Source-linked AI summary
Distributionally Robust Chance Constrained Data-enabled Predictive Control
Jeremy Coulson, John Lygeros, Florian Dörfler
TL;DR
The paper addresses finite-horizon constrained control of unknown stochastic LTI systems using noisy input/output data without explicitly identifying a model. It combines a non-parametric DeePC representation with distributionally robust optimization and reports high-confidence out-of-sample performance and constraint guarantees.
Problem
The challenge is to ensure performance and safety when data-driven control operates with unknown stochastic dynamics, corrupted data, and noise.
Method
The method combines raw-data DeePC and Page/Hankel behavioural representations with distributionally robust optimization and output chance constraints.
Results
The tractable robust DeePC formulation provides an upper bound on expected cost and satisfies the CVaR constraint with probability 1 − β under the theorem’s assumptions.
Takeaways & Limitations
The method provides end-to-end control design for unknown stochastic constrained LTI systems using raw data, without explicit model identification.
Abstract
from arXiv · showhide
We study the problem of finite-time constrained optimal control of unknown stochastic linear time-invariant systems, which is the key ingredient of a predictive control algorithm -- albeit typically having access to a model. We propose a novel distributionally robust data-enabled predictive control (DeePC) algorithm which uses noise-corrupted input/output data to predict future trajectories and compute optimal control inputs while satisfying output chance constraints. The algorithm is based on (i) a non-parametric representation of the subspace spanning the system behaviour, where past trajectories are sorted in Page or Hankel matrices; and (ii) a distributionally robust optimization formulation which gives rise to strong probabilistic performance guarantees. We show that for certain objective functions, DeePC exhibits strong out-of-sample performance, and at the same time respects constraints with high probability. The algorithm provides an end-to-end approach to control design for unknown stochastic linear time-invariant systems. We illustrate the closed-loop performance of the DeePC in an aerial robotics case study.
I. INTRODUCTION
The paper develops an end-to-end data-driven approach for controlling unknown stochastic constrained systems without explicitly identifying a predictive model. It combines DeePC with distributionally robust optimization to provide performance and safety guarantees.
- Motivation: Data-driven control designs controllers directly from data rather than first identifying a predictive model.This is useful when first-principles models are unavailable, overly complex, or costly to build.
- Motivation: A central challenge is guaranteeing performance and safety under uncertainty, corrupted data, and noise.
- Approach: The proposed method combines DeePC with distributionally robust optimization for unknown stochastic constrained linear systems.It uses raw input/output data and remains agnostic to the particular probabilistic uncertainty.
- Guarantees: The distributionally robust formulation supports high-confidence out-of-sample performance guarantees and chance-constraint satisfaction.The method can be robust against compatible systems involving non-Gaussian, non-additive, and weakly nonlinear noise.
- Novelty: The paper introduces a Page-matrix formulation of the behavioural fundamental lemma, contrasting with the Hankel matrices used in prior behavioural control work.The authors report tighter guarantees and better performance from this alternative data structure.
- Organization: The paper reviews behavioural systems, develops deterministic and stochastic DeePC algorithms, and evaluates them in a quadcopter simulation.
C. Hankel and Page Matrices
The section defines Hankel and Page data structures and their associated excitation conditions. Page matrices avoid repeated entries within columns, which supports favourable statistical and algorithmic properties under noisy measurements.
- Matrix structures: Hankel and Page matrices organize sequential input data into two alternative matrix structures.Hankel matrices use overlapping shifted segments, whereas Page matrices arrange non-overlapping blocks by depth.
- Excitation: Hankel exciting inputs require the corresponding Hankel matrix to have full row rank.
- Excitation: L-Page exciting inputs require the corresponding Page matrix to have full row rank and depend on depth L and order M.
- Excitation: A Hankel-exciting sequence requires T ≥ L(m + 1) − 1, while an L-Page-exciting sequence requires T ≥ L((mL + 1)M − 1).
- Noise properties: Page matrices have independent entries within their data matrices because they contain no repeated entries in each matrix.This differs from the repeated-entry structure underlying Hankel matrices.
