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Data-Driven Model Predictive Control with Stability and Robustness Guarantees

Julian Berberich, Johannes Köhler, Matthias A. Müller, Frank Allgöwer

arXiv:1906.04679v3eess.SY

TL;DR

Data-driven MPC lacks established closed-loop guarantees despite avoiding explicit system identification. This paper uses behavioral trajectory representations and terminal equality constraints, extending the nominal scheme to bounded measurement noise with regularized slack. It proves exponential stability without noise and practical exponential stability relative to noise under multi-step application.

  • Problem

    Few data-driven MPC methods provide theoretical guarantees on closed-loop stability, recursive feasibility, or constraint satisfaction.

  • Method

    The scheme uses a persistently exciting measured input-output trajectory as an implicit behavioral model, with terminal equality constraints nominally and a regularized slack variable for bounded noise.

  • Results

    The paper proves exponential stability for the noise-free nominal scheme and practical exponential stability relative to the noise level for the robust multi-step scheme.

  • Takeaways & Limitations

    The results provide a theoretical analysis of closed-loop properties for a simple purely data-driven MPC scheme using initially measured data.

Abstract

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We propose a robust data-driven model predictive control (MPC) scheme to control linear time-invariant (LTI) systems. The scheme uses an implicit model description based on behavioral systems theory and past measured trajectories. In particular, it does not require any prior identification step, but only an initially measured input-output trajectory as well as an upper bound on the order of the unknown system. First, we prove exponential stability of a nominal data-driven MPC scheme with terminal equality constraints in the case of no measurement noise. For bounded additive output measurement noise, we propose a robust modification of the scheme, including a slack variable with regularization in the cost. We prove that the application of this robust MPC scheme in a multi-step fashion leads to practical exponential stability of the closed loop w.r.t. the noise level. The presented results provide the first (theoretical) analysis of closed-loop properties, resulting from a simple, purely data-driven MPC scheme.

I. INTRODUCTION

The paper addresses the open problem of obtaining stability and robustness guarantees for purely data-driven MPC. It proposes an LTI-system MPC scheme based on measured trajectories and behavioral systems theory, avoiding prior system identification.

  • Few data-driven control methods provide theoretical guarantees such as closed-loop stability or constraint satisfaction.
  • The proposed scheme uses a Hankel matrix built from a previously measured persistently exciting input-output trajectory to represent LTI-system trajectories.
  • The method requires an initially measured persistently exciting trajectory and an upper bound on system order, without set-based modeling or online estimation.
  • Terminal equality constraints support analysis of recursive feasibility, constraint satisfaction, and exponential stability in the nominal scheme.
  • The robust extension addresses bounded additive measurement noise and yields practical exponential stability when applied in a multi-step fashion.

II. PRELIMINARIES

The paper formulates control of an unknown LTI system using only measured input-output data and characterizes its trajectories through persistence of excitation and behavioral systems theory.

  • Persistence of excitation of order L means the corresponding Hankel matrix has rank mL.
  • The objective is to control an unknown order-n LTI system with m inputs and p outputs using only measured input-output data.
  • A trajectory is defined as an input-output sequence generated by a minimal realization, which entails controllability and observability.
  • The behavioral result states that a Hankel matrix from one persistently exciting trajectory spans all trajectories of the unknown LTI system.
  • Input-output data of length at least n induces a unique internal state, enabling equilibrium-based MPC without an explicit system model.
  • The framework imposes pointwise input and output constraints and uses past n measurements to specify the prediction initial condition.

III. NOMINAL DATA-DRIVEN MPC

The nominal scheme uses noise-free measurements and terminal equality constraints to predict data-consistent trajectories, with proofs of recursive feasibility, constraint satisfaction, and exponential stability.

  • The nominal data-driven MPC scheme predicts future trajectories from noise-free measurements using the behavioral systems result.
  • Under mild assumptions, the paper proves recursive feasibility, constraint satisfaction, and exponential closed-loop stability.
  • The scheme includes terminal equality constraints within the data-driven MPC formulation.

A. Nominal MPC scheme

The nominal MPC replaces an explicit plant model with data-consistency constraints, optimizes constrained predicted trajectories, and uses a terminal equality constraint to reach the desired equilibrium.

