={vj(tki)−vj(t),j∈Ni0,otherwise\begin{aligned} e_{xi}(t) &= x_i(\red{t_k^i}) - x_i(t) \\ e_{vi}(t) &= v_i(\red{t_k^i}) - v_i(t) \\ e_{xij}(t) &= \left\{\begin{aligned} &x_j(\red{t_k^i}) - x_j(t), & j \in N_i \\ &0, & \text{otherwise} \end{aligned}\right. \\ e_{vij}(t) &= \left\{\begin{aligned} &v_j(\red{t_k^i}) - v_j(t), & j \in N_i \\ &0, & \text{otherwise} \end{aligned}\right. \end{aligned}exi(t)evi(t)exij(t)evij(t)=xi(tki)−xi(t)=vi(tki)−vi(t)={xj(tki)−xj(t),0,j∈Niotherwise={vj(tki)−vj(t),0,j∈Niotherwise

分布式事件触发采样规则为
Ei(t)=(bi+di)∥exi(t)+evi(t)∥+∥Ai∥∥e~xi(t)+e~vi(t)∥+bi∥exi0(t)+evi0(t)∥−βiHi(t)=0(4)\begin{aligned} E_i(t) &= (b_i+d_i) \|e_{xi}(t)+e_{vi}(t)\| + \|\mathcal{A}_i\| \|\tilde{e}_{xi}(t)+\tilde{e}_{vi}(t)\| \\ &+ b_i \|e_{xi0}(t)+e_{vi0}(t)\| - \beta_i H_i(t) = 0 \end{aligned}\tag{4}Ei(t)=(bi+di)∥exi(t)+evi(t)∥+∥Ai∥∥e~xi(t)+e~vi(t)∥+bi∥exi0(t)+evi0(t)∥−βiHi(t)=0(4)

结合论文中图 3 展示的效果，我猜测这里的触发条件应该是 Ei(t)>0E_i(t) > 0Ei(t)>0。

y1i(t)=(bi+di)∥exi(t)+evi(t)∥+∥Ai∥∥e~xi(t)+e~vi(t)∥+bi∥exi0(t)+evi0(t)∥y2i(t)=βiHi(t)\begin{aligned} y_{1i}(t) &= (b_i+d_i) \|e_{xi}(t)+e_{vi}(t)\| + \|\mathcal{A}_i\| \|\tilde{e}_{xi}(t)+\tilde{e}_{vi}(t)\| \\ &+ b_i \|e_{xi0}(t)+e_{vi0}(t)\| \\ y_{2i}(t) &= \beta_i H_i(t) \end{aligned}y1i(t)y2i(t)=(bi+di)∥exi(t)+evi(t)∥+∥Ai∥∥e~xi(t)+e~vi(t)∥+bi∥exi0(t)+evi0(t)∥=βiHi(t)

A. Fixed Topology

B. Switching communication topologies

IV. Illustrative example

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