Structural-Identification Aspects of Decision-Making in Systems with Bouc-Wen Hysteresis ()
1. Introduction
Various models are used to describe hysteresis [1] . The Bouc-Wen model (BWM) is widely used to describe hysteresis. BWM is proposed by R. Bouс [2] and is generalized by Y.K. Wen [3] (system SBW)
(1)
, (2)
, (3)
where
is mass,
is damping,
is the recovering force,
,
,
,
,
is exciting force,
are some numbers. Equation (3) is the BWM.
Many modifications of BWM [4] are proposed. Each model considers the features of the considered object. The BWM successful application depends on the identification of its parameters. The solution of the nonlinear Equation (3) is the main problem of BWM identification. The methods of identification and control Bouc-Wen hysteresis systems are given in [5] [6] . Adaptive modelling methods [6] are used to analyze the state of structural dynamics objects. An approach to the BWH identification based on analysis of a priori information and some heuristic procedures is proposed in [7] . Adaptive algorithms are proposed in [8] [9] for the BWM parameters estimation with the data forgetting [10] . It is assumed that are available for measurement
and z, and
, x are obtained by integration. The approach to adaptive identification [11] [12] is based on the least-squares method application and correction of the gain matrix. Change areas
, x and the parameter n value are set. The adaptive observers use for the BWH identification is considered in [13] . The analysis of other approaches to the BWH parameters identification is given in [4] [14] [15] [16] [17] . Most procedures are based on measuring derivatives x. This possibility does not always exist when solving practical problems.
Examples [18] are known when BWM parameters estimations do not coincide with results obtained for other inputs. Such examples speak about the ambiguity of identification, which causes the instability of the model. Explain it with the fact that the Bouc-Wen model should be stable and ensure the adequacy of a physical process [19] . Requirements for BWM [19] : 1) adequacy of the mathematical model to the physical process; 2) BWM stability. Stability conditions impose restrictions on the changing area of model parameters. The choice of parameters belonging to the stability domain does not always give the adequate BVM [16] under uncertainty. Therefore, the approach [18] to the hysteresis; 3) parameters identification based on the BWH approximation of polynomial is proposed.
So, the analysis of publications shows that many algorithms and procedures for the BWM parameters identification propose. Proposed models consider features of the system. As a rule, the area of the BWM parameters changing sets a priori. Some parameters, such as n, are considered known. It is often assumed that derivatives of the system are measured. This situation does not always occur in practice and gives to the non-realizability of algorithms. The structure choice of the system (1)-(3) (structural identification (SI)) is the use result of the researcher’s knowledge and intuition. This approach does not always give an adequate choice of the BWM structure under uncertainty. Often the structural identification problem is reduced to the parametric identification problem [20] [21] . This approach is laborious under uncertainty.
So, we see that BWH structural identification problem has not been developed. The SBW-system modeling effectiveness depends on the choice of its parameters due to its initial instability. The input perturbation choice is significant for obtaining adequate results. The incorrect choice of input can lead to a system’s non-identifiability. These problems require a solution for BWH. Some results on these problems are presented in [22] .
The systematic approach proposed in this work gives the problem solve of identification systems with Bouc-Wen hysteresis. It includes: 1) the method for the input affect estimation on the SBW-system identifiability; 2) a hierarchical immersion method that allows you to decide on the BWH structure under uncertainty; 3) the adaptive identification of BWH parameters based on input-output data; 4) the method for estimating the identifiability degree based on the analysis of GF and the phase portrait of the SBW-system.
Work structure: 1) problem statement; 2) the method of SBW-system adaptive identification; 3) modifications of the SBW-system to simplify it and ensure stability; 4) the estimation method of structural identification and identifiability SBW-systems; 5) the properties analysis of the input
, which guarantees the SI and identifiability of the SBW-system; 6) BMW modifications guarantee its stability.
Remark 1. The parametric approach does not allow estimating of the BWH structure under uncertainty. The proposed approach is based on the properties analysis of geometric frameworks.
2. Problem Statement
Consider SBW-system. We have information on the input and the output
, (4)
where
,
are limited functions of time.
