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Data-Enabled Predictive Control: In the Shallows of the DeePC
Jeremy Coulson, John Lygeros, Florian Dörfler
TL;DR
The paper addresses optimal, constrained trajectory tracking for unknown systems when explicit models and full-state measurements are unavailable. It proposes DeePC, which learns system behaviour from finite input/output data and uses regularized formulations for nonlinear stochastic systems. DeePC is equivalent to classical MPC for deterministic LTI systems and outperforms identification-based MPC in the reported quadcopter simulations.
Problem
Optimal trajectory tracking for unknown systems requires safe output-feedback control, while classical MPC depends on an accurate parametric model and state estimation.
Method
DeePC learns a non-parametric representation of system behaviour from finite data and computes constrained controls using real-time output feedback; regularizations address nonlinear stochastic settings.
Results
DeePC has equivalent closed-loop behaviour to MPC for deterministic LTI systems and performed better than sequential identification followed by MPC in nonlinear stochastic quadcopter simulations.
Takeaways & Limitations
Finite data can support predictive control without explicit system identification or state estimation, within the demonstrated LTI equivalence and quadcopter simulation settings.
Abstract
from arXiv · showhide
We consider the problem of optimal trajectory tracking for unknown systems. A novel data-enabled predictive control (DeePC) algorithm is presented that computes optimal and safe control policies using real-time feedback driving the unknown system along a desired trajectory while satisfying system constraints. Using a finite number of data samples from the unknown system, our proposed algorithm uses a behavioural systems theory approach to learn a non-parametric system model used to predict future trajectories. The DeePC algorithm is shown to be equivalent to the classical and widely adopted Model Predictive Control (MPC) algorithm in the case of deterministic linear time-invariant systems. In the case of nonlinear stochastic systems, we propose regularizations to the DeePC algorithm. Simulations are provided to illustrate performance and compare the algorithm with other methods.
I. INTRODUCTION
Optimal trajectory tracking for unknown systems requires output-feedback control that follows references, minimizes cost, and respects constraints without relying on an expensive accurate model. DeePC addresses this gap by learning system behaviour directly from data and is formally equivalent to MPC for deterministic LTI systems.
- Optimal trajectory tracking combines reference following, stage-cost minimization, and input/output safety constraints.
- Accurate parametric state-space models are central to MPC but can be costly and time-consuming to obtain for complex systems.
- Existing data-driven and learning-based approaches do not jointly solve constrained optimal tracking for unknown black-box systems with partial output observations.
- DeePC learns system behaviour rather than a parametric model and computes constrained optimal controls using real-time output feedback.
- For deterministic LTI systems, DeePC is formally equivalent to classical MPC under the paper’s stated assumptions.
III. MPC: A BRIEF OVERVIEW
Classical MPC repeatedly optimizes predicted trajectories over a finite horizon using a known system model, state estimates, costs, and constraints. Its model dependence motivates DeePC as a model-free alternative for complex unknown systems.
- MPC solves a finite-horizon optimization problem with system dynamics, input constraints, output constraints, and tracking costs.
- The optimization uses predicted input, state, and output sequences over horizon N, with Q and R weighting output and control costs.
- When the state is unavailable, MPC estimates it from past input/output data using an observer based on the known system model.
- In receding-horizon operation, MPC estimates the state, solves the optimization, applies part of the control sequence, updates measurements, and repeats.
- Under standard assumptions, MPC is recursively feasible and stabilizing, but accurate modeling and state estimation can be cumbersome for complex systems.
IV. PRELIMINARIES
Behavioural systems theory describes dynamical systems through the set of signal trajectories they permit rather than through a chosen parametric representation. The preliminaries define behaviours, system properties, truncated trajectories, and persistently exciting inputs.
- A behavioural system is defined by a discrete-time axis, a signal space, and a behaviour containing admissible trajectories.
- Linearity, time invariance, and completeness characterize the system class used for behavioural analysis.
