Consensus in topologically interacting swarms under communication constraints and time-delays

Abstract

The emergence of collective decision in swarming systems underscores the central role played by information transmission. Using network-control- and information-theoretic elements applied to a group of topologically interacting agents seeking consensus under switching topologies, the effects of constraints in the information capacity of the communication channel are investigated. This particular system requires us to contend with constantly reconfigurable and spatially embedded interaction networks. We find a sufficient condition on the information data rate guaranteeing the stability of the consensus process in the noiseless case. This result highlights the profound connection with the topological structure of the underlying interaction network, thus having far-reaching implications in the nascent field of swarm robotics. Furthermore, we analyze the more complex case of combined effect of noise and limited data rate. We find that the consensus process is degraded when decreasing the data rate. Moreover, the relationship between critical noise and data rate is found to be in good agreement with information-theoretic predictions. Lastly, we prove that with not-too-large time-delays, our system of topologically interacting agents is stable, provided the underlying interaction network is strongly connected. Using Lyapunov techniques, the maximum allowed time-delay is determined in terms of linear-matrix inequalities.

Appendices

Appendices

1.1 Proof of Theorem 2

The following Lemma is instrumental to establish the proof of Theorem 2.

Lemma 2

Consider the following linear system

$$\begin{aligned} \left\{ \begin{array}{l}\varvec{\dot{x}}(t)=\mathbf {A}^0x(t)+\mathbf {A}^1(t)x(t-\tau ),\\ \mathbf {x}(s)=\varvec{\phi }(s), \forall s\in [-\tau ,0]\end{array}\right. , \end{aligned}$$

where $\mathbf {x}\in \mathbb {R}^N$ is the state variable, $\mathbf {A}^0$ is a constant matrix and $\phi (s)$ corresponds to the set of initial conditions considered over the interval $[-\tau ,0]$. The system is asymptotically stable for all values $\tau$ in the interval $[0,\bar{\tau }]$ if

• (i) $1+\lambda (\mathbf {A}^1(t))\frac{1-e^{s\bar{\tau }}}{s}\not =0$ holds for all $t\ge 0$, $s\in \mathbb {C}^+$, and $\lambda (\mathbf {A}^1(t))$ are the eigenvalues of $\mathbf {A}^1(t)$;

  and

• (ii) there exist two positive-definite matrices $(\mathbf {P},\mathbf {Q})$ such that the LMI below holds:

  $$\begin{aligned}&\left[ \begin{array}{c}(\mathbf {A}^0+\mathbf {A}^1(t))^T\mathbf {P}+\mathbf {P}(\mathbf {A}^0+\mathbf {A}^1(t))+\bar{\tau }\mathbf {Q}\\ \bullet \end{array}\right. \\&\left. \begin{array}{c}(\mathbf {A}^0+\mathbf {A}^1(t))^T\mathbf {PA}^1\\ -\bar{\tau }\mathbf {Q}\end{array}\right] <0. \end{aligned}$$

Proof

It can be proved as in [22, p. 222] by considering the following Lyapunov candidate

$$\begin{aligned} \mathbf {V}(t)= & {} \bigg (\mathbf {x}^T(t)+\int _{t-\tau }^t\mathbf {x}^T(s)ds\mathbf {A}^1(t)^T\bigg )\mathbf {P}\\&\cdot \,\bigg (\mathbf {x}(t)+\mathbf {A}^1(t)\int _{t-\tau }^t\mathbf {x}(s)ds\bigg )\\&+\int _{-\tau }^0\int _{t+r}^t\mathbf{x}^T(s)\mathbf{Q}{} \mathbf{x}(s)dsdr. \end{aligned}$$

$\square$

We now prove the statement of Theorem 2. First, note that all eigenvalues of the matrix $\mathbf {B}(t)$ are located in the open-right plane given that the interaction network is strongly connected. Therefore $\mathbf {B}^T(t)\mathbf {P}+\mathbf {PB}(t)$ can be negative definite, and the linear-matrix inequality (24) is feasible for small enough $\bar{\tau }$. Next, using Lemma 1, stability of the systems (17) or (18) is equivalent to the stability of (20) and (21). By exploiting Lemma 2 with $\mathbf {A}^0=-\mathbf {I}$ and $\mathbf {A}^1(t)=\mathbf {B}(t)+\mathbf {I}$, it is observed that the solution of system (20) tends to zero, i.e., $\lim _{t\rightarrow \infty }\mathbf {x}_1(t)=0$.

Finally, the Laplace transform of (21) yields

$$\begin{aligned} \mathbf {X}_2(s)=\frac{\mathbf {x}_2(0)+\int _{-\tau }^0\mathbf {x}_2(u)e^{-(u+\tau )s}du}{s+1-e^{-\tau s}}, \end{aligned}$$

where s is a complex variable. The stability of (21) is defined by the denominator of the Laplace transform ($s+1-e^{-\tau s}=0$). Letting $s=\sigma +j\omega$ with $\sigma ,\omega \in \mathbb {R}$ and $j^2=-1$, we have

$$\begin{aligned}&\sigma +1-e^{-\sigma \tau }\cos (\omega \tau )=0, \end{aligned}$$

(25)

$$\begin{aligned}&\omega +e^{-\sigma \tau }\sin (\omega \tau )=0. \end{aligned}$$

(26)

The system (21) is stable for any time-delay $\tau$, i.e., there exists a $\mathbf {x}_2^*$ such that $\lim _{t\rightarrow \infty }\mathbf {x}_2(t)=\mathbf {x}_2^*$ if solution of (25) leads to $\sigma <0$. We consider the following cases:

• (i) $\sigma =0\Rightarrow \cos (\omega \tau )=e$ which is impossible since $-1\le \cos (\omega \tau )\le 1$.

• (ii) $\sigma>0\Rightarrow e^{-\sigma \tau }\cos (\omega \tau )=\sigma +1>1$ which is again impossible since $e^{-\sigma \tau }<1$ and $-1\le \cos (\omega \tau )\le 1$, then their product cannot be larger than 1.

Therefore, the value of $\sigma$ obtained from (25) is strictly negative ($\sigma <0$) which guarantees the stability of system (21). This concludes our proof.

1.2 Remarks about Theorem and Corollary 1 and Theorem 2

It is worth noting that Theorem 1 and Corollary 1 are based on the Gelfand spectral radius formula and on the construction of the joint spectral radius. The special structure of the matrix $\tilde{\mathbf {L}}(t)=L(t)/k$, originating from the topologically interacting agents, plays a key role here. These considerations are very different from the result in Ref. [26].

The key challenge in proving Theorem 2 mainly lies in the construction of the LMI and the appropriate application of the Laplace transform. Another key simplification is obtained by means of the coordination transformation in Eq. (19), which simplifies our equation and facilitates the analytical treatment for the time-delay system.