Assessment of Computational Tools for Analysing the Observability and Accessibility of Nonlinear Models

Abstract

Accessibility and observability are two properties of dynamic models that provide insights into the structural relationships between their input, output, and state variables. They are closely related to controllability and structural local identifiability, respectively. Observability and identifiability determine, respectively, the possibility of inferring the unmeasured state variables and parameters of a model from output measurements; accessibility and controllability describe the possibility of driving its state by changing its input. Analysing these structural properties in nonlinear models of ordinary differential equations can be challenging, particularly when dealing with large systems. Two main approaches are currently used for their study: one based on differential geometry, which uses symbolic computation, and another one based on sensitivity calculations that uses numerical integration. These approaches are implemented in two MATLAB (R2024b) software tools: the differential geometry approach in STRIKE-GOLDD, and the sensitivity-based method in StrucID. These toolboxes differ significantly in their features and capabilities. Until now, their performance had not been thoroughly compared. In this paper we present a comprehensive comparative study of them, elucidating their differences in applicability, computational efficiency, and robustness against computational issues. Our core finding is that StrucID has a substantially lower computational cost than STRIKE-GOLDD; however, it may occasionally yield inconsistent results due to numerical issues.

1. Introduction

Structural local identifiability and observability are theoretical properties that characterize the ability to infer, respectively, the parameters and the states of a dynamic model from knowledge of its inputs and outputs [1,2,3]. Structural identifiability is a necessary but not sufficient condition for model calibration, i.e., for parameter estimation [4]. Similarly, observability is a prerequisite for designing state observers and, consequently, for real-time monitoring of the model’s internal states. In the linear case, observability is analysed by computing the rank of the observability matrix, which can be readily obtained through matrix multiplications [5]. In nonlinear systems, observability depends on the operating point and can also be assessed via the rank of an observability matrix [6]. However, the computation of nonlinear observability in this way becomes more involved, as it requires Lie derivatives [7]. Observability and structural identifiability are closely related concepts; in fact, by treating model parameters as constant state variables, the identifiability problem can be reformulated as an observability problem [8]. As a result, both properties can be analysed jointly; we will refer to Structural Identifiability and Observability with the acronym SIO. Note that in this work we consider these properties from a structural viewpoint, which is sometimes also referred to as “a priori”. Their practical counterparts can be approached using methods for practical identifiability analysis [9,10] and uncertainty quantification [11,12].

Historically, the main challenges in analysing SIO have been the limited applicability of some methods for specific models and problem conditions, and the computational burden associated with large-scale models. However, over the past decade, several methods for performing structural identifiability analysis along with publicly available implementations have been developed [13]. Due to these recent advances, it has been argued that the analysis of SIO no longer represents an obstacle in the development of useful mathematical models [10]. The availability of fast and reliable methods for detecting structurally unidentifiable parameters enables researchers to perform this analysis repeatedly throughout the model development process and across various programming environments. One approach, previously mentioned, relies on constructing an observability and identifiability matrix and computing its rank. This test, known as the Observability Rank Condition (ORC), draws on concepts from differential geometry, namely Lie derivatives, and it is implemented in the MATLAB toolbox STRIKE-GOLDD [14,15]. While the ORC is theoretically powerful, its computational cost can be high, especially for complex models with a large number of unknowns (parameters and states) relative to the number of outputs. As an alternative, numerical approaches have been proposed, such as the one presented in [16], which is based on computing the sensitivity matrix and applying singular value decomposition (SVD). A MATLAB implementation is available as the StrucID toolbox [17].

Controllability, like observability, is a property that depends on a model’s structure [7,18,19,20]. Broadly speaking, it describes the possibility of driving a system from an initial state to a final state within a specified period of time. A related property, local accessibility, refers to the possibility of driving a system from an initial state to a full-dimensional target set, i.e., it means that the system can be steered from an initial state to any point in its neighbourhood [21]. For linear models at equilibrium points, controllability and accessibility coincide, and they can be determined by checking whether the controllability matrix has full rank, which implies that the reachable set at time T has a non-empty interior. In contrast, for nonlinear models, these concepts may differ. In particular, nonlinear controllability at non-equilibrium points is associated with a property sometimes called strong accessibility, which means that the accessible set has a non-empty interior exactly at time T. Accessibility has also been referred to in the literature as local weak controllability [7] or simply accessibility.

The analysis of Nonlinear Accessibility and Controllability (NAC) resembles that of structural identifiability and observability (SIO) in that it can also be performed with methods based on differential geometry or sensitivities [22,23]. Except for the simplest cases, such analyses typically involve complex computations and require algorithmic implementations of the associated tests. The differential geometric approach to NAC analysis involves the computation of Lie brackets, instead of the Lie derivatives used for SIO analysis. A number of conditions can be used to analyse NAC (see [15,24]); like the ORC test for SIO analyses, they are also implemented in the aforementioned STRIKE-GOLDD toolbox. Thus, this software tool provides a comprehensive differential geometry approach to SIO and NAC analysis. Likewise, the StrucID toolbox was initially designed for SIO analysis and later extended to NAC [25], thus providing a unified implementation for performing sensitivity-based analyses of SIO and NAC. It should be noted that, although many specialised software tools exist for testing SIO [13], there are comparatively fewer tools for analysing NAC. This gap may be one of the reasons why the controllability of nonlinear models remains underexplored in domains such as biological modelling, biomedicine, or bioprocesses [26].

Until now, the trade-off between computational accuracy and efficiency in these tools has not yet been evaluated in sufficient detail. Some partial results have been published: for example, Wang and Qi [27] presented a method based on automatic differentiation and compared its performance on two models with STRIKE-GOLDD and StrucID; however, they did not share a software implementation, nor did they disclose key details of the comparisons. Rey Barreiro and Villaverde [13] conducted an extensive comparison of SIO analysis tools but limited to symbolic computation methods, and therefore not considering StrucID. Heinrich and colleagues [28] reviewed recent advances in identifiability analysis and compared a number of tools, including StrucID and STRIKE-GOLDD, but focusing only on their computational speed. Moreover, we have also performed preliminary comparisons of these two tools, as well as of the empirical Gramian framework [29], although we only examined their performance for SIO analysis [30].

To fill this gap, our aim in this paper is to compare the performance of the two main tools that are currently available for the analysis of structural local identifiability, observability, accessibility and controllability of nonlinear models: STRIKE-GOLDD and StrucID. We consider models described by ordinary differential equations (ODEs), and examine three criteria: the applicability of the methods, the reliability of their results, and the computational time taken by the calculations. The remainder of this paper is organised as follows: First, in Section 2, we introduce the modelling framework and the methods. Then, in Section 3 we present the results of the evaluations and discuss them. Lastly, we provide our conclusions in Section 4.