- Noise properties: The independence of Page-matrix entries enables statistically and algorithmically favourable operations such as singular-value-thresholding denoising.The paper later connects Page matrices with tighter robustness and optimality guarantees and superior control performance.
D. Non-parametric System Representation
The paper represents finite-length trajectories of an unknown controllable LTI system by columns of a sufficiently rich Page data matrix. This non-parametric representation simultaneously supports state estimation, trajectory prediction, and optimal control.
- Fundamental lemma: The Page-matrix fundamental lemma states that every L-length trajectory is representable by a linear combination of Page-matrix columns under suitable controllability, horizon, and excitation conditions.The input must be L-Page exciting of order n(B) + 1 with L ≥ n(B).
- Non-parametric representation: The Page matrix replaces a model or system-identification process by providing a raw-data trajectory library whose linear combinations recover the system’s trajectory space.Each column can be interpreted as a motion primitive.
- Non-parametric representation: For L ≥ ℓ(B), the data-matrix column span coincides with the restricted behaviour and has rank mL + n(B).An upper bound on the unknown system order can replace n(B) in the excitation condition.
- Prediction: DeePC uses measured past input/output trajectories to implicitly estimate the initial state and predict future outputs from data matrices.The predicted output is formed as bY_fg, and it is unique when Tini ≥ ℓ(B).
- DeePC: The deterministic DeePC procedure collects sufficiently exciting offline data, forms past/future data matrices, and solves an online input/output optimization problem.Its online optimization requires input/output measurements and does not require an identified model.
- DeePC and MPC: For deterministic LTI systems, DeePC and model-based MPC produce equivalent and unique open-loop or receding-horizon behaviour under the stated conditions.This equivalence follows from the correspondence between their feasible input/output trajectories.
IV. DISTRIBUTIONALLY ROBUST DEEPC
The distributionally robust DeePC method handles stochastic disturbances and unknown distributions by replacing deterministic objectives and constraints with a tractable robust formulation. It uses repeated input/output data, Wasserstein ambiguity sets, and CVaR constraints to obtain robustness and probabilistic guarantees.
- Robust formulation: Stochastic DeePC replaces the deterministic formulation with a finite, convex, tractable program that robustifies objectives and constraints against disturbances.The reformulation begins as a semi-infinite optimization problem and is made tractable under reasonable assumptions.
- Robust formulation: Noise makes future trajectory predictions uncertain, can destroy the low-dimensional data-subspace property, and leaves the true distribution unknown.The method addresses these issues without explicitly identifying a predictive model.
- Robust formulation: The method minimizes worst-case expected objective values over an ambiguity set while imposing worst-case CVaR output constraints.CVaR relaxes almost-sure output constraints that may be infeasible under stochastic disturbances.
- Wasserstein ambiguity: The ambiguity set is a Wasserstein ball centered on the empirical distribution, with radius ϵ controlling the desired robustness level.This choice supports tractability and performance guarantees without requiring prior knowledge of the true data-generating distribution.
- Data collection: Offline data collection repeats an identical Page-exciting input experiment, builds data matrices and empirical samples, and permits larger N to reduce uncertainty and improve performance.The experiments use a common unknown initial state and identical inputs so the output matrices yield independent and identically distributed samples.
B. Main results
The paper reformulates distributionally robust DeePC as a tractable optimization problem and establishes high-confidence guarantees for expected cost and CVaR constraints. Its robustness depends on Wasserstein ambiguity sets, with conservativeness and sample-size limitations remaining practical considerations.
- Tractable reformulation: The distributionally robust DeePC problem admits a finite, convex, tractable reformulation under convexity and Lipschitz assumptions.Separate reformulations handle the objective and constraints.
- Tractable reformulation: The reformulated objective equals a sample-average objective plus dual-norm regularization weighted by the objective Lipschitz constant and Wasserstein radius.Robustness in trajectory space under the r-norm induces dual q-norm regularization.
- Probabilistic guarantees: The Wasserstein radius is selected so the true distribution lies in the empirical-data ball with confidence at least 1−β, under a light-tailedness assumption.The concentration result requires a finite exponential moment for some a > 1.
