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Robust data-driven state-feedback design

Julian Berberich, Anne Romer, Carsten W. Scherer, Frank Allgöwer

arXiv:1909.04314v3eess.SY

TL;DR

The paper tackles robust state-feedback design from a single finite noisy open-loop trajectory without model knowledge. It parametrizes all data-consistent closed-loop matrices, applies robust control methods for guaranteed stability and performance, and illustrates the approach with an unstable example whose H∞ performance is close to the model-based ideal.

  • Problem

    Finite noisy data do not generally provide non-conservative end-to-end closed-loop guarantees for LTI systems.

  • Method

    The paper parametrizes uncertain closed-loop matrices directly from one noisy open-loop trajectory and applies robust control methods, with an extension for partial model knowledge.

  • Results

    2.3 closed-loop H∞-norm was achieved versus 2.2 for the minimal nominal model-based state-feedback value, despite noisy measurements.

  • Takeaways & Limitations

    The approach provides end-to-end guarantees using a single finite noisy open-loop trajectory and offers an alternative to sequential identification and robust control.

Abstract

from arXiv · show

We consider the problem of designing robust state-feedback controllers for discrete-time linear time-invariant systems, based directly on measured data. The proposed design procedures require no model knowledge, but only a single open-loop data trajectory, which may be affected by noise. First, a data-driven characterization of the uncertain class of closed-loop matrices under state-feedback is derived. By considering this parametrization in the robust control framework, we design data-driven state-feedback gains with guarantees on stability and performance, containing, e.g., the $\mathcal{H}_\infty$-control problem as a special case. Further, we show how the proposed framework can be extended to take partial model knowledge into account. The validity of the proposed approach is illustrated via a numerical example.

I. INTRODUCTION

The paper addresses the open problem of obtaining non-conservative closed-loop guarantees from finite noisy data. It extends direct data-driven control toward robust stability and performance guarantees without prior system identification.

  • Finite noisy data generally do not yet provide non-conservative end-to-end closed-loop guarantees for LTI systems.
  • Behavioral systems theory shows that one persistently exciting measured trajectory can characterize an LTI system without prior identification.
  • Earlier data-driven state-feedback parametrizations addressed stabilization and linear-quadratic regulation but lacked meaningful guarantees for noisy data.
  • The proposed framework uses one finite noisy input-state trajectory to design controllers guaranteeing stability and performance for every system consistent with the data and noise bound.
  • The approach extends noisy-data closed-loop parametrization with robust control techniques and also accommodates partial model knowledge.

II. PRELIMINARIES

The preliminaries specify the system, data, disturbance, and controller setting for direct robust design. The disturbance is constrained by a known quadratic matrix inequality that can represent practical noise bounds.

  • The system includes state x_k, disturbance w_k, input u_k, and performance output z_k, with state-feedback controllers u_k = Kx_k.
  • The true system matrices A_tr and B_tr are unknown, while B_w, C, D_w, and D are assumed known for disturbance modeling and performance specification.
  • Persistence of excitation is distinguished from the input-only notion in earlier work, and the proposed results generally do not require it explicitly.
  • The controller is designed directly from a single open-loop input-state trajectory without prior system identification.
  • The unknown disturbance realization is assumed to belong to a known set described by a quadratic matrix inequality.
  • A bounded disturbance singular value is a special case of the quadratic description, obtained with Q_w = −I, S_w = 0, and R_w = w̄^2I.

III. DATA-DRIVEN STATE-FEEDBACK

The data-driven state-feedback section characterizes uncertain closed-loop matrices from noisy measured data, then applies robust control methods to design gains with stability and performance guarantees.

  • A single open-loop trajectory is used to derive a data-driven characterization of the uncertain closed loop under state-feedback.
  • Known robust control methods are applied to the resulting parametrization to design controllers guaranteeing stability and performance for all consistent closed-loop matrices.
  • The design is based directly on measured data perturbed by a disturbance satisfying Assumption 3.

A. Uncertain closed-loop parametrization

The paper exactly parametrizes the closed-loop uncertainty induced by noisy open-loop data and a fixed state-feedback gain. Under persistent excitation, this parametrization captures all possible closed-loop matrices; without it, robust design remains valid but can become conservative.

  • Data-consistent system class: The uncertain system class ΣX,U contains all pairs (A, B) consistent with the measured trajectory and admissible disturbance realization W.It is represented using fixed data matrices X and U together with the disturbance set W.
  • Closed-loop parametrization: For a fixed state-feedback gain K, the paper constructs the corresponding set of data-consistent closed-loop matrices directly from a single open-loop trajectory.The construction requires neither closed-loop measurements nor knowledge of the true system matrices.
  • Closed-loop parametrization: Theorem 4 gives an exact parametrization of the uncertain closed loop by matrices G satisfying the data constraint XG = I and an affine disturbance constraint.The disturbance parameter is restricted both by W ∈ W and by the constraint ensuring consistency with some system matrices A and B.
  • Persistent excitation: If the measured input-state trajectory is persistently exciting, every possible closed-loop matrix under state feedback can be constructed through the G parametrization.Equivalently, the parametrized set equals the full set of possible closed-loop matrices under state feedback.
  • Persistent excitation: Without persistent excitation, the parametrized set may be a strict subset of the possible closed-loop matrices, increasing conservatism even though it remains exact for each fixed gain K = UG.A stabilizing controller may exist outside the class representable by a feasible G satisfying the data constraint.
  • Computational consideration: Computing the exact uncertain class requires a kernel calculation, so the paper later uses a superset of the class to obtain simpler robust stability and performance procedures.The kernel computation may be undesirable numerically.