2. Materials and Methods

2.1. Modelling Framework

We study dynamic models described by systems of ordinary differential equations (ODEs) of the general form

$$\mathcal{M}:\left\{\begin{array}{c}\dot{x}\left(t\right)=f(x\left(t\right),u\left(t\right),\theta ),\hfill \\ y\left(t\right)=h\left(x\right(t),\theta ).\hfill \end{array}\right.$$

(1)

where f and h are analytic functions. Here, $\theta \in {\mathbb{R}}^p$ denotes the parameter vector, $u\left(t\right)\in {\mathbb{R}}^r$ represents the input vector, $x\left(t\right)\in {\mathbb{R}}^n$ the state vector, and $y\left(t\right)\in {\mathbb{R}}^m$ the output vector. In this paper, we may sometimes omit the dependence on time to simplify the notation, i.e., we write $\dot{x}=f(x,u,\theta )$ and $y=h(x,\theta ),$ for example.

Very often, models are affine in the inputs, i.e., their dynamics can be expressed in the following form:

$$\dot{x}=f(x,\theta )+\sum _{i=1}^r{u}_i{g}_i(x,\theta ),$$

(2)

where $u_i$ are the control inputs and, as before, f and $g_i$ are analytic vector fields. This type of model can adequately describe many systems and processes, including biological ones [31,32]. Many theoretical conditions about accessibility and controllability obtained with differential geometric approaches—including those reviewed in Section 2.4.2—are not applicable to models of the general nonlinear form (1), and instead require that the model under study is affine in the inputs.

2.2. Concepts

2.2.1. Structural Identifiability and Observability (SIO)

A parameter ${\theta }_i\in \theta$ is structurally locally identifiable [33,34] if, for almost every ${\theta }^{\ast }\in {\mathbb{R}}^q$ (except for a set of measure zero), there exists a neighbourhood $\mathcal{N}\left({\theta }^{\ast }\right)$ such that for all $\widehat{\theta }\in \mathcal{N}\left({\theta }^{\ast }\right)$,

$$y(t,{\theta }^{\ast })=y(t,\widehat{\theta })⟺{\theta }_i^{\ast }={\widehat{\theta }}_i,$$

(3)

If this condition does not hold in any neighbourhood of ${\theta }^{\ast }$, then ${\theta }_i$ is structurally unidentifiable. In this work, we will use the word “identifiability” to refer specifically to structural local identifiability.

Similarly, a state variable $x_i\left(\tau \right)$ is said to be observable if it can be locally determined from the output $y\left(t\right)$ and the input $u\left(t\right)$ over a finite interval $t_0\le \tau \le t\le t_f$ ($x_i\left(\tau \right)$ is the ith state variable evaluated at time $\tau$, and ${\theta }_i\in \theta$ denotes the ith model parameter. The symbols $t_0$ and $t_f$ correspond to the initial and final times of the observation interval) as discussed in [3]. The observability of a state variable is equivalent to the identifiability of its initial condition, if the latter is treated as a parameter. (Note that observability is also a structural and local property, in the same way as the version of identifiability that we study here.) A model as a whole is said to be identifiable (respectively, observable) if all its parameters (respectively, state variables) satisfy the corresponding property. Otherwise, it is called unidentifiable (respectively, unobservable) [10]. We use the acronym SIO to refer to structural local identifiability and observability.

2.2.2. Nonlinear Accessibility and Controllability (NAC)

A system has the accessibility property (i.e., it is called accessible) if it can be steered from an initial state $x_0$ to a full-dimensional final set $x\left(t_f\right)=x_f$, using appropriate control inputs $u\left(t\right)$ [35]. A system is said to be locally controllable if it can be steered from an initial state $x_0$ to any neighbouring point and back. The definition of controllability entails that the initial state lies within the accessible neighbourhood [24]. Thus, accessibility is a less demanding property than controllability, which is why it is sometimes referred to as weak local controllability [7]. Therefore, while every controllable system is necessarily accessible, the reverse is not always true [24]. We will use the acronym NAC to refer to nonlinear accessibility and controllability.

2.3. StrucID

StrucID is a MATLAB toolbox for which the source code, as well as a compiled version that enables its use without a MATLAB installation, is available on Github (https://github.com/jdstigter/StrucID, accessed on 1 November 2025). It is also integrated in the D2D-framework (https://github.com/Data2Dynamics/d2d, accessed on 1 November 2025). StrucID is designed for the structural analysis of nonlinear dynamic models. It evaluates structural identifiability, observability, and accessibility (for which it uses the equivalent term “reachability”). It can test for the identifiability of unknown initial conditions (i.e., observability), while known initial conditions can also be provided in the input file and are used to inform the numeric integration. The input to a StrucID analysis is provided in a single .txt file [17,36]. Figure 1 shows the interface of this software.

The tests provided by StrucID are based on sensitivity calculations. Sensitivities have the advantageous property of being governed by linear dynamics, even when the system itself is nonlinear. By integrating these linear dynamics over a short time interval and sampling the resulting trajectories, a sensitivity matrix can be formed. If this matrix satisfies a specific rank condition, the local structural property of the system under investigation is confirmed. Moreover, performing a singular value decomposition (SVD) on the sensitivity matrix not only determines its rank but also pinpoints the exact system components responsible for any failure to satisfy the local structural property.

2.3.1. SIO Analysis with StrucID

The parametric state and output sensitivity matrix functions, denoted by ${\displaystyle \frac{\partial x}{\partial \theta }}\left(t\right)$ and ${\displaystyle \frac{\partial y}{\partial \theta }}\left(t\right)$, respectively, play a central role. For convenience of notation, these matrix functions are represented as $x_{\theta }\left(t\right)$ and $y_{\theta }\left(t\right)$, having dimensions $n\times p$ and $m\times p$, respectively. At any given time t, $x_{\theta }\left(t\right)$ and $y_{\theta }\left(t\right)$ quantify the sensitivity of the state vector $x\left(t\right)$ and the output vector $y\left(t\right)$ with respect to the parameter vector $\theta$. By differentiating Equation (1) with respect to the parameter vector $\theta$, and by interchanging differentiation with respect to the independent parameters, it can be shown that the dynamics of these sensitivity matrices are governed by the following relations:

$${\displaystyle \frac{d{x}_{\theta }}{dt}}={\displaystyle \frac{\partial f}{\partial x}}{x}_{\theta }+{\displaystyle \frac{\partial f}{\partial \theta }},$$

(4)

$$y_{\theta }={\displaystyle \frac{\partial h}{\partial x}}{x}_{\theta }+{\displaystyle \frac{\partial h}{\partial \theta }}.$$

(5)

