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Data informativity: a new perspective on data-driven analysis and control

Henk J. van Waarde, Jaap Eising, Harry L. Trentelman, M. Kanat Camlibel

arXiv:1908.00468v3math.OCmath.DS

TL;DR

Data-driven analysis and control often rely on persistently exciting data, which can also uniquely identify the underlying system. This paper develops a general informativity framework for non-persistently-exciting data and derives necessary and sufficient conditions for analysis and control. It finds that some problems are solvable without system-identifying data, while LQR requires informativity for system identification.

  • Problem

    Existing data-driven analysis and control approaches use persistently exciting data, leaving open whether controllers can be obtained when the data do not uniquely identify the system.

  • Method

    The paper introduces a general informativity framework and derives necessary and sufficient conditions for several analysis and data-driven control problems.

  • Results

    For many analysis and control problems, including controllability analysis and stabilization, data need not be informative for system identification; for LQR, such informativity is necessary.

  • Takeaways & Limitations

    Data-driven analysis and control can be possible with weaker information than unique system identification, but persistently exciting data are theoretically justified for LQR.

Abstract

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The use of persistently exciting data has recently been popularized in the context of data-driven analysis and control. Such data have been used to assess system theoretic properties and to construct control laws, without using a system model. Persistency of excitation is a strong condition that also allows unique identification of the underlying dynamical system from the data within a given model class. In this paper, we develop a new framework in order to work with data that are not necessarily persistently exciting. Within this framework, we investigate necessary and sufficient conditions on the informativity of data for several data-driven analysis and control problems. For certain analysis and design problems, our results reveal that persistency of excitation is not necessary. In fact, in these cases data-driven analysis/control is possible while the combination of (unique) system identification and model-based control is not. For certain other control problems, our results justify the use of persistently exciting data as data-driven control is possible only with data that are informative for system identification.

I. INTRODUCTION

The paper develops data informativity as a framework for data-driven analysis and control without explicitly identifying a system model. It distinguishes problems solvable with non-identifying data from those requiring data informative for system identification.

  • Model-based control typically follows system identification because the system model is rarely known a priori.This creates a two-step procedure: identify the system from measured data, then design the controller.
  • Data-driven control instead constructs controllers directly from data, potentially when system identification is difficult or impossible.The approach avoids explicitly identifying a model and can operate when the data provide insufficient information for identification.
  • The fundamental lemma links persistently exciting trajectories to generation of all possible trajectories and implicit system representation through Hankel matrices.It also supplies conditions under which data uniquely identify the system model within a given model class.
  • The paper introduces a general informativity framework with necessary and sufficient conditions for system-theoretic analysis and several data-driven control problems.Studied control problems include state-feedback stabilization, dynamic measurement feedback, deadbeat control, and linear quadratic regulation.
  • For stabilization and related problems, data-driven analysis or control can use data that are not informative for system identification, whereas LQR informativity is essentially the same as system identification.This provides a theoretical justification for persistently exciting data in LQR while identifying cases where such excitation is unnecessary.

II. PROBLEM FORMULATION

The framework characterizes what can be inferred or guaranteed from the set of systems consistent with measured data. Informativity requires properties or controller objectives to hold for every system that remains compatible with those data.

  • The unknown true system belongs to a model class, and the paper uses data to assess its properties and design controllers.The true system is not known, but the data are generated by it.
  • The consistency set ΣD contains all systems in the model class that could have generated the data.This set is the basis for both data-driven analysis and control guarantees.
  • For analysis, data are informative for property P exactly when every system consistent with the data has P.Formally, informativity is expressed as ΣD ⊆ ΣP.
  • For stabilizability, the condition means that all systems compatible with the input/state measurements are stabilizable.The example states this as Σ(U−,X) ⊆ Σstab.
  • For control, informativity requires existence of a controller whose specified closed-loop property holds for every system in ΣD.The associated problems separate deciding whether such a controller exists from finding one under that assumption.
  • The framework assumes that the given data are not corrupted by noise.

III. DATA-DRIVEN ANALYSIS

The paper develops data-driven analysis of controllability, stabilizability, and stability using sets of systems consistent with input/state or state measurements. It gives necessary and sufficient informativity conditions and shows that some properties can be certified without uniquely identifying the true system, whereas stability requires unique identification.

