文章目录
I. INTRODUCTION
II. PRELIMINARIES
A. Notation
B. Algebraic Graph Theory
C. Assumptions
D. Problem Formulation
The following NMASs with N N N following agents are considered in this article. This dynamics fo the k k kth agent are given as
{ x ˙ k ( t ) = A x k ( t ) + B f ( x k ( t ) ) + C f ( x k ( t − φ ( t ) ) + u k ( t ) + B w w k ( t ) ψ k ( t ) = D x k ( t ) (2) \left\{\begin{aligned} \dot{x}_k(t) &= A x_k(t) + B f(x_k(t)) + C f(x_k(t-\varphi(t)) + u_k(t) + B_w w_k(t) \\ \psi_k(t) &= D x_k(t) \end{aligned} \right. \tag{2} { x˙k(t)ψk(t)=Axk(t)+Bf(xk(t))+Cf(xk(t−φ(t))+uk(t)+Bwwk(t)=Dxk(t)(2)
The dynamics of the leader are described as
{ x ˙ 0 ( t ) = A x 0 ( t ) + B f ( x 0 ( t ) ) + C f ( x 0 ( t − φ ( t ) ) ψ 0 ( t ) = D x 0 ( t ) (3) \left\{\begin{aligned} \dot{x}_0(t) &= A x_0(t) + B f(x_0(t)) + C f(x_0(t-\varphi(t)) \\ \psi_0(t) &= D x_0(t) \end{aligned} \right. \tag{3} {
x˙0(t)ψ0(t)=Ax0(t)+Bf(x0(t))+Cf(x0(t−φ(t))=Dx0(t)(3)
To save network communication resources and make the NMASs reach consensus, a distributed ETS is equipped with each agent. If the ETS is satisfied, the agent could receive the sampled data from its neighbors
[ y k ( t p k + β h ) ] T Ω y k ( t p k + β h ) ≤ θ k [ ω k ( t p k + β h ) ] T Ω ω k ( t p k + β h ) (4) \begin{aligned} [y_k (t^k_p + \beta h)]^\text{T} \Omega y_k(t^k_p+\beta h) \le \theta_k [\omega_k(t^k_p + \beta h)]^\text{T} \Omega \omega_k(t^k_p + \beta h) \end{aligned} \tag{4} [yk(tpk+βh)]TΩyk(tpk+βh)≤θk[ωk(tpk+βh)]TΩωk(tpk+βh)(4)
where
θ k > 0 \red{\theta_k}>0 θk>0 means the threshold parameter.
Ω \red{\Omega} Ω means the event-triggered matrix.
h \red{h} h denotes the sampling period, and
t p k \red{t^k_p} tpk denotes the p p pth event-triggered instant of the agent k k k.
t p k + β h \red{t^k_p + \beta h} tpk+βh represents the currently sampled instant and
y k ( t p k + β h ) = x k ( t p k ) − x k ( t p k + β h ) ω k ( t p k + β h ) = ∑ l ∈ N k a k l [ x k ( t p k ) − x l ( t p ^ l ) ] + γ k [ x k ( t p k ) − x 0 ( t p k + β h ) ] t p ^ l = max { t ∣ t ∈ { t p l , p = 1 , 2 , 3 , ⋯ } , t ≤ t p l + β h } \begin{aligned} y_k(t^k_p + \beta h) &= x_k(t^k_p) - x_k(t^k_p+\beta h) \\ \omega_k(t^k_p + \beta h) &= \sum_{l \in \N_k} a_{kl} [x_k(t^k_p) - x_l(t^l_{\hat{p}})] + \gamma_k [x_k(t^k_p) - x_0(t^k_p+\beta h)] \\ t^l_{\hat{p}} &= \max \{ t | t \in \{t^l_p, p=1,2,3,\cdots \}, t \le t^l_p + \beta h \} \end{aligned} yk(tpk+βh)ωk(tpk+βh)tp^l=xk(tpk)−xk(tpk+βh)=l∈Nk∑akl[xk(tpk)−xl(tp^l)]+γk[xk(tpk)−x0(tpk+βh)]=max{ t∣t∈{ tpl,p=1,2,3,⋯},t≤tpl+βh}
u k ( t ) = − K ( r t ) { ∑ l ∈ N k a k l ( r t ) [ x k ( t p k ) − x l ( t p ^ l ) ] + γ k ( r t ) [ x k ( t p k ) − x 0 ( q h ) ] } (5) \begin{aligned} u_k(t) = -K(r_t) \{ \sum_{l \in N_k} a_{kl} (r_t) [x_k(t^k_p) - x_l(t^l_{\hat{p}})] + \gamma_k(r_t) [x_k(t^k_p) - x_0(qh)] \} \end{aligned} \tag{5} uk(t)=−K(rt){ l∈Nk∑akl(rt)[xk(tpk)−xl(tp^l)]+γk(rt)[xk(tpk)−x0(qh)]}(5)
where
t ∈ [ t p k , t p + 1 k ] \red{t} \in [t^k_p, t^k_{p+1}] t∈[tpk,tp+1k]
q \red{q} q denotes an integral.
K ( r t ) \red{K(r_t)} K(rt) is the feedback gain matrix.
a k l ( r t ) \red{a_{kl}(r_t)} akl(rt) is the element of leader adjacency matrix.
γ k ( r t ) > 0 \red{\gamma_k(r_t)} > 0 γk(rt)>0 if the k k kth agent could receive information from the leader;
III. MAIN RESULTS
A. Consensus Analysis
B. Leader-Following Dissipativity Consensus Control
IV. SIMULATION
Example 1
程序 main.m，效果如下
程序 main_Event.m，效果如下
但是论文中没有给出 Omega 的具体值，事件触发我测试了很多值，就是两种情况，要么不触发，要么一直触发。
我尝试去调整 theta，这个也是影响事件触发的一个值，但是无论怎么调，都是不行。
