=Paper=
{{Paper
|id=Vol-1623/papermp2
|storemode=property
|title=Predictive Model's Development Based on the Wavelet Analysis Technique
|pdfUrl=https://ceur-ws.org/Vol-1623/papermp2.pdf
|volume=Vol-1623
|authors=Natalia Bakhtadze,Valery Pyatetsky,Ekaterina Sakrutina
|dblpUrl=https://dblp.org/rec/conf/door/BakhtadzePS16
}}
==Predictive Model's Development Based on the Wavelet Analysis Technique==
https://ceur-ws.org/Vol-1623/papermp2.pdf
Predictive Model’s Development
Based on the Wavelet Analysis Technique
Natalia N. Bakhtadze1, Valery E. Pyatetsky2, Ekaterina A. Sakrutina1
1
V.A. Trapeznikov Institute of Control Sciences, Russia
sung7@yandex.ru, kathrina@inbox.ru
2
National University of Science and Technology “MISIS”, Russia
7621496@gmail.com
Abstract. In the paper, associative search identification models in variable struc-
ture control systems are considered. The methods of non-linear systems identifica-
tion by using the multiple-scale wavelet transform are presented. A methodology
of a variable structure control system’s synthesis is presented based on information
models. Issues of the stability of a model built by use of the associative search are
considered, in the aspect of the spectrum analysis of the multi-scale wavelet ex-
pansion. Methods based on the wavelet analysis are characterized with a unique
possibility to select ''frequency-domain windows''.
Keywords: system identification, predictive model, information model, virtual
model, associative search, fuzzy model, knowledge base, wavelets.
1 Preliminaries
Variable structure control systems (VSCS) are a class of control systems, providing an
effective possibility to solve main problems of the control theory – stabilization problem
and tracking problem – by use of automatically switched algorithms.
VSCS provide a rational system performance in accordance to a set dynamics without
constructing an adaptive system model (structure and parametric one). In paper [1], the
control invariance with respect to parametric disturbances was proven.
The methodology of the synthesis of variable structure systems is based on introduc-
ing a fundamental notion of new kind feed-backs (operator and coordinate-operator
ones). The approach is forming an operator (an algorithm of forming a control imple-
mented by a controller) transforming signals-coordinates (functions in the time) as an
element of a set of stabilizing feed-backs. A parameterization of the set enables one to
impose a one-to-one relationship between the signals-coordinates and signals-operators.
In paper [1], a constructive approach to the design of such algorithms is presented,
enabling the sliding mode, for second order linear systems. However, the VSCS ideolo-
Copyright © by the paper's authors. Copying permitted for private and academic purposes.
In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
Predictive Model’s Development Based on the Wavelet Analysis Technique 139
gy provides a possibility to design control systems of such a kind under the conditions of
the parametric uncertainty not for linear dynamic plants only. In the present paper, a
possibility of applying the principle of the operator feed-backs for non-linear as well as
time-varying systems is demonstrated.
To solve such a problem, one assumes under forming the control algorithm at each
time instant (what corresponds to the definition of a variable structure system) applying
information on the system status/state (current and archive one), in other words: applica-
tion of all its dynamic previous history. Meanwhile, not a conventional control system
with a feed-back identifier is formed: a system with operator feed-back is formed, in
which the signal-operator is formed by use of a virtual plant model created at each time
instant on the basis of intelligent analysis of persistently updated date on the plant dy-
namics.
2 Variable structure systems using inductive knowledge
Let us consider a scheme of a control system with additional dynamic error feed-back,
displayed in Fig. 1.
In systems investigated in [1], the operator produced the sign change. As to the
general case, in this block some, generically, non-linear transformation of the signal is
implemented:
(1)
Here is a non-linear operator that is formed on the basis of analysis a specific con-
trol problem. In the most general case, may be, for instance, logic production operator
implementing both conventional and fuzzy control.
To form at each time instant the signal-operator producing varying the structure
of the regulator , besides the dynamic error one should use additional a priori
information on the system. In this connection, let us recollect that the unique information
code sets the system state, its information ''portrait''. How can one use this information
(interpreted as the signal ), and how is formed?
is to provide to an intelligent (based on knowledge, that is regularities, formed
and updated at each time instant on the basis of the history data analysis) prediction in-
formation model of the plant. This model provides to an ''information support'' to make
a decision by the system on changing the controller structure. Based on this support, the
signal-operator is formed.