Determine conditions of the system SBW identifiability and structural components of the Bouc-Wen model (3) based on analysis of the sets (4).
Solving this problem is answered to the question: can we get an estimate of the system (3) structure under uncertainty?
Consider the identification of system parameters (1)-(3) by
.
3. Adaptive Identification of BWH
3.1. Problem Statement
Consider the system SBW. Let
be the output of the system. The set of the experimental data is
, where
is the specified time interval.
Designate by the parameters vector of the system as
.
Problem: an adaptive observer to design for the evaluation of vector A such that
, (5)
where
is the output of the adaptive observer,
.
Remark 2. The identification effectiveness of the system SBW depends on features of the input
. Requirements to
in identification problems are known. The force
must satisfy the constant excitation (CE) condition. This condition is necessary but not enough [23] . The input having the CE property cannot ensure the identifiability of the hysteresis structure. The structural identifiability of hysteresis is possible if
has the S-stabilization property of the system [23] .
3.2. Adaptive System Identification
The set
has the form (4). Therefore, it is not applicable for estimating the parameters of the system SBW. Design an adaptive observer for the system (1)-(3).
Consider the simplified system (1)-(3) when
,
. Substitute
in (1), and divide it by
, where
does not coincide with roots of the polynomial
,
. Then
, (6)
(7)
where
,
,
.
Equations (6), (7) contain only measurable variables except z. It complicates the identification process of the system SBW parameters. Apply the model
(8)
to the estimation of the system (6) parameters, where
is the specified number;
,
and
are adjusted parameters.
Designate
. Obtain the equation for the identification error from (6), (8)
, (9)
where
,
,
,
.
(9) is not solvable as the variable z is unknown in (7). Obtain the current estimation for
. Consider model
. (10)
Determine the misalignment
and use it for the variable z estimation. Let
is the current estimation z. Apply the model to the estimation z
, (11)
where
;
is the given number
,
are the hysteresis (3) parameters estimations;
is the integration step.
Introduce the misalignment
and obtain the equation for
, (12)
,
,
where
,
,
.
Present (8) as
, (8a)
where
. (13)
Then (9) rewrite as
, (14)
and adaptive algorithms describe as
(15)
where
;
.
Tuning algorithms for
and
in (11) have the form
(16)
where
are parameters ensuring a convergence of algorithms.
Several algorithms are applicable for the indicator n estimation in (11). The effectiveness of their work depends on several factors. The simple algorithm has the form
(17)
where
are set positive numbers,
.
Remark 3. The identification procedure stability is the main problem of the system synthesis with BWM. We propose the method based on adaptive observer application.
3.3. Properties of Adaptive System
We estimate the adaptive system stability by applying Lyapunov vector functions. Consider the subsystem
described by (14), (15). Let
,
, (18)
, (19)
where
. Next, we give a results generalization [23] [24] .
Assumption 1. The input
is constantly exciting and limited.
Theorem 1. Let 1) functions (19),
are positive definite and satisfy conditions
,
; 2) assumption 1 for the system (1)-(3) is satisfied. Then all trajectories of the system
are limited belong area
and the estimation
is fair.
Theorem 1 shows the limitation of adaptive system trajectories. The asymptotical stability ensuring the system demands to impose additional conditions.
Let
.
Definition 1. The vector P is constantly excited with a level
or have property
if
:
fairly for
and
on some interval
, where
is the unity matrix.
If the vector
has property
then we will write
.
The system
is stable, and the input
is restricted. Therefore, present the property
for the matrix
as
,
, (20)
where
is some number.
Let the estimation to
be fair
, (21)
where
,
are minimum and maximum eigenvalues of the matrix
.
