S-synchronization and Excitation Constancy in Structural Identifiability Problem of Nonlinear Systems

An approach to analysis the structural identifiability (SI) of nonlinear dynamical systems under uncertainty was proposed. S-synchronizability condition of an input is the basis for the structural identifiability estimation of the nonlinear system. A method for obtaining a set containing information about the nonlinear part of the system wasproposed. The decision on SI of the system was based on the analysis of geometric frameworks reflected the state of the system nonlinear part. Geometric frameworks were defined on the specified set. Conditions for structural indistinguishability of geometric frameworks and local identifiability of the nonlinear part were obtained. It shown that a non-S-synchronizing input gives an insignificant geometric framework. This input is a sign of structural non-identifiability of the nonlinear system. The method for estimating the structural identifiability of the nonlinear system was proposed. We show that the structural identifiability is the basis for structural identification of the system. The structural identifiability degree was introduced, and the method of its estimation was proposed.


Introduction
Analysis of recent publications shows that the system identifiability is performed in a parametric space. Many publications are devoted to the study of this problem. Identifiability results are typically presented in the form accepted in the parametric estimation problem. Various methods were proposed for estimating the structural identifiability [1,2].
Many authors study the parametric identifiability of nonlinear systems (see, e.g. [2][3][4][5]). In [2], an approach based on the sensitivity analysis of system output was applied to study identifiability and the analysis of experimental data was used for obtaining parametric identifiability conditions. The critical analysis of approaches applied to the estimate the identifiability of biological models is given in [3]. Models for estimating the identifiability of nonlinear systems are typically based on Taylor series expansion, identifiability tables, and Lie algebra study. Research on practical identifiability was discussed in [4].
The analysis shows that the identifiability of the model is normally understood as the possibility of estimating its parameters and the proposed methods use the nondegeneracy estimation of the information matrix. Similar results were obtained in the parametric estimation theory as the condition of nondegeneracy (completeness of rank) of the matrix input. As a rule, the model structure is set a priori, and, thus, the use of local structural identifiability is not obvious. The concept of structure is widely used in problems of identifiability. The identifiability of a nonlinear system reduces to the parametric identifiability problem and is based on the application of various linearization methods. This extensive field of research does not cover the structural identifiability problem of nonlinear dynamical systems in regarding to deciding on the structure (form, dependence) of the nonlinear system under uncertainty. This statement corresponds to the analysis of structural aspects of the identifiability (identification) system. Also, the question not considered which an input having the excitement constancy property guarantees the structural identifiability of the system. This problem proposed in [6] first.
Also, an important question: "Which input having the excitement constancy property can provide guarantees the structural identifiability of the system" was not considered [6].
In this paper, we study the structural identifiability problem of a nonlinear system. It is important to note that this problem is very complex problem and that methods of formalizing the system structure have not been developed yet. The concept of the structural identifiability ( h -identifiability) was introduced in [6].
We propose the approach for solving the structure estimating problem of a nonlinear dynamical system. The approach is based on the analysis of a specific class of frameworks that adequately describe the state of the nonlinear system. The main results obtained in this study can be considered as a generalization of those obtained in [6,7].
where u R ∈ , y R ∈ are the input and output, ( ) y ϕ is scalar nonlinear function, The information set is known for the system (1). Problem: use the analysis I o and estimate the structural identifiability (SI) of the system (1).

Method of design S ey -framework
The proposed approach based on the analysis of geometric framework ey S . The ey S -framework design based on the formation of a set , I N g contained information about the function ( ) y ϕ [9]. Apply the differentiation operation to ( ) y t and denote the obtained variable as 1 x . Find the forecast for variable 1 x by applying model (4) I g t ∀ ∈ , and generate an error Remark 1. Choosing the model structure (4) is the stage for structural identification of the system (1). Simulation results show that the model (4) is applicable in identification systems of objects with static nonlinearity. The model (4) structure choice proposed for another nonlinearity in [9].
The phase portrait S described by function described on the plane ( , ) y e the change in the framework ey S . Apply the models (4) and present the system (1) as ( ) 1 , : , is the variable describing the general solution of the system (1); R ζ ∈ is a limited disturbance, which is the result of a procedure applied for the variable definition e .

Structural identifiability of system (1)
Consider the system S ϕ and properties , I N g that allow solving the problem of structural identification ( hidentifiability). Let the following conditions be satisfied. В1. The input ( ) u t is constantly excited at the interval J .
A "bad" input that is constantly excited exists. This input gives a so-called "insignificant" ey S -framework ( ey NS -framework). But the ey NS -framework can be hidentifiable. Identification the system (1) with ey NSframework gives results not typical in this case.
Give the conditions for the existence of the framework. Consider a class of nonlinear functions to which the homotopy operation is applies [10]. Let where l a , r a are estimations obtained least-squares method (LSM Remark 2. , where h S is the phase portrait of the system (1).
So, we present two criteria (7) and (10) for the existence of the insignificant framework ey S . The system S ϕ structure and hence the system (1)  Remark 4. The described approach applies to the nonlinear system with a dynamic law of the nonlinearity changing, where multi-level structure identifiability analysis used.
The identifiability of the system y S considered in [11].

On constant excitation influence on system identifiability
In [11], the influence of the excitation constancy (EC) condition on the system identifiability estimation was shown. It is noted that not every input had the ECproperty guarantees the structural identifiability of the system. We present results allowing us to evaluate the effect of this condition. The property of excitation constancy for input ( ) u t d -optimal on the set U h , then the fulfilment of condition (10) follows from inequality (15). The condition (15) shows that the system (1) nonlinear part structure with the input k u does not coincide with structurally identifiable parameters with h u . Theorem 5. Let 1) the input k u satisfy the condition (12); 2) the framework k ey S corresponds to the input k u ; 3) condition (13) held for S h u ∈ ; 4) conditions (14) and (15) held. Then the S ϕ -system is structurally unidentifiable by input k u , and structural parameters of the S ϕ -system do not correspond to the system (1) with the identifiable framework h ey S .
Estimate the degree of non-identifiability of the S ϕsystem. Let the phase portrait S constructed for the system (1 Then (i) the S ϕ -system is structurally non-identifiable by input k u ; (ii) structural parameters of the S ϕ -system do not correspond to the system (1) with the identified framework h ey S .
Example. Consider a system with Bouc-Wen hysteresis (system BW S ) [12] ( , , )     Fig. 1 We see that the frequency properties of the input significantly affect the identifiability of the system. It is for the nonlinear part of the system where the change in input properties affects the estimate to structural parameters of the hysteresis. This effect presented in Fig.  2, where the output of the Bouc-Wen model (19) shown for various i u . The identifiability area D Q showed in Fig. 3 for the BW S -system. It confirms the made conclusions. So, we propose the method for quantifying estimate the degree identifiability of the system (1). The area of system identifiability obtained.

Conclusion
The concept of S-synchronizability of the input was introduced. Meeting the input S-synchronizability condition is the basis for structural identifiability and structural identification of the nonlinear system. The input influence on the estimating possibility of the structural parameters nonlinear system has been studied and conditions for the local identifiability and nonidentifiability of the system have been obtained. The identifiability degree the system under uncertainty has been introduced. The influence of the constant excitation of input on the possibility of the system structural identifiability has been studied. The method for calculating the identifiability domain under uncertainty has been proposed.