- Finite trajectory windows are represented as truncated behaviours, with trajectories partitioned into input and output signals.
- A controllable behavioural system permits trajectories to be connected through finite-time transitions.
- Persistently exciting inputs have sufficiently rich Hankel matrices to generate output sequences representative of system behaviour.
B. Parametric system representation
Behavioural systems can also be represented by classical input/output/state models, whose minimal order and lag characterize state dimension and observability-related memory. The Fundamental Lemma connects sufficiently rich finite data to trajectory construction without explicit identification.
- B. Parametric system representation: A behavioural system may be represented by state-space matrices, with a minimal representation using the smallest state dimension n(B).
- B. Parametric system representation: The lag ℓ(B) is the smallest horizon associated with the observability matrix having rank n(B).
- B. Parametric system representation: Given sufficiently long initial input/output data with Tini ≥ ℓ(B), the state driven by the initial input sequence is unique.
- B. Parametric system representation: For controllable systems, persistently exciting data allow the columns of a Hankel matrix to span all length-t trajectories.
- B. Parametric system representation: The Hankel matrix thereby acts as a non-parametric predictive model that can implicitly estimate state, predict behaviour, and support optimal feedforward control.
A. Data collection
DeePC collects persistently exciting input/output data, partitions it into past and future blocks, and uses those data to represent and predict system trajectories without knowing the system representation.
- Data collection: The unknown LTI system generates offline input/output data with inputs persistently exciting of order Tini+N+n(B).The data length must satisfy T ≥ (m + 1)(Tini + N + n(B)) − 1.
- Data partitioning: The collected sequences are partitioned into past and future input/output blocks.Up and Yp contain the first Tini block rows, while Uf and Yf contain the last N block rows.
- Trajectory prediction: Past data estimate the underlying initial condition, while future data predict future trajectories.The initial input/output trajectory implicitly fixes the state without requiring a known input/output/state representation.
- Trajectory representation: Any Tini + N-length system trajectory can be constructed from the collected data through a coefficient vector g.A trajectory belongs to the system behavior exactly when such a g exists.
C. DeePC algorithm
DeePC formulates constrained trajectory tracking directly from past input/output data and future data matrices, replacing the model and state estimate used by MPC. Under stated deterministic LTI assumptions, it has equivalent feasible sets and closed-loop behavior, while requiring sufficient excitation and data.
- C. DeePC algorithm: DeePC optimizes tracking and control costs subject to input and output constraints using a coefficient vector g and fixed data matrices.The future inputs and outputs are determined by Uf, Yf, and g rather than being independent decision variables.
- C. DeePC algorithm: The algorithm repeatedly solves for g⋆, applies an initial segment of the optimal input sequence, updates recent measurements, and resolves.At each iteration, u⋆ = Ufg⋆ and the past data are replaced by the Tini most recent measurements.
- D. Equivalence of DeePC and MPC: Under controllability, sufficient horizon length, and persistently exciting data, DeePC and MPC have equal feasible sets.Theorem 5.1 requires Tini ≥ ℓ(B) and excitation of order Tini + N + n(B).
- D. Equivalence of DeePC and MPC: With Q ⪰ 0, R ≻ 0, and convex non-empty constraints, DeePC and MPC produce identical optimal control sequences and outputs at every iteration.The equivalence follows from equal feasible sets and coincident costs under the theorem assumptions.
- D. Equivalence of DeePC and MPC: DeePC combines system identification, state estimation, and trajectory prediction in one linear equation and a quadratic program.The formulation has T − Tini − N + 1 decision variables, with a minimum data requirement determined by m, Tini, N, and n(B).
- D. Equivalence of DeePC and MPC: If n(B) is underestimated, insufficient data produce a reduced-order approximate input/output behavior whose implications remain unresolved.In practice, the authors suggest collecting enough data to exceed the persistency-of-excitation requirement.
