$\tilde{h}_1 = h_2 = x_2$

Line 12 provides: $^m\mu_0^0 = 1$, $^m\mu_1^1 = 0$, $^m\mu_1^0 = 0$, $^m\mu_1^1 = x_3$ and $^m\nu_0^0 = 1$, $^m\nu_0^1 = 0$, $^m\nu_1^0 = 0$, $^m\nu_1^1 = 1/x_3$. obtained from (3.6).

In addition, from (3.7) we obtain: $^m\tilde{g}^0 = g^0$ and $^m\tilde{g}^1 = g^1/x_3$. Line 13 computes

$$ \Omega = \left\langle \left\{ \begin{array}{cc} g^0 & g^1/x_3 \\ g^2 & g^3 \end{array} \right\} \middle| \operatorname{span}\{\nabla h_1, \nabla h_2\} \right\rangle, $$

By a direct computation, we obtain:

$$ \Omega = \operatorname{span}\{\nabla h_1, \nabla h_2, \nabla \mathcal{L}_{g^0} h_1\} = \operatorname{span}\{\nabla x_1, \nabla x_2, \nabla x_3\}. $$

We have $\deg_w(\Omega) = 2$. Hence, it is larger than the previous $m (= 1)$ but still smaller than $m_w (= 3)$. As a result, the loop at lines 9-14 is repeated again.

Line 11 provides:

$\tilde{h}_2 = x_3.$

The tensor $^m\mu$ has 9 components. In particular, the values of the new components are: $^m\mu_2^0 = ^m\mu_2^1 = ^m\mu_2^0 = ^m\mu_1^2 = 0$, and $^m\mu_2^2 = 1$. The values of the new components of its inverse are: $^m\nu_2^0 = ^m\nu_2^1 = ^m\nu_2^0 = ^m\nu_1^2 = 0$, and $^m\nu_2^2 = 1$.

As result, $^m\tilde{g}^0 = ^m\nu_0^0 g^0 + ^m\nu_1^0 g^1 + ^m\nu_2^0 g^2 = g^0$, $^m\tilde{g}^1 = ^m\nu_0^1 g^0 + ^m\nu_1^1 g^1 + ^m\nu_2^1 g^2 = g^1/x_3 = [0, 1, 0]^T$, and $^m\tilde{g}^2 = ^m\nu_0^2 g^0 + ^m\nu_1^2 g^1 + ^m\nu_2^2 g^2 = g^2$. Line 13 provides:

$$ \Omega = \left\langle \left\{ \begin{array}{c} ^m\tilde{g}^0, \ ^m\tilde{g}^1 \ ^m\tilde{g}^2 \\ g^3 \end{array} \right\} \middle| \operatorname{span}\{\nabla x_1, \nabla x_2, \nabla x_3\} \right\rangle = \operatorname{span}\{\nabla x_1, \nabla x_2, \nabla x_3\}. $$

As a result, $\deg_w(\Omega) = 2 = m$ and the loop at Lines 9-14 is interrupted. As $m_u = 0$, Line 21 sets $\tilde{k}^m = 0$.

Then, the nested loop is executed. The state is augmented, by including the third UI ($w_3$). The vectors that characterize the dynamics of the augmented state are:

$$ g^0 = \begin{bmatrix} x_3 \\ 0 \\ x_1 w_3 \\ 0 \end{bmatrix}, \quad g^1 = \begin{bmatrix} 0 \\ x_3 \\ 0 \\ 0 \end{bmatrix}, \quad g^2 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad g^3 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}. \qquad (9.7) $$

Line 27 provides the new $^m\mu$. It only changes its 02 component, which becomes: $^m\mu_2^0 = x_1 w_3$. Similarly, the new $^m\nu$ only differs for its 02 component, which becomes: $^m\nu_2^0 = -x_1 w_3$. Finally, we obtain:

$$ ^m\tilde{g}^0 = ^m\nu_0^0 g^0 + ^m\nu_1^0 g^1 + ^m\nu_2^0 g^2 = g^0 - x_1 w_3 g^2 = [x_3, 0, 0, 0]^T, \quad ^m\tilde{g}^1 = ^m\nu_0^1 g^0 + ^m\nu_1^1 g^1 + ^m\nu_2^1 g^2 = g^1/x_3 = [0, 1, 0, 0]^T, \quad \text{and} \quad ^m\tilde{g}^2 = ^m\nu_0^2 g^0 + ^m\nu_1^2 g^1 + ^m\nu_2^2 g^2 = g^2 = [0, 0, 1, 0]^T. $$

