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Linear tracking MPC for nonlinear systems Part II: The data-driven case

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

arXiv:2105.08567v4math.OCeess.SY

TL;DR

Data-driven MPC for unknown nonlinear systems lacks a unifying framework without assuming known basis functions. This paper develops a locally linear, data-updating MPC scheme with practical closed-loop stability under stated assumptions, while extending affine-system parametrization and robustness analysis.

  • Problem

    Existing nonlinear data-driven MPC approaches assume known basis functions, leaving no unifying framework for nonlinear systems.

  • Method

    The scheme uses measured input-output data, local linearizations, the Fundamental Lemma extended to affine systems, and strictly convex quadratic programs.

  • Results

    The MPC scheme practically stabilizes the closed loop under suitable design parameters, nearby initial conditions, and sufficiently close initialization data.

  • Takeaways & Limitations

    The affine-system extension and robustness bounds are directly applicable to other data-driven MPC formulations and support stronger guarantees for a robust affine tracking scheme.

  • Takeaways & Limitations

    The paper does not address how to ensure closed-loop persistence of excitation and therefore proves only practical stability.

Abstract

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We present a novel data-driven model predictive control (MPC) approach to control unknown nonlinear systems using only measured input-output data with closed-loop stability guarantees. Our scheme relies on the data-driven system parametrization provided by the Fundamental Lemma of Willems et al. We use new input-output measurements online to update the data, exploiting local linear approximations of the underlying system. We prove that our MPC scheme, which only requires solving strictly convex quadratic programs online, ensures that the closed loop (practically) converges to the (unknown) optimal reachable equilibrium that tracks a desired output reference while satisfying polytopic input constraints. As intermediate results of independent interest, we extend the Fundamental Lemma to affine systems and we derive novel robustness bounds w.r.t. noisy data for the open-loop optimal control problem, which are directly transferable to other data-driven MPC schemes in the literature. The applicability of our approach is illustrated with a numerical application to a continuous stirred tank reactor.

I. INTRODUCTION

The paper addresses the lack of a unifying framework for data-driven control of unknown nonlinear systems by updating input-output data online and exploiting local linearization. It develops an MPC scheme with closed-loop stability guarantees, based on an affine extension of the Fundamental Lemma and illustrated numerically.

  • Research gap: Data-driven MPC has strong motivation when accurate system models are difficult to obtain, but nonlinear-system stability guarantees remain limited.Existing nonlinear extensions often assume known basis-function parametrizations, restricting practical applicability.
  • Approach: The proposed MPC updates the data-driven system parametrization online and uses local linearization to control unknown nonlinear systems.The approach uses past input-output measurements rather than an available nonlinear model or linearization.
  • Control objective: The controller targets the optimal reachable output equilibrium for a desired setpoint, which may lie outside the output equilibrium manifold.An artificial equilibrium is optimized online to support tracking and stability.
  • Guarantees: The analysis establishes practical stability when data and predictions remain sufficiently accurate during closed-loop operation.The result relies on assumptions including slow system evolution during initial data collection and persistently exciting closed-loop inputs.
  • Implementation: The scheme requires solving strictly convex quadratic programs and only a potentially rough upper bound on the nonlinear system order.The Fundamental Lemma is extended to affine systems because local linearization generally produces affine dynamics.
  • Evaluation and contributions: The paper presents the method in a nonlinear numerical example and concludes with robustness bounds that can transfer to other data-driven MPC formulations.The paper also discusses directness and simplicity relative to sequential identification and model-based MPC.

II. FUNDAMENTAL LEMMA FOR AFFINE SYSTEMS

The paper extends Willems et al.’s Fundamental Lemma from linear to affine systems, requiring a stronger data-richness condition and an affine-specific coefficient constraint.

