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Formulas for Data-driven Control: Stabilization, Optimality and Robustness

Claudio De Persis, Pietro Tesi

arXiv:1903.06842v3eess.SYmath.OC

TL;DR

The paper asks how linear control systems can be designed directly from data without identifying explicit system matrices. It uses persistently exciting data to parametrize feedback systems and formulate data-dependent LMIs, obtaining stabilization and regulation results while extending the approach to noisy data and nonlinear equilibria.

  • Problem

    The paper addresses how to design control systems directly from process data without an intermediate system-identification step.

  • Method

    The approach parametrizes linear feedback systems from persistently exciting data and formulates control designs as data-dependent Linear Matrix Inequalities.

  • Results

    The approach covers state- and output-feedback stabilization, linear quadratic regulation, noisy measurements, and local stabilization of nonlinear equilibria.

  • Takeaways & Limitations

    Persistently exciting data can replace explicit system matrices for several control-design problems, including stabilization and regulation.

Abstract

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In a paper by Willems and coauthors it was shown that persistently exciting data can be used to represent the input-output behavior of a linear system. Based on this fundamental result, we derive a parametrization of linear feedback systems that paves the way to solve important control problems using data-dependent Linear Matrix Inequalities only. The result is remarkable in that no explicit system's matrices identification is required. The examples of control problems we solve include the state and output feedback stabilization, and the linear quadratic regulation problem. We also discuss robustness to noise-corrupted measurements and show how the approach can be used to stabilize unstable equilibria of nonlinear systems.

I. INTRODUCTION

The paper addresses direct control design from process data without intermediate system identification. It develops data-based feedback parametrizations and applies them to LMI-based stabilization and regulation problems.

  • Data-driven control seeks to design control systems directly from process data with no intermediate system identification.
  • Willems et al.’s fundamental result represents all trajectories of a linear system through finitely many sufficiently excited system trajectories.
  • The paper revisits this result using classic state-space descriptions and derives data-dependent representations of open-loop and closed-loop dynamics.
  • The closed-loop parametrization represents the feedback system transition matrix while parametrizing the controller itself through data.
  • The resulting framework casts state-feedback stabilization as a data-dependent Linear Matrix Inequality and solves linear quadratic regulation through convex optimization.
  • Notation: The notation defines finite signal restrictions and Hankel matrices for organizing experimentally collected input, output, and state samples.

A. Persistently exciting data and the fundamental lemma

Persistently exciting inputs make collected data rich enough to represent every finite input/output trajectory of the linear system. This fundamental lemma supports data-based open-loop and feedback representations without requiring a parametric system model.

  • The paper imposes a rank condition on collected data because it ensures that the data encode all information needed for direct control-law design.
  • An input is persistently exciting of order L when its associated data matrix has the required rank, with signal length T ≥ (σ + 1)L − 1.
  • If the input is persistently exciting of order n + t, the collected data satisfy the rank condition required for trajectory representation.
  • Any t-long input/output trajectory of the system can then be expressed as a linear combination of collected input/output data.
  • This representation replaces a parametric system description with data and enables data-dependent representations of open-loop and closed-loop dynamics.
  • The required excitation can be enforced by applying an input signal of sufficiently high order, although the condition is not directly assessable using only input/output data.
  • Theorem 1 gives an equivalent data-based system representation, interpretable as the solution of a least-squares problem and connected to Dynamic Mode Decomposition.

B. Data-based closed-loop representation

Theorem 2 parametrizes closed-loop state-feedback dynamics directly from data, allowing stabilization tests and controller design without identifying a parametric system model.

