Event-Driven Communication and Control in Multi-Agent Networks

Abstract

Event-triggered/driven control is a measurement-based (e.g., system state or output) sampling control whereas the time instants for sampling and control actions should be determined by a predefined triggering condition (i.e., a measurement-based condition). It thus can be viewed as a type of hybrid control. In a network environment, an important issue in the implementation of distributed algorithms is the communication and control actuation rules. An event-driven scheme would be more favorable in the communication and control actuation for MANs, especially for embedded, interconnected devices with limited resources.

Appendix

Proof of Lemma 9.2

Proof

It is easy to observe that V (y, z) is nonnegative continuous. The reminder is to verify that $\frac {d V(y(t),z(t))}{dt}\mid _{\text{(9.12)}}\leq 0$, ∀t ≥ 0.

For t ∈ [kh, (k + 1)h), taking time derivative of V (y(t), z(t)) along with the trajectory of system (9.12) yields

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{d V(y(t),z(t))}{dt}\mid_{\text{(9.12)}} &\displaystyle = &\displaystyle y(t)^\top \dot{y}(t)+ z(t)^\top \dot{z}(t) \\ &\displaystyle = &\displaystyle -y(t)^\top \varPhi_1(kh)-z(t)^\top \varPhi_2(kh), \end{array} \end{aligned} $$

where Φ ₁(kh) = (L ₁ + p ₁ D ₁)(y(kh) + e _y(kh)) + p ₁ M ₁(z(kh) + e _z(kh)), Φ ₂(kh) = p ₂ M ₂(y(kh) + e _y(kh)) + (L ₂ + p ₂ D ₂)(z(kh) + e _z(kh)). Then it follows

$$\displaystyle \begin{aligned} \frac{d V(y(t),z(t))}{dt}\mid_{\text{(9.12)}} \leq \varOmega_1(kh)+\varOmega_2(kh), \end{aligned} $$

(9.15)

where Ω ₁(kh) = −y(kh)^⊤ Φ ₁(kh) + hΦ ₁(kh)^⊤ Φ ₁(kh), Ω ₂(kh) = −z(kh)^⊤ Φ ₂(kh) + hΦ ₂(kh)^⊤ Φ ₂(kh).

Under the condition $2h\lambda _{\max }(L_r+p_rD_r)\leq 1$, r = 1, 2, one has

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varOmega_1 (kh) &\displaystyle \leq &\displaystyle -\frac{1}{2}y(kh)^\top (L_1+p_1D_1)y(kh) +\frac{1}{2} e_y(kh)^\top (L_1+p_1D_1) e_y(kh) \\ &\displaystyle &\displaystyle -p_1y(kh)^\top M_1 \big(z(kh)+e_z(kh)\big) \\ &\displaystyle &\displaystyle +p_1 \big(z(kh)+e_z(kh)\big)^\top M_1^\top M_1 \big(z(kh)+e_z(kh)\big) \\ &\displaystyle &\displaystyle +2p_1 \big(y(kh)+e_y(kh)\big)^\top (L_1+p_1D_1)^\top M_1 \big(z(kh)+e_z(kh)\big), \\ \varOmega_2 (kh) &\displaystyle \leq &\displaystyle -\frac{1}{2} z(kh)^\top (L_2+p_2D_2)z(kh) +\frac{1}{2} e_z(kh)^\top (L_2+p_2D_2) e_z(kh) \\ &\displaystyle &\displaystyle -p_2z(kh)^\top M_2 \big(y(kh)+e_y(kh)\big) \\ &\displaystyle &\displaystyle +p_2 \big(y(kh)+e_y(kh)\big)^\top M_2^\top M_2 \big(y(kh)+e_y(kh)\big) \\ &\displaystyle &\displaystyle +2p_2 \big(z(kh)+e_z(kh)\big)^\top (L_2+p_2D_2)^\top M_2 \big(y(kh)+e_y(kh)\big). \end{array} \end{aligned} $$

