Geometrical Frameworks in Identification Problem

The purpose of this review 
is to apply geometric frameworks in identification problems. In contrast to the 
qualitative theory of dynamical systems (DSQT), the chaos and catastrophes, 
researches on the application of geometric frameworks have not been performed in 
identification problems. The direct transfer of DSQT ideas is inefficient through 
the peculiarities of identification systems. In this paper, the attempt is made based on the 
latest researches in this field. A methodology for the synthesis of geometric 
frameworks (GF) is proposed, which reflects 
features of nonlinear systems. Methods based on GF analysis are developed for the 
decision-making on properties and structure of nonlinear systems. The problem 
solution of structural identifiability is obtained for nonlinear systems under uncertainty.


Introduction
The framework (FR) concept is applied in control, identification, and analysis and data processing tasks. FR is the synonym of such concepts as a frame, structure, system, platform, concept, a basis, and set of approaches. The term "framework" is used in two directions in scientific research. The first direction of FR application represents the term integrating a set of method approaches or procedures. So, FR in [1] is interpreted as the set of mathematical and technical procedures and methods for identification of the automobile battery control process. The approach to the identification is based on the Bayesian framework. In [2], this concept combines the set of identification methods based on predic-tion error computing. Proposed methods show that such procedures allow obtaining estimations in some optimum sense. The key moment in this parametric paradigm is the choice of a necessitated reference structure. The same paradigm based on the creation of the new concept to system identification is proposed in [3] [4]. It is based on the compilation of existing approaches.
The framework can be interpreted as the theoretical model structure for the analysis of a content transmitted to video [5].
So, we have the system of theoretical provisions which is applied for the solution of a specific problem. The hybrid system identification scheme (methodology) based on the continuous optimization application is proposed in [6]. In [7], the uniform theoretical concept (framework) is proposed for nonlinear discrete dynamic system identification. It is based on the application of neural networks. The procedure (framework) is proposed in [8] for the identification of functional refusal. It is the basis of the new approach for functional refusal risk estimation in physical systems. The framework is based on the integration of functionality hierarchical systemic models and behavioral simulation. Such interpretation of FR is dominating (see, for example [9] [10] [11] [12] [13]).
The second interpretation of FR is based on the application of mapping describing processes and properties of the system in the generalized view. Bases of such approach are proposed in the qualitative theory of dynamic systems [14] [15] [16]. Some geometrical framework corresponds to such mapping. This approach is widely applied in chaos research. The attractor is the framework example in identification problems (see for example [17] [18] [19]). The framework equation is specified as a priori within unknown parameters in these works. Further, the identification problem is solved to obtain a required form of the attractor. Another approach [20] [21] [22] [23] is based on the geometrical frameworks (GF) application for analyzing the system under uncertainty. GF gives the solution to the structural identification problem. Further, we will interpret this approach as the methodology based on the design and the analysis GF. The main difference between proposed by GF and frameworks in [17] [18] [19]: mathematical mapping (GF) is not postulated a priori, and is determined based on data processing. The GF is the main object of the analysis and allows deciding on system behaviour and properties. They contain the following areas of the identification theory.
1) Structural identification of the nonlinear system.
2) The estimation of Lyapunov exponents.
3) Structural identifiability of the nonlinear system. 4 ) The system phase portrait reconstruction on the time series. 5) The system structure estimation with lag variables. The structure of the paper. It is a review of the application of GF in identification problems. Section 2 contains the problem statement. The methodology for geometrical frameworks design in identification problems states in Section 3. We are showed that GF for static and dynamic systems differ significantly. The special class of mappings is applied to decision-making on the linear dynamic system structure. We show the GF application to the estimation of Lyapunov exponents. The significant geometrical framework obtaining depends on the structural identifiability of the dynamic system. The structural identifiability of nonlinear dynamic systems is presented in Section 4. It is showed that the system input should be S-synchronizing for the obtaining of significant GF. Reconstruction of the system phase portrait or attractor is also the identification problem. This problem is discussed in Section 5. The system structure choice with lag variables is discussed in Section 6. Two approaches to the choice of the system structure are considered. The first approach is based on statistical methods application. The second approach is founded on the Lyapunov exponents estimation. The proposed approach implementation example is described. The conclusion contains the main inferences and results.

