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# Local identifiability of nonlinear ODE models with time-varying parameters: The general algebraic condition

Agostino Martinelli

► To cite this version:

Agostino Martinelli. Local identifiability of nonlinear ODE models with time-varying parameters: The general algebraic condition. IEEE Transactions on Automatic Control, 2025, pp.1-16. 10.1109/TAC.2025.3627899. hal-05373956

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Submitted on 20 Nov 2025

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# Local identifiability of nonlinear ODE models with time-varying parameters: The general algebraic condition

Agostino Martinelli

*Abstract*—Identifiability describes the possibility of determining the values of the unknown parameters that characterize a dynamic system from the knowledge of its inputs and outputs. This paper finds the general algebraic condition that characterizes this property locally. It is the first-ever necessary and sufficient condition that can be successfully applied in the presence of time-varying parameters. The only requirement is that the time dependence of the parameters is analytical. The use of this condition requires no inventiveness from the user as it simply needs to follow the steps of a systematic procedure that only requires performing the calculation of derivatives and matrix ranks. Time-varying parameters are treated as unknown inputs and the condition is constructed from the recent analytical solution of the unknown input observability problem [1], [2]. The condition is used to study two well-known nonlinear biological models. The former characterizes the dynamics of the HIV, the latter is commonly used to characterize the genetic toggle switch. In both cases, the condition allows us to obtain new results that contradict the current state of the art. The paper also analyzes these discrepancies, showing that they arise from intrinsic difficulties faced by existing methods when applied to identifiability problems involving time-varying parameters.

*Index Terms*—Identifiability; time-varying parameters; Non-linear Unknown Input Observability; Unknown Input Reconstruction; Biological systems; HIV; Genetic toggle switch

## I. INTRODUCTION

Ordinary Differential Equations (ODEs) are used to model a plethora of phenomena in many scientific domains, ranging from the natural and applied sciences up to the social sciences. An ODE model is characterized by a state that consists of several scalar quantities. Its time evolution is precisely described by a set of ordinary differential equations (one for each state component). In addition to the state components, an ODE model can also include a set of parameters which may be known or unknown. Finally, an ODE model is characterized by a set of outputs and, very often, by a set of inputs. The inputs are functions of time that act on the state dynamics and that can be assigned (with some restrictions that depend on the specific case). The outputs are available quantities and can be expressed in terms of the state and, in some cases, also in terms of the above inputs and the above model parameters.

An ODE model may also include the presence of *unknown* inputs, i.e., inputs that act on the system dynamics precisely as the aforementioned inputs. However, they differ from them because they are unknown. From a mathematical perspective,

an unknown input serves the same role as an unknown time-varying parameter (from now on TV-parameter).

*State observability and parameter identifiability* are two structural properties of an ODE model. The former characterizes the possibility of inferring the values that the state components take, starting from the knowledge of the system inputs and outputs (e.g., [3]–[7]). The latter characterizes the possibility of inferring the values of the model parameters, again, from the knowledge of the system inputs and outputs (e.g., [8]–[10]). Finally, the possibility of inferring the values of the TV-parameters has been called in different manners: *input observability, input reconstruction, and sometimes system invertibility* (e.g., [11]–[18]).

Throughout this paper, we adopt the following terminology. *Observability* refers to the components of the state. *Unknown Input Observability* (abbreviated UIO) still refers only to the state components, but when the ODE model also includes the presence of unknown inputs (or TV-parameters). *Identifiability* refers to all the unknown parameters. When a parameter is time-varying we also use the term *unknown input reconstruction* to mean its identifiability. Finally, we use the terms *unknown input* and *TV-parameter* interchangeably.

In this paper, we study the notion of local identifiability, according to the definition provided in Section III (Definition 3). ODE identifiability analysis is present in a large variety of scientific domains and several interesting methods have been proposed. An exhaustive review of the methods proposed up to 2011 can be found in [19]. A more recent overview on definitions and approaches is available in [20]. Moreover, several recent software tools for identifiability analysis are reviewed in [21]. Finally, a recent approach was suggested in [22]. All these methods (in [19], [20], [22]) are interesting and, in many cases, they can successfully be used to detect all the identifiability properties of a given ODE model. However, none of these methods can determine identifiability in the presence of TV-parameters. Furthermore, they cannot be used if the system is time-varying and, in most cases, they can only be used if the system is linear or in the presence of a specific type of nonlinearity (e.g., polynomial). Note that in [19] the authors used one of these methods to study the identifiability of a well-known ODE model in the framework of HIV dynamics. In particular, in Section 6.2 of [19], the study accounts for the presence of a TV-parameter. The results of this study contain an error (the reader is addressed to Section VI where we study exactly the same model and we also analyze the error made by the authors of [19]). Also note that the methods in
[19] can only be applied for a specific type of nonlinearity (polynomial).

In the last decade, thanks to the progress in UIO, several new interesting approaches have been proposed. Specifically, the nonlinear UIO problem was approached by introducing an extended state that includes the original state together with the unknown inputs and their time derivatives up to a given order [23]–[28]. In [25]–[28], this was used to study identifiability in the presence of unknown TV-parameters. In particular, in [25]–[28], several automatic iterative algorithms were introduced. A comparative analysis on the performance of these algorithms (and more recent algorithms based on the same idea) is available in [21]. These algorithms work automatically and are able to check identifiability in the presence of TV-parameters. On the other hand, they suffer from the following limitations:

* They do not converge, automatically. In particular, at each iterative step, the state is extended by including new time derivatives of the unknown inputs. Consequently, if the extended state is observable at a given step, convergence is achieved. However, if this is not the case, the algorithm provides no indication of whether or when observability will eventually be achieved. In other words, in the presence of unobservability, these algorithms remain inconclusive due to the lack of a convergence criterion.

* Due to the previous state augmentation, the computational burden can easily become prohibitive after a few steps.

Recently, we introduced the algebraic solution of the nonlinear UIO [1], starting from the results/derivations presented in [2]<fnref n="1" />. In particular, the systematic procedure introduced in [1] does not encounter the aforementioned limitations<fnref n="2" /> and provides a full answer to the problem of the state observability for nonlinear ODE models in the presence of unknown inputs (or TV-parameters). A MATLAB toolbox implementing the solution from [1] for systems with unknown inputs and no control inputs is available in [31], thus facilitating its practical adoption. The theoretical basis for this simplified case is presented in [32], where a significantly more intuitive and accessible procedure is developed. This latter contribution may serve as a stepping stone for readers interested in understanding the more general theory in [1]. This paper builds upon [1] and introduces the following two new contributions:

1) Introduction of the algebraic condition that characterizes local identifiability in the presence of TV-parameters.

2) Determination of the identifiability properties of a well-known HIV model and a genetic toggle switch model, along with an analysis of the discrepancies with previous studies in the scientific literature.

The main novelties of the first contribution, compared to [1], are twofold: they lie in both the different problem addressed and solved, as well as in the novel derivations

presented. While [1] focused on the state observability problem, this paper investigates the identifiability problem. In [1], the identifiability issue was only marginally addressed, with a sufficient condition that applied exclusively when the state was observable (Theorem 4 in [1]). In contrast, this paper presents a necessary and sufficient condition that holds independently of state observability. The solution to the identifiability problem builds upon the observability solution from [1]. As detailed in the first part of Section V, applying the solution from [1] to the observability problem directly leads to the necessary and sufficient condition for parameter identifiability (Eq. (15)). This is achieved by extending the state to include the model parameters. From this condition, more efficient computational conditions are derived (since they do not require state extension). These include: (i) the necessary and sufficient condition that applies when the state is observable (Theorem 1, extending Theorem 4 from [1], where it was shown that the condition was only sufficient); (ii) the necessary and sufficient condition when the system is in canonical form (Theorem 2); and (iii) the universally applicable necessary condition (Theorem 3). Compared to the condition expressed in Eq. (15), these theorems offer more efficient conditions, as they do not require state extension. As further discussed in the Conclusions (Section X), the computational cost introduced by state extension can become critical in models with multiple unknown inputs. While in the examples considered in this paper the impact is limited, in more complex systems such as those arising in visual-inertial sensor fusion the use of the proposed theorems allows a substantial reduction in dimensionality and computation time.

The paper is organized as follows. Section II provides a basic mathematical characterization of the problem together with the assumptions. In Section III we introduce the definition of identifiability. Section IV reminds the reader of the basic concepts and definitions related to nonlinear unknown input observability, together with the solution in the case when the system is in canonical form (the solution of the general case can be found in Section 5 of [1], while a simplified derivation for systems without control inputs is presented in [32], and it is unnecessary for the derivations in this paper). Section V provides the algebraic condition for the local identifiability. Sections VI, and VII use the algebraic condition to study the local identifiability of two biological models. These sections provide new results that contradict the results available in the current scientific literature. To illustrate the reasons for these discrepancies, these sections provide, from one side, an unequivocal proof about the validity of our results. From another side, they provide an analysis of the discrepancies, and, in this manner, these sections illustrate that the state-of-the-art methods are not fully equipped to handle the complexities of determining identifiability in the presence of TV-parameters. Note that the system in Section VII is not affine in the inputs and illustrates how the algebraic condition can automatically be used also in this case. Section VIII provides an elementary example that illustrates the application of the algebraic condition in the case of time-varying systems and Section IX presents additional elementary models to cover all possible scenarios. Finally, Section X provides our conclusion.

<footnote n="1">¹This solution also relies on results from [29] and [30], which address driftless systems with a single unknown input.</footnote>

<footnote n="2">²Note that, the solution introduced in [1] could need the inclusion and the redefinition of some of the UIs in the state. However, this inclusion/redefinition is targeted and terminates in a finite number of steps, as soon as the extended system achieves its highest unknown input degree of reconstructability (see Section 5 in [1]).</footnote>
## II. SYSTEM AND PROBLEM STATEMENT

### A. System characterization

We refer to the systems characterized by the following equations:

$$ egin{cases} 
\dot{x}(t) = g^0(x, t) + \sum_{k=1}^{m_u} f^k(x, t)u_k(t) + \\ 
\qquad + \sum_{j=1}^{m_w} g^j(x, t)w_j(t) \\ 
y(t) = [h_1(x, t), \dots, h_p(x, t)], 
\end{cases} 
\quad (1) $$

where:

* $x \in \mathcal{M}_n$ is the state and $\mathcal{M}_n$ is a $\mathcal{C}^\infty$-manifold of dimension $n$.

* $u_1(t), \dots, u_{m_u}(t)$ are the known inputs.

* $w_1(t), \dots, w_{m_w}(t)$ are the TV-parameters.

* $f^1, \dots, f^{m_u}, g^0, g^1, \dots, g^{m_w}$ are $\mathcal{C}^\infty$-vector functions of $x$ and $t$, defined on $\mathcal{M}_n \times \mathcal{I}$, where $\mathcal{I} \subseteq \mathbb{R}$ is an open time interval.

* $h_1, \dots, h_p$ are $\mathcal{C}^\infty$-scalar functions of $x$ and $t$, defined on $\mathcal{M}_n \times \mathcal{I}$.

Note that $w_1(t), \dots, w_{m_w}(t)$ can be regarded as system inputs that, precisely as $u_1(t), \dots, u_{m_u}(t)$, act on the system dynamics. However, they differ from them in two fundamental respects: (i) they cannot be assigned, and (ii) they are unknown. We make the following assumption:

**Assumption 1 (Analyticity of $w_1, \dots, w_{m_w}$)** The functions $w_1(t), \dots, w_{m_w}(t)$, are analytic in $t$.

**Remark 1** Although analyticity of the time-varying parameters is formally assumed, it is not essential for the validity of the results presented here or in [1]. In both cases, the derivations only require differentiability up to a sufficiently high finite order. The analyticity assumption is retained mainly for consistency with the formulation in [1]. This perspective is also consistent with our earlier work [29], where differentiability up to a given order was explicitly stated as sufficient.

Note that, for our purposes, the above characterization can easily account for the presence of constant parameters:

$$ Q = [Q_1, \dots, Q_q]^T \in \mathbb{R}^q. $$

This is done by including all of them in the state $x$ and appropriately setting $f^1, \dots, f^{m_u}, g^0, g^1, \dots, g^{m_w}$. Specifically, this involves setting to zero all components that will give $\dot{Q}$ in (1). Note that many nonlinear systems have the structure in (1). In addition, we demonstrate that seemingly more general systems can be characterized by (1). Specifically, let us consider the following more general characterization:

$$ egin{cases} 
\dot{x}(t) = f(x(t), t, u(t), Q, w(t)) \\ 
y(t) = h(x(t), t, u(t), Q, w(t)), 
\end{cases} 
\quad (2) $$

with $u = [u_1, \dots, u_{m_u}]^T$, and $w = [w_1, \dots, w_{m_w}]^T$. Furthermore, the functions $f$ and $h$ are defined on $\mathcal{M}_n \times \mathcal{I} \times \mathcal{M}_{m_u} \times \mathbb{R}^q \times \mathcal{M}_{m_w}$, with $\mathcal{M}_{m_u}$, and $\mathcal{M}_{m_w}$, $\mathcal{C}^\infty$-manifolds of dimension $m_u$, and $m_w$, respectively. The characterization in (2) is more general than the one in (1), since $f$ and $h$ can account for any nonlinearity also with respect to the

TV-parameters and the inputs (which can also appear in the output). We can easily convert (2) to (1), provided that, if $f$ is not affine in $u$ and/or $h$ depends on $u$, the following additional assumption is satisfied:

**Assumption 2 (Only needed if $f$ is not affine in $u(t)$ and/or the output depends on $u(t)$)** The function $u(t)$ is differentiable.