Line 32 provides $\Omega_1 = \Omega_0$. As a result, the condition at Line 33 is met, the nested and the main loop are terminated.

$$ \mathcal{O} = \operatorname{span}\{\nabla x_1, \nabla x_2, \nabla x_3\} \qquad (9.8) $$

The system is not canonic.

### 9.2.2 Identifiability analysis

First of all, the observability codistribution of the extended state that includes $w_3$, is given in (9.8). Hence, the identifiability of $w_3$ is directly obtained by checking if $\nabla w_3 \in \mathcal{O}$. This is not the case and, as a result, $w_3$ is not identifiable, even locally.

Let us check the identifiability of $w_1$, $w_2$, and $\dot{w}_3$. We execute Algorithm 4. First, as the system is not canonic with respect to its unknown inputs, we have symmetries of the unknown input due to the system non canonicity ($^c\chi$). In particular, from (4.5). we have a single symmetry: $^c\chi = [0, 0, 1]^T$, which expresses the unidentifiability of the third UI, i.e., $\dot{w}_3$.
For the remaining parameters, we compute the distribution orthogonal to the observable codistribution. We have:

$$ \mathcal{O}^{\perp} = \operatorname{span} \left\{ \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}, \qquad (9.9) $$

We have a single symmetry of the state. It is immediate to compute the symmetry of the unknown input.

In accordance with Equation (4.6), we need to compute $\xi_1^0, \xi_1^1, \xi_1^2$ for $w_1$, and $\xi_2^0, \xi_2^1, \xi_2^2$ for $w_2$. By a direct computation, we obtain: $\xi_1^0 = \xi_1^1 = \xi_1^2 = 0$, $\xi_2^0 = x_1$, and $\xi_2^1 = \xi_2^2 = 0$. As a result:

$$ ^u\chi_1 = 0, \quad ^u\chi_2 = -x_1 \neq 0. $$

Therefore, only $w_1$ is locally identifiable.

## 9.3 Third example

We study the system characterized by the following equations:

$$ \left\{ \begin{aligned} \dot{x}_1 &= x_3 + w_1 + w_2 \\ \dot{x}_2 &= u \\ \dot{x}_3 &= x_1 + u \\ y &= [x_1, x_3]. \end{aligned} \right. \qquad (9.10) $$

It is a special case of (2.2) with: $m_u = 1$, $m_w = 2$, $p = 2$, $h_1(x) = x_1$, $h_2(x) = x_3$,

$$ g^0 = \begin{bmatrix} x_3 \\ 0 \\ x_1 \end{bmatrix}, \quad g^1 = g^2 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad f^1 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \qquad (9.11) $$

### 9.3.1 Observability analysis

We run Algorithm 1 to obtain the observability codistribution. Line 2 provides:

$$ \Omega = \operatorname{span}\{\nabla h_1, \nabla h_2\} = \operatorname{span}\{[1, 0, 0], [0, 0, 1]\}. $$

$\deg_w(\Omega) = 1$ ($< m_w$), as $\mathcal{L}_{g^1}h_1 = \mathcal{L}_{g^2}h_1 = 1$, and $\mathcal{L}_{g^1}h_2 = \mathcal{L}_{g^2}h_2 = 0$ (and the rank of the unknown input reconstructability matrix from $h_1, h_2$ is 1). The system is not in canonical form with respect to its unknown input. We enter the loop at lines 9-14.

Line 11 provides:

$$ \tilde{h}_1 = h_1 = x_1 $$

Line 12 provides: $^m\mu_0^0 = 1$, $^m\mu_0^1 = 0$, $^m\mu_1^0 = x_3$, $^m\mu_1^1 = 1$ and $^m\nu_0^0 = 1$, $^m\nu_0^1 = 0$, $^m\nu_1^0 = -x_3$, $^m\nu_1^1 = 1$. obtained from (3.6).

