# An Algebraic Approach to Identifiability
## 2 Preliminaries
### Indistinguishability of States  
First consider an autonomous system  
\[\dot{x}=f(x),\ y=h(x) \tag{2}\]  
with \(f:\mathcal{M}\to\mathrm{T}\mathcal{M},h:\mathcal{M}\to\mathbb{R}^{l}\), where the system state and parameters are condensed in the variables \(x\in\mathcal{M}\) for a moment. As previously noted, two system states \(x,\tilde{x}\in\mathcal{M}\) are said to be distinguishable if and only if there exits an interval \([0,T]\) such that the output trajectories \(h(x(t))\) corresponding to initial conditions \(x(0)=x\) and \(x(0)=\tilde{x}\), respectively, differ in that interval.  
As the output trajectories are locally analytic, there is a one-to-one correspondence between the trajectory and the coefficients of the Lie-series. Thus, the distinguishability can be equivalently expressed in terms of the _observability map_  
\[q(x)=\begin{pmatrix}h(x)\\ \mathrm{L}_{f}h(x)\\ \mathrm{L}_{f}^{2}h(x)\\ \vdots\end{pmatrix}\]  
as follows: Two states \(x,\tilde{x}\in\mathcal{M}_{p}\) are indistinguishable if and only if  
\[q(x)=q(\tilde{x})\implies x=\tilde{x}. \tag{3}\]  
This formula, however, is to some extend unsatisfying as it contains an infinite number of equations. In [32], this problem has been handled for the case of polynomial vector fields and output maps. The key of this approach is to use two copies of Equation (2) with a vector field  
\[F:\mathcal{M}\times\mathcal{M}\to\mathrm{T}(\mathcal{M}\times \mathcal{M})\] \[F(x,\tilde{x})=f(x)\tfrac{\partial}{\partial x}+f(\tilde{x}) \tfrac{\partial}{\partial\tilde{x}}\]  
and output map  
\[H(x,\tilde{x})=h(x)-h(\tilde{x}).\]  
This way the components of the residuum observability map  
\[Q(x,\tilde{x})=q(x)-q(\tilde{x}) \tag{4}\]  
are precisely the Lie derivatives \(\mathrm{L}_{F}^{i}H(x,\tilde{x})\), \(i=0,1,2,\dots\). The common zero-set of \(Q\) is an algebraic variety; thus, the set of polynomials in \(\mathbb{R}[x,\tilde{x}]\) that vanish on that variety is finitely generated by [PERSON]'s basis theorem. This polynomial ideal is generated by the polynomials \(H_{k}\in\mathbb{R}[x,\tilde{x}]\), \(k=1,\dots,l\) as well as their (higher-order) Lie derivatives along the vector field \(F\). As \(\mathrm{L}_{F}\) is a derivative operator \(\mathrm{L}_{F}\colon\mathbb{R}[x,\tilde{x}]\to\mathbb{R}[x,\tilde{x}]\), a generating set for the ideal generated by the components of the extended observability map \(Q\) can be computed in a finite number of steps, see [31; 32; 35].  
A very simple yet computationally suboptimal algorithm starts with a generating set  
\[G_{0}=\{H_{1},\dots,H_{l}\}\]of the output maps and iteratively adds their (extended) Lie derivatives along \(F\):  
\[G_{k+1}=G_{k}\cup\Big{\{}\mathrm{L}_{F}^{k}H_{1},\ldots,\mathrm{L}_{F}^{k}H_{l} \Big{\}}.\]  
The such generated ideals form an ascending chain  
\[\langle G_{0}\rangle\subseteq\langle G_{1}\rangle\subseteq\cdots\subseteq \langle G_{N}\rangle=\langle G_{N+1}\rangle=\cdots\stackrel{{ \mathrm{def}}}{{=}}\mathrm{L}_{F}^{\infty}\langle G_{0}\rangle \tag{5}\]  
that terminates after a finite number of steps by the ascending chain condition, yielding a finite generating set \(G_{N}\). Now, Equation (3) can be written using a finite number of atomic formulas:  
\[\bigwedge_{g\in G_{N},\langle G_{N}\rangle=\mathrm{L}_{F}^{\infty}\langle G_{0 }\rangle}g(x,\bar{x})=0\implies x=\bar{x}.\]