140 N. N. Bakhtadze, V. E. Pyatetsky and E. A. Sakrutina
System
y Dynamics x
g Knowledge
Base
s
D . σ S
r
G g e R
f u
y W
∆W
Fig. 1. Scheme of control system with additional dynamic error feed-back
Taking into account the fact that the same value of the dynamic error may take
place under different sets of signal values (involving operator ones), we come to the
inference that the signal may be represented as a regression (in the general case,
nonlinear) of the following variables:
(2)
A principal distinction of the intelligent model from a conventional predicting model
is the following. The intelligent (that is based on knowledge revealed from the data anal-
ysis) model is, in its entity, a situational (''virtual'') one, that is accounting particularities
of the current state, hence, at each time instant this model may have a new structure.
This its attribute enables one to synthesize VSCS for non-linear, and, involving, time-
varying systems.
The operator (also, in the general case, being non-linear) is to be formed by use an
approach based on the data analysis. The history system data analysis will enable one to
reveal certain regularities (inductive knowledge). An example of such knowledge may
consist in relating the system input (a vector in corresponding vector space) at each cur-
rent time instant to a certain domain of this space (clustering).
To determine , in particular, one may apply the technique of the associative
search [2], being a search method based on the analysis of the previous history of state
dynamics of a plant under study and constructing virtual models.
Predictive Model’s Development Based on the Wavelet Analysis Technique 141
3 The associative search method of virtual model development
Algorithms based on knowledge revealed from history system data (inductive
knowledge, persistently enriched) implement an intelligent approach to constructing
identification models. The intelligence is applying knowledge (Knowledge Based) re-
vealed from history data on the basis of their analysis (Data Mining).
The process of knowledge processing in the intelligent system is reduced to recover-
ing (associative search of) knowledge over its fragment. Meanwhile, the knowledge may
be interpreted as associative connections between images. As an image, we will use ''sets
of indicators'', that is components of input vectors, input variables.
The criterion of closeness between images may be formulated in very different man-
ners. In the most general case, it may be represented as a logic function, the predicate. In
a particular case, when sets of indicators are vectors in n-dimensional space, the criterion
of closeness may be a distance in this space.
The associative search process may be implemented either as a process of recovering
the image over partially given indicators (or recovering a knowledge fragment under the
conditions of incomplete information; as a rule, just this process is simulated in different
models of the associative memory), or as a process of searching another images that are
associatively connected with the given one, attached to other time instants.
In papers [2-4], an approach to form the support on decision making on the control is
proposed, based on dynamic modeling the associative search procedure. Results of adop-
tion of the associative search algorithms developed by the authors for industrial process-
es of the chemical and petroleum manufacturing, processes of control in intelligent pow-
er networks (smart grids), trading processes, transport logistic processes.
The method of the associative search consists in constructing virtual predicting mod-
els. The method assumes constructing predicting model of a dynamic plant, being new
under each t, by use of a set of history data (''associations'') formed at the stage of learn-
ing and adaptively corrected in accordance to certain criteria, rather than approximating
real process in the time.
Within the present context, linear dynamic model is of the form:
(3)
where: is the prediction of the output plant at the time instant , is the vector
of input actions, is the memory depth in the output, is the memory depth in the
input, is the dimension of the input vectors, and are the tuned coefficient,
meanwhile are selected disregarding the order of the chronological decreasing,
have been referred as the associative pulse.
Let us note that this model is not classical regression one: there are selected certain
inputs in accordance to a certain criterion, rather than all chronological ''tail''.
The algorithm of deriving the virtual model consists in constructing at each time in-
stant an approximating hypersurface of the space of input vectors and single-dimensional
outputs. To construct the virtual model, corresponding to some time instant, from the
archive there are selected input vectors being in a certain sense close to the current input
vector. An example of selecting the vectors is described below. The dimension of this
142 N. N. Bakhtadze, V. E. Pyatetsky and E. A. Sakrutina
hypersurface is selected heuristically. Again, by use of classical (non-recursive) least
squares (LS) method there is determined the output value (modeled signal) in the next
time instant.