Apply inequalities (20), (21) and obtain estimations for
, (22)
, (23)
Theorem 2. Let conditions be satisfied 1) positive definite Lyapunov functions
and (18) allow the indefinitely small highest limit at
,
; 2)
; 3) equality
is fair in the area
with
, where
,
is some neighborhood of the point O; 4) the function
satisfies (21); 5)
satisfy the system of inequalities
; (24)
6) the upper solution for
satisfies to the comparison equation
if
,
, (25)
where
,
,
is M-matrix. Then the system
is exponentially stable with the estimation
, (26)
if
. (27)
Theorem 2 shows that the adaptive system gives the estimates for system (1) parameters. It is fair at the fulfillment of conditions (27). We supposed that the variable
restricted. The boundedness of the variable
follows from the boundedness of the system
trajectories.
Consider subsystem
described by Equations (12), (16). Introduce Lyapunov functions
(28)
Theorem 3. Let 1) functions
,
are positive definite and satisfy condition
,
; (29)
2) the function
has the form (28); 3) the function
,
, (30)
exists, where
is the definition range of the subsystem
; 4)
,
; 5)
,
; 6) the assumption 1 holds for the system (1)-(3). Then all trajectories of the system
are bounded, belong in the area
, and the estimation
(31)
is fair if
. (32)
So, the boundedness of trajectories in the adaptive system is proved. The analysis showed that the subsystem
is asymptotically stable. The prove of trajectories boundedness for the subsystem
is a more complex problem. The estimation (31) shows that the quality of processes in the
-system depends on the output derivative of the
-system. The following result give more exact estimations for processes in the
-system.
Theorem 4. Let 1) positive definite Lyapunov functions
and
exist and have the indefinitely small higher limit at
to
; 2)
; 3) such
exist that conditions
(33)
are satisfied in the area
, where
,
is some neighbourhood of the point O; 4) inequality
holds for almost all t where
; 5) such
and
exist that
and
; 6) the function
,
(34)
exists, where
the subsystem
definition domain; 7)
,
satisfy the system of inequalities
(35)
8) the upper solution for
satisfies to the equation
(36)
if
,
, (37)
where
,
,
is M-matrix. Then the system
is exponentially dissipative with the estimation
, (38)
if
,
,
,
(39)
M-matrix is considered in [25] .
So, the system
is exponentially dissipative. The dissipativity area depends on the informational set
of the
-system.
3.4. Simulation Results
Consider the system (1)-(3) with parameters
,
,
,
,
,
,
. Parameters are selected based on simulation. The exciting force
. The system is modeled with initial conditions
,
,
. Form the set
. The system phase portrait and output of the hysteresis shown in Figure 1.
Figure 1. System phase portrait and hysteresis change.
Consider the identification of the system parameters. Determine the parameter
of the system (13) using the transient process analysis for
and
. Calculate Lyapunov exponents (LE) [26] . The estimation for the maximum LE is −0.9. Therefore, we set
. Initial conditions in (7) are equal to zero.
Adaptive system work results presented in Figures 2-4. Parameters
,
are equal to 2.5 and 0.75. The tuning process of
-systems (the model (8)) parameters shown in Figure 2. Figure 3 shown the model (11) parameters tuning.
Show the modification of identification errors
in Figure 4. We see that the accuracy of obtained estimations depends on the numbers of tuned parameters and the level
and properties
. Obtained results confirm statements of theorems 3, 4. The
-system work results influence the tuning processes in the
-system. Gain coefficients in (15), (16) and (17) are
,
,
,
,
. The hysteresis output estimation shown in Figure 5.
So, simulation results confirm the exponential dissipativity of the designed system.
4. Modification SBW-Systems
Various modifications of BWM have been proposed (see, for example, [4] [5] [26] [27] [28] ). They reflect the features and properties of the control object. System (1)-(3) is the basis for modifications. The BWM modification proposes for the case of asymmetric hysteresis in [29] . The model has the form
. (40)
The BWM modifications set is based on the introduction of new multipliers in (3) [4] [30] . They reflect requirements to the system. BWM considering the degradation and clamping of reinforced concrete structures has the form [30]
, (41)
Figure 2. Tuning of model (8) parameters.
Figure 3. Tuning of model (11) parameters: 1 is tuning
, 2 is tuning
.
Figure 4. Outputs modification of systems
,
.
