Source-linked AI summary
Fast-DeePC: Fast Data-Enabled Predictive Control using Soft-constrained Scheme
Terry Zhang, Jan Drgona, Malika Sader, Di Liu, Liang Wu
TL;DR
DeePC’s online optimization grows with the stored data because it retains a data-length coordinate and enforces behavioral consistency online. Fast-DeePC relaxes that consistency into a quadratically regularized objective and eliminates the coordinate offline, reducing the problem to a fixed-dimensional QP. Across three benchmarks, its dedicated solver matches a general-purpose solver on the identical soft-constrained problem while reducing online solve time by up to two orders of magnitude.
Problem
DeePC’s online problem grows with the amount of stored data because its coordinate dimension grows and behavioral consistency must be enforced during each optimization.
Source: A. Related Work · p. 1, C. The DeePC Algorithm · p. 3Method
Fast-DeePC relaxes behavioral consistency into quadratic penalties with a ridge on g, eliminates g offline, and solves a fixed-dimensional box-constrained QP.
Source: B. Contribution · p. 2, §III · p. 3Results
Across three benchmarks, the proposed solver achieves solution quality comparable to a general-purpose solver on the same soft-constrained problem while reducing online solve time by up to two orders of magnitude.
Source: B. Contribution · p. 2, §V · p. 7Takeaways & Limitations
The reduced problem has a unique minimizer for arbitrary recorded data whenever it is feasible, independent of data rank.
Source: B. Contribution · p. 2
Abstract
from arXiv · showhide
Data-enabled predictive control (DeePC) replaces the explicit system model of model predictive control with a data-driven predictor built directly from recorded input-output trajectories, but the original hard-equality formulation keeps its coordinate vector as an online decision variable whose size grows with the offline data length, so the online problem grows more expensive as more data is used to enrich the behavioral representation. This article proposes Fast-DeePC, a framework which relaxes this consistency relation into the control objective with a quadratic ridge on the coordinate vector in place of the customary sparsity-promoting penalty, admitting a closed-form, offline elimination of the coordinate vector and reducing the online problem to a fixed-size box-constrained quadratic program solved by a structure-exploiting interior-point method. On identical soft quadratically regularized problems, the result matches a general-purpose solver's solution and is up to two orders of magnitude faster.
I. INTRODUCTION
DeePC uses recorded input–output trajectories instead of an explicit plant model, but its online problem grows with the stored data. Fast-DeePC addresses this computational bottleneck by eliminating the behavioral coordinate while retaining a soft-constrained formulation.
I. INTRODUCTION
- Accurate plant models can be difficult to obtain for systems with strong nonlinearities, large state dimensions, or costly identification experiments.The paper motivates using routinely collected operational data instead.Source: §I · p. 1
A. Behavioral System Representation
- DeePC predicts from measured input–output trajectories represented through a Hankel matrix, avoiding explicit parametric model identification.Under noise-free measurements and persistent excitation, this behavioral formulation is equivalent to model-based MPC with an exact state estimate,.Source: §I · p. 1, A. Behavioral System Representation · p. 2
A. Related Work
- DeePC’s online optimization grows with stored data and must enforce behavioral consistency constraints at every step.Model-based MPC instead compresses data into an offline model, keeping online complexity largely independent of the identification dataset.Source: A. Related Work · p. 1
- Prior predictor-based reductions enable fixed-dimensional online optimization, but do not eliminate the coordinate from an exact soft-constrained DeePC objective while guaranteeing a unique solution independent of data rank.Other computational approaches include offline neural policies, trajectory-segment selection, SVD, and range-space reformulation, [26],,.Source: A. Related Work · pp. 1–2
B. Contribution
- Fast-DeePC penalizes behavioral consistency with quadratic terms and a ridge, eliminates the coordinate by partial minimization, and solves a fixed-dimensional QP with a dedicated interior-point method.The reduced Hessian is positive definite for arbitrary recorded data, and the solver reduces online solve time by up to two orders of magnitude while matching a general-purpose solver on the same soft-constrained problem.Source: B. Contribution · p. 2
B. Data Collection and the Consistency Equation
The data collection step partitions a persistently exciting recorded trajectory into past and future Hankel blocks. At each control step, the measured initialization window and candidate future inputs and outputs are linked to those blocks through a behavioral coordinate.