**Algorithm 1 Transform system into the form (1)**

**Require:** Analytic expressions of the dynamics $f$ and outputs $h$ of the system in the form (2).

**Ensure:** Equivalent system expressed in the form (1).

1. Define the extended state as:

$$ S = [x^T \quad u^T \quad w^T \quad Q^T]^T $$

2. Consider the system driven by $\dot{u}$ and $\dot{w}$.

3. Define the vector fields:

$$ g^0(S, t) = \begin{bmatrix} f(S, t) \\ 0_{m_u} \\ 0_{m_w} \\ 0_q \end{bmatrix}, \quad f^k = \begin{bmatrix} 0_n \\ e_{m_u}^k \\ 0_{m_w} \\ 0_q \end{bmatrix}, \quad g^j = \begin{bmatrix} 0_n \\ 0_{m_u} \\ e_{m_w}^j \\ 0_q \end{bmatrix} $$

4. Define the outputs $h_i(S, t)$ as the components of the function $h$ in (2).

The conversion is achieved by following the steps of Algorithm 1, where $0_N$ denotes the zero column vector of dimension $N$, and $e_N^i$ is the unit vector of dimension $N$ with a 1 in the $i$-th position and zeros elsewhere.

In the rest of this paper, we directly refer to the characterization given in (1). The reader is addressed to Section VII for the analysis of a system characterized by (2) and not directly by (1). Note that all the functions in (1) can explicitly depend on time (i.e., $g^0, g^1, \dots, g^{m_w}, f^1, \dots, f^{m_u}, h_1, \dots, h_p$). In other words, we also account for non-autonomous systems, in addition to considering the presence of TV-parameters.

### B. Problem statement

Given the system in (1) with Assumption 1 (which is equivalent to the system in (2) when also Assumption 2 is honoured), the goal of this paper is to provide the general algebraic condition to check the local identifiability of all its unknown parameters.

## III. DEFINITION OF IDENTIFIABILITY IN THE PRESENCE OF TV-PARAMETERS

We introduce the definition of identifiability for the model in (2). The definition will be based on the concept of indistinguishability. We must account for (i) the explicit time dependence of the model, and (ii) the presence of TV-parameters. As the TV-parameters act on the system dynamics as unknown inputs, we use the same concepts which were adopted to introduce the concept of indistinguishability in the presence of unknown inputs (e.g., see Section II in [30] or Definition 6.15 in [2]). In addition, we account for the explicit time
dependence of the model in (2) by proceeding as in [38]. On the other hand, in contrast with these previous works, we are here interested in parameter identifiability instead of state observability. However, state observability and parameter identifiability are interconnected. To account for this, we start by introducing an extended state that includes the state and all the parameters. We use the symbol $\theta$ to denote all the parameters, i.e., $\theta = [Q^T, w^T(t)]^T$. We introduce the $\theta$-state, defined as follows:

$$S = [x^T, \theta^T]^T = [x^T, Q^T, w^T]^T. \quad (3)$$

Note that, for the model in (1) we have $S = [x^T, w_1, \dots, w_{m_w}]^T$, as the constant parameters are already included in $x$.

We denote by $x(t; t_0, \bar{x}; \bar{Q}, \bar{w}; u)$ the state at the time $t \ge t_0$ that satisfies the dynamics in (2) when the state at time $t_0$ is $\bar{x}$, the model parameters are $\bar{Q}, \bar{w}$, and the known input is $u$ (note that $\bar{w}$ and $u$ are functions of time).

Let $\mathcal{V}$ be the Banach function space of all possible first order time derivative of the unknown input functions ($\dot{w}$).

We define indistinguishability as follows:

**Definition 1 (Indistinguishable $\theta$-states)** Given the system in (2), two $\theta$-states, $S_a = [x_a^T, Q_a^T, w_a^T]^T$, $S_b = [x_b^T, Q_b^T, w_b^T]^T$, are indistinguishable at $t_0 \in \mathcal{I}$ if, for any time assignment $u(t)$, $t \in \mathcal{I} \cap [t_0, \infty)$, there exists a nonshy$^3$ subset $\mathcal{V}_a \subseteq \mathcal{V}$ such that, for any unknown input $w_A(t)$ with $\dot{w}_A \in \mathcal{V}_a$ and $w_A(t_0) = w_a$, there exists an unknown input $w_B(t)$ with $w_B(t_0) = w_b$ such that, $\forall t \in \mathcal{I} \cap [t_0, \infty): h(x(t; t_0, x_a; Q_a, w_A; u), t, u(t), Q_a, w_A(t)) = h(x(t; t_0, x_b; Q_b, w_B; u), t, u(t), Q_b, w_B(t))$.

The above definition takes into account that we do not know the time behaviour of the unknown input $w(t)$. In particular, Definition 1 does not exclude the possibility that the outputs produced by $S_a$ and $S_b$ can differ even for many time profiles $w_A(t)$ and $w_B(t)$. However, Definition 1 establishes that the outputs coincide for all the profiles that belong to a nonshy set. In other words, there exists a non-vanishing probability that the outputs coincide$^4$. This will make conservative the following definitions of identifiability and local identifiability (Definitions 2 and 3), because they are based on Definition 1.

**Definition 2 (Identifiability)** The system in (2) is identifiable at a given $x \in \mathcal{M}_n$, $Q \in \mathbb{R}^q$, $w \in \mathcal{M}_{m_w}$, and $t \in \mathcal{I}$ if all the $\theta$-states $[\bar{x}^T, \bar{Q}^T, \bar{w}^T]^T$ indistinguishable from $[x^T, Q^T, w^T]^T$ at $t$, are such that:

$$\bar{Q} = Q, \text{ and } \bar{w} = w. \quad (4)$$

It is identifiable on a subset $\mathcal{G} \subseteq \mathcal{M}_n \times \mathbb{R}^q \times \mathcal{M}_{m_w} \times \mathcal{I}$, if it is identifiable at any $(x, Q, w, t) \in \mathcal{G}$.

**Definition 3 (Local identifiability)** The system in (2) is locally identifiable at a given $x \in \mathcal{M}_n$, $Q \in \mathbb{R}^q$, $w \in \mathcal{M}_{m_w}$, and $t \in \mathcal{I}$ if there exists an open neighbour $\mathcal{B}$ of $[x^T, Q^T, w^T]^T$ such that all the $\theta$-states $[\bar{x}^T, \bar{Q}^T, \bar{w}^T]^T \in \mathcal{B}$ indistinguishable from $[x^T, Q^T, w^T]^T$ at $t$, are such that the same equalities in (4) hold. It is locally identifiable on a subset $\mathcal{G} \subseteq \mathcal{M}_n \times \mathbb{R}^q \times \mathcal{M}_{m_w} \times \mathcal{I}$, if it is locally identifiable at any $(x, Q, w, t) \in \mathcal{G}$.

In the case where the set of all points outside $\mathcal{G}$ has zero Lebesgue measure, we state that the model is identifiable/locally identifiable.

### A. Identifiability of a single parameter

The above definitions can be extended in order to characterize identifiability (or local identifiability) of a single parameter. Let us denote by $\theta_l, l = 1, \dots, q+m_w$, one of the parameters of our model in (2). In particular, $\theta_1, \dots, \theta_q$ are the constant parameters $Q_1, \dots, Q_q$, and $\theta_{q+1}, \dots, \theta_{q+m_w}$ are the TV-parameters $w_1, \dots, w_{m_w}$. The definition of identifiability (or local identifiability) of the parameter $\theta_l$ is obtained by simply replacing the equalities in (4) with the single equality $\bar{\theta}_l = \theta_l$.

### B. Comparison with previous definitions of identifiability

We compare our definitions of identifiability and local identifiability with various definitions in the scientific literature. Unfortunately, no definition in the scientific literature takes into account for the unknown time behaviour of the TV-parameters and an explicit time dependence of the system. As a result, the comparison can only be done by referring to autonomous systems in the absence of TV-parameters. We refer to the definitions summarized in [20]. Our definition regards a continuous system. Hence, we need to compare our definition with Definitions 1-11 in [20] (Definitions 12-21 in [20] hold for a discrete system). Finally, from Table 1 in [20], we remark that Definitions 1-7 hold for a specific input, and Definitions 8-10 hold for a specific initial state. As a result, our definition can only be compared with Definition 11, and, as we aforementioned, the comparison must be done by only referring to autonomous systems in the absence of TV-parameters.

The main difference between our definition and Definition 11 in [20] resides in the manner to deal with the initial state. For simplicity sake, we highlight this aspect by referring to the case in which there are no inputs. Furthermore, we compactly denote by $y(x, Q)$ the output obtained with the initial state $x$ and the parameter vector $Q$.

Let us suppose that the system is identifiable in accordance with Definition 11 in [20]. Then, the following condition holds:

$$\bar{Q} \neq Q \implies y(x, Q) \neq y(x, \bar{Q}), \forall x \in \mathcal{M}_n^0 \quad (5)$$

with $\mathcal{M}_n^0$ a dense subset of $\mathcal{M}_n$.

<footnote>$^3$See [40] for a definition of shy set.</footnote>

<footnote>$^4$For finite-dimensional spaces, the probability is defined by using the concept of measure. Unfortunately, the concept of measure in spaces with infinite dimensions is not trivial. In particular, there is no analogue of Lebesgue measure on an infinite-dimensional Banach space, such as our function space $\mathcal{V}$. One possibility, which is frequently adopted, is to use the concept of prevalent and shy sets [40]. The probability that a given $\dot{w}$ belongs to a shy subset of $\mathcal{V}$, is 0. The probability that a given $\dot{w}$ belongs to a prevalent subset of $\mathcal{V}$, is 1. Finally, the probability that a given $\dot{w}$ belongs to a nonshy subset of $\mathcal{V}$, is strictly larger than 0.</footnote>
Let us consider now our definition. First of all, in this case without inputs, our definition of identifiability reads as follows: The system is identifiable at $x \in \mathcal{M}_n$ and $Q \in \mathbb{R}^q$ if the following condition holds:

$$ \bar{Q} \neq Q \Rightarrow y(x, Q) \neq y(\bar{x}, Q), \forall \bar{x} \in \mathcal{M}_n \quad (6) $$

It is identifiable on a subset $\mathcal{G} \subseteq \mathcal{M}_n \times \mathbb{R}^q$, if it is identifiable at any $(x, Q) \in \mathcal{G}$.

Our condition is clearly stronger because in the inequality in (6) we do not constrain the initial state to be the same at the two members (as it is the case in (5)). Hence, our definition of identifiability is more restrictive than Definition 11 in [20]. In other words, if a system is identifiable according to our definition it is certainly identifiable according to Definition 11 in [20]. Note that Definition 11 in [20] corresponds to the definition of identifiability proposed in [34] (Definition 2 in [34]). One of the contributions of [34] is the proof that this notion of identifiability is equivalent to identifiability with known initial conditions. In contrast, our definition assumes that the initial state is unknown. As a result, it is not surprising that our condition is stronger. If we can determine the model parameters without knowing the initial state (system identifiable with our definition) we can certainly determine them if we also know the initial state (system identifiable with Definition 2 in [34]).

Note that Definition 2 in [34] concerns actually the local identifiability (the inequality $y(x, Q) \neq y(x, \bar{Q})$ holds not only if $Q \neq \bar{Q}$ but also if $Q$ and $\bar{Q}$ are "close"). In our definition of local identifiability, the locality also regards the initial state. Hence, the two initial states, $x$ and $\bar{x}$ in the inequality $y(x, Q) \neq y(\bar{x}, \bar{Q})$, must be "close".

## IV. NONLINEAR UNKNOWN INPUT OBSERVABILITY

We remind the reader of some concepts and definitions introduced in [1], [2], [29], [30], together with a description of the solution of the unknown input observability problem introduced in [1]. For the sake of clarity, we adopt two simple examples (the case studies) to better illustrate these concepts.

### A. Case studies

They are both characterized by the same state, $[x_1, x_2]^T$, and the same dynamics:

$$ \left\{ \begin{aligned} \dot{x}_1(t) &= x_2(t)w_1(t) \\ \dot{x}_2(t) &= x_1(t) + x_2(t). \end{aligned} \right. \quad (7) $$

The first, called CSA, has output

$$ y = h_1(x) = x_1^2 + x_2^2 $$

The second, called CSB, has outputs

$$ y = [h_1(x), h_2(x)] = [x_1^2 + x_2^2, x_2]. $$

They are special cases of (1) with: $m_u = 0, m_w = 1$,

$$ g^0 = \begin{bmatrix} 0 \\ x_1 + x_2 \end{bmatrix}, \quad g^1 = \begin{bmatrix} x_2 \\ 0 \end{bmatrix}. \quad (8) $$

In addition, CSA is characterized by $p = 1$, and $h_1(x) = x_1^2 + x_2^2$, while CSB is characterized by $p = 2$, $h_1(x) = x_1^2 + x_2^2$, $h_2(x) = x_2$.

### B. Basic concepts and definitions in [1]

1) Unknown input reconstructability matrix: Given a system characterized by (1) and $k$ scalar and smooth functions of the state, $\lambda_1(x), \dots, \lambda_k(x)$, the unknown input reconstructability matrix of the system from $\lambda_1, \dots, \lambda_k$ is:

$$ \mathcal{RM}(\lambda_1, \dots, \lambda_k) := \begin{bmatrix} \mathcal{L}_{g^1} \lambda_1 & \mathcal{L}_{g^2} \lambda_1 & \dots & \mathcal{L}_{g^{m_w}} \lambda_1 \\ \mathcal{L}_{g^1} \lambda_2 & \mathcal{L}_{g^2} \lambda_2 & \dots & \mathcal{L}_{g^{m_w}} \lambda_2 \\ \vdots & \vdots & \ddots & \vdots \\ \mathcal{L}_{g^1} \lambda_k & \mathcal{L}_{g^2} \lambda_k & \dots & \mathcal{L}_{g^{m_w}} \lambda_k \end{bmatrix} $$

where $\mathcal{L}_\tau$ is the Lie derivative operator along the vector $\tau$. Note that, for time-varying systems, $\lambda_1, \dots, \lambda_k$ can also depend on time (as in the example given in Section VIII).

