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Data-driven stabilization of nonlinear polynomial systems with noisy data

Meichen Guo, Claudio De Persis, Pietro Tesi

arXiv:2011.07833v1math.OCeess.SY

TL;DR

The paper addresses stabilization of unknown nonlinear polynomial systems from noisy data without requiring explicit model identification. It formulates data-dependent SOS stability certificates whose solutions provide stabilizing controllers and Lyapunov functions, then develops alternative parameterizations. The resulting designs support global asymptotic stabilization, reduce assumptions or computation in variants, and connect with least-square-based controller design.

  • Problem

    The paper addresses how to stabilize unknown nonlinear polynomial systems using noisy input-state data alone.

  • Method

    The method uses data-based closed-loop representations and SOS relaxations to jointly obtain a state-dependent stabilizing controller and Lyapunov function.

  • Results

    The proposed conditions provide globally asymptotically stabilizing controllers, while alternative parameterizations remove an assumption or improve computational efficiency.

  • Takeaways & Limitations

    Changing the data parameterization can improve stabilization-design efficiency even when the same underlying design method is used.

Abstract

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In a recent paper we have shown how to learn controllers for unknown linear systems using finite-sized noisy data by solving linear matrix inequalities. In this note we extend this approach to deal with unknown nonlinear polynomial systems by formulating stability certificates in the form of data-dependent sum of squares programs, whose solution directly provides a stabilizing controller and a Lyapunov function. We then derive variations of this result that lead to more advantageous controller designs. The results also reveal connections to the problem of designing a controller starting from a least-square estimate of the polynomial system.

I. INTRODUCTION

The paper develops direct data-driven stabilization for unknown nonlinear polynomial systems using noisy data, extending earlier linear-system results. Its main tools are data-dependent SOS conditions that yield stabilizing controllers and Lyapunov functions, with alternative parameterizations improving assumptions or computational efficiency.

  • Direct data-driven control synthesizes controllers from measured data without explicitly identifying a system model.
  • The paper extends noisy-data stabilization results from unknown linear systems to nonlinear polynomial systems.
  • The main design uses data-based closed-loop representations, matrix-valued Young’s inequality, and SOS relaxations to obtain stabilizers and Lyapunov functions.
  • Alternative closed-loop parameterizations produce stabilization results requiring fewer assumptions or offering improved computational efficiency.
  • The results connect data-driven controller synthesis with controller design based on least-square estimates of system dynamics.

II. PROBLEM FORMULATION

The problem formulation represents an unknown polynomial system in a linear-like monomial basis and designs a state-dependent feedback gain from noisy input-state data. A reduced polynomial basis supports the Lyapunov analysis and helps address SOS computational growth.

  • Unknown polynomial vector fields are expressed in a linear-like form using monomial vectors Z(x) and W(x) with unknown constant matrices.
  • Including all high-order monomials in Z(x) can make the SOS controller design computationally difficult or prevent a solution.
  • A smaller, lower-degree basis ˆZ(x) is introduced for controller design and stability analysis.
  • The Lyapunov function is chosen as V(x) = ˆZ(x)⊤P^-1ˆZ(x), with P positive definite and ˆZ(x) radially unbounded and zero only at the origin.
  • An offline experiment collects sampled input-state data, which are organized into Hankel matrices for the data-driven design.
  • The objective is to use noisy experimental input-state data alone to design a state-dependent gain yielding global asymptotic stability at the origin.

III. DATA-DRIVEN STABILIZATION WITH NOISY DATA

This section develops data-driven stabilization conditions for unknown polynomial systems using noisy data, under a structured noise assumption. It relates the assumption to prior linear-system results and previews nonlinear extensions and controller-design variants.

  • Arbitrary noise cannot generally support stabilization, motivating a structured assumption on the data disturbance.
  • The noise is constrained by the matrix inequality 0 ⪯ R_D D R_D^⊤ D for known R_D ∈ R^{n×T}.
  • With R_D = γX_1 for γ > 0, the assumption reduces to the signal-to-noise-ratio form used in prior data-driven control work.
  • The section extends noisy-data stabilization results from linear systems to nonlinear polynomial systems and introduces modified data-based closed-loop designs.

A. Data-driven stabilization with noisy data

The paper designs state-feedback stabilizers for unknown polynomial systems from noisy input-state data using SOS conditions and a Lyapunov function. The result establishes stability and, under strict positivity, global asymptotic stability while identifying data and rank requirements.

  • A. Data-driven stabilization with noisy data: The closed-loop representation uses measured data through the relation Z0Y(x)=H(x)P and rewrites the dynamics in terms of X1, disturbance data, and input terms.This parameterization connects the unknown system matrices to the data-based controller design.
  • A. Data-driven stabilization with noisy data: SOS relaxations replace generally intractable positivity conditions on matrix polynomials, enabling computational synthesis with tools such as SOSTOOLS.The SOS decision variables include P, the coefficients of Y(x), and optionally the certificate coefficients.
  • A. Data-driven stabilization with noisy data: If ϵ1(x)>0 for every x≠0, the resulting closed-loop system is globally asymptotically stable at the origin.Radial unboundedness of V(x)=Ẑ(x)ᵀP^-1Ẑ(x) supports the global conclusion.
  • A. Data-driven stabilization with noisy data: When the input vector field is independent of x, stabilization can be achieved without Assumption 2 on the unknown input matrix B.The special case admits a linear-like system representation and a corresponding data-based closed-loop expression.
  • A. Data-driven stabilization with noisy data: Feasibility imposes rank conditions on Z0, and practical instances may require full row rank, making the number of samples T sufficiently large.These requirements follow from the rank compatibility needed for Z0Y(x)=H(x)P.