Meanwhile, each point of the global non-linear surface of the regression is formed in
the result of using linear ''local'' models, in each new time instant.
In contrast to classical regression models, for each fixed time instant from the archive
there are selected input vectors being close to the current input vector in the sense of a
certain criterion (rather than the chronological sequence as it is done in regression mod-
els). Thus, in equation (3) is the number of vectors from the archive (from the time
instant 1 to the time instant ), selected in accordance to the associative search criterion.
At each time segment there is selected a certain set of vectors, .
The criterion of selecting the input vectors from the archive to derive the virtual model
in the given time instant over the current plant state may be as follows.
Let us introduce as a distance (a norm in ) between points of the S-dimensional
space of inputs the value:
(4)
where are components of the input vector at the current time instant .
By virtue of a property of the norm (''the triangle inequality'', we have:
(5)
Let for the current input vector :
(6)
To derive an approximating hypersurface for the vector let us select from the ar-
chive of the input data such vectors , that for a set the condition will be
hold:
(7)
where may be selected, for instance, from the condition:
(8)
If in the selected domain there will be not enough quantity of inputs to apply the LS
method, that is the corresponding system of linear equations will be unsolvable, then the
selected criterion of selecting points in the space of inputs may be weakened due to in-
creasing the threshold .
Under the assumptions that the input actions meet the Gauss-Markov conditions, the
estimates obtained via the LS method are unbiased and statistically effective.
Predictive Model’s Development Based on the Wavelet Analysis Technique 143
4 Solving the system of linear equations for the LS method
procedure
However, solving the problem of constructing virtual closed-loop models (as well as
the conventional identification synthesis) are characterized by considerable difficulties.
Under the closed-loop case, the control process is with depended values, and the ques-
tion on the existence of identifying strategies is not trivial. Optimal controllers generate
linear state feed-backs; this leads to an degenerate problem [5].
For the identification of dynamic plants by use of the associative search technique in
the degenerate case, Moore-Penrose method [6, 7] is applicable to obtain pseudo-
solutions to the system of linear equations under using the least squares procedure. Ap-
plying the method to solve the identification problem is presented in paper [8].
5 Clustering based associative search
In order to increase the computation capability at the stage of learning and under sub-
sequent real plant performance, one of the data mining methods – clustering (dynamic
classification, automated grupping data, unsupervised learning) is used. There are known
numerous such methods: hierarchical algorithms, -means algorithm, minimal covering
tree algorithm, nearest neighbor method, etc. All they determine (in the dependence on
the time, in contrast to the classification) the belonging of the point in the multi-
dimensional space by one of the domain, which the space is partitioned on.
As a result, each investigated point in the multi-dimensional space may be related to
a group by assigning a cluster label to it. In the problem of the associative search to se-
lect input vectors being ''close'' to the current one, the cluster label is determined in ac-
cordance to the associative pulse (the criterion of the associative selection), and to derive
the virtual models the vectors are selected inside the corresponding cluster.
In the problem of determining the signal-operator, the method of the associative
search enables one to predict on the basis of data on belonging the point to one
of the clusters of the space
(9)
Such an approach enables one to avoid the need of information on the structure of the
control plant . The plant is even not needed to be linear. The control quality will de-
pend on the amount of analyzed and clustering criterion.
6 Data Mining techniques in associative search tasks
The use of a priori information (process knowledge) plays the key role in the intelli-
gent approach to predictive model building described above. Therefore, the intelligent
data analysis, in particular, the selection of a data set meeting a certain criterion (“asso-
ciative impulse”) at each algorithmic step, plays the key role. In the simplest case, the
associative impulse presumes the selection of the next vector from the history (this selec-
144 N. N. Bakhtadze, V. E. Pyatetsky and E. A. Sakrutina
tion operation is called association) in such a way that this vector belongs to a certain
domain in the space of vectors stored in the process history. A certain metric is intro-
duced respectively.
Associative search algorithm discussed above is effective for the case, when the con-
trol plant is nonlinear, high response speed is not required, and the computational re-
sources enable the exhaustive search of the process history. Such an approach is quite
satisfactory for the identification of a rather broad class of control plants, for example,
continuous and semi batch processes in chemical and oil refining industries.