  • A. Nominal MPC scheme: The behavioral trajectory characterization replaces model dynamics with a constraint requiring predicted input-output trajectories to be data-consistent.
  • A. Nominal MPC scheme: At each time step, the scheme uses the past n input-output pairs, solves the optimization problem, applies the first optimal input, and repeats.
  • A. Nominal MPC scheme: The optimization minimizes a quadratic stage cost relative to a desired equilibrium while enforcing input and output constraints.
  • A. Nominal MPC scheme: Without terminal ingredients or a sufficiently long horizon, data-driven MPC may fail to guarantee stability and constraint satisfaction.
  • A. Nominal MPC scheme: The terminal equality constraint aligns the predicted internal state after L steps with the steady-state associated with the desired equilibrium.

B. Closed-loop guarantees

Under persistently exciting data and suitable horizon and cost assumptions, the nominal data-driven MPC scheme is recursively feasible, satisfies constraints, and yields exponential closed-loop stability. The scheme uses terminal equality constraints and requires no prior system identification.

  • Assumptions: The prediction input is assumed persistently exciting of order L + 2n, with prediction horizon L ≥ n.The stronger excitation order accounts for reconstructed trajectories of length L+n, while the horizon condition follows from the terminal constraints.
  • Closed-loop guarantees: If the initial MPC problem is feasible, recursive feasibility holds at every future time.The proof uses a shifted previously optimal solution with an appended zero input.
  • Closed-loop guarantees: The closed loop satisfies the input and output constraints for all times.Constraint satisfaction follows from the recursive-feasibility argument and the admissibility of the shifted candidate.
  • Closed-loop guarantees: The equilibrium x_s = 0 is exponentially stable on the feasible initial-state set X_L.A Lyapunov function combining the optimal cost with an IOSS Lyapunov function is bounded above and below quadratically.
  • Analysis and implementation: The nominal analysis applies model-based MPC arguments to an exact data-driven system description, using detectability through an IOSS Lyapunov function.In the noise-free case, the data-driven formulation is equivalent to the corresponding model-based input-output description.
  • Analysis and implementation: A single measured trajectory can directly formulate the MPC problem without prior identification, with online complexity similar to classical MPC.The data-driven formulation uses trajectory coefficients rather than an identified model.

IV. ROBUST DATA-DRIVEN MPC

The robust extension addresses bounded additive output measurement noise in both the data used for prediction and online measurements. Applied in a multi-step fashion, it yields practical exponential closed-loop stability, with the attraction region approaching all initially feasible points as the noise bound vanishes.

  • Robust scheme: The robust scheme uses a slack variable with cost regularization to compensate bounded output noise in prediction data and online measurements.The slack variable is regularized in the cost as part of the robust modification.
  • Stability result: The multi-step robust data-driven MPC scheme leads to practical exponential stability in the presence of bounded additive output measurement noise.The result concerns the closed loop produced by applying the scheme over multiple steps.
  • Stability result: As the noise bound tends to zero, the closed-loop region of attraction approaches the set of all initially feasible points.The robust section does not impose output constraints; an extension with tightened output constraints is cited separately.

A. Robust MPC scheme

The robust MPC formulation relaxes noisy trajectory matching with a bounded slack variable and regularizes trajectory coefficients. An n-step implementation supports stronger theoretical guarantees under bounded noise, while the formulation involves non-convexity and assumptions on data richness, horizon, and noise magnitude.

  • Noise model: Bounded noise perturbs both the prediction data and online output measurements, affecting the Hankel-based trajectory representation and initial conditions.The two perturbations correspond to multiplicative model uncertainty in the data and a noisy output-feedback setting online.
  • Robust MPC scheme: The robust problem relaxes nominal equality constraints with a bounded slack variable and adds quadratic regularization of the trajectory coefficient α.The slack accounts for noisy data and measurements, while regularization limits the influence of the noisy Hankel matrix on predictions.
  • Robust MPC scheme: Increasing the α-regularization weight trades tracking performance against overfitting by reducing the complexity of the data-driven system description.The slack penalty instead favors small slack values and can improve prediction accuracy.
  • Implementation boundary: The paper leaves the effect of alternative regularization norms on practical performance as an open question.Theoretical results are conjectured to extend to general norm penalties, but only quadratic penalties are analyzed.
  • Multi-step implementation: The n-step scheme applies the first n optimized inputs before shifting the horizon and resolving the robust MPC problem.The algorithm repeats this procedure using the latest past n input-output measurements.
  • Multi-step implementation: With terminal equality constraints, the n-step scheme admits practical exponential stability, whereas the corresponding one-step scheme guarantees recursive feasibility only locally.The reported numerical performance of the two schemes is almost indistinguishable despite the stronger theoretical result for the n-step version.
  • Implementation boundary: The slack constraint is non-convex because its bound depends on ∥α(t)∥1, although a sufficiently large slack penalty can allow the constraint to be neglected.Without that constraint and with polytopic input constraints, the problem becomes a strictly convex quadratic program.
  • Assumptions: The robust analysis requires persistent excitation of order L + 2n, prediction horizon L ≥ 2n, and a sufficiently small persistence-of-excitation-to-noise bound.The noise is assumed bounded, and the relevant bound can depend on data length and excitation magnitude.