  • II. FUNDAMENTAL LEMMA FOR AFFINE SYSTEMS: The extension is nontrivial because unknown offsets cannot be set to zero without loss of generality.The affine dynamics include unknown offsets e and r that must be represented through the data.
  • II. FUNDAMENTAL LEMMA FOR AFFINE SYSTEMS: The required persistence-of-excitation condition is stronger than the condition used for the original Fundamental Lemma.Under controllability, sufficiently rich input data with full row rank can enforce the stronger condition.
  • II. FUNDAMENTAL LEMMA FOR AFFINE SYSTEMS: The affine extension parametrizes trajectories using one measured input-output trajectory and a coefficient vector α.The trajectory characterization is stated as an if-and-only-if condition involving α and the measured data.
  • II. FUNDAMENTAL LEMMA FOR AFFINE SYSTEMS: Unlike the linear case, the affine parametrization requires the coefficients to satisfy Σα_i = 1.This constraint carries the unknown affine offsets from the data into the newly generated trajectory.
  • II. FUNDAMENTAL LEMMA FOR AFFINE SYSTEMS: The affine result also connects to prior offset-free data-driven control and to difference trajectories that correspond to an LTI system.These alternatives provide related ways to handle affine offsets in different settings.

III. NONLINEAR DYNAMICS AND LINEARIZATION

The paper first establishes the problem setup and assumptions, then derives a technical bound on how nonlinearities influence the data before presenting the control scheme.

  • III. NONLINEAR DYNAMICS AND LINEARIZATION: Section III introduces the problem setup, theoretical assumptions, and a bound on the influence of nonlinearity in the data.The bound is presented as a technical intermediate result for the subsequent guarantees.

A. Problem setup

The problem concerns data-driven MPC for an unknown control-affine nonlinear system, using recent input-output measurements to track a desired setpoint under input constraints.

  • A. Problem setup: The system is unknown, control-affine, and subject to pointwise convex compact-polytopic input constraints.The vector fields and system matrices or offsets used in linearized descriptions are not assumed known.
  • A. Problem setup: The proposed MPC tracks a desired output reference using only the last N input-output measurements and no explicit knowledge of the system vector fields.The scheme does not require state measurements, so the analysis uses an extended state vector.
  • A. Problem setup: The target is the optimal reachable equilibrium ysr, whose corresponding input-state pair is unique under later assumptions.The optimal reachable equilibrium is defined through a steady-state optimization with a positive-definite weighting matrix.
  • A. Problem setup: Linearizing at a point produces affine dynamics with matrices A˜x, B, C˜x, D and offsets e˜x, r˜x.The linearized model is used to formulate the optimal steady-state problem and data-driven predictions.
  • A. Problem setup: The analysis assumes a persistently exciting input-state trajectory and a uniformly bounded minimum-norm coefficient vector αsrLin(D).The optimal coefficient vector depends on both the linearization point and the data set.

B. Assumptions

The theoretical guarantees rely on smooth, controllable, observable linearizations, compact steady-state manifolds, local tracking conditions, and assumptions linking extended-state observations to steady-state behavior.

  • B. Assumptions: The system is assumed to satisfy the companion paper’s Assumptions 1–5, including smoothness, controllability, tracking, nonsingularity, and compactness conditions.The vector fields are twice continuously differentiable, linearizations are controllable, I − A˜x is nonsingular, and the union of steady-state manifolds is compact.
  • B. Assumptions: An observability assumption requires a locally Lipschitz map that reconstructs the state from finite input-output information.The corresponding affine map is defined for the dynamics linearized at the current state.
  • B. Assumptions: A further assumption bounds the relation between an extended state, its reconstructed state, and the optimal reachable steady-states on compact sets.The constants in this bound are denoted ceq,1 and ceq,2.
  • B. Assumptions: Uniform boundedness of αsrLin(D) can follow in closed loop from uniformly persistently exciting data and compactness of the steady-state manifold.Incremental input-output data are proposed as an alternative when this boundedness is not guaranteed a priori.
  • B. Assumptions: The setpoint is required to be reachable for one sufficient condition ensuring the extended-state assumption, with ysr = yr and m = p.This condition is given together with assumptions inherited from the companion paper.

