# Willems' Lemma Reformulations: Which Operators Preserve LTI System Behavior?

Alexandre Faye-Bédrin<fnref n="1" />, Paul Chauchat<fnref n="2" />, Stanislav Aranovskiy<fnref n="1" />, and Romain Bourdais<fnref n="1" />

**Abstract**— In the behavioral approach, dynamical systems are abstracted as sets of trajectories. This approach gave birth to the celebrated Willems' Fundamental Lemma, for which various reformulations have been proposed in the literature, for instance, using frequency-domain data. In this note, we show that all reformulations are necessarily based on linear shift-invariant transformations, which have the fundamental property that they preserve the trajectory space of all LTI systems. Our results unify the existing reformulations and provide guidelines for designing novel ones.

**Index Terms**— Data driven control, Linear systems

## I. INTRODUCTION

THE behavioral approach to system theory [1] defines a dynamical system as a set of trajectories called behavior. This abstract perspective allows to separate the representation of systems (e.g., input-output transfer function, state-space representation, convolution) from the study of their fundamental properties (e.g., linearity, complexity, dissipativity). Extensive literature revolves around linear time-invariant (LTI) systems, which are defined as shift-invariant subspaces, as detailed in Section II-A.

A result from the behavioral approach has recently gained interest: the Fundamental Lemma [2] (which we recall in Section II-B) allows for the behavior (set of possible trajectories) of any controllable LTI system to be described solely with raw records of input-output trajectories organized in a Hankel matrix. This simple and powerful data-driven representation is at the core of a model-free predictive control method [3], referred to as DeePC (Data-EnablEd Predictive Control). Alternative formulations of the Fundamental Lemma have been developed in the literature, using transformations of input-output sequences instead of raw data. Authors in [4] propose two formulations: in the first one, the data matrix is based on cross-correlation of input-output signals, and the second one uses cross-spectrum (i.e., the Fourier transform of cross-correlation). In [5], the authors propose a formulation where the data matrix contains frequency-domain data. In both cases, authors prove the correctness of their formulations by applying a similar procedure as for the multiple dataset case [6]. The similarity in the proofs is a crucial observation, as it suggests a fundamental property of the data transformations: they preserve the relation between input-output signals induced by system dynamics.

**Novelty and Contribution.** This note aims to generalize the aforementioned reformulations. To this objective, we address the following question: what is the set of transformations that preserve the trajectory space of any LTI system? Characterizing these transformations, we clarify the relationship between the Fundamental Lemma and its reformulations. We show that the set of data transformations that preserve the behavior of all LTI systems is and only is the set of linear shift-invariant (LSI) operators. We also suggest a compact reformulation of Willems' Lemma for a class of transformations, including the Fourier transform. These results are obtained without state-space representation or stability assumptions. They also allow for novel applications of the Fundamental Lemma, e.g., applying a bank of filters to the system trajectories.

The paper has the following structure. First, Section II presents some preliminaries. Next, we show our main results in Section III and study its connection to a family of operators related to the Fourier transform in Section IV. A perspective on continuous-time LTI systems is then presented in Section V, and some applications of our results are given in Section VI, which include the three reformulations of the Fundamental Lemma (with frequency domain data, cross-correlation, and cross-spectrum). Finally, the conclusion is in Section VII.

## Notation

* The operator $\text{col}(\cdot)$ stacks up its vector arguments.

* A set of numbers excluding 0 is denoted with $(\cdot)^*$, e.g. $\mathbb{N}^* = \mathbb{N} \setminus \{0\}$ and $\mathbb{C}^* = \mathbb{C} \setminus \{0\}$.

* The space of sequences (discrete-time signals) on time axis $\mathbb{Z}$ with values in $\mathbb{R}^q$ is denoted $(\mathbb{R}^q)^{\mathbb{Z}} = \{\mathbb{Z} \to \mathbb{R}^q\}$. For $q = 1$, we write $\mathbb{R}^{\mathbb{Z}}$ or $\mathbb{C}^{\mathbb{Z}}$.

* The set of discrete-time LTI systems of order $n$ with $m$ inputs and $q-m$ outputs is $\mathbb{B}_m^{q,n}$. When the input-output partition is not specified, we write $\mathbb{B}^{q,n}$, and $\mathbb{B}_m^{q,n} \subseteq \mathbb{B}^{q,n}$. When not relevant to the discussion, $n$ is not specified, e.g., $\mathbb{B}^q$ or $\mathbb{B}_m^q$.

* For a sequence $s$ with $s(t) \in \mathbb{R}^q$ or $s(t) \in \mathbb{C}$ and $a \le b$ we use the following notation for a stacked window:

$$s_{[a,b]} = \text{col}(s(a), s(a+1), \dots, s(b)).$$

* For a finite sequence $\{s(t)\}_{t=1}^T$ with $s(t) \in \mathbb{R}^q$, we denote the Hankel matrix $H_L(s) \in \mathbb{R}^{qL \times (T-L+1)}$ as

$$H_L(s) = [s_{[1,L]} \ s_{[2,L+1]} \ \dots \ s_{[T-L+1,T]}].$$

* The range space of a matrix or linear operator $A$ is denoted $\text{im}(A)$, and its null space is $\ker(A)$.