However, the unpractical use of computational resources within this approach is ob-
vious. Worse yet, each data search cycle requires to pick out the number of vectors suffi-
cient for solving a system of equations, which, according to the LSM, allows to predict
the output at the next time step. It is not guaranteed that such selection (even with high
redundancy) is attainable in a single step.
7 Application of traditional clusterization methods for the
associative search
To increase the algorithm speed (which is a key performance index of such algorithms
for certain applications) and computational resource saving, it is proposed to teach the
system. Clustering (learning without a teacher) looks an effective technique.
When using any of the known clustering algorithms (“crisp” case), the original set of
objects is split into several disjoint subsets. Here, any object from X
belongs to a single class only.
When using fuzzy clustering techniques, it is allowed for one object to belong to sev-
eral (or all) clusters simultaneously but with different degrees of certainty determined by
the selected membership function. Here, the clusters are fuzzy sets. Fuzzy clustering
may often be more preferable than crisp, for example, for the objects located at cluster
borders.
Associative search problem is solved by clustering technique (both crisp and fuzzy)
in the following way.
The current vector under investigation is attributed to a certain cluster per the criteri-
on of minimum distance to the center: where is the current
input vector of the control plant under investigation.
Further, the vectors are picked out from this cluster, which meet the selected associa-
tive search criterion. If the vectors picked out from the cluster are not enough for solving
the prediction problem by means of LMS, the cluster can be enlarged by one of the
known single-link methods which combine 2 clusters with minimum distance between
any 2 of their members.
This naturally ensures a huge saving of computational resources as against the ex-
haustive search over the whole process history at each step. However, this aggregation
into a new cluster presumes that many objects deliberately not meet the associative
search criterion.
The approach described below looks the most reasonable.
Predictive Model’s Development Based on the Wavelet Analysis Technique 145
Virtual clustering (“impostor” method). For each time step N, the current vector
under investigation is attributed to a certain cluster per the criterion of the minimum
distance to the center – in the same way as with the traditional method.
Assume is attained for k=r. Further, is assigned to be the
center of the cluster Ar. If additional selection is needed from the archive of vectors
meeting the associative search criterion, then the clusters are chosen for aggregation with
the minimum distance between their centers and the vector . In such case not only a
considerable number of vectors distant from will be discarded, but also the maximum
possible number of vectors meeting the associative search criterion will appear.
After the associative search is completed, the assignment of as the center of the
cluster Ar is cancelled, and the process is continued in the same way as in the traditional
algorithm.
8 The methodology of the intelligent VSCS synthesis
The sets of values of signals, forming virtual model (9) in different time instants, and
values of , corresponding to them, fill in the Knowledge Base of the system (see Fig.
1). Under accounting the content of the knowledge base, involving values of the signal
one may obtain (by use of the LS method) statistically effective unbiased prediction
of the system output. However, under the closed-loop description, it is reasonable the
operator , whose main function is to generate the system dynamics, to implement the
multiplication of and values of a limited signal , where and are sto-
chastically independent.
The virtual model that is synthesized by virtue of changing the internal system dy-
namics is the generalized information model of the control plant.
In the event, when data on the coordinate signals are taken into account only, that is
in the model coefficients at , are set to be equal to zero, we come to the conventional
closed-loop control scheme with an identifier accompanied with all its problems.
In the event, when, wise versa, in the model the coefficients at , are not equal to
zero, while the coefficients at , , are equal to zero, we obtain the information model
of the internal system dynamics. Such a model will provide vanishing the dynamic error
, . However, in the general case, meeting certain control properties is not guaranteed,
and, first of all, the stability. In other words, the information plant model permits not
only to provide vanishing the dynamic error, but also to keep certain quality of the sys-
tem performance.
In papers [8, 9], sufficient criteria of the stability of time-varying dynamic plants in
the terms of the spectrum of the multi-scale wavelet expansion of inputs and outputs of
the system. And since under deriving virtual predicting models the associative search
procedure is used, one may say about the method of the synthesis of VSCS for a broad
class of non-linear systems.
More flexible control (in particular, providing the stability), achieved by use of the
system dynamics generator, may be obtained due to using the generalized information
model under forming and .
