Constricting Tubes for Prescribed-Time Safe Control

1. Problem Statement

The paper addresses the challenge of prescribed-time control for control-affine systems subject to input constraints. The specific goal is to drive a dynamical system from an initial state $x(0)$ (which lies outside a target safe set $\mathcal{C}$) into $\mathcal{C}$ by a user-specified deadline $T$, and maintain it there for all $t \geq T$.

Key Challenges:

• Input Constraints: Existing prescribed-time methods often rely on time-varying gains that diverge (go to infinity) as $t \to T$. This makes it impossible to guarantee feasibility under hard input limits (e.g., $|u| \leq u_{max}$).
• Scalability: Traditional methods like Hamilton-Jacobi (HJ) reachability provide formal guarantees but suffer from the "curse of dimensionality," requiring grid-based discretization that scales exponentially with state dimension.
• Asymptotic vs. Finite-Time: Standard Control Barrier Functions (CBFs) ensure safety (forward invariance) but only achieve recovery asymptotically, not within a fixed time.

2. Methodology: The Constricting Tube Framework

The authors propose a novel framework that transforms the prescribed-time recovery problem into a forward invariance problem of a time-varying "safety tube."

A. Constricting CBF Definition

Instead of a static safe set $\mathcal{C} = \{x \mid h(x) \geq 0\}$, the authors define a time-dependent relaxation:
$$\tilde{h}(x, t) = h(x) + r(x(0), t)$$
where $r(x(0), t)$ is a constriction schedule satisfying:

1. Initial Containment: $r(x(0), 0) = \max(0, -h(x(0)))$, ensuring the initial state is inside the relaxed set $\tilde{\mathcal{C}}(0)$.
2. Target Recovery: $r(x(0), T) = 0$, ensuring the tube collapses exactly to the target set $\mathcal{C}$ at time $T$.
3. Monotonicity: $\dot{r} \leq 0$, meaning the tube shrinks over time.

The "tube" is the set $\tilde{\mathcal{C}}(t) = \{x \mid \tilde{h}(x, t) \geq 0\}$. If a controller keeps the system inside this shrinking tube, the system is guaranteed to be in $\mathcal{C}$ at time $T$.

B. Control Synthesis

The control input $u$ is synthesized by solving a Quadratic Program (QP) at each timestep to minimize control effort while satisfying the Constricting CBF condition:
$$L_f h(x) + L_g h(x) u + \dot{r}(x(0), t) \geq -\gamma(\tilde{h}(x, t))$$
Here, $\dot{r}$ acts as an "inward pressure" driving the state toward the target. Unlike other methods, this constraint is affine in $u$ and does not depend on the state dimension for the number of constraints (only one per timestep).

C. Feasibility Analysis under Input Constraints

A critical contribution is the derivation of a closed-form condition for feasibility. The authors define Barrier Authority $\sigma(x)$, which represents the maximum rate at which the barrier function can be increased given the input limit $u_{max}$:
$$\sigma(x) = \max_{\|u\| \leq u_{max}} \dot{h}(x, u) = \|L_g h(x)\| u_{max} + L_f h(x)$$

Theorem 2 (Feasibility & Minimum Recovery Time):
For a linear constriction schedule $r(t) = r_0(1 - t/T)$, the control problem is feasible if and only if:
$$T \geq T_{min} = \frac{r_0}{\sigma_{min}}$$
where $r_0$ is the initial violation and $\sigma_{min}$ is the worst-case barrier authority over the tube. This provides a design-time certificate: if the deadline $T$ is too short relative to the system's control authority, the problem is provably infeasible.

D. Extensions

• High-Order CBFs (HOCBF): The framework extends to systems with relative degree $k > 1$ by propagating the constriction schedule through the higher-order derivatives.
• Stochastic Systems: The framework is noted to generalize to stochastic settings (e.g., generative sampling models).

3. Key Contributions

1. Constricting CBF Framework: A method to guarantee prescribed-time recovery via forward invariance of a shrinking tube, avoiding the need for singular time-varying gains.
2. Bounded Control Effort: Unlike existing prescribed-time methods where control effort diverges near the deadline, this framework ensures bounded control effort by design, making it compatible with hard input constraints.
3. Analytic Feasibility Certificate: Derivation of a verifiable condition ($T \geq T_{min}$) that links the deadline, initial state, and control authority without requiring offline PDE solving or Lyapunov analysis.
4. Scalability: The method imposes a single affine constraint per timestep regardless of state dimension, scaling linearly ($O(n)$) compared to the exponential scaling of HJ reachability.

4. Experimental Results

The framework was validated on three distinct scenarios:

• 16-Dimensional Multi-Agent System:

  • Setup: 8 decoupled agents (16D state) with a joint barrier function.
  • Result: The method successfully drove all agents into the target set by $T=6.4$s.
  • Comparison: Compared against a standard Prescribed-Time CLF method [5]. The proposed method maintained smooth, bounded control effort (peak $\approx 3.66$), whereas the baseline method saturated persistently near the limit ($\approx 28.28$) due to diverging gains. HJ reachability was deemed intractable for this dimension.

• Unicycle Reach-Avoid (Nonlinear):

  • Setup: A unicycle model (nonlinear dynamics) tasked with reaching a target while avoiding an obstacle on the direct path.
  • Result: Integrated into a Model Predictive Control (MPC) framework. The planner successfully navigated around the obstacle and reached the target exactly at the deadline, respecting both the reach and avoid constraints simultaneously.

• Double Integrator (HOCBF):

  • Setup: A relative-degree-2 system.
  • Result: Demonstrated the HOCBF extension, showing that the controller could enforce the deadline even when a nominal PD controller failed to do so, while respecting input bounds.

5. Significance and Impact

This paper bridges a critical gap between safety guarantees and timing constraints in constrained control systems.

• Practicality: By avoiding diverging control gains, the method is directly applicable to real-world systems with actuator limits (e.g., robots, satellites, autonomous vehicles).
• Computational Efficiency: It offers a scalable alternative to HJ reachability, enabling real-time synthesis for high-dimensional systems where traditional methods fail.
• Design Certainty: The closed-form $T_{min}$ allows engineers to determine a priori whether a specific mission deadline is achievable given the hardware's control authority, eliminating the need for trial-and-error tuning or post-hoc verification.

In summary, the "Constricting Tube" approach provides a rigorous, scalable, and computationally efficient solution for forcing systems into safe sets within a fixed time, even under strict physical limitations.