<footnote n="1">¹ IETR, CentraleSupélec, Rennes, France
surname.name@centralesupelec.fr</footnote>

<footnote n="2">² Aix-Marseille Univ, CNRS, LIS, Marseille, France
paul.chauchat@lis-lab.fr</footnote>
* Application of operator A to s is denoted $A[s]$, or $As$ when there is no ambiguity.

* The shift operator $\sigma$ is defined, with any sequence $s$:

$$ \forall t, \sigma[s](t) = s(t+1) $$

## II. PRELIMINARIES

### A. LTI systems in the behavioral framework

In the behavioral framework, a discrete-time LTI system (or behavior) $\mathcal{B} \in \mathbb{B}^q$ is a shift-invariant subspace of the space of signals $(\mathbb{R}^q)^Z$. In other words, it is a space of trajectories with values in $\mathbb{R}^q$, with the property that $w \in \mathcal{B} \iff \sigma w \in \mathcal{B}$, where $w \in \mathcal{B}$ means that $w$ is a trajectory of $\mathcal{B}$. Such a system $\mathcal{B}$ can be described with its so-called kernel representation

$$ \mathcal{B} = \{w \in (\mathbb{R}^q)^Z \mid R(\sigma)w = 0\} = \ker(R(\sigma)) $$

where $R(\xi)$ is a matrix polynomial with indeterminate $\xi$. An LTI system with input-output partitioning $\mathcal{B} \in \mathbb{B}_m^q$ can also be described as

$$ \mathcal{B} = \{u \in (\mathbb{R}^m)^Z, y \in (\mathbb{R}^{q-m})^Z \mid R_y(\sigma)y = R_u(\sigma)u\}. $$

The restricted behavior $\mathcal{B}|_L$ of a system $\mathcal{B} \in \mathbb{B}^q$ is defined as the set of trajectories of $\mathcal{B}$ with length $L$

$$ \mathcal{B}|_L = \{w_{[1,L]} \mid w \in \mathcal{B}\} $$

or with an input-output partition

$$ \mathcal{B}|_L = \left\{ \begin{bmatrix} u_{[1,L]} \\ y_{[1,L]} \end{bmatrix} \mid (u, y) \in \mathcal{B} \right\}. $$

### B. Willems' Fundamental Lemma

We recall Willems' Fundamental Lemma extended for multiple datasets [6], which mobilizes the idea of collective persistence of excitation.

**Definition 1:** A collection of sequences $s = \{s_i = \{s_k(t)\}_{t=1}^{T_k}\}_{k=1}^N$ with $s_k(t) \in \mathbb{R}^q$ is said to be collectively persistently exciting of order $M$ if

$$ \operatorname{rank}([H_M(s_1) \ H_M(s_2) \ \dots \ H_M(s_N)]) = qM. $$

We then write $s \in \text{CPE}(M)$.

Let $\mathcal{B} \in \mathbb{B}_m^{q,n}$ a controllable<fnref n="1" />. LTI system of order $n$. Let $(u^D = \{\{u_k^D(t)\}_{t=1}^{T_k}\}_{k=1}^N, y^D = \{\{y_k^D(t)\}_{t=1}^{T_k}\}_{k=1}^N)$ be a collection of finite-length input-output sequences of $\mathcal{B}$.

**Theorem 1 ([6]):** If $u^D \in \text{CPE}(L+n)$ then

$$ \mathcal{B}|_L = \operatorname{im} \begin{bmatrix} H_L(u_1^D) & \dots & H_L(u_N^D) \\ H_L(y_1^D) & \dots & H_L(y_N^D) \end{bmatrix}. \quad (1) $$

**Remark 1:** The proof of Theorem 1 was derived in [6] using state-space representation. However, it is also possible to use the same proof as in the original Lemma [2, Theorem 1] by simply replacing the single Hankel matrices with mosaic-Hankel matrices. This modification enables to use Theorem 1 without assuming a state-space representation.

**Remark 2:** Theorem 1 has been extended to the case of uncontrollable systems and under looser conditions [8]. It

would be straightforward to connect this extended theorem to our results, however the more complicated formulation would hinder clarity while not adding significant value. The same can be said for other kinds of data matrices or different definitions of PE, such as in [9].

## III. OPERATORS AND LTI SYSTEMS

In this section, we state our main result: the characterization of operators that preserve the behavior of all LTI systems.