Figure 5. Hysteresis estimation at adaptation of
-system.
where
and
are parameters reflecting the decrease in rigidity and strength of the structure,
considers the pinching effect.
The analysis showed that the last term in (3) is responsible for “fine-tuning” the hysteresis in the saturation or switching areas. If this is not critical for the object, then by selecting the parameters of the SW-system, this term in Equation (3) can compensate. In addition, some modifications simplify and increase the system (1)-(3) stability. They have the form [32]
, (42)
, (43)
. (44)
The introduction in (42) of the linear component of z increases the feasibility of the BWM and the
-system stability. As the system is nonlinear, the function
introduces to ensure the required hysteresis state. It guarantees a change z in the specified boundaries. Parameters
are some numbers.
A comparison of the models (42)-(44) and BWM is shown in Figure 6. The representation allows comparing model properties by generalized indicators in the “minimum-maximum” space. Notation in Figure 6: z is model (3), z1 is model
, z2 is model
, z3 is model
; t is average value; — is median; ¡ is the extreme value (end of the “saturation” region).
So, BWM modifications are considered. The application of proposed models depends on the object properties. The parameters influence analysis of models (42)-(44) give in [31] .
5. Theoretical Foundations of SI
5.1. Preliminary
The modern direction of structural identification is based on the parametric
Figure 6. Comparison of hysteresis models (3), (42)-(44).
paradigm. It is explained by the formation and development of the theory of identification. Nonlinear systems SI methods are based on the approximation of nonlinearity by parametric models (see, for example, [32] [33] [34] [35] [36] ). This approach leads to levelling of the nonlinearity structure. The second direction of structural identification is related to the analysis of geometric frameworks (GF). GF reflect the state of the system nonlinear part. It is the new direction in the identification theory. This approach proposes in [22] [37] . The statement of this approach gives below.
5.2. Problem Statement
Consider dynamic system
(45)
where
,
are input and output system;
,
,
;
is the scalar nonlinear function belonging to the class of the hysteresis
;
. We suppose that A is the Hurwitz matrix.
Various assumptions are made about the structure of the function
. They determine by the level of a priori information. Under a priori definiteness, apply the methods based on linearization [38] . In the absolute stability study of nonlinear systems, suppose [28]
, (46)
where
is the nonlinearity input.
is a linear combination of the state variables (the vector X). The sector condition is used for approximation of function
, (47)
Static nonlinearity often applies in control systems. Therefore, next, we consider the static (algebraic) functions which describe a hysteresis. For system (45), we have a set of the data
. (48)
Problem: determine a form and parameters of function
based of the analysis and a processing of the set
.
The problem solution is based on the formation of the set
contained data about
.
5.3. Formation of Set
The differentiation operation applies to
and designates the obtained variable as
. Generate informational the set
. Select the data
subset described the particular solution (steady state) of the system (45). The mathematical model
(49)
applies to obtain
. Model (49) is determined on the time gap
and gives the linear component
estimation.
is a parameters model vector. The choice of an interval
depends on the value of criterion
.
Determine a vector H as
, (50)
where
.
Apply the model (49) and determine the forecast for the variable
. Compute the error
.
depends on nonlinearity
in the system (45). Obtain set
, (51)
which we will use next. We will apply the designation
, supposing that
.
The further problem solution is based on the analysis of frameworks
,
which reflect the state of the nonlinearity.
Remark 4. Choice of the model (49) structure is one of the stages of structural identification. Simulation results show that the model (49) is used in identification systems of plants with static nonlinearity. For other classes of nonlinearity, this problem demands further research.
5.4. Frameworks
,
Go into space
and construct the phase portrait
of the system (45). The framework
corresponds to a phase portrait
[37] .
describes function
.
must have a closed form. This property
differs from frameworks
.
is applied for the analysis of statics systems [39] [40] .
For decision making, we will use also
-framework.
is described by function
, where
is a coefficient of structural properties [39] systems (45) in space
. (52)
Next, we construct sector sets for system (45) in the space
and will be decision-making on the class
. The solution to this problem is based on the analysis of proposed frameworks.