B. Data-driven stabilization without bounds on the input matrix B

A second data parameterization removes the bound on B and yields the same stabilization guarantee under the noisy-data assumption. The paper also relates this design to least-squares estimation and introduces a formulation whose SOS condition is independent of the sample count.

  • B. Data-driven stabilization without bounds on the input matrix B: Theorem 2 stabilizes the polynomial system under Assumption 1 alone and guarantees global asymptotic stability when ϵ1(x)>0 for all x≠0.Its proof uses the alternative data-based closed-loop representation and an SOS condition involving ϵ2(x).
  • B. Data-driven stabilization without bounds on the input matrix B: Compared with Theorem 1, Theorem 2 removes Assumption 2 by grouping the unknown dynamics A and B into a single matrix S.A change of decision variable also makes the SOS condition more computationally efficient.
  • B. Data-driven stabilization without bounds on the input matrix B: A least-squares estimate S*=X1W0† provides an alternative model representation whose analogous stability result does not require Assumption 2.The estimate minimizes the Frobenius norm of the data residual, while its error depends on D0W0† under full-row-rank W0.
  • B. Data-driven stabilization without bounds on the input matrix B: The least-squares-based representation preserves the same stability result while relaxing the requirement concerning the unknown matrix B.This connects direct data-driven stabilization with model-based design using an estimated system.

C. Computationally more efficient stabilization conditions

The paper reformulates stabilization using a data parameterization whose SOS condition is independent of the number of samples, yielding a more computationally efficient design. The resulting polynomial controller and Lyapunov certificate preserve global asymptotic-stability guarantees under the stated conditions.

  • C. Computationally more efficient stabilization conditions: Corollary 1 searches for P ≻0, polynomial K(x), and SOS multipliers satisfying the data-dependent matrix condition, then constructs the state-feedback controller.The controller is expressed using K(x)P −1 ˆZ(x).
  • C. Computationally more efficient stabilization conditions: If ϵ1(x) > 0 for every x ≠ 0, the resulting closed-loop polynomial system is globally asymptotically stable.The corollary otherwise guarantees stabilization under its stated assumptions.
  • C. Computationally more efficient stabilization conditions: Theorem 2’s sample-dependent decision variable is replaced by K(x), making the SOS condition independent of T and reducing stabilization-design computation.This modification changes the closed-loop parameterization while retaining the SOS-based design framework.
  • C. Computationally more efficient stabilization conditions: The SOS formulation is a relaxation of a pointwise necessary-and-sufficient condition based on the Lyapunov function V(x) = ˆZ(x)⊤P −1 ˆZ(x).The pointwise matrix condition is converted into a tractable polynomial SOS requirement.
  • C. Computationally more efficient stabilization conditions: The approach extends noisy-data stabilization to nonlinear polynomial systems while using quadratic noise constraints and Lyapunov-based certificates.This connects the construction to prior quadratic-constraint and SOS approaches.

IV. EXAMPLE

The Van der Pol oscillator serves as a benchmark for the noisy-data stabilization method, with controllers designed by Theorems 1 and 2 and Corollary 1. All designs stabilize the oscillator at the origin, while Corollary 1 requires substantially less computation.

  • IV. EXAMPLE: The Van der Pol oscillator is used as a benchmark, with simulations conducted in MATLAB using SOSTOOLS.The experiment applies the data-driven control method to a standard nonlinear test system.
  • IV. EXAMPLE: The experiment uses noisy input-state data generated with u = sin t over t = 0 to 10, sampling period 0.5, T = 12, and D0 = 0.05X1.The polynomial basis contains monomials of degrees 1 to 3, while ˆZ(x) = [x1 x2]⊤.
  • IV. EXAMPLE: Corollary 1 designs a degree-2 state-dependent gain K(x) through an SOS program, providing an alternative to the Theorem 2 design.The example compares the resulting controller with those obtained from the theorem-based formulations.
  • IV. EXAMPLE: 146.5957s, 151.3056s, and 10.6662s are the computational times for Theorem 1, Theorem 2, and Corollary 1, respectively.The reported times include SOS formulation, solving, and obtaining F(x).
  • IV. EXAMPLE: All three controllers stabilize the Van der Pol system at the origin using the same data set, but their transient performances differ.The phase portraits in Figures 1–3 display the respective closed-loop trajectories.

V. CONCLUSION

The paper demonstrates data-driven synthesis of stabilizers and Lyapunov functions for unknown nonlinear polynomial systems from noisy data. It also identifies improved computational efficiency through alternative data parameterizations and points toward future extensions in nonlinear control and SOS computation.

  • V. CONCLUSION: The method synthesizes stabilizers and Lyapunov functions for unknown nonlinear polynomial systems using noisy data and Lyapunov’s second theorem.The noise need only satisfy a known quadratic bound over the experiment.
  • V. CONCLUSION: Changing the closed-loop data parameterization can improve stabilization-design efficiency even when the same design method is used.The conclusion identifies parameterization as a computational-design lever.
  • V. CONCLUSION: The results provide a foundation for future nonlinear polynomial-system work on local control with guaranteed domains of attraction and quadratic-cost optimal control.These directions are presented as ongoing work rather than established results of this paper.
  • V. CONCLUSION: Further work is needed on SOS-programming computation and improving the efficiency of data-driven designs.The conclusion also situates the results in applications where polynomial systems and SOS optimization are used.
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