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Safety-Aware Performance Boosting for Constrained Nonlinear Systems

Danilo Saccani, Haoming Shen, Luca Furieri, Giancarlo Ferrari-Trecate

Abstract

We study a control architecture for nonlinear constrained systems that integrates a performance-boosting (PB) controller with a scheduled Predictive Safety Filter (PSF). The PSF acts as a pre-stabilizing base controller that enforces state and input constraints. The PB controller, parameterized as a causal operator, influences the PSF in two ways: it proposes a performance input to be filtered, and it provides a scheduling signal to adjust the filter's Lyapunov-decrease rate. We prove two main results: (i) Stability by design: any controller adhering to this parametrization maintains closed-loop stability of the pre-stabilized system and inherits PSF safety. (ii) Trajectory-set expansion: the architecture strictly expands the set of safe, stable trajectories achievable by controllers combined with conventional PSFs, which rely on a pre-defined Lyapunov decrease rate to ensure stability. This scheduling allows the PB controller to safely execute complex behaviors, such as transient detours, that are provably unattainable by standard PSF formulations. We demonstrate this expanded capability on a constrained inverted pendulum task with a moving obstacle.

Safety-Aware Performance Boosting for Constrained Nonlinear Systems

Abstract

We study a control architecture for nonlinear constrained systems that integrates a performance-boosting (PB) controller with a scheduled Predictive Safety Filter (PSF). The PSF acts as a pre-stabilizing base controller that enforces state and input constraints. The PB controller, parameterized as a causal operator, influences the PSF in two ways: it proposes a performance input to be filtered, and it provides a scheduling signal to adjust the filter's Lyapunov-decrease rate. We prove two main results: (i) Stability by design: any controller adhering to this parametrization maintains closed-loop stability of the pre-stabilized system and inherits PSF safety. (ii) Trajectory-set expansion: the architecture strictly expands the set of safe, stable trajectories achievable by controllers combined with conventional PSFs, which rely on a pre-defined Lyapunov decrease rate to ensure stability. This scheduling allows the PB controller to safely execute complex behaviors, such as transient detours, that are provably unattainable by standard PSF formulations. We demonstrate this expanded capability on a constrained inverted pendulum task with a moving obstacle.
Paper Structure (12 sections, 3 theorems, 28 equations, 3 figures)

This paper contains 12 sections, 3 theorems, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Safe Performance Boosting
  4. PSF with scheduled Lyapunov-decrease
  5. Behavioral characterization and strict dominance in $\ell_2$
  6. Parameterization of the PB controller
  7. Training procedure
  8. Numerical Example
  9. Conclusions
  10. Proof of Theorem \ref{['prop:ell2_stability']}
  11. Proof of Lemma \ref{['lem:monotone_Ut']}
  12. Proof of Theorem \ref{['thm:inclusion_strict']}

Key Result

Theorem 1

Suppose Assumptions ap:cost function--ap: terminal set hold and let the PSF eq:PSF use the scheduled stability constraint eq: new-PSF-stability where $\rho_t = \psi(\|u_{\mathrm{L},t}\|)$ is generated by a tightening schedule $\psi$ satisfying Definition def:psi. If this problem is feasible at time

Figures (3)

  • Figure 1: The proposed framework. The PB operator $\mathbfcal M_\theta(\cdot)$ generates $\mathbf u_L$, which feeds both the scheduler $\boldsymbol{\psi}(\cdot)$ and the PSF. Using the resulting rate $\boldsymbol{\rho}$ and the state $\mathbf x$, the PSF filters $\mathbf u_L$ into the safe applied input $\mathbf u$.
  • Figure 2: Angle trajectories with obstacle avoidance. Lines with varying opacity correspond to multiple initial conditions. Blue lines: the proposed approach. Orange lines: the PSF with $\rho_t\equiv\bar{\rho}$ and $\mathbf{u_\mathrm{L}}\equiv \mathbf{0}$. The translucent red region shows the moving obstacle in the $(t,\theta)$ plane; the dotted black line marks the lower and upper bounds, while the dashed black line marks the unstable equilibrium ($\theta=0$).
  • Figure 3: Time evolution of the PSF certificate $J_t^*$ for different initial conditions. Blue lines: the proposed approach. Orange lines: the PSF with $\rho_t\equiv\bar{\rho}$ and $\mathbf{u_\mathrm{L}}\equiv \mathbf{0}$.

Theorems & Definitions (6)

  • Definition 1
  • Definition 2: Tightening schedule
  • Theorem 1
  • Definition 3
  • Lemma 1
  • Theorem 2