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Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions

Charles Dawson, Zengyi Qin, Sicun Gao, Chuchu Fan

arXiv:2109.06697v2eess.SYcs.RO

TL;DR

Nonlinear robotic control must satisfy safety and stability despite model uncertainty, yet automatically synthesizing such controllers remains difficult. The paper learns robust control Lyapunov barrier functions and derives feedback controllers using robust optimization. In simulation, the controllers match or exceed robust MPC across several tasks while reducing computational cost by an order of magnitude.

  • Problem

    Automatically synthesizing safe, stable, and robust controllers for nonlinear reach-avoid tasks under model uncertainty remains difficult.

  • Method

    The paper learns robust control Lyapunov barrier functions with a model-based supervised framework and derives controllers using robust convex optimization.

  • Results

    Across challenging simulated safe-control tasks, the method matches or exceeds robust MPC while reducing computational cost by an order of magnitude.

  • Takeaways & Limitations

    The approach provides robust nonlinear feedback controllers with explicit robustness guarantees and real-time execution in the reported car-tracking experiment.

  • Takeaways & Limitations

    The learned robust CLBF conditions are enforced on finite training samples rather than exhaustively verified across the state space.

Abstract

from arXiv · show

Safety and stability are common requirements for robotic control systems; however, designing safe, stable controllers remains difficult for nonlinear and uncertain models. We develop a model-based learning approach to synthesize robust feedback controllers with safety and stability guarantees. We take inspiration from robust convex optimization and Lyapunov theory to define robust control Lyapunov barrier functions that generalize despite model uncertainty. We demonstrate our approach in simulation on problems including car trajectory tracking, nonlinear control with obstacle avoidance, satellite rendezvous with safety constraints, and flight control with a learned ground effect model. Simulation results show that our approach yields controllers that match or exceed the capabilities of robust MPC while reducing computational costs by an order of magnitude.

1 Introduction

The paper addresses the open problem of automatically synthesizing safe, stable, and robust controllers for nonlinear reach-avoid tasks under model uncertainty. It proposes a learning-based framework with explicit robustness guarantees and evaluates it against robust MPC across challenging control problems.

  • Motivation: Model uncertainty and task complexity make robotic controllers difficult to design when safety and stability must be satisfied simultaneously.The physical plant may differ from the engineering model through uncertain mass, friction, or unmodeled effects.
  • Related approaches: Robust MPC offers general-purpose synthesis but has recognized disadvantages, including computational complexity and limited guarantees for safety, stability, or recursive feasibility.These issues are especially relevant for resource-constrained robotic systems.
  • Related approaches: CLFs and CBFs provide certificates for stability and safety, but constructing them is difficult for nonlinear systems.Certificate construction becomes particularly challenging with complex specifications such as obstacle avoidance.
  • Related approaches: Learning-based control raises questions about soundness, robustness, and generalization, despite incorporating certificates such as Lyapunov and barrier functions.Many neural certificate methods do not explicitly address robustness to model uncertainty.
  • Contribution: The proposed framework learns robust control Lyapunov barrier functions and derives robust nonlinear feedback controllers from safety and stability specifications.The approach uses robust convex optimization and targets generalization beyond parameters seen during training.
  • Contribution: Across trajectory tracking, obstacle avoidance, flight control, and satellite rendezvous, the method matches or exceeds robust MPC while reducing computational cost.The experiments compare the proposed method with robust MPC on multiple safe-control tasks.

3 Preliminaries and Background

The paper formulates robust safe control for uncertain control-affine nonlinear systems and uses a single control Lyapunov barrier function to encode reachability and safety. An admissible-control condition and QP-based controller then provide certified behavior.

  • Problem formulation: The system is modeled as a continuous-time control-affine nonlinear dynamical system with uncertain parameters entering smooth dynamics.The dynamics are affine in the uncertain parameters and may include inertia, damping, friction, or bounded disturbances.
  • Problem formulation: The robust safe control problem seeks to reach a goal configuration while avoiding unsafe states under uncertain system parameters.Reachability is used instead of asymptotic stability to permit small steady-state error.
  • CLBF formulation: A control Lyapunov barrier function unifies stability and safety certificates for reach-avoid tasks.Safe and unsafe regions are represented through sub- and super-level sets of one certificate.
  • CLBF formulation: The admissible control set K(x) contains inputs satisfying the CLBF decrease condition L_fV + L_gV u + λV ≤ 0.Any policy selecting controls from this set is covered by the CLBF safety and stability theorem.
  • Controller construction: The CLBF-based controller selects a feasible control near a nominal controller, encouraging smoothness while preserving the certificate conditions.The nominal controller is particularly relevant near the desired fixed point, where V̇ becomes small.

4 Robust CLBF Certificates for Safe Control

The paper extends CLBF theory to bounded parametric uncertainty by enforcing the certificate condition across uncertainty scenarios. The resulting theorem guarantees safety and stability for policies satisfying the robust admissible-control constraints.