VI. BEYOND DETERMINISTIC LTI SYSTEMS
The paper extends DeePC beyond deterministic LTI systems by introducing regularizations for nonlinear stochastic settings and evaluating their utility numerically.
- VI. BEYOND DETERMINISTIC LTI SYSTEMS: The extension beyond deterministic LTI systems uses regularizations and a numerical study to assess their utility.The paper also provides plausible reasoning for the proposed regularizations.
A. Regularized DeePC Algorithm
For nonlinear systems with measurement noise, regularized DeePC modifies the optimization with slack, one-norm, and low-rank terms to address inconsistency, robustness, and noisy data representations.
- A. Regularized DeePC Algorithm: The considered nonlinear system has nonlinear state and output maps and white measurement noise, with process noise excluded from the analysis.Focusing on measurement noise isolates its effect on DeePC.
- A. Regularized DeePC Algorithm: The regularized problem introduces an output slack variable, one-norm penalties, and a low-rank approximation of the data matrix.σy handles output-constraint inconsistency, while λy and λg weight regularization terms.
- A. Regularized DeePC Algorithm: A sufficiently large λy makes nonzero slack indicate that the data are inconsistent and preserves feasibility when noisy constraints cannot be satisfied exactly.The slack is penalized with a weighted one-norm.
- A. Regularized DeePC Algorithm: The low-rank approximation retains dominant sub-behavior, filters measurement noise, and represents the closest deterministic LTI system under the SVD Frobenius-norm criterion.For nonlinear dynamics, the construction is motivated by lifting to infinite-dimensional linear dynamics with a nonlinear output map.
B. Aerial Robotics Case Study
The regularized DeePC algorithm was evaluated on nonlinear, stochastic quadcopter tracking tasks and compared with identification-based MPC. DeePC showed better tracking, constraint satisfaction, and cost, while regularization improved performance but low-rank approximation remained sensitive.
- Experimental setup: The study simulated regularized DeePC on a high-fidelity nonlinear quadcopter model and compared it with system identification followed by MPC.The model included spatial and angular coordinates, velocities, rotor-thrust inputs, full-state measurements, and additive white measurement noise.
- Tracking comparison: DeePC performed better than sequential identification and MPC in reference tracking and constraint satisfaction during 60-second figure-eight tracking.The comparison used a nonlinear and stochastic quadcopter model commanded to follow a series of figure-eight trajectories.
- Cost and constraints: DeePC consistently outperformed identification-based MPC in cost and constraint satisfaction across 30 step-trajectory repetitions.The repetitions used different data sets and measurement-noise random seeds; constraint-violation duration and cost were measured.
- Interpretation: The authors attribute DeePC’s superior performance intuitively to simultaneous optimization of the system model, state estimation, and control policy.Conventional MPC instead fixes a system model and performs these tasks independently.
- Regularization: Regularizations improved DeePC performance in an eight-data-set sensitivity analysis of λy and λg.The analysis measured averaged constraint-violation duration and cost for a repeated step trajectory without low-rank approximation.
- Regularization: One-norm regularization of λg appeared more effective and robust than low-rank Hankel-matrix approximation in preliminary simulations.The low-rank approximation appeared sensitive and required case-by-case treatment to avoid unstable behaviour.
VII. CONCLUSION
The paper presents DeePC for unknown systems, establishes equivalence to classical MPC for unknown LTI systems, and demonstrates regularized DeePC on stochastic nonlinear quadcopter dynamics. The simulations report superior performance versus system identification followed by MPC, while robustness remains future work.
- DeePC applies finite data to unknown LTI systems and formally establishes equivalence with classical MPC.
- DeePC learns unknown-system behaviour and uses real-time feedback to compute constrained controls for desired-trajectory tracking.
- Regularized DeePC was simulated on stochastic nonlinear quadcopter dynamics, extending the evaluation beyond deterministic LTI systems.
- The simulated DeePC performance was superior to system identification followed by MPC.
- Robustness of DeePC and its regularization for stochastic and nonlinear dynamics remains ongoing and future work.