Technical Summary: Limits of Learning Linear Dynamics from Experiments

1. Problem Statement

The paper addresses the fundamental problem of identifying the parameters $(A, B)$ of an unknown continuous-time linear time-invariant (LTI) system from observed state-input trajectories. The system is governed by the ordinary differential equation:
$$\dot{x}(t) = Ax(t) + Bu(t), \quad x(0) = x_0$$
where $x(t) \in \mathbb{R}^n$ is the state, $u(t) \in \mathbb{R}^m$ is the control input, and $(A, B)$ are the unknown parameters.

Classical identifiability theory for controlled LTI systems establishes that controllability of the system and a persistently exciting (PE) input are sufficient conditions to uniquely recover $(A, B)$ from a single experiment. However, in many practical settings—particularly sparse and high-dimensional systems—controllability often fails. When controllability is absent, it remains unclear whether a single experiment can uniquely determine the full system or if only parts of the system are recoverable. Furthermore, existing literature lacks a closed-form characterization of identifiability for non-autonomous LTI systems that does not assume controllability.

The core problem is to characterize the experiment-conditional identifiability: determining which parts of the system are uniquely determined by a specific experimental setup (defined by an initial state $x_0$ and a control input $u$), and which parts remain ambiguous.

2. Methodology and Theoretical Framework

The authors develop a geometric and algebraic framework to analyze the limits of information recoverable from a single experiment $e = (x_0, u)$.

2.1 The Visible Subspace

The central concept introduced is the Visible Subspace, denoted $\mathcal{V}(x_0)$. This is defined as the smallest $A$-invariant subspace containing the initial state $x_0$ and the image of the input matrix $B$ (i.e., $\text{Im}(B)$). Formally, it is the Krylov span:
$$\mathcal{V}(x_0) := \mathcal{K}_n(A, [x_0 \ B]) = \text{span}\left\{ [x_0 \ B], A[x_0 \ B], \dots, A^{n-1}[x_0 \ B] \right\}$$
The authors prove a Orbit Confinement Lemma: regardless of the complexity of the input $u$, the resulting state trajectory $\phi(t)$ is strictly confined to $\mathcal{V}(x_0)$. Consequently, the system dynamics outside this subspace (in the "blind" complement) cannot influence the observed trajectory.

2.2 Experiment-Consistent vs. Design-Consistent Sets

The paper distinguishes between two sets of systems consistent with observations:

1. Design-Consistent Set $[A, B]_E$: The set of all systems consistent with the experiment class (all possible inputs $u \in U$). This represents the structural limit of information given the initial state and input capabilities.
2. Experiment-Consistent Set $[A, B]_e$: The set of all systems consistent with a single realized experiment $(x_0, u)$.

The authors establish a "Bridge" theorem: if the realized trajectory is informative (meaning the joint regressor of the projected state and input has a full-rank Gramian), then the single-experiment set collapses to the design-consistent set ($[A, B]_e = [A, B]_E$). They show that a persistently exciting input of sufficient order guarantees this informativeness.

2.3 Characterization of Identifiability

Under an informative experiment, the authors derive a closed-form parametrization of all systems consistent with the data. Using a coordinate transformation $T$ that separates the visible subspace $\mathcal{V}(x_0)$ from its complement $\mathcal{W}$, any consistent system $(\tilde{A}, \tilde{B})$ must satisfy:

• $\tilde{B} = B$
• The restriction of $\tilde{A}$ to the visible subspace equals the restriction of $A$: $\tilde{A}|_{\mathcal{V}(x_0)} = A|_{\mathcal{V}(x_0)}$
• The dynamics on the complement $\mathcal{W}$ and the cross-coupling terms can be arbitrary.

Thus, the system is fully identifiable if and only if $\mathcal{V}(x_0) = \mathbb{R}^n$ (i.e., the visible subspace spans the entire state space). Otherwise, only the restricted dynamics on the visible subspace are uniquely determined.

2.4 Algebraic Criteria

To quantify proximity to non-identifiability, the paper proposes two algebraic tests based on the augmented pair $(A, [x_0 \ B])$:

1. Left-Eigenvector Test: Full identifiability fails if there exists a left eigenvector $w$ of $A$ such that $w^\top x_0 = 0$ and $w^\top B = 0$. A margin $\mu_{\min}$ is defined to measure the alignment of eigenvectors with the excitation sources.
2. Fixed-Experiment PBH Test: A modified Popov–Belevitch–Hautus (PBH) margin $d_{\text{PBH}}(A, B | x_0)$ is derived, measuring the distance to a loss of identifiability given the fixed $x_0$.

3. Key Contributions

1. Partial Identifiability Theorem: The paper proves that even when the full system is not identifiable, the dynamics restricted to the visible subspace $\mathcal{V}(x_0)$ are uniquely determined by a single informative experiment.
2. Closed-Form Parametrization: It provides an explicit description of the set of all systems consistent with a given experiment, characterizing the degrees of freedom in the unidentifiable subspace.
3. Algebraic Criteria for Identifiability: It introduces computable metrics ($\mu_{\min}$ and $d_{\text{PBH}}$) to test for full identifiability and quantify the robustness of the identification problem.
4. Extension to Controlled Systems: Unlike prior work focused on autonomous systems, this framework explicitly disentangles the roles of the initial state and control input in controlled LTI systems.

4. Experimental Results

The authors validate their theoretical findings on synthetic sparse LTI systems, which are prone to controllability failures.

• Controllability vs. Identifiability: Experiments show that as sparsity increases, the fraction of controllable systems drops significantly. However, identifiability is shown to be experiment-conditional: dense initial states can render an uncontrollable system fully identifiable, while sparse initial states may lead to partial identifiability.
• Subsystem Recovery: The paper benchmarks four identification methods (DMDc, MOESP, SINDy, and Neural ODEs) on sparse systems.
  • Result: All methods successfully recover the visible subsystem (the restriction to $\mathcal{V}(x_0)$) with near-zero error, even when the full system is not identifiable.
  • Contrast: The error in recovering the full system ($REE_{\text{full}}$) remains bounded away from zero, reflecting the structural ambiguity in the unexcited subspace.
  • Noise Robustness: As observation noise increases, the error in the visible subsystem remains lower than the full-system error, though Neural ODEs (NODEs) exhibit higher errors overall due to optimization challenges compared to closed-form estimators.

5. Significance and Claims

The paper claims to resolve a gap in the literature regarding the identifiability of non-autonomous LTI systems without assuming controllability. Its primary significance lies in shifting the perspective from "is the system identifiable?" to "what is identifiable given this specific experiment?"

The authors argue that:

• Classical guarantees (controllability + PE) are often too restrictive for sparse, high-dimensional systems.
• Practitioners should interpret mechanistic conclusions relative to the visible subspace induced by their experimental design.
• Any claim about dynamics outside the visible subspace is unwarranted, regardless of how well a model fits the training data.
• The proposed algebraic criteria allow practitioners to quantify the "epistemic uncertainty" of their identification task before or after running an experiment.

The work does not propose new estimation algorithms but rather provides a theoretical lens to understand the limits of existing methods and the fundamental information content of experimental data.