  • Data-based representation: Theorem 2 gives an equivalent data-based representation of a state-feedback closed-loop system.It uses data-dependent matrices to represent the closed-loop dynamics under u = Kx.
  • Data-based representation: The controller K can be selected through a matrix GK satisfying X0,T GK = I_n, with X1,T GK representing A + BK.The resulting feedback matrix is K = U0,1,T GK.
  • Identification-free design: Theorem 2 enables identification-free stabilization design by treating GK as a decision variable subject to stability and performance requirements.This avoids reconstructing the system matrices A and B before controller design.
  • Identification-free design: A candidate controller can be tested without placing it in feedback by computing the eigenvalues of X1,T GK.Any solution GK to the data constraint yields X1,T GK = A + BK.
  • LMI-based stabilization: State-feedback stabilization is formulated as a data-dependent LMI, and every stabilizing gain can be represented through a feasible Q.The gain is recovered as K = U0,1,T Q(X0,T Q)^−1.
  • Continuous-time extension: The continuous-time formulation requires derivative samples, which may necessitate filters because derivatives are usually unavailable.The resulting gain stabilizes the continuous-time system rather than only its sampled-data model.

B. Linear quadratic regulation

The data-based parametrization extends to linear quadratic regulation by converting the optimal state-feedback problem into a data-dependent convex optimization. The paper also discusses data requirements and numerical boundaries for unstable systems.

  • LQR formulation: The H2 state-feedback problem designs u = Kx to stabilize A + BK while minimizing the H2 norm from external input ξ to performance output z.The objective uses weighting matrices Qx ⪰ 0 and R ≻ 0, with (Qx, A) observable.
  • Data-dependent optimization: The data-dependent formulation is obtained by applying suitable variable changes to the standard convex formulation of the regulation problem.The paper relates equivalent formulations through variables such as GK, W, and X.
  • Data-dependent optimization: The optimal controller is computed directly from data as K = U0,1,T Q(X0,T Q)−1 through a convex optimization problem.Theorem 4 states that Q optimizes a trace-based data-dependent formulation.
  • Illustrative example: For the batch-reactor example, the data-driven controller coincides with the controller obtained using MATLAB’s dare command.The experiment uses random data of length T = 15 with Qx = I_n and R = I_m.
  • Data requirements and boundaries: The method requires only T = (m+1)n+m open-loop samples when a persistently exciting input of order n + 1 ensures the data condition.This gives an a priori duration for open-loop data collection and supports moderate-size systems that are not strongly unstable.
  • Data requirements and boundaries: Large-scale and highly unstable systems create greater numerical difficulty, for which closed-loop experiments with added persistent excitation may be needed.A known stabilizing controller can be combined with a persistently exciting signal to collect data while preserving the results’ assumptions.

V. ROBUSTNESS: NOISE-CORRUPTED DATA AND NONLINEAR SYSTEMS

The paper extends LMI-based, data-driven stabilization to noisy measurements by replacing noise-free data with measured data and imposing robustness conditions. Under explicit assumptions, the resulting controller is stabilizing, while simulations show the theoretical noise bound can be conservative.

  • Stabilization with noisy data: Noisy state measurements are represented by Z0,T := X0,T + W0,T and Z1,T := X1,T + W1,T, replacing the noise-free data matrices in the stabilization design.The available data contain the system-state measurements perturbed by noise.
  • Stabilization with noisy data: The noisy-data formulation searches for Q and α > 0 through LMI constraints, then computes K = U0,1,TQ(Z0,TQ)−1.The equality constraint can be enforced after solving the inequality constraints.
  • Stabilization with noisy data: Assumptions 1 and 2 require that noise does not cause substantial information loss and that the data satisfy a signal-to-noise-ratio condition.Assumption 1 concerns rank and information content; Assumption 2 restricts the effect of noise through γ.
  • Stabilization with noisy data: Any solution to (35) returns a stabilizing controller when γ < α2/(4+2α), provided Assumptions 1 and 2 hold.The controller is K = U0,1,T Q(Z0,T Q)−1.
  • Illustrative example: In simulations, controllers were systematically stabilizing for noise of order 10−2 and stabilized more than half of cases for noise of order 10−1.The theoretical condition was indicated as satisfied for noise of order 10−4, showing conservatism of the sufficient bound.
  • Limitations: Checking the sufficient robustness condition requires prior knowledge of an upper bound on R0,T, which depends on bounds related to the noise and the system matrix A.The paper notes that the direction of noise can matter in addition to its magnitude.