2) Observable function and observability codistribution: The definition of an observable function can be found in [29] and is reported here for convenience.

**Definition 4 (Observable function)** A scalar function $\phi: \mathcal{M}_n \times \mathcal{I} \to \mathbb{R}$ is called observable if it is $\mathcal{C}^\infty$ on $\mathcal{M}_n \times \mathcal{I}$ and remains constant over indistinguishable sets.

The observability codistribution is defined as the span of the differentials of all observable functions.

3) Unknown input degree of reconstructability: Given a system characterized by (1), and functions $\lambda_1, \dots, \lambda_k$, the unknown input degree of reconstructability of the system from $\lambda_1, \dots, \lambda_k$ is the rank of $\mathcal{RM}(\lambda_1, \dots, \lambda_k)^5$. In addition, given a system characterized by (1), we will denote by $\mathcal{O}$ its observability codistribution. When the set of functions $\lambda_1, \dots, \lambda_k$ consists of all the independent observable functions (i.e., $\mathcal{O} = \text{span}\{d\lambda_1, \dots, d\lambda_k\}$) we omit to specify the functions and we refer to the unknown input degree of reconstructability of the system. We introduce the following definition:

**Definition 5 (UI identifiers)** Given a system characterized by (1), scalar fields $\tilde{h}_1, \dots, \tilde{h}_m$ are m-UI identifiers if they are observable and rank $\{\mathcal{RM}(\tilde{h}_1, \dots, \tilde{h}_m)\} = m$.

We illustrate these concepts through our case studies. In particular, we compute the unknown input degree of reconstructability from the outputs. For CSA, we need to compute the matrix $\mathcal{RM}(\lambda_1, \dots, \lambda_k)$ for $k = 1$ and $\lambda_1 = x_1^2 + x_2^2$. By using (8), we obtain $\mathcal{RM}(x_1) = [2x_1x_2]$ and the unknown input degree of reconstructability from the outputs is 1. For CSB, we need to compute the matrix $\mathcal{RM}(\lambda_1, \dots, \lambda_k)$ for $k = 2$, $\lambda_1 = x_1^2 + x_2^2$, and $\lambda_2 = x_2$. By using (8), we obtain $\mathcal{RM}(x_1) = [2x_1x_2]$ and the unknown input degree of reconstructability from the outputs is 1. Finally, for both cases, $\tilde{h}_1 \triangleq h_1 = x_1^2 + x_2^2$ is a UI identifier.

<footnote n="5">⁵As in [1], in this paper we always refer to an open set of $\mathcal{M}_n$ where the above rank takes a constant value.</footnote>
4) *Canonic system with respect to its unknown input and canonical form:* By construction, given a system characterized by (1), its unknown input degree of reconstructability from any set of scalar functions cannot exceed $m_w$. A system is canonic with respect to its unknown inputs if its unknown input degree of reconstructability is $m_w$. In general, at the beginning, the observable functions are not available, with the exception of the system outputs, $h_1, \dots, h_p$, (which are certainly observable functions). We say that the system has been set in canonical form with respect to its unknown inputs as soon as the unknown input degree of reconstructability from all the available observable functions is $m_w$.$^6$ Note that there are systems that are not canonic with respect to their unknown inputs and, consequently, cannot be set in canonical form.

Let us refer to our case studies. The unknown input degree of reconstructability from the outputs is $1(= m_w)$, in both cases. Hence, they are both canonic and in canonical form.

### C. State observability when the system is in canonical form

We provide a summary of the solution of the unknown input observability problem for a system in canonical form. The reader can find a proof of its validity in Section 4 of [1] (the special case when $m_w = 1$ and without drift is also addressed in [29] and [30]).

Given a vector field $\phi$, we introduce the operator:

$$ \mathcal{L}_{\phi} \triangleq \mathcal{L}_{\phi} + \frac{\partial}{\partial t}. $$

Let us denote by $\Omega$ the span of the differentials of all the available observable functions. As the system is in canonical form, we can select from the generators of $\Omega$ a set of $m_w$ UI identifiers, which we denote by $\tilde{h}_1, \dots, \tilde{h}_{m_w}$. Starting from these, we introduce the $(m_w + 1) \times (m_w + 1)$ matrix $\mu$, its inverse $\nu$, and the set of vector fields $\tilde{g}^0, \dots, \tilde{g}^{m_w}$. Defining $\tilde{h}_0 = t$, the matrix $\mu$ is defined as follows$^7$:

$$ \mu_{\alpha}^{\beta} = \begin{bmatrix} \mathcal{L}_{g^0} \tilde{h}_{\alpha} & \text{if } \beta = 0 \\ \mathcal{L}_{g^\beta} \tilde{h}_{\alpha} & \text{if } \beta = 1, \dots, m_w \end{bmatrix} \quad (9) $$

where each index, i.e., $\alpha$ and $\beta$, runs from 0 to $m_w$, with the lower index corresponding to the row and the upper index to the column. Finally, $\tilde{g}^\alpha$ are the following vector fields:

$$ \tilde{g}^{\alpha} = \sum_{\beta=0}^{m_w} \nu_{\beta}^{\alpha} g^{\beta}, \quad \alpha = 0, 1, \dots, m_w. \quad (10) $$

The observability codistribution is (see Theorem 1 in Section 4.3 of [1]):

$$ \mathcal{O} = \langle \mathcal{L}_{f^1}, \dots, \mathcal{L}_{f^{m_w}}, \mathcal{L}_{\tilde{g}^0}, \mathcal{L}_{\tilde{g}^1}, \dots, \mathcal{L}_{\tilde{g}^{m_w}} \mid \Omega + \tilde{\mathcal{O}} \rangle \quad (11) $$

where:

* The angle brackets denote the minimal codistribution that includes $\Omega + \tilde{\mathcal{O}}$ and that is simultaneously invariant with respect to $\mathcal{L}_{f^1}, \dots, \mathcal{L}_{f^{m_w}}, \mathcal{L}_{\tilde{g}^0}, \mathcal{L}_{\tilde{g}^1}, \dots, \mathcal{L}_{\tilde{g}^{m_w}}$.

* $\tilde{\mathcal{O}}$ is the codistribution defined as follows:

$$ \tilde{\mathcal{O}} \triangleq \sum_{q=1}^{m_w} \sum_{j=0}^{2n-m_w} \sum_{\alpha_1=0}^{m_w} \dots \sum_{\alpha_j=0}^{m_w} \sum_{i=1}^{m_w} \operatorname{span} \{ d\mathcal{L}_{[f^i]^{(\alpha_1, \dots, \alpha_j)}} \tilde{h}_q \}. $$

The reader can find, in Appendix A, the definition of the algebraic operation, called *Autobracket*, to compute the vector fields $[f^i]^{(\alpha_1, \dots, \alpha_j)}$.

As an illustration, we study the state observability of our case studies. Let us begin with CSA. As mentioned at the end of Section IV-B3, $h_1$ is a UI identifier. From (9), we obtain: $\mu_0^0 = 1$, $\mu_1^0 = 0$, $\mu_1^0 = 2x_2(x_1+x_2)$, and $\mu_1^1 = 2x_1x_2$. Hence, we have: $\nu_0^0 = 1$, $\nu_0^1 = 0$, $\nu_1^0 = -\frac{x_1+x_2}{x_1}$, and $\nu_1^1 = \frac{1}{2x_1x_2}$. From (10), we obtain:

$$ \tilde{g}^0 = \nu_0^0 g^0 + \nu_1^0 g^1 = g^0 - \frac{x_1 + x_2}{x_1} g^1 = \begin{bmatrix} -x_2(x_1 + x_2)/x_1 \\ x_1 + x_2 \end{bmatrix}, $$

$$ \tilde{g}^1 = \nu_0^1 g^0 + \nu_1^1 g^1 = \frac{1}{2x_1x_2} g^1 = \begin{bmatrix} 1/(2x_1) \\ 0 \end{bmatrix}. $$

As $m_w = 0$ the codistribution $\tilde{\mathcal{O}}$ vanishes. In addition, $\Omega = \operatorname{span}\{dh_1\} = \operatorname{span}\{x_1dx_1+x_2dx_2\}$. It is also immediate to check that $\Omega$ is invariant with respect to $\mathcal{L}_{\tilde{g}^0}$, and $\mathcal{L}_{\tilde{g}^1}$. Therefore, $\mathcal{O} = \Omega = \operatorname{span}\{x_1dx_1 + x_2dx_2\}$. Its rank is $1 < 2 = n$, and the state is unobservable.

For CSB, we note that the span of the differentials of the two outputs has rank 2. Hence, the state is observable.

### D. The case of systems that are not in canonical form

We do not provide the solution in this case. The reader can find it in Section 5 of [1]. For systems without control inputs, an equivalent and more concise derivation is given in [32].

## V. MODEL IDENTIFIABILITY

The definition of identifiability provided in Section III, coincides with the definition given for the observability in the presence of unknown inputs and for time-varying systems (e.g., see Definition 6.15 in [2] or Section II in [30]), when applied to the corresponding component of the $\theta$-state. This allows us to study the identifiability of a given parameter by studying the observability codistribution of the $\theta$-state. In our system characterization given in (1), the constant parameters $Q_1, \dots, Q_q$ are already in the state. The algebraic condition that characterizes their local identifiability is trivially $(i = 1, \dots, q)$:

$$ dQ_i \in \mathcal{O}. \quad (12) $$

<footnote>$^6$When we perform an observability analysis we can somehow get new observable functions, in addition to the outputs. This is precisely the case when we follow the steps of Algorithm 9 in [1] (see also Algorithm 1 in [32], which provides an equivalent but more direct derivation for systems without control inputs).</footnote>

<footnote>$^7$Setting $\tilde{h}_0 = t$ reflects a deeper conceptual point, thoroughly discussed in [2], and can be explained as follows. Time influences a dynamical system in two distinct ways. On one hand, it behaves like an uncontrollable input – a property it shares with unknown inputs. On the other hand, it also acts as an output, since sensors typically provide both the measurement and the time at which it is taken. A powerful approach to modeling dynamical systems is to treat time as an unknown input. However, since time is actually known, we already possess a corresponding UI identifier – namely, $t$ itself. Therefore, the choice of setting $\tilde{h}_0 = t$ is both valid and conceptually elegant.</footnote>
Regarding the TV-parameters, the brute force solution con-
sists of including all of them in the state and checking whether
their differential belongs to the observability codistribution.
In other words, we study the system characterized by the
following state:

$$
S \triangleq [x^T, w_1, \dots, w_{m_w}]^T \quad (13)
$$

From the characterization given in (1), we can easily deter-
mine the dynamics of S. We obtain:

$$
\dot{S} = g_S^0 + \sum_{k=1}^{m_u} f_S^k u_k + \sum_{j=1}^{m_w} g_S^j W_j
$$

with

$$
g_S^0 \triangleq \left[ g^0 + \sum_{j=1}^{m_w} g^j w_j, 0, \dots, 0, 0 \right]^T, \quad (14)
$$

$$
g_S^1 \triangleq \begin{bmatrix} 0_n \\ 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \quad g_S^2 \triangleq \begin{bmatrix} 0_n \\ 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \quad \dots, \quad g_S^{m_w} \triangleq \begin{bmatrix} 0_n \\ 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix},
$$

$$
f_S^1 \triangleq \begin{bmatrix} f^1 \\ 0_{m_w} \end{bmatrix}, \quad f_S^2 \triangleq \begin{bmatrix} f^2 \\ 0_{m_w} \end{bmatrix}, \quad \dots, \quad f_S^{m_w} \triangleq \begin{bmatrix} f^{m_w} \\ 0_{m_w} \end{bmatrix},
$$

$0_n$ and $0_{m_w}$ denote the zero column vectors of dimension $n$
and $m_w$, respectively, and $W_j = \dot{w}_j$, for $j = 1, \dots, m_w$.

Therefore, the dynamics of *S* have the same affine struc-
ture in both the *m*<sub>*u*</sub> known inputs (which remain the same,
*u*<sub>1</sub>, . . . , *u*<sub>*m*<sub>*u*</sub></sub>), and in the *m*<sub>*w*</sub> unknown inputs (which become
the first order time derivatives of the original unknown inputs).
Consequently, we can use the solution of the unknown input
observability problem to determine the observability codistri-
bution.

Let $\mathcal{O}^S$ denote the observability codistribution of the ex-
tended state in (13). This notation will be used throughout the
article. The condition to check the local identifiability of the
original TV-parameters is $(j = 1, \dots, m_w)$:

$$
dw_j \in \mathcal{O}^S. \tag{15}
$$

There are two particularly significant cases that frequently
arise in which it is unnecessary to include the TV-parameters
in the state, allowing us to directly obtain the condition for
the identifiability of these parameters by referring only to the
original system. These two cases are:

1) When *x* is observable.

2) When the original system is in canonical form.

Section V-A provides the condition in the first case (Theo-
rem 1) and Section V-B provides the condition in the second
case (Theorem 2). Finally, Section V-C provides a simple
necessary condition for local identifiability that applies to all
systems (Theorem 3).

Note that for systems that are not affine in some of the inputs
(i.e., those not directly characterized by (1)), as mentioned in
Section II-A, it is necessary to include these inputs in the state.
For these systems, we must directly study the observability of

the $\theta$-state. Consequently, the model identifiability is deter-
mined by the condition in (15). An example can be found in
Section VII.

A. Identifiability when the state is observable

In [1], we proved that, when the state that characterizes the
system is observable, if the system is canonic with respect
to its unknown inputs, then all the unknown inputs can be
reconstructed (Theorem 4 in [1]). It also holds the viceversa
and, consequently, when the state is observable, we obtain a
complete answer to our problem. We have the following result:

**Theorem 1** Let us consider a system characterized by (1) and let us suppose that the state is observable. The system is canonic with respect to its unknown inputs if and only if all the unknown inputs are locally identifiable.

