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From noisy data to feedback controllers: non-conservative design via a matrix S-lemma

Henk J. van Waarde, M. Kanat Camlibel, Mehran Mesbahi

arXiv:2006.00870v2math.OC

TL;DR

The paper addresses non-conservative feedback control for unknown dynamical systems using noisy data, where finite samples and uncertainty make robust guarantees difficult. It introduces a matrix S-lemma with strict and non-strict forms, then derives data-based LMI methods for stabilization and H2/H∞ control. The resulting designs provide necessary and sufficient conditions, are computationally tractable, and use decision variables independent of the experiment horizon.

  • Problem

    Finite noisy data create an open problem of obtaining non-conservative control-design strategies that guarantee stability and performance for unknown systems.

  • Method

    The paper generalizes the classical S-lemma to matrix variables and formulates data-driven control as implications between quadratic matrix inequalities converted into data-guided LMIs.

  • Results

    The paper derives necessary and sufficient, non-conservative LMI-based methods for quadratic stabilization, H2 control, and H∞ control from noisy data.

  • Takeaways & Limitations

    The designs are tractable with modern LMI solvers, and their decision variables are independent of the experiment time horizon.

Abstract

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We propose a new method to obtain feedback controllers of an unknown dynamical system directly from noisy input/state data. The key ingredient of our design is a new matrix S-lemma that will be proven in this paper. We provide both strict and non-strict versions of this S-lemma, that are of interest in their own right. Thereafter, we will apply these results to data-driven control. In particular, we will derive non-conservative design methods for quadratic stabilization, H_2 and H_inf control, all in terms of data-based linear matrix inequalities. In contrast to previous work, the dimensions of our decision variables are independent of the time horizon of the experiment. Our approach thus enables control design from large data sets.

I. INTRODUCTION

The paper develops direct feedback-controller design from noisy data, addressing the open problem of non-conservative guarantees from finitely many samples. It extends the S-lemma to matrix variables and applies it to tractable data-driven control.

  • The paper targets robust control laws for unknown dynamical systems using noisy input/state data, without an intermediate system-identification step.
  • Review of the S-lemma: A new matrix S-lemma provides necessary and sufficient conditions for deciding implications between quadratic matrix inequalities.The paper gives strict and non-strict versions and uses the result as its central technical ingredient.
  • Control applications: The resulting data-guided LMI designs cover quadratic stabilization, H2 control, and H∞ control while remaining computationally tractable.The approach formulates data-driven control through quadratic matrix inequalities and their LMI equivalents.
  • Noise model: The method assumes general bounded disturbances rather than noise statistics and can represent energy, covariance, norm, and subspace constraints on noise.The noise model is expressed through a quadratic matrix inequality; negative definiteness of Φ22 ensures bounded admissible noise matrices.
  • Computational scope: Decision-variable dimensions are independent of the experiment time horizon, making the approach applicable to large data sets.This is presented as an advantage over closed-loop system parameterizations that become computationally intractable for big data.

B. Problem formulation

The problem formulation collects finite measured state and input data and considers every system consistent with those data and an assumed noise bound. The goal is to determine when the data support a common quadratic stabilizing controller for that uncertainty set.

  • Problem formulation: The explanation set Σ contains all system matrices (A, B) that fit the measured data for some noise matrix satisfying the assumed bound.It is defined as Σ := {(A, B) | (7) holds for some W− satisfying (5)}.
  • Informativity: Data are informative for quadratic stabilization when one feedback gain K and one positive-definite matrix P certify stabilization for every (A, B) ∈Σ.The paper specifically requires a common Lyapunov matrix across the uncertainty set.
  • Informativity: The informativity problem asks for necessary and sufficient conditions under which the measured data enable such a stabilizing controller.
  • Controller design: The associated design problem seeks a feedback that stabilizes all systems in Σ, thereby guaranteeing stability of the unknown true system.The paper also considers extensions with performance specifications, including H2 and H∞ control.

C. Our approach

The approach characterizes systems explaining noisy data through a quadratic matrix inequality, then reduces stabilization design to deciding when one quadratic matrix inequality implies another using a matrix-valued S-lemma.

  • Data-driven formulation: The unknown systems explaining the data are equivalently characterized by a quadratic matrix inequality in A and B.
  • Data-driven formulation: Quadratic stabilization requires determining when a design quadratic matrix inequality holds for every system satisfying the data-consistency inequality.
  • Data-driven formulation: The central question is when one quadratic matrix inequality implies another.
  • Matrix-valued S-lemma: This paper generalizes the S-lemma from vector variables to matrix variables, providing the theoretical tool for the data-driven control formulation.
  • Matrix-valued S-lemma: The classical S-lemma converts implication between quadratic inequalities into feasibility of a linear matrix inequality in a nonnegative scalar multiplier.