B. Data Collection and the Consistency Equation
- The recorded trajectory is partitioned into past and future Hankel blocks, with each column collecting a past window and an N-step future segment.The data length is chosen to meet a necessary condition for persistent excitation of order Tini + N + n.Source: B. Data Collection and the Consistency… · p. 2
- At time t, the recent measured inputs and outputs form the initialization window, while u, y, and r stack future inputs, outputs, and references over the prediction horizon.Source: B. Data Collection and the Consistency… · p. 2
- The stacked signal is admissible exactly when a behavioral coordinate g matches its past window and candidate future inputs and outputs to the Hankel blocks.When Tini is at least the system lag, the measured window fixes a unique state and each candidate future input determines a unique predicted output without explicit state estimation.Source: B. Data Collection and the Consistency… · p. 2
C. The DeePC Algorithm
DeePC optimizes tracking and input effort subject to behavioral consistency and input/output constraints. It applies the optimized input sequence in receding-horizon fashion, but its coordinate decision variable grows with the offline data length.
C. The DeePC Algorithm
- The DeePC objective balances output tracking and input effort subject to behavioral consistency and the input and output constraint sets.Source: C. The DeePC Algorithm · p. 3
- DeePC solves for g, applies the first s optimized inputs, updates the measured window, and repeats the optimization.Here s is a design parameter taken as 1.Source: C. The DeePC Algorithm · p. 3
- The coordinate g has dimension ng, which grows with offline data length T and therefore inflates the online QP as more data are added.The proposed scheme removes this coupling by relaxing behavioral consistency into the objective and eliminating g offline.Source: C. The DeePC Algorithm · p. 3
III. METHOD
Fast-DeePC relaxes the hard behavioral constraint into weighted quadratic penalties and adds a ridge on the behavioral coordinate. Partial minimization eliminates that coordinate offline, leaving a fixed-dimensional box-constrained QP for online control.
III. METHOD
- The method moves the data-length-dependent computation for g offline, leaving an online dense box-constrained QP in z = col(u, y).The QP dimension is fixed by the prediction horizon and input/output dimensions.Source: §III · p. 3
- The formulation groups past-input, past-output, and future-input consistency in one λ-weighted term, while separately weighting future-output mismatch by ρ.The cost also includes tracking and input effort terms.Source: §III · p. 3
- Fast-DeePC penalizes past-and-input mismatch and output-prediction mismatch, adding a quadratic ridge on g in place of the customary ℓ1 penalty.Larger λ and ρ enforce consistency more tightly, while smaller values favor tracking and constraint satisfaction.Source: 1) The penalized program and offline… · p. 3
- The inner quadratic problem has a unique minimizer because its matrix M is positive definite, and M depends only on offline data and fixed weights.The matrix or its Cholesky factor is computed before control begins.Source: 1) The penalized program and offline… · p. 3
- Substitution of the affine forcing term and future-input block removes both the per-step full-regressor product and the ng-dimensional solve from the online phase.The forcing vector depends on the measured window, while the future-input block is U_f.Source: 1) The penalized program and offline… · p. 3
- Quadratic ridge regularization enables closed-form elimination of g, unlike the ℓ1-regularized formulation, and the two formulations can produce different closed-loop trajectories.The distinction is the availability of elimination, not merely the nonsmoothness of the ℓ1 penalty.Source: 1) The penalized program and offline… · pp. 3–4
A. Substituting g⋆back
Substituting the optimal behavioral coordinate into the objective yields a quadratic cost in the physical input-output variables. A Woodbury reformulation moves data-length-dependent construction offline, leaving a fixed-dimensional online BoxQP whose specialized solver and constraint scope are specified.
