Distributed Safe Consensus Under Asymmetric Input and Time-Varying Output Constraints

Abstract: This paper studies safe distributed consensus for single-integrator multi-agent systems over connected undirected graphs under simultaneous asymmetric actuator constraints and output safety constraints. Each agent is equipped with a continuously differentiable asymmetric actuator dynamics that maps a commanded control signal to the realized plant input while keeping the latter strictly inside a prescribed admissible interval. To address output safety, a barrier-coordinate transformation is introduced over a common time-varying safe interval, and a distributed synchronization law is designed in the transformed coordinates. The resulting controller integrates a graph-based coordination layer with an actuator-side tracking layer, thereby enabling simultaneous enforcement of input admissibility, forward invariance of the safe output set, and asymptotic synchronization. For compact admissible sets of initial conditions, it is shown that the closed-loop solution is complete, all signals remain bounded, the actuator inputs remain strictly within their asymmetric bounds, and the agent outputs remain inside the prescribed safe interval for all time. Moreover, the transformed synchronization errors converge exponentially to zero, and the original agent outputs asymptotically synchronize to a designer-selected admissible trajectory embedded in the common safe interval. Numerical simulations validate the proposed framework and demonstrate safe consensus under both asymmetric actuation bounds and time-varying output constraints.

• The paper develops a distributed control framework for multi-agent systems, ensuring robust consensus under asymmetric actuator dynamics and time-varying output constraints.
• The framework features a two-layer architecture: an actuator-tracking layer to maintain input feasibility and a coordination layer using logarithmic barrier coordinates for safety.
• The framework guarantees exponential decay of synchronization errors and strict adherence to safety constraints, with explicit gain and interiority conditions for practical implementation.

This paper develops a distributed control framework for single-integrator multi-agent systems over connected undirected graphs that simultaneously enforces strict asymmetric actuator constraints and time-varying output safety constraints while achieving asymptotic synchronization (2606.16116). The central architectural idea is to separate the problem into two cascaded layers: an actuator-tracking layer that guarantees the realized plant input never leaves its admissible interval, and a coordination layer operating in logarithmic barrier coordinates over a common time-varying safe interval. The analysis is semiglobal, self-contained, and yields explicit a priori gain and interiority conditions.

Problem formulation

The network consists of $N$ single-integrator agents, $\dot{x}_i = u_i$, communicating over a connected undirected graph $\mathcal{G}$ with Laplacian $L$. Two features distinguish the setup from standard consensus formulations:

• Asymmetric actuator dynamics: the commanded signal $v_i$ is not the plant input; instead, each agent has first-order actuator dynamics $\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$, where the regularization factor $\sigma_i(u_i)$ is continuously differentiable on the admissible interval $(\underline{u}_i,\overline{u}_i)$ and vanishes only at its boundaries, with even-power exponents $\gamma_i = 2n_i$. The bounds may be asymmetric ($|\underline{u}_i| \neq |\overline{u}_i|$) and heterogeneous across agents.
• Time-varying output constraints: each agent must satisfy $\dot{x}_i = u_i$0, with feasibility requiring a nonempty common intersection of all individual intervals (a uniform width condition $\dot{x}_i = u_i$1).

The paper poses two problems: consensus under asymmetric bounded inputs, and safe consensus under simultaneous input and output constraints, both in a semiglobal sense over compact admissible initial sets. A notable remark clarifies that the objective is asymptotic agreement, not average consensus, since the actuator dynamics preclude preservation of the initial-state average.

Consensus under asymmetric input constraints

The controller combines a nominal graph-based law $\dot{x}_i = u_i$2 with an exact actuator-tracking inversion: the commanded input cancels the known actuator dynamics and imposes $\dot{x}_i = u_i$3 on the tracking error $\dot{x}_i = u_i$4, so that $\dot{x}_i = u_i$5 exactly. On the disagreement subspace, the closed loop reduces to $\dot{x}_i = u_i$6.

Using a quadratic Lyapunov function in $\dot{x}_i = u_i$7, the main theorem establishes exponential decay at rate $\dot{x}_i = u_i$8, provided the gain condition $\dot{x}_i = u_i$9 holds. Completeness and strict admissibility follow from an explicit interiority condition $\mathcal{G}$0, which guarantees $\mathcal{G}$1 remains in a symmetric compact subinterval of the generally asymmetric set $\mathcal{G}$2. The consensus value $\mathcal{G}$3 is computed explicitly as the initial average plus an integral contribution from the exponentially decaying tracking errors. An important concession is made here: because the interiority test uses a symmetric inner bound inside an asymmetric interval, it is sufficient but conservative; one-sided refinements are acknowledged but not pursued.