In addition, from (3.7) we obtain: $^m\hat{g}^0 = g^0 - x_3g^1 = [0, 0, x_1]^T$ and $^m\hat{g}^1 = g^1$. Line 13 computes

$$ \Omega = \left\langle f^1, \begin{Bmatrix} m\hat{g}^0 & m\hat{g}^1 \\ g^2 & g^2 \end{Bmatrix} \middle| \operatorname{span}\{\nabla x_1, \nabla x_3\} \right\rangle = \operatorname{span}\{\nabla x_1, \nabla x_3\}. $$

because $\mathcal{L}_{f^1}x_1 = 0$, $\mathcal{L}_{f^1}x_3 = 1$, $\mathcal{L}_{m\hat{g}^0}x_1 = 0$, $\mathcal{L}_{m\hat{g}^0}x_3 = x_1$, $\mathcal{L}_{m\hat{g}^1}x_1 = 1$, and $\mathcal{L}_{m\hat{g}^1}x_3 = 0$.

As a result, $\deg_w(\Omega) = 1 = m$, and the loop at Lines 9-14 is interrupted.

As $m_u = 1 > 0$, Line 19 runs Algorithms 2 and 3 to determine $\hat{k}^m = s_x^m + r^m$. In particular, by running Algorithm 2 we obtain $s_x^m = 2$ and by running Algorithm 3 we obtain $r^m = 2$. Therefore, $\hat{k}^m = 4$.

Then, the nested loop is executed. The state is augmented, by including the second UI ($w_2$). The vectors that characterize the dynamics of the augmented state are:
$$ g^0 = \begin{bmatrix} x_3 + w_2 \\ 0 \\ x_1 \\ 0 \end{bmatrix}, \quad g^1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \quad g^2 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \quad f = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}. \qquad (9.12) $$

Line 27 provides the new $^m\mu$. It only changes its 01 component, which becomes: $^m\mu_1^0 = x_3 + w_2$. Similarly, the new $^m\nu$ only differs for its 01 component, which becomes: $^m\nu_1^0 = -(x_3 + w_2)$. Finally, we obtain:

$$ ^m\hat{g}^0 = ^m\nu_0^0 g^0 + ^m\nu_1^0 g^1 = g^0 - (x_3 + w_2)g^1 = [0, 0, x_1, 0]^T, \text{ and } ^m\hat{g}^1 = ^m\nu_0^1 g^0 + ^m\nu_1^1 g^1 = g^1. $$

Line 32 provides $\Omega_1 = \Omega_0$.

The condition at Line 33 is not met because $k = 1 < \hat{k}^m = 4$. On the other hand, it is immediate to realize that the codistribution $\tilde{\mathcal{O}}$ in (3.5) is empty. As a result, the convergence is achieved and the nested and the main loop are terminated.

$$ \mathcal{O} = \operatorname{span}\{\nabla x_1, \nabla x_3\} \qquad (9.13) $$

The system is not canonic.

### 9.3.2 Identifiability analysis

First of all, the observability codistribution of the extended state that includes $w_2$, is given in (9.13). Hence, the identifiability of $w_2$ is directly obtained by checking if $\nabla w_2 \in \mathcal{O}$. This is not the case and, as a result, $w_2$ is not identifiable, even locally.

Let us check the identifiability of $w_1$, and $\dot{w}_2$. We execute Algorithm 4. First, as the system is not canonic with respect to its unknown inputs, we have symmetries of the unknown input due to the system non canonicity ($^c\chi$). In particular, from (4.5). we have a single symmetry: $^c\chi = [0, 1]^T$, which expresses the unidentifiability of the second UI, i.e., $\dot{w}_2$.

For the remaining parameter, we compute the distribution orthogonal to the observable codistribution. We have:

$$ \mathcal{O}^{\perp} = \operatorname{span}\left\{ \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}, \qquad (9.14) $$

In accordance with Equation (4.6), we need to compute $\xi_1^0, \xi_1^1$. By a direct computation, we obtain, for the first generator: $\xi_1^0 = \xi_1^1 = 0$, and, consequently $^u\chi_1 = 0$. For the second generator we obtain: $\xi_1^0 = 1, \xi_1^1 = 0$, and, consequently $^u\chi_1 = -1 \neq 0$. Therefore, $w_1$ is not locally identifiable.
# Chapter 10

## Conclusion

In this paper we provided the following four main contributions:

* **First contribution:** Compact presentation of the systematic procedure introduced in [2] to determine the observability of the state of any nonlinear system in the presence of time-varying parameters (or unknown inputs). In particular, the systematic procedure was presented to make it usable by a non specialist user.

