Signal-Guided Control

• Signal-guided control is a coordinated system where agents use minimal, quantized signals to maintain prescribed distances in a rigid one-dimensional formation.
• It employs a control law based on binary feedback, ensuring convergence through non-smooth dynamics analysis, with specific gains and sign functions governing agent motion.
• The method guarantees geometric determinacy and efficient bandwidth usage while addressing pathological cases with Krasowskii solutions and Lyapunov stability analysis.

Signal-guided control refers to distributed or centralized control strategies in which agents, actuators, or controllers adjust their behavior in real time exclusively according to explicit low-bandwidth “signals” (often extremely coarse, such as quantized states) transmitted by a guidance system. In the context of one-dimensional formation control, this paradigm achieves robust agent coordination in scenarios where communication resources are severely restricted. The defining characteristics are: control policies that use minimal, quantized guidance; strict distance constraints enforcing structural rigidity; rigorous non-smooth analysis for convergence guarantees; and explicit identification of pathological failure cases.

1. Control Law Formulation Using Coarse Signals

The core of the approach is a control policy for $n$ mobile agents aligned along a one-dimensional axis (the $x$-axis), with the desired formation defined by strict pairwise distance constraints: $$|x_i - x_{i+1}| = d_i, \qquad i = 1, \ldots, n-1, \qquad d_i > 0.$$ This system is rigid since, modulo translation, these constraints uniquely determine the agents' configuration. The guidance system transmits only the signs of inter-agent separations relative to these prescribed distances via a quantizer

$$\operatorname{sgn}(z) = \begin{cases} +1 \quad &z \geq 0 \ -1 \quad &z < 0 \end{cases}$$

to assess whether the current distance is above/below the target.

The resulting agent-level continuous-time dynamics are:

• For agent 1 (the leader):

$$\dot{x}_1 = -k_1\, \operatorname{sgn}(x_1 - x_2)\, \operatorname{sgn}(|x_1 - x_2| - d_1)$$

• For agent $i$, $2 \leq i \leq n-1$ (interior agents):

$$\dot{x}_i = \operatorname{sgn}(x_{i-1} - x_i)\, \operatorname{sgn}(|x_{i-1} - x_i| - d_{i-1}) - k_i\, \operatorname{sgn}(x_i - x_{i+1})\, \operatorname{sgn}(|x_i - x_{i+1}| - d_i)$$

• For agent $n$ (the tail):

$$\dot{x}_n = \operatorname{sgn}(x_{n-1} - x_n)\, \operatorname{sgn}(|x_{n-1} - x_n| - d_{n-1})$$

The gains $k_i > 0$ are design parameters. This law aligns each agent in accordance with coarse positional information—no more than 2 bits per distance per agent are required.

2. Communication Constraints and Bandwidth Quantization

A salient property of this scheme is its reliance on extremely low-rate, quantized guidance. Each intermediate agent depends on two bits per update (one for each adjacent distance), but due to the structure of the law, up to four bits bandwidth per agent suffice. End agents require only two bits each. Thus, for $n$ agents, the guidance system uses*

$4n - 2 \quad \text{bits per guidance cycle}.$

*This decouples system performance from the data rate of the channel, which is essential in emerging intelligent transport or vehicular networks where severe communication bottlenecks are common.

3. Rigidity and Uniqueness in Formation

The formation is “rigid” in the sense of graph theory: the set of pairwise distances $d_i$ uniquely determines all agent positions (up to a global translation—a one-dimensional gauge). The equilibrium set for the closed-loop system is characterized as all vectors $(z_1, \ldots, z_{n-1})$ satisfying either $z_i = 0$ (a degenerate case) or $|z_i| = d_i$ for all $i$. Formally: $\sum_{i=1}^{n-1} |z_i|\, ||z_i| - d_i| = 0$ This property ensures that, absent degeneracy, the only equilibria correspond to agents at the prescribed separations, fulfilling the geometric specification.