  • Robust CLBF definition: A robust CLBF extends the standard CLBF by requiring its conditions to hold for uncertain parameters in the convex hull of specified scenarios.The dynamics must be affine with respect to the uncertain parameters.
  • Robust CLBF definition: The robust CLBF decrease condition is enforced simultaneously for every uncertainty scenario θ_i.This produces one set of constraints covering the selected parameter scenarios.
  • Robust controller: The robust admissible-control set K_r(x) consists of inputs satisfying the scenario-wise robust CLBF inequalities.The corresponding controller is implemented through a quadratic program.
  • Guarantee: If V is a robust CLBF, any policy π(x) ∈ K_r(x) is safe and stable for systems whose uncertain parameters lie in the prescribed convex hull.This is the soundness guarantee for robust CLBF-based control.
  • Guarantee: The theorem establishes soundness and robustness but does not provide a construction method for a valid robust CLBF.The paper addresses construction separately through automated model-based learning.

5 Learning Robust CLBFs

The paper learns robust CLBFs from sampled state-space data and uses them to construct an online rCLBF-QP controller. Across simulated tracking, stabilization, and navigation tasks, the controller maintains safety while reducing computation and handling nonlinear dynamics.

  • Learning Robust CLBFs: The model-based training framework learns a neural rCLBF by sampling states and optimizing losses linked to safety, stability, and robust derivative conditions.The loss uses positive tuning parameters, a strictness parameter ϵ, and ReLU penalties for violated conditions.
  • Learning Robust CLBFs: The online rCLBF-QP controller solves a small quadratic program over admissible controls, while the learned neural policy primarily demonstrates feasibility.Execution need not use the neural policy directly; any control in the admissible set can be selected.
  • Learning Robust CLBFs: Finite sampled-state training does not enforce dense satisfaction of the robust CLBF condition, leaving sparse violation regions that may require shrinking the certified invariant set.The authors state that empirical performance does not substitute for rigorous verification.
  • Car trajectory tracking: ≈10 ms: the rCLBF-QP controller runs in real time on a laptop for car tracking, whereas robust MPC required slower-than-real-time execution and became unstable at dt = 0.25 s.For the sideslip model, rCLBF-QP uses 2 variables and 2 constraints versus robust MPC's 35 variables and 23 constraints.
  • UAV stabilization: Factor of 10: the controller reduces evaluation time relative to MPC in UAV stabilization while maintaining safety and achieving much lower goal error for the neural lander.The rCLBF-QP incorporates the full nonlinear dynamics, including the learned ground effect, whereas robust MPC cannot drive the neural lander to the goal.
  • Navigation with non-convex safety constraints: In non-convex navigation, rCLBF-QP successfully navigates the Segway and satellite environments where robust MPC fails to track the reference path.HJ preserves satellite safety but fails to reach the goal, while CPO reaches the satellite goal without safety.

7 Discussion & Conclusion

The proposed controllers match MPC on safety and goal error when MPC is stable, while reducing evaluation time by an order of magnitude. Across simulations, they satisfy safety constraints despite model-parameter variation.

  • Our controller’s safety rate and goal-relative error are comparable to MPC when the MPC controller is stable.The method sometimes achieves lower steady-state error under highly nonlinear dynamics, while robust MPC can perform better when linearization is accurate.
  • An order-of-magnitude reduction in evaluation time trades increased offline computation for faster controller execution than traditional MPC.Robust MPC was highly sensitive to control frequency and was stable only at frequencies that could not run in real time on a laptop.
  • In all cases, the learned controller satisfies safety constraints despite variation in model parameters.
  • The paper identifies scalable verification, sample complexity, convergence rates, and hardware deployment with delay and state-estimation uncertainty as open questions.

Supplementary Materials

The supplementary material establishes the stability and safety guarantees underlying CLBF-based control and describes training, specification, feasibility, and implementation choices for learned controllers.

  • Proof of Theorem 1: A CLBF-based policy exponentially stabilizes the system to xgoal.The result follows from the CLBF condition on the closed-loop derivative.
  • Proof of Theorem 1: Any trajectory initialized in Xsafe never enters Xunsafe under a CLBF-based policy.The proof uses the strict decrease of V from its initial safe-set bound.
  • Robustness: Satisfying the robust CLBF constraint at each sampled parameter scenario extends satisfaction to every parameter in their convex hull.This follows because the constraint mapping is affine in the uncertain model parameters.
  • Training: Training uses tanh neural networks, batched stochastic gradient descent, weight decay, and targeted sampling from goal, safe, and unsafe regions.A representative sampling scheme allocates 0.1Ntrain points each to the goal, unsafe, and safe regions, with 0.7Ntrain points sampled from the full state space.
  • Training: Some experiments initialize the CLBF network from a quadratic Lyapunov function obtained by linearization about the goal, then alternate training V and uNN.
  • Problem specification: Reach-avoid specifications require a region between safe and unsafe sets so the neural rCLBF can adjust its boundary.A constraint preventing departure from the training-data region also improves the controller.
  • QP relaxation: The rCLBF-QP relaxes derivative constraints to preserve feasibility, penalizing relaxation and using it as a training signal for V.

