Event-Triggered Adaptive Taylor-Lagrange Control for Safety-Critical Systems

This paper addresses safety-critical control for nonlinear systems under sampled-data implementations by proposing an adaptive Taylor–Lagrange Control (aTLC) framework with an event-triggered implementation. The core contribution is dynamically selecting the discretization parameter online as a state-dependent variable to balance feasibility and safety, resulting in a controller that improves feasibility, guarantees safety, and achieves smoother control actions compared to fixed-parameter methods like non-adaptive Taylor–Lagrange Control (TLC).

The gist

The proposed adaptive Taylor–Lagrange Control (aTLC) framework defines the discretization time scale as a state-dependent variable selected online, enabling the controller to dynamically balance feasibility and safety by adjusting the effective time scale of the Taylor expansion.

Problem Formulation and Approach

The objective is to generate a control strategy for an affine control system, defined by its dynamics in equation (1), that ensures convergence to a desired equilibrium while satisfying safety requirements and respecting input constraints. The problem involves minimizing a cost functional (6) subject to safety constraints of the form h(x) ≥ 0 (7) and input limitations U (2). Existing methods like non-adaptive TLC rely on a fixed time scale, which can lead to infeasibility or unsafety in the presence of input constraints and inter-sampling effects. To address this, the approach introduces an adaptive framework where the time scale is treated as a statedependent variable selected online. This scheme combines event-triggered implementation with an adaptive selection rule to choose the discretization parameter from a finite candidate set, aiming to improve feasibility under input constraints.

Adaptive Taylor–Lagrange Control (aTLC)

The key idea of aTLC is to treat the time scale appearing in the Taylor–Lagrange expansion as a state-dependent parameter that can be adjusted online. The condition for the time scale parameterized adaptive TLC (8) involves a supremum over admissible control inputs u(ξ) within an interval defined by the state and time scale τ. The corresponding control input is computed using this condition with τ selected from a set of candidate values at each update instant, denoted as τ = K(x(t0)) (9). Theorem 2 proves that if the input satisfies this condition for a time scale τ selected by (9), the safe set C is forward invariant.

Event-Triggered Implementation and Feasibility Characterization

To mitigate inter-sampling effects, an event-triggered implementation is used, where control updates occur only when the system state exits a prescribed neighborhood S(x(tk)) (12). At each event time tk, a local set S(xk) is constructed. The robust aTLC condition (16) is defined using quantities like hratlc and Gratlc to provide a valid lower bound on the safety expression for all admissible control inputs. Feasibility is characterized by defining the set of admissible controls U(x, τ) based on the margin function M(x, τ), where feasibility is equivalent to M(x, τ) ≥ 0 (22). The minimal feasible time scale τ∗(x) is defined as the infimum of time scales that admit a non-empty set of admissible controls.

Adaptive Selection Rule and Simulation Results

To select the time scale online, a rollout-based algorithm (Algorithm 1) is proposed. This algorithm evaluates candidate values of τ from a finite set by solving the QP constraint (17), simulating the system forward over a horizon Tlook with constant input u(τ), and computing h predmin(τ) to select the time scale that maximizes this predicted safety margin. Simulation results on an adaptive cruise control (ACC) problem demonstrate that aTLC achieves improved feasibility, guaranteed safety, and smoother control actions compared to non-adaptive TLC while requiring a single automatically tuned parameter. Furthermore, it shows that aTLC maintains QP feasibility and guarantees safety even when the input constraints are tight, unlike TLC which can become infeasible under certain conditions. The method is shown to be comparable in performance to well-tuned High-Order Control Barrier Functions (HOCBFs) while tuning only one parameter.

Complexity Analysis

The computational complexity per event at each time tk is analyzed as O(Nτ (TQP + Tsim)), where Nτ is the number of candidate values, TQP is the time required to solve one QP, and Tsim is the cost of one rollout simulation. Since Nτ and both QP and rollout are computed over short horizons, the method remains computationally efficient for real-time implementation. The evaluations for different candidate time scales are independent and can be parallelized to reduce effective computation time per event.

Conclusion

The proposed adaptive Taylor–Lagrange Control (aTLC) framework successfully addresses the limitations of fixed-parameter methods by treating the discretization time scale as a state-dependent variable selected online.

Here are the specific improvements that can be made to AI systems based on this scientific paper, along with what those improved systems could achieve:

) Improved AI Systems and Capabilities:

1. Incorporate an event-triggered adaptive Taylor–Lagrange Control (aTLC) framework for safety-critical control of nonlinear agents.

2. Implement a rollout-based selection rule to dynamically choose the optimal Taylor expansion time scale online, balancing feasibility and safety constraints.

3. Integrate a value-function based characterization of feasibility (via the margin function) into the control policy selection process to ensure continuous QP feasibility under input constraints.

4. Improved AI System Capabilities:

The resulting AI system can perform high-precision, real-time autonomous control in dynamic, safety-critical environments where traditional fixed-parameter controllers fail due to input limitations or time delays (inter-sampling effects). Specifically:

1. In a self-driving vehicle (like the ACC case study), the system will maintain guaranteed safe distances and adhere strictly to physical limits (speed, acceleration) even when encountering unpredictable road conditions or sudden braking requirements.

2. The AI will demonstrate superior robustness against input constraints, avoiding situations where a fixed controller becomes infeasible under tight control bounds by dynamically adjusting its internal safety margin based on the current state of the environment.

3. The system will exhibit smoother control actions compared to non-adaptive methods, leading to less jerky and more energy-efficient maneuvers (lower control effort) while strictly guaranteeing that safety requirements (e.g., maintaining a minimum safe distance) are met over extended periods without violating input limits.

4. The AI can operate effectively in scenarios where the system is sampled or implemented discretely, as it handles inter-sampling effects by only updating its control strategy when necessary, leading to more efficient real-time computation compared to continuously re-solving complex optimization problems.