### A. Characterization of behavior-preserving operators

Let $\mathcal{A} \subseteq \{\mathbb{R}^Z \to \mathbb{R}^Z\}$ be the set of linear shift-invariant (LSI) operators on real-valued sequences. For $A \in \mathcal{A}$, the linearity implies that for any sequences $x, y \in \mathbb{R}^Z$ and any scalars $a, b \in \mathbb{R}$ we have

$$ A(ax + by) = aA(x) + bA(y), $$

and the shift-invariance implies that $A$ commutes with the shift operator, i.e. for any $x \in \text{dom } A$

$$ A[\sigma x] = \sigma[Ax]. $$

Our main result is the following theorem, where we highlight a fundamental relationship between LTI systems and LSI operators.

**Theorem 2:** An operator preserves the behavior of all LTI systems if and only if it is in $\mathcal{A}$:

$$ \mathcal{A} = \{A | \forall q \in \mathbb{N}^*, \forall \mathcal{B} \in \mathbb{B}^q, w \in \mathcal{B} \Rightarrow Aw \in \mathcal{B}\} $$

where $Aw$ is the component-wise application of $A$ to $w$, i.e., $Aw = [Aw^1 \ \dots \ Aw^q]^\top$, where $w^i$ is the scalar sequence corresponding to the $i$-th component of $w$.

**Proof:** First, we show that all operators in $\mathcal{A}$ preserve all LTI behaviors. Take $w \in \mathcal{B}$:

$$ R(\sigma)[Aw] = A[R(\sigma)w] = 0 \Rightarrow Aw \in \mathcal{B} $$

due to linearity and shift-invariance of $A$.

Now, we prove that all LTI behavior-preserving operators belong to $\mathcal{A}$. To this end, we show that any operator $A$ such that

$$ \forall q \in \mathbb{N}^*, \forall \mathcal{B} \in \mathbb{B}^q, (w \in \mathcal{B} \Rightarrow Aw \in \mathcal{B}) \quad (2) $$

is both (a) linear and (b) shift-invariant, and thus $A \in \mathcal{A}$.

(a) **Linearity:** If $A$ is not linear, then there exist constants $(a, b) \in \mathbb{R}$ and sequences $(\bar{u}_1, \bar{u}_2) \in \mathbb{R}^Z$ such that

$$ A[a\bar{u}_1 + b\bar{u}_2] \neq aA\bar{u}_1 + bA\bar{u}_2. $$

With such $a$ and $b$, let $\mathcal{B} \in \mathbb{B}_2^3$ defined by $y = au_1 + bu_2$. Then, with $u_1 = \bar{u}_1$ and $u_2 = \bar{u}_2$,

$$ (Au, Ay) \neq (Au, aAu_1 + bAu_2) \text{ i.e. } (Au, Ay) \notin \mathcal{B} $$

contradicting (2), so $A$ must be linear.

(b) **Shift-invariance:** If $A$ is not shift-invariant, then there exists a sequence $\bar{y} \in \mathbb{R}^Z$ such that

$$ A[\sigma \bar{y}] \neq \sigma[A \bar{y}]. $$

<footnote n="1">¹Controllability has a specific definition in the behavioral framework [1], equivalent to the usual definition for state-space systems [7, Section 5.2]</footnote>
Let $\mathcal{B} \in \mathbb{B}_1^2$, defined by $\sigma y = u$. Then, with $u = \sigma \bar{y}$, we
get $y = \bar{y}$ and

$$
(Au, Ay) \neq (\sigma Ay, Ay) \text{ i.e. } (Au, Ay) \notin \mathcal{B}
$$

contradicting (2), so A must be shift-invariant.

<figure><img src="image_1.png" /></figure>

Theorem 2 gives the whole set of operators that preserve the trajectory space of all LTI systems: there cannot exist any shift-variant or non-linear operator that would satisfy this property. It is tempting to conclude that the three reformulations of Willets' Lemma cited in Section I (using cross-correlation, cross-spectrum, and frequency-domain data) must be based on LSI operators. However, these reformulations are about restricted behaviors: we still need to clarify the link between restricted and unrestricted behaviors.

B. Operators that preserve restricted behaviors

We introduce a new set of operators that, when applied to trajectories of any given LTI system, yield restricted trajectories of the same system. We then look at the relationship between these new operators and behavior-preserving operators. Let $L \in \mathbb{N}^*$, we denote $A|_L \subseteq \{\mathbb{R}^z \to \mathbb{R}^L\}$ the set of operators defined such that, if $\bar{A} \in A|_L$, for all $q \in \mathbb{N}^*$ and any LTI system $B \in \mathbb{B}^q$

$$
w \in \mathcal{B} \Rightarrow \bar{A}w = \begin{bmatrix} \bar{A}[w](1) \\ \vdots \\ \bar{A}[w](L) \end{bmatrix} \in \mathcal{B}|_L. \quad (3)
$$

Now we formulate the question: what is the link between $A|_L$ and $\mathbb{A}$? The answer is given in the following theorem.

