# An Algebraic Approach to Identifiability
## 3 Observability and Identifiability
### Global Identifiability and Observability  
We discuss the most general case with an extended state \(z=(x,\bar{x};p,\bar{p})\) containing two copies of the state and parameter tuple first. The space \(\mathcal{Z}=\mathcal{M}_{p}\times\mathcal{M}_{p}\times\Theta^{2}\) of the extended states is a semi-algebraic set; thus, its Zariski closure is the real variety of an ideal \(\mathfrak{Z}\). This ideal is generated by all equations governing the manifolds \(\mathcal{M}_{p}\) and \(\mathcal{M}_{p}\) and possibly the parameter space \(\Theta^{2}\) if the parameters are not independent of each other. Within the set of extended states, one may identify equal extended states, e.g., the subset  
\[\mathcal{J}=\left\{z=(x,\bar{x};p,\bar{p})\in\mathcal{Z}=\mathcal{M}_{p}\times \mathcal{M}_{p}\times\Theta^{2}\,|\,x=\bar{x}\wedge p=\bar{p}\right\} \tag{6}\]  
of equal states and parameters for each system copy. The main question is if that set of equal extended states matches the set \(\mathcal{I}\subseteq\mathcal{Z}\) of indistinguishable states. As the equal states are obviously indistinguishable, one has \(\mathcal{I}\supseteq\mathcal{J}\), and it suffices to test if the reverse inclusion holds. Assume for a moment that \(\mathcal{J}\) in Equation (6) is a real variety, i.e., the parameter set \(\Theta\) is an algebraic yet not semi-algebraic set. This means that the space \(\mathcal{Z}\) of extended states is the real variety of a real ideal \(\mathfrak{Z}\) and that the varieties in Equation (6) correspond to a real ideal  
\[\mathfrak{Z}=\mathfrak{Z}+\langle x_{1}-\bar{x}_{1},\ldots,x_{n}-\bar{x}_{n}, p_{1}-\bar{p}_{1},\ldots,p_{s}-\bar{p}_{s}\rangle\subset\mathbb{R}[x,\bar{x},p, \bar{p}].\]  
Furthermore the set \(\mathcal{I}\) of indistinguishable states equals the real variety of the stabilized ideal \(\mathrm{L}_{F}^{\infty}\langle H_{1},\ldots,H_{l}\rangle\) of the extended output maps, where \(\mathcal{F}\) is the linear span of the extended vector fields  
\[F_{k}(x,\bar{x};p,\bar{p})=f_{k}(x;p)\tfrac{\partial}{\partial x}+f_{k}(\bar{ x};\bar{p})\tfrac{\partial}{\partial\bar{x}}. \tag{7}\]  
The equality of the indistinguishable and equal states can be tested by comparing the corresponding real ideals  
\[\mathfrak{Z}=\mathfrak{Z}+\mathbf{rad}^{\mathbb{R}}(\mathrm{L}_{\mathcal{F}}^ {\infty}\langle H_{1},\ldots,H_{l}\rangle) \tag{8}\]  
and \(\mathfrak{Z}\). Note that the ideal sum with \(\mathfrak{Z}\) is neccesary to restrict the states to the manifold they are contained in. In many cases, the parameter space is not algebraic because of physical restrictions, e.g., a mass is usually restricted to positive values. In this case, \(\mathfrak{Z}\) is taken to be the real ideal corresponding to the Zariski closure of \(\mathcal{Z}\). Then, however, the variety \(\mathbf{var}^{\mathbb{R}}(\mathfrak{Z})\) may contain irreducible components that do not respect these restrictions, e.g., lie in a plane where the mass vanishes. In such cases, the comparison of the ideals \(\mathfrak{Z}\) and \(\mathfrak{Z}\) is not sufficient to test global identifiability. Therefore, we compute the decomposition of \(\mathfrak{Z}\) into its minimal associated primes  
\[\mathfrak{Z}=\mathfrak{Z}_{1}\cap\cdots\mathfrak{Z}_{k}\]and check if the corresponding irreducible varieties lie in the extended state and parameter space:  
\[\exists x,\bar{x},p,\bar{p}:(x,\bar{x},p,\bar{p})\in\mathcal{Z}\cap\mathbf{var}^ {\mathbb{R}}(\mathfrak{I}_{i})\,\quad i=1,\ldots,k. \tag{9}\]  
The set can be described by a boolean combination of equations and inequalities. Note that the equations are the collections of the form \(g(x,\bar{x},p,\bar{p})=0\) with all polynomials \(g\) in the generators of \(\mathfrak{I}_{i}\supseteq\mathfrak{I}_{3}\). The quantified Equation (9) can always be transformed into a quantifier-free formula [37; 38].