Simultaneous integration of the differential Equations (1) and (4), starting from initial conditions $x\left(t_0\right)$, yields the trajectories $x\left(t\right)$ and $x_{\theta }\left(t\right)$. Substituting these into Equation (5) provides the corresponding output sensitivity matrix $y_{\theta }\left(t\right)$. A necessary and sufficient condition for unique parameter determination from the measured output $y\left(t\right)$ of the system (Equation (1)), under a given set of initial conditions, is that $y_{\theta }\left(t\right)$ possesses the full column rank, i.e., p linearly independent columns. This rank condition can be verified by evaluating $y_{\theta }\left(t\right)$ at discrete time instants $t_0<t_1<\cdots <t_N$, forming the parametric output sensitivity matrix:

$$Y_{\theta }=\left[\begin{array}{c}{y}_{\theta }\left(t_0\right)\\ y_{\theta }\left(t_1\right)\\ ⋮\\ y_{\theta }\left(t_N\right)\end{array}\right]\in {\mathbb{R}}^{(N+1)m\times p}.$$

(6)

The rank condition, referred to as the Sensitivity Equation Rank Condition (SERC), is

$$\mathrm{rank}\left(Y_{\theta }\right)=p.$$

StrucID evaluates the SERC by performing a singular value decomposition (SVD) of the sensitivity matrix $Y_{\theta }$. If this matrix is rank deficient, its SVD will yield one or more zero singular values. Each zero singular value corresponds to a singular vector obtained from the decomposition, and the nonzero elements of this vector identify the parameters that are mutually correlated and therefore cannot be uniquely estimated.

By interpreting the initial state $x\left(t_0\right)$ as a set of parameters, the local observability of the state can be analysed as the local structural identifiability of the initial condition $x\left(t_0\right)$ from the system output $y\left(t\right)$ for $t\ge t_0$. Thus, the state vector $x\left(t_0\right)$ plays the same role as the parameter vector $\theta$ in Equations (1)–(8). Since in Equations (7) and (8) the functions f and h do not explicitly depend on $x_0$, these equations simplify to

$${\displaystyle \frac{d{x}_{x_0}}{dt}}={\displaystyle \frac{\partial f}{\partial x}}{x}_{x_0},$$

(7)

$$y_{x_0}={\displaystyle \frac{\partial h}{\partial x}}{x}_{x_0},$$

(8)

where $x_{x_0}\left(t\right)$ and $y_{x_0}\left(t\right)$ denote the parametric sensitivity matrix functions of $x\left(t\right)$ and $y\left(t\right)$ with respect to the initial state $x_0$. The simultaneous integration of the differential Equations (1) and (7), subject to the initial condition and

$$x_{x_0}\left(t_0\right)=I_n,$$

(9)

yields the trajectories $x\left(t\right)$ and $x_{x_0}\left(t\right)$. In Equation (9), $I_n$ represents the $n\times n$ identity matrix, corresponding to the sensitivity of $x\left(t_0\right)=x_0$ with respect to itself. Substituting $x\left(t\right)$ and $x_{x_0}\left(t\right)$ into Equation (8) yields $y\left(t\right)$ and $y_{x_0}\left(t\right)$. The sensitivity matrix function $y_{x_0}\left(t\right)$ having n linearly independent columns is a necessary and sufficient condition to uniquely determine $x_0$ from the system output $y\left(t\right)$, given a choice of the initial conditions. Analogously to Equation (6), to verify this rank condition, the sensitivity matrix function $y_{x_0}\left(t\right)$ is evaluated at discrete time instants $t_0<t_1<\cdots <t_N$ to form the concatenated sensitivity matrix:

$$Y_{x_0}=\left[\begin{array}{c}{y}_{x_0}\left(t_0\right)\\ y_{x_0}\left(t_1\right)\\ ⋮\\ y_{x_0}\left(t_N\right)\end{array}\right]\in {\mathbb{R}}^{(N+1)m\times n}.$$

(10)

Thus, the Sensitivity Equation Rank Condition (SERC) for local observability is

$$\mathrm{rank}\left(Y_{x_0}\right)=n.$$

The SERC is necessary and sufficient. If $Y_{x_0}$ does not have full rank n, the nonzero components of the singular vectors associated with zero singular values indicate state variables that are mutually correlated and therefore cannot be uniquely identified.

2.3.2. NAC Analysis with StrucID

By considering sensitivities, an important equivalence was established between the observability of the initial state $x_0$ from those trajectories and the observability of the linear dynamics along trajectories. This dual relationship extends naturally to controllability, showing that accessibility of a nonlinear system can likewise be analysed through the controllability of the linearised dynamics along trajectories. The duality between observability and controllability is established by linearising the nonlinear system around a trajectory [37], giving the perturbation dynamics

$$\delta \dot{x}={\displaystyle \frac{\partial f}{\partial x}}\delta x+{\displaystyle \frac{\partial f}{\partial u}}\delta u.$$

(11)

By applying the substitutions

$$t\to t_N-t,y\to u,Y\to U,x\left(t_0\right)=x_0\to x\left(t_N\right)=x_N,{\displaystyle \frac{\partial f}{\partial x}}\to {\left({\displaystyle \frac{\partial f}{\partial x}}\right)}^T,{\displaystyle \frac{\partial h}{\partial x}}\to {\left({\displaystyle \frac{\partial f}{\partial u}}\right)}^T,$$

the dual system is obtained as

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_{xN}}{dt}}& ={\left({\displaystyle \frac{\partial f}{\partial x}}\right)}^T{x}_{xN},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill u_{xN}& ={\left({\displaystyle \frac{\partial f}{\partial u}}\right)}^T{x}_{xN},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill x_{xN}\left(t_N\right)& =I_n.\hfill \end{array}$$

(14)

The resulting sensitivity matrix,

$$U_{xN}=\left[\begin{array}{c}{u}_{xN}\left(t_N\right)\\ u_{xN}\left(t_{N-1}\right)\\ ⋮\\ u_{xN}\left(t_0\right)\end{array}\right]\in {\mathbb{R}}^{(N+1)r\times n},$$

(15)

serves as the dual counterpart of the output sensitivity matrix $Y_{x0}$ introduced for observability analysis [25]. Its structure captures how infinitesimal variations in the input sequence affect the final state $x_N$, thus providing a basis for evaluating the local controllability and strong accessibility of the nonlinear system. For both local structural identifiability and local observability, the singular value decomposition (SVD) of the sensitivity matrix $Y_{x_0}$ plays a central role in the analysis. Analogously, for the assessment of local strong accessibility, an SVD of the sensitivity matrix $U_{x_N}$ serves this same fundamental purpose. The SVD of this matrix can be expressed as

$$U_{x_N}=\sum _{i=1}^n{u}_i{r}_i{v}_i^T,U_{x_N}\in {\mathbb{R}}^{(N+1)r\times n},u_i\in {\mathbb{R}}^{(N+1)r},r_i\in \mathbb{R},v_i\in {\mathbb{R}}^n.$$

(16)

This decomposition represents $U_{x_N}$ as a sum of equally sized matrices $u_i{v}_i^T$, each weighted by its corresponding singular value $r_i\ge 0$. If one or more of the singular values $r_i$ are (numerically) zero, the matrix $U_{x_N}$ becomes rank deficient. The column vectors $v_i$ associated with zero singular values form a basis for the null space of $U_{x_N}$. These vectors define directions in the state space along which the terminal state $x_N$ cannot be locally controlled.