Safe consensus via barrier coordinates

For the safe-consensus problem, the authors strengthen Assumption on output constraints by postulating a common time-varying admissible core $\mathcal{G}$4 of uniformly positive width $\mathcal{G}$5 and bounded derivatives up to second order. The key device is the logarithmic barrier coordinate

$\mathcal{G}$6

shown to be a $\mathcal{G}$7-diffeomorphism from $\mathcal{G}$8 onto $\mathcal{G}$9 for every fixed $L$0. Boundedness of $L$1 is therefore equivalent to uniform strict interiority of the physical output constraint—a property that distinguishes this construction from funnel or prescribed-performance transformations applied to error variables rather than to outputs directly.

In transformed coordinates, the synchronization input includes three terms: a feedforward $L$2 tracking the prescribed trajectory's transformed representation, a graph coupling term $L$3, and a pinning term $L$4. The pinning term eliminates the neutral consensus mode, yielding synchronization to the designer-selected admissible trajectory $L$5—an objective strictly stronger than agreement to an unspecified limit. A remark verifies that $L$6, needed for the actuator inversion, is computable from local information only (own state, neighbor-relative quantities, and the known functions $L$7 and their derivatives), so the commanded signal remains distributed and requires no derivatives of $L$8 or neighbors' commanded inputs.

The main safe-consensus theorem proves, under the gain condition $L$9 (where $v_i$0 bounds the barrier Jacobian over the invariant slab $v_i$1) together with an interiority condition analogous to the input-constrained case:

• forward invariance of $v_i$2, with states confined to the compact strip $v_i$3, $v_i$4;
• strict actuator admissibility and completeness of solutions;
• exponential decay of the joint transformed-synchronization/tracking error at rate $v_i$5;
• asymptotic convergence $v_i$6 for all $v_i$7, using the mean-value bound $v_i$8.

A corollary treats the special case of a constant transformed reference $v_i$9, showing that if the safe core itself is time-invariant, the network converges to a fixed admissible point $\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$0 determined by $\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$1 through the inverse barrier map. This provides an interpretable tuning knob: the designer selects the limiting consensus value by choosing its logit-ratio position within the safe interval.

Numerical validation

Simulations use a five-agent cycle graph with heterogeneous asymmetric actuator intervals (e.g., agent 1 admits $\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$2 while agent 5 admits $\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$3) and a moving asymmetric safe core built from multi-frequency sinusoidal components with lower/upper half-widths satisfying $\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$4, $\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$5. Three reference profiles are tested via a normalized location variable $\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$6: a multi-frequency signal ($\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$7), a biased sinusoid ($\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$8), and a smooth transition emulating relocation of the operating point ($\dot{u}_i = p_{1,i}\sigma_i(u_i)v_i - p_{2,i}u_i$9). In all cases, the state-safety margin $\sigma_i(u_i)$0 remains strictly positive and the regularization factor $\sigma_i(u_i)$1 stays positive—the latter also confirming nonsingularity of the actuator-compensating command, since $\sigma_i(u_i)$2 appears in the denominator of $\sigma_i(u_i)$3. A four-agent case with constant barrier-coordinate reference confirms the corollary's predictions. The simulations support all three theoretical conclusions but do not stress-test the conservatism of the interiority condition near actuator boundaries.

Limitations and open questions

Several restrictions are explicit in the development. The framework covers scalar single-integrator agents; extension to higher-order or nonlinear agent dynamics is not addressed. The semiglobal guarantee requires compactness of the admissible initial set, and both theorems depend on interiority conditions that are admittedly conservative—symmetric inner bounds within asymmetric actuator intervals—with tighter one-sided conditions left as an open refinement. Feasibility rests on strong structural assumptions: a common safe core with bounded derivatives up to second order, a $\sigma_i(u_i)$4 transformed reference with known bounds, and perfect knowledge of the core boundaries and their derivatives at every agent (the design is not robust to boundary estimation error). The actuator model, while differentiable and asymmetric, is a specific first-order parametrization; whether the guarantees extend to other regularity structures or to input disturbances is not examined. Finally, no delay, switching topology, or directed-graph generalizations are considered.

Conclusion

The paper contributes a unified distributed architecture in which asymmetric actuator feasibility, forward invariance of a time-varying output-safe set, and synchronization to a designer-selected admissible trajectory are established in a single closed-loop analysis, without auxiliary a priori boundedness assumptions. Its main technical assets are the exact actuator-tracking inversion producing closed-form exponential tracking errors and the output-level barrier transformation whose boundedness certifies physical safety. The explicit gain and interiority conditions make the guarantees checkable a priori, though at the cost of documented conservatism that subsequent work could reduce.