* **Second contribution:** Introduction of the systematic procedure to determine the identifiability of all the time-varying parameters of any ODE model.

* **Third contribution:** Introduction of the continuous transformations (one parameter Lie groups) that allow us to build the set of indistinguishable states and indistinguishable unknown inputs in the presence of unobservability and unidentifiability.

* **Fourth contribution:** Introduction of new important results about the observability and the identifiability of a very popular HIV model, a Covid-19 model, and a genetic toggle switch model. The results on the HIV model and the genetic toggle switch model are in contrast with the current state of the art.

### 10.1 First contribution

The systematic procedure was provided in Chapter 3 and is Algorithm 1. This algorithm can be executed by a non specialist user. All the functions/operations that appear in the algorithm were defined in Sections 3.1-3.3. Algorithm 1 can be executed without restrictions on the system investigated. It provides the so-called *observability codistribution* (denoted by $\mathcal{O}$) for any ODE model in the presence of time-varying parameters, independently of its complexity and type of nonlinearity. Note that the system was modelled by Equation (2.2), and, in Section 2.1, we showed that any system modelled by (2.1) can be easily converted to a system modelled by (2.2).

Once $\mathcal{O}$ is computed, we can easily check whether a given physical quantity, which can be analytically expressed in terms of the state, is observable or not. It suffices to compute its gradient with respect to the state and verify if this gradient belongs to $\mathcal{O}$. This holds for any physical quantity that can be analytically expressed in terms of the state. In particular, it holds for the unknown constant parameters, which were included in the state. To check their identifiability it suffices to compute their gradient. This is trivial. The gradient of a given parameter is a row vector with zero everywhere with the exception of the entry that has the same index of the parameter in the state, which is 1.

The theoretical foundation that ensures the general validity of Algorithm 1 is available in [2].

### 10.2 Second contribution

The systematic procedure was provided in Chapter 4 and is Algorithm 4. Also in this case, to obtain the identifiability of all the time-varying parameters, is sufficient to simply follow the
steps of this systematic procedure (Algorithm 4). The validity of the algorithm was ensured by
two new theoretical results which are Theorems 4.2 and 4.3.

Algorithm 4 determines the identifiability of the unknown time-varying parameters without restrictions on the system investigated. It can handle any system, regardless of its complexity and type of nonlinearity (as for Algorithm 1, the system was modelled by Equation (2.2), which is equivalent to (2.1)).

For the special case when the system is characterized by a state that is observable, Chapter 4 provided a simpler theoretical result, which was Theorem 4.1. In this special case, the identifiability can be easily verified by checking if the system is canonic with respect to its unknown inputs. Chapter 4 showed that the results stated by Theorems 4.2 and 4.3 include the result of Theorem 4.1, as a special case.

10.3 Third contribution

When the state is unobservable and/or a set of parameters (constant and/or time-varying) is
unidentifiable, there are more values for all these unobservable and unidentifiable quantities
that reproduce exactly the same outputs (and also agree with the same known inputs, when
present). Chapter 5 introduced the mathematical tool that quantitatively allows us to determine
all these indistinguishable values. This tool consists of two sets of first order ordinary differential
equations. Specifically, they are Equations (5.5) and (5.7). The former is a consequence of the
system non-canonicity with respect to its unknown inputs and only regards the time-varying
parameters (and not the state). The solution of this first set of differential equations is trivial
and can be obtained analytically (and automatically) for any system. Note that this set of
equations exists if and only if the system is not canonic with respect to its unknown inputs.
In addition, if the system is not canonic with respect to its unknown inputs, we certainly have
simple analytical solutions of (5.5). In other words, we can certainly generate values of the time-
varying parameters that agree with the same system known inputs and outputs. In particular,
this regards the last $m_w - m$ time-varying parameters, where $m_w$ is the total number of them
and $m$ the highest unknown input degree of reconstructability (note that the order of the $m_w$
time-varying parameters may have changed after the execution of Algorithm 1 and, moreover,
after this execution some of these parameters may have been redefined as the time derivatives of
the original ones, of a given order).

