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Data-driven control via Petersen's lemma
Andrea Bisoffi, Claudio De Persis, Pietro Tesi
TL;DR
The paper asks how to stabilize unknown linear or polynomial dynamics directly from noisy open-loop measurements, when several systems may fit the data. It represents those dynamics with a matrix ellipsoid and applies Petersen’s lemma, obtaining exact LMI conditions for linear systems and sufficient SOS conditions for polynomial systems. The framework also supports interpretations of the controller design and is illustrated numerically.
Problem
Noisy measurements can correspond to multiple data-consistent dynamics, requiring a controller that robustly stabilizes all such systems.
Method
The paper represents uncertainty in data-consistent linear and polynomial dynamics with a matrix ellipsoid and applies Petersen’s lemma to derive stabilization conditions.
Results
The paper obtains necessary and sufficient linear-system stabilization conditions as linear matrix inequalities and sufficient polynomial-system conditions through sum-of-squares programs.
Takeaways & Limitations
The matrix-ellipsoid representation supports interpretations of the designed controllers and extends robust data-driven stabilization from linear to polynomial systems.
Takeaways & Limitations
The polynomial sum-of-squares feasibility condition is bilinear in decision variables, so the paper adopts alternating optimization as a suboptimal strategy.
Abstract
from arXiv · showhide
We address the problem of designing a stabilizing closed-loop control law directly from input and state measurements collected in an open-loop experiment. In the presence of noise in data, we have that a set of dynamics could have generated the collected data and we need the designed controller to stabilize such set of data-consistent dynamics robustly. For this problem of data-driven control with noisy data, we advocate the use of a popular tool from robust control, Petersen's lemma. In the cases of data generated by linear and polynomial systems, we conveniently express the uncertainty captured in the set of data-consistent dynamics through a matrix ellipsoid, and we show that a specific form of this matrix ellipsoid makes it possible to apply Petersen's lemma to all of the mentioned cases. In this way, we obtain necessary and sufficient conditions for data-driven stabilization of linear systems through a linear matrix inequality. The matrix ellipsoid representation enables insights and interpretations of the designed control laws. In the same way, we also obtain sufficient conditions for data-driven stabilization of polynomial systems through (convex) sum-of-squares programs. The findings are illustrated numerically.
1 Introduction
The paper studies data-driven controller design when noisy measurements leave multiple data-consistent dynamics possible. It advocates Petersen’s lemma to robustly stabilize these dynamics, yielding LMI-based linear results and SOS-based polynomial results.
- Motivation: Noisy data induce a set C of systems that could have generated the measurements, so control design must stabilize every system in C.This replaces exact system representation with robust design over data-consistent dynamics.
- Method: Petersen’s lemma eliminates norm-bounded uncertainty from robust inequalities, enabling equivalent conditions that do not explicitly contain the uncertain matrices.The paper uses this elimination principle as the basis for its data-driven control results.
- Contributions: For linear systems, the paper provides necessary and sufficient quadratic-stabilization conditions in the form of linear matrix inequalities.These conditions are presented as an alternative to existing data-driven stabilization conditions.
- Contributions: For polynomial systems, the paper derives sufficient data-driven stabilization conditions that can be relaxed into convex sum-of-squares programs.The polynomial-system results extend the matrix-ellipsoid and Petersen’s-lemma approach beyond linear dynamics.
- Contributions: The paper also interprets the resulting design conditions through connections with certainty equivalence and robust indirect control.These connections are discussed for stochastic noise models in related control research.
Relations with the literature
The paper positions its approach within robust and data-driven control under unknown-but-bounded noise. It distinguishes its Petersen’s-lemma formulation, linear-system conditions, and polynomial-system results from related methods while noting numerical illustration of the findings.
- Noise model: The approach assumes an upper bound on the noise norm, following an unknown-but-bounded noise paradigm rather than stochastic noise descriptions.This places the method near set-membership identification and control.