5.5. About Properties
Consider the set
properties ensured the solution of hysteresis F1 structural identification. Let fulfill to conditions
(i) the set
ensures the solution of the model (49) parametric identification problem.
(ii) the input
ensures obtaining informative framework
or
.
If
has properties (i), (ii), then input
is representative.
Let the framework
is closed and its area is not zero. Designate altitude
as
, where the altitude is the distance between two points of the opposite sides of framework
. Then the framework
is identified on set
.
Let
, where
is the constancy excitation property
(53)
fair for
and
on some interval
.
Statement 1 [37] . Let (i) the linear part of system (45) is stable and nonlinearity satisfies the condition (47); (ii) the input
is piecewise continuous, limited and constantly exciting; (iii) exists
such that
. Then the framework
is identified on set
.
Proof of Statement 1. Consider input
satisfied to condition 1).
corresponds Fourier series containing a sinusoid with frequency
. The output
contains components of this spectrum and has a phase shift. The variable
is the result of the differentiation
. Hence,
contains components with this frequent spectrum. Therefore, the framework
(phase portrait) on a phase plane
has a closed form. The
-framework has the same form. Determine the distance
between opposite points of the framework
.
satisfy the condition 2) statement 1. Therefore, for almost all
. n
The framework
which has referred properties, we will name h-identified. Further, we believe that
has the specified properties.
Features of the h-identifiability.
1) h-identifiability is a concept not parametric identification, and structural identification;
2) The demand for parametric identifiability is the base h-identifiability;
3) h-identifiability determines more stringent demands to the system input.
Feature 3 means that “the bad” input can satisfy a constancy excitation condition. Such input can give a so-called “insignificant”
-framework (framework
) [37] which will have property h-identifiability.
5.6. Framework
Consider the framework
. Let
, where
are left and right fragments
. Determine for
secants
,
, (54)
where
are the numbers determined using the method of least squares (LSM).
Theorem 5 [37] . Let (i) the framework
is h-identifiable; (ii) the framework
have the form
, where
are left and right fragments
; (iii) for
secants (54) are obtained. Then
is
-framework, if
, (55)
where
is some number.
Theorem 5 proves based on sets homothety.
Remark 5.
-frameworks are characteristic for systems with many-valued nonlinearities. They are the input inadequate use result.
6. Structural Identification and Structural Identifiability BWH
We have noted (see introduction) that structural identifiability (SID) is the result of structural identification. Therefore, we will consider the SID basics guaranteed SI.
Apply the SI model (49) and represent the system (45) as (system
)
(56)
where
is a variable describing the general solution of the system (45),
is a bounded perturbation appearing as the analysis result of the variable e.
6.1. System
Consider the identifiability problem system
. Let conditions hold.
B1. The input is constantly excited at the interval J.
B2. The analysis of
gives the solution to the estimation problem the nonlinear properties of the system
.
We state the basic concepts, generalizing the results [22] .
Definition 2. If
satisfies В1 and В2 conditions, then the input
is representative.
Let the framework
closed, and the area
is not zero. Denote height
as
where height is the distance between two points opposite sides of the framework
.
Theorem 6 [37] . Let (i) the linear part of the system (45) is stable; (ii) the nonlinearity
satisfies the condition (47); (iii) the input is bounded and constantly excited; 4)
, where
. Then the framework
is identified on the set
.
Definition 3. The framework
is called h-identifiable if theorem 6 holds for
.
Let
be h-identifiable. Introduce designations:
is definition range of the framework
,
is diameter
. Let
, where U is an acceptable set of inputs for the system (45). The set U contains representative inputs.
Definition 4. If
of the framework
has a maximum diameter
, the input S-synchronizes the system (45).
Consider a reference framework
.
is the framework
reflecting all properties of the function
. Designate by the diameter
as
.
exists if the input the system (45) is S-synchronizing.