In contrast, the second set of differential equations (i.e., (5.7)) regards the time-varying pa-
rameters and the state, simultaneously, and provides their indistinguishable values provided that
the condition in (5.6) is honoured. This set of equations exists only when the state is unobserv-
able (i.e., when the dimension of $\mathcal{O}$ is strictly smaller than the dimension of the state). The state
unobservability is a necessary but not a sufficient condition for the existence of indistinguishable
time-varying parameters of this second type. In other words, there exist cases for which the set
of equations in (5.7) only regards the state (i.e., the quantities $^u\chi^i$ in (5.7) vanish for all the
values of $i$ (i.e., for $i = 1, \dots, S$). This was for instance the case of the first scenario of our case
study (see Section 4.4.3).

Unfortunately, in contrast with the equations system in (5.5), the system in (5.7) is complex
and its solution cannot be obtained automatically. Even worse, in many cases the solution can
only be obtained numerically. On the other hand, when it is possible to obtain an analytical
solution, we obtain very useful properties about the ODE model under investigation. This was
the case of our viral applications in Chapters 6 and 7. On the other hand, the solution of both the
sets of differential equations in (5.5) and (5.7) was not requested to determine the observability of
the state and the identifiability of all the time-varying parameters. The determination of them
was obtained by simply following the steps of the two systematic procedures (i.e., Algorithm
1 and Algorithm 4, respectively). This is always the case and makes possible to obtain the
observability and the identifiability of any ODE model, automatically.

10.4 Fourth contribution

Chapters 6, 7, and 8 provided a detailed study of the observability and identifiability properties
of two viral ODE models and a genetic toggle switch model. In particular, the ODE model
investigated in Chapter 6 is very popular and regards the HIV dynamics. It was studied in many previous works (e.g., [3, 35, 40, 41, 42]). The second ODE model, studied in Chapter 7, was a Covid-19 model adopted in [43, 44]. The third model was investigated also in [35].

We started by simply executing Algorithm 1 and 4 to obtain the observability of the state and the identifiability of the time-varying parameters. In the first two models, the state is unobservable and the time-varying parameters unidentifiable. In the third case, all the model parameters are unidentifiable. In the case of the HIV ODE model, and in the case of the genetic toggle switch model, the results are in contrast with the current state of the art<fnref n="1" />.

For all these ODE models, it was possible to obtain an analytical solution of the differential equations in (5.7) (the set of equations in (5.5) does not exist because these systems are canonic with respect to their unknown input). This allowed us to determine indistinguishable unknown inputs that agree with the same outputs of the model (these models do not contain known inputs and, consequently, the only source of information about the state and the model parameters comes from the outputs).

Regarding the HIV ODE model, the differential equations in (5.7) become the system in (6.11). This system is complex and it is not surprising that, in the state of the art, no value of the time-varying parameter, different from the true one but indistinguishable from it, was detected, and the model was classified *identifiable*. In other words, the solution of (6.11), although obtained analytically, is complex and not possible to be determined by following an intuitive reasoning, as for the second scenario of our case study. In this last case, the differential equations in (5.7) become the system in (5.9), which is trivial. Its solution is (5.10) and could have been obtained with intuitive reasoning (basically, the physical meaning of this solution is that we cannot reconstruct the absolute scale, which was intuitive).

In Section 6.6, we used a data set available in the literature and, by using the solution of (5.7), we generated infinite different unknown inputs that produce the same outputs, starting from the true unknown input and the true states. This unequivocally showed the unidentifiability of the system, in contrast with the results available in the state of the art. Finally, we determined the minimal external information (external to the knowledge of the outputs) requested to uniquely determine the system parameters. In Section 7.6, we performed a similar analysis for the Covid-19 ODE model here investigated.

Finally, for the first two ODE models, we provided a description based on observable states (Eq. (6.9) and (7.10), respectively). This may be useful for future investigations on the HIV and Covid-19 dynamics.