*Theorem 3:* The set of operators $\mathbb{A}|_L$ that preserve restricted behaviors is equivalent to the set of LSI operators $\mathbb{A}$ in the sense that

$$
\mathbb{A}|_L = \{ (s \mapsto A[s]_{[1,L]}) \mid A \in \mathbb{A} \}. \quad (4)
$$

Proof: First, it is obvious that if we take $A \in \mathbb{A}$

$$
w \in \mathcal{B} \Rightarrow A[w]_{[1,L]} \in \mathcal{B}|_L
$$

which implies the right inclusion $\supseteq$ in (4). We will show that
each $\bar{A} \in A|_L$ has a corresponding extension $A \in \mathbb{A}$. Take
$\bar{A} \in A|_L$. Due to the properties of $A$ and $B$, we have

$$
w \in \mathcal{B} \Rightarrow \forall t \in \mathbb{Z}, \sigma^t w \in \mathcal{B} \Rightarrow \forall t \in \mathbb{Z}, \bar{A}[\sigma^t w] \in \mathcal{B}|_L.
$$

Now define the operator $A : \mathbb{R}^z \to \mathbb{R}^z$ as

$$
\forall t \in \mathbb{Z}, \forall s \in \mathbb{R}^z, A[s](t) = \bar{A}[\sigma^{t-1}s](1).
$$

By this definition, $A$ is shift-invariant, and it is easy to check
that

$$
\forall s \in \mathbb{R}^z, \bar{A}[s] = A[s]_{[1,L]}.
$$

Finally, we prove that $A$ must be linear: to this end, we show
that $\bar{A}$ is linear. Let $\mathcal{B} \in \mathbb{B}_2^3$, defined by $y = au_1 + bu_2$. If $\bar{A}$ is
not linear, then there exists constants $(a, b) \in \mathbb{R}$ and sequences
$(\bar{u}_1, \bar{u}_2) \in \mathbb{R}^z$ such that

$$
\bar{A}[a\bar{u}_1 + b\bar{u}_2] \neq a\bar{A}\bar{u}_1 + b\bar{A}\bar{u}_2.
$$

With such $a$ and $b$, let $\mathcal{B} \in \mathbb{B}_2^3$ defined by $y = au_1 + bu_2$.
Then, if we take $u_1 = \bar{u}_1$ and $u_2 = \bar{u}_2$

$$
\begin{bmatrix} \bar{A}u \\ \bar{A}y \end{bmatrix} \neq \begin{bmatrix} \bar{A}u \\ a\bar{A}u_1 + b\bar{A}u_2 \end{bmatrix} \quad \text{i.e.} \quad \begin{bmatrix} \bar{A}u \\ \bar{A}y \end{bmatrix} \notin \mathcal{B}|_L
$$

contradicting $A \in \mathbb{A}|_L$ with $A|_L$ defined in (3), so $\bar{A}$ must be linear. It follows that $A$ is also linear, so we have $A \in \mathbb{A}$. Hence, we have the left inclusion $\subseteq$ in (4).

<figure><img src="image_2.png" /></figure>

Theorem 3 makes clear that for each reformulation of the Fundamental Lemma cited in Section I, as well as any potential new future reformulation, there are corresponding LSI operators. This means that separate, specific theoretical studies of each reformulation can be avoided since they are natural declinations of Theorem 1 that arise from the fundamental properties of LSI operators.

IV. THE FUNDAMENTAL LEMMA WITH GEOMETRIC SEQUENCES

In this section, we show how Theorem 3 can be applied to
get the frequency-domain reformulation of Willets' Lemma,
i.e., for the Fourier transforms of trajectories. To this end,
we introduce a specific set of operators based on geometric
sequences and show that they are LSI. Whereas this result,
together with Theorem 3, is sufficient, we provide a more
compact formulation using reduced matrices thanks to the
properties of these operators.

Let us introduce a family of linear forms $\mathbb{G}^z \subseteq \{\mathbb{C}^z \to \mathbb{C}\}$ that maps trajectories to complex numbers, with the special property that any $G \in \mathbb{G}^z$ satisfies

$$
\forall s \in \mathbb{C}^z, G[\sigma s] = zGs
$$

where $z \in \mathbb{C}^*$. Note that this is a partial characterization of
the Z-transform, which is defined as

$$
\forall s \in \mathbb{C}^z, \mathcal{Z}[s](z) = \sum_{k \in \mathbb{Z}} z^{-k} s(k).
$$

By definition, we have $(s \mapsto \mathcal{Z}[s](z)) \in \mathbb{G}^z$. Then, an example of such a linear form is the discrete-time Fourier transform discussed in Section VI-B.

Let $G \in \mathbb{G}^z$ and $\bar{z} \in \mathbb{C}^z$ a geometric sequence of common ratio $z$ (i.e. $\sigma \bar{z} = z \bar{z}$). Consider the operators

$$
\mathcal{R} : s \mapsto \Re(\bar{z}Gs) \quad \text{and} \quad \mathcal{I} : s \mapsto \Im(\bar{z}Gs)
$$

where $\Re$ and $\Im$ denote, respectively, the real and imaginary
parts of their argument.