- Linear systems: Petersen’s lemma differentiates the paper’s linear data-driven stabilization approach from several related robust-control formulations.The authors contrast their use of the lemma with prior approaches addressing data-driven stabilization and related performance problems.
- Linear systems: The paper provides necessary and sufficient discrete-time linear stabilization conditions under an easily enforceable condition related to persistence of excitation.The supplied passage identifies this as a central difference from prior work.
- Polynomial systems: For polynomial systems, the paper uses Lyapunov methods to obtain sufficient conditions for data-based global asymptotic stabilization, followed by tractable relaxations.The broader paper describes these relaxations as convex sum-of-squares programs.
- Evaluation: All results are illustrated numerically after the paper develops its linear and polynomial data-driven control results.The numerical examples are presented in the paper’s final section.
2 Preliminaries and problem setting
The paper formulates data-driven stabilization as robust control over all dynamics consistent with noisy open-loop data. It models bounded-energy disturbances, represents the consistent set through a matrix ellipsoid, and targets quadratic stability under data-dependent conditions.
- Petersen’s lemma: Petersen’s lemma provides strict and nonstrict matrix-inequality versions for eliminating norm-bounded uncertainty in robust control conditions.The paper reports both versions and discusses assumptions needed for the nonstrict form.
- Data collection: The data experiment collects input, state, shifted-state, or state-derivative measurements while an unknown disturbance makes the observations noisy.The setup covers discrete-time and continuous-time linear time-invariant systems.
- Noise model: The disturbance model assumes that the full disturbance sequence has bounded energy rather than imposing only an instantaneous disturbance bound.This constraint captures signal-to-noise-ratio conditions and related disturbance descriptions.
- Consistent dynamics: The data-consistent set C contains all matrix pairs (A, B) that reproduce the measured data with a disturbance sequence in the prescribed set.The true pair belongs to C when the disturbance assumptions hold.
- Problem formulation: The design objective is to find state feedback that robustly stabilizes every closed-loop matrix A + BK for (A, B) ∈ C.In discrete time stability means Schur stability, while in continuous time it means Hurwitz stability.
- Set reformulation: Under the data rank assumption, the consistent set has a matrix-ellipsoid representation with A ≻ 0 and Q ⪰ 0, and it is bounded in any matrix norm.The rank condition is related to persistence of excitation and can often be enforced by collecting more data.
3 Data-driven control for linear systems
The linear-system results reformulate the data-consistent dynamics as a matrix ellipsoid and apply Petersen’s lemma to obtain equivalent controller-design conditions. The resulting feasibility problems use positive-definite variables and recover the gain as K = Y P^-1.
- Uncertainty parameterization: The matrix-ellipsoid parameterization represents uncertainty through a center and a norm-bounded factor, enabling Petersen’s lemma to handle all consistent dynamics.The parameterization uses Z = Zc + A^-1/2ΥQ^1/2 with Υ^⊤Υ ⪯ I.
- Discrete time: For discrete-time quadratic stabilization, feasibility of the original robust condition is equivalent to feasibility of the Petersen’s-lemma-based condition in Theorem 1.When feasible, the controller gain is recovered as K = Y P^-1.
- Convexification: The change of variables Y = KP is preferred because using K directly makes the matrix inequality nonlinear.The resulting conditions are expressed with P ≻ 0 and Y as decision variables.
- Continuous time: For continuous-time quadratic stabilization, Theorem 2 gives the analogous equivalent feasibility condition using the same uncertainty elimination strategy.Its controller gain is likewise K = Y P^-1.
- Ellipsoidal over-approximation: The same feasibility equivalences apply when the consistent set is supplied directly as a matrix-ellipsoid over-approximation satisfying A ≻ 0 and Q ⪰ 0.The resulting controller gain remains K = Y P^-1.