We conclude by emphasizing the generality of the two algorithms (Algorithms 1 and 4) introduced in this paper. They provide, automatically, the state observability and the parameters identifiability, of any nonlinear system, independently of its complexity and type of nonlinearity and in the presence of time-varying parameters (or unknown inputs). This generality also regards the two sets of equations in (5.5) and (5.7) here introduced, which allow us to build the set of indistinguishable states and indistinguishable unknown inputs in the presence of unobservability and unidentifiability. This paper provided a first application of all these theoretical results on two viral models and several important new properties (and also an error in the state of the art) were automatically discovered. Due to the aforementioned generality, we strongly believe that our theoretical results can be very useful in many other scientific domains, basically to investigate any system that can be described by an ODE model with time-varying parameters. This occurs in many scientific domains, ranging from the natural and applied sciences up to the social sciences. In particular, we are interested in studying the identifiability of different astrophysical models in the context of the G-dwarf problem [52]. To solve this problem, several models have been proposed since more than half century. Many of them are based on time-varying parameters. For instance, these time-varying parameters characterize the star formation rate, the presence of gas radial flows through our galaxy, and even the possibility that the Initial Mass Function

<footnote n="1">¹The interested reader can find a detailed explanation of the serious error made by the authors of [3] in [5]. Note that Section 6.2 of [3] contains two distinct errors. The former, which is a simple typo, was recently acknowledged by the authors (see [4]). However, in [4], the authors still concluded that the system is observable/identifiable. The very serious error committed by the authors is highlighted in Section 3.2 of [5] and lies in an incorrect use of the Inverse Function Theorem. In particular, when using this theorem, the authors did not take into account the constraints due to the dynamics of the system, as shown in Section 3.2 of [5].</footnote>
(IMF) is time-varying, as we suggested long time ago, in [53]. It would be interesting to use the
results introduced by this paper to investigate the identifiability of the time-varying parameters
that characterize these models in order to determine which measurements are suitable to provide
sufficient information that makes these parameters identifiable and, eventually, to discriminate
between all the models proposed so far.
# Appendix A

## Proof of Theorem 4.2

We start by proving the following property:

**Lemma A.1.** Let us consider the system returned by Algorithm 1 and let us suppose that $m < m_w$. We have:

$$ \mathcal{L}_{g^{m+j}} \theta = 0, \quad j = 1, \dots, m_w - m, $$

for any observable function $\theta$ (i.e., for any $\theta$ such that $\nabla \theta \in \mathcal{O}$).

*Proof.* We remind the reader that, when the system is not canonic, the observability codistribution is the limit codistribution of the series $\mathcal{O}_k^\infty$ defined in Section 6.2 of [2], which are computed by Algorithm 1 in this paper and denoted by $\Omega_k$. From Proposition 2 in [2] we know that the series of codistributions $\mathcal{O}_k^\infty$ converges and the limit codistribution is precisely $\mathcal{O}$ (see also Line 36 of Algorithm 1). Hence, the codistribution $\mathcal{O}$ is invariant under the Lie derivatives along the vector fields $m\hat{g}_{d\infty}^\alpha$, which are defined by Equation (31) in [2], for any $\alpha = 0, 1, \dots, m$. As a result, for any function $\theta$, such that $\nabla \theta \in \mathcal{O}$, we obtain that $\mathcal{L}_m \hat{g}_{d\infty}^\alpha \theta$ is independent of $w_{m+j}$, for any $j = 1, \dots, m - m_w$ and for any $\alpha = 0, 1, \dots, m$ (note that, for a time varying system, and for $\alpha = 0$, this result actually regards $\dot{\mathcal{L}}_m \hat{g}_{d\infty}^0 \theta$, with $\dot{\mathcal{L}}_m \hat{g}_{d\infty}^0 := \mathcal{L}_m \hat{g}_{d\infty}^0 + \frac{\partial}{\partial t}$). Let us consider the above function when $\alpha = 0$. From Equation (31) in [2], and by knowing that $m\nu_0^0 = 1$, we have:

$$ m\hat{g}_{d\infty}^0 = g_{d\infty}^0 + \sum_{j=1}^{d} m\nu_j^0 g_{d\infty}^j, $$

with $d = m_w - m$. From Equations (27) and (28) in [2], we remark that, in the above expression, only $g_{d\infty}^0$ depends on $w_{m+j}$ ($j = 1, \dots, m_w - m$). On the other hand:

$$ \mathcal{L}_m \hat{g}_{d\infty}^0 \theta = \mathcal{L}_{g_{d\infty}^0} \theta + \sum_{j=1}^{d} m \nu_j^0 \mathcal{L}_{g_{d\infty}^j} \theta $$

and, in this function, only the first term, $\mathcal{L}_{g_{d\infty}^0} \theta$, can depend on $w_{m+j}$.