*Lemma 1:* The operators $\mathcal{R}$ and $\mathcal{I}$ are linear and shift-invariant.

*Proof:* Take $s \in \mathbb{C}^z$, then for the real part

$$
\sigma[\Re(\bar{z}Gs)] = \Re(\sigma[\bar{z}Gs]) = \Re(z\bar{z}Gs) = \Re(\bar{z}G[\sigma s])
$$

and the same holds for the imaginary part $\Im(\bar{z}Gs)$. Hence the
operators $\mathcal{R}$ and $\mathcal{I}$ belong to $\mathbb{A}$.

<figure><img src="image_3.png" /></figure>

Let $N$ scalars $\{z_k \in \mathbb{C}^*\}_{k=1}^N$, their corresponding linear forms $\{G_k \in \mathbb{G}^{z_k}\}_{k=1}^N$ and geometric sequences $\{\bar{z}_k \in \mathbb{C}^z : \bar{z}_k(t) = z_k^{t-1}\}_{k=1}^N$. Let $(u = \{u_k\}_{k=1}^N, y = \{y_k\}_{k=1}^N)$ be a collection of $N$ input-output sequences of a controllable
system $\mathcal{B} \in \mathbb{B}^{q,n}_m$, i.e., for any $k$, $u_k \in (\mathbb{R}^m)^Z$. For a vector-valued sequence $s \in (\mathbb{R}^r)^Z$, $G$ is applied component-wise, i.e. $Gs = [Gs^1 \ \dots \ Gs^r]^\top \in \mathbb{C}^r$. We read $\bar{z}Gs$ as such: at any time $t$, $(\bar{z}Gs)(t) = [\bar{z}(t)Gs^1 \ \dots \ \bar{z}(t)Gs^r]^\top \in \mathbb{C}^r$. For $s_k \in \{u_k, y_k\}$, we then denote

$$ \xi_k^s = \bar{z}_k G_k s_k \in (\mathbb{C}^r)^Z \quad (5) $$

and

$$ \mathcal{H}_L(s) = [\xi_{1,[1,L]}^s \ \dots \ \xi_{N,[1,L]}^s] \in \mathbb{C}^{r \times N}. $$

The following theorem is the main result of this section.

**Theorem 4:** Suppose that

$$ \mathrm{rank} [\Re(\mathcal{H}_{L+n}(u)) \ \Im(\mathcal{H}_{L+n}(u))] = (L+n)m \quad (6) $$

then

$$ \mathcal{B}|_L = \mathrm{im} \left[ \Re(\mathcal{H}_L(u)) \ \Im(\mathcal{H}_L(u)) \ \Re(\mathcal{H}_L(y)) \ \Im(\mathcal{H}_L(y)) \right]. \quad (7) $$

*Proof:* Theorem 3 and Lemma 1 ensure that $\mathrm{im} \left[ \Re(\mathcal{H}_L(u)) \ \Im(\mathcal{H}_L(u)) \ \Re(\mathcal{H}_L(y)) \ \Im(\mathcal{H}_L(y)) \right] \subseteq \mathcal{B}|_L$. Condition (6) implies

$$ \{\Re(\xi_{k,[1,L+n]}^u)\}_{k=1}^N \cup \{\Im(\xi_{k,[1,L+n]}^u)\}_{k=1}^N \in \mathrm{CPE}(L+n), $$

see Definition 1. Then according to Theorem 1

$$ \mathcal{B}|_L = \mathrm{im} \left[ \begin{bmatrix} \Re(H_L(\xi_{k,[1,T]}^u)) & \Im(H_L(\xi_{k,[1,T]}^u)) \\ \Re(H_L(\xi_{k,[1,T]}^y)) & \Im(H_L(\xi_{k,[1,T]}^y)) \end{bmatrix} \right]_{1 \le k \le N} \quad (8) $$

with $T = L + n$. This readily proves a reformulation of Willems' Lemma using operators $\mathcal{R}$ and $\mathcal{I}$. The difference with (7) is the size of the involved matrix, which is $qL \times 2N$ in (7) and $qL \times 2(n+1)N$ in (8). To conclude, we only need to show that we can simply consider the first column of each $H_L(\xi_{k,[1,T]}^s)$. To do so, it is easy to check that

$$ H_L(\xi_{k,[1,T]}^s) = \begin{bmatrix} \xi_k^s(1) \\ \xi_k^s(2) \\ \vdots \\ \xi_k^s(L) \end{bmatrix} \begin{bmatrix} 1 & z_k & \cdots & z_k^n \end{bmatrix} $$

and so

$$ \mathrm{im} \left[ \Re \left( H_L \left( \xi_{k,[1,L+n]}^s \right) \right) \ \Im \left( H_L \left( \xi_{k,[1,L+n]}^s \right) \right) \right] \\ = \mathrm{im} \left[ \Re \left( \xi_{k,[1,L]}^s \right) \ \Im \left( \xi_{k,[1,L]}^s \right) \right]. $$

Hence, (8) is equivalent to (7).