Definitions 2, 3 show if
, then
, where
,
is the proximity sign. Elements of the subset
have property
. (57)
Synchronization
is the choice of such input
that reflects all features
in
. It is true if
ensures
and
. We interpret the choice
as ensuring synchronization between structures of a model and the system.
is the condition of h-identifiability which can represent as
. (58)
The condition for
. (59)
(58) can be interpreted as proximity domain
, (60)
which is understood as
for almost
.
We will write
if considered frameworks are close.
Domain
is the S-synchronizability area on
or the structural identifiability domain on
, where
is the phase portrait of the system (45) if the condition
is true for
almost
.
So, two criteria (55) and (59) presented for the existence of the insignificant framework
. Structure of systems
and (45) are structurally non-identifiable in this case.
Let the input
synchronize the system (45). If
is S-synchronizing, then we will write
. Note that a finite set
exists for the system (45). The choice of optimal
depends on
and (58). The hold of the condition (58) is one of the prerequisites for SI of the system (45).
Definition 5. If framework
is h-identified and conditions
, (8) are satisfied, then the framework
or the system (56) (system (45)) is structurally identified or
-identifiable.
Remark 6. Conditions specified in definition 5 are the conditions for the structural identification of systems (45), (56).
Definition 5 shows if the system (45) is
-identified then the framework
has the maximum diameter of area
.
Definition 6. The model (49) is SM-identifying if the framework
is
-identifiable.
The framework
is defined on
and
satisfies condition B1. Therefore,
corresponds to the nonlinearity
defined on the class
, (61)
where
are some numbers.
Note that the term SM-identifying does not coincide with the concept proposed in [41] .
Theorem 7 [42] . Let (i) the input
is constantly excited; (ii) the system (45) phase portrait have m features; (iii)
-framework is
-identified and contains fragments corresponding to features of the system (45). Then the model (49) is SM-identifying.
The theorem 7 shows if the model (49) is not SM-identifying then model (49) structure or the informational set (48) need to change.
Let
is the center of the framework
on the set
,
is the center of the area
.
Theorem 8. Let the set
given for the system
and (i) exists
such that
; (ii)
, where
are coefficients of secants (54) for
. Then the system (56) is
-identifiable, the input
, and the framework
defines the class
.
Proof of Theorem 8. Consider the input
. Since condition
is satisfied, the framework
is symmetric concerning the point
plane
. Consequently, definitional domains diameters of the fragments
for the framework
coincide up to a certain value
on the set
, i.e.
, (62)
where
are definitional domains
. Then the framework
centre is equal to
. Since
, there exist
such that
. The fulfillment of conditions (i), (ii) guarantees
and
. Therefore, the framework
contains all
the features characteristic of the function
at
. So,
, and system (45) is
-identifiable. n
As
, then the area
have center
,
is some interval.
Len subset
(
) which elements have the property of S-synchronizability exists. The framework
has the diameter
and corresponds to every
. As
the diameter
has the property
-optimality.
Let the hypothetical framework
(a framework
) of the system (45) have diameter
.
Definition 7. The framework
has
-optimality property on the set
if
such that
.
Definition 8. If
,
and frameworks
have
-optimality property, then frameworks
are structurally indiscernible on sets
.
So, the
-identifiability estimate can be obtained from any input, following definitions 6, 7.
Definition 9. If frameworks
have
-optimality property, then
is locally structurally identifiable on the set
.
Let the framework
having
-optimality property is
, and the locally structurally identified framework
is
.
The framework
is locally structurally identifiable on the set
if
. (63)
Remark 7. We consider nonlinearities satisfying condition (47). Therefore, notes made above are valid.
Definition 10. The framework
that does not have the
-optimality property is locally structurally non-identifiable on the set
.
The framework
that is structurally non-identifiable on the set
defines a class
.
Remark 8. The described approach applies to the nonlinear system with a dynamic law of nonlinearity change. In this case, the hierarchical immersion method [43] is used for the structure estimation of the nonlinearity.
The identifiability of system
considered in [44] . Let the phase portrait
constructed for the system.