**Proof:** The proof is given in Appendix B. ▲

B. Identifiability when the system is in canonical form

Let us denote by $\mathcal{O}^{\perp}$ the orthogonal distribution of $\mathcal{O}$.

**Theorem 2** The unknown input $w_j$ ($j = 1, \dots, m_w$) is locally identifiable if and only if, for any $\xi \in \mathcal{O}^{\perp}$ and for each $\alpha = 0, 1, \dots, m_w$, the $(m_w + 1)$-dimensional vector $\xi^{\alpha}$ with components indexed by $\beta = 0, 1, \dots, m_w$, defined as

$$
\xi_{\beta}^{\alpha} = \begin{cases} \mathcal{L}_{\xi} \dot{\mathcal{L}}_{g^0} \tilde{h}_{\beta}, & \alpha = 0, \\ \mathcal{L}_{\xi} \mathcal{L}_{g^{\alpha}} \tilde{h}_{\beta}, & \alpha = 1, \dots, m_w, \end{cases} \quad (16)
$$

satisfies the following condition: the $j^{th}$ component of the
product $\nu\xi^{\alpha}$ (where $\nu$ is the inverse of $\mu$ as defined in (9))
vanishes, i.e.,

$$
[\nu\xi^{\alpha}]_j = 0. \tag{17}
$$

In particular, if for any $\xi \in \mathcal{O}^{\perp}$ and for each $\alpha = 0, 1, \dots, m_w$, $\xi^{\alpha} \in \ker(\nu)$, then all the unknown inputs are locally identifiable.

**Proof:** The proof is given in Appendix C.

We conclude by applying these results to our case studies.
For CSA, since the system is in canonical form, we apply
the result stated by Theorem 2. The distribution orthogonal
to $\mathcal{O}$ is: $\mathcal{O}^{\perp} = \operatorname{span}\{\xi\}$ with $\xi = [-x_2, x_1]^T$. We obtain:
$\xi_1^0 = 2x_1^2 + 4x_1x_2 - 2x_2^2$, and $\xi_1^1 = 2x_1^2 - 2x_2^2$. For $\alpha = 0$, by
substituting the above $\xi_1^0$ and $\nu_1^1 = \frac{1}{2x_1x_2}$ in (15), we obtain:
$\nu_1^1\xi_1^0 = x_1/x_2 + 2 - x_2/x_1 \neq 0$. Therefore, $w$ is unidentifiable,
even locally. For CSB, since the state is observable, we apply
the result stated in Theorem 1. Given that the system is
canonic, we conclude that $w$ is locally identifiable.

C. *Necessary condition for local identifiability*

Independently of the observability of the state, we have the
following result:

**Theorem 3** If all the TV-parameters are locally identifiable
the system is canonic.

**Proof:** The proof is given in Appendix D.
### D. Algebraic condition recap: a step-by-step procedure

We conclude this section by outlining the steps to determine the local identifiability of the parameters for any ODE model characterized by (2).

### Algorithm 2 Check Local Identifiability of a Nonlinear ODE Model with Time-Varying Parameters

**Require:** Analytic expressions of the system dynamics $f$ and output functions $h$ as in (2).

**Ensure:** Assessment of local identifiability for constant and time-varying parameters.

1. **Include Constant Parameters:** Augment the state by including all constant parameters as additional state variables.

2. **if** the system equations do not satisfy the form (1) **then**

3. Transform the model using Algorithm 1.

4. **Compute Observability Codistribution:**

5. **if** the system is in canonical form **then**

6. Compute the observability codistribution $\mathcal{O}$ using (11).

7. **else**

8. Compute $\mathcal{O}$ using Algorithm 9 in [1] (see also Algorithm 1 in [32] for an equivalent but more direct derivation for systems without control inputs).

9. **Check Identifiability of Constant Parameters:** Use the condition in (12) to assess local identifiability.

10. **Check Identifiability of TV-Parameters:**

11. **if** the system is canonical and the state is observable **then**

12. The model is locally identifiable (Theorem 1).

13. **if** the system is canonical but the state is unobservable **then**

14. Use Theorem 2 to determine the identifiability of the TV-parameters.

15. **if** the system is not canonical **then**

16. The model is unidentifiable (Theorem 3).

Note that Algorithm 9 in [1], or Algorithm 1 in [32], may redefine TV-parameters; in this case, for parameters included in the state of the new system, it suffices to check whether their differential belongs to the observability codistribution.

## VI. FIRST EXAMPLE: HIV INFECTION

We study the observability and the identifiability properties of the ODE model characterized as follows:

$$ \left\{ \begin{aligned} \dot{T}_U(t) &= \lambda - \rho T_U(t) - \eta(t) T_U(t) V(t) \\ \dot{T}_I(t) &= \eta(t) T_U(t) V(t) - \delta T_I(t) \\ \dot{V}(t) &= N \delta T_I(t) - cV(t) \\ y(t) &= [V(t), T_U(t) + T_I(t)], \end{aligned} \right. \quad (18) $$

This model includes one TV-parameter, $\eta$, and five constant parameters, $\lambda, \rho, \delta, N, c$. All of them are assumed to be unknown. This model is commonly adopted to describe the HIV dynamics in HIV-infected patients with antiretroviral treatment [19], [28], [33]–[36] [41]. The interested reader is referred to [41] for a biological description of this system, which includes the physical meaning of all the aforementioned states and parameters. Additionally, in [41], we study other

variants of the HIV model and characterize the minimal information that must be added to the outputs to make all model parameters identifiable and ensure that the state is observable. Finally, for each variant, we provide a complete description of the system solely in terms of an observable state.

### A. Observability and identifiability analysis

We run Algorithm 2. Line 1 sets:

$$ x = [T_U, T_I, V, \lambda, \rho, \delta, N, c]^T \quad (19) $$

The system is directly a special case of (1). In particular, $m_u = 0, m_w = 1, p = 2, h_1(x) = V, h_2(x) = T_U + T_I, g^0 = [\lambda - \rho T_U, -\delta T_I, N \delta T_I - cV, 0, 0, 0, 0, 0]^T$, and

$$ g^1 = [-T_U V, T_U V, 0, 0, 0, 0, 0, 0]^T. \quad (20) $$

We have: $\mathcal{L}_{g^1} h_1 = \mathcal{L}_{g^1} h_2 = 0$. Consequently, the rank of the unknown input reconstructability matrix from $h_1, h_2$ vanishes. In particular, it is smaller than $m_w(=1)$. The system is not in canonical form with respect to its unknown input. On the other hand, it is immediate to realize that the following two scalar fields are observable: $h_3 \triangleq \dot{h}_1 = N \delta T_I - cV$, and $h_4 \triangleq \dot{h}_2 = \lambda - \rho T_U - \delta T_I$. Note that these two scalar fields are automatically determined by executing Algorithm 9 in [1], or Algorithm 1 in [32] (see for instance Section 6.1 of [37]). Now, the system is in canonical form. In particular, $\mathcal{L}_{g^1} h_4 \neq 0$ and we introduce the following UI identifier:

$$ \tilde{h}_1 = h_4 = \lambda - \rho T_U - \delta T_I \quad (21) $$

(we would obtain the same final result by setting $\tilde{h}_1 = h_3$, as also $\mathcal{L}_{g^1} h_3 \neq 0$). From (9), we obtain:

$$ \mu_1^0 = T_I \delta^2 - \rho(\lambda - T_U \rho), \quad \mu_1^1 = -T_U V(\delta - \rho), $$

$$ \nu_1^0 = \frac{T_I \delta^2 + T_U \rho^2 - \lambda \rho}{T_U V(\delta - \rho)}, \quad \nu_1^1 = -\frac{1}{T_U V(\delta - \rho)} \quad (22) $$

$\nu_0^0 = 1$, and $\nu_0^1 = 0$. Hence, from (10) we obtain:

$$ \hat{g}^0 = \left[ -\frac{\delta(T_I \delta - \lambda + T_U \rho)}{\delta - \rho}, \frac{\rho(T_I \delta - \lambda + T_U \rho)}{\delta - \rho}, NT_I \delta - Vc, 0, 0 \right]^T, $$
and $\hat{g}^1 = \left[ \frac{1}{\delta - \rho}, -\frac{1}{\delta - \rho}, 0, 0 \right]^T$. Finally, as $m_u = 0$,

the codistribution $\tilde{\mathcal{O}}$ vanishes. From (11), we have: $\mathcal{O} = \langle \dot{\mathcal{L}}_{\hat{g}^0}, \mathcal{L}_{\hat{g}^1} \mid \operatorname{span}\{dh_1, dh_2, dh_3, dh_4\} \rangle$.

By a direct computation, we finally obtain: $\mathcal{O} = \operatorname{span}\{dh_1, dh_2, dh_3, dh_4, d\mathcal{L}_{\hat{g}^0} h_3, d\mathcal{L}_{\hat{g}^1} h_3, d\mathcal{L}_{\hat{g}^0}^2 h_3\}$.

The rank of $\mathcal{O}$ is 7, which is smaller than the dimension of the state in (19). Its orthogonal distribution is:

$$ \mathcal{O}^{\perp} = \operatorname{span}\{[T_I \delta, -T_I \delta, 0, 0, 0, \delta(\delta - \rho), N \rho, 0]^T\}. \quad (23) $$

It is immediate to verify the following:

$$ d\lambda \in \mathcal{O}, \ d\rho \in \mathcal{O}, \ d\delta \notin \mathcal{O}, \ dN \notin \mathcal{O}, \ dc \in \mathcal{O}. $$

(it suffices to check whether the above differentials are or not orthogonal to $\mathcal{O}^{\perp}$). Therefore, regarding the constant parameters, two of them ($\delta$ and $N$) are unidentifiable, even
locally. This result contradicts the result available in the state of the art (e.g., see Section 6.2 of [19] or Section 3.3 of [28]). In Section VI-B, we provide an unequivocal proof that validates our result and, in Section VI-C, we analyze the origin of these discrepancies.

We investigate the identifiability of the TV-parameters. The state is unobservable and we use Theorem 2. In accordance with Equation (17), we need to compute $\xi_1^0$ and $\xi_1^1$. From (20) and (21) we obtain: $\mathcal{L}_{g^0}\tilde{h}_1 = T_I\delta^2 - \rho(\lambda - T_U\rho)$, $\mathcal{L}_{g^1}\tilde{h}_1 = -T_UV(\delta - \rho)$. From Equation (16), by using the generator of $O^\perp$ in (23) for $\xi$, we obtain: $\xi_1^0 = T_I\delta(\delta - \rho)^2$, $\xi_1^1 = -V\delta(T_I + T_U)(\delta - \rho)$. The condition in (17) is not honoured (for both $\alpha = 0, 1$). This means that the TV-parameter $\eta(t)$ is not identifiable, even locally. This result contradicts the result available in the state of the art (e.g., see Section 6.2 of [19] or Section 3.3 of [28]). In Section VI-B, we provide an unequivocal proof that validates our result and, in Section VI-C, we analyze the origin of these discrepancies.

## B. Direct proof

We provide an unequivocal proof of the validity of the results obtained in Section VI-A. In particular, we directly prove the non uniqueness of $T_U, T_I, \delta, N$, and $\eta$ by exhibiting infinitely many distinct but indistinguishable values of these quantities. The non uniqueness is even local.

Let us denote by $T_U(t), T_I(t), V(t), \lambda, \delta, \rho, N, c$, and $\eta(t)$ the true state and parameters. Starting from them, we introduce the following new states and parameters that depend on a given parameter, denoted by $\tau$, that can take all the values in a given interval $(-\epsilon, \epsilon)$ (for a given $\epsilon \in \mathbb{R}^+$):

$$ \begin{aligned} \bar{T}_U &= \bar{T}_U(t, \tau) &= T_U(t) + T_I(t) - \frac{T_I(t)}{\rho} (\delta e^{-\rho\tau} + \rho - \delta) \\ \bar{T}_I &= \bar{T}_I(t, \tau) &= \frac{T_I(t)}{\rho} (\delta e^{-\rho\tau} + \rho - \delta) \\ \bar{V} &= \bar{V}(t) &= V(t) \\ \bar{\lambda} &= \bar{\lambda}(\tau) &= \lambda \\ \bar{\delta} &= \bar{\delta}(\tau) &= \frac{\delta\rho}{(\rho-\delta)e^{\rho\tau}+\delta} \\ \bar{\rho} &= \bar{N}(\tau) &= \rho \\ \bar{c} &= \bar{N}(\tau) &= c, \end{aligned} \quad (24) $$

$$ \bar{\eta} = \bar{\eta}(t, \tau) = \quad (25) $$

$$ \frac{\eta(t)T_U(t)V(t)e^{\rho\tau} + [T_I(t)\delta^2 - T_I(t)\delta\rho - \eta(t)T_U(t)V(t)\delta][e^{\rho\tau} - 1]}{V(t)[T_I(t)\delta + T_U(t)\rho]e^{\rho\tau} - V(t)T_I(t)\delta} $$

We claim that $\bar{T}_U, \bar{T}_I, \bar{V}, \bar{\eta}, \bar{\lambda}, \bar{\delta}, \bar{\rho}, \bar{N}$, and $\bar{c}$ are indistinguishable from $T_U, T_I, V, \eta, \lambda, \delta, \rho, N$, and $c$, respectively. We prove this claim in two distinct manners: analytical (Section VI-B1) and numerical (Section VI-B2).

