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Data-Driven Controlled Invariant Sets for Gaussian Process State Space Models

Paul Griffioen, Bingzhuo Zhong, Murat Arcak, Majid Zamani, Marco Caccamo

TL;DR

This work addresses safety guarantees for nonlinear systems modeled by Gaussian Process State Space Models by deriving probabilistic controlled invariant (PCI) sets and an SDP-based controller design that maximizes the probability of staying inside a safety set under input constraints. It develops an MLI/SDP framework that verifies and enlarges PCI ellipsoids, anchored by the ellipsoid $\mathcal{E}(0,P^{-1})$ and probability level $p$, while linking robust and probabilistic invariance through confidence regions and LMIs. The contributions include formalizing the relationship between RIS and PIS for GPSSMs, providing verification and design procedures for PCI sets via LMIs/SDPs, and validating the approach on a quadrotor in both high-fidelity simulations and a physical testbed with probabilistic safety guarantees (e.g., $p$ near $99.98\%$ in simulation and $97.36\%$ in real experiments). The results demonstrate practical data-driven safety guarantees for nonlinear, uncertain dynamics and highlight potential to scale to more complex systems and larger PCI sets using reachability analysis and performance-aware objectives.

Abstract

We compute probabilistic controlled invariant sets for nonlinear systems using Gaussian process state space models, which are data-driven models that account for unmodeled and unknown nonlinear dynamics. We investigate the relationship between robust and probabilistic invariance, leveraging this relationship to design state-feedback controllers that maximize the probability of the system staying within the probabilistic controlled invariant set. We propose a semi-definite-programming-based optimization scheme for designing the state-feedback controllers subject to input constraints. The effectiveness of our results are demonstrated and validated on a quadrotor, both in simulation and on a physical platform.

Data-Driven Controlled Invariant Sets for Gaussian Process State Space Models

TL;DR

This work addresses safety guarantees for nonlinear systems modeled by Gaussian Process State Space Models by deriving probabilistic controlled invariant (PCI) sets and an SDP-based controller design that maximizes the probability of staying inside a safety set under input constraints. It develops an MLI/SDP framework that verifies and enlarges PCI ellipsoids, anchored by the ellipsoid and probability level , while linking robust and probabilistic invariance through confidence regions and LMIs. The contributions include formalizing the relationship between RIS and PIS for GPSSMs, providing verification and design procedures for PCI sets via LMIs/SDPs, and validating the approach on a quadrotor in both high-fidelity simulations and a physical testbed with probabilistic safety guarantees (e.g., near in simulation and in real experiments). The results demonstrate practical data-driven safety guarantees for nonlinear, uncertain dynamics and highlight potential to scale to more complex systems and larger PCI sets using reachability analysis and performance-aware objectives.

Abstract

We compute probabilistic controlled invariant sets for nonlinear systems using Gaussian process state space models, which are data-driven models that account for unmodeled and unknown nonlinear dynamics. We investigate the relationship between robust and probabilistic invariance, leveraging this relationship to design state-feedback controllers that maximize the probability of the system staying within the probabilistic controlled invariant set. We propose a semi-definite-programming-based optimization scheme for designing the state-feedback controllers subject to input constraints. The effectiveness of our results are demonstrated and validated on a quadrotor, both in simulation and on a physical platform.
Paper Structure (15 sections, 11 theorems, 61 equations, 11 figures, 1 table, 1 algorithm)

This paper contains 15 sections, 11 theorems, 61 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Gaussian Process State Space Models
  3. System Model
  4. Problem Formulation
  5. Robust and Probabilistic Invariance
  6. Robust Positive Invariance
  7. Confidence Regions
  8. Probabilistic Positive Invariance
  9. Probabilistic Controlled Invariant Sets
  10. Controller Verification
  11. Controller Design
  12. Case Studies
  13. High Fidelity SIMULINK Model
  14. Experimental Physical Testbed
  15. Conclusion

Key Result

Corollary 1

Let $\bar{\Sigma}_1=\bar{\Sigma}_1^T\succeq0$ and $\bar{\Sigma}_2=\bar{\Sigma}_2^T\succeq0$. Then

Figures (11)

  • Figure 1: A quadrotor moving on an $\mathsf{x}$-$\mathsf{y}$ plane.
  • Figure 2: High fidelity SIMULINK for the quadrotor.
  • Figure 3: Simulation results for the high fidelity SIMULINK model of the quadrotor case study. Figure (a) denotes the projection of the PCI set and some of the state trajectories of the quadrotor onto the $x_1$-$x_3$ plane. Figure (b) demonstrates a few sequences of the velocity of the quadrotor. Figure (c) illustrates sequences of the control inputs of the quadrotor.
  • Figure 4: Left: Physical quadrotor for the experiment. Right: Test field equipped with a motion capture system and a ground control station.
  • Figure 5: Top: Evolution of the quadrotor's positions in the real-world experiment. Bottom: Values of $x_k^TPx_k$ along the trajectory of the quadrotor.
  • ...and 6 more figures

Theorems & Definitions (17)

  • Definition 1
  • Corollary 1: seeger1990direct
  • Definition 2: alessandri2004estimation
  • Theorem 1
  • Definition 3: hewing2018correspondence
  • Corollary 2
  • Definition 4: hewing2018correspondence
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • ...and 7 more