B. Stabilization of nonlinear systems

The paper extends data-driven robust stabilization to unstable equilibria of nonlinear systems by treating higher-order nonlinear terms as small disturbances near equilibrium. Under rank and small-disturbance conditions, the resulting data-based controller locally stabilizes the equilibrium, with an inverted-pendulum example illustrating the approach.

  • Nonlinear-system representation: A nonlinear system near an equilibrium is represented by its linearization plus higher-order terms that vanish faster than state and input deviations.The higher-order remainder can be treated as a small process disturbance when experiments remain sufficiently close to equilibrium.
  • Data-based stabilization: The approach provides sufficient conditions for designing a stabilizing gain directly from data when the linearized pair is stabilizable.A controller stabilizing the linearization also exponentially stabilizes the original nonlinear equilibrium under the stated local conditions.
  • Data-based stabilization: The assumptions require full row rank and an experiment sufficiently close to equilibrium so nonlinear disturbances remain small relative to state deviations.These assumptions parallel the noisy-data conditions used for robust stabilization.
  • Illustrative example: In the inverted-pendulum example, random data and CVX produce a controller that stabilizes the unstable upright equilibrium in agreement with Theorem 6.With α = 0.0422, the condition holds because γ is approximately 10^-6 while α^2/(4+2α) is approximately 10^-4.
  • Illustrative example: The method also returned stabilizing controllers for initial conditions and inputs in [−0.5, 0.5], although the sufficient inequality did not always hold.This interval corresponds to an initial displacement of about 28 degrees from equilibrium.

VI. INPUT-OUTPUT DATA: THE CASE OF SISO SYSTEMS

For SISO systems with only input-output measurements, the paper converts an input-output representation into a nonminimal state-space realization using lagged outputs and inputs. Persistently exciting data and a full-rank condition then support a data-based representation of the system.

  • SISO realization: A SISO system is written in left difference-operator form, relating current output to past outputs and inputs through coefficients a_i and b_i.The representation corresponds to the paper’s state-space model when the direct term D is zero.
  • SISO realization: The constructed state χ(k) stacks n past outputs followed by n past inputs, producing a state-space system of order 2n.This realization is not minimal but permits the state-measurement arguments to be reused.
  • Data conditions: The data matrix must be full row rank, with experiment length T satisfying T ≥ 2n + 1 for the lifted realization.The rank condition is the analogue of the condition used for systems with measured states.
  • Data conditions: Persistently exciting input data of order 2n + 1, together with coprime system polynomials, ensures controllability of the lifted model and enables the data representation.The argument applies the fundamental lemma to the 2n-dimensional realization.

A. Data-based open-loop representation

The paper shows that a full-rank input-state data matrix lets any compatible input-state pair be expressed through collected data, yielding an equivalent open-loop representation based only on measurements. This representation supports analysis and output-feedback design without identifying a parametric model.

  • Data-based representation: The full-rank property enables the system to be expressed through data for any input-state pair.The coefficient vector g can be solved from the data equation and substituted into the trajectory representation.
  • Data-based representation: Theorem 7 gives an equivalent representation of the lifted SISO system under the full-rank condition.The representation uses the measured input-output data encoded in the lifted state.
  • Data-based representation: A minimal order-n representation can be extracted from the lifted model, which depends only on measured input-output data.The extracted model is then used for analysis and output-feedback design without identifying a parametric system model.

B. Design of output feedback controllers

The paper parametrizes output-feedback controllers directly from input-output data by working with a lifted state of past inputs and outputs. Under a full-rank condition, the parametrization characterizes stabilizing controllers and produces a controller realization, illustrated on a coupled-cart system.