Hence, the inequality (9.15) implies

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle &\displaystyle \frac{d V(y(t),z(t))}{dt}\mid_{\text{(9.12)}} \\ &\displaystyle \leq &\displaystyle -\frac{1}{2}y(kh)^\top (L_1+p_1D_1)y(kh) +\frac{1}{2} e_y(kh)^\top (L_1+p_1D_1) e_y(kh) \\ &\displaystyle &\displaystyle -p_1y(kh)^\top M_1 \big(z(kh)+e_z(kh)\big) \\ &\displaystyle &\displaystyle +p_1 \big(z(kh)+e_z(kh)\big)^\top M_1^\top M_1 \big(z(kh)+e_z(kh)\big) \\ &\displaystyle &\displaystyle +2p_1 \big(y(kh)+e_y(kh)\big)^\top (L_1+p_1D_1)^\top M_1 \big(z(kh)+e_z(kh)\big) \\ &\displaystyle &\displaystyle -\frac{1}{2} z(kh)^\top (L_2+p_2D_2)z(kh) +\frac{1}{2} e_z(kh)^\top (L_2+p_2D_2) e_z(kh) \\ &\displaystyle &\displaystyle -p_2z(kh)^\top M_2 \big(y(kh)+e_y(kh)\big) \\ &\displaystyle &\displaystyle +p_2 \big(y(kh)+e_y(kh)\big)^\top M_2^\top M_2 \big(y(kh)+e_y(kh)\big) \\ &\displaystyle &\displaystyle +2p_2 \big(z(kh)+e_z(kh)\big)^\top (L_2+p_2D_2)^\top M_2 \big(y(kh)+e_y(kh)\big). \end{array} \end{aligned} $$

(9.16)

Next, according to the repulsion mechanism given in Definition 9.3, we discuss the relationship (9.16) from the following two cases. To the point, the first one is devoted to clarifying $\frac {d V(y(t),z(t))}{dt}\mid _{\text{(9.12)}}\leq 0$ in the case that there exists at least one extra-subgroup link, while the other is to deal with the case that all extra-subgroup links are reset to 0.

A contradiction is carried out first. Suppose there exists a large enough real number T ₀ > 0 such that the rule (9.8) holds for $t_k^i>T_0$, namely there exist at least one pair of link (i ₁, i ₂), i ₁ ∈ N ₁, i ₂ ∈ N ₂, such that $|x_{i_1}(t_k^{i_1i_2})-x_{i_2}(t_k^{i_1i_2})|$ is lower bounded by a positive threshold. This hypothesis contradicts with the fact that MAN (9.1) reach consensus under the updating rule (9.9) if the network topology G is connected and remains unchanged.

Case 1 Assume there exists at least one linked pair (i ₁, i ₂), i ₁ ∈ V ₁, i ₂ ∈ V ₂, and a large enough real number T ₁ > 0 such that the relationship (9.8) holds for $t_k^{i_1i_2} \leq T_1$, then it follows for kh ∈ [0, T ₁) that

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \left( \begin{array}{c} y(kh) \\ z(kh) \\ \end{array} \right)^\top \left( \begin{array}{cc} D_1 & M_1 \\ M_2 & D_2 \\ \end{array} \right) \left( \begin{array}{c} y(kh) \\ z(kh) \\ \end{array} \right) \geq \sigma_1 y(kh)^\top L_1y(kh) \\ \quad +\sigma_2 z(kh)^\top L_2z(kh). \end{array} \end{aligned} $$

(9.17)

Based on the updating rule (9.9), one can get

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta V &\displaystyle = &\displaystyle V(y((k+1)h),z((k+1)h))-V(y(kh),z(kh))\qquad \\ &\displaystyle = &\displaystyle -\frac{1}{2}\parallel y(kh)\parallel^2+\frac{1}{2}\parallel y(kh)-h(L_1+p_1D_1)y(kh)-h p_1M_1z(kh)\parallel^2 \\ &\displaystyle &\displaystyle -\frac{1}{2}\parallel z(kh)\parallel^2 +\frac{1}{2}\parallel z(kh)-h p_2M_2y(kh)-h(L_2+p_2D_2)z(kh)\parallel^2. \end{array} \end{aligned} $$

Then combining with the inequality (9.17), one has

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta V &\displaystyle \leq &\displaystyle -h y(kh)^\top L_1y(kh)-hz(kh)^\top L_2z(kh) \\ &\displaystyle &\displaystyle -h\left( \begin{array}{c} y(kh) \\ z(kh) \\ \end{array} \right)^\top \left( \begin{array}{cc} p_1D_1 &\displaystyle p_1M_1 \\ p_2M_2 &\displaystyle p_2D_2 \\ \end{array} \right) \left( \begin{array}{c} y(kh) \\ z(kh) \\ \end{array} \right) \\ &\displaystyle &\displaystyle + h^2 \left( \begin{array}{c} y(kh) \\ z(kh) \\ \end{array} \right)^\top \bigg[ \left( \begin{array}{cc} L_1^2 &\displaystyle 0 \\ 0 &\displaystyle L_2^2 \\ \end{array} \right)+ \left( \begin{array}{cc} p_1D_1 &\displaystyle p_1M_1 \\ p_2M_2 &\displaystyle p_2D_2 \\ \end{array} \right)^2 \bigg]\left( \begin{array}{c} y(kh) \\ z(kh) \\ \end{array} \right), \end{array} \end{aligned} $$

which, again with the condition $2h\lambda _{\max }(L_r+p_rD_r)\leq 1$ (r = 1, 2), implies