B. Robust state-feedback for stability

The paper characterizes uncertain closed-loop systems from one noisy open-loop trajectory and applies robust control methods to design stabilizing state-feedback gains. The resulting conditions are computationally tractable but can be conservative because they use a closed-loop superset and a common Lyapunov function.

  • Robust stabilization: Robust control methods design K = UG to stabilize every closed-loop matrix consistent with the measured data.The guarantee follows when the parameter G satisfies the data-dependent constraints used in the robust design.
  • Robust stabilization: Stabilizing controllers may be found even without persistence of excitation, although persistence of excitation is required for exact closed-loop characterization and improves feasibility.Full row rank of X can suffice for designing stabilizing controllers from data in the stated setting.
  • Conservatism and computation: The tractable design is conservative because the used set can strictly contain the exact uncertain closed loop and stabilization employs a common Lyapunov function.The superset has a simpler robust-control representation, while the exact conservatism remains to be quantified.
  • Conservatism and computation: The nonlinear feasibility condition can be converted into an LMI and solved as a semidefinite program using standard solvers.The reformulation uses a congruence transformation, Schur complements, and variables Y = X^-1 and M = GX^-1.

C. Robust state-feedback for performance

The framework extends data-driven robust state-feedback design from stability to quadratic closed-loop performance. Using a finite noisy data trajectory, it guarantees performance over an infinite horizon for all data-consistent closed loops, including H∞ control as a special case.

  • Performance specifications: Quadratic performance includes H∞ control with Q = −γ^2I, S = 0, R = I and strict passivity with Q = 0, S = −I, R = 0.The disturbance appearing in the noisy data and the disturbance in the performance objective play distinct roles.
  • Robust performance design: The performance design uses measured data to construct K so that all data-consistent closed-loop matrices satisfy stability and quadratic performance conditions.The design addresses the performance channel w 7→z under state feedback K = UG.
  • Performance guarantees: A finite-length data trajectory and a finite-horizon noise bound suffice to guarantee quadratic performance over an infinite time horizon for arbitrary ℓ2 disturbances.The performance-channel disturbances need not satisfy the same bound used to model the noise in the initial data trajectory.
  • Computational solution: The performance feasibility problem can be converted into an LMI for a fixed multiplier and solved by line-search over that multiplier.The reformulation uses variables Y = X^-1 and M = GX^-1.

D. Systems with partial model knowledge

The framework is extended to systems combining unknown data-driven components with known model-based components. A single open-loop trajectory replaces the unknown matrices in the closed-loop dynamics, after which robust performance design proceeds as in the fully data-driven case.

  • Problem setting: The mixed configuration assumes A1 and B1 are unknown while all other system matrices are known.The available data trajectory includes the measured states, known-component states, and inputs, subject to the assumed disturbance description.
  • Data/model integration: Matrices G1 and G2 are used with state feedback uk = K1xk + K2x̃k to replace the unknown matrices in the closed-loop dynamics.The resulting relations produce an LFT depending only on known matrices and the open-loop data trajectory.
  • Robust design: Robust controllers for the mixed system are derived using the same robust performance procedure as for the fully data-driven performance LFT.The mixed-system LFT parametrizes a superset of the uncertain closed-loop dynamics.
  • Data requirements: The mixed-system construction requires X to have full row rank and N ≥ n + ñ; persistent excitation allows any controller K1 and K2 to be constructed.The stated rank and data-length conditions apply specifically to the mixed configuration.
  • Application to loop-shaping: The framework supports iterative H∞ loop-shaping without knowledge of the unknown plant matrices by refining filter dynamics and repeatedly solving the robust performance problem.For the loop-shaping case, A2 = 0 means measurements of the filter state x̃ are not required.

IV. EXAMPLE

The example applies the data-driven robust state-feedback design to an unstable system using noisy open-loop data, achieving feasible designs and performance close to nominal model-based control. Increasing data length improves feasibility, while computational cost scales cubically with data length.

  • IV. EXAMPLE: The example uses an unstable system with known disturbance and performance matrices while the true system matrices are unavailable.A noisy trajectory of length N = 20 is generated for controller design via Corollary 10.
  • IV. EXAMPLE: γ = 2.4 yields a feasible design with closed-loop H∞-norm 2.3, compared with 2.2 for nominal model-based state-feedback.The data-driven controller therefore achieves performance close to the nominal full-model case despite noisy measurements.
  • IV. EXAMPLE: N ≥15 produces successful controller designs in 100 out of 100 experiments, while N = 4 succeeds in more than 50% of scenarios.The experiments vary random inputs and disturbances across 100 trials for each data horizon.
  • IV. EXAMPLE: Increasing N enhances feasibility because the set of data-consistent systems decreases as more data points become available.The robust guarantees cover a superset of the uncertain closed-loop matrices consistent with the data.
  • IV. EXAMPLE: The feasibility problem has 2(n + mw) + pz + N LMI rows and scales cubically with N and proportionally to n^6 in system dimension n.For the example with N = 20, the LMI has size 35 × 35 and the equality constraint has size 9.

V. CONCLUSION

The paper develops direct data-driven state-feedback procedures that provide robust closed-loop stability and performance guarantees from a single finite noisy open-loop trajectory. It also incorporates partial model knowledge, while identifying robust data-driven output-feedback control as future work.

  • V. CONCLUSION: The proposed procedures design state-feedback gains with guaranteed closed-loop stability and performance using noisy input-state data.The guarantees are obtained through a data-driven parametrization of closed-loop matrices consistent with the data and robust control methods.
  • V. CONCLUSION: The framework extends to systems combining data-driven and model-based components.The extension allows partial model knowledge to be incorporated into the controller-design approach.
  • V. CONCLUSION: Future work should extend the results to robust data-driven output-feedback control.

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