Problem Statement
The system (1) nonlinear part is described by static (algebraic) equations often.
Therefore, further, we consider the case when ( ) y ϕ describe by the algebraic equation.
The informational set be known for the system (1) Problem: evaluate the class of nonlinear function ( ) y ϕ in (1) and characterise the matrix A on the basis of the data processing (3).

ey S -Frameworks
The geometrical framework ey S design is one of the solution main stages in the structural identification problem. The method for the framework ey S design is defined by the estimation possibility of system structural parameters. The framework ey S is derivative from a phase portrait S . S is the starting point for further researches on the formation ey S under uncertainty. The GF design approach depends on system properties and the considered problem of structure identification. The synthesis ey S method is proposed in [21] and is generalized on dynamic systems in [20] [22]. The approach based on the forming of a subset I GF which allows obtaining a mapping for the design ey S . I GF is the result of the set I o analysis. I GF may contain data on the transient process or the steady motion in the system, which contains the information about system nonlinear properties is formed.
The set , I N g is identified as follows. Apply to ( ) y t the differentiation operation and designate by the obtained variable as 1 x . Determine the model Construct the phase portrait S and GF ey S described by functions . ey S is the basis for the analysis and the identification system design? The framework ey S should have specified properties [20]. Properties of structural identifiability and S-synchronizability [23] [20]. The framework ey NS is the result of nonfulfillment of the condition S-synchronizability (SS) (see Section 4). S-synchronizability of the system (1) (framework ey S ) gives the excitation constancy condition fulfilment for the input ( ) u t . The significance ey S estimation algorithm is based on the sector set properties analysis for ey S [21] if the SS-condition is satisfied. Definition 1. The framework ey S is called the regular if the condition S-synchronizability is satisfied for the system (1).
The example of the regular framework ey S for the system with a static hysteresis is presented in Figure 1 [21].
If the function ( ) y ϕ has the complex law of change, the approach application described above can give a "false" framework ey NS . The example of such framework for the system describing processes in RC-OTA the chaotic oscillator [24]    ( ) The obtaining of the regular framework gives to the application of the hierarchical immersion method [21] in state-space. This method provides the model (4) structure choice for each layer of hierarchy.
The example of the regular framework ey S for the system (5) is shown in Figure 3. Designations showed in Figure 3 given in [21]. Example 1. Consider a mechanical system with Bouc-Wen hysteresis [25]. This has the form a β γ are some numbers. Denote by the system (6) as SBW.
SBW-system parameters for controlling the actuator are 5, 6 are d a m = = , The model (4) has the form: ˆ0.199 0.471 The application of the proposed method gives ey S -frameworks ( Figure 4). Ranges of definition y and z match. Analysis of the ey S -structure shows that the system (6) is nonlinear.  apply to the structure choice for the system (1) linear part. This task differs from the problem considered above. Therefore, mappings allowing making decisions should have another form [22] [23]. They are based on the analysis of the Lyapunov exponent (LE) dynamics change. Apply the model

3.2.
to the particular solution estimation of the system (1) on the output y, where is the estimation of the system output and its derivative. The choice of the interval q J J ⊂ depends on system (1) properties.
Further, we obtain the estimation for the system (1) general solution on the basis ˆq X is determined on the basis by the LE theory [26].
reflects the change dynamics of indexes depending on LE. Consider also the function describing the first difference   ( ) i s k t ∆ is determined by analogy with (10), and i desig- The application (8)- (11) allows to obtain the LE set and to estimate their type. The approach generalization on periodic dynamic systems is given in [27].