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \varDelta V \leq -h\big(\frac{1}{2}+\min(p_1,p_2)\sigma_1-\frac{p_1+p_2}{2}\big)y(kh)^\top L_1y(kh) \\ -h\big(\frac{1}{2}+\min(p_1,p_2)\sigma_2-\frac{p_1+p_2}{2}\big)z(kh)^\top L_2z(kh). \end{array} \end{aligned} $$

(9.18)

According to the ZOH rule, it follows

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \frac{d V(y(t),z(t))}{dt}\mid_{\text{(9.12)}} \leq -\big(\frac{1}{2}+\min(p_1,p_2)\sigma_1-\frac{p_1+p_2}{2}\big)y(kh)^\top L_1y(kh) \\ -\big(\frac{1}{2}+\min(p_1,p_2)\sigma_2-\frac{p_1+p_2}{2}\big)z(kh)^\top L_2z(kh).\\ \end{array} \end{aligned} $$

(9.19)

Since $\sigma _1,\sigma _2\geq \frac {p_1+p_2-1}{2\min (p_1,p_2)}$ and , (9.19) ensures $\frac {d V(y(t),z(t))}{dt}\mid _{\text{(9.12)}} \leq ~0$.

Case 2 Contrarily, when $t_k^i>T_1$, the relationship (9.8) breaks for all i ₁ ∈ V ₁, i ₂ ∈ V ₂. Namely, $a_{i_1i_2}(t_k^{i_1i_2})$ is reset to 0, which means M ₁ = 0, M ₂ = 0, D ₁ = 0, D ₂ = 0. Then the relationship (9.16) gives rise to

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{d V(y(t),z(t))}{dt}\mid_{\text{(9.12)}} &\displaystyle \leq &\displaystyle -\frac{1}{2}y(kh)^\top L_1y(kh)+\frac{1}{2} e_y(kh)^\top L_1e_y(kh) \\ &\displaystyle &\displaystyle -\frac{1}{2} z(kh)^\top L_2z(kh)+\frac{1}{2} e_z(kh)^\top L_2e_z(kh). \end{array} \end{aligned} $$

Substituting the inequality (9.13), it follows

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \frac{d V(y(t),z(t))}{dt}\mid_{\text{(9.12)}} & \leq & -\frac{1}{2}(1-\alpha_1 \lambda_{\max}(L_1+p_1D_1))y(kh)^\top L_1y(kh) \\ && -\frac{1}{2} (1-\alpha_2 \lambda_{\max}(L_2+p_2D_2)) z(kh)^\top L_2z(kh) \\ && -\frac{1}{2} \left( \begin{array}{c} y(kh) \\ z(kh) \\ \end{array} \right)^\top \varTheta \left( \begin{array}{c} y(kh) \\ z(kh) \\ \end{array} \right), \end{array} \end{aligned} $$

(9.20)

where $\varTheta =\left ( \begin {array}{cc} p_1D_1 & 0 \\ 0 & p_2D_2 \\ \end {array} \right ) -\sum _{r=1}^2 \alpha _r\lambda _{\max }(L_r+p_rD_r)\left ( \begin {array}{cc} D_1 & 0 \\ 0 & D_2 \\ \end {array} \right )$.

Noting that $\alpha _r \leq \frac {\min (p_1,p_2)}{2\lambda _{\max }(L_r+p_rD_r)}$, r = 1, 2, one has $\sum _{r=1}^2 \alpha _r\lambda _{\max }(L_r+p_rD_r)\leq \min (p_1,p_2)$, which implies Θ ≥ 0. Then from (9.20) one can get

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \frac{d V(y(t),z(t))}{dt}\mid_{\text{(9.12)}} &\displaystyle \leq &\displaystyle -\frac{1}{2}\big(1-\frac{\min(p_1,p_2)}{2}\big) y(kh)^\top L_1y(kh) \\ &\displaystyle &\displaystyle -\frac{1}{2} \big(1-\frac{\min(p_1,p_2)}{2}\big) z(kh)^\top L_2z(kh), \end{array} \end{aligned} $$

(9.21)

which with 0 < p ₁, p ₂ < 2 ensures $\frac {d V(y(t),z(t))}{dt}\mid _{\text{(9.12)}}\leq 0$.

Therefore, combining Case 1 and Case 2, it follows $\frac {d V(y(t),z(t))}{dt}\mid _{\text{(9.12)}}\leq 0$, ∀t ≥ 0, in the sense that the relationships (9.19) and (9.21) hold. This completes the proof. □