4 Discussion and interpretations
The discussion interprets the data-driven stabilization conditions through uncertainty-set geometry, least-squares estimation, and comparisons with alternative robust-control formulations. It also explains how ellipsoidal over-approximations extend the approach to data-consistent sets that are not themselves ellipsoids.
- Ellipsoidal uncertainty and control interpretation: The stability inequalities separate the uncertainty center from its size, enabling an interpretation as robust stabilization around the center dynamics.Zc represents the center, while A and Q determine the uncertainty size; the uncertainty grows with the noise bound.
- Ellipsoidal uncertainty, least squares and certainty-equivalence control: The uncertainty-set center coincides with the ordinary least-squares estimate of the system dynamics.This connects the design to certainty-equivalence control when uncertainty is small.
- Ellipsoidal uncertainty, least squares and certainty-equivalence control: With noisy data, robust control is generally needed, whereas solving the proposed conditions directly avoids explicitly constructing an intermediate model.The paper contrasts this direct approach with robust indirect control under stochastic noise descriptions.
- Comparison with alternative conditions in [41]: The Petersen’s-lemma conditions and the matrix S-procedure provide alternative necessary and sufficient conditions for quadratic stabilization, but rely on different assumptions.The Slater condition can capture unbounded uncertainty sets, while Assumption 1 excludes them; ideal data require different arguments under the Slater-based approach.
- C as an ellipsoidal over-approximation: A matrix ellipsoid can over-approximate non-ellipsoidal data-consistent sets, making the robust-control results applicable at the cost of possible conservatism.For instantaneous bounded disturbances, the ellipsoid can be computed by an optimization problem, and its size generally decreases as more data are collected.
5 Data-driven control for polynomial systems
The paper extends data-driven robust stabilization to polynomial systems by representing data-consistent dynamics with a matrix ellipsoid and applying Petersen’s lemma pointwise. Sum-of-squares conditions then provide sufficient guarantees for global or local asymptotic stability across all compatible systems.
- Polynomial system model: The polynomial system uses known monomial regressors Z and W, while the coefficient matrices A⋆ and B⋆ are unknown.Regressor selection affects feasibility of the optimization-based control law.
- Data and uncertainty: Noisy experiments define a set of polynomial dynamics consistent with measured states, state derivatives, inputs, and an unknown-but-bounded disturbance model.The compatible set is written as X1 = AZ0 + BV0 + D with D belonging to the disturbance set.
- Robust stabilization conditions: Petersen’s lemma is applied pointwise to the matrix-ellipsoid uncertainty set, converting robust Lyapunov inequalities into tractable polynomial positivity conditions.The approach uses the nonstrict lemma because the subsequent sum-of-squares relaxation and numerical implementation handle nonstrict inequalities.
- Global stabilization: A Lyapunov function V, controller k, and multiplier λ certify global asymptotic stability for every data-consistent pair (A, B), including the true system.The proof establishes positive definiteness and radial unboundedness of V together with decrease along all compatible closed-loop systems.
- Sum-of-squares implementation: Theorem 3 expresses these requirements through sum-of-squares constraints, but bilinear products among V, k, and λ make the feasibility program nonconvex.The paper describes alternating optimization as a suboptimal strategy for handling this nonconvexity.
- Local stabilization: A related corollary extends the Lyapunov argument to local asymptotic stability on a specified domain Dc.The conditions impose Lyapunov bounds and derivative decrease over the region defined by ℓ0(x) ≤ c.
6 Numerical examples
The numerical examples apply the proposed data-based stabilization methods to discrete-time linear, continuous-time linear, and polynomial systems under bounded disturbances. The experiments illustrate controller certification through Lyapunov functions, while the polynomial case requires careful regressor selection, richer data, and alternating sum-of-squares optimization.
- Experimental setup: The examples use unknown systems only for data generation, with bounded disturbances converted into uncertainty descriptions for the stabilization methods.For linear systems, the disturbance bound is converted into Δ; for the polynomial system, an ellipsoidal over-approximation is retained.