1) Analytical proof: Our statement can be easily checked by a direct substitution. First, the outputs $y_1(t) = T_U(t) + T_I(t)$ and $y_2(t) = V(t)$ remain the same because $\bar{T}_U(t, \tau) + \bar{T}_I(t, \tau) = T_U(t) + T_I(t)$ and $\bar{V}(t) = V(t)$. Second, from (18) and (24-25) it is possible to check, by a simple substitution, the validity of the following equations:

$$ \left\{ \begin{aligned} \frac{d}{dt}\bar{T}_U(t, \tau) &= \bar{\lambda} - \bar{\rho}\bar{T}_U(t, \tau) - \bar{\eta}(t, \tau)\bar{T}_U(t, \tau)\bar{V}(t) \\ \frac{d}{dt}\bar{T}_I(t, \tau) &= \bar{\eta}(t, \tau)\bar{T}_U(t, \tau)\bar{V}(t) - \bar{\delta}\bar{T}_I(t, \tau) \\ \frac{d}{dt}\bar{V}(t) &= \bar{N}\delta T_I(t, \tau) - \bar{c}\bar{V}(t) \end{aligned} \right. \quad (26) $$

In other words, $\bar{T}_U, \bar{T}_I$, and $\bar{V}$ satisfy exactly the same dynamical equations in (18) with the new parameters.

We conclude that $\bar{T}_U, \bar{T}_I, \bar{V}, \bar{\eta}, \bar{\lambda}, \bar{\delta}, \bar{\rho}, \bar{N}$, and $\bar{c}$ cannot be distinguished from $T_U, T_I, V, \eta, \lambda, \delta, \rho, N$, and $c$, respectively. As $\bar{T}_U \neq T_U$, and $\bar{T}_I \neq T_I$, we conclude that $T_U$, and $T_I$ are not observable. Regarding the model parameters, as $\bar{\eta} \neq \eta, \bar{\delta} \neq \delta$, and $\bar{N} \neq N$, we conclude that $\eta, \delta$, and $N$, are not identifiable. Finally, we remark that $\tau$ can take infinitesimal values and that, in the limit of $\tau \to 0$, we have: $\bar{T}_U \to T_U, \bar{T}_I \to T_I, \bar{\eta} \to \eta, \bar{\delta} \to \delta$, and $\bar{N} \to N$. Therefore, we conclude that the unobservability and unidentifiability of $T_U, T_I, \eta, \delta$, and $N$, is even local. This validates the results in Section VI-A.

2) Numerical proof: We refer to the same data available in the literature (e.g., see the electronic supplementary material of [28]). The data set is characterized as follows. The time interval is $\mathcal{I} = [0, 201]$ in days. The initial conditions are$^8$ $T_U(0) = T_I(0) = 300 \text{ CD4/mm}^3$, $V(0) = 10^5 \text{ copies/ml}$, $\lambda = 36 \text{ mm}^{-3} \text{d}^{-1}$, $\rho = 0.108 \text{ d}^{-1}$, $\delta = 0.5 \text{ d}^{-1}$, $N = 10^3 \text{ mm}^3 \text{ ml}^{-1}$, and $c = 3 \text{ d}^{-1}$. The TV-parameter is:

$$ \eta(t) = k \left(1 - 0.9 \cos\left(\frac{\pi}{1000} t\right)\right), \quad (27) $$

with $k = 0.00009$.

Figure 1 displays the two outputs, $y_1 = V$ and $y_2 = T_U + T_I$. The blue solid line displays the outputs obtained with the true state. The true state at time $t$ is obtained by directly integrating the differential equations in (18) with the constant parameters equal to the true values, the TV-parameter $\eta(t)$ set as in (27), and the initial conditions set to the true values. Red circles display the outputs obtained by generating indistinguishable states as follows. The state at time $t$ is obtained by directly integrating the differential equations in (18) with the constant parameters equal to the values in (24) for $\tau = 1$ (i.e., $\bar{\lambda} = \lambda = 36 \text{ mm}^{-3} \text{d}^{-1}$, $\bar{\rho} = \rho = 0.108 \text{ d}^{-1}$, $\bar{\delta} = \delta\rho / ((\rho-\delta)e^{\rho\tau} + \delta) \approx 0.8532 \text{ d}^{-1}$, $\bar{N} = Ne^{\rho\tau} \approx 1114 \text{ mm}^3 \text{ ml}^{-1}$, and $\bar{c} = c = 3 \text{ d}^{-1}$), TV-parameter $\eta(t)$ set as in (25) with $\tau = 1$, and the initial conditions equal to the values obtained by using the first three equations in (24) at $t = 0$ and with $\tau = 1$ (i.e., $\bar{T}_U(0) = T_U(0) + T_I(0) - \frac{T_I(0)}{\rho} (\delta e^{-\rho\tau} + \rho - \delta) \approx 442.184 \text{ CD4/mm}^3$, $\bar{T}_I(0) = \frac{T_I(0)}{\rho} (\delta e^{-\rho\tau} + \rho - \delta) \approx 157.816 \text{ CD4/mm}^3$, $\bar{V}(0) = V(0) = 10^5 \text{ copies/ml}$). From the profiles displayed in Figure 1 it is possible to conclude that the outputs obtained with the true state coincide with the outputs built by using (24) and (25) with $\tau = 1$, and this directly and unequivocally proves their indistinguishability. Note that we obtained the same result by using other values of $\tau$ and also by using different data sets.

These direct proofs (analytical and numerical) are trivially obtained by simple substitution (Section VI-B1) and numerical integration (Section VI-B2), once the indistinguishable states and parameters in (24-25) are determined. The determination of these quantities may seem complex, but they are generated by a systematic procedure. Its presentation is beyond the scope of this paper; the reader can find it in Section 5.1 of

<footnote>$^8$Note that in the original data set $T_U(0) = 600 \text{ CD4/mm}^3$ and $T_I(0) = 0 \text{ CD4/mm}^3$.</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. The outputs $y_1 = V$ (left side) and $y_2 = T_U + T_I$ (right side). The blue solid line displays the output obtained with the true state while red circles display the outputs obtained by generating the indistinguishable state with $\tau = 1$ as explained in the text.</figcaption></figure>

[37] or Sections 2.4 and 2.7 of [41]. A more detailed and comprehensive treatment of this procedure will appear in the related monograph [32].

### C. Comparison with state-of-the-art methods

We examine the analysis of observability and identifiability provided by current state-of-the-art methods. To the best of our knowledge, all observability/identifiability analyses of this system are given in [19] (Section 6.2) and in [28] (Section 3.3). In both cases, the authors concluded that the system is observable and identifiable. Sections VI-C1 and VI-C2 analyze the origin of these discrepancies.

1) Error in [19]: The derivation given in Section 6.2 of [19] is based on an intermediate result that is correct. This result states that the TV-parameter $\eta(t)$ is identifiable if all the constant parameters are identifiable. The error is in the derivation that proves that the constant parameters are identifiable.

This derivation is based on two steps. In the first step the authors obtain Eq. (6.23), which is:

$$ \ddot{y}_1 y_2 \dot{y}_2 - \dot{y}_1 y_2 \ddot{y}_2 - \delta y_1 y_2 \ddot{y}_2 + \lambda y_2 \ddot{y}_2 + (\rho - c) \dot{y}_1 y_2 \dot{y}_2 \quad (28) \\ + (\delta \rho - \delta^2 - \delta c) y_1 y_2 \dot{y}_2 + \lambda c y_2 \dot{y}_2 + \rho c \dot{y}_1 y_2^2 + (\rho \delta c - \delta^2 c) y_1 y_2^2 - N \delta y_1 \dot{y}_1 y_2 \\ + c \dot{y}_1 y_2^2 - N \delta (\rho + \delta) y_1 \dot{y}_1 y_2 - N \delta^2 \rho y_1^2 y_2 + N \delta^2 \lambda y_1 y_2 = 0. $$

Note that the expression in (6.23) provided in [19] contains two typos that have recently been corrected. The reader can verify that (28) matches (6.23) given in [39]. Let us compactly denote the expression in (28) by $\phi = \phi(y_1, \ddot{y}_1, \ddot{y}_2, y_2, \ddot{y}_2, \ddot{y}_2, \lambda, \delta, \rho, N, c)$. In this notation, (28) becomes: $\phi = 0$. The authors suggested two methods to prove the local identifiability of the constant parameters: (i) by using the third statement of Theorem 3.8 in [19], and (ii) by using the implicit function Theorem (in accordance with the procedure given in Section 3.4 of [19])<fnref n="9" />. We cannot comment the first method because the authors only provided the last polynomial (i.e., $B_n(u, y, \theta_1, \dots, \theta_q)$) of Theorem 3.8. They did not provide the remaining polynomials mentioned by the theorem (i.e., $B_1, \dots, B_{n-1}$).

Regarding the second method, the error is pointed out as follows. In accordance with the procedure in Section 3.4 of

[19], we need, first of all, to compute the first 4 time derivatives of $\phi$. In this manner, we obtain four additional equations, i.e., $\phi^1 \equiv \frac{d}{dt}\phi = 0, \dots, \phi^4 \equiv \frac{d^4}{dt^4}\phi = 0$. Then, we include all the above functions in the vector function $\Phi$. In other words, we introduce the vector function $\Phi \equiv [\phi, \phi^1, \phi^2, \phi^3, \phi^4]^T$. As the parameters $\lambda, \delta, \rho, N, c$ are constant, $\Phi$ is a function of them and of the two outputs ($y_1, y_2$) and their time derivatives up to the sixth order. To check the identifiability of $\lambda, \delta, \rho, N, c$, the procedure in Section 3.4 of [19] uses the following equation:

$$ \Phi = 0 \quad (29) $$

Now, by using the implicit function Theorem, we need to compute the Jacobian of $\Phi$ with respect to $\lambda, \delta, \rho, N, c$. By proceeding in this manner, we obtain a 5 × 5 matrix that we denote by $M$. The entries of $M$ are expressed in terms of $\lambda, \delta, \rho, N, c, y_1, y_2$, and the time derivatives of $y_1, y_2$, up to the sixth order.

The error made by the authors of [19] was to consider independent all the above quantities, i.e., $\lambda, \delta, \rho, N, c, y_1, y_2$, and the time derivatives of $y_1, y_2$, up to the sixth order. By doing this we obtain that the rank of $M$ is equal to 5 (and the authors concluded that the model parameters are identifiable).

On the other hand, the two output functions, $y_1(t), y_2(t)$, and their time derivatives up to any order, cannot take arbitrary values. They must satisfy the constraints due to the system dynamics. In [19], neither in Section 3.4, nor in Section 6.2, this fundamental aspect was mentioned. In accordance with the dynamics constraints, we must express the two outputs ($y_1, y_2$) and their time derivatives up to the sixth order, in terms of the state ($T_U, T_I, V$), the constant model parameters $\lambda, \delta, \rho, N, c$, and the TV-parameter $\eta$ and its time derivatives up to the fifth order. We obtain all these expressions by repetitively using (18). Finally, we compute the rank of $M$ and, by a direct computation, we obtain the following fundamental property. Independently of the values of $\lambda, \delta, \rho, N, c, T_U, T_I, V, \eta, \eta^{(1)}, \eta^{(2)}, \eta^{(3)}, \eta^{(4)}, \eta^{(5)}$ (with $\eta^{(k)} \equiv \frac{d^k \eta}{dt^k}$), this rank cannot exceed 4 < 5. As a result, it is erroneous to conclude that Equation (29) allows us to locally identify all the five parameters $\lambda, \delta, \rho, N, c$.

It is also possible to determine the null space of $M$. Specifically, by a direct computation, we obtain:

$$ \mathrm{null}(M) = \mathrm{span}\{\zeta\}, \quad \zeta = [0, \delta(\delta - \rho), 0, N\rho, 0]^T. $$

The vector field $\zeta$, appearing here in a different context, consists of components four through eight of the generator of the Lie algebra associated with the one-parameter Lie group underlying Eqs. (24)–(25); details are in Sections 5.1 and 6.5 of [37], with a more complete discussion in the monograph [32]. In particular, this can be verified by noting that the last five equations in (24), in the limit $\tau = \epsilon \to 0$, become:

$$ \begin{bmatrix} \bar{\lambda} \\ \bar{\delta} \\ \bar{\rho} \\ \bar{N} \\ \bar{c} \end{bmatrix} = \lambda \\ = \delta + \epsilon \delta(\delta - \rho) \\ = \rho \\ = N + \epsilon N \rho \\ = c \Rightarrow \begin{bmatrix} \bar{\lambda} \\ \bar{\delta} \\ \bar{\rho} \\ \bar{N} \\ \bar{c} \end{bmatrix} = \begin{bmatrix} \lambda \\ \delta \\ \rho \\ N \\ c \end{bmatrix} + \epsilon \zeta. $$

<footnote n="9">⁹The implicit function Theorem is used in [19] to prove the algebraic identifiability (Definition 2.9 in [19]), which implies the local identifiability (Definition 2.6 in [19]) on the basis of Theorem 3.9 in [19].</footnote>
This is a further proof of the indistinguishability of $\bar{\delta}$ and $\bar{N}$ from $\delta$ and $N$.

Further details related to the analysis presented in this section can be found in the preprint [42]. In particular, [42] provides MATLAB codes that reproduce the computations shown here, as well as a simplified version of the model in (18). For this simplified case, the same type of error found in [19] is demonstrated by systematically mapping each step of the analysis to the reduced setting.

2) Discrepancy with the results in [28]: Regarding the analysis provided in [28], the origin of the discrepancy lies in the assumption made in Section 2 of [28] (Assumption 2.5). This assumption states that the time derivatives of the time-varying parameters vanish above a given order. This assumption is notably restrictive, as it does not hold in many common cases (e.g., the function $\sin(t)$ does not satisfy this assumption). Furthermore, this assumption may be misunderstood: one might mistakenly consider it equivalent to the reasonable approximation of time-varying parameters by polynomials. However, this is not the case. Approximating time-varying parameters by polynomials is a much less restrictive condition and does not permit substituting them directly with unknown polynomials to assess model identifiability. We illustrate this important distinction through a simple example. We consider the elementary system characterized by the constant parameter $p$, the TV-parameter $w(t)$, and the following dynamics and output:

$$ \begin{bmatrix} \dot{x} = px + w \\ \dot{p} = 0 \\ y = x. \end{bmatrix} \qquad (30) $$

By using the solution in Section V, or through intuitive reasoning, we obtain that both the parameters $w$ and $p$ are unidentifiable, even locally. On the other hand, they are identifiable if Assumption 2.5 is used. Let us analyze this in detail and determine their values for a specific case.