2.4. STRIKE-GOLDD

STRIKE-GOLDD is a MATLAB toolbox originally developed for analysing structural identifiability and observability. Since version 4.1.0 [38], it can also analyse accessibility and controllability, thanks to the functions in the module NLControllability. It can be downloaded from GitHub at https://github.com/afvillaverde/strike-goldd, (accessed on 1 November 2025). Figure 2 illustrates the main interfaces of this software.The analysis of SIO with STRIKE-GOLDD can be performed with three algorithms—FISPO [39], ORC-DF [40], and ProbObsTest [15]—of which we use FISPO and ProbObsTest in this study. For analysing NAC, up to four algorithms are available [15]: LC, ARC, LARC, and GSC. Of these, the latter two are used in this study. These methods are described in the remainder of this section.

2.4.1. SIO Analysis in STRIKE-GOLDD

• FISPO

The FISPO algorithm is an advanced implementation of the ORC test mentioned in the Introduction, which can be applied to rational and non-rational models. The ORC, which was introduced in [7], provides the theoretical differential geometric basis for determining observability as well as—by considering the parameters as state variables—structural local identifiability. The FISPO algorithm is an extended computational implementation of the ORC. One key feature is its iterative nature, i.e., only a minimum number of Lie derivatives is calculated initially, with higher-order derivatives being computed only if necessary [14]. Another important extension of FISPO with respect to the original ORC is that it can take into account the effect of time-varying inputs explicitly [41], as well as analysing models with unknown inputs [39].

To analyse SIO, the parameters ${\theta }_i$ are considered constant state variables, thus extending the state vector as $\tilde{x}=\tilde{x}(x,\theta )$. The algorithm builds the observability-identifiability matrix:

$$\mathcal{OI}\left(x\right)=\left(\begin{array}{c}{\displaystyle \frac{\partial h\left(\tilde{x}\right)}{\partial \tilde{x}}}\\ {\displaystyle \frac{\partial \left(L_{f}h\left(\tilde{x}\right)\right)}{\partial \tilde{x}}}\\ {\displaystyle \frac{\partial \left(L_f^{2}h\left(\tilde{x}\right)\right)}{\partial \tilde{x}}}\\ ⋮\\ {\displaystyle \frac{\partial \left(L_f^{n+q-1}h\left(\tilde{x}\right)\right)}{\partial \tilde{x}}}\end{array}\right).$$

(17)

which is obtained from Lie derivatives that, in their general form, can be calculated as

$$L_{f}h\left(\tilde{x}\right)={\displaystyle \frac{\partial h\left(\tilde{x}\right)}{\partial \tilde{x}}}f(\tilde{x},u)$$

(18)

$$L_f^{i}h\left(\tilde{x}\right)={\displaystyle \frac{\partial \left(L_f^{i-1}h\left(\tilde{x}\right)\right)}{\partial \tilde{x}}}f(\tilde{x},u)+\sum _{j=0}^{i-2}{\displaystyle \frac{\partial \left(L_f^{i-1}h\left(\tilde{x}\right)\right)}{\partial u^{\left(j\right)}}}{u}^{(j+1)},i>1.$$

(19)

The $\mathcal{OI}$ matrix must have full rank, i.e., equal to $n+q,$ for the system to be fully identifiable and observable. If rank < $n+q$, the identifiability/observability of each variable is determined by removing each column and re-calculating the rank. If this operation decreases the rank, the variable associated to the column is identifiable/observable; otherwise, it is not [39].

The calculation of Lie derivatives is performed symbolically, and its computational cost increases steeply with $i.$ One alternative to alleviate this computational complexity is the ProbObsTest algorithm.

• ProbObsTest

This algorithm speeds up the ORC test by avoiding the symbolic calculations of Lie derivatives. It is based on the algorithm proposed by Sedoglavic [42] and, like it, is applicable to rational models. Unlike classical methods based on algebraic symbolic theory, which typically have exponential time complexity, this algorithm uses semi-numerical methods and performs the analysis in polynomial time. Sedoglavic’s variational system is used to avoid symbolic computations, and symbolic variables are replaced with random numerical values [15]. The output derivatives $Y^{\left(j\right)}$ are generated up to a certain order $\nu$, using derivatives with respect to the vector field F. The partial derivatives of the outputs with respect to the state variables and parameters form a Jacobian matrix:

$$J={\displaystyle \frac{\partial \left(Y^{\left(j\right)}\right),0\le j\le \nu }{\partial (X,\Theta )}}$$

Power series of the output are obtained, and the Jacobian is computed numerically. As in FISPO, variables and parameters that reduce the rank are considered unobservable. The algorithm uses modular arithmetic, and its probability of success can be made arbitrarily large by tuning a precision parameter.

2.4.2. NAC Analysis with STRIKE-GOLDD

STRIKE-GOLDD provides four algorithms for testing NAC conditions: LC, ARC, LARC, and GSC. In this study, we focus exclusively on the LARC and GSC algorithms, which test, respectively, for accessibility and controllability. The other two, LC and ARC, can be useful in specific case studies but are not essential and thus not considered here for simplicity. We describe the LARC and GSC algorithms below.

• Lie Algebra Rank Condition (LARC)

The LARC test computes a control distribution ${\Delta }_c$ recursively, starting with ${\Delta }_0=\mathrm{span}\left(\mathcal{F}\right),$ where $\mathcal{F}=\{f,g_1,\dots ,g_m\},$ and updating it at subsequent steps as ${\Delta }_{i+1}=\mathrm{span}\left({\Delta }_i\cup \{[v,h]\mid v\in \mathcal{F},h\in {\Delta }_i\}\right).$ After each step, it checks whether the distribution is invariant with respect to the set $\mathcal{F}$, i.e., if for every $h\in {\Delta }_i$ and $v\in \mathcal{F}$, the Lie bracket $[v,h]\in {\Delta }_i$. If this condition is satisfied, the algorithm stops, and ${\Delta }_i$ is the final control distribution, ${\Delta }_c$. Then, the system is accessible at $x_0$ if and only if, at the end of the process, ${\Delta }_c\left(x_0\right)$ has full rank (i.e., $dim\left({\Delta }_c\left(x_0\right)\right)=n$).

