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arxiv: 2407.11256 · v2 · submitted 2024-07-15 · 📡 eess.SY · cs.SY

Controlled Invariant Sets for Gaussian Process State Space Models

Paul Griffioen , Bingzhuo Zhong , Murat Arcak , Majid Zamani , Marco Caccamo This is my paper

Pith reviewed 2026-05-23 22:33 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords gaussian processcontrolled invariant setssemidefinite programmingstate feedbacknonlinear systemsprobabilistic safetyquadrotor
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The pith

A semidefinite programming scheme designs state-feedback controllers that maximize the probability trajectories remain inside probabilistic controlled invariant sets for Gaussian process state-space models while respecting input constraints

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes probabilistic controlled invariant sets for nonlinear systems whose unknown dynamics are captured by data-driven Gaussian process state-space models. It formulates a semidefinite program that synthesizes state-feedback controllers to maximize the probability that trajectories stay inside these sets. The same program enforces input constraints. The approach is tested on a quadrotor both in simulation and on physical hardware.

Core claim

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 propose a semidefinite programming scheme for designing state-feedback controllers that maximize the probability of the trajectories staying within a probabilistic controlled invariant set while satisfying input constraints. The results are validated on a quadrotor, both in simulation and on a physical platform.

What carries the argument

semidefinite programming scheme that maximizes the probability of remaining inside a probabilistic controlled invariant set defined from a Gaussian process state-space model

Load-bearing premise

The Gaussian process state-space model learned from data sufficiently captures the unmodeled nonlinear dynamics so that the computed probabilistic invariant sets remain valid for the true system.

What would settle it

Run the designed controller on the physical quadrotor and measure the empirical fraction of time trajectories exit the computed set; a large mismatch with the designed probability would indicate the GP model does not support the invariance claim.

Figures

Figures reproduced from arXiv: 2407.11256 by Bingzhuo Zhong, Majid Zamani, Marco Caccamo, Murat Arcak, Paul Griffioen.

Figure 2
Figure 2. Figure 2: to validate the probabilistic guarantee proposed in this paper via Monte Carlo analysis. We consider state constraints xx, xy ∈ [−5, 5] m and vx, vy ∈ [−7, 7] m/s, and input constraints ux, uy ∈ [−5, 5] m/s2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: A quadrotor moving on an x-y plane. model the quadrotor dynamics as a GPSSM accord￾ing to (1) with states and inputs described according to [13]. The system states and control inputs are given by x ≜ h xx vx xy vy iT and u ≜ h ux uy iT , where xi , vi , and ui are the position, velocity, and acceleration of the quadrotor on the i th axis, respectively, with i ∈ {x, y}. 5.1 High Fidelity SIMULINK Model Firs… view at source ↗
Figure 3
Figure 3. Figure 3: Simulation results for the high fidelity [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: Physical quadrotor for the experiment. Right: [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Top: Evolution of the quadrotor’s positions in the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sequences of the quadrotor’s velocity and control [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original 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 propose a semidefinite programming scheme for designing state-feedback controllers that maximize the probability of the trajectories staying within a probabilistic controlled invariant set while satisfying input constraints. The results are validated on a quadrotor, both in simulation and on a physical platform.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes computing probabilistic controlled invariant sets for nonlinear systems via Gaussian process (GP) state-space models that capture unmodeled dynamics from data. It introduces a semidefinite programming (SDP) formulation to synthesize state-feedback controllers maximizing the probability that closed-loop trajectories remain inside the probabilistic invariant set while respecting input constraints. Validation is reported on a quadrotor both in simulation and on hardware.

Significance. If the probabilistic invariance guarantees transfer from the GP model to the true nonlinear plant, the SDP-based design would offer a principled data-driven route to safe control under uncertainty, with direct relevance to robotics applications. The combination of GP uncertainty quantification with SDP optimization and the inclusion of physical-platform experiments are strengths that could support broader adoption if the model-mismatch issue is resolved.

major comments (2)
  1. [Abstract (and implied theoretical sections)] The SDP designs invariance and probability bounds explicitly for the learned GP state-space model, yet the central claim asserts validity for the underlying nonlinear system. No section supplies a rigorous bound on the GP approximation error, a robustness margin, or a mismatch analysis that would guarantee the computed probabilities remain valid on the true dynamics; this gap is load-bearing for the transfer of the invariance result.
  2. [Abstract (validation paragraph)] The physical quadrotor validation is cited, but the manuscript does not report quantitative comparison between observed trajectory frequencies and the GP-predicted probabilities, nor does it include an error analysis or statistical test confirming that the empirical invariance rates align with the computed values. This leaves the hardware claim unsupported for the probabilistic guarantee.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below, with proposed revisions to improve clarity and validation.

read point-by-point responses

  1. Referee: [Abstract (and implied theoretical sections)] The SDP designs invariance and probability bounds explicitly for the learned GP state-space model, yet the central claim asserts validity for the underlying nonlinear system. No section supplies a rigorous bound on the GP approximation error, a robustness margin, or a mismatch analysis that would guarantee the computed probabilities remain valid on the true dynamics; this gap is load-bearing for the transfer of the invariance result.

    Authors: The probabilistic invariance guarantees and SDP formulation are derived explicitly for the GP state-space model. The manuscript does not provide a rigorous bound on GP approximation error or a formal transfer result to the true nonlinear dynamics. We will revise the abstract and introduction to state that the theoretical results hold for the learned GP model, with the physical experiments serving as empirical validation rather than a formal guarantee. revision: yes

  2. Referee: [Abstract (validation paragraph)] The physical quadrotor validation is cited, but the manuscript does not report quantitative comparison between observed trajectory frequencies and the GP-predicted probabilities, nor does it include an error analysis or statistical test confirming that the empirical invariance rates align with the computed values. This leaves the hardware claim unsupported for the probabilistic guarantee.

    Authors: We agree that a direct quantitative comparison would strengthen the hardware results. In the revised manuscript we will add an analysis comparing the empirical frequency of invariant trajectories from the hardware experiments to the probabilities obtained from the SDP, including basic error measures. revision: yes

Circularity Check

0 steps flagged

No circularity: SDP scheme computes invariants directly on the GP model without self-referential reduction

full rationale

The derivation consists of formulating an SDP that optimizes a state-feedback law to maximize an invariance probability under the posterior of a learned GP state-space model, subject to input constraints. The probabilistic sets and controller are defined and solved explicitly from the GP mean and covariance; the result is not obtained by fitting a parameter to a subset and relabeling it as a prediction, nor does any load-bearing step invoke a self-citation chain or uniqueness theorem that collapses to the authors' prior ansatz. External quadrotor validation supplies an independent check. The central claim therefore remains a constructive computational procedure rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5598 in / 844 out tokens · 17114 ms · 2026-05-23T22:33:38.405626+00:00 · methodology

discussion (0)

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