Structural Identifiability and Structural Identification of Nonlinear Dynamic System
In Section 3, it is noted that the structure estimation of nonlinear dynamic systems depends on the system identifiability.
Many publications (see for example [28] [29] [30]) are devoted to the dynamic systems parametric identifiability problem. The structural identifiability of nonlinear dynamic systems reduced to the parametrical identifiability based on various approximation methods application [29] [30] [31] [32]. Proposed approaches are generalized to the case when not all system parameters can identify In [23], structural identifiability is considered in the following aspect: determine conditions in which the nonlinear system structure estimation is possible under uncertainty. The solution to this problem for the system (1) is given in [23] when the nonlinear function ( ) . It means that the analysis S1 gives the estimation problem solution of the system (1) properties.
Remark 2. The excitation constancy property, which is the basis for the parametric identifiability estimation, is affected by the identifiability problem solution.
Let the framework ey S be closed and the area ey S is not zero. Designate by height ey where the height is the distance between two points of opposite sides of the framework ey S . Statement 1 [21]. Let i) the linear part of the system (1) is stable, and the nonlinearity ( ) ϕ ⋅ satisfies the condition (2); ii) the input ( ) u t is limited, piecewise continuous and constantly excited; iii) 0 . Then the framework ey S is identifiable on the set , I N g . Definition 2. The framework ey S having the specified properties in the statement 1 is h-identified.
Statement 1 conditions fulfillment can give "insignificant" ey S -framework ( ey NS -framework). Therefore, h-identifiability is a sufficient but necessary condition of structural identifiability (SI). Such a condition is S-synchronizability is an admissible set of inputs for the system (1).
S-synchronizes the system (1) if the framework ey S definitional domain has the maximum diameter y D on the set  (4) is Theorem 1 [20]. Let 1) the input ( ) u t is constantly excited and ensures the system (1) S-synchronization; 2) the phase portrait S of the system (1) has features; 3) the ey S -framework is h h δ -identifiable and contains fragments corresponding to features of the system (1). Then the model (4) is SM-identifying.
Remark 3. According to the results of Section 4, the process design of the model (4) structure can have a hierarchical form. It is rightly for nonlinearities, which do not satisfy the condition (2).
Consider the framework ey S . Designate by the center ey S on the set as с S , and the center of the area y D as y D с .
Theorem 2 [23]. Let on the set U of representative inputs ( ) u t of the system (1): i) exists  [23]. The approach bases on the integral indicator application for the framework ey S analysis and is the development of results obtained in [21].
Example 2. Consider the system (6). The structure ey S is shown in Figure 4.
Models (13) structurally coincide with (14). These results confirm the fulfill- is the rotation angle of the object shaft, u is excitation current of the actuator winding, y is output, 1 Measurements set is The frameworks , ey S S are presented in Figure 5. Apply the proposed approach to SI estimation and obtain the structural identifiability of the system  gives the final estimate for nonlinearity. We found that the influence degree of the 2 2 x x on system properties is 97%. The framework 2 2 , x x ε S ( Figure 6) confirms the properties of the system ST S .
So, the analysis confirms the possibility of the system ST S structural identification estimation and its identifiability at the interval y J . The model (4) application depends on the system structure (framework ey S ). The general approach to the choice of the model (4) structure not succeeds. The nonlinearity structure depends on the specifics of the system. This conclusion illustrates this example. It confirms the versatility and complexity of the considered problem. The system with several nonlinearities requires the development of proposed approaches.
Example 3. System for generating self-oscillations  ( ) 5 y t affects all processes in the system.
2) The indirect effect of variables on each other. It is a fundamental feature of systems with multiple nonlinearities. This feature levels the influence of some variables on system properties. Estimation of leveling is not always possible under uncertainty.
The compensation for these difficulties. First, build a tree of relationships. The example relationships 1 y , 2 y tree with other variables are shown in Figure 7.
Markers highlight significant relationships that exceed the 80% level. Such a layered tree is obtained for the system state vector. Apply the approach described in Section 4. The analysis showed that the object is described by the linear equation (variables 1 2 , y y ). Variables 1 y , 5 y impact the variable 3 y (the amplifier-gauge 1 output), and variables 2 y , 4 y , 5 y are impacted variable 3 y . The phase portrait of the amplifier-gauge 1 is shown in Figure 8. We see that the amplifier-gauge 1 is nonlinear. Choose the model similar to (4) and variables to estimate the nonlinear function. Analyze the relationships for this element and obtain the model  (Figure 9). We see that the framework      So, we see that the possibility of structural identifiability of a nonlinear system depends on the interaction of its elements. Just the structural organization of the system determines the ability to solve the structural identifiability problem.
Therefore, we see that the possibility of the structural identifiability estimation of the nonlinear system depends on the interaction of its elements. Just the structural organization of the system determines the ability of structural identifiability problem solves.
In the appendix, we state the problem of structural identification on a set of model structures. Next, we introduce the concept of structural identifiability at the set level.