The LARC test can be computationally intensive due to the exponential growth in the number of Lie brackets and the need to check the non-invariance condition at every step. The LARC implementation in STRIKE-GOLDD includes modifications to alleviate its computational complexity [15].

• General Sufficient Condition (GSC)

Sussmann’s GSC [43] assigns to each Lie bracket generated from the set $F=\{f,g_1,\dots ,g_m\}$ a type in the form of a numerical vector $(k,{\ell }_1,\dots ,{\ell }_m)$, where k denotes the number of occurrences of the vector f, and ${\ell }_i$ denotes the number of occurrences of $g_i$. Brackets in which f appears an odd number of times and each $g_i$ appears an even number of times are called “bad” brackets. If the system is at equilibrium at the point $x_0$ (i.e., $f\left(x_0\right)=0$) and the LARC condition holds, then the system is STLC if each bad bracket can be expressed as a smooth linear combination of brackets of the lower type. This lower type is defined in weighted terms, $\theta k_j+{\ell }_{j1}+\cdots +{\ell }_{jm}<\theta k+{\ell }_1+\cdots +{\ell }_m,$ where $\theta \in [0,1]$ is an arbitrary weight (typically chosen close to zero to minimize the weight of f).

In STRIKE-GOLDD, the GSC is verified through a sequence of steps. First, the control distribution is generated by applying the LARC algorithm. Next, “bad” Lie brackets are detected. Lastly, it is verified whether each bad bracket can be expressed as a linear combination of brackets of the lower type; this is checked using algebraic operations in vector spaces such as computing the null space.

2.5. Case Studies

In this study, we analysed the SIO and NAC of a set of 32 nonlinear models that cover a wide range of processes, including gene regulatory networks, enzymatic reactions, signalling pathways, metabolic systems, and synthetic biology circuits. The name of each model, a brief description, and the corresponding references are summarised in

Table 1. Summary of the 32 biological models analysed in this study. The table includes each model’s short name, its description, and the original reference.

Model	Description	Ref.
1_A_integral	Linear circuit with integral feedback	[44]
1B_Prob_integral	Linear circuit with proportional-integral feedback	[44]
1C_nonlinear	Nonlinear circuit model of hormonal reactions	[44]
2DOF	A mechanical system with two independent motions	[40]
Arabidopsis	Circadian rhythm in the plant Arabidopsis thaliana	[45]
Bolie	Glucose-insulin regulatory system in humans	[46]
C2M	Compartmental model with two compartments	[41]
Fujita	Epidermal Growth Factor-dependent Protein Kinase B signalling pathway	[47]
HIV_known input	HIV infection dynamics	[4]
Bioprocess	Biochemical reactions	[48]
TS_Known input	Genetic toggle switch	[49]
1D_BIG	Glucose-insulin regulation via $\beta$-cell and insulin	[44]
NFKB (Merkt)	NF-$\kappa$B (Nuclear Factor kappa B) signalling pathway	[50]
PC (hSRPsIPr)	A phage cocktail, describing bacteriophage-bacteria interactions and the host immune response	[38]
Raia_jakstat	JAK/STAT signalling pathway in lymphoma cells	[51]
MAPK	Mitogen Activated Protein Kinase signalling pathway	[52]
BioSD_I	Two-species synthetic biology circuit for signal differentiation	[53]
BioSD_II	Three-species synthetic biology circuit for signal differentiation	[53]
BioSD_III	Three-species synthetic biology circuit for signal differentiation with positive and negative feedback	[53]
BioSD_II_MM_simple	Three-species synthetic biology circuit for signal differentiation with simple kinetics	[53]
BioSD_II_MM_complex	Three-species synthetic biology circuit for signal differentiation with complex kinetics	[53]
Dichotomous_Feedback_BettaRR	Synthetic biology circuit implementing dichotomous feedback—RR molecules (Response Regulator)	[54]
Dichotomous_Feedback_BettaSR	Synthetic biology circuit implementing dichotomous feedback—SR molecules (Sequestering Regulator)	[54]
Dichotomous_Feedback_kap	Synthetic biology circuit with input signal $kap$	[54]
Dichotomous_Feedback_I_BettaRR	Synthetic biology circuit with dichotomous feedback, input I, and RR molecules	[54]
Dichotomous_Feedback_I_BettaSR	Synthetic biology circuit with dichotomous feedback, input I, and SR molecules	[54]
Dichotomous_Feedback_I	Synthetic biology circuit with dichotomous feedback, with input signal I	[54]
Kelly_1	Synthetic biology circuit implementing negative feedback via sRNA-tuned autorepression	[55]
Kelly_1_gr	Synthetic biology circuit with generalised regulatory dynamics	[55]
Kelly_2	Synthetic biology circuit implementing negative feedback via closed-loop sRNA regulation	[55]
Kelly_2_gxgr	Synthetic biology circuit with generalised regulatory dynamics	[55]
CSTR_observer	Continuous Stirred Tank Reactor with High Gain Observer	[56]

. More detailed information can be found in the original references listed in the table. We analysed the SIO of all of these models (for one of them, the “Bioprocess”, we analysed several versions with different number of experiments, thus increasing the number of cases to 35). However, we did not analyse the NAC of all models because some of them do not have control inputs.

3. Results

In this study, to provide a clear and comprehensive understanding of the investigation, when discussing the results of both families of analyses, SIO and NAC, we focused on three key aspects:

(i) Applicability: we assessed whether each model could be fully analysed by both tools, and identified which algorithms were capable of performing the analyses.

(ii) Reliability of the results: we compared the results obtained from the two software tools, to detect differences in their outputs, and investigated the potential reasons behind such discrepancies.

(iii) Computational efficiency: we measured the time required to complete the analyses.

The results of the analyses are presented and discussed below.

3.1. Observability and Identifiability Analyses

We first analysed the SIO of all the models with StrucID and STRIKE-GOLDD (in the latter case with the ProbObsTest and FISPO algorithms).

3.1.1. Applicability

According to the applicability of the methods, the models can be classified into four categories.

First category: models that can be successfully analysed by all three algorithms. Of the 35 models considered, 14 fell into this category.

Second category: models that can be analysed by StrucID and ProbObsTest but not by FISPO due to their computational cost. Of the 35 models considered, 13 fell into this category. The main reason behind the failure of FISPO in these cases is the high dimensionality of the models, characterised by a large number of state variables, parameters, and equations.

Third category: models that can be analysed by StrucID and the FISPO algorithm but not by ProbObsTest due to the models being non-rational. Of the 35 models considered, 2 fell into this category.

Fourth category: models that can only be analysed using StrucID. Of the 35 models considered, 6 fell into this category. Similar to the second category, the excessive computational load is the primary factor preventing the algorithms in STRIKE-GOLDD from completing the analysis.

