Formation Under Communication Constraints: Control Performance Meets Channel Capacity

TL;DR

The paper addresses how wireless channel capacity constrains formation control in second-order MASs with bounded noises, and develops a joint control-communication framework. It introduces a guaranteed communication region (GCR) that quantifies feasible receiver locations given data rate, bandwidth, transmit power, noise, and transmitter uncertainty, and proves the GCR radius is a concave function of the transmission time $\tau$, revealing a fundamental trade-off between region size and data rate. It derives fundamental data-rate limits for achieving a prescribed formation accuracy, and presents an integrated design combining an estimation-based controller with a distributed transmit-power strategy to preserve connectivity and achieve bounded formation under communication constraints. The framework is validated with a six-UAV numerical example showing formation convergence under realistic data-rate and power constraints, illustrating practical impact for cooperative aerial networks under spectrum limitations.

Abstract

In wireless communication-based formation control systems, the control performance is significantly impacted by the channel capacity of each communication link between agents. This relationship, however, remains under-investigated in the existing studies. To address this gap, the formation control problem of classical second-order multi-agent systems with bounded process noises was considered taking into account the channel capacity. More specifically, the model of communication links between agents is first established, based on a new concept -- guaranteed communication region, which characterizes all possible locations for successful message decoding in the present of control-system uncertainty. Furthermore, we rigorously prove that, the guaranteed communication region does not unboundedly increase with the transmission time, which indicates an important trade-off between the guaranteed communication region and the data rate. The fundamental limits of data rate for any desired accuracy are also obtained. Finally, the integrated design to achieve the desired formation accuracy is proposed, where an estimation-based controller and transmit power control strategy are developed.

Figures (9)

• Figure 1: Illustration of transmitters and receivers. Consider agent $i$ with the transmission time $T = \tau h$ can successfully receive messages transmitted by agents $j_1$ and $j_2$. At time step $m\tau$, agent $i$ broadcasts message $\mathcal{M}_{i,m\tau}$ and receives messages $\mathcal{M}_{j_1,(m-1)\tau}$ and $\mathcal{M}_{j_2,(m-1)\tau}$ from agents $j_1$ and $j_2$.
• Figure 2: Illustration of the communication region. The channel capacity from agent $2$ to agent $1$ at time $k$ is $C_{1,2,k}=100\mathrm{kbps}$. If agent $2$ uses the data rate $\mu=64\mathrm{kbps} < C_{1,2,k}=100\mathrm{kbps}$, agent $1$ can successfully receive (decode) the transmitted messages; the corresponding communication region is the (lighter blue) circle with the dashed boundary. If agent $2$ uses the data rate $\mu=128\mathrm{kbps} > C_{1,2,k}=100\mathrm{kbps}$, the communication link from agent $2$ to agent $1$ breaks; the corresponding communication region is the (darker blue) circle with the dashed boundary. It is clear that agent $1$ is within the communication region of agent $2$ if $\mu=64\mathrm{kbps}$, but it falls outside the region if $\mu=128\mathrm{kbps}$.
• Figure 3: Illustration of the guaranteed communication region of agent $i$ with position range $\llbracket \mathbf{p}_{i,k}\rrbracket^{[i]}=\left\{ p'_{i,k},p"_{i,k},p"'_{i,k} \right\}$. The communication regions of agent $i$ located at $p'_{i,k}, p"_{i,k}, p"'_{i,k}$ are $\Omega_{p'_{i,k}}, \Omega_{p"_{i,k}}, \Omega_{p"'_{i,k}}$, respectively. The guaranteed communication region for agent $i$ with position range $\llbracket \mathbf{p}_{i,k}\rrbracket^{[i]}$ is the intersection of these (blue) circles, i,e., $\Omega _{\llbracket \mathbf{p}_{i,k} \rrbracket^{[i]}}=\Omega _{p'_{i,k}}\cap \Omega _{p"_{i,k}}\cap \Omega _{p"'_{i,k}}$ (the region enclosed by red dashed lines).
• Figure 4: The relationship between the communication region and the data rate. Consider two agents $i$ (as a transmitter) and $j$ (as a receiver) moving in X-Y plane. The sampling period $h=0.05\mathrm{s}$ and the radii of both position and velocity noise ranges for each agent are $0.5\mathrm{m}$. The communication parameters are: $M=64*8\mathrm{bits},~B_w=10^6\mathrm{Hz},~d_0=1\mathrm{m},~g_{d_0}=1/(16\pi)^2,~P_{i,k}^{\mathrm{tx}}=1\mathrm{w},~\psi=2$. The noise power of each agent at time instant $k \in \mathbb{N}_0$ is the sum of channel noise and jamming noise, i.e., $W=(N_0+N_{\rm{jam}})B_w$, where the channel and jamming noise power spectrum density are $N_0=10^{-11}\mathrm{w}/\mathrm{Hz}$ and $N_{\rm{jam}}=2.5*10^{-10}\mathrm{w}/\mathrm{Hz}$, respectively. Take the location of agent $i$ as the origin of the 2-D plane. (a) The illustration of the maximal achievable data rate w.r.t. the relative position between agents $i$ and $j$. (b) The communication radii corresponding to different data rates.
• Figure 5: Illustration of Proposition \ref{['prop:Concave Property']} for agent $i$. The parameters are the same as those in Fig. \ref{['fig:CapacityCurves']}. (a) The relationship between the GCR, the transmit power, and $\tau$. (b) Curve of the GCR w.r.t $\tau$, where the transmit power is $P_{i,k}^{\mathrm{tx}}=1\mathrm{w}$.

Theorems & Definitions (26)

• Definition 1: Set of Neighbors
• Definition 2: Communication Region
• Remark 1
• Definition 3: Guaranteed Communication Region
• Definition 4: Edge Set
• Lemma 1: Set of Neighbors
• Definition 5: Bounded Formation Stability
• Definition 6: Power Control Strategy
• Proposition 1: Position Range
• Remark 2