C. Bounding the influence of the nonlinearity on the data

The paper compares measured nonlinear-system data with data generated by a local affine linearization. Under smoothness, their output mismatch is bounded, becoming small when the state trajectory changes slowly.

  • Linearized data construction: The analysis constructs artificial input-output data by applying the measured inputs to the nonlinear system’s affine linearization at the current state.The linearized trajectory starts from the nonlinear system’s past state and uses the same input sequence.
  • Mismatch bound: The output difference between the nonlinear system and its local linearization is bounded on any compact state set under the stated smoothness assumption.Lemma 1 introduces Δ_t,k as the difference between the known nonlinear output and the unknown affine-linearized output.
  • Bound derivation: The state and output recursions isolate nonlinear deviations as Δx,k and Δy,k, which are then propagated through the linearized dynamics.Uniformly bounded Jacobians and smoothness yield a constant c_Δ for the resulting mismatch bound.
  • Interpretation: The bound depends on the squared distance between the current state and the past states used for data collection.Thus, slowly evolving state trajectories make the measured nonlinear data close to data from the linearized dynamics.

IV. DATA-DRIVEN MPC FOR NONLINEAR SYSTEMS

This section introduces the data-driven MPC scheme for unknown nonlinear systems. It then establishes continuity of the underlying optimal-control solution and uses it to prove practical exponential closed-loop stability.

  • Section overview: The proposed scheme controls unknown nonlinear systems with stability guarantees using a data-driven MPC formulation.The formulation is developed before the continuity and closed-loop stability analyses.
  • Section overview: The analysis establishes a continuity property for the solution of the underlying optimal control problem.This property is used as an intermediate step in the later stability proof.
  • Section overview: The closed loop under the proposed scheme is proved to be practically exponentially stable.The result is presented after introducing the MPC scheme and analyzing optimal-solution continuity.

A. MPC scheme

The MPC scheme updates recent nonlinear-system measurements online and uses Hankel-based predictions as local linear approximations. It optimizes an artificial equilibrium through strictly convex quadratic programs and applies inputs in a multi-step fashion.

  • MPC formulation: At each time t, the scheme uses the last N nonlinear-system input-output measurements to formulate a robust open-loop MPC problem.The robust formulation accounts for the fact that measured nonlinear data are not exact trajectories of the current linearization.
  • Tracking objective: The cost tracks an online-optimized artificial equilibrium while penalizing its distance from the desired setpoint.Tracking and input terms use Q and R, while S penalizes deviation of the artificial equilibrium from the reference.
  • Prediction model: Hankel-matrix predictions are updated online and approximate the affine dynamics obtained by linearizing the nonlinear system at the current state.The data-driven prediction model therefore changes with the measurement window and current operating point.
  • Optimization structure: The optimization is a strictly convex quadratic program with initial-condition constraints, terminal equality constraints, equilibrium and trajectory input constraints, and an order upper-bound requirement.The prediction initialization uses past measurements, and the terminal constraint spans n + 1 steps.
  • Online implementation: Algorithm 1 computes the equilibrium representation, solves the MPC problem, applies the first n inputs, and repeats after advancing time by n steps.Only measured data and a potentially rough upper bound on system order are required; no nonlinear model is needed.

B. Continuity of the optimal input and cost

The continuity analysis compares the robust data-driven problem using nonlinear-system data with an ideal problem using data from the current linearization. Under persistent excitation and regularity assumptions, it bounds feasibility, optimal-cost, and optimal-input discrepancies.

