Weighted Averaged Behavior of Synchronization among Heterogeneous Agents in a Sampled data Setting

Published in: ACC 2020
Authors: Jiyeon Nam*, Taekyoo Kim, Gyunghoon Park, and Hyungbo Shim
Abstract: In this paper, we address the problem of synchronization among heterogeneous agents in a sampled-data setting. The key observation is that the sample-and-hold process introduces additional node-wise weights on the Laplacian matrix in the discrete-time domain. We then show that in the sampled-data setting, all of the agents’ states approach the state of the so-called weighted averaged dynamics, an auxiliary dynamics that represents the collective behavior of heterogeneous agents when they are synchronized. In addition, admissible ranges of the sampling period and the diffusive coupling gain are computed, with which the consensus among the agents is achieved in a practical sense.

Our research about the Weighted Averaged Behavior of Synchronization among Heterogeneous Agents in a Sampled-data Setting will be presented. The highlight of this work is ”in a sampled-data setting.”

Previously on continuous time domain, there are studies about the behavior of heterogeneous multi-agent system. The point is that, when the coupling gain is large, the multi-agent system is easily analyzed through the blended dynamics. Let us consider the heterogeneous N multi agents with diffusive coupling.

$$\dot{x}_i = f_i(t, x_i) + \kappa \sum_{j \in {\mathcal{N}}_i} (x_j – x_i ) \quad \in \mathbb{R}^n, \quad i \in \{1,2,\cdots, N\}$$

Here, agent $i$ has state $x_i$ and its own dynamics $f_i$ with input given by the diffusive coupling, $\mathcal{N}_i$ denotes its neighboring agents and $\kappa$ is the coupling gain.

We define the blended dynamics as a virtual system with “average of all agent’s vector fields”.

$$\dot{s} = \frac{1}{N} \sum_{i=1}^N f_i(t, s) \quad \in \mathbb{R}^n$$

And $s$ is the state of the blended dynamics.

If the blended dynamics is stable, then, over an undirected and connected graph, there exists $\bar{\kappa}$ and class $\mathcal{K}$ function $\gamma$ such that for all $\kappa$ bigger than $\bar{\kappa}$ , every trajectory approaches the trajectory of blended dynamics practically.

$$\limsup_{t \rightarrow \infty} \|x_i(t) – s(t)\| \leq \gamma(1/\kappa), \quad \forall \kappa > \bar{\kappa}, \quad \forall i$$

Here, “practically” means that as you can make the synchronization error arbitrarily small by enlarge $\kappa$ .

Let’s just call this property as blending principle. Blending principle, of course, sufficiently large $\kappa$ , has lots of utilities such as first, analysis of heterogeneous multi-agent system. You get approximate behavior of multi-agent system. Second, Synthesis of multi-agent system. You can design blended dynamics first and then split the task into individual agents. Third, plug and play operation. Agents can join or leave during the operation due to the stability of blended dynamics. So, the blended has been utilized for analysis of coupled oscillator, distributed optimization, and distributed estimation.

The blended dynamics has many uses. So, Its engineering applications with digital communication should be considered. Let us see this picture.

The plant of each agent is continuous-time, and each agent communicates discrete-time samples. We present the following observations for a simple first order linear system.

First, discrete-time communication does not cause any trouble if the sampling period $T$ is sufficiently small. Second, if $T$ is not small, the coupling gain $\kappa$ cannot be increased arbitrarily large, and so, arbitrarily small error cannot be achieved. And the blended dynamics is not the simple average of individual vector field, but a weighted average depending on $T$ .

$$\dot{x}_i(t) = a_i x_i(t) + b_i (t) + u_i (t), \quad i \in \{1, \cdots, N\}$$

$$u_i(t) = \kappa \sum_{j \in \mathcal{N}_i} (x_j (kT)-x_i (kT)), \quad \forall t \in [kT, (k+1)T), \quad k \in \mathbb{Z}$$

Consider $N$ heterogeneous agents. Each dynamics is $a_i x_i + b_i$ , $b_i (t)$ represents possible exciting signals, and $a_i$ is not necessarily negative, in other words, it allows instable agents. Their input is diffusive coupling with coupling gain $\kappa$ over an undirected connected network and it is zero order hold interaction. Its time index is $k$ and the sampling period is $T$ .

The zero order hold equivalent model is obtained from the variations of constants formula.

$$\bar x_i[k+1] = e^{a_i T} x_i(kT) + \int_{kT}^{(k+1)T} e^{a_i ((k+1)T-\tau)} b_i(\tau) d\tau + \int_{0}^{T} e^{a_i (T-\tau)} d\tau u_i(kT) = e^{a_i T} \bar x_i[k] + \bar b_i[k] + T \theta_i(T) u_i(kT) $$

where $\theta_i(T)$ is $\frac{e^{a_iT}-1}{a_iT}$ if $a_i \not = 0$ , or 1 if $a_i = 0$ .

Let us denote the discrete time signal by the bar notation. The zero order hold equivalent model exactly captures the system properties along time stamps. We observe $T\theta$ here, next to the input. This theta plays important role in this paper because it acts like a input gain of each agent to the value of diffusive coupling. It has many form depending on the agent’s dynamics $a_i$ but goes to 1 as sampling period $T$ approaches to zero.