- 6.1 Linear system in discrete time: The discrete-time linear example uses T = 100 samples, δ = 0.1, uniform random input, and a controller certified by a quadratic Lyapunov function.The resulting closed-loop solutions and Lyapunov level sets are shown in Fig. 2.
- 6.2 Linear system in continuous time: The continuous-time linear example uses T = 100 samples, δ = 0.1, and a sweeping-sine input spanning frequencies 0 to 0.8 with amplitude 2.Theorem 2 produces a controller whose stabilization is certified by a quadratic Lyapunov function, with closed-loop behavior shown in Fig. 4.
- 6.3 Polynomial system: The polynomial example uses T = 1000 samples and δ = 0.01, producing a different state evolution because the system is nonlinear.The resulting controller, Lyapunov function, and multiplier are reported alongside closed-loop solutions in Fig. 6.
- 6.3 Polynomial system: Polynomial stabilization depends on parsimonious monomial regressors because excessive regressors enlarge coefficient uncertainty and can critically reduce sum-of-squares feasibility.The numerical procedure also uses alternating sum-of-squares programs, fixing the Lyapunov function while solving for the controller and multiplier, and vice versa.
- 6.3 Polynomial system: When the uncertainty ellipsoid is formed from richer data, its size and required robustness generally decrease with T, enlarging the sum-of-squares program’s feasibility set.The experiment therefore uses a sweeping sine to obtain richer data.
A Proof of Fact 1
The proof of Fact 1 establishes an equivalence between a robust quadratic inequality over norm-bounded matrix uncertainty and a multiplier-based matrix condition. It combines a maximization lemma with a standard separation argument to obtain the result.
- Auxiliary results: Fact 4 supplies a matrix multiplier condition for combining positive-semidefinite and negative-definite quadratic terms.If the stated definiteness and strict negativity condition hold, some λ > 0 makes the combined matrix negative definite.
- Auxiliary results: Lemma 4 evaluates the maximum of (xᵀFy)^2 over matrices satisfying FᵀF ⪯ ΦᵀΦ as |x|²|Φy|².The proof uses Cauchy–Schwarz and constructs an uncertainty matrix attaining equality.
- Proof of Fact 1: The proof applies Lemma 4 to replace maximization over uncertain matrices with a norm expression involving ΦGx.This establishes the necessity of C ≺ 0 before the multiplier argument is applied.
- Proof of Fact 1: Fact 4 then yields a positive multiplier λ satisfying the matrix inequality, while the reverse implication follows directly from the uncertainty bound.The argument does not require E, F, or G to be nonzero or positive definite in the final statement.
B Proof of Fact 2
The proof of Fact 2 converts a robust inequality over norm-bounded matrix uncertainty into an equivalent quadratic condition and then uses the S-procedure to obtain a multiplier certificate. The converse constructs an admissible uncertainty matrix from arbitrary compatible vectors.
- Proof of Fact 2: The proof begins from a quadratic condition in x and F with FᵀF ⪯ ΦᵀΦ, then rewrites it using the relation y = FGx.This produces a condition over vectors satisfying yᵀy ≤ xᵀGᵀΦᵀΦGx.
- Proof of Fact 2: The reverse direction uses the multiplier inequality and the uncertainty bound to recover the original robust matrix inequality.The proof explicitly notes that the implication remains valid without requiring E ≠ 0, F ≻ 0, or G ≠ 0 for the final statement.
- Proof of Fact 2: For every compatible pair x and y, the proof constructs an F satisfying y = FGx and FᵀF ⪯ ΦᵀΦ.This construction establishes the direction from the uncertain-matrix inequality to the quadratic condition.
- Proof of Fact 2: The resulting quadratic implication is lifted to a variable z = (x, y) and treated with the nonstrict S-procedure.The separation argument supplies a nonnegative multiplier, which is shown to be positive under the stated assumptions.