Source-linked AI summary
Regularized and Distributionally Robust Data-Enabled Predictive Control
Jeremy Coulson, John Lygeros, Florian Dörfler
TL;DR
DeePC is extended to unknown stochastic systems, where noisy input/output data and unknown disturbances complicate trajectory prediction and expected-cost control. The paper uses distributionally robust optimization to connect worst-case performance to regularized DeePC, and establishes a probabilistic performance guarantee.
Problem
DeePC for stochastic systems must handle noise-corrupted measurements and unknown disturbances while controlling an unknown system.
Source: §I · p. 1, §II · p. 2Method
The paper robustifies DeePC using noisy input/output data and distributionally robust optimization to connect worst-case control to regularization.
Source: §I · p. 1Results
The robust problem’s optimal value upper-bounds out-of-sample performance with high confidence, and robustifying DeePC yields principled regularization.
Source: §IV · p. 4, §VI · p. 8Takeaways & Limitations
The results support regularized DeePC for stochastic systems by providing probabilistic performance guarantees and a principled basis for its regularizers.
Source: §I · p. 2, §VI · p. 8
Abstract
from arXiv · showhide
In this paper, we study a data-enabled predictive control (DeePC) algorithm applied to unknown stochastic linear time-invariant systems. The algorithm uses noise-corrupted input/output data to predict future trajectories and compute optimal control policies. To robustify against uncertainties in the input/output data, the control policies are computed to minimize a worst-case expectation of a given objective function. Using techniques from distributionally robust stochastic optimization, we prove that for certain objective functions, the worst-case optimization problem coincides with a regularized version of the DeePC algorithm. These results support the previously observed advantages of the regularized algorithm and provide probabilistic guarantees for its performance. We illustrate the robustness of the regularized algorithm through a numerical case study.
I. INTRODUCTION
The paper targets control of unknown systems from input/output data when measurements are noisy. It extends DeePC to stochastic systems using distributionally robust optimization, giving probabilistic guarantees that justify regularization.
I. INTRODUCTION
- Growing system complexity makes accurate model identification a crucial, time-consuming, and expensive step that data-driven control seeks to bypass with measurements.The paper frames direct use of system measurements as the goal of data-driven control.Source: §I · p. 1
- Related approaches may require a known model, full-state measurements, disturbance knowledge, or many data measurements, while DeePC uses raw input/output data.The introduction notes that these approaches are not suitable for real-time, constrained, optimal control based only on input/output samples.Source: §I · p. 1
- DeePC was extended from deterministic systems to stochastic systems after heuristic regularizations performed well under disturbances but lacked robustness guarantees.The paper gives those regularizations rigorous meaning through probabilistic guarantees.Source: §I · p. 1
- Distributionally robust optimization yields a robust DeePC algorithm for noisy input/output data, with probabilistic performance guarantees that justify regularization.The authors also report new insights into regularizer design.Source: §I · pp. 1–2
II. PROBLEM STATEMENT
The problem is finite-horizon control of an unknown stochastic LTI system using input/output measurements. The controller seeks an input sequence minimizing expected objective cost, but unknown dynamics and disturbances make prediction and expectation difficult.
II. PROBLEM STATEMENT
- The stochastic LTI system has unknown dynamics and an unknown disturbance distribution, while only input/output measurements are available.The paper assumes controllability of (A, B) and observability of (A, C).Source: §II · p. 2
- The controller chooses a constrained finite-horizon input sequence to minimize the expected objective over the resulting stochastic output trajectory.The input sequence belongs to the constraint set U.Source: §II · p. 2
- Unknown system dynamics prevent future-output prediction, and an unknown disturbance distribution prevents computing the expectation; joint output chance constraints are outside scope.Even when the system and distribution are known, the paper notes that high-dimensional integration can make the problem computationally intractable.Source: §II · p. 2
A. DeePC Algorithm for Deterministic Systems
Deterministic DeePC uses raw input/output data to build a non-parametric predictive model and optimize future inputs. It represents trajectories through Hankel matrices and applies the resulting control in a receding-horizon loop.
A. DeePC Algorithm for Deterministic Systems
- DeePC uses raw input/output data to construct a non-parametric predictive model for deterministic systems.The method is grounded in behavioural system theory.Source: A. DeePC Algorithm for Deterministic… · p. 2
- With Tini ≥ ℓ, the measured past input/output trajectory fixes the initial state, and a coefficient vector g represents the resulting future input/output trajectory.The trajectory is represented through the Hankel matrices built from measured data.Source: A. DeePC Algorithm for Deterministic… · p. 3
- The optimizer selects a future input sequence from the Hankel representation, applies an initial portion, then updates recent measurements and resolves.The procedure repeats after advancing time by s steps, for some s ≤ Tf − 1.Source: A. DeePC Algorithm for Deterministic… · p. 3
- The input data must be persistently exciting of order Tini + Tf + n, requiring a sufficiently rich and long signal.Persistent excitation means the corresponding Hankel matrix has full row rank; the passage gives T ≥ (q + 1)L − 1 for order L.Source: A. DeePC Algorithm for Deterministic… · pp. 2–3
- Regularized DeePC adds terms with nonnegative λini and λg, and the paper states this problem coincides with a distributionally robust variation of the stochastic control problem.This provides a robust-optimization interpretation for the regularizations.Source: A. DeePC Algorithm for Deterministic… · p. 3
B. Distributionally Robust DeePC
This section formulates DeePC with noisy data as a distributionally robust optimization. It connects the robust problem to regularized DeePC and a high-confidence bound on out-of-sample performance.