From Equation (27) in [2], we have:

$$ \mathcal{L}_{g_{d\infty}^0} \theta = \mathcal{L}_{g^0} \theta + \sum_{j=1}^{d} (\mathcal{L}_{g^{m+j}} \theta) w_{m+j} $$

Since this function must be independent of $w_{m+j}$, for any $j = 1, \dots, m - m_w$, we obtain:

$$ \mathcal{L}_{g^{m+j}} \theta = 0 $$

for any $j = 1, \dots, m - m_w$ and for any function $\theta$ such that $\nabla \theta \in \mathcal{O}$. $\blacksquare$

Now, we are ready to prove Theorem 4.2.
Proof. To check the reconstructability of the $m_w$ unknown inputs, we consider the extended state:

$$X := [x^T, w_1, \dots, w_{m_w}]^T \qquad (A.1)$$

and we compute its observability codistribution. We denote it by $O^X$. To obtain $O^X$ we could run again Algorithm 1 for the extended system characterized by $X$. However, as we will see, it is unnecessary because we already have the observability codistribution $O$ of the system characterized by $x$.

Starting from the expression of the vector fields that characterize the dynamics of $x$, it is immediate to obtain the expression of the vector fields that characterize the dynamics of $X$. In particular, the drift, which will be denoted by $G^0$, has the following expression:

$$G^0 = \left[ \begin{array}{c} g^0 + \sum_{j=1}^{m_w} g^j w_j \\ 0_{m_w} \end{array} \right] \qquad (A.2)$$

where $0_{m_w}$ is the zero $m_w$-column vector.

We know that, for any observable function of $x$, namely a function $\theta(x)$ such that $\nabla\theta \in O$, its first order Lie derivative along $G^0$ is observable. In other words, we have:

$$\nabla \dot{\mathcal{L}}_{G^0} \theta \in O^X$$

with $\dot{\mathcal{L}}_{G^0} := \mathcal{L}_{G^0} + \frac{\partial}{\partial t}$.

We denote the dimension of $O$ by $n_o(\le n)$. Algorithm 1 builds the $n_o$ generators of $O$. They will be denoted by $\theta_1(x), \dots, \theta_{n_o}(x)$. We have:

$$O = \operatorname{span}\{\nabla\theta_1, \dots, \nabla\theta_{n_o}\}.$$

Algorithm 1 also provides the functions $\tilde{h}_1, \dots, \tilde{h}_m$, which are $m$ observable functions of $x$ such that the reconstructability matrix has full rank. As they are observable, $\nabla\tilde{h}_i \in O, i = 1, \dots, m$, and, consequently,

$$\nabla \dot{\mathcal{L}}_{G^0} \tilde{h}_i \in O^X, \quad i = 1, \dots, m.$$

In addition, as the reconstructability matrix from them has full rank, the covectors $\nabla \dot{\mathcal{L}}_{G^0} \tilde{h}_1, \dots, \nabla \dot{\mathcal{L}}_{G^0} \tilde{h}_m$ are independent and they are also independent from the generators of $O$ (because the latter are independent of the unknown inputs). As a result, the $n_o + m$ covectors $\nabla\theta_1, \dots, \nabla\theta_{n_o}, \nabla \dot{\mathcal{L}}_{G^0} \tilde{h}_1, \dots, \nabla \dot{\mathcal{L}}_{G^0} \tilde{h}_m$ are independent.

On the other hand, we know that the functions $\tilde{h}_1, \dots, \tilde{h}_m$ are observable functions, i.e., their gradient belongs to $O$. From Lemma A.1, and from the structure of $G^0$ in (A.2), we obtain that all the functions $\dot{\mathcal{L}}_{G^0} \tilde{h}_1, \dots, \dot{\mathcal{L}}_{G^0} \tilde{h}_m$ are independent of the last $m_w - m$ unknown inputs (i.e., the last $m_w - m$ entries of $X$). As a result, the dimension of $O^X$ cannot exceed $n_o + m$.