B. S-lemma with matrix variables

The paper develops homogeneous, non-strict, and strict S-lemmas for quadratic matrix inequalities, establishing multiplier-based equivalent conditions under generalized Slater-type assumptions.

  • Homogeneous matrix S-lemma: The homogeneous matrix S-lemma gives equivalent conditions for nonnegativity of X⊤MX whenever X⊤NX is positive semidefinite or positive definite.
  • Non-strict matrix S-lemma: Under the generalized Slater condition, the non-strict matrix implication is equivalent to existence of α ≥ 0 satisfying M − αN ≥ 0.
  • Assumptions: The generalized Slater condition is necessary for the full equivalence, although the equivalence between statements (i) and (iii) can use the standard Slater condition.
  • General matrix S-lemma: The general matrix S-lemma extends the classical result to quadratic functions of matrix variables and recovers the vector-variable case when k = 1.
  • Strict matrix S-lemma: The strict matrix S-lemma handles strict inequalities involving M when the feasible set SN is bounded and a generalized Slater condition holds.
  • Strict matrix S-lemma: Under additional matrix-structure assumptions, boundedness can be removed while the strict implication remains equivalent to existence of α ≥ 0 and β > 0.

IV. DATA-DRIVEN STABILIZATION REVISITED

The matrix S-lemma yields a necessary-and-sufficient LMI characterization for obtaining quadratically stabilizing feedback controllers directly from noisy data.

  • Problem formulation: The stabilization problem asks whether the closed-loop condition holds for every system in the data-consistent set Σ.
  • Necessary-and-sufficient design: Theorem 14 characterizes informative data for quadratic stabilization by feasibility of an LMI in P, L, α, and β.
  • Controller recovery: If P and L satisfy the stabilization LMI, K := LP⁻¹ stabilizes every system in Σ.
  • Assumptions: The generalized Slater condition is satisfied whenever N has at least n positive eigenvalues, and the stated theorem assumes this condition.
  • Design features: The resulting procedure is non-conservative because the LMI condition is necessary and sufficient for obtaining quadratically stabilizing controllers from data.
  • Design features: The decision variables and LMI dimension are independent of the experiment time horizon T, supporting design from large data sets.

V. INCLUSION OF PERFORMANCE SPECIFICATIONS

The stabilization framework is extended to performance specifications by treating H_2 and H_∞ control.

  • Performance specifications: The paper extends data-driven stabilization to H_2 and H_∞ control to illustrate the general applicability of the matrix S-lemma theory.

A. H2 control

The H2 design characterizes informativity through data-based LMIs whose feasible variables yield a controller guaranteeing stability and H2 performance for every system consistent with the data.

  • H2 control: The H2 requirement uses a common positive-definite matrix P for all systems in Σ, together with the trace bound tr P < γ2.The common-matrix restriction converts the robust performance requirement into a single data-driven design condition.
  • H2 control: The derivation transforms the performance inequalities using Y := P −1 and L := KY, then applies Schur complements and the matrix S-lemma theorem.The converse direction reconstructs Y, L and Z from a valid common P and controller K.
  • H2 control: The data are informative for H2 control with performance γ exactly when matrices Y, Z, L and scalars α, β satisfy (H2).The equivalence assumes the generalized Slater condition for N.
  • H2 control: A feasible pair Y and L produces the feedback gain K := LY −1, guaranteeing stability and ∥G(z)∥H2 < γ for all (A, B) ∈Σ.The guarantee applies to every system explaining the measured data.
  • H2 control: Prior knowledge that disturbances lie in im E can be incorporated by modifying the LMI involving Z, although experimental noise and attenuated disturbances need not share a subspace.The same E may be used in the noise model and H2 LMI, but this is not required.

B. H∞control

For H∞ control, the paper formulates informativity as robust stability and performance over all data-consistent systems, and characterizes it by data-based LMIs.

  • H∞ control: The H∞ formulation uses the same transformed variables Y := P −1 and L := KY as the H2 design.The transformed inequalities again place the unknown system matrices in a form suitable for the paper’s matrix S-lemma.
  • H∞ control: The data are informative for H∞ control with performance γ exactly when Y, L, α and β satisfy the H∞ LMIs.The equivalence assumes the generalized Slater condition for N.
  • H∞ control: A feasible Y and L produce K := LY −1, guaranteeing A + BK stability and ∥G(z)∥H∞ < γ for every (A, B) ∈Σ.The controller guarantee covers all systems explaining the data.
  • H∞ control: The proof of the H∞ characterization follows the same steps as the H2 proof and relies on Theorem 13.The paper does not report the proof in detail.

VI. EXAMPLES

This section illustrates the paper’s theoretical results with examples and numerical simulations.