  • Assumptions: Persistent excitation of order L + n + 1 and LICQ are imposed to obtain the continuity result for the optimal solution.LICQ requires linear independence of equality and active inequality constraint rows.
  • Continuity bounds: For sufficiently small data and initial-condition mismatch, the robust problem remains feasible whenever the nominal problem is feasible and its optimal cost stays close to the nominal cost.The proof uses a candidate solution for feasibility and a bound parameterized by the mismatch size.
  • Proof mechanism: The optimal input difference is bounded through strong convexity of the nominal quadratic program and properties of multiparametric QPs.The result also supports a separation-type robustness analysis for affine systems with noisy data and perturbed initial conditions.
  • Problem comparison: The analysis compares the robust MPC problem using nonlinear-system data with a nominal problem using ideal data from the dynamics linearized at the current state.With zero slack, the nominal constraints represent trajectories of the current affine linearization.
  • Scope of the assumption: The proposition’s mismatch condition is needed only for the open-loop analysis because the proposed closed-loop MPC scheme implicitly enforces a corresponding bound.This connects the technical open-loop assumption to the later closed-loop stability analysis.

C. Closed-loop guarantees

The data-driven MPC is recursively feasible and practically exponentially stabilizing under persistence-of-excitation and local-data assumptions. Its guarantees depend on sufficiently accurate predictions, suitable regularization, and conservative design conditions.

  • Uniform closed-loop persistence of excitation is crucial because online data updates can otherwise produce inaccurate predictions.The assumption requires full row rank and a uniform bound on the relevant Hankel-matrix pseudoinverse.
  • Theorem 2 guarantees recursive feasibility and closed-loop stability when Assumptions 1–5 and sufficiently small prediction-error bounds hold.The result applies for θ in a sufficiently small interval and suitable regularization parameters.
  • The Lyapunov-based result shows convergence close to the optimal reachable equilibrium rather than necessarily exact asymptotic convergence.The bound depends on the cost weight S, regularization parameters, and the initial distance between stored and current states.
  • Prediction accuracy depends on the distance between past states and the current state, so sufficiently small initial θ prevents excessively rapid closed-loop motion.The bound is propagated recursively during operation.
  • The stability theorem is a practical result: fixed persistence-of-excitation bounds prevent θ from becoming arbitrarily small, ruling out general asymptotic stability.This differs from the companion model-based result, which assumes an exact linearization and does not require closed-loop persistence of excitation.
  • With bounded output noise, the closed loop converges to a region whose size increases with both the prediction-error parameter θ and the noise bound.Small noise preserves qualitative guarantees but can worsen tracking error and reduce the region of attraction.

V. NUMERICAL EXAMPLE

The numerical study applies the data-driven MPC to an unknown continuous stirred tank reactor using online-updated input-output data. The controller tracks the reachable equilibrium, while performance depends strongly on data updating and tuning choices.

  • Setup: The study evaluates the proposed MPC on a continuous stirred tank reactor with an unknown model and input-output data only.The implementation uses prediction horizon L = 40 and data length N = 120.
  • Setup: The controller uses incremental inputs and extended output constraints to enforce the original input constraints while preserving the theoretical guarantees.The data provide an exact prediction model for the mapping ∆u 7→u.
  • Closed-loop behavior: Online data updating is crucial: using only the initial data can produce a huge tracking error, with output converging to 1.14 instead of yr = 0.6519.The reported failure occurs when k = 0 data are used for prediction at all times.
  • Closed-loop behavior: The proposed data-driven MPC converges very closely to the optimal reachable equilibrium, and persistence of excitation can be maintained by stopping data updates.The data-driven input is more unsteady because of less accurate predictions and terminal equality constraints.
  • Comparison: An identification-based MPC with incremental parameter updates achieves comparable performance, whereas simple least-squares parameter estimation does not steer the system successfully to the setpoint.The comparison reports no corresponding closed-loop stability theory under similar assumptions.
  • Parameter study: The normalized closed-loop cost is generally worse than model-based MPC because of initial excitation, smaller S, and slower convergence.A broad acceptable performance corridor is reported for S ∈ [7, 30] over a wide range of λα.
  • Parameter study: Smaller S improves prediction accuracy by limiting state motion, while excessively large S makes predictions inaccurate and can worsen performance.The example uses S = 10, λα = 3 · 10^-6, and λσ = 10^7.
  • Scope: The proof relies on conservative estimates, so the reported stability analysis is qualitative and the example’s tuning values need not satisfy the proof’s sufficient bounds.Other optimization parameters also affect performance, motivating broader studies for different applications.