  • Controller parametrization: The output-feedback controller is defined for the lifted state containing past outputs and inputs, and stabilization means asymptotic stability of the resulting closed-loop system.The lifted representation allows output-feedback design to reuse the state-feedback construction.
  • Controller parametrization: Under condition (61), the closed-loop system has an equivalent data-based representation parametrized by a matrix associated with the controller coefficients.Theorem 8 establishes the closed-loop representation and its relation to the controller realization.
  • Controller parametrization: Any matrix Q satisfying the stated data-based condition yields controller coefficients that stabilize the SISO system, and every stabilizing controller has this form.This gives a characterization of the whole set of stabilizing controllers within the chosen controller structure.
  • Controller realization: The controller coefficients can be arranged into a state-space realization of order n.The realization is obtained from the coefficient ordering specified after Theorem 8.
  • Scope and assumptions: The method requires knowledge of the system order n, which may come from first principles or be estimated from data using subspace identification.Estimating the order is separate from the remaining algorithmic procedure.
  • Illustrative example: In the coupled-cart example, a controller designed from a length-T = 9 random experiment stabilizes the closed-loop dynamics.The example considers uncertain spring stiffness, while the design assumes γ = 1 unknown to the controller designer.

C. A remark on the case of MIMO systems

For MIMO systems, the paper extends the data-driven feedback parametrization under a persistent-excitation condition. The resulting controller characterization is equivalent to stabilization and uses data-dependent matrices.

  • MIMO representation: The MIMO input-output dynamics are represented by a difference equation with matrix coefficients A_i and B_i.The input and output dimensions are m and p, respectively.
  • MIMO representation: Data must be collected with an input sequence persistently exciting of order (m + p)n + 1.The stated data-length condition is T ≥ ((m + p)n + 1)(m + 1).
  • Feedback parametrization: Under the required condition, the analysis used for the preceding feedback result extends to the MIMO setting.The section explicitly states that the same analysis can be repeated.
  • Stabilization: Any matrix Q satisfying condition (74) produces a controller that stabilizes the MIMO system.The controller coefficients are given by the corresponding parametrization.
  • Stabilization: Conversely, every controller of the stated form that stabilizes the MIMO system can be represented through coefficients derived from a solution Q to (74).This establishes a converse characterization of stabilizing controllers.
  • Implications: Persistently exciting data yield data-dependent matrices that replace system models and reduce stabilization to an equivalent data-dependent LMI.The paper identifies this parametrization as a basis for further data-driven control problems.

APPENDIX

The appendix establishes the trajectory representation used by the data-driven framework. It shows how coefficients multiplying collected data generate system-consistent input-output trajectories.

  • Trajectory representation: A rank condition, together with the Rouché–Capelli theorem, guarantees existence of a vector g solving the required linear equations.The proof then substitutes this representation into the system relation.
  • State-space construction: The state-space construction uses shift-like identity blocks and the system matrices −A_i and B_i to encode the dynamics.The displayed block matrices organize delayed outputs and inputs.
  • Trajectory representation: For any compatible coefficient vector g, U0,t,T−t+1g represents a t-long input sequence of the system.The corresponding output is Y0,t,T−t+1g = O_tx0 + T_tu[0,t−1].
  • Solution structure: The appendix also characterizes the solutions of the relevant equations as an infinite family parameterized by g.The orthogonal projector onto the kernel of S describes the associated freedom in the solution.

C. Proof of Corollary 1

The proof derives Corollary 1 by transforming matrix inequalities and bounding cross terms. Parameter relationships then establish the assumptions needed to transfer the preceding theorem’s result.

  • Proof strategy: The proof begins from Assumption 3 and shows that it implies Assumption 2 with an appropriate choice of γ.This is identified as the central proof strategy.
  • Matrix transformation: Pre- and post-multiplication by [B A] and [B A]⊤ transforms condition (44) into an inequality involving the compact matrix V0,T.The proof defines V0,T := AZ0,T + BU0,1,T.
  • Parameter selection: The parameter γ1 must lie in (0, 0.5), and each such γ1 uniquely determines a positive δ1.This relationship is used to rewrite condition (89).
  • Inequality bounding: A standard matrix inequality for X, Y, F with F ≻ 0 bounds the cross terms after setting ε = 0.5, X = V0,T, F = I, and Y = AW0,T.The resulting δ1 and δ2 terms are used to verify the target condition.
  • Conclusion: The proof combines the transformed inequalities with the preceding relations to verify Assumption 2 and complete the corollary’s claim.The final verification uses the stated γ and the bounds involving W1,T, Z1,T, and X1,T.
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