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Predictive stability filters for nonlinear dynamical systems affected by disturbances

Alexandre Didier, Andrea Zanelli, Kim P. Wabersich, Melanie N. Zeilinger

TL;DR

The paper addresses ensuring safety and stability for nonlinear dynamical systems under disturbances when inputs may come from humans or learning-based controllers. It extends predictive safety filters by imposing a Lyapunov decrease constraint using an implicit Lyapunov function $V((x,\mathbf{u}))$ derived from predicted trajectories, and augments the state with a warmstart input sequence to enable robust stability guarantees. The main theoretical result proves robust asymptotic stability of the augmented difference inclusion $s^+ \in H_{\rho}(s,w)$ for all disturbances in $\mathcal{W}$, leveraging an ISS Lyapunov function framework. It also provides a design method for linear systems with polytopic constraints and demonstrates the approach on a lane-keeping automotive example, highlighting practical viability and tunable stability via the parameter $\rho$.

Abstract

Predictive safety filters provide a way of projecting potentially unsafe inputs, proposed, e.g. by a human or learning-based controller, onto the set of inputs that guarantee recursive state and input constraint satisfaction by leveraging model predictive control techniques. In this paper, we extend this framework such that in addition, robust asymptotic stability of the closed-loop system can be guaranteed by enforcing a decrease of an implicit Lyapunov function which is constructed using a predicted system trajectory. Differently from previous results, we show robust asymptotic stability with respect to a predefined disturbance set on an extended state consisting of the system state and a warmstart input sequence. The proposed strategy is applied to an automotive lane keeping example in simulation.

Predictive stability filters for nonlinear dynamical systems affected by disturbances

TL;DR

The paper addresses ensuring safety and stability for nonlinear dynamical systems under disturbances when inputs may come from humans or learning-based controllers. It extends predictive safety filters by imposing a Lyapunov decrease constraint using an implicit Lyapunov function derived from predicted trajectories, and augments the state with a warmstart input sequence to enable robust stability guarantees. The main theoretical result proves robust asymptotic stability of the augmented difference inclusion for all disturbances in , leveraging an ISS Lyapunov function framework. It also provides a design method for linear systems with polytopic constraints and demonstrates the approach on a lane-keeping automotive example, highlighting practical viability and tunable stability via the parameter .

Abstract

Predictive safety filters provide a way of projecting potentially unsafe inputs, proposed, e.g. by a human or learning-based controller, onto the set of inputs that guarantee recursive state and input constraint satisfaction by leveraging model predictive control techniques. In this paper, we extend this framework such that in addition, robust asymptotic stability of the closed-loop system can be guaranteed by enforcing a decrease of an implicit Lyapunov function which is constructed using a predicted system trajectory. Differently from previous results, we show robust asymptotic stability with respect to a predefined disturbance set on an extended state consisting of the system state and a warmstart input sequence. The proposed strategy is applied to an automotive lane keeping example in simulation.
Paper Structure (10 sections, 32 equations, 1 figure, 1 algorithm)

This paper contains 10 sections, 32 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work and contribution
  3. Notation
  4. Preliminaries
  5. Predictive Stability Filter
  6. Main theoretical result
  7. Example and linear filter design
  8. Design for linear systems and polytopic constraints
  9. Stability filter for motion control
  10. Conclusion

Figures (1)

  • Figure 1: Numerical example of vehicle motion control according to Section \ref{['subsec:motion_control']} for different values of $\rho$ in \ref{['eq:Lyapunov_decrease']} are shown. Linear (top) and nonlinear (bottom) vehicle simulations are performed with the task of recovering from a destabilizing initial state. The lateral offset with respect to a straight road is shown in the left column, where stability filters are applied to mitigate constraint violations. The middle column shows the impact of reducing the stability filter interventions by relaxing the stability filter constraint for larger values of $\rho$. The right column displays the corresponding impact on the Lyapunov function.