B. Distributionally Robust DeePC
- Measured outputs are treated as random-variable realizations, with uncertainty organized row-wise and the joint distribution represented as a product of row distributions.The row distributions are determined by the system dynamics and the unknown disturbance distribution.Source: B. Distributionally Robust DeePC · pp. 3–4
- Noise-corrupted outputs can make the past-output consistency equation infeasible, so the method softens it using a slack variable with a 1-norm penalty.With a sufficiently large penalty, the original equality is violated only when it is infeasible.Source: B. Distributionally Robust DeePC · p. 3
- Because the in-sample optimum may perform poorly out of sample, the method uses a robust objective designed to upper-bound out-of-sample performance with high confidence.The ambiguity set is constructed to contain the actual distribution with high confidence.Source: B. Distributionally Robust DeePC · p. 4
- The ambiguity set uses a Wasserstein ball around the empirical distribution, with distributional distance defined through an optimal mass-transport plan.The plan couples distributions with the specified marginals.Source: B. Distributionally Robust DeePC · p. 4
- The robust problem is computationally tractable and yields regularized DeePC whose optimal value upper-bounds out-of-sample performance with high confidence.The bound follows from constructing the ambiguity set to contain the actual data distribution with high confidence.Source: B. Distributionally Robust DeePC · p. 4
IV. MAIN RESULTS
For certain objective functions, distributionally robust DeePC reduces to a tractable convex program equivalent to regularized DeePC. The robust optimum also upper-bounds out-of-sample performance with high confidence under the stated assumptions.
IV. MAIN RESULTS
- With high confidence, the robust problem’s optimal value upper-bounds out-of-sample performance when the disturbance distribution is light-tailed.The theorem provides a Wasserstein-ball radius ε(β); compactly supported distributions satisfy the light-tailed assumption, as do Gaussian and exponential distributions.Source: §IV · p. 4
- For separable convex objectives with the stated bounded-conjugate condition, the semi-infinite robust problem reduces to a tractable convex program equivalent to regularized DeePC.The reformulation’s equality conditions specify the uncertainty supports for which the equivalence is exact.Source: §IV · p. 4
- The reformulation’s equality condition holds for unbounded disturbance support when F is full row rank, including the Gaussian case.Under these conditions, disturbances affect all entries of the output vector.Source: §IV · pp. 4–5
- The chosen trajectory norm sets the regularization’s robustness target, and the decision-variable regularizer’s coefficient depends on the cost function and Wasserstein radius.For example, an l1 trajectory norm makes λ_g play the role of either ε times a cost-dependent bound or ελ_ini.Source: §IV · p. 5
- The Wasserstein-ball radius can be larger than necessary, so the authors recommend choosing it in a data-driven fashion.The stated coverage probability may be much lower than β even when the radius is ε(β).Source: §IV · p. 5
V. SIMULATIONS
The quadcopter simulations examine Robust DeePC on a figure-8 tracking task and assess performance across Wasserstein radii. The results favor a moderate radius over both zero and large radii.
V. SIMULATIONS
- The test used a quadcopter model linearized around hover, with full-state measurements, additive zero-mean Gaussian state noise, and four rotor-thrust inputs.The model state comprises three spatial coordinates, their velocities, three angular coordinates, and their velocities.Source: §V · p. 7
- The data-collection input was uniformly random for persistent excitation, and 214 input/output measurements were used to populate the data matrices.The output data were corrupted by additive zero-mean Gaussian noise; 214 measurements were identified as the minimum needed for persistency of excitation.Source: §V · p. 7
- For figure-8 tracking, the algorithm showed desirable behavior despite measurement and process noise, with the first input constrained by red dashed bounds.The input bounds mimic normalized rotor-thrust constraints; the figures show the trajectory, position, and first input signal.Source: Figure 1 · p. 7, Figure 2 · p. 7, Figure 3 · p. 7
- Figure 4 supports choosing a Wasserstein radius ε in [0.001, 0.01]; ε = 0 and large radii show poor performance.The zero-radius case optimizes in-sample performance, while large radii are over-conservative.Source: Figure 4 · p. 8, §V · p. 8
VI. CONCLUSION
The conclusion addresses optimal control of unknown stochastic systems using input/output trajectories without knowing the nature of the stochasticity. It proposes robust DeePC with noise-corrupted data and identifies future directions in multiple data sets and output constraints.
VI. CONCLUSION
- The control problem is to minimize an objective depending on input/output trajectories of an unknown stochastic system without knowledge of the stochasticity's nature.Source: §VI · p. 8
- The proposed robust DeePC algorithm uses noise-corrupted input/output data to compute optimal and robust control inputs.The authors report that robustification yields principled regularization terms, supporting the regularized algorithm's observed superior performance on stochastic systems.Source: §VI · p. 8
- Future work includes incorporating multiple measured data sets to improve performance and adding output constraints.Source: §VI · p. 8