Gaussian behaviors and stochastic data-driven control

Abstract: We propose a stochastic behavioral modeling framework, termed Gaussian behaviors, which augments a deterministic linear time-invariant (LTI) behavior with a Gaussian noise component. We show that this notion is a tractable subclass of stochastic behaviors and encompasses classical parametric stochastic LTI state-space system models as special cases. Analogously to deterministic LTI behaviors, the framework enables simple and tractable stochastic data-driven control methods. To this end, we obtain a method for prediction by conditioning the Gaussian behavior on the known part of the trajectory, which is identified directly from the sample covariance of trajectory data. Building on this method, we develop predictive control formulations that optimize over feedforward or disturbance affine feedback policies. The resulting formulations are shown to be convex. We further derive a finite-sample confidence bound on the prediction accounting for both aleatoric and epistemic uncertainty, and incorporate it into a robust control method, for which a tractable convex upper bound is obtained. Within this framework, subspace predictive control is recovered when only the mean prediction is used, while data-enabled predictive control is shown to account for the prediction uncertainty in an optimistic fashion. Numerical case studies illustrate the benefits of the proposed methods.

• The paper develops Gaussian behaviors that combine deterministic linear time-invariant dynamics with Gaussian exogenous noise, enabling predictive distributions to be identified directly from sample covariances.
• The framework converts chance-constrained control with causal disturbance-feedback policies into convex quadratic and second-order cone programs, while robust formulations account for finite-data uncertainty and poorly excited directions.
• Case studies show that robust control nearly halves realized cost versus baseline methods, while disturbance-feedback control reduces HVAC control cost by 77% compared with feedforward chance-constrained control.

Overview and motivation

The paper "Gaussian behaviors and stochastic data-driven control" (2607.15949) by Sasfi, Padoan, Markovsky, and Dörfler develops a stochastic extension of behavioral systems theory tailored to data-driven predictive control. The authors observe that existing data-driven predictive control methods—DeePC, SPC, and their variants—are built on a fundamentally deterministic behavioral model, so that uncertainty is handled only a posteriori through regularization or robustification. Prior stochastic extensions of behavioral theory (Willems' static formulation and its dynamical generalization) are mathematically general but have not yielded actionable control algorithms, while polynomial-chaos-based approaches scale poorly with expansion order. Parametric stochastic MPC methods, in turn, are risk-neutral, optimize over feedforward inputs only, or require known noise covariances. The paper addresses this gap with a modeling framework that is deliberately parsimonious: it augments a deterministic LTI behavior with a Gaussian noise component, yielding tractable prediction, chance-constrained control over disturbance affine feedback policies, and a robust min-max formulation with convex surrogates.

Gaussian behaviors

A finite-horizon LTI Gaussian behavior is defined as a pair $(\mathcal{B}_L, \Sigma_e)$, where $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$ is a deterministic restricted behavior (a shift-invariant subspace of dimension $d = mL + n$) and $\Sigma_e \succeq 0$ is a covariance describing exogenous stochasticity. Every random length-$L$ trajectory decomposes as $w = \bar{w} + e$ with $\bar{w} \in \mathcal{B}_L$ and $e \sim \mathcal{N}(0, \Sigma_e)$. Two types of randomness are distinguished: endogenous stochasticity, arising from random initial states or inputs but supported on $\mathcal{B}_L$, whose law is left unspecified; and exogenous stochasticity, which pushes trajectories off the nominal subspace.