3.1.2. Reliability

Table 2. Results of the SIO analyses using the FISPO and Prob_Obs_Test algorithms from the STRIKE-GOLDD and the StrucID toolboxes. The table classifies model parameters and state variables into four categories: Identifiable, Non-identifiable (Non-Id), Observable (Obs), and Non-observable (Non-Obs).

Models	FISPO (STRIKE-GOLDD)				Prob_Obs_Test (STRIKE-GOLDD)				StrucID
	Identifiable	Non-Id	Obs	Non-Obs	Identifiable	Non-Id	Obs	Non-Obs	Identifiable	Non-Id	Obs	Non-Obs
1_A_integral	All		All		All		All		All		All
1B_Prob_integral	All		All		All		All		All		All
2DOF	All		All		All		All		All		All
C2M	All		All		All		All		x4	x3, x5, x6	x1	x2
Fujita	All		All		All		All		R_2_k2, R_3_k1, R_4_k1, R_5_k2, R_6_k1, R_7_k1, R_8_k1	SF_p, SF_pAkt, SF_pS6,EGFR_turnover, R_1_k1, R_1_k2, R_2_k1, R_5_k1, R_9_k1		pS6, EGFR, pEGFR, pEGFR_Akt, Akt, pAkt, S6, pAkt_S6, EGF_EGFR
MAPK	All		All		not rational				All		All
HIV	All		All		All		All		All		All
PC	All		All		All		All		All		All
1C-nonlinear		p1, p2	x1	x2, x3		p1, p2	x1	x2, x3		p1, p2	x1	x2, x3
Bolie	Vp, p1, p3	p2, p4	q1	q2	p1, p3, Vp	p2, p4	q1	q2		p1, p2, p3, p4, Vp		q1, q2
Raia-jakstat	t1, t10, t12, t13, t14, t16, t18, t19, t2, t20, t3, t4, t5, t6, t7, t8, t9	t11, t15, t17, t21, t22	x1, x11, x2, x3, x4, x5, x6, x8	x10	t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t12, t13, t14, t16, t18, t19, t20	t11, t15, t17, t21, t22	x1, x2, x3, x4, x5, x6, x8, x11, x13	x10	t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t12, t13, t14, t16, t18, t19, t20	t11, t15, t17, t21, t22, x10	x1, x2, x3, x4, x5, x6, x8, x11, x13	x10
BioSD-I	p1, p2	p3, p4, p5, p6	x1	x2	p1, p2	p3, p4, p5, p6	x1	x2		p1, p2, p3, p4, p5, p6		x2
BioSD-II	p1, p2, p4	p3, p5, p6, p7	x1	x2, x3	p1, p2, p4	p3, p5, p6, p7	x1	x2, x3	p4	p1, p2, p3, p5, p6, p7	x1	x2, x3
BioSD-III	p1, p2	p3, p4, p5, p6, p7	x1	x2, x3	p1, p2	p3, p4, p5, p6, p7	x1	x2, x3		p1, p2, p3, p4, p5, p6, p7		x2, x3
TS-Known input	All		All		not rational				k01, k1, ntetr, k02, k2, nlaci	tatc, natc, tiptg, niptg	x1, x2
1D-BIG	p3, p4, p5	p1, p2	x1	x2, x3	p3, p4, p5	p1, p2	x1	x2, x3	p3, p4, p5	p1, p2	x1	x2, x3
NFKB					t1, t2, c3a, c4a, c5, k1, k3, kprod, kdeg, i1, e2a, i1a, a1, a2, a3, c1a, c2a, c5a, c6a, c1, c2, c3, kv, e1a	k2, c4, c1c, c2c, c3c	x1, x3, x4, x5, x6, x10, x11, x13, x14	x8, x15	t1, t2, c3a, c4a, c5, k1, k3, kprod, kdeg, i1, e2a, i1a, a1, a2, a3, c1a, c2a, c5a, c6a, c1, c2, c3, kv, e1a	k2, c4, c1c, c2c, c3c	x1, x3, x4, x5, x6, x10, x11, x13, x14	x8, x15
BioSD-II-MM-simple					p1, p2, p4, p6	p3, p5, p7, p8	x1	x2, x3	p4, p6	p1, p2, p3, p5, p7, p8	x1	x2, x3
BioSD-II-MM-complex					p1, p2, p4, p5, p7	p3, p6, p8, p9	x1	x2, x3	p4, p5, p7	p1, p2, p3, p6, p8, p9	x1	x2, x3
Dichotomous-F-BettaRR					All		All		All		All
Dichotomous-F-BettaSR					All		All		All		All
Dichotomous-F-kap					All		All		All		All
Dichotomous-F-I					All		All		p1, p2, p3, p4, p5, p6, p7, p8, p9	p10, p11	All
Kelly-1					All		All		p1, p5, p6, p7, p10, p13, p14	p2, p3, p4, p8, p9, p11, p12	t, s	c, T
CSTR-observer					All		All		All		All
Dichotomous-F-I-BettaRR					p1, p2, p4, p5, p7, p8, p9	p3, p10, p11	All		p1, p2, p4, p5, p7, p8, p9	p3, p10, p11	All
Dichotomous-F-I-BettaSR					p1, p2, p4, p5, p6, p7, p9	p3, p10, p11	All		p1, p2, p4, p5, p6, p7, p9	p3, p10, p11	All
Kelly-2					p2, p3, p4, p8, p10, p11	p1, p5, p6, p7, p9		r, s, c	p4, p8, p11	p1, p2, p3, p5, p6, p7, p9, p10	R	r, s, c
Kelly-2-gxgr					p4, p5, p7, p8, p11	p6	r, s	c	p4, p5, p7, p8, p11	p6	r, s, R	c
Arabidopsis					not rational				a, n1, r3, k1, k4, m1, m4, n2, q2, r1, r2, r4	g1, g2, k2, k3, k5, k6, k7, m2, m3, m5, m6, m7, p1, p2, p3, q1	x1, x4	x2, x3, x5, x6, x7
Bioprocess					not rational				p1, p2, p3, p4, p5, p6, p7, p8		x1, x2, x3, x4, x5
Bioprocess2xp					not rational					p1, p2, p3, p4, p5, p6, p7, p8	x1E1, x2E1, x3E1, x4E1, x5E1, x1E2, x2E2, x3E2, x4E2, x5E2
Bioprocess3xp					not rational				p1, p2, p3	p4, p5, p6, p7	x1E1, x2E1, x3E1, x4E1, x5E1, x1E2, x2E2, x3E2, x4E2, x5E2, x1E3, x2E3, x3E3, x4E3, x5E3
Bioprocess4xp					not rational				All		All
Kelly-1-gr					not rational				p1, p5, p6, p7, p10, p13, p14	p2, p3, p4, p11, p12	t, s	c, T

summarizes the results of the SIO analyses. For 11 models, we obtained different results with StrucID and STRIKE-GOLDD. Specifically, STRIKE-GOLDD classified certain parameters as identifiable, while StrucID reported them as non-identifiable. This pattern was consistent across the eleven models. To address these discrepancies, we used a third tool, the SIAN algorithm, implemented in Julia [57]. The corresponding results are presented in

Table 3. Comparison of structural identifiability and observability results for models with discrepancies between STRIKE-GOLDD and StrucID analyses, including the results obtained using the Julia-based SIAN algorithm as a third reference. For the Kelly_1 model, SIAN cannot analyse it and returns an error because it contains non-rational functions, specifically exponential or power terms with parameters in the exponents.