  • VI. EXAMPLES: The paper includes examples to illustrate its theoretical results.The section introduces the examples and simulations without specifying their outcomes in this passage.
  • VI. EXAMPLES: The section also includes numerical simulations.No simulation setup or quantitative result is given in the supplied passage.
  • VI. EXAMPLES: The examples and simulations serve as the section’s stated means of illustrating the theoretical results.This restates the explicit purpose of the section without adding an outcome.

A. Stabilization using bounds on the noise samples

The stabilization experiment evaluates data-driven controller synthesis under bounded noise across six noise levels, finding that feasibility declines as noise increases but remains substantial at the largest tested level.

  • A. Stabilization using bounds on the noise samples: The noise samples are bounded by ∥w(t)∥2 ≤ ε and drawn uniformly from the corresponding Euclidean ball.This prior knowledge is encoded with Φ11 = TεI, Φ12 = 0 and Φ22 = −I.
  • A. Stabilization using bounds on the noise samples: Six noise levels ε ∈ {0.5, 1, 1.5, 2, 2.2, 2.4} are tested using 100 randomly generated data sets per level.The experiment uses a time horizon T = 20 and checks the generalized Slater condition for every data set.
  • A. Stabilization using bounds on the noise samples: Increasing the noise level decreases the percentage of data sets for which the LMI (FS) is feasible.The paper attributes this to enlargement of the set of explaining systems Σ, making simultaneous stabilization harder.

B. H2 control of a fighter aircraft

The fighter-aircraft experiment shows that noisy-data H2 controllers can approach model-based performance with sufficient data, while higher noise can degrade performance or make design infeasible. The comparison also demonstrates stabilization guarantees unavailable from the related procedures in this example.

  • H2 controller design: The unstable fighter-aircraft model is discretized and used to collect 750 noisy input and state samples for data-driven H2 control.Inputs, initial state, and noise are sampled from Gaussian distributions; the experiment assumes a known energy bound on the noise.
  • H2 controller design: The model-based H2 benchmark is γ = 1.000, while the data-driven controller achieves γ_s^2 = 1.007 on the true system.The data-driven result is described as almost identical to the smallest possible H2 norm.
  • Effect of data volume: Using only the first i samples, controller performance is poor for i < 500 but becomes close to the true system’s optimal performance from i = 500 onward.The experiment repeats controller synthesis for i = 50, 100, ..., 750 samples and reports the results in Figure 1.
  • Effect of noise: With σ = 0.5, the H2 controller achieves γ_s^2 = 3.579, while σ = 1 makes the H2 LMI infeasible for every γ.Increasing noise variance enlarges the set of explaining systems, making it harder to control all systems in that set.
  • Effect of noise: A tighter noise bound at σ = 0.5 improves performance to γ_s^2 = 2.706, showing that the controller depends on prior noise knowledge as well as the design strategy.The tighter bound is also satisfied by the example’s noise sequence.
  • Comparison with related results: For a three-sample scalar example, Theorem 14 yields K = L/P = −1.5, which stabilizes every explaining system and gives true closed-loop matrix −0.5.The selected solution is (P, L, α, β) = (0.9, −1.35, 1.1, 0.18), and the Slater condition holds.
  • Comparison with related results: The compared procedures in and have no feasible solutions for this example, whereas Theorem 14 produces a controller guaranteed to stabilize the true system.The infeasibility arguments are established separately for the formulations associated with and.

VII. DISCUSSION AND CONCLUSIONS

The paper recasts noisy-data control as a quadratic matrix-inequality implication and resolves it with a matrix S-lemma. This yields non-conservative, tractable LMI designs for stabilization and H2/H∞ control, while several extensions remain future work.

  • Main contribution: The matrix S-lemma converts implication between quadratic matrix inequalities into feasibility of a linear matrix inequality in a scalar variable.This generalizes the classical vector-variable S-lemma to matrix variables.
  • Main contribution: Necessary and sufficient noisy-data conditions are provided for obtaining stabilizing, H2, and H∞ controllers through data-guided LMIs solvable by modern LMI solvers.The resulting control design is described as non-conservative and computationally tractable.
  • Scalability: The decision variables are independent of the experiment’s time horizon, making the approach applicable to large data sets.This property is identified as an attractive feature alongside non-conservatism.
  • Scope and future work: The paper applies only the strict-inequality matrix S-lemma to data-driven control, leaving the non-strict version for possible applications such as dissipativity verification.The non-strict version is identified as potentially useful but is not applied here.
  • Scope and future work: More specialized techniques for norm-bounded noise may be possible, and the authors defer a detailed treatment of that setting to future work.The current noise model can describe norm bounds, among other noise descriptions.
  • Scope and future work: Extending the state-feedback results to data-driven dynamic output feedback from finite noisy input/output samples is another proposed direction.
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