DUST: A Framework for Data-Driven Density Steering

TL;DR

The paper addresses data-driven density steering for unknown LTI systems with Gaussian disturbances by proposing DUST, a framework that fuses behavioral system theory with covariance steering to separately address mean and covariance steering using offline data. It develops indirect (mean) and direct (covariance) data-driven formulations, and introduces noise-estimation methods (MLE and NN) with high-confidence uncertainty sets to enable robust control via SLS and SDP-based LMIs. The work also presents a parametric-uncertainty variant (PU-DD-DS) and provides extensive numerical experiments comparing CE, robust, and model-based baselines, showing improved constraint satisfaction and performance under data noise. These contributions offer a principled, scalable approach to density steering in the presence of data imperfections, with potential impact on aerospace, autonomous systems, and any domain requiring reliable distributional control under uncertainty.

Abstract

We consider the problem of data-driven stochastic optimal control of an unknown LTI dynamical system. Assuming the process noise is normally distributed, we pose the problem of steering the state's mean and covariance to a target normal distribution, under noisy data collected from the underlying system, a problem commonly referred to as covariance steering (CS). A novel framework for Data-driven Uncertainty quantification and density STeering (DUST) is presented that simultaneously characterizes the noise affecting the measured data and designs an optimal affine-feedback controller to steer the density of the state to a prescribed terminal value. We use both indirect and direct data-driven design approaches based on the notions of persistency of excitation and subspace identification to exactly represent the mean and covariance dynamics of the state in terms of the data and noise realizations. Since both the mean and the covariance steering sub-problems are plagued with stochastic uncertainty arising from noisy data collection, we first estimate the noise realization from this dataset and subsequently compute tractable upper bounds on the estimation errors. The first and second moment steering problems are then solved to optimality using techniques from robust control and robust optimization. Lastly, we present an alternative control design approach based on the certainty equivalence principle and interpret the problem as one of CS under multiplicative uncertainty. We analyze the performance and efficacy of each of these data-driven approaches using a case study and compare them with their model-based counterparts.

Figures (10)

• Figure 1: Breakdown of DUST framework into data-collection, noise estimation, uncertainty set construction, and robust control. The noisy dataset $\mathbb{D}$ is used to estimate the past noise realization $\hat{\Xi}_{0,T}$, which is subsequently used to generate norm-bounded uncertainty sets $\Delta_{\mathrm{model}}$ for the (indirect) DD-MS and $\Delta_{\mathrm{noise}}$ for the (direct) DD-CS problems. The end result is optimal moment trajectories that satisfy terminal distributional constraints with high probability.
• Figure 2: Empirical distribution of norm of joint estimation errors $\Delta\Xi_{0,T}$, for varying sampling horizons $T$.
• Figure 3: MSE distribution $\mathbb{E}[\|\Xi_{0,T} - \hat{\Xi}_{0,T}\|^{2}]$ of estimation errors between neural network estimation (black) and maximum likelihood estimation (blue).
• Figure 4: Comparison of (left) DD-MS and (right) R-DD-MS terminal splashpoints for 500 randomized trials of data collected from the true model over a horizon $T=30$, with empirical error distribution (bottom).
• Figure 5: Comparison of (R)DD-MS optimal trajectories for 500 randomized trials of data collected from random models with $\|[\Delta B \ \Delta A]\| \leq \alpha$ over a horizon $T = 30$.