<figure><img src="image_4.png" /></figure>

# V. CONTINUOUS TIME LTI SYSTEMS

## A. In the behavioral framework

Up to this point, we have focused exclusively on discrete-time (DT) LTI systems, but it is possible to transpose some results to the continuous-time (CT) setup. We write $\bar{\mathbb{B}}^q \subseteq (\mathbb{R}^q)^\mathbb{R}$ the set of $q$-variate differential CT LTI systems. In the behavioral framework, such a system $\mathcal{B} \in \mathbb{B}^q$ can be described with its kernel representation (see [10])

$$ \mathcal{B} = \left\{ w \in (\mathbb{R}^q)^\mathbb{R} \middle| R \left( \frac{d}{dt} \right) w = 0 \right\} $$

where $R$ is, as in the DT case, a matrix polynomial. It is worth mentioning that other, larger classes of CT LTI systems exist, such as systems with delays [11] or leveraging the idea of distributional behavior [12]. For the sake of conciseness, we only consider differential CT systems, but we infer that our approach would yield analogous results.

Compared to the DT setup, the signal space changes from $(\mathbb{R}^q)^Z$ to $(\mathbb{R}^q)^\mathbb{R}$ (the time axis becomes the set of real numbers), and the shift operator $\sigma$ is substituted with the time derivative $\frac{d}{dt}$. We elude on the precise definition of CT signals: we only assume they are compatible with the operations we use, so our propositions are not impacted. Depending on the context, signals may be considered as (infinitely) differentiable functions with respect to time, but they can also be defined as distributions (to allow for discontinuities), or (locally) square integrable functions. Some discussion can be found in [12] and references therein.

There exist different versions of the Fundamental Lemma for CT systems [13], [14], [15], but a detailed discussion on the differences between the DT and the several CT cases is out of scope for this paper. Still, using our results, we infer that these CT Lemmas could be reformulated, for instance in the frequency domain.

## B. Behavior-preserving operators, complex exponentials

We can draw similar results to those of Theorem 2 and Lemma 1 in the CT setup.

For the first result, we introduce the CT equivalent of DT LSI operators. Let $\bar{A} \subseteq \{\mathbb{R}^\mathbb{R} \to \mathbb{R}^\mathbb{R}\}$ be the set of linear operators such that any $A \in \bar{A}$ satisfies

$$ \forall s \in \mathbb{R}^\mathbb{R}, \forall \tau \in \mathbb{R}, A[\sigma^\tau s] = \sigma^\tau[As] $$

where $\sigma$ denotes, with an abuse of notation, the delay operator in the CT setup:

$$ \forall s \in \mathbb{R}^\mathbb{R}, \forall (\tau, t) \in \mathbb{R}, \sigma^\tau[s](t) = s(t + \tau). $$

For any signal $s \in \mathbb{R}^\mathbb{R}$ that is differentiable with respect to time, the definition of $A$ implies that it commutes with $\frac{d}{dt}$:

$$ A \left[ \frac{d}{dt} s \right] = A \left[ \lim_{t \to 0} \frac{\sigma^t - 1}{t} s \right] = \frac{d}{dt} [As]. \quad (9) $$

Conversely, if $A$ commutes with $\frac{d}{dt}$, then it is time-invariant.

**Theorem 5:** The set $\bar{A}$ is the set of operators that preserve the behavior of all CT LTI systems:

$$ \bar{A} = \{A | \forall q \in \mathbb{N}^*, \forall \mathcal{B} \in \bar{\mathbb{B}}^q, w \in \mathcal{B} \Rightarrow Aw \in \mathcal{B}\}. $$

For the second result, we introduce the CT equivalent of $\mathbb{G}^z$. Let $z \in \mathbb{C}^*$ and let $\bar{\mathbb{G}}^z \subseteq \{\mathbb{C}^\mathbb{R} \to \mathbb{C}\}$ the set of linear forms such that any $G \in \bar{\mathbb{G}}^z$ satisfies

$$ \forall s \in \mathbb{C}^\mathbb{R}, G \left[ \frac{d}{dt} s \right] = zGs. $$

**Lemma 2:** Let $\mathcal{B}$ a CT LTI system and $A \in \bar{\mathbb{G}}^z$. Let $\bar{z} \in \mathbb{C}^\mathbb{R}$ satisfying the differential equation $\frac{d}{dt} \bar{z} = z\bar{z}$. Then

$$ w \in \mathcal{B} \Rightarrow \left\{ \begin{array}{l} \Re(\bar{z}Aw) \in \mathcal{B} \\ \Im(\bar{z}Aw) \in \mathcal{B} \end{array} \right. . $$
Proof: The proofs of Theorem 5 and Lemma 2 follow the exact same structure as the proof of Theorem 2 and Lemma 1, where the shift operator $\sigma$ is substituted with the time derivative $\frac{d}{dt}$.