By differentiating the first equation in (30) multiple times, we obtain, for any integer $k$:

$$ y^{(k+1)} = py^{(k)} + w^{(k)}. \qquad (31) $$

Let us set:

$$ w(t) = A_0 + A_1 t + A_2 t^2 + A_3 t^3 + \dots + A_i t^i, \qquad (32) $$

and then attempt to identify the coefficients $A_0, A_1, \dots, A_i$ for a specific setting. We assume that the true parameters are: $w^{\text{True}}(t) = e^{2t}$, $p^{\text{True}} = 1$. We also assume that the true initial state is $x(0) = 1$. Note that $w^{\text{True}}(t) = e^{2t}$ is analytic and can be approximated by a polynomial. By integrating (30) with the true values we easily obtain: $y(t) = x(t) = e^{2t}$. Therefore, the time derivatives of the output at $t = 0$ are:

$$ y^{(k)} := \frac{d^k y}{dt^k} \Big|_{t=0} = 2^k. \qquad (33) $$

Let us use Assumption 2.5. We consider $k \ge i + 1$ and differentiate (32) $k$ times. We obtain: $w^{(k)} = 0$. From (31), we easily obtain $p = \frac{y^{(k+1)}}{y^{(k)}}$. Finally, by using (33), we obtain: $p = \frac{2^{k+1}}{2^k} = 2 \ne p^{\text{True}} = 1$. Then, by using this value for

$p$ in the first equation of (30), we obtain: $w = \dot{x} - px = \dot{y} - 2y = 2e^{2t} - 2e^{2t} = 0 \ne w^{\text{True}}(t) = e^{2t}$. Therefore, by using Assumption 2.5 we determine the following values for our polynomial in (32): $A_0 = A_1 = A_2 = \dots = A_i = 0$. The result is catastrophic. The above values have nothing to do with the coefficients of a polynomial that approximates $w^{\text{True}}(t) = e^{2t}$, regardless of the chosen $i$.

3) Comparison with state-of-the-art methods: As we stated in the introduction, the methods available in the state of the art are not fully equipped to handle the complexities of determining identifiability/observability in the presence of TV-parameters. As a result, for these systems, it is not surprising to find discrepancies with previous studies.

In the presence of TV-parameters, and when the system is unobservable or unidentifiable, all the recursive algorithms based on the aforementioned state augmentation (e.g., [23]–[26], [28]) do not converge, automatically. As a result, all these algorithms remain inconclusive. Trying to stop their execution with artificial assumptions can produce wrong results, as we highlighted in Section VI-C2.

Similarly, the method adopted in [19] cannot deal with systems characterized by TV-parameters. To account for the presence of the TV-parameter $\eta(t)$, the authors of [19] adopted a heuristic and ad hoc procedure. Unfortunately, they achieved an erroneous result, as we highlighted in Section VI-C1. Note that, even if the procedure had been applied correctly, it would still have been inconclusive, as it merely shows that a sufficient condition for local identifiability does not hold. We also emphasize that the method adopted in [19] can only deal with polynomial nonlinearities.

## VII. SECOND EXAMPLE: GENETIC TOGGLE SWITCH

This example is provided to illustrate the use of our results when the dynamics of the system are not affine in the inputs. We study the observability and the identifiability properties of the ODE model characterized by as follows:

$$ \begin{cases} \dot{x}_1(t) = k_{01} + \frac{k_1}{1 + \left(\frac{x_2(t)}{V_1(t)}\right)^{n_1}} - x_1(t) \\ \dot{x}_2(t) = k_{02} + \frac{k_2}{1 + \left(\frac{x_1(t)}{V_2(t)}\right)^{n_2}} - x_2(t), \end{cases} \qquad (34) $$

and $y(t) = [x_1(t), x_2(t)]$. This model includes two TV-parameters, $V_1$ and $V_2$, and six constant parameters, $k_{01}, k_{02}, k_1, k_2, n_1, n_2$. All of them are assumed to be unknown. This model is commonly adopted to characterize the genetic toggle switch. The interested reader is addressed to Section 3.2 of [28] to find a biological description of this system, which includes the physical meaning of all the above states and parameters. Note that, (34) coincides with Eq. (3.5) in [28] with $n_1 = \eta_{TetR}$, $n_2 = \eta_{LacI}$, $V_1 = 1 + w_1$, and $V_2 = 1 + w_2$.

### A. Observability and identifiability analysis

We run Algorithm 2. The model in (34) is a special case of (2) but not directly a special case of (1). Following the steps of Algorithm 1, we introduce a state that, in addition to the six constant parameters, also includes $V_1$ and $V_2$. In other words:

$$ S = [x_1, x_2, V_1, V_2, k_{01}, k_1, n_1, k_{02}, k_2, n_2]^T \qquad (35) $$
and the dynamics are:

$$
\left\{
\begin{aligned}
\dot{x}_1(t) &= k_{01} + \frac{k_1}{1+\left(\frac{x_2(t)}{V_1(t)}\right)^{n_1}} - x_1(t) \\
\dot{x}_2(t) &= k_{02} + \frac{k_2}{1+\left(\frac{x_1(t)}{V_2(t)}\right)^{n_2}} - x_2(t) \\
\dot{V}_1(t) &= w_1(t) \\
\dot{V}_2(t) &= w_2(t) \\
\dot{k}_{01} &= \dot{k}_1 = \dot{n}_1 = \dot{k}_{02} = \dot{k}_2 = \dot{n}_2 = 0. \\
y(t) &= [x_1(t), x_2(t)],
\end{aligned}
\right.
\tag{36}
$$

Note that we denote the state by $S$ because it is the $\theta$-state of
the original system. The system is now a special case of (1). In
particular, $m_u = 0, m_w = 2, p = 2, h_1(S) = x_1, h_2(S) = x_2,$

$$
g^0 = \left[ k_{01} + \frac{k_1}{1+\left(\frac{x_2}{V_1}\right)^{n_1}} - x_1, k_{02} + \frac{k_2}{1+\left(\frac{x_1}{V_2}\right)^{n_2}} - x_2, 0_8 \right]^T,
$$
$$
g^1 = [0, 0, 1, 0, 0_6]^T, \quad g^2 = [0, 0, 0, 1, 0_6]^T. \quad (37)
$$

We have: $\mathcal{L}_{g^1}h_1 = \mathcal{L}_{g^1}h_2 = \mathcal{L}_{g^2}h_1 = \mathcal{L}_{g^2}h_2 = 0$. As a result, the rank of the unknown input reconstructability matrix from $h_1$, $h_2$ vanishes. In particular, it is smaller than $m_w(= 2)$. The system is not in canonical form with respect to its unknown input. On the other hand, it is immediate to realize that the following two scalar fields are observable: $h_3 \triangleq \dot{h}_1 = k_{01} + \frac{k_1}{1+\left(\frac{x_2}{V_1}\right)^{n_1}} - x_1$, and $h_4 \triangleq \dot{h}_2 = k_{02} + \frac{k_2}{1+\left(\frac{x_1}{V_2}\right)^{n_2}} - x_2$.

Note that these two scalar fields are automatically determined
by executing Algorithm 9 in [1], or Algorithm 1 in [32] (see
for instance Section 8.1 of [37]). It is immediate to check
that rank {$\mathcal{RM}(h_3, h_4)$} = 2. As a result, the system is
in canonical form and we introduce the following two UI
identifiers $\tilde{h}_1 = h_3$, and $\tilde{h}_2 = h_4$. From (9) and (37) we
compute all the components $\mu_1^0, \mu_2^0, \mu_1^1, \mu_2^1, \mu_2^2$, and $\mu_2^3$.
By inverting $\mu$, we determine $\nu_1^0, \nu_2^0, \nu_1^1, \nu_2^1, \nu_2^2$, and $\nu_2^3$.
Then, by using (10), we determine $\tilde{g}^0, \tilde{g}^1$, and $\tilde{g}^2$. We do not
provide their expressions, for the sake of brevity. Finally, as
$m_u = 0$, the codistribution $\tilde{\mathcal{O}}$ vanishes. From (11), we have:
$$ \mathcal{O}^S = \langle \mathcal{L}_{\tilde{g}^0}, \mathcal{L}_{\tilde{g}^1}, \mathcal{L}_{\tilde{g}^2} \mid \operatorname{span}\{dh_1, dh_2, dh_3, dh_4\} \rangle, \text{and,} $$
by a direct computation, we finally obtain: $\mathcal{O}^S =
\operatorname{span}\{dh_1, dh_2, dh_3, dh_4\} $$

It is immediate to verify that the differential of all the
six constant parameters does not belong to O<sup>S</sup>. No constant
parameter is therefore identifiable, not even locally. This result
contradicts the result available in Section 3.2 of [28] according
to which all parameters, constants and TV, are identifiable. In
Section VII-B, we provide an unequivocal proof that validates
our result and, in Section VII-C, we analyze the origin of
the discrepancy.

We remark that the two original TV-parameters, $V_1$ and $V_2$, were included in the state (defined in Eq. (35)). As result, their identifiability is obtained by checking if their differential belongs to $\mathcal{O}^S$. It is immediate to check that this is not the case. This result contradicts the result available in Section 3.2 of [28]. In Section VII-B, we provide an unequivocal proof that validates our result and, in Section VII-C, we analyze the origin of the discrepancy.

B. Direct proof

We provide an unequivocal proof of the validity of the
results obtained in Section VII-A. In particular, we directly

prove the non uniqueness of $V_1(t)$, $V_2(t)$, $k_{01}$, $k_{02}$, $k_1$, $k_2$, $n_1$, and $n_2$ by exhibiting six distinct sets of infinitely many distinct but indistinguishable values of these quantities. The non uniqueness is even local.

Let us denote the true state and parameters as fol-
lows: $x_1(t)$, $x_2(t)$, $V_1(t)$, $V_2(t)$, $k_{01}$, $k_1$, $n_1$, $k_{02}$,
$k_2$, and $n_2$. The new parameters will be denoted by
$\overline{V}_1$, $\overline{V}_2$, $\overline{k}_{01}$, $\overline{k}_1$, $\overline{n}_1$, $\overline{k}_{02}$, $\overline{k}_2$, $\overline{n}_2$. They will depend on a
given parameter, denoted by $\tau$, that can take all the values
in a given interval $(-\epsilon, \epsilon)$ (for a given $\epsilon \in \mathbb{R}^+$). The
indistinguishability will be proved by showing that, for any
$\tau$, the new parameters provide exactly the same dynamics of
$x_1$ and $x_2$, obtained with the true parameters, i.e, from (34),
by proving the following two equalities:

$$
\overline{k}_{01} + \frac{\overline{k}_1}{1 + \left(\frac{x_2}{V_1}\right)^{\overline{n}_1}} = k_{01} + \frac{k_1}{1 + \left(\frac{x_2}{V_1}\right)^{n_1}} \quad (38)
$$

$$
\overline{k}_{02} + \frac{\overline{k}_2}{1 + \left(\frac{x_1}{\overline{V}_2}\right)^{\overline{n}_2}} = k_{02} + \frac{k_2}{1 + \left(\frac{x_1}{\overline{V}_2}\right)^{n_2}} \quad (39)
$$

1) First set:

$$
\bar{k}_{01} = k_{01} + n_1 k_1 \tau, \quad \bar{V}_1 = \bar{V}_1(t, \tau), \qquad (40)
$$

with $\overline{V}_1(t, \tau) = x_2(t) \left( \frac{1-n_1(1+\left(\frac{x_2(t)}{V_1(t)}\right)^{n_1})\tau}{\left(\frac{x_2(t)}{V_1(t)}\right)^{n_1}+n_1\left(\frac{x_2(t)}{V_1(t)}\right)^{n_1}\tau+n_1\tau} \right)^{\frac{1}{n_1}}$.
The remaining parameters do not change, i.e.: $\overline{V}_2 = V_2$, $\overline{k}_1 = k_1$, $\overline{n}_1 = n_1$, $\overline{k}_{02} = k_{02}$, $\overline{k}_2 = k_2$, $\overline{n}_2 = n_2$. By a direct substitution, we obtain the validity of (38). The validity of (39) is trivial. The existence of this set of indistinguishable parameters proves the unidentifiability of $V_1(t)$ and $k_{01}$. Finally, as $\tau$ can take infinitesimal values, and $\lim_{\tau \to 0} \overline{V}_1 = V_1$, and $\lim_{\tau \to 0} \overline{k}_{01} = k_{01}$, the unidentifiability is even local.

2) Second set:

$$
\bar{k}_1 = k_1 e^{n_1 \tau}, \quad \bar{V}_1 = \bar{V}_1(t, \tau), \qquad (41)
$$

with $\overline{V}_1(t, \tau) = x_2(t) \left( \frac{1}{1+\left(\frac{x_2(t)}{V_1(t)}\right)^{n_1}} e^{n_1\tau} - 1 \right)^{\frac{1}{n_1}}$. The remaining parameters do not change. The existence of this set of indistinguishable parameters proves the unidentifiability of $V_1(t)$, and $k_1$. Finally, as $\tau$ can take infinitesimal values, and $\lim_{\tau \to 0} \overline{V}_1 = V_1$, and $\lim_{\tau \to 0} \overline{k}_1 = k_1$, the unidentifiability is even local.

