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A Continuous-Time Generalization of the LPV Fundamental Lemma

Philipp Schmitz, Chris Verhoek

arXiv:2609.07241v2eess.SYcs.SYmath.OC

TL;DR

Direct data-driven methods have largely focused on discrete-time LTI systems, leaving continuous-time LPV behavioral representations beyond LTI undeveloped. This paper uses an LTI identifiability result and a temporary embedding to generalize the LPV Fundamental Lemma to a specified continuous-time LPV class. It presents a behavioral data-driven representation for that class beyond continuous-time LTI systems.

  • Problem

    Direct data-driven methods have mainly addressed discrete-time LTI systems, while continuous-time LPV behavioral representations had not been developed.

    Source: §I · p. 1

  • Method

    The paper reformulates the continuous-time LTI Fundamental Lemma as an identifiability condition and generalizes it to Leibniz-affine LPV systems through a temporary LTI embedding.

    Source: §I · pp. 1–2

  • Results

    The generalized result provides a behavioral data-driven representation for continuous-time LPV systems beyond the LTI class, for systems with Leibniz-affine scheduling dependence.

    Source: §VI · p. 6

  • Takeaways & Limitations

    The result supplies a continuous-time data-driven representation for the specified LPV class, which the paper presents as its first such behavioral representation beyond LTI.

    Source: §VI · p. 6

    The result is limited to continuous-time LPV systems representable by a kernel representation with Leibniz-affine scheduling dependence.

    Source: §VI · p. 6

Abstract

from arXiv · show

Controller design for physical systems directly from data is of tremendous interest in both the industry and academia. However, the research on direct data-driven control has thus far mainly focused on discrete-time linear time-invariant systems. To make the highly non-trivial step to continuous-time nonlinear systems, this paper explores an intermediate route through the framework of linear parameter-varying (LPV) systems. LPV systems can be used as a convenient surrogate for nonlinear systems to achieve systematic analysis and controller design. In this work, we generalize the LPV Fundamental Lemma for discrete-time systems towards a class of continuous-time LPV systems, achieving data-driven representations that can be used to design continuous-time LPV controllers.

I. INTRODUCTION

Direct data-driven system methods have largely centered on discrete-time LTI systems, while continuous-time representations beyond LTI remain undeveloped. The paper takes an intermediate route through continuous-time LPV systems and targets a special class whose behavior can be represented from measurement data.

I. INTRODUCTION

  • The Fundamental Lemma represents horizon-L behavior of discrete-time LTI systems from a single sufficiently informative measurement trajectory.Related data-based methods use this representation for simulation, performance analysis, and control design, [4],,.Source: §I · p. 1
  • Earlier continuous-time input–state predictors assume measurable state derivatives and do not use input–output data,.The paper notes that measuring state derivatives is generally restrictive in practice.Source: §I · p. 1
  • A continuous-time formulation avoids discretization approximation because LPV dynamics depend on the scheduling trajectory within each sampling interval.Direct continuous-time methods also apply to physical-system dynamics without a discretization step, which is desirable in applications such as power electronics.Source: §I · p. 1
  • The paper uses continuous-time LPV systems as an intermediate route toward data-driven representations for nonlinear dynamics.LPV systems have parameters varying with a measurable scheduling signal and can serve as surrogates for nonlinear dynamics,.Source: §I · p. 1
  • The contribution is a continuous-time LPV Fundamental Lemma for systems with Leibniz-affine scheduling dependence, using a temporary LTI embedding.The paper first reformulates the continuous-time LTI Fundamental Lemma as an identifiability condition on input–output data, then generalizes that result to the stated LPV class; it identifies this as the first continuous-time behavioral data-driven representation beyond LTI.Source: §I · pp. 1–2

II. PROBLEM DESCRIPTION

  • The problem section focuses on whether behaviors described by the specified Leibniz-affine kernel representation can be characterized on an interval from measurement data.This dependence structure includes examples such as RC circuits with variable capacitance and mass-spring-damper systems with velocity-dependent damping; it also admits static affine state-space realizations for LMI-based analysis and control synthesis.Source: §II · pp. 2–3

III. LEIBNIZ-AFFINE LPV BEHAVIORS &

This section frames the considered LPV behavior as a function space and recalls its structural quantities and alternative representations.