The paper establishes two structural results. First, any observable stochastic state-space model induces a Gaussian behavior, with $\Sigma_e$ given explicitly in terms of the process and measurement noise covariances propagated through the Toeplitz noise map; conversely, no structure beyond positive semi-definiteness is imposed on $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$0, making the class more expressive than single-step models. Second, via a kernel representation $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$1 of $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$2, the decomposition of Definition 1 is shown equivalent to the condition $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$3 with $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$4, i.e., Gaussian behaviors form a tractable subclass of the abstract stochastic dynamical systems of Baggio et al., with the probability measure implicitly placed on the quotient space $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$5. Deterministic behaviors are recovered at $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$6. A consequence worth noting is non-uniqueness: because $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$7 and the support of $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$8 may intersect, the pair $\mathcal{B}_L \subseteq \mathbb{R}^{qL}$9 is not identifiable from data—an issue the authors resolve by shifting the identification target to the predictive distribution, which is unique.

Identification and prediction

Under an experiment-design assumption—that the data consist of $d = mL + n$0 trajectories whose nominal components are zero-mean Gaussians independent of the exogenous noise—the joint trajectory distribution is Gaussian with covariance $d = mL + n$1. Partitioning each trajectory into free variables $d = mL + n$2 (initial conditions plus future input) and dependent variables $d = mL + n$3 (future output), conditioning yields a Gaussian predictive distribution $d = mL + n$4, identified from the sample covariance as

$d = mL + n$5

These are maximum likelihood estimates under independence, and $d = mL + n$6 is the minimum-variance linear predictor. With the standard partitioning, $d = mL + n$7 coincides exactly with the classical subspace predictor used in SPC, so SPC's mean prediction is reinterpreted as the conditional mean of a Gaussian behavior. The authors are explicit that this sidesteps the errors-in-variables problem: rather than recovering the "true" system, they identify the optimal predictor directly.

Two extensions follow. First, a finite-sample confidence ellipsoid (Lemma 4) guarantees that, with probability $d = mL + n$8, the true output satisfies $d = mL + n$9, where $\Sigma_e \succeq 0$0 involves an inverse F-distribution quantile. This bound captures both aleatoric uncertainty (through $\Sigma_e \succeq 0$1) and epistemic uncertainty (through the data-dependent term), and shrinks directionally toward regions of the free-variable space that were well excited during data collection. Second, uncertain forecasts of exogenous variables (e.g., weather) are incorporated by augmenting the free variables with forecast values and lumping the forecast error into the exogenous noise. Because the multi-step model treats the horizon jointly rather than recursively, arbitrarily correlated, non-stationary forecast errors—such as uncertainty growing over the horizon—are captured without structural assumptions on their temporal correlation, a capability single-step recursive models lack.

Control under aleatoric uncertainty

Treating $\Sigma_e \succeq 0$2 and $\Sigma_e \succeq 0$3 as exact, the future output obeys $\Sigma_e \succeq 0$4 with $\Sigma_e \succeq 0$5. Minimizing expected quadratic cost subject to individual chance constraints reduces, via the standard Gaussian quantile reformulation, to a deterministic QP; setting $\Sigma_e \succeq 0$6 recovers SPC exactly, since the constraint then acts on the mean and the trace term is constant. CVaR constraints fit the same structure with a different weighting on the standard deviation.

The more substantive contribution is optimization over disturbance affine feedback policies $\Sigma_e \succeq 0$7 with $\Sigma_e \succeq 0$8 restricted to strictly block lower triangular matrices for causality. Theorem 1 shows the resulting chance-constrained problem is equivalent to a convex program: the cost becomes an LQG-type expression penalizing $\Sigma_e \succeq 0$9 and $L$0, while the constraints become second-order cone constraints. Optimizing $L$1 actively shapes the predicted output covariance $L$2, counteracting the open-loop accumulation of uncertainty over the horizon—at the price of making the applied input stochastic. This brings the disturbance feedback machinery of classical stochastic MPC into a fully data-driven setting without requiring known noise statistics.