A. Substituting g⋆back
- Substituting g⋆ produces a quadratic cost in z = col(u, y), with terms independent of z discarded.The tracking-cost terms contribute H0 and h0,t.Source: A. Substituting g⋆back · p. 4, 1) T-independent online construction… · p. 4
- The reduced objective’s +λI and +ρI terms come from squared targets, while its negative terms encode curvature from eliminating g.These components appear in the reduced Hessian blocks.Source: A. Substituting g⋆back · p. 4
- A Woodbury reformulation makes the offline factorization dimension independent of data length T without changing the soft quadratic DeePC problem or its optimizer.The construction leaves the soft problem unchanged.Source: 1) T-independent online construction… · p. 4
- The online matrices have dimensions independent of T, so the online phase does not solve an ng-dimensional system or reconstruct g⋆.Dimensions depend on N, Tini, m, and p.Source: 1) T-independent online construction… · p. 4
- The online problem is a box-constrained quadratic program solved with a feasible Mehrotra predictor-corrector interior-point method.The variables are rescaled to componentwise bounds −1 ≤ z ≤ 1.Source: 1) T-independent online construction… · p. 4
- The reduced problem remains quadratic only for polyhedral constraint sets, while this specialized solver requires componentwise boxes.Coupled polyhedral or nonlinear constraints require a more general QP or nonlinear-programming solver.Source: 1) T-independent online construction… · pp. 4–5
- The generally dense Newton matrix is factored by Cholesky at each interior-point iteration, with the factor reused for predictor and corrector directions.A diagonal-block Schur elimination is not applicable to the generally dense reduced Hessian.Source: 1) T-independent online construction… · p. 5
- Rank-one inverse updates could reduce per-iteration linear-algebra cost from O(n^3) to O(n^2), but this work retains dense Cholesky.The proposed substitution is left for future work.Source: 1) T-independent online construction… · p. 5
2) Strong convexity and penalty consistency:
The reduced problem has a unique minimizer under the stated positive-weight and feasible-box conditions, and its optimizer varies proportionally with changes in measured windows or references. Penalty residuals are bounded, and increasing both consistency penalties recovers the hard quadratically regularized problem in the limit.
2) Strong convexity and penalty consistency:
- The reduced QP has a unique minimizer for arbitrary recorded data under the stated positive-weight and nonempty closed-box conditions.The reduced objective is strongly convex and coercive.Source: 2) Strong convexity and penalty… · p. 5
- Small changes in the measured window or reference produce proportionally small changes in the optimizer and applied input.The applied input is the first block of the optimizer.Source: 2) Strong convexity and penalty… · p. 5
- A feasible zero-penalty point bounds both behavioral-consistency and output-prediction residuals at any minimizer.The residual bounds follow by comparing the full objective with the feasible point’s objective value.Source: 2) Strong convexity and penalty… · p. 5
- As both consistency penalties tend to infinity, every limit point satisfies the hard behavioral relation and solves its hard quadratically regularized counterpart.This limit result assumes compact input and output boxes and fixed λg.Source: 2) Strong convexity and penalty… · p. 5
IV. NUMERICAL EXAMPLES
The numerical examples compare the original hard-equality ℓ1 controller as a control-performance reference against alternative solvers for the same soft quadratic problem. They evaluate three benchmarks and use a predictor–corrector interior-point method for the reduced BoxQP.
IV. NUMERICAL EXAMPLES
- Each benchmark compares the original hard-equality ℓ1 controller as a performance reference with full CVXPY/OSQP, reduced BoxQP with OSQP, and the proposed algorithm on the same soft quadratic problem.The comparisons assess both the formulation and solver acceleration.Source: §IV · pp. 5–6
- Algorithm 1 applies a Mehrotra predictor–corrector interior-point method to the reduced BoxQP and returns an approximate minimizer.The supplied pseudocode initializes the decision vector inside its box and factorizes an adjusted Hessian during the affine predictor step.Source: §IV · p. 6
- Table I lists benchmark dimensions, horizons, and penalty weights; simulations run in Python on an Intel i7-11800H desktop with an NVIDIA GeForce RTX 3060 GPU.The code is reported as publicly available.Source: Table I · p. 6, §IV · p. 6
A. Nonlinear rocket landing
In the rocket-landing simulation, all three controllers land successfully, while Fast-DeePC reproduces the full soft-constrained solution to a very small trajectory error. The benchmark uses full-state feedback, making it a state-feedback predictive-control test rather than an output-only state-estimation test.