III. LEIBNIZ-AFFINE LPV BEHAVIORS &

  • The section defines the behavior associated with (2) as a function space.It also recalls structural quantities of behaviors and alternative representation forms.Source: §III · p. 3
  • The section recalls structural quantities used to characterize LPV behaviors.The passage identifies these quantities as part of the section’s treatment of behavior structure.Source: §III · p. 3
  • The section also recalls alternative representation forms for the behavior associated with (2).Source: §III · p. 3

A. Kernel Representations via Weak Solutions

The paper defines LPV behavior through weak solutions, then characterizes its complexity using structural quantities tied to minimal representations.

A. Kernel Representations via Weak Solutions

  • Weak solutions are defined in L2(I, Rnw) × L∞(I, Rnp), and Lemma 1 establishes that behavior B is a closed subset of that product space.The paper says this choice ensures B is a Banach space; alternatively, behavior can be defined using locally integrable function spaces.Source: A. Kernel Representations via Weak… · p. 3

B. Complexity of LPV Behaviors

  • The complexity triple comprises the behavior’s order, lag, and input dimension.Order is the smallest state dimension for a static-scheduling state-space realization; lag is the smallest kernel-representation order; input dimension is the dimension of a maximally free input.Source: B. Complexity of LPV Behaviors · p. 3
  • For a partition w = col(u, y), the input dimension is the dimension of u when no output component can be chosen freely for all scheduling signals.Source: B. Complexity of LPV Behaviors · p. 3
  • These structural quantities are linked to minimality, including the relation nk = nw − m(B) for a minimal representation (2).Source: B. Complexity of LPV Behaviors · p. 3

C. Input–Output Representations

The paper introduces input–output representations by partitioning trajectories into maximally free inputs and outputs, then moves to LTI embeddings for its data-driven result.

C. Input–Output Representations

IV. LTI EMBEDDINGS OF LPV BEHAVIORS

The section embeds LPV behavior in an augmented LTI behavior by temporarily treating scheduling-weighted signals as free auxiliary inputs. The embedding yields an exact trajectory characterization and preserves the behavior’s order and lag while increasing its input dimension.

IV. LTI EMBEDDINGS OF LPV BEHAVIORS

  • Treating the scheduling-weighted signals as free auxiliary inputs forms an augmented LTI behavior B′ that contains the LPV behavior B.Fig. 2 depicts the LPV behavior, the augmented behavior, and their embedding.Source: §IV · p. 4, Figure 2 · p. 4
  • A pair (w, p) belongs to B if and only if its augmented trajectory w′ = col(w, p ⊗ w) belongs to B′.Source: §IV · p. 4
  • B′ is a strict over-approximation of B because its auxiliary inputs are free, whereas LPV trajectories require them to equal p ⊗ w.Source: §IV · p. 4
  • The embedding preserves McMillan degree n(B) and lag l(B), while increasing input dimension by npnw: m(B′) = m(B) + npnw.Source: §IV · p. 4

V. A FUNDAMENTAL LEMMA

This section reformulates the continuous-time LTI Fundamental Lemma as an identifiability condition and uses that step to formulate an LPV Fundamental Lemma.

A. An LTI Fundamental Lemma

  • Unlike the continuous-time LTI Fundamental Lemma’s sufficient, a-priori input-design condition, the reformulation gives necessary-and-sufficient, a-posteriori conditions on input–output data.The authors identify proving necessity of the LPV rank condition as future work.Source: A. An LTI Fundamental Lemma · p. 4, B. An LPV Fundamental Lemma · p. 5
  • Proposition 4 makes the LTI trajectory representation equivalent to a Gramian rank condition, without requiring controllability of the behavior.The condition is necessary and sufficient for representing the behavior from measured input–output data.Source: A. An LTI Fundamental Lemma · p. 4

B. An LPV Fundamental Lemma

  • The LPV theorem requires L ≥ l(B) + 1 and a rank condition on a Gramian formed from w′ = col(w, p ⊗ w).The condition is evaluated on the full input–output data trajectory rather than specified as an input-design criterion.Source: B. An LPV Fundamental Lemma · p. 5
  • For the considered LPV class, a trajectory is characterized by the existence of a coefficient function satisfying a differential-algebraic equation and a compatibility condition.The theorem gives an if-and-only-if data-driven representation.Source: B. An LPV Fundamental Lemma · p. 5
  • The same data matrix Γ can describe trajectories on a different time interval after embedding the LPV behavior in a continuous-time LTI behavior.The coefficient function is then defined on the new interval.Source: B. An LPV Fundamental Lemma · p. 5
  • The continuous-time scheduling matrix has lower block-triangular structure populated by scheduling-signal derivatives, unlike the discrete-time block-diagonal matrix.Projecting the DAE solution space onto compatible trajectories remains an open topic.Source: B. An LPV Fundamental Lemma · p. 5
  • The data matrix uses higher-order derivatives of input, output, and scheduling signals, which are typically noise-sensitive.The paper points to algebraic differentiators as a proposed robust technique.Source: B. An LPV Fundamental Lemma · p. 5