VI. CONCLUSION

The paper establishes a data-driven MPC method for unknown nonlinear systems and proves practical closed-loop stabilization under explicit design, initialization, and data-variation assumptions.

  • VI. CONCLUSION: The proposed MPC updates an affine-system data parametrization online to approximate unknown nonlinear dynamics through local linearization.The method uses input-output data rather than a model of the nonlinear system.
  • VI. CONCLUSION: The MPC scheme practically stabilizes the closed loop when design parameters are suitably chosen, the initial condition is near the steady-state manifold, and initialization data are sufficiently consistent.These conditions define the stated scope of the stability guarantee.
  • VI. CONCLUSION: The paper extends the Fundamental Lemma to affine systems and derives robustness bounds for data-driven MPC that also apply to other formulations.The bounds support a robust data-driven tracking MPC scheme for affine systems with stronger guarantees than existing approaches.

B. PROOF OF THEOREM 2

The proof propagates local discrepancies between nonlinear and linearized trajectories into bounded data perturbations, then combines open-loop robustness with MPC stability arguments to obtain practical convergence.

  • B. PROOF OF THEOREM 2: The proof first shows that bounded discrepancies in earlier extended states remain bounded at the current time under suitable conditions.This provides the perturbation bound required for the data-driven MPC analysis.
  • B. PROOF OF THEOREM 2: Local linearization and Lipschitz continuity bound the differences between nonlinear trajectories, linearized trajectories, and their outputs.Controllability and bounded inverse controllability matrices are used in constructing comparison inputs.
  • B. PROOF OF THEOREM 2: The resulting data perturbation is interpreted as bounded noise, enabling robustness bounds for the open-loop data-driven optimization problem.The linear-system robustness result remains applicable to affine systems because it depends on trajectory differences.
  • B. PROOF OF THEOREM 2: Combining the perturbation bound with nominal MPC Lyapunov analysis yields practical stability of the nonlinear closed loop.The closed loop converges to a neighborhood whose size depends on the disturbance or data-error bound.
  • B. PROOF OF THEOREM 2: The affine-system extension provides a data-driven tracking MPC scheme with closed-loop stability and robustness results under its stated assumptions.The analysis requires conditions including horizon length, LICQ, and parameter inequalities.

D. Proof of Theorem 3

The proof of Theorem 3 constructs feasible successor trajectories and uses Lyapunov bounds to establish recursive feasibility and practical exponential stability under input disturbances.

  • D. Proof of Theorem 3: The proof establishes lower and upper bounds for the Lyapunov function candidate over feasible extended states.These bounds relate the candidate function to the distance from the optimal reachable equilibrium.
  • D. Proof of Theorem 3: One candidate shifts the previous optimal trajectory, steers the system toward its previous equilibrium, and preserves terminal constraints.Controllability and sufficiently small disturbances ensure input feasibility.
  • D. Proof of Theorem 3: The candidate constructions yield a Lyapunov decrease inequality despite bounded input disturbances.The proof bounds effects of disturbed initial conditions and predicted input-output trajectories.
  • D. Proof of Theorem 3: A second candidate interpolates between the previous artificial equilibrium and the optimal reachable equilibrium to obtain a feasible trajectory near the target.Small interpolation and disturbance parameters preserve input constraints.
  • D. Proof of Theorem 3: For sufficiently small disturbances, recursive feasibility is preserved and the closed loop practically exponentially stabilizes the optimal reachable equilibrium.The guaranteed region is defined by V(ξ0) ≤ VROA, while the limiting neighborhood grows with the disturbance bound.
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