B. Local upper bound of Lyapunov function

The robust MPC value function is locally bounded above by a quadratic expression in an augmented state that captures recent inputs and outputs. This construction addresses the fact that the optimal cost may not be quadratically bounded by a minimal realization state alone.

  • The optimal cost can be arbitrarily large for a zero current state because it depends on potentially large past inputs and outputs.Therefore, a quadratic upper bound in the minimal realization state is unavailable in general.
  • The analysis introduces a non-minimal state ξ that collects the information needed to represent the current data-driven prediction.The minimal realization state is related to ξ through a generally non-invertible linear transformation.
  • An IOSS Lyapunov function W(ξ)=∥ξ∥_P^2 is combined with the optimal cost as V_t:=J*_L(ξ̃_t)+γW(ξ_t).This yields a Lyapunov function suitable for bounding the augmented data-driven state.
  • Locally, the robust MPC problem is feasible and V admits a quadratic upper bound under the stated assumptions.Lemma 1 establishes this result for ξ_t in a neighborhood of the origin.
  • The upper-bound proof constructs a feasible input-output trajectory that drives the state and output to zero within L steps, with a slack variable compensating measurement noise.The resulting slack satisfies ∥σ(t)∥∞≤ε̄(∥α(t)∥1+1).
  • Observability and norm bounds control the trajectory coefficient α and the slack contribution, leading to the quadratic upper bound on V.The term c4 in the bound arises solely from the slack variable.

C. Prediction error bound

The prediction-error analysis bounds the mismatch between the predicted output and the output generated by open-loop application of the optimal input. This bound depends on the data-driven coefficients, slack, and system parameters and supports later feasibility and stability arguments.

  • The predicted output and the actual open-loop output differ because the prediction uses measured initial conditions and a data-driven trajectory representation.The paper analyzes their difference by exploiting the LTI system structure and observability.
  • Lemma 2 provides inequalities bounding the prediction mismatch for every prediction step k∈[0,L−1].The bounds are expressed using constants defined for the relevant time index and the coefficient-dependent slack bound.
  • The proof identifies the data-driven trajectory component generated by the optimal input and compares it with the output arising from the actual initial conditions.The difference is represented through a zero-input trajectory and the observability matrix.
  • The mismatch bound is the key quantitative link between the predicted trajectory and the actual output produced by applying the optimal input open loop.It is subsequently used to establish recursive feasibility and practical stability.

D. Recursive feasibility

Recursive feasibility is established for the robust multi-step MPC scheme when the noise bound is sufficiently small relative to a Lyapunov sublevel set. The proof shifts the previous solution, uses the prediction-error bound, and appends a local deadbeat control segment.

  • If the robust MPC problem is feasible at time t, it remains feasible at time t+n when the noise level is sufficiently small.This is the main recursive-feasibility result for the n-step scheme.
  • The candidate input is formed by shifting the previously optimal input, while the candidate output follows the open-loop output for the initial portion of the horizon.The construction preserves the relevant initial-data constraints at the shifted time.
  • The feasibility proof verifies the remaining constraints using the slack definition and the noise bound.The constructed trajectory satisfies the terminal and constraint conditions after the appended control segment.
  • The required noise threshold decreases as the Lyapunov sublevel set grows because the noise acts as a multiplicative uncertainty in prediction.Standard recursive feasibility additionally requires invariance of the sublevel set, which is established later.
  • For any chosen VROA, there is a noise threshold ε̄0 such that feasibility holds at t+n for states satisfying V_t≤VROA and ε̄≤ε̄0.The initial-state set defined by V_0≤VROA becomes the guaranteed region of attraction in the main result.
  • Terminal equality constraints make the final n predicted outputs zero, and the prediction-error bound implies that the corresponding open-loop state is close to the origin.A local deadbeat controller can then steer the state and output to zero over n steps.
  • For a 1-step MPC scheme, the same construction is only local and requires the relevant open-loop outputs to be close to zero.This condition is met, for example, when the initial state is close to the origin.
  • The stated guarantees extend to nonzero equilibria, but the corresponding bounds can worsen quantitatively with the equilibrium and noise level.The paper omits this extension because it requires substantially more notation.