3) Third set:

$$
\bar{n}_1 = n_1 e^{\tau}, \quad \bar{V}_1 = \bar{V}_1(t, \tau) = x_2(t) e^{\left(e^{-\tau} \log \frac{V_1(t)}{x_2(t)}\right)}, \quad (42)
$$

and the remaining parameters do not change. By a direct
substitution, we obtain the validity of the equality in (38).
The validity of (39) is trivial.

The existence of this set of indistinguishable parameters
proves the unidentifiability of $V_1(t)$, and $n_1$. Finally, as $\tau$
can take infinitesimal values, and $\lim_{\tau \to 0} \overline{V}_1 = V_1$, and
$\lim_{\tau \to 0} \overline{n}_1 = n_1$, the unidentifiability is even local.
4) Fourth set: As the first set, with the change 1 $\leftrightarrow$ 2. In this case the validity of (38) is trivial while to prove the validity of (39) we need to use (40) with the change 1 $\leftrightarrow$ 2.

The existence of this set of indistinguishable parameters proves the unidentifiability of $V_2(t)$, and $k_{02}$. Finally, as $\tau$ can take infinitesimal values, and $\lim_{\tau \to 0} \bar{V}_2 = V_2$, and $\lim_{\tau \to 0} \bar{k}_{02} = k_{02}$, the unidentifiability is even local.

5) Fifth set: As the second set, with the change 1 $\leftrightarrow$ 2. In this case the validity of (38) is trivial while to prove the validity of (39) we need to use (41) with the change 1 $\leftrightarrow$ 2.

The existence of this set of indistinguishable parameters proves the unidentifiability of $V_2(t)$, and $k_2$. Finally, as $\tau$ can take infinitesimal values, and $\lim_{\tau \to 0} \bar{V}_2 = V_2$, and $\lim_{\tau \to 0} \bar{k}_2 = k_2$, the unidentifiability is even local.

6) Sixth set: As the third set, with the change 1 $\leftrightarrow$ 2. In this case the validity of (38) is trivial while to prove the validity of (39) we need to use (42) with the change 1 $\leftrightarrow$ 2.

The existence of this set of indistinguishable parameters proves the unidentifiability of $V_2(t)$, and $n_2$. Finally, as $\tau$ can take infinitesimal values, and $\lim_{\tau \to 0} \bar{V}_2 = V_2$, and $\lim_{\tau \to 0} \bar{n}_2 = n_2$, the unidentifiability is even local.

As in Section VI-B, the above direct proof is trivially obtained by simple substitutions, once the six sets of indistinguishable states and parameters are determined. These sets are automatically generated by using the procedure introduced in Section 5.1 of [37]; a more detailed description of the general procedure will appear in the monograph [32].

### C. Comparison with state-of-the-art methods

As in Section VI-C, we examine the analysis of observability and identifiability provided by current state-of-the-art methods. To the best of our knowledge, the observability/identifiability analysis of the system in (34) was only given in [28] (Section 3.2). The authors concluded that the system is observable and identifiable. The origin of the discrepancy is the same as highlighted in Section VI-C2, i.e., it is due to Assumption 2.5 in [28]. Due to the presence of time-varying parameters, no state-of-the-art method can correctly determine identifiability (apart from possible heuristic and ad hoc procedures).

## VIII. THIRD EXAMPLE: A SIMPLE TIME-VARYING SYSTEM

The following elementary example illustrates the use of the algebraic condition in the case of time-varying systems:

$$ \begin{bmatrix} \dot{x}_1(t) & = & 1 \\ \dot{x}_2(t) & = & w_1(t) \\ y(t) & = & x_1(t) + x_2(t) \end{bmatrix} \quad (43) $$

### A. Observability and identifiability analysis

We run Algorithm 2. The system has not constant parameters and it is directly a special case of (1). In particular, $m_u = 0$, $m_w = 1$, $p = 1$, $h_1(x) = x_1 + x_2 t$,

$$ g^0 = [1, 0]^T, \quad g^1 = [0, 1]^T. \quad (44) $$

We have $\mathcal{L}_{g^1} h_1 = t \neq 0$. Hence, $\text{rank}\{\mathcal{RM}(h_1)\} = 1 (= m_w)$, and the system is directly in canonical form. We introduce the following UI identifier:

$$ \tilde{h}_1 = h_1 = x_1 + x_2 t. \quad (45) $$

From (9), by using (44) and (45) we obtain $\mu_1^1 = t$ and, $\mu_1^0 = \dot{\mathcal{L}}_{g^0} h_1 = \mathcal{L}_{g^0} h_1 + \frac{\partial h_1}{\partial t} = 1 + x_2$. Hence, $\nu_1^1 = 1/t$, and $\nu_1^0 = -\frac{1+x_2}{t}$. By using (10), we obtain: $\tilde{g}^0 = \nu_0^0 g^0 + \nu_1^0 g^1 = [1, -(1+x_2)/t]^T$, $\tilde{g}^1 = \nu_0^1 g^0 + \nu_1^1 g^1 = [0, 1/t]^T$. From (11), we have: $\mathcal{O} = \langle \dot{\mathcal{L}}_{\tilde{g}^0}, \mathcal{L}_{\tilde{g}^1} | \text{span}\{dh_1\} \rangle$, and, by a direct computation, we finally obtain: $\mathcal{O} = \text{span}\{dh_1\}$. The rank of $\mathcal{O}$ is 1, which is smaller than the dimension of the state. In particular, its orthogonal distribution is not empty. We have:

$$ \mathcal{O}^{\perp} = \text{span}\{[-t, 1]^T\}, \quad (46) $$

from which it is immediate to conclude that neither $x_1$, nor $x_2$ are observable.

We use Theorem 2. In accordance with Equation (17), we need to compute $\xi_1^0$ and $\xi_1^1$. From (44) and (45) we obtain: $\dot{\mathcal{L}}_{g^0} \tilde{h}_1 = \mathcal{L}_{g^0} \tilde{h}_1 + \frac{\partial \tilde{h}_1}{\partial t} = 1 + x_2$, $\mathcal{L}_{g^1} \tilde{h}_1 = t$. From Equation (16), by using the generator of $\mathcal{O}^{\perp}$ in (46) for $\xi$, we obtain: $\xi_1^0 = 1$ and $\xi_1^1 = 0$. The condition in (17) is not honoured for $\alpha = 0$. This means that the TV-parameter $w_1(t)$ is not identifiable, even locally.

### B. Direct proof and comparison with state-of-the-art methods

The output together with all its time derivatives are:

$$ \begin{align} y &= x_1 + x_2 t, \nonumber \\ y^{(1)} &= 1 + w_1 t + x_2, \tag{47} \\ y^{(k)} &= k w_1^{(k-2)} + t w_1^{(k-1)}, \quad k \ge 2, \nonumber \end{align} $$

with $y^{(k)} := \frac{d^k y}{dt^k}$, and $w_1^{(k)} := \frac{d^k w_1}{dt^k}$.

It is immediate to conclude that, from the outputs and all its time derivatives it is not possible to determine the components of the state ($x_1$ and $x_2$) and the model parameter ($w_1$). In other words, the state is unobservable and the parameter is unidentifiable. All the recursive algorithms based on a state augmentation (e.g., [23]–[26], [28]) do not converge for the system in (43). Additionally, by using Assumption 2.5 in [28], all these algorithms would conclude that $w_1(t)$ is locally identifiable. Let us study the appropriateness of this assumption in this specific case. Let us set $w_1^{(N)} = 0$, for a given integer $N \ge 1$. The model observability and identifiability totally relies on the knowledge of the $(N+1)^{th}$ order time derivative of the output. Specifically, from (47) and $w_1^{(N)} = 0$ we obtain $y^{(N+1)} = (N+1)w_1^{(N-1)} + tw_1^{(N)} = (N+1)w_1^{(N-1)}$, which allows us to determine $w_1^{(N-1)}$. Then, from $w_1^{(N-1)}$ and $y^{(N)} = N w_1^{(N-2)} + tw_1^{(N-1)}$ we determine $w_1^{(N-2)}$. By proceeding iteratively, we finally determine $w_1^{(0)} (= w_1)$ from $y^{(2)}$. Then, by using the second equation in (47), from $y^{(1)}$ we determine $x_2$. Finally, by using the first equation in (47), from $y$ we determine $x_1$. We note that the values of $x_1$, $x_2$, and $w_1$ are determined by the value of $w_1^{(N)}$, which we arbitrarily set to zero. By setting $w_1^{(N)}$ to a different value the result changes
completely (i.e., we obtain entirely different values for $x_1$, $x_2$, and $w_1$). Therefore, Assumption 2.5 renders the result of the determination spurious.

## IX. COMPREHENSIVE MODEL OVERVIEW

The three examples discussed in the previous sections include both time-invariant systems (Examples 1 and 2) and time-varying systems (Example 3), as well as dynamics that are affine in the inputs (Examples 1 and 3) and those that are not (Example 2). However, all of these systems are canonic with respect to their unknown inputs, and the state is unobservable. We conclude this paper by presenting additional elementary ODE models to cover all possible systems, distinguishing whether they are canonic with respect to their unknown inputs, whether the state is observable, and whether the model is locally identifiable. All possibilities are displayed in Table I. We have 5 possibilities instead of 8, because of the results stated by Theorem 1 and Theorem 3. All the examples given in Sections VI, VII, and VIII belong to the fourth category. An example of system that belongs to the third category is the system in Section VIII, with $y = x_1 + x_2$.

<table><caption>TABLE I
OVERVIEW OF MODELS.</caption><thead><tr><th>Category</th><th>State observability</th><th>Canonicity</th><th>Identifiability</th></tr></thead><tbody><tr><td>1</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>2</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>3</td><td>No</td><td>Yes</td><td>Yes</td></tr><tr><td>4</td><td>No</td><td>Yes</td><td>No</td></tr><tr><td>5</td><td>No</td><td>No</td><td>No</td></tr></tbody></table>

### Further models

The following three elementary examples belong to the first, to the second, and to the fifth category, respectively:

$$ \begin{bmatrix} \dot{x}_1 \\ y \\ \end{bmatrix} = \begin{bmatrix} w_1 \\ x_1 \\ \end{bmatrix}, \quad \begin{bmatrix} \dot{x}_1 \\ y \\ \end{bmatrix} = \begin{bmatrix} w_1 + w_2 \\ x_1 \\ \end{bmatrix}, \quad \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ y \\ \end{bmatrix} = \begin{bmatrix} w_1 \\ w_2 \\ x_1 \\ \end{bmatrix} $$

## X. CONCLUSION

In this paper we provided the following two main contributions:

* Introduction of the general algebraic condition that characterizes identifiability in the presence of TV-parameters.

* Determination of the identifiability properties of a well-known HIV model and a genetic toggle switch model, along with an analysis of the discrepancies with previous studies in the scientific literature.

The algebraic condition was provided in Section V. It is the first ever condition that can be applied even in the presence of TV-parameters. In addition, the model can also be time-varying. Specifically, it can be applied to the general model in (2), under Assumptions 1, and 2. Its application requires no inventiveness from the user. It is sufficient to carry out basic and automatic computation (derivatives and matrix ranks).

It is worth noting that the observability problem for systems with unknown inputs had already been solved in [1] (or, more recently, in [32], a simpler approach for a less complex

scenario). This solution could, in principle, be used to study the identifiability of time-varying parameters by including them in the state (see the first part of Section V and Equation (15)). However, the main novelty of this paper is that it provides a set of theorems that allow one to infer identifiability properties directly from the observability codistribution of the original state, thus avoiding any state extension. While the computational benefits may appear marginal in the systems investigated here – where only one or two time-varying parameters are involved – they become substantial in high-dimensional settings. For example, in the sensor fusion problems analyzed in [2, Section 8.2.3], the system has 11 state variables and 5 unknown inputs. Studying identifiability by extending the state would lead to a 16-dimensional system, which is already challenging for standard tools. The situation is even more critical in the more complex scenario discussed in [2, Section 8.2.4], where the state dimension increases to 18 variables. In that case, extending the state with unknown inputs would result in a 23-dimensional system, which is practically intractable with existing computational methods. Thanks to the results presented in this paper, one can instead work with the original state dimensions and still derive identifiability results for the time-varying parameters, greatly reducing computational complexity.

Regarding the second contribution, note that the inaccuracies detected by this paper have had damaging consequences in the scientific literature. In particular, several authors have spent a great deal of effort to perform the impossible identification of parameters that are actually unidentifiable. For instance, in [43], the authors introduced methods based on the Extended Kalman Filter and on the Particle Filter to estimate the state and the parameters of the HIV model that we studied in Section VI, when $\eta$ is constant and when is time-varying. Not surprisingly, they obtained good performance only in the first case (i.e., when all the parameters are constant and the model is really observable and identifiable), while, in the second case (time-varying $\eta$), the performance was poor (especially in correspondence with variations in $\eta$).

We conclude by noting that the algebraic condition introduced here is a deterministic condition, obtained under the assumption - commonly made in this research domain - that the measurements are noiseless. Recent approaches have been introduced to identify missing physics through machine learning (e.g., [44]). These methods are not deterministic and are not entirely model-based; in particular, they allow for model refinement if necessary.

## APPENDIX A

### AUTOBRACKET

Given a system in canonical form, and a set of UI identifiers $\tilde{h}_1, \dots, \tilde{h}_{m_w}$, we compute $\nu$ (the inverse of $\mu$ in (9)). We introduce the following operation.

**Definition 6 (Autobracket)** Given the vector field $\phi$, we define its autobracket the following $m_w + 1$ vector fields:

$$ [\phi]^{\alpha} = \sum_{\beta=0}^{m_w} \nu_{\beta}^{\alpha} [g^{\beta}, \phi] + \nu_{0}^{\alpha} \frac{\partial \phi}{\partial t} \quad (48) $$
where, the square brackets in the sum denote the Lie brackets.