System Attractor Reconstruction
Reconstruction (restoration) of the phase portrait (PP) or a system attractor can be performed on the basis of time series analysis. The proof of this approach is given in [33], and the practical application is based on Wolf and Rosenstein algorithms [34] [35]. This problem can be interpreted as the system structure restoration task in the phase space. Many authors (see reviews in [36] [37] [38]) have studied this problem. Reconstruction attractor procedures are heuristic [37]. The phase portrait construction depends on the choice to recover optimum parameters of reconstruction. The main parameter is the choice time delay for generating new variables on base the available time series. To solve this problem, various approaches (see references in [38] [39]) use: the autocorrelation and cross-correlation, the choice of the attractor shape, the method of the neighbor, and also the prediction statistics based on various models. Recommendations about the choice of the delay value estimation method are not provided. It is explained by the complexity and the variety of considered objects. The second problem is concatenated to the quality criteria choice [37] for the estimation of the PP reconstruction. Unfortunately, this problem has not obtained the final solution. Some recommendations are given for solving this problem in [37].
The choice of an attractor dimension [36] [40] [41] is also an important task. An attempt to resolve this problem is made in [36]. In [38], a statistic is proposed for the choice of the delay value and the attractor dimension. It is shown that these statistics can be applied to the attractor creation for multidimensional systems. The reconstructed attractor further analysis problem is not completed at this stage. As a rule, the designed attractor not always satisfies the requirements of the researcher. The attractor is not smooth. Therefore, smoothing various methods [36] [42] apply to obtain a smooth mapping.
Identification of the dynamic system was considered in [36] [37] based on the obtained set of state space variables. This issue is discussed in the review [43] in more detail. Various approximation methods of the operator describing the system state are applied to the model design. The basis of identification methods is interpolation procedures decomposition of a nonlinear function on the specified basis, the application of spline-functions and neural networks, and many other approaches [43].
Remark 4. As noted in [37], none of the considered identification methods is efficient. The major role is played by heuristics, the researcher experience and the prior information. This remark is true for PP restoration methods [36]. As a rule, at first, the data approximation is performed on the given class of functions.
Then the phase portrait, topologically the equivalent to an initial system, is construction. Next, unknown parameters are introduced in the obtained model that the properties of the obtained mapping improve. For this purpose, various heuristics and procedures are applied for additional information accounting on a system. Obtained models are very unwieldy and inconvenient for the application. Therefore, in [36], it is noted that the use of complex models is not always justified in practical applications.