E. Practical exponential stability

Under bounded measurement noise, the robust data-driven MPC scheme yields a closed loop that converges exponentially to a noise-dependent neighborhood, establishing practical exponential stability under suitable conditions.

  • E. Practical exponential stability: Theorem 3 establishes practical exponential stability for sufficiently small noise, sufficiently large persistence of excitation, and suitably selected regularization parameters.The result applies to the n-step application of the robust MPC scheme.
  • E. Practical exponential stability: The invariant sublevel set V_t ≤ V_ROA is preserved, and V_t converges exponentially to the set V_t ≤ β(ε̄) for initially feasible states within the region of attraction.The function β satisfies β(0)=0, so the limiting neighborhood shrinks as the noise level decreases.
  • E. Practical exponential stability: The resulting Lyapunov function converges to a set whose size shrinks with the noise level, implying practical exponential stability of the equilibrium ξ = 0.The result requires small noise, persistence of excitation large relative to noise, and suitable regularization.
  • E. Practical exponential stability: The regularization parameters must balance stability and noise amplification: λ_α cannot be too small or too large, while λ_σ must be sufficiently large but not arbitrarily large.For fixed c_pe, overly large regularization parameters deteriorate robustness because λ_αc_peε̄ and λ_σc_peε̄^2 must remain small.
  • E. Practical exponential stability: Increasing the region of attraction requires smaller noise or larger persistence of excitation relative to noise, while lower c_pe also reduces β(ε̄).As c_peε̄ approaches zero, the region of attraction approaches the set of all initially feasible points.
  • E. Practical exponential stability: Computing the parameter bounds required by the theorem is difficult in practice, and the proof uses conservative estimates that may weaken quantitative conditions.The paper identifies less conservative, verifiable noise bounds as a topic for future research.

V. EXAMPLE

The four-tank example evaluates the robust data-driven MPC scheme using only measured input-output data and bounded noisy measurements. Terminal equality constraints achieve tracking, whereas the corresponding unconstrained scheme becomes unstable with the selected parameters.

  • V. EXAMPLE: The four-tank system is a linearized example with unknown system matrices, and the controller uses only measured input-output data to track a specified setpoint.The system is open-loop stable but can be destabilized by short-horizon MPC without terminal constraints.
  • V. EXAMPLE: N = 400 samples were collected with random inputs in [−1, 1]^2, while online and recorded outputs had uniformly distributed additive noise bounded by ε̄ = 0.002.The same noise type affects the measurements used to update the MPC initial conditions.
  • V. EXAMPLE: With L = 30, Q = 3·I_p, R = 10^-4I_m, λ_σ = 1000, and λ_αε̄ = 0.1, the terminal-constrained scheme tracks the setpoint with only slight deviations.The reported configuration uses a 1-step MPC application of the robust problem.
  • V. EXAMPLE: Without terminal constraints, the closed loop is unstable and diverges for both 1-step and n-step MPC under the chosen parameters.This example contrasts the terminal-constrained scheme with the unconstrained scheme labeled UCON in Figure 2.
  • V. EXAMPLE: The regularization range 0.05 ≤ λ_αε̄ ≤ 0.5 yields desirable closed-loop performance, whereas choosing λ_α too low makes the closed loop unstable through noise amplification.The example supports the qualitative tuning guidelines associated with Theorem 3.
  • V. EXAMPLE: For the four-tank system, using an upper bound n = 10 is sufficient when the true order is n = 4, but assuming an order below 4 can destabilize the closed loop.The prediction horizon can be chosen roughly within 7 ≤ L ≤ 70; tracking error generally increases with noise level.

VI. CONCLUSION

The paper develops and analyzes a purely data-driven MPC scheme using past measured data, establishing practical closed-loop guarantees under bounded noise. It also identifies conservative bounds and extensions that remain open.

  • The scheme uses only past measured data and requires no prior system identification step.
  • The analysis provides qualitative guidance for selecting design parameters and shows how persistence-of-excitation bounds influence the region of attraction.
  • The practical example shows stability with the proposed terminal-constraint scheme, whereas an existing scheme without terminal constraints produces an unstable closed loop.
  • Many proof bounds are conservative, motivating less conservative verifiable conditions on admissible noise levels for closed-loop stability.
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