Therefore:

$$O^X = \operatorname{span}\{\nabla\theta_1, \dots, \nabla\theta_{n_o}, \nabla\dot{\mathcal{L}}_{G^0}\tilde{h}_1, \dots, \nabla\dot{\mathcal{L}}_{G^0}\tilde{h}_m\}. \qquad (A.3)$$

Now, by definition, any vector field that belongs to the null space of $O^X$ is a simultaneous symmetry of both the state and the unknown input. Let us study the structure of this null space. As all the $n_o + m$ functions $\theta_1, \dots, \theta_{n_o}, \dot{\mathcal{L}}_{G^0} \tilde{h}_1, \dots, \dot{\mathcal{L}}_{G^0} \tilde{h}_m$ are independent of the last $m_w - m$ unknown inputs, any vector with dimension $n + m_w$, with the first $n + m_w - m$ entries equal to zero, certainly belongs to the null space of $O^X$. Therefore, the vectors $c\chi$ in (4.5) are $m_w - m$ independent symmetries of the unknown input. $\blacksquare$
# Appendix B

## Proof of Theorem 4.3

*Proof.* We proceed as in the proof of Theorem 4.2. We need to prove that the vector field:

$$ \xi^X := \begin{bmatrix} \xi \\ u \chi \end{bmatrix}, \qquad (B.1) $$

with $^u\chi$ defined in (4.6), belongs to the null space of $\mathcal{O}^X$ in (A.3).

Let us consider the first $n_o$ generators of $\mathcal{O}^X$. As $\xi \in \mathcal{O}^T$, we have:

$$ \frac{\partial \theta_i}{\partial x} \cdot \xi = 0, \quad i = 1, \dots, n_o. $$

Hence,

$$ \begin{aligned} \nabla \theta_i \cdot \xi^X &= \frac{\partial \theta_i}{\partial x} \cdot \xi + \frac{\partial \theta_i}{\partial w} \cdot {}^u\chi = \\ & \qquad 0 + \underbrace{[0, \dots, 0]}_{m_w} \cdot {}^u\chi = 0, \quad i = 1, \dots, n_o. \end{aligned} $$

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

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

$$ \nabla \dot{\mathcal{L}}_{G^0} \tilde{h}_l \cdot \xi^X = \frac{\partial \dot{\mathcal{L}}_{G^0} \tilde{h}_l}{\partial x} \cdot \xi + \sum_{k=1}^{m} \frac{\partial \dot{\mathcal{L}}_{G^0} \tilde{h}_l}{\partial w_k} {}^u\chi_k \qquad (B.2) $$

On the other hand,

$$ \dot{\mathcal{L}}_{G^0} \tilde{h}_l = \dot{\mathcal{L}}_{g^0} \tilde{h}_l + \sum_{j=1}^{m} (\mathcal{L}_{g^j} \tilde{h}_l) w_j, \quad l = 1, \dots, m \qquad (B.3) $$

By using (B.3) in (B.2) and by using (4.7) we obtain:

$$ \begin{aligned} \nabla \dot{\mathcal{L}}_{G^0} \tilde{h}_l \cdot \xi^X &= \xi_l^0 + \sum_{j=1}^{m} \xi_l^j w_j + \sum_{k=1}^{m} \mathcal{L}_{g^k} \tilde{h}_l \ {}^u\chi_k = \\ & \qquad \xi_l^0 + \sum_{j=1}^{m} \xi_l^j w_j + \sum_{k=1}^{m} {}^m\mu_l^k \ {}^u\chi_k. \end{aligned} $$

Now, by using the expression of $^u\chi_k$ in (4.6) and that $^m\mu$ is the inverse of $^m\nu$ (i.e., $\sum_{k=1}^m {}^m\mu_l {}^m\nu_k^i = \delta_l^i$), we finally obtain:

$$ \nabla \dot{\mathcal{L}}_{G^0} \tilde{h}_l \cdot \xi^X = 0. $$

<figure><img src="image_17.png" /></figure>
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