VI. CONCLUSION AND FUTURE WORKS

The paper extends the continuous-time LPV Fundamental Lemma to systems with Leibniz-affine scheduling dependence, using an LTI embedding to generalize an LTI identifiability condition.

VI. CONCLUSION AND FUTURE WORKS

  • The generalized lemma gives behavioral data-driven representations for continuous-time LPV systems with kernel representations and Leibniz-affine scheduling dependence.The authors describe this as the first such representation beyond continuous-time LTI systems.Source: §VI · p. 6
  • Necessary identifiability conditions, broader continuous-time LPV classes, and robust numerical methods for solving the differential-algebraic equation with its compatibility condition remain future work.The authors identify these methods as needed to make the results actionable in data-driven optimal control or simulation and to demonstrate them in realistic numerical examples.Source: §VI · p. 6

APPENDIX

The appendix establishes closure and embedding properties for the behavioral representations, then constructs a minimal LPV state-space realization used in the proof of Lemma 3.

A. Proof of Lemma 1

  • Continuity of the functional defined by (4) shows that limits of trajectories and scheduling signals in B remain in B.The proof applies this to convergence in L2 for trajectories and L∞ for scheduling signals.Source: A. Proof of Lemma 1 · p. 6

B. Proof of Lemma 2

  • The proof of Lemma 2 establishes that (w, p) belongs to B exactly when col(w, p ⊗ w) belongs to B′.It obtains each direction by applying (4) and (9) to the corresponding embedded signal.Source: B. Proof of Lemma 2 · p. 6

C. Proof of Lemma 3

  • For Lemma 3, the cut-and-shift method realizes the input–output form as a minimal LPV state-space model with static scheduling dependence.The construction uses integration under Assumption 1; an LPV Kalman decomposition reduces state dimension while retaining affine scheduling dependence.Source: C. Proof of Lemma 3 · p. 7
  • The proof relates the embedded behavior’s input, state, and observability dimensions to those of the original behavior.When p ⊗ w is free, it shows m(B′) = m(B) + npnw, while n(B′) = n(B) and l(B′) = l(B).Source: C. Proof of Lemma 3 · p. 7

D. Proof of Proposition 4

The proof connects a Gramian rank condition to the rank of Γ, using an observable LTI realization and implications among the proposition’s statements.

D. Proof of Proposition 4

  • With L ≥ l(B′) + 1, observability gives the map S_L full column rank, and the Gramian factorization links Γ to Γ0.A permutation matrix separates derivatives of inputs and outputs in the factorization.Source: D. Proof of Proposition 4 · p. 7
  • The proof shows that rank(Γ) = Lm(B′) + n(B′) is equivalent to rank(Γ0) = Lm(B′) + n(B′).It uses the factorization through S_L and equality of the corresponding kernels and images.Source: D. Proof of Proposition 4 · p. 7
  • The proof establishes equivalence of (i), (ii), and (iii): (iii) implies (ii) by adapting [7, Thm. 22], while (ii) implies (i) via an arbitrary state–input trajectory, yielding rank(Γ) = Lm(B′) + n(B′).Source: D. Proof of Proposition 4 · p. 8

E. Proof of Theorem 5

For the embedded LPV behavior, the proof establishes that membership is equivalent to the representation in (14) by a function g.

E. Proof of Theorem 5

  • For L ≥ l(B′) + 1, (w̄, p̄) belongs to B exactly when some g satisfies (14).The proof applies Lemma 2 and Proposition 4 in both directions, using w̄′ = col(w̄, p̄ ⊗ w̄).Source: E. Proof of Theorem 5 · p. 8

F. Proof of Corollary 6

The proof establishes the equivalence of (14) and equations (16)–(18) by deriving each direction.

F. Proof of Corollary 6

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