Models	STRIKE_GOLDD				StrucID	SIAN
	FISPO		Prob_Obs_Test		Non-Id.	Id.	Non-Id.
	Id.	Non-Id.	Id.	Non-Id.
C2M	All		All		x3, x5, x6, x2	All
Fujita	All		All		scaleFactor_p, EGFR, scaleFactor_p, Akt, scaleFactor_pS6, EGFR_turnover, R_1_k1, R_1_k2, R_2_k1, R_5_k1, R_9_k1, EGFR, pEGFR, pEGFR_Akt, Akt, pAkt, S6, pAkt_S6, pS6, EGF_EGFR	All
Bolie	Vp, p1, p3, q1	p2, p4, q2	Vp, p1, p3, q1	p2, p4, q2	p1, p2, p3, p4, Vp, q1, q2	p1, p3, Vp, q1	q2, p4, p2
BioSD_I	p2, p1, x1	p4, p3, p6, p5, x2	p2, p1, x1	p4, p3, p6, p5, x2	p1, p2, p3, p4, p5, p6, x2	p2, p1, x1	p4, p3, p6, p5, x2
BioSD_II	p2, p1, p4, x1	p3, p6, p7, p5, x3, x2	p2, p1, p4, x1	p3, p6, p7, p5, x3, x2	p1, p2, p3, p5, p6, p7, x2, x3	p2, p1, p4, x1	p3, p6, p7, p5, x3, x2
BioSD_III	p2, p1, x1	p4, p3, p6, p7, p5, x3, x2	p2, p1, x1	p4, p3, p6, p7, p5, x3, x2	p1, p2, p3, p4, p5, p6, p7, x2, x3	p2, p1, x1	p4, p3, p6, p7, p5, x3, x2
BioSD_II_MM_simple			p2, p1, p4, p6, x1	p3, p5, p7, p8, x3, x2	p1, p2, p3, p6, p8, p9, x2, x3	p2, p1, p4, p6, x1	p3, p5, p7, p8, x3, x2
BioSD_II_MM_complex			p2, p4, p5, p1, p7, x1	p3, p6, p8, p9, x3, x2	p1, p2, p3, p6, p8, p9, x2, x3	p2, p4, p5, p1, p7, x1	p3, p6, p8, p9, x3, x2
Dichotomous_Feedback_I			All		p10, p11	All	-
Kelly_1			All		p2, p3, p4, p8, p9, p11, p12, c, T	error
Kelly_2			p2, p3, p4, p8, p10, p11	p9, p5, p1, p6, p7, s, c, r	p1, p2, p3, p5, p6, p7, p9, p10, r, s, c	p10, p4, p2, p3, p11, p8	p9, p5, p1, p6, p7, s, c, r

, which shows that the results obtained with SIAN are in full agreement with those obtained with the FISPO and ProbObsTest algorithms within the STRIKE-GOLDD toolbox (1 of the 11 models could not be analysed using SIAN).

These findings hint at the existence of numerical issues in StrucID. One explanation is that the sensitivity calculations performed by the methods in this toolbox rely on numerical integrations, without mechanisms to control the propagation of numerical errors due to double precision operations. Furthermore, the rank of the resulting observability matrix is determined by calculating its singular values numerically, which leads to a well-known issue: this procedure cannot result in singular values that are exactly equal to zero; instead, the singular values will always be positive, and it is necessary to choose a threshold to decide which of them correspond to the zeros of a symbolic algorithm. A recent discussion of this issue can be found, for example, in [58].

3.1.3. Computational Efficiency

Figure 3 plots the CPU time required to analyse each model with the three SIO methods considered. A clear conclusion can be extracted: StrucID is consistently faster than both of the STRIKE-GOLDD methods, FISPO and ProbObsTest. While the difference can be small in absolute terms for small models, for larger models it becomes substantial. Furthermore, some models could only be analysed with StrucID.

3.2. Accessibility and Controllability Analyses

We analysed the NAC of the models with control inputs using the LARC and GSC algorithms from STRIKE-GOLDD, along with the sensitivity-based algorithm in StrucID.

Table 4. Comparison of the results of different Controllability and local Accessibility analysis methods for the case studies. The columns show the case name, Lie algebraic rank condition (LARC), generalised Sussmann condition (GSC), and the outcome of the StrucID tool. “not Affine in ’U”’ refers to models not affine in the input, which makes them impossible to analyse with the algorithms included in STRIKE-GOLDD.

Case Study	LARC	GSC	StrucID
1_A_integral	Accessible	STLC	Accessible
1B_Prob_integral	Accessible	STLC	Accessible
1C_nonlinear	Accessible	STLC	Accessible
2DOF	Accessible	STLC	Accessible
Arabidopsis	Accessible	—	Accessible
Boile	Accessible	—	Accessible
C2M	Accessible	STLC	Accessible
CSTR_observer	Accessible	STLC	Accessible
BioSD_I	Accessible	STLC	Accessible
BioSD_II	Accessible	STLC	Accessible
BioSD_II_MM_simple	Accessible	STLC	Accessible
Kelly_1_gr	Accessible	STLC	inAccessible
Kelly_2_gxgr	Accessible	STLC	inAccessible
BioSD_III	Accessible	STLC	inAccessible
Dichotomous_Feedback_BettaRR	Accessible	STLC	Accessible
Dichotomous_Feedback_BettaSR	Accessible	STLC	Accessible
Dichotomous_Feedback_kap	Accessible	STLC	Accessible
Dichotomous_Feedback_I_BettaRR	Accessible	STLC	Accessible
Dichotomous_Feedback_I_BettaSR	Accessible	STLC	Accessible
Raia_jakstat	Accessible	—	inAccessible
Fujita	inAccessible	—	inAccessible
HIV	—	—	Accessible
Dichotomous_Feedback_I	not Affine in “U”		Accessible
BioSD_II_MM_complex	not Affine in “U”		Accessible
Kelly_1	not Affine in “U”		inAccessible
Kelly_2	not Affine in “U”		inAccessible
Bioprocess	not Affine in “U”		inAccessible
Bioprocess 2xp	not Affine in “U”		Accessible
Bioprocess 3xp	not Affine in “U”		inAccessible
Bioprocess 4xp	not Affine in “U”		Accessible
TS	not Affine in “U”		inAccessible
1D_BIG	computation limit		inAccessible
Bachman jackstat	computation limit		inAccessible
NFKB (Merkt)	computation limit		Accessible
PC	computation limit		Accessible

summarizes the results.