6.2. Non-Identifiability Degree
Obtain the non-identifiability degree estimate of the system (56). Definitional domains of
and
are coincident. Therefore, the diameter
of the framework
is known. Consider the set
having the property
. Determine the framework
for each
and obtain
. Suppose
and denote the corresponding input as
. Determine diameters
for all inputs
. Since
, therefore
. As
, therefore
. Then evaluate the non-identifiability degree as
(64)
shows that
-system (1) is structurally identifiable if
.
If estimates for the fragments
of the phase portrait
are known, then the identifiability degree is defined as
, (65)
where
are diameters of fragments
. The system
is structurally identifiable if
where
is neighborhood 1.
Example 1. Consider the BWH from Section 3.4. Consider four variant inputs
(66)
Calculate diameters for the phase portrait definitional domain
,
,
,
. (67)
Results obtained for the system
steady state. The analysis showed
. We assume that the system
with the phase portrait
is the standard and
. The degree of non-identifiability of the system
for various
,
,
. (68)
We see that the
-system is structurally non-identifiable with
, and the
-system with input
is structurally indistinguishable from input
. So, frameworks
are frameworks of class
, and the framework
belongs to class
.
6.3. Hierarchical Immersion Method
If nonlinear processes are complex, then the model (49) will be inadequate. Then the hierarchical immersion method (HIM) [44] is used to design the
-framework. HIM realizes the subsequent stages of synthesis
if the model (47) is inadequate. The method is based on the application (49) in a new structural space and the synthesis for
new framework. If the new model (49) is significant, HIM stops. Otherwise, a new iteration is implemented.
6.4. SI and SID Bouc-Wen Hysteresis
Consider the BWH from section 3.4. Introduce the framework
to estimate the
-system structural identifiability.
is the hysteresis estimation in the structural space
. Apply the model
(69)
and calculate the error
.
The framework
described by the mapping
and is showed in Figure 7.
Apply the approach proposed in [45] . Draw the straight line parallel to the ordinate axis through point
. Obtain two fragments
. Determine secants for the left
and right
fragment
(70)
Let
be the distance between the opposite sides of the framework
. The framework
satisfies conditions of theorem 6. The height
, and the input
is constantly excited and S-synchronized. Therefore, the
-framework (system
) is
-identifiable. Figure 8 confirms this conclusion. Models (69) is SM-identifying.
Consider the structural identification of BWH. Apply the hierarchical immersion method for estimating the BWH structure. Calculate the derivative for e applying numerical derivation. This procedure is sensitive to calculation errors. Therefore, perform smoothing
applying the Fourier transform.
Denote the obtained variable as
. Further analysis has shown that
did not depend on x (see Figure 8). Thus,
depends on
or z.
Consider the framework
described by the mapping
, where
is the estimate of the derivative
. Determine the secant
for
:
. (71)
The model (71) presents in Figure 8. Therefore, Figure 8 and the model (71) confirm effect
on hysteresis properties.
Estimate the relationship between variables z and
. Use the variable e as the estimation z. Apply the denote
. The analysis shows
and
not relates by the linear dependence. Therefore, the correlation between
and the combination
and
exists. Eliminate the effect of the linear component
on
. Obtain the variable
. Go to into the space
,
,
.
The example of the relation estimation is shown in Figure 9, where
. The secant
framework
has the form
, coefficient of determination
. The parameter h cannot correspond to the parameter n in (3). The cause of such discrepancy follows from the proposed approach. True of BWM parameters estimates based on the use of the parametric identification.
Figure 8. Framework for effect
estimation of
-system.
Figure 9. Estimation of correlation
and
.
The effect estimates of the variable
can be obtained in space
. This conclusion follows from Figure 9, where the dependency
is presented.
Remark 9. Secant (71) can use as the output for estimating structural relationships in BWM.
Figure 10 confirms the validity of the proposed approach. The framework reflects the relationship between the reference and received hysteresis estimates. The secant
has the form
,
. (72)
So, the structure analysis has shown that the hysteresis dynamics depends on variables z and
. The system (1), (2) output does not influence the change of the hysteresis. The structure analysis is based on the application of adequate mathematical methods and guaranteed decision-making on the structure of the system
.