  • Framework: Data-driven analysis uses input/state measurements and the set of all systems compatible with those data.The compatible-system set captures every model that could have generated the measurements.
  • Informativity: The data are informative for controllability or stabilizability when every consistent system has the corresponding property.For input/state data, these notions are formalized by requiring the consistent-system set to lie within the controllable or stabilizable systems.
  • Controllability and stabilizability: Theorem 8 provides necessary and sufficient data-matrix conditions for informativity in controllability and stabilizability.The conditions are data-driven Hautus tests and can be checked using ranks for finitely many complex values of λ.
  • Controllability and stabilizability: Controllability and stabilizability can be inferred even when the true system cannot be uniquely identified.An example concludes that all systems explaining the data are controllable although the data are not informative for system identification.
  • Stability: For autonomous-system data, stability can be concluded only when the true system is uniquely identified.The stability characterization requires full row rank of X− and yields the unique compatible matrix As = X+X†.
  • Stability: The stability characterization differs because it concerns data generated by an autonomous system, whereas the preceding characterization assumes inputs.This distinction separates the data settings underlying the two results.

IV. CONTROL USING INPUT AND STATE DATA

The paper studies state-feedback control problems from input/state measurements, beginning with data-driven stabilization and then considering deadbeat control and linear quadratic regulation. It separates informativity and controller-design questions within this framework.

  • State-feedback control: The control section studies static state-feedback stabilization using input/state measurements and treats informativity and controller design separately.The same techniques are then applied to deadbeat control.
  • Linear quadratic regulation: The section later considers linear quadratic regulation, seeking a stabilizing feedback that also minimizes a specified quadratic cost.The LQR problem adds a cost-minimization objective to stabilization.

A. Stabilization by state feedback

The paper characterizes when input/state data are informative for stabilization by state feedback and provides controllers directly from those data. It also shows that stabilization can be possible without unique system identification, while stabilizability alone does not guarantee a common stabilizing controller.

  • Informativity for stabilization: Data are informative for stabilization when one feedback gain stabilizes every system consistent with the measurements.This common-gain requirement distinguishes stabilization from merely establishing that each compatible system is stabilizable.
  • Stabilizability versus stabilization: A scalar example shows that all compatible systems may be stabilizable while no single state-feedback gain stabilizes them all.The systems (−1, 1) and (1, 1) cannot be stabilized by the same controller u(t) = Kx(t).
  • Necessary and sufficient conditions: Theorem 16 requires X− to have full row rank and a right inverse X† such that X+X† is stable.Every stabilizing gain is then given by K = U−X†.
  • Controller design: Theorem 17 reformulates the stabilization conditions as feasible linear matrix inequalities and yields K = U−Θ(X−Θ)−1.The LMI feasibility can be checked using standard methods, and these conditions do not require persistently exciting inputs.
  • Deadbeat control: For deadbeat control, informativity requires full row rank of X− and a suitable right inverse, with K := U−X† as the resulting feedback gain.Deadbeat control means the closed-loop state reaches zero in finite time.

B. Informativity for linear quadratic regulation

The paper studies whether input/state data can determine an optimal LQR controller without uniquely identifying the system. It finds that, apart from a pathological stable zero-cost case, LQR informativity essentially requires system-identifying data.

  • LQR formulation: The LQR problem seeks an input that drives the state to zero while minimizing the quadratic cost functional.Solvability requires finite cost, convergence to zero, and optimality for every initial condition.
  • Known-system LQR: For a known system, LQR solvability is equivalent to stabilizability and observability of every unit-circle eigenvalue through (Q, A).Under these conditions, the optimal input is generated by a stabilizing state-feedback law.
  • Informativity conditions: Theorem 26 states that data are informative for LQR only in two cases: unique system identification with solvable LQR, or a pathological stable case with QA = 0 and K = 0.In the first case, the optimal gain is the usual Riccati-based feedback gain.
  • Implications for data richness: Except for the pathological case, LQR-informative data are also informative for system identification.This provides theoretical justification for persistently exciting data assumptions used in prior data-driven LQR work.
  • Data-based characterization: The paper derives an equivalent data-based characterization using matrices constructed from the measured trajectories and Riccati-related conditions.Theorem 28 gives conditions involving a matrix Θ and the data matrices X− and X+.

C. From data to LQ gain

This section develops a data-driven method for computing the optimal LQ feedback gain directly from input/state data. Under informativity, the value matrix is obtained uniquely from a data-only optimization problem, after which the gain is recovered from linear equations.

  • Method: The method computes the optimal feedback gain directly from data using Riccati-inequality ideas.The approach is designed to avoid first identifying a system model.
  • Main result: P+ is the unique solution of a data-only optimization problem, and the optimal gain can subsequently be found from linear equations.The theorem establishes both uniqueness of P+ and a constructive route to K.
  • Design procedure: The design procedure first solves the semidefinite program and then computes a right inverse of X− to obtain K = U−X†−.The right inverse satisfies the theorem's stated constraints.
  • Comparison: The data-driven formulation uses an optimization variable of dimension n×n, whereas an earlier formulation used variables of dimensions T×n and m×m.The paper notes that T is generally much larger than n.
  • Comparison: Unlike an approach whose Riccati approximation becomes exact only as data points tend to infinity, this method obtains P+ from the data.The comparison concerns the exactness and construction of the Riccati solution.

