formally-guaranteed-control-adaptation

name

formally-guaranteed-control-adaptation

description

Control adaptation with formal guarantees for ODD-resilient autonomous systems. Combines barrier functions, adaptive control, and reachability analysis to provide safety certificates during online controller parameter adjustment. Use when designing adaptive controllers with runtime safety guarantees, ODD-resilient autonomous systems, barrier function-based adaptive control, reachability analysis for control systems, verified parameter adaptation. Trigger: control adaptation, ODD resilience, barrier functions, formal safety guarantees, adaptive control verification, reachable set, design domain, autonomous system safety.

Formally Guaranteed Control Adaptation

Core methodology from arXiv 2604.07414. Provides formal safety guarantees for control adaptation in ODD (Operational Design Domain) resilient autonomous systems.

When to Use

• Adaptive controllers requiring runtime safety certificates
• Autonomous systems operating in variable/uncertain conditions
• ODD boundary detection and safe adaptation
• Verified online parameter tuning
• Safety-critical control with formal guarantees

Step 1: Define Nominal System Model

Define the nominal dynamics:

dx/dt = f(x, u, theta_nom)

Where theta_nom are nominal parameters for the design ODD.

Step 2: Define ODD and Boundary Conditions

Characterize the operational design domain:

• Environmental conditions (weather, lighting, road type)
• Sensor performance bounds
• Actuator capability limits
• Dynamic constraints (speed, acceleration limits)

Define ODD boundary function B(x, env) = 0.

Step 3: Design Barrier Function

Construct a Control Barrier Function (CBF) h(x):

h(x) >= 0  =>  x is in safe set

The barrier condition:

dh/dt + alpha(h(x)) >= 0

Where alpha is an extended class K function.

Step 4: Adaptive Control Law

Design adaptive control law with parameter estimation:

u = u_nom + u_adapt(theta_hat)
d(theta_hat)/dt = adaptation_law(x, u)

The adaptation law must preserve the barrier condition.

Step 5: Safety Projection

When parameters adapt, project onto safe set:

theta_safe = argmin ||theta - theta_hat|| s.t. h(x, theta) >= 0

Use quadratic programming for real-time projection.

Step 6: Reachability Verification

For critical transitions (ODD violation), compute reachable set:

R(t) = {x(t) : x(0) in X0, u in U_admissible}

Verify R(t) stays within safe envelope for all t in [0, T].

Implementation Patterns

Pattern A: QP-based Safety Filter

import cvxpy as cp
import numpy as np

# State, dynamics, barrier function
x = ...  # current state
theta_hat = ...  # adapted parameters

u_nom = nominal_controller(x)
# Safety filter via QP
u = cp.Variable(2)  # control input
obj = cp.Minimize(cp.sum_squares(u - u_nom))

# Barrier condition: Lf h + Lg h * u + alpha(h) >= 0
constraints = [
    Lf_h(x, theta_hat) + Lg_h(x) @ u + alpha * h(x) >= 0,
    u_min <= u,
    u <= u_max
]

prob = cp.Problem(obj, constraints)
prob.solve()
u_safe = u.value

Pattern B: ODD Monitor with Mode Switching

while running:
    env = sense_environment()
    if in_ODD(env):
        u = nominal_controller(x)
    else:
        if not safe_switch_verified(x, env):
            u = emergency_safe_controller(x)
        else:
            u = adaptive_controller(x, theta_hat)
            theta_hat = update_parameters(x, u)
            theta_hat = project_to_safe_set(theta_hat, x)
    apply_control(u)

Pattern C: Reachability-based Safety Check

# Compute over-approximated reachable set
from reachability_tool import compute_reach_set

# Over next T seconds
R = compute_reach_set(
    dynamics=f,
    x0=current_state,
    u_bounds=(u_min, u_max),
    theta_bounds=(theta_min, theta_max),
    time_horizon=T,
    method="hamilton_jacobi"  # or "polytope"
)

# Check if reachable set intersects unsafe set
if intersects(R, unsafe_set):
    # Take corrective action
    apply_emergency_controller()

Key Pitfalls

• Barrier functions must be continuously differentiable -- non-smooth barriers break the certificate
• Adaptive law must be designed WITH the barrier condition, not after
• Reachability computation can be conservative -- use tight over-approximations
• ODD boundary detection must have bounded false positive/negative rates
• Parameter adaptation rate must be slow enough for the barrier certificate to hold
• Multi-agent systems require decentralized barrier functions

References

• Control Barrier Functions (Ames et al., 2019)
• Reachability Analysis (Mitchell et al., 2005)
• arXiv 2604.07414: Full technical derivations and proofs