Verification of Learned CLBFs

The learned rCLBFs satisfy their conditions across most of the state space and produce successful simulated controllers, but they are not exhaustively verified.

  • The learned rCLBFs satisfy their conditions in the vast majority of the state space.
  • The paper does not claim exhaustive verification of the learned rCLBFs.Scalable verification for learned certificate functions remains an open problem.
  • Future verification methods include neural-network reachability analysis, SMT solvers, Lipschitz-informed sampling, and probabilistic learning-theory claims.These methods could also inform training and provide counterexamples for augmenting training data.

Implementation of Robust MPC

The robust MPC baseline uses a constant or trajectory-based linearization and solves a compiled robust quadratic program with YALMIP, Gurobi, and Matlab timing.

  • The robust MPC algorithm relies on linearized system dynamics, using goal-state linearization or reference-trajectory linearization for tracking.
  • YALMIP formulates the robust MPC problem and Gurobi serves as the underlying QP solver.
  • Evaluation timing measures only the solve time of a precompiled robust QP.The paper notes that further optimizations such as explicit MPC could reduce evaluation time and could also be applied to the QP solution method.

Implementation of Hamilton Jacobi Control Synthesis

The Hamilton-Jacobi baseline uses helperOC and toolboxLS to compute a value function, approximates parametric uncertainty additively, and applies the optimal HJ controller forward.

  • The HJ baseline computes a value function with helperOC, which wraps toolboxLS.
  • Parametric uncertainty is over-approximated as additive uncertainty before HJ synthesis.
  • The resulting value function is paired with the optimal HJ controller applied forwards.

Details on Simulation Experiments

The experiments evaluate uncertain car-tracking models using path-centric dynamics, neural rCLBF controllers, and simulation-based checks of decrease-condition violations.

  • Kinematic car: The kinematic model uses error states and controls for steering-angle rate and longitudinal acceleration.
  • Kinematic car: The kinematic controller uses a nominal LQR policy and two-layer tanh networks for V and πNN.
  • Kinematic car: 0.0225 was the maximum observed violation of the rCLBF decrease condition for the kinematic car over the plotted range.
  • Sideslip car: The sideslip model adds sideslip angle and heading-error rate, while reverting to the kinematic model when |v| < 0.1 because its dynamics become singular at low speeds.
  • Sideslip car: No decrease-condition violation was detected for the sideslip model on the xe, ye cross-section, although other cross-sections may contain violations.

3D Quadrotor

The 3D quadrotor and neural lander experiments test safe hovering under mass uncertainty, including a learned nonlinear ground-effect disturbance.

  • 3D quadrotor: The 3D quadrotor has nine states for position, velocity, and orientation, with mass treated through its affine dependence on 1/m.
  • 3D quadrotor: The quadrotor safe-hover task constrains altitude and state norm while testing masses uniformly sampled from m ∈[1.0, 1.5].
  • 3D quadrotor: 1.9×10−4 was the maximum rCLBF-condition violation observed in the 3D quadrotor contour evaluation.
  • Neural lander: The neural lander includes a learned ground-effect disturbance represented by a four-layer neural network, alongside mass uncertainty.
  • Neural lander: The neural lander safe-hover task samples mass uniformly from m ∈[1.47, 2.00].

2D Quadrotor

The obstacle-avoidance and rendezvous experiments apply learned rCLBFs to nonlinear systems with uncertain parameters and nonconvex safety constraints, while contrasting HJ, MPC, and CPO baselines.

  • 2D quadrotor: The 2D quadrotor models mass and inertia uncertainty through affine dependence on 1/m and 1/I.
  • 2D quadrotor: The obstacle-safe region is offset 0.1 m from obstacle boundaries and uncertainty spans (m, I) ∈[1.0, 1.05] × [0.01, 0.0105].
  • 2D quadrotor: The learned rCLBF level set aligns well with obstacle boundaries, while derivative-condition violation is at most 7.3 × 10−2 in a small region.
  • Satellite: The satellite task requires rendezvous within a line-of-sight sector, with uncertain chaser mass and target mean-motion parameters.
  • Satellite: The satellite rCLBF distinguishes safe and unsafe sets, with violation below 5.5 × 10−2 for most plotted samples.
  • Segway: CPO stabilized the Segway without obstacles but failed to stabilize it when an obstacle was included during training.

Pandemic Considerations

The paper reports simulation-only experiments because COVID-19 limited facility access, while taking steps to assess real-time feasibility and model uncertainty. It also identifies actuator-limit variation, state estimation, and delay as gaps between simulation and hardware deployment.

  • COVID-19 facility-access limitations prevented hardware experiments, so the evaluation focuses on simulation.
  • The study reports controller evaluation times against control frequency to assess potential real-time deployment.
  • Randomly varying model parameters during safety and error evaluations simulates uncertainty in physical-system models.
  • The paper does not thoroughly evaluate varying actuator limits, assumes high-quality full-state estimates, and does not study delay effects on safety or stability.
  • The authors hope future hardware demonstrations will test these assumptions and address the simulation-to-reality gaps.
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