3.2.1. Applicability

A fundamental difference between the capabilities of the StrucID and STRIKE-GOLDD toolboxes in analysing NAC stems from the fact that STRIKE-GOLDD cannot analyse models that are not affine with respect to the inputs. In STRIKE-GOLDD, models are first checked for this characteristic; if a model is not affine, the toolbox automatically attempts to convert it into an affine form using an internal algorithm in order to proceed with the analysis. If this conversion fails, STRIKE-GOLDD reports that it is unable to analyse the controllability and accessibility of the model. In contrast, StrucID is capable of analysing all models regardless of their structure.

Additionally, some affine models could not be analysed with STRIKE-GOLDD due to their excessive complexity, given by a high number of state variables or the complexity of their equations. This limitation depends on the available computing resources; we set a predefined time limit of 3600 s, beyond which we reported that a method has reached its computation limit.

3.2.2. Reliability

Among the models that were successfully analysed by both toolboxes, similarities and differences were observed in the results. STRIKE-GOLDD and StrucID agreed on the results of 17 models, most of which (all but one, the ‘Fujita’ model) were classified as accessible. For other models, however, discrepancies arose. The biosd_III, Kelly_1_gr, Kelly_2_gxgr and Raia_JAKSTAT models were classified as accessible by the LARC algorithm and as inaccessible by StrucID. However, in all cases in which a model was classified as accessible by StrucID, it was also accessible according to LARC.

Unlike in the SIO case, we could not find a third tool on which to rely to verify the results of the two approaches. However, since in each tool the SIO and NAC analyses share the same methodological foundations, we hypothesize that the discrepancies between both tools regarding NAC can be explained in the same way as those regarding SIO—that is, they might be caused by numerical issues affecting the algorithms in StrucID.

3.2.3. Computational Efficiency

The time required for the NAC analyses of the models included in this study is shown in Figure 4. As for SIO, the results indicate that StrucID performs the analysis in a shorter time. An important remark is that, within the STRIKE-GOLDD framework, there exists some dependency between the NAC algorithms, and several computational steps are shared among them: specifically, the LARC test is a prerequisite for checking the GSC.

3.3. Discussion

In our analyses, we focused on three key aspects: applicability, reliability, and computational efficiency. Our results allow us to derive clear takeaways about them.

In regard to the first aspect, StrucID is the most widely applicable tool since it can handle nonlinear ODE models of a very general type. In contrast, STRIKE-GOLDD does provide one algorithm with the same applicability for SIO analyses (FISPO), but all of their algorithms for NAC analysis are limited to systems that are affine in the inputs. Furthermore, its other SIO algorithm, ProbObsTest, which is much faster than FISPO, requires models to be rational.

In regard to the second aspect, reliability, we found some interesting results. For SIO analysis, StrucID and STRIKE-GOLDD usually produced consistent results, but we found some cases in which they differ. All parameters classified as identifiable by StrucID were also classified as identifiable by STRIKE-GOLDD. However, some parameters classified as identifiable by STRIKE-GOLDD were classified as unidentifiable by StrucID. Given the different methodological foundations of their algorithms, it would seem reasonable to assume that STRIKE-GOLDD yields more accurate results than StrucID since it is based on symbolic computations instead of numerical ones. We confirmed this intuition by using a third tool, SIAN, which follows a different approach (differential algebra) than the other toolboxes examined here. The results obtained with SIAN coincided with those yielded by STRIKE-GOLDD, thus reinforcing the conclusion that it is more reliable than StrucID. For NAC analyses, we do not have an alternative verification tool, unlike for SIO. However, it should be noted that for each toolbox the foundations of the NAC and SIO analyses are the same: STRIKE-GOLDD relies on symbolic calculations of Lie derivatives (for SIO) and Lie brackets (for NAC), which are obtained in a similar way. Likewise, StrucID decides SIO and NAC based on the same type of sensitivity-based numerical calculations. Hence, it is reasonable to assume that the discrepancies in the NAC results produced by both approaches have the same causes as the discrepancies in the SIO results, i.e., they can be attributed to numerical issues that affect the reliability of the sensitivity-based calculations.

In regard to the third aspect, computational efficiency, StrucID is the clear winner. Its numerical approach yields extremely fast calculations, and for most models the computation time is at least an order of magnitude lower than the STRIKE-GOLDD algorithms. For this latter toolbox, there is a clear difference between the two SIO algorithms: ProbObsTest is more efficient, with computation times that are sometimes comparable to StrucID, while FISPO is slower and more demanding in terms of computational resources. However, this superior efficiency of ProbObsTest with respect to FISPO comes at the expense of inferior applicability since it cannot handle rational models. As for the NAC tests, again we conclude that the StrucID sensitivity-based approach is faster than the geometric control conditions that can be checked by STRIKE-GOLDD.

4. Conclusions

In this article, we report the results of a systematic analysis of the performance of two computational tools for analysing control-theoretic properties of nonlinear models, StrucID and STRIKE-GOLDD. While a few studies have already been published in this regard [27,28,30], our motivation for the work presented in this paper was to extend them in a number of ways, namely by applying the tools to a much larger set of models, discussing their applicability in detail, comparing their computational efficiency both for identifiability/observability and accessibility/controllability, assessing the correctness of their results, and providing practical guidance for tool selection in complex nonlinear models.

Some remarks are in order regarding the set of case studies selected for this assessment. Obviously, the choice of models to be analysed can and will affect the results. However, we note that the aim of this study was to obtain clear guidance in the form of qualitative conclusions, as opposed to quantitative measures, about the performance of the two tools being examined. That is, we did not intend to quantify exactly how much faster a tool is when compared to another, nor did we attempt to calculate the percentage of models for which their results disagree. Instead, our aim was to establish if a given tool is clearly faster than the other (as we indeed found to be the case, often by several orders of magnitude); or, regarding reliability, to find cases in which a tool gives wrong results and to explain their cause (as we have done here). Thus, the set of models that we used should not be considered a representative selection of those found in the literature—they just need to be realistic and sufficiently diverse to account for the different situations that can arise.

In short, our results show the existence of a trade-off: STRIKE-GOLDD is more reliable than StrucID, but StrucID is more generally applicable and computationally more efficient than STRIKE-GOLDD. Model users should choose between these tools accordingly when working on a particular application.

It should be noted that the two tools evaluated here are implemented in MATLAB. To the best of our knowledge, there are no other open research tools capable of performing these analyses in other programming languages. Their development would be a valuable addition to the set of resources available to the user.