## VI. EXAMPLES OF LSI OPERATORS

This section discusses examples of LSI operators used for the Lemma reformulations. We make connections to existing results, provide new perspectives, such as banks of filters, and discuss the use of LSI operators with non-LTI systems. Note that these results neither assume system stability nor a state-space representation (see Remark 1), contrary to [4], [16].

### A. Convolution and cross-correlation

Convolution $f*g$ between scalar-valued signals $(f,g) \in \mathbb{R}^Z$ is defined as

$$f * g = \sum_{k=-\infty}^{\infty} f(-k)\sigma^k g.$$

For any given $f \in \mathbb{R}^Z$, $(s \mapsto f*g)$ is linear and shift-invariant. Take any sequence $f \in \mathbb{R}^Z$ (e.g., a trajectory of an arbitrary non-linear system), then Theorem 2 implies that if $w$ is a trajectory of any LTI system $B$, $f*g$ is also a trajectory of $B$. Note that any LSI operator can be represented as a convolution, for instance with $A \in A$:

$$\forall s \in \mathbb{R}^Z, A[s] = \sum_{k \in \mathbb{Z}} A[\delta](-k) \sigma^k s = A[\delta] * s$$

where $\delta(k) = 0$ for all $k$ except $\delta(0) = 1$ and $A[\delta](-k)$ is the value of $A[\delta]$ at the time instance $-k$.

Cross-correlation between two signals is similar to convolution: for $(f,g) \in \mathbb{R}^Z$ it is defined as

$$f \star g = \sum_{k=-\infty}^{\infty} f(k)\sigma^k g.$$

Again, given $f \in \mathbb{R}^Z$, $(s \mapsto f*g)$ is linear and shift-invariant. In the first reformulation given in [4, Theorem 1], the input-output signals are correlated with the system inputs. Then, the correlations are used in place of raw data in the mosaic-Hankel matrices of Theorem 1.

### B. Discrete-time Fourier transform

The Fourier transform of a sequence $s \in \mathbb{C}^Z$ at frequency $\omega$ is defined as

$$\mathcal{F}(\omega)[s] = \sum_{k=-\infty}^{+\infty} s(k)e^{-j\omega k}$$

where $j$ is the imaginary unit, then we note that

$$\mathcal{F}(\omega)[\sigma s] = e^{j\omega}\mathcal{F}(\omega)[s] \Rightarrow \mathcal{F}(\omega) \in \mathbb{G}^{e^{j\omega}}.$$

Using Theorem 4 with $z = e^{j\omega}$, we recover Theorem 1 of [5], which is reproduced in the following Corollary.

**Corollary 1:** Define $\xi_k^s$ in (5) using $\mathcal{F}(\omega_k)$ as $G_k$, $(e^{j\omega_k t})_{t \in \mathbb{Z}}$ as $\bar{z}_k$. Then, if (6) is satisfied, then (7), which we recall here, holds:

$$B|_L = \mathrm{im} \begin{bmatrix} \Re(\mathcal{H}_L(u)) & \Im(\mathcal{H}_L(u)) \\ \Re(\mathcal{H}_L(y)) & \Im(\mathcal{H}_L(y)) \end{bmatrix}$$

Note that we express the persistence of excitation condition (6) by separating the real and imaginary part of $\mathcal{H}_{L+n}(u)$, while authors in [16] equivalently use the complex values and their complex conjugates. Our version of the result also allows for complex values of $\omega$.

Successive application of cross-correlation and Fourier transform yields a formulation with cross-spectrums, similar to the one exposed in [4, Theorem 2]. While we only allow for real-valued signals, [4] considers complex-valued trajectories.

### C. Continuous-time linear sampling functional

Authors in [17] propose a “linear sampling functional”: a CT signal $w$ is then “sampled” as $\bar{w}(t) = \int w(\tau + t)g(\tau)d\tau$ where $g$ is a user-defined function or distribution. Ordinary sampling is recovered when $g$ is the Dirac distribution. This sampling method is motivated by the easy computation of time derivatives: $\frac{d\bar{w}}{dt}(t) = -\int w(\tau + t)\frac{dg}{d\tau}(\tau)d\tau$. From our standpoint, this is equivalent to using an LTI operator (convolution with $\tau \mapsto g(-\tau)$) before sampling, and thus preserves the behavior.