V. CONTROL USING INPUT AND OUTPUT DATA

This section considers stabilization by dynamic measurement feedback when input and output measurements are available. The controller uses measured output to update an internal state and generates the control input from that state.

  • Problem: The problem is to stabilize a system using input and output measurements only.The data are collected as input/output trajectories over a single time interval.
  • Controller: The dynamic controller updates its internal state through w(t + 1) = Kw(t) + Ly(t) and applies u(t) = Mw(t).The resulting closed-loop system must be stable.

A. Stabilization using input, state and output data

This section characterizes stabilization by dynamic measurement feedback from input, state, and output data. It gives necessary and sufficient informativity conditions and a constructive controller design, including a reduction when the input data lack full row rank.

  • Rank reduction: When U− lacks full row rank, the data can be reduced to an equivalent full-row-rank problem without changing informativity.The reduction factors U− as S ˆU− with ˆU− of full row rank.
  • Rank reduction: The reduced-data and original-data problems are equivalent for stabilization by dynamic measurement feedback.The corresponding controllers are transformed using S and a left inverse S†.
  • Necessary and sufficient conditions: The data are informative for stabilization by dynamic measurement feedback exactly when the theorem's stated conditions hold.Under full row rank of U−, Theorem 34 provides the necessary and sufficient characterization.
  • Controller design: The controller design selects M and L to stabilize two data-dependent matrices, then defines K := (X+ − LY−)(V1 + V2M).The construction uses stabilizability of (X+V1, X+V2) and detectability of (Y−V1, X+V1).
  • Identification requirement: Under full-row-rank input data, stabilization requires unique identification of the system matrices As, Bs, Cs, and Ds.The theorem's conditions force the consistent-system set to contain only the true system.
  • Implementation: The resulting controller is observer-based, with M and L computable by standard methods such as pole placement or LMIs.The construction stabilizes the system through state-feedback and observer-like components.

B. Stabilization using input and output data

This section studies stabilization by dynamic measurement feedback using input/output data alone. It provides sufficient conditions based on state reconstruction, while noting that the characterization is not yet necessary and sufficient.

  • Problem: The data are defined as informative when one dynamic controller stabilizes every system consistent with the input/output data.This is the paper's data-driven stabilization notion for measurement feedback.
  • State reconstruction: Under suitable Hankel-based conditions, the compatible state sequence can be reconstructed up to an unknown invertible similarity transformation.This converts input/output data into an input/state/output trajectory.
  • Sufficient condition: A sufficient condition combines a rank condition with informativity for stabilization using the reconstructed input/state/output data.A stabilizing controller can then be obtained by applying the corresponding input/state/output result.
  • Scope: The proposed sufficient conditions are not necessary, and data satisfying them are also informative for system identification.In that case, the consistent-system set contains only the true system and its similar representations.
  • Scope: No conclusive necessary-and-sufficient characterization is given, paralleling the absence of such conditions for subspace identification.The paper notes that several sufficient conditions are known for subspace identification.

VI. CONCLUSIONS AND FUTURE WORK

The paper introduces a general framework for data informativity and applies it to system-theoretic analysis and data-driven control. It shows that data-driven methods can outperform the identification-then-control procedure in some problems, while LQR requires identification-informative data.

  • These results distinguish data-driven approaches from the classical combination of system identification and model-based control.
  • The paper introduces a comprehensive framework for studying informativity in data-based system analysis and data-driven control.
  • For controllability analysis and stabilization, data-driven methods can use data that are not informative for system identification.
  • For data-driven linear quadratic regulation, the data must be informative for system identification.

Future work

Future work extends the framework to additional data types, system properties, model classes, disturbances, and experiment design. These directions include settings where system identification is difficult or where inputs must be selected to make data informative.

  • Future studies could examine input-output data, passivity, and robust control problems within the framework.
  • The model class could be generalized to differential algebraic or polynomial systems, or narrowed using structural or parametric prior knowledge.
  • Data-driven control under disturbances is a practical direction because disturbances make system identification less straightforward.
  • Existing work considers data-driven stabilization under measurement noise and unknown disturbances, while data-driven LQR often assumes Gaussian process noise.
  • Experiment design remains an open direction in which inputs are chosen so the resulting data become informative.
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