Coordination of Multi-agent Systems with Intermittent Access to a Cloud Repository

Abstract

A cloud-supported multi-agent system is composed of autonomous agents required to achieve a common coordination objective by exchanging data over a shared cloud repository. The repository is accessed asychronously by different agents, and direct inter-agent commuication is not possible. This model is motivated by the problem of coordinating a fleet of autonomous underwater vehicles , with the aim to avoid the use of expensive and power-hungry modems for underwater communication. For the case of agents with integrator dynamics, a control law and a rule for scheduling the cloud access are formally defined and proven to achieve the desired coordination. A numerical simulation corroborate the theoretical results.

Appendix

Lemma 4

The agents’ dynamics (15) and the objective function (16) satisfy Assumption 4 with $\beta =\Vert B_\mathcal {T}\sqrt{P}\Vert$, $\gamma =\frac{2\min {{\mathrm{eig}}}(Q)}{\max {{\mathrm{eig}}}(P)}$, and $v_i(x)$ given by (17).

Proof

Taking the derivative of (16), we have

$$\begin{aligned} \frac{\partial V(x)}{\partial x} = (B_\mathcal {T}P B_\mathcal {T}^\top \otimes I_2) (x-b), \end{aligned}$$

which, taking norms of both sides and applying the triangular inequality, gives

$$\begin{aligned} \bigg \Vert \frac{\partial V(x)}{\partial x}\bigg \Vert \le \Vert B_\mathcal {T}\sqrt{P}\Vert \sqrt{V(x)}. \end{aligned}$$

Moreover, if we take the feedback law $v_i(x)=\sum _{j\in \mathcal {V}_i} (x_j(t)-b_j-x_i(t)+b_i)$, then we have $v(x)=-(CB^\top \otimes I_2)(x-b)$, and

$$\begin{aligned} \begin{aligned} \frac{\partial V(x)}{\partial x}^\top F(x,v(x),0_{n_d}) =&-(x-b)^\top (B_\mathcal {T}P B_\mathcal {T}^\top CB^\top \otimes I_2)(x-b)\\ =&-(x-b)^\top (B_\mathcal {T}PR B_\mathcal {T}^\top \otimes I_2) (x-b)\\ =&-(x-b)^\top (B_\mathcal {T}Q B_\mathcal {T}^\top \otimes I_2) (x-b)\\ \le&-\min {{\mathrm{eig}}}(Q)\Vert (B_\mathcal {T}\otimes I_2)(x-b)\Vert ^2. \end{aligned} \end{aligned}$$

(40)

Noting that $V(x)\le \frac{1}{2}\max {{\mathrm{eig}}}(P)\Vert (B_\mathcal {T}\otimes I_2)(x-b)\Vert ^2$, we can further bound (40) as

$$\begin{aligned} \begin{aligned} \frac{\partial V(x)}{\partial x}^\top F(x,v(x),0_{n_d}) \le&-\frac{2\min {{\mathrm{eig}}}(Q)}{\max {{\mathrm{eig}}}(P)}V(x). \end{aligned} \end{aligned}$$

which completes the proof.$\square$