- Probabilistic guarantees: With probability 1−β, the robust DeePC optimal value upper-bounds the true expected cost, and the CVaR constraint holds under the true distribution.The probability is with respect to the N-fold product distribution of the collected data.
- Practical considerations: The radius decreases with sample size, but halving it requires N to increase by 2^⌊T/(Tini+Tf)⌋, and its constants are difficult to quantify in data-driven settings.The authors recommend data-driven tuning or cross-validation for practical radius selection.
- Algorithm: The robust DeePC optimization is implemented receding-horizon style by solving for an optimizer, applying part of the input sequence, updating recent measurements, and repeating.The procedure applies ν ≤ Tf inputs before recomputing the solution.
A. Nonlinear and Stochastic Aerial Robotics Case Study
The quadcopter case study evaluates distributionally robust DeePC under nonlinear stochastic dynamics and examines how data structure and hyperparameters affect tracking. The method achieves constrained reference tracking, while larger datasets and suitable horizons improve or stabilize performance.
- Case-study setup: The nonlinear stochastic quadcopter simulation uses three-dimensional position outputs, rotor thrust and body-rate inputs, and receding-horizon distributionally robust DeePC.The setup uses 25-step prediction, Tini = 6, constrained inputs and outputs, and a Wasserstein radius of ϵ = 0.003.
- Closed-loop performance: The representative closed-loop trajectory reaches the reference while satisfying output constraints.The simulation uses control horizon ν = 1 and implements the optimization in receding-horizon fashion.
- Hyperparameter selection: Stable-flight experiments for selecting the Wasserstein radius may not always be feasible, motivating data-based approximation of an optimal radius.Increasing the number of data batches broadens the range of radii associated with satisfactory tracking because the empirical distribution approaches the true distribution.
- Data quantity: Increasing data-matrix size significantly improves tracking until performance becomes approximately constant beyond 200 Page or 300 Hankel columns.The authors attribute this threshold behavior to the data matrices capturing a sufficiently rich trajectory subspace approximating the nonlinear dynamics.
- Data quantity: With T = 527, the Page representation yields poor performance and quadcopter crashes, whereas T = 15407 gives good performance for both structures.The larger-data Hankel case has a much larger solve time, and tracking errors remain approximately constant beyond a data threshold.
- Data-matrix structure: Across 273 stable-flight simulations, the Page matrix significantly outperforms the Hankel matrix in tracking performance.Possible explanations include tight reformulation, non-repeated Page entries, reduced noise sensitivity, and SVD-based preprocessing; Page matrices require more samples.
APPENDIX
The appendix establishes behavioural and optimization results supporting DeePC, including data-based characterization of feasible trajectories and tractable reformulations of robust objectives and constraints. It also identifies conditions under which these reformulations are exact or conservative.
- Behavioural-system results: The fundamental-lemma proof shows that sufficient Page excitation and system assumptions yield the required full-row-rank property.The argument constructs left-kernel vectors and uses excitation, controllability, and Cayley–Hamilton reasoning to establish linear dependence and rank.
- Behavioural-system results: Under Page excitation, the data-driven feasible set equals the set of input-output pairs compatible with the system and initial trajectory.The initial state is uniquely determined from the measured past inputs and outputs, coinciding with the current state.
- Robust reformulations: Convex Lipschitz objective and constraint functions admit distributionally robust reformulations based on duality and Wasserstein ambiguity sets.The appendix separates objective and constraint reformulations and uses Lipschitz constants Lobj and Lcon.
- Robust reformulations: The objective reformulation is exact for the full trajectory uncertainty set, while constraint reformulation exactness requires boundedness conditions.The appendix states equality for the objective under Ξ = R^{p(Tini+Tf)⌊T/(Tini+Tf)⌋} and gives a corresponding condition for constraints.
- Robust reformulations: For non-constant constraint functions, the CVaR constraint reformulation is generally an inner approximation under the stated unbounded uncertainty set.Exact coincidence would require h(Yfg) to be bounded on the uncertainty set for all g, which can occur only when h is constant.