A. Nonlinear rocket landing
- With full-state feedback and Tini = 1, this benchmark tests state-feedback predictive control rather than state estimation from output data alone.The simulator's state comprises planar position, velocity, pitch, and pitch rate; its inputs are normalized main thrust, side thrust, and gimbal angle.Source: A. Nonlinear rocket landing · p. 6
- The predicted pitch constraint is |θ| ≤ 0.45 rad, a safety margin below the physical tipping limit of |θ| ≤ 0.6 rad.The original DeePC-Hunt controller retains its own formulation, tracking weights, and trained regularizers as the performance baseline.Source: A. Nonlinear rocket landing · p. 6
- 7×10^-5 closed-loop trajectory RMSE: Fast-DeePC matches CVXPY/OSQP on the identical soft-constrained problem, and all three controllers land successfully.The result supports that offline elimination of g reproduces the full penalized solution up to solver tolerance.Source: A. Nonlinear rocket landing · p. 6
B. Two-tank system
In the two-tank experiment, Fast-DeePC improves tracking error and common closed-loop cost relative to the original hard-ℓ1 baseline. Both controllers keep realized outputs within their prescribed bounds, and the reduced native-OSQP and C implementations agree closely.
B. Two-tank system
- The linear two-state plant uses two flow-command inputs and two measured outputs, with equal channel tracking weights, no input-effort term, and input and output bounds of [-1, 1].The hard-ℓ1 baseline uses λg = 2.Source: B. Two-tank system · p. 6
- Both controllers' realized outputs remain within prescribed bounds, while mean behavioral residuals stay below 10^-6.The reduced native-OSQP and C solutions have a closed-loop trajectory RMSE of 0.007, providing a numerical check of the C implementation.Source: B. Two-tank system · p. 6
- Tracking RMSE fell from 0.209 to 0.184 and common closed-loop cost from 2.63 to 2.04 with Fast-DeePC versus the hard-ℓ1 baseline.Figure 2 shows closer reference tracking by Fast-DeePC in this experiment.Source: B. Two-tank system · p. 6, Figure 2 · p. 7
C. Mixed-traffic platoon
Across the benchmarks, the C BoxQP solver is much faster than full soft OSQP, with the largest reported speedup on mixed traffic. In that platoon experiment, Fast-DeePC avoids collision using its optimized inputs alone, unlike the hard-ℓ1 baseline that required emergency-braking interventions.
C. Mixed-traffic platoon
- The eight-vehicle platoon has CAVs at positions 3 and 6 and six HDVs following nonlinear optimal-velocity dynamics.The output combines all vehicles' velocity errors with the two CAV spacing errors; CAV accelerations are constrained to [-5, 2] m/s².Source: C. Mixed-traffic platoon · p. 7
- Fast-DeePC avoided collision using optimized inputs alone with automatic emergency braking disabled; the hard-ℓ1 baseline required 28 emergency-braking interventions.The emergency-braking layer triggered when the baseline's optimized inputs alone were about to violate the spacing constraint.Source: C. Mixed-traffic platoon · p. 7
- The C solver is approximately 19× faster on two-tank, more than two orders of magnitude faster on mixed traffic, and approximately 8.5× faster on rocket than full soft OSQP.Figure 4 and Table II separate the effects of eliminating g and using the specialized solver.Source: Figure 4 · p. 7, Table II · p. 7, C. Mixed-traffic platoon · p. 7
V. CONCLUSION
Fast-DeePC softens the behavioral constraint and eliminates its coordinate vector offline, yielding a fixed-dimensional online quadratic program. On three benchmarks, its interior-point solver closely tracks a general-purpose solver while cutting online solve time by up to two orders of magnitude.
V. CONCLUSION
- On rocket-landing, two-tank, and mixed-traffic benchmarks, the proposed solver closely tracks a general-purpose solver on the identical soft problem while reducing online solve time by up to two orders of magnitude.The comparison concerns the same soft-constrained problem.Source: §V · p. 7
- Fast-DeePC relaxes the behavioral constraint into the objective with a quadratic ridge and exactly eliminates the coordinate vector by partial minimization.This reduces the online problem to a fixed-dimensional box-constrained quadratic program.Source: §V · p. 7
- The reduced problem has a unique minimizer for any recorded data, and its consistency residuals decay as penalty weights increase.Its solution approaches hard quadratically regularized DeePC in the limit.Source: §V · p. 7
- Future work will address incremental offline-matrix updates as trajectory data arrive and execution-time certification of the online solver.Source: §V · p. 7