Now for the sampled data system we define a weighted average dynamics as follows.

$$\bar s[k+1] = \frac{\sum_{i=1}^N e^{a_i T}/\theta_i(T)}{\sum_{i=1}^N 1/\theta_i(T)} \bar s[k] + \frac{\sum_{i=1}^N \bar b_i[k]/\theta_i(T)}{\sum_{i=1}^N 1/\theta_i(T)}$$

Its dynamics has the form of weighted average of all agents dynamics with weighting numbers given by the inverse of $\theta_i$ ‘s. Based on these settings, our main result is to show that each agent’s state is practically synchronized to the state of the weighted average dynamics. Let us look for appropriate conditions.

$$\limsup_{k \rightarrow \infty}\|\bar x_i[k] – \bar s[k]\| \leq \gamma(1/\kappa), \quad \gamma \in {\mathcal K}$$

Since this work has real life motivation, we show the theoretical convergence and the explicit condition of the sampling period $T$ and the coupling gain $\kappa$ that gurantees the practical synchronization. We assume three conditions. First, the graph is undirected and connected. Second, the time varying signal $b_i (t)$ is uniformly bounded in $t$ . Third, The summation of $a_i$ for all $i$ is negative. This condition is equivalent to that the weighted averaged dynamics is Schur stable for any $T$ .

Then, there exists two continuous function $\kappa_m$ and $\kappa_M$ . And as $\tau$ goes to infinity, $\kappa_M$ goes to infinity and $\kappa_m$ converges to constant value. Therefore, there exists $T^\star$ such that makes $\kappa_m$ becomes smaller that $\kappa_M$ . Furthermore, for any $T$ bigger than 0, smaller than $T^\star$ , and for any $\kappa$ bigger than $\kappa_m$ and smaller than $\kappa_M$ , the solutions of the overall system practically converge to the weighted averaged dynamics practically.

Theorem:
1. $\exists$ two continuous functions $\kappa_{\rm m}(\cdot)$ and $\kappa_{\rm M}(\cdot)$ such that
$$ \kappa_{\rm M}(\tau) \to \infty \; \text{as} \; \tau \to 0, \quad \text{and} \quad \lim_{\tau \to 0} \kappa_{\rm m}(\tau) < \infty$$
and therefore,
$$\exists \; T^\star \quad \text{s.t.} \quad \kappa_{\rm m}(\tau) \leq \kappa_{\rm M}(\tau), \; \forall \tau \in (0, T^\star]$$

2. For any $T \in (0, T^\star]$ , $\exists \gamma \in {\mathcal K}$ s.t.
$$ \limsup_{k \rightarrow \infty} \|\bar{x}_i[k]-\bar{s}[k]\| \leq \gamma(1/\kappa), \quad \forall \kappa \in [\kappa_{\rm m}(T), \kappa_{\rm M}(T)], \; \forall i.$$

Let us compare with continuous time case. In the continuous time case, the blending principle ensures the practical convergence to the blended dynamics as coupling gain enlarges. When it comes to discrete time communication cases, it is not possible to achieve the practical converge to blended dynamics by strong coupling gain. Instead, we have the practical converge to the weighted averaged dynamics for all $\kappa$ in the valid interval. If $T$ goes to zero, weighted averaged dynamics tends to going to blended dynamics. However, sampling period $T$ cannot be zero, so the weighted averaged dynamics is clearly different the blended dynamics.

It’s time to see numerical example. Consider a group of 5 agents. Each system perturbes nearby 0 and external inputs are sinusoidal signal. Varying two design variables do three sets of simulations.

$$\dot{x}_i(t) = a_i x_i(t) + \psi_i \sin(\nu_i t + \phi_i) + \kappa \sum_{j\in \mathcal{N}_i} (x_j(kT) – x_i(kT))$$
where $a_i \in (-1, 1)$ , $\psi_i \in [0.5,2]$ , $\nu_i \in [10,20]$ , and $\phi_i \in [0,3]$ .

Firstly, we vary coupling gain $\kappa$ at fixed sampling period. Then we vary sampling period $T$ and $\kappa$ is given by corresponding maximum $\kappa$ .

$T=T_1$ and $\kappa = \kappa_{\rm m}(T_1)$
$T=T_1$ and $\kappa = \kappa_{\rm M}(T_1)$
$T= T_2<T_1$ and $\kappa = \kappa_{\rm M}(T_2)$

Here, sampling period $T_2$ is smaller than $T_1$ .

See this figure, please.

The blue solid line is the state of each agent. The red solid line is the error between each state and the solution ot the weighted averaged dynamics. Increasing kappa at the same $T_1$ is not enough to achieve desired error bound of the practical consensus because there is an upper bound of $\kappa$ and the class $\mathcal{K}$ function is dependent on $\kappa$ . To achieve desired error bound of practical consensus, lessen $T$ firstly, and then increase $\kappa$ up to the $\kappa_M$ .

In a sampled-data setting, the collective behavior emerges as the weighted averaged dynamics as we revealed. And we propose the explicit relation between the sampling period and the coupling gain for practical synchronization.