### D. Bank of filters

We now suggest a novel application of the Fundamental Lemma. Recall that the behavior of a system can be represented by its spectrum, see Corollary 1. More generally, Theorem 1 and Theorem 3 open the way to using any filter bank that is adapted to a given system. That is, consider $N$ operators $\{F_k \in A|_{T_k}\}_{k=1}^N$, with $T_k \ge 1$. For $s \in \mathbb{R}^Z$, we consider the following extended Hankel matrix

$$H_L^F(s) = [H_L(s_{[1,T]}) \quad H_L(F_1[s]) \quad \dots \quad H_L(F_N[s])]$$

Applying the filters to both input and output signals, i.e., substituting $s$ for input-output data $(u, y)$ of a system $B \in \mathbb{B}_m^{q,n}$, we have:

$$\mathrm{rank} H_{L+n}^F(u) = (L+n)m \implies B|_L = \mathrm{im} \begin{bmatrix} H_L^F(u) \\ H_L^F(y) \end{bmatrix}.$$

The choice of such filters is application-specific, and the main advantage is that one is free to choose filters adapted to the particular scenario: noise reduction, work on different frequency bands in parallel (and not just some specific frequencies), etc. We can for instance draw inspiration from Dynamic Regressor Extension used in parameter estimation schemes [18], and choose first-order filters as

$$F_k[s] = \left( \frac{1}{\sigma - \lambda_k} s \right)_{[1, T_k]}, \quad 0 < \lambda_k < 1.$$

### E. Non-LTI systems with LTI embeddings

Theorem 2, Theorem 3 and Theorem 4 only apply to the behavior of LTI systems. Reformulations of Willems' Lemma were proposed for non-LTI systems through LTI embeddings, e.g. linear parameter-varying systems [19], affine time-invariant systems [20], [21], and bilinear systems [22].

Let $B_{nl} \subset (\mathbb{R}^q)^Z$ a non-LTI system, we say that $B_{emb} \in \mathbb{B}^Q$ is an LTI embedding of $B_{nl}$ if there exists an embedding map $E: (\mathbb{R}^q)^Z \to (\mathbb{R}^Q)^Z$ such that

$$B_{nl} = \{w | E(w) \in B_{emb}\}.$$
Note that $\mathcal{B}_{emb}$ is an LTI system, and that in general the embedding map $E$ is not surjective:

$$ \bar{w} \in \mathcal{B}_{emb} \Leftrightarrow \exists w \in \mathcal{B}_{nl}, E(w) = \bar{w}. $$

Applying LSI transformations to $\mathcal{B}_{emb}$ is allowed, but does not guarantee that the transformed trajectories can be mapped back to the original system $\mathcal{B}_{nl}$: take any $A \in \mathbb{A}$ then

$$ \begin{aligned} w \in \mathcal{B}_{nl} &\Rightarrow E(w) \in \mathcal{B}_{emb} \\ &\Rightarrow A[E(w)] \in \mathcal{B}_{emb} \\ &\Rightarrow \exists \bar{w} \in \mathcal{B}_{nl}, \bar{w} = A[E(w)]. \end{aligned} $$

To ensure the existence of a reverse mapping from $\mathcal{B}_{emb}$ to $\mathcal{B}_{nl}$, one needs to enforce additional constraints on $A$.

Consider the example of affine time-invariant (ATI) systems. Willems' Lemma was adapted to these systems in [20], with applications to non-linear system control [21], [23]. An ATI system of order $n$ with $q$ variables can be defined, with the state-space formalism, as

$$ \mathcal{B} = \left\{ (u, y) \middle| \exists x, \begin{cases} \sigma x = Fx + Bu + e \\ y = Cx + Du + r \end{cases} \right\} \quad (10) $$

where (like in LTI systems) $x \in (\mathbb{R}^n)^Z$ is the state, $u \in (\mathbb{R}^m)^Z$ the input and $y \in (\mathbb{R}^{q-m})^Z$ the output. $F, B, C, D$ are matrices of suitable dimensions, and $e, r$ are constant vectors.

An LTI embedding for (10) can be defined as an LTI system $\bar{B}$ with an extended input $\bar{u}$, with $\forall t, \mathbb{1}(t) = 1$:

$$ \left\{ \begin{array}{l} \sigma x = Fx + \bar{B}\bar{u}, \\ y = Cx + \bar{D}\bar{u}, \end{array} \quad \bar{u} = \begin{bmatrix} u \\ \mathbb{1} \end{bmatrix}, \bar{B} = [B \ e], \bar{D} = [D \ r]. \right. $$

Let $A \in \mathbb{A}$. $A$ preserves $\bar{B}$, but for $A$ to also preserve $\mathcal{B}$, it has to preserve sequences of ones:

$$ \begin{bmatrix} A[u] \\ \mathbb{1} \end{bmatrix} = A[\bar{u}] = \begin{bmatrix} A[u] \\ A[\mathbb{1}] \end{bmatrix}. $$

## VII. CONCLUSION

We have shown that the set of transformations that preserve the behavior of all DT LTI systems is the set of LSI operators $\mathbb{A}$. We demonstrated how a family LSI operators based on geometric sequences, including the Fourier transform, allow a more compact representation of the behavior. These results unify the three reformulations of the Fundamental Lemma (using correlation, cross-spectrum, and frequency domain data) and provide new ones based on filter banks, simplifying the analysis for each case. Regarding the continuous time, we introduced a family of transformations $\bar{\mathbb{A}}$ in CT that also preserves behaviors, opening new directions for research.

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