4. Non-Smooth Dynamics and Krasowskii Solutions

The discontinuous nature of the sign function induces non-smooth vector fields, rendering classical ODE techniques inapplicable. The system must be analyzed through the lens of differential inclusions—specifically, Krasowskii solutions: $\dot{x} \in \mathcal{K}(f(x))$ where $\mathcal{K}(f(x))$ is the convex hull of vector fields “near” $x$, closure taken. To analyze convergence, a Lyapunov function

$$V(z) = \frac{1}{4} \sum_{i=1}^{n-1} (z_i^2 - d_i^2)^2$$

is constructed. Along Krasowskii solutions, $V$ is non-increasing away from switching surfaces, and strictly decreases except when all pairs are at prescribed distances. In neighborhoods of discontinuity (e.g., as an agent crosses the $|z_i| = d_i$ surface), solutions follow sliding modes captured by the differential inclusion rather than single-valued vector fields. This analysis is crucial for correctly establishing stability and convergence in systems with quantized guidance.

5. Convergence Properties and Thin Set of Failures

For all but a measure-zero set of initial conditions, robust convergence to the equilibrium formation is established:

• For three agents ($n=3$) with, e.g., $k_1 = k_2 = 1$, the dynamics in $z$-coordinates are:

$$\dot{z}_1 = -2\, \operatorname{sgn}(z_1)\, \operatorname{sgn}(|z_1| - d_1) + \operatorname{sgn}(z_2)\, \operatorname{sgn}(|z_2| - d_2)$$

$$\dot{z}_2 = \operatorname{sgn}(z_1)\, \operatorname{sgn}(|z_1| - d_1) - 2\, \operatorname{sgn}(z_2)\, \operatorname{sgn}(|z_2| - d_2)$$

For initial conditions where neither inter-agent separation is zero, all Krasowskii solutions converge in finite time to the set $|z_1|=d_1,\,|z_2|=d_2$.

• For the pathological case where $z_1$ or $z_2$ is initially zero (i.e., some agents coincide), the system may experience “sliding modes” or ambiguous behaviors due to the multi-valued nature of the differential inclusion at these surfaces. This can result in non-unique or chattering behaviors, which are detrimental in practical deployments. Such initial conditions constitute a measure-zero “thin” set, and outside these, the convergence is global.

6. Analytical Summary and Design Implications

The signal-guided control paradigm for rigid 1-D formations demonstrates that:

Principle	Property/Guarantee	Implementation Feature
Quantized Guidance	2–4 bits per agent per cycle; uses only signs of state errors	Extreme communication sparsity, simple hardware
Rigidity	Locally unique formation shape, up to translation	Geometric determinacy
Non-Smooth Analysis	Krasowskii solutions, Lyapunov argument for differential inclusions	Handles discontinuities, guarantees stability
Failure Modes	“Thin” set with sliding modes or non-uniqueness	Pathological only for degenerate initial states

This approach enables bandwidth-efficient distributed control in vehicular platoons, satellite arrays, and resource-constrained robotic swarms, where robust geometric coordination via limited signaling is critical. Design choices (e.g., sign quantization, choice of gains $k_i$) must balance convergence speed and sensitivity to initial agent overlap, as the system is provably robust except in degenerate scenarios characterized by coincident initial agent positions.

7. Broader Implications and Extensions

Signal-guided control as articulated in this formation context provides a constructive answer to the question of how much information is truly needed for distributed coordination. It validates that, for one-dimensional rigid formations, real-time guidance with only binary quantized feedback is sufficient for almost-global stabilization. The analytical toolbox—primarily non-smooth and set-valued analysis—applies equally to higher-dimensional extension, quantized consensus, and broader classes of non-smooth, signal-limited multi-agent systems.

Thus, the signal-guided control law forms a canonical paradigm for emergent distributed control in networks with severe information constraints, providing strong guarantees on geometric coordination, convergence, and communication efficiency, together with an explicit demarcation of its boundary of applicability.