# An Algebraic Approach to Identifiability
###### Abstract  
This paper addresses the problem of identifiability of nonlinear polynomial state-space systems. Such systems have already been studied via the input-output equations, a description that, in general, requires differential algebra. The authors use a different algebraic approach, which is based on distinguishability and observability. Employing techniques from algebraic geometry such as polynomial ideals and Grobner bases, local as well as global results are derived. The methods are illustrated on some example systems.  
Keywords:identification; identifiability; observability; polynomial dynamical systems; algebraic methods; Lie derivative +
Footnote †: journal: Applied Soft  
also addresses the quality of the measurements and its implication on the quality of the identified parameters.  
Identification and identifiability are related to several other problems in control theory. The most obvious connection is to observer design and observability, which concerns the estimation of the system's state [9; 12; 13; 14; 15]. In [16; 17], the authors carry out the identification by means of adaptive control. The combination with the previous approach results in adaptive observes [18; 19]. A further relation concerns the field of fault detection, i.e., the detection of changing system properties over the time, see [14; 20].  
In the literature, identifiability is defined in different ways depending on the context [1; 9; 20]. As this article deals with structural algebraic methods, the definitions that seem most common are used herein. Note that what is called locally identifiability herein is sometimes referred to as structural identifiability, see [21]. The algebraic identifiability approaches mostly examine the differential input-output relations by eliminating the state variables over a differential ring [22; 23; 24; 25]. This leaves one with a set of equations of the known input and output trajectories and the parameters. Identifiability then asks for the solvability for the parameters. This elimination can be done over the differential polynomial ring using the well-known Ritt algorithm [26].  
This article considers a method that does not rely on a differential ring and can determine the identifiability using computations in the ordinary polynomial ring [27; 28; 29] alone. The computation in this simpler domain comes at the cost of more variables that are required to formulate the problem. This approach is based on an observability criterion [30; 31; 32] and is extended to an identifiability criterion for polynomial systems. [A note to the difference and extension from the observability criterion as suggested by reviewer 1 and reviewer 3 point 1.] In contrast to the observability test, the distinguishability is concerned differently for state variables and parameters. This will be translated into the algebraic context.  
As suggested by the academic editor, a note to the polynomial system class is inserted. Many chemical reactions or biological systems are described by such polynomial differential equations. Even for linear electrical networks, the physical parameters, such as resistance and capacitance, usually appear in products, yielding polynomials of the parameters in the differential equations. Some other systems can be recast into a polynomial form using an embedding into a higher-dimensional space. Those embeddings can be handled as well.  
The paper structures as follows: In Section 2, we provide the required mathematical preliminaries. The main results are stated in Section 3. Our approach is illustrated on several example systems in Section 4. Finally, the results obtained will be discussed in Section 5.