System Structure Choice with Lag Variables
Models with distributed lags (DL) are widely applied in various areas [44]- [50]. S . Therefore, previously considered methods do not apply to DL analysis. The structure estimation of the system with DL is based on the analysis using secants [38].
Further, the estimation method of the DL system structure based on Lyapunov exponent identification is stated. This method is the development of the approach described in subsection 3.2. The direct transfer of results [22] [23] on the considered system class is impossible since these systems have the specifics.
Consider the system where n y R ∈ is output; is input vector which elements are limited extremely nondegenerate functions; , , , Let the informational set I o for the system (15) is the parameter vector. The system (15) is not dynamic in the standard sense. Assumption 1. Let the system (15) contain the variable where i α , κ are some numbers, h < ∞ , n R ζ ∈ is some limited function for all N n J ∈ .
Let the system (18) be stable, i.e. Allocate the transient process (the system (18) general solution) for the application of LE to the S π -system. Localize a space in (15) to which the variable , n j n u π = belongs. π-steady state eliminates the interval , where the interval g J corresponds to the π-state in the S π -system. Consider the set I o (16). Apply the model (17) where ( ) Theorem 3 [22]. If function n b changes the sign h times on the interval In [22], it is shown, if Theorem 3 conditions are satisfied, then the local minima of the framework , then S π -system have π-state.
The proof of theorem 4 is obvious. The local minima quantity corresponds to the lag structure of the system (15) on the variable , j n u . So, we showed that the discrete informational set I o modification is based on the approach [22]. This modification allows extending the methodology of geometrical frameworks application to systems with the distributed lags of input variables.
Consider the identifiability problem of Lyapunov exponents. Let the vector n U is limited constantly excited for some is the unit matrix.
If (20) is satisfied, then we will write n U α ∈PE . As shown in Section 4, the fulfilment (20) is sufficient for the S π -system π-state estimation. , \ n n i n U U u = and n U α ∈ PE , 0 α > . S π -system with π-state corresponds to the system (15 Figure 11. Figure 12 shows processes are non-smooth in the S π -system. Results of the lag structure estimation are presented in Figure 12 where frameworks The set S π M of Lyapunov exponent is shown in Figure 12. The analysis of results shows that the system (18) describing the change 1,n u has the order 2. Example 5. Consider the control system for supplying cars to the Vladivostok transport hub (Russia). Study the case of 6 cars simultaneous giving from railway tracks on berth tracks. The maximum capacity of the hub is 175 cars. Let 4 N is the number of cars from the railway; 5 N is the number of cars received The determination coefficient of the model (23) is 0.964. The simulation showed good predictive properties of the model (23).
So, modelling results confirm the approach efficiency to the structure estimation of the system (15).

Conclusions
The analysis of the concept "framework" application in identification problems is fulfilled. It is showed that this concept is widely used in parametric estimation problems. The term "framework" can be interpreted as a frame, a structure, the system, a platform, the concept, the basis, the system of approaches. It is shown that framework can be used in two directions: 1) the conceptual idea integrating the number of methods, approaches or procedures; 2) the mapping describing in the generalized form processes and properties in the system. The second direction is closer to methods that are applied in the qualitative theory of dynamic systems. In work, this approach is interpreted as the methodology based on the analysis of virtual geometrical frameworks (GF). In work, this approach is interpreted as the methodology based on the analysis of virtual geometrical frameworks. The main difference GF: they did not postulate a prior, and they are determined based on the experimental data processing. GF is the main object of analysis. They allow the decision-making about the properties and features of the system. The review contains the identification theory areas where this methodology is applicable.
1) Structural identification of the nonlinear system. 2) Lyapunov exponent estimation of the system. 3) Structural identifiability of the nonlinear system. 4) The system phase portrait reconstruction on the time series. 5) The system structure estimation with lag variables. We consider the application of Lyapunov exponents for the decision-making on static systems structure with lag variables.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper. is a set contained available information about the structure of the vector function S S F ∈ , parameters ( ) , G S A B ⊂ , characteristics of the input, output, and perturbation.

Appendix. Structural Identifiability at Structures Set of System Operator
The set S S can contain information about the class of operators, describing the system (A.1) dynamics and some of its structural parameters A S . The cardinality of set A S determines by the level of a priori information. In identification problems, the S S and A S formation based on the researcher intuition.