The HIM stop rule. Let
is an informational set on which the framework
is defined, where i is the hierarchical immersion level. Examples of sets
and frameworks are presented above. Let
is the insignificant framework, and at the level i the system is structurally identifiable.
Let
is the insignificant framework, and the system
is structurally identifiable at the level i.
is a sign of structural non-identifiability the system at the level
.
is a sign of the system (1)-(3) structural non-identifiability at level
.
Theorem 9. The system
is structurally identifiable on the set
if
at the level
.
The proof of theorem 9 follows from the analysis of secant for framework at each step i.
Figure 11 represents the framework
and the secant, where
,
. Obtain the model for the variable
on the set
, (73)
Figure 10. Estimation of proximity z and
.
Figure 11. Insignificant framework
.
and introduce the misalignment
. An approximation
by model
shows that this relationship is insignificant. This conclusion confirms the presence of the third term in the right part of the Equation (3).
So, we propose the approach for structure estimating of the Bouc-Wen model based on the set
analysis. The approach is based on the hierarchical immersion method and the analysis of geometrical frameworks. Frameworks describe the state of the system nonlinear part at each SI stage.
7. Excitation Constancy Effect on System Identifiability
Let input
of the system (56) have the property
, (74)
where
,
, (75)
is a model for
based on the Fourier series and given on the set of frequencies
.
Let
,
. Consequently,
. For
is hold
,
, (76)
where
.
Compare (75), (76) and obtain
. (77)
From (77) have
. (78)
The definitional domain of frameworks
do not coincide, and
is
-optimal on the set
. Therefore, the fulfillment of condition (58) follows from inequality (78). Consequently, the structure of the system (45) nonlinear part with
has indicators that do not coincide with the structurally identifiable parameters of the system (45) with
.
So, the CE condition of the input affects the
-system
-identifiable and, consequently, the system (56).
The statement is true.
Theorem 10 [37] . Let (i) the input
to condition (75); (ii) the
-framework corresponds to the input
; (iii) there is the input
such that the condition (76) satisfied; (iv) conditions (77), (78) holds. Then (a) the
-system is structurally non-identifiable by the input
; (b) structural parameters of the
-system do not correspond to the system
with the identifiable framework
.
The input amplitude can influence on the SI of nonlinear systems. Modify conditions (75), (76)
,
, (79)
,
, (80)
where
,
are model
,
parameter vectors.
Present models
,
as
,
, (81)
,
are modifications of models (78), (76);
,
,
is an element
;
,
.
(
) denotes the generalized amplitude of the input.
Condition (77) transformed into the form
. (82)
Since
then
. This conclusion follows from
, (83)
and the model
approximates the input ensuring S-synchronization of the system
.
Obtain
-optimality of the diameter
from
. The framework
does not have this property (see (80)). Therefore, the input
, which has a smaller generalized amplitude, gives the diameter
.
Theorem 11 [43] . Let (i) the input
of the system (45) satisfies the condition (79); (ii) the framework
corresponds to input
; (iii) there is the input
such that the condition (80) holds; (iv) conditions (77), (78) are hold. Then (a) the
-system is structurally non-identifiable by the input
; (b) structural parameters of the system
do not correspond to the system (45) with the identifiable framework
if
.
So, the properties influence of input on SI and the structural identifiability of the system with BWH show.
8. Conclusion
The estimate problem of Bouc-Wen hysteresis parameters is relevant under uncertainty. The existing approaches to the identification are based on the parametric paradigm and consider a priori information. Under uncertainty, the BWM synthesis requires time-consuming research. The parametric approach plays an approximative role for a given a priori model structure. It allows you to describe the behaviour of the system or set trends in its development. The structure is a hidden and non-formalized property of the system. Therefore, indirect and object methods should be used that reveal the features (structure) of the system. The paper proposes a structural-identification approach (CIA) for analyzing features of the Bouc-Wen hysteresis under uncertainty. Geometric frameworks are the basis of the CIA. The GF analysis allows for the evaluation of the BWH structure and identifiability. The proposed approaches demonstrate the possibilities of the stated paradigm.