We can apply the autobracket repetitively. We denote by $[\phi]^{(\alpha_1, \dots, \alpha_k)}$ the vector fields obtained by applying the autobracket repetitively $k$ consecutive times, first along $\alpha_1$ and last along $\alpha_k$ (these operations are not commutative).

## APPENDIX B
### PROOF OF THEOREM 1

**Proof:** Based on the result stated by Theorem 4 in [1], it remains to prove the viceversa, i.e., if the unknown inputs are locally identifiable then the system is canonic with respect to its unknown inputs. We proceed by contradiction. We assume that the system is not canonic with respect to its unknown inputs. As the state is observable, all its components belong to the observation space and we have $\mathcal{O} = \text{span}\{dx_1, \dots, dx_n\}$, at least locally. As we assumed that the system is not canonic, the unknown input degree of reconstructability is strictly less than $m_w$. Therefore, the matrix $\mathcal{R}\mathcal{M}(x_1, \dots, x_n)$ has rank smaller than $m_w$. On the other hand, by an explicit computation

we obtain: $\mathcal{R}\mathcal{M}(x_1, \dots, x_n) = \begin{bmatrix} [g^1]_1 & [g^2]_1 & \dots & [g^{m_w}]_1 \\ [g^1]_2 & [g^2]_2 & \dots & [g^{m_w}]_2 \\ \vdots & \vdots & \ddots & \vdots \\ [g^1]_n & [g^2]_n & \dots & [g^{m_w}]_n \end{bmatrix} = [g^1 \ g^2 \ \dots \ g^{m_w}]$, where $[g^j]_i$ is the $i^{th}$ component of the vector field $g^j$ ($j = 1, \dots, m_w$ and $i = 1, \dots, n$). Therefore, as $\text{rank}(\mathcal{R}\mathcal{M}(x_1, \dots, x_n)) < m_w$, the vectors $g^1 \ g^2 \ \dots \ g^{m_w}$ are linearly dependent. We denote by $\hat{n}$ a non trivial vector that belongs to the null space of the above matrix. Let us consider the dynamics $\dot{x} = g^0 + \sum_{k=1}^{m_w} f^k u_k + \sum_{j=1}^{m_w} g^j w_j$. For any $w = [w_1, \dots, w_{m_w}]$, $w$ and $w' = w + \hat{n}$ yield identical dynamics and are thus indistinguishable. ◀

## APPENDIX C
### PROOF OF THEOREM 2

**Proof:** We prove that, when the system is in canonical form with respect to its unknown inputs, the condition in (15) coincides with the condition in (17).

Let us compute $\mathcal{O}^S$. We know $\mathcal{O}$, which is the observability codistribution of the original system. We know that, for any scalar function $\gamma(x)$ such that $d\gamma \in \mathcal{O}$, $d\dot{\gamma} \in \mathcal{O}^S$, where $\dot{\gamma}$ is the time derivative of $\gamma$ obtained by setting all the known inputs to zero ($u_1 = \dots = u_{m_w} = 0$). In other words, $\dot{\gamma} = \dot{\mathcal{L}}_{g_S^0} \gamma$, with $g_S^0$ given in (14). Consequently: $d\dot{\mathcal{L}}_{g_S^0} \gamma \in \mathcal{O}^S$. We denote the rank of $\mathcal{O}$ by $n_o$ ($\le n$, where $n$ is the dimension of $x$). Let us denote by $\gamma_1(x), \dots, \gamma_{n_o}(x)$ a set of generators of $\mathcal{O}$. We have: $\mathcal{O} = \text{span}\{d\gamma_1, \dots, d\gamma_{n_o}\}$. We also know that $d\tilde{h}_i \in \mathcal{O}$, $i = 1, \dots, m_w$. As a result: $d\dot{\mathcal{L}}_{g_S^0} \tilde{h}_i \in \mathcal{O}^S$, $i = 1, \dots, m_w$. In addition, as the reconstructability matrix from them has full rank, the covectors $d\dot{\mathcal{L}}_{g_S^0} \tilde{h}_1, \dots, d\dot{\mathcal{L}}_{g_S^0} \tilde{h}_{m_w}$ are independent and they are also independent from the generators of $\mathcal{O}$ (because the latter are independent of the unknown inputs). As a result, the $n_o + m_w$ covectors $d\gamma_1, \dots, d\gamma_{n_o}, d\dot{\mathcal{L}}_{g_S^0} \tilde{h}_1, \dots, d\dot{\mathcal{L}}_{g_S^0} \tilde{h}_{m_w}$ are independent. On the other hand, since $\mathcal{O}$ is the observability codistribution and its rank is $n_o$, the rank of $\mathcal{O}^S$ cannot exceed $n_o + m_w$. Therefore:

$$ \mathcal{O}^S = \text{span}\{d\gamma_1, \dots, d\gamma_{n_o}, d\dot{\mathcal{L}}_{g_S^0} \tilde{h}_1, \dots, d\dot{\mathcal{L}}_{g_S^0} \tilde{h}_{m_w}\}. \quad (49) $$

Let us study the structure of the null space of $\mathcal{O}^S$. Let us denote it by $\Delta^S$ (i.e., $\Delta^S := (\mathcal{O}^S)^\perp$). $\Delta^S$ is involutive ($\mathcal{O}^S$ is an integrable codistribution by construction).

We prove that $\forall \xi \in \mathcal{O}^\perp$, the vector field:

$$ \xi^S := \begin{bmatrix} \xi \\ \chi \end{bmatrix}, \quad (50) $$

with $\chi = [\chi_1, \dots, \chi_{m_w}]^T$ and $\chi_j, j = 1, \dots, m_w$,

$$ \chi_j = \chi_j(x, w) = -\sum_{i=1}^{m_w} \nu_j^i \left( \xi_i^0 + \sum_{k=1}^{m_w} \xi_i^k w_k \right), \quad (51) $$

belongs to the null space of $\mathcal{O}^S$ in (49).

Let us consider the first $n_o$ generators of $\mathcal{O}^S$. As $\xi \in \mathcal{O}^\perp$, we have: $\frac{\partial \gamma_i}{\partial x} \cdot \xi = 0$, $i = 1, \dots, n_o$. Hence, $\frac{\partial \gamma_i}{\partial S} \cdot \xi^S = \frac{\partial \gamma_i}{\partial x} \cdot \xi + \frac{\partial \gamma_i}{\partial w} \cdot \chi = 0 + \underbrace{[0, \dots, 0]}_{m_w} \cdot \chi = 0$, $i = 1, \dots, n_o$, as

$\gamma_i$ is independent of $w$ (it only depends on $x$).

Let us consider the remaining $m_w$ generators. We have:

$$ \frac{\partial \dot{\mathcal{L}}_{g_S^0} \tilde{h}_l}{\partial S} \cdot \xi^S = \frac{\partial \dot{\mathcal{L}}_{g_S^0} \tilde{h}_l}{\partial x} \cdot \xi + \sum_{k=1}^{m_w} \frac{\partial \dot{\mathcal{L}}_{g_S^0} \tilde{h}_l}{\partial w_k} \chi_k \quad (52) $$

with $l = 1, \dots, m_w$. On the other hand,

$$ \dot{\mathcal{L}}_{g_S^0} \tilde{h}_l = \dot{\mathcal{L}}_{g^0} \tilde{h}_l + \sum_{j=1}^{m_w} (\mathcal{L}_{g^j} \tilde{h}_l) w_j. \quad (53) $$

By using (53) in (52) and by using (9) and (16) we obtain: $\frac{\partial \dot{\mathcal{L}}_{g_S^0} \tilde{h}_l}{\partial S} \cdot \xi^S = \xi_l^0 + \sum_{j=1}^{m_w} \xi_l^j w_j + \sum_{k=1}^{m_w} \mathcal{L}_{g^k} \tilde{h}_l \chi_k = \xi_l^0 + \sum_{j=1}^{m_w} \xi_l^j w_j + \sum_{k=1}^{m_w} \mu_l^k \chi_k$. Now, by using the expression of $\chi_k$ in (51) and that $\mu$ is the inverse of $\nu$ (i.e., $\sum_{k=1}^m \mu_l^k \nu_k^l = \delta_l^j$), we finally obtain: $\frac{\partial \dot{\mathcal{L}}_{g_S^0} \tilde{h}_l}{\partial S} \cdot \xi^S = 0$. Hence, we proved that $\xi^S \in \Delta^S$. Checking the local identifiability of a given parameter is equivalent to check whether or not its differential belongs to $\mathcal{O}^S$. But this is equivalent to check if the component of $\chi$ that corresponds to this parameter vanishes for all the vector fields that belong to the null space of $\mathcal{O}^S$. In other words, regarding the $j^{th}$ TV-parameter, its local identifiability is equivalent to the condition $\chi_j = 0$, for any $\xi \in \mathcal{O}^\perp$, with $\chi_j$ given in (51). As all the coefficients $\xi_i^\alpha$ that appear in (51) are independent of $w$ (they only depend on $x$), $\chi_j = 0$ if and only if we have the condition in (17), for any $\alpha = 0, 1, \dots, m_w$. This proves the theorem. ◀

## APPENDIX D
### PROOF OF THEOREM 3

**Proof:** The proof is by contradiction. By proceeding as in Appendix C, we obtain that $\mathcal{O}^S$ is generated by the following covectors: $d\gamma_1, \dots, d\gamma_{n_o}$, $d\dot{\mathcal{L}}_{g_S^0} \gamma_1, \dots, d\dot{\mathcal{L}}_{g_S^0} \gamma_{n_o}$ (note that in Appendix C, the scalar fields $\tilde{h}_1, \dots, \tilde{h}_{m_w}$ are UI identifiers, i.e., can be selected from the generators $\gamma_1, \dots, \gamma_{n_o}$). As remarked in Appendix C, the rank of $\mathcal{O}^S$ cannot exceed $n_o + m_w$. Let us proceed by contradiction and let us assume that the system is not canonic with respect to its unknown inputs. We obtain that the rank of $\mathcal{O}^S$ is strictly smaller
than $n_o + m_w$. But this ensures that there exists at least one unknown input for which the condition in (15) is not honoured. Consequently, there exists at least one unknown input that is not locally identifiable, in contrast with the assumption. ◀

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[31] A. Martinelli, Nonlinear Unknown Input Observability Toolbox, GitHub repository, 2025. Available: https://github.com/agomartinelli/TUIO6

[32] A. Martinelli, "Algebraic Criterion for Nonlinear Unknown Input Observability in Systems Without Control Inputs," *IEEE Transactions on Automatic Control*, under review, 2025. See also the related monograph *Observability and Identifiability of Nonlinear Systems Driven by Unknown Inputs: Algebraic Solution and Practical Implementation in MATLAB*, in preparation, SpringerBriefs.

[33] A. S. Perelson, and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 1999, 41, 3-44.

[34] X. Xia, and C. H. Moog, Identifiability of nonlinear systems with applications to HIV/AIDS models, in IEEE Transactions on Automatic Control, 2003, 48, 330-336.

[35] J. Chen, and H. Wu, Estimation of time-varying parameters in deterministic dynamic models, Stat. Sinica, 2008, 18, 987-1006.

[36] M. J. Mhawej, C. H. Moog, F. Biafore, and C. Brunet-Franois, Control of the HIV infection and drug dosage, Biomedical Signal Processing and Control, 2010, 5(1), 45-52, ISSN 1746-8094, https://doi.org/10.1016/j.bspc.2009.05.001.

[37] A. Martinelli, Identifiability of nonlinear ODE Models with Time-Varying Parameters: the General Analytical Solution and Applications in Viral Dynamics, arXiv:2211.13507 [eess.SY].

[38] A. Martinelli, Extension of the Observability Rank Condition to Time-Varying Nonlinear Systems, in IEEE Transactions on Automatic Control, 2022, 67(9), 5002-5008, doi: 10.1109/TAC.2022.3180771.

[39] H. Miao, A. S. Perelson, and H. Wu, Erratum: On Identifiability of Nonlinear ODE Models and Applications in Viral Dynamics, SIAM Review, 2023, 65(3), 732-732.

[40] B. R. Hunt, T. Sauer and J. A. Yorke, Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 1992, 27(2).

[41] A. Martinelli, Revisiting the observability and identifiability properties of a popular HIV model. J Theor Biol. 2024, Vol 584, ISSN 0022-5193.

[42] A. Martinelli. Detection of a serious error in the paper: "On identifiability of nonlinear ODE models and applications in viral dynamics". 2024. hal-04419008

[43] A. Hartmann, S. Vinga, and J. M. Lemos, Online Bayesian Time-varying Parameter Estimation of HIV-1 Time-series, 16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012

[44] P. Shah, M. Z. Sheriff, M. S. Faizan Bangi, C. Kravaris, J. Sang-II Kwon, C. Botre, J. Hirota, Deep neural network-based hybrid modeling and experimental validation for an industry-scale fermentation process: Identification of time-varying dependencies among parameters, Chemical Engineering Journal, Volume 441, 2022, 135643

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Agostino Martinelli received the M.Sc. degree in theoretical physics from the University of Rome Tor Vergata, Rome, Italy, in 1994 and the Ph.D. degree in astrophysics from the University of Rome La Sapienza, in 1999. While working toward the Ph.D. degree, he spent one year at the University of Wales, Cardiff, U.K., and one year with the Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy. His research focused on the chemical and dynamical evolution in elliptical galaxies, in quasars, and in the intergalactic medium. After receiving the Ph.D. degree, his interests moved to the problem of autonomous navigation.

He was with the University of Rome Tor Vergata for two years, and, in 2002, he moved to the Autonomous Systems Laboratory, Ecole Polytechnique Federale de Lausanne, Switzerland, as a Senior Researcher. Since 2006, he has been a Researcher with the Institut National de Recherche en Informatique et en Automatique (INRIA), Grenoble, France. His current research focuses on Visual-Inertial Structure from Motion and Non linear observability.