Robust and optimistic control under epistemic uncertainty

When epistemic uncertainty from finite data is included, the confidence ellipsoid radius depends on the input magnitude, so optimizing over feedback gains would destroy convexity; the paper therefore restricts to feedforward inputs here. The robust formulation minimizes the worst-case cost over the ellipsoid $L$3. Theorem 2 derives, via Lagrangian duality, a tractable convex upper bound parameterized by a multiplier $L$4: the resulting cost adds to the nominal tracking term a variance penalty weighted by $L$5 and a term $L$6 times the squared norm of the decision vector under $L$7, which penalizes large inputs in poorly excited directions. Treating $L$8 as a tuning knob trades tracking performance against conservatism; joint optimization over $L$9 would be exact but non-convex. The worst-case objective is also shown to upper-bound the VaR of the control cost at level $w = \bar{w} + e$0 with respect to both aleatoric and epistemic uncertainty.

The complementary optimistic stance—minimizing over the best-case realization within the ellipsoid—recovers regularized DeePC with projection-based regularization (Theorem 3), with the Lagrange multiplier matching DeePC's regularization weight up to a factor $w = \bar{w} + e$1. This gives regularization in DeePC a precise stochastic interpretation: DeePC optimizes against the best-case prediction and accounts only for aleatoric noise, ignoring epistemic uncertainty. The authors argue this explains the empirical observation that closed-loop DeePC performance improves monotonically with the regularization weight, and note that optimism can severely degrade performance when the ellipsoid is large—a claim borne out in the first case study.

Case studies

Two studies support the theory. On a fourth-order double spring-mass-damper system with unevenly excited offline data (input variances $w = \bar{w} + e$2 versus $w = \bar{w} + e$3), GB-Robust ($w = \bar{w} + e$4, $w = \bar{w} + e$5) achieves a realized cost of $w = \bar{w} + e$6 over 1000 noise realizations, compared with $w = \bar{w} + e$7 for SPC, $w = \bar{w} + e$8 for MSM-SMPC, and $w = \bar{w} + e$9–$\bar{w} \in \mathcal{B}_L$0 for DeePC across regularization weights. Notably, GB-Robust routes actuation through the well-excited second channel, whereas SPC, DeePC, and MSM-SMPC exploit the poorly excited first channel and suffer large deviations between predicted and realized outputs. DeePC's cost decreases monotonically in $\bar{w} \in \mathcal{B}_L$1, converging toward SPC, consistent with the optimistic interpretation.

In an HVAC building control study with uncertain outside-temperature forecasts (cumulative-sum error model mimicking growing forecast uncertainty), the disturbance affine feedback policy reduces predicted output variance enough to plan trajectories closer to the comfort constraints; in closed-loop simulation over two days, it reduces the control cost by 77% relative to the feedforward chance-constrained formulation. The authors candidly note that because these formulations account only for aleatoric uncertainty, the true output realizations slightly exceed the nominal confidence bounds.

Limitations and open questions

Several assumptions constrain applicability. The confidence bound requires mutually independent data trajectories, i.e., separate experiments per column of the data matrix; Hankel matrices built from a single continuous stream violate this, and relaxing the independence assumption is left open. The epistemic analysis assumes invertibility of $\bar{w} \in \mathcal{B}_L$2 and Gaussian endogenous excitation. In the robust formulation, $\bar{w} \in \mathcal{B}_L$3 is fixed a priori rather than optimized, so the min-max problem is solved only through a conservative dual upper bound; whether an efficient exact reformulation exists remains unanswered. Feedback policies are excluded once epistemic uncertainty is considered, due to the induced non-convexity. Finally, the framework is offline: online updates of the Gaussian behavior for adaptive data-driven control are identified as future work.

Conclusion

The paper contributes a pragmatic stochastic behavioral framework in which prediction, chance-constrained control with disturbance feedback, and robust min-max control all reduce to convex programs identified directly from sample covariances. It unifies SPC as mean-prediction and regularized DeePC as optimistic uncertainty handling within a single probabilistic formalism, and demonstrates substantial empirical gains—roughly halving realized cost relative to SPC and DeePC in the presence of unevenly excited data, and a 77% cost reduction from feedback policies in building control.