146 N. N. Bakhtadze, V. E. Pyatetsky and E. A. Sakrutina
Forming the information model of kind (3), in particular, we obtain the associative
virtual model of the dynamics generator.
9 Wavelet approach application to the analysis of non-stationary
processes
Within the last two decades, applying the wavelet-transform to the analysis of non-
stationary processes has been widely used. The wavelet-transform of signals is a general-
ization of the spectral analysis, for instance, with regard to the Fourier transform.
First papers on the wavelet analysis of time (spatial) series with a pronounced heter-
ogeneity have appeared in the middle of 1980s [10]. The method was positioned as an
alternative to the Fourier transform, localizing the frequencies but not providing the time
extension of a process under study. In sequel, the theory of wavelets has appeared and is
developed, as well as its numerous applications.
Today the wavelet analysis is used for processing and synthesis of non-stationary
signals, solving problems of compressing and coding information, image processing, in
the theory and practice of the pattern recognition, in particular, in the medicine, and in
many other branches. The practice of applying wavelets was found effective for studying
geophysical fields, time meteorological series, forecasting earth quakes. The approach is
effective for research of functions and signals being non-stationary in the time are heter-
ogeneous in the space, when results of the analysis are to contain not only the frequency
signal characteristics (distribution of the signal power over frequency components), but
also information on local coordinates, at which certain groups of the frequency compo-
nents are manifested, or at which fast changes of the frequency components of the signal
are the case.
The wavelet analysis is based on applying a special linear transform of processes to
study real data interpreted by these processes, characterizing processes and physical
properties of real plants, in particular, technological processes. Such a linear transform is
implemented by use of special soliton-like functions (wavelets) forming an orthonormal
basis in L2.
The wavelet transform (WT) of a one-dimensional signal is its representation in the
form of the generalized Fourier series or Fourier integral over a system of basis func-
tions. The method is based on the fundamental conception of representation of arbitrary
functions on the basis of shifts and extensions of a one localized small wave, or the
wavelet function, that quickly decays in the direction to zero. The wavelet is formed in
such a manner that the function forming it (the wavelet forming function, or the mother
wavelet) is characterized by a certain scale (frequency) and localization in the time due
to the operations of the time shift and changing the time scale. The time scale is analo-
gous to the oscillation period, i.e. it is inverse one with regard to the frequency, and the
shift interprets the displacement of the signal over the tine axis.
The wavelet transform maps a one-dimensional process into a two-dimensional sur-
face in the three-dimensional space (the frequency and time are considered as independ-
ent variables). Thus, properties of the process are studied both in the time and frequency
domains, providing a possibility to investigate the dynamics of the frequency process
Predictive Model’s Development Based on the Wavelet Analysis Technique 147
make-up and its local particularities. This enables one to reveal coordinates at which
certain frequencies are manifested most considerably.
A graphical interpretation of the wavelet transform may be demonstrated by project-
ing onto the plane and emphasizing iso-lines characterizing changing intensities of coef-
ficients of the wavelet transform under different time scales, as well as to obtain disposi-
tion of local extrema of surfaces.
In comparison to the Fourier transform apparatus, when a function is used generating
the orthonormal basis of the space by use of the scale transform, the wavelet transform is
formed by use of a basis function localized in a bounded domain and belonging to the
space, i.e. to the all numerical axis. In comparison to the «window» Fourier transform, to
obtain the transformation on one frequency all time information is not already required.
The Fourier transform does not provide information on local properties of a signal under
fast enough changes in the time of its spectral make-up. Thus, the wavelet transform may
provide one with frequency-time information on a function, which in many practical
situations is more actual than information obtained by the standard Fourier analysis.
Wavelets also provide a powerful tool of approximation, which may be used for syn-
thesis of functions that are poorly approximated by other methods, with a minimal num-
ber of basis functions.
The wavelet transform, as a mathematical tool, serves, basically, to analyze data in
the time and frequency domains. The wavelet theory may be used for the system identi-
fication in different aspects. An investigation of the interaction of the identification theo-
ry and wavelet analysis was, to some extent, presented in [11]. It was pointed out that
wavelets are used, mainly, for the identification of non- linear systems with a particular
structure, where unknown time-varying coefficients may be represented as a linear com-
bination of basis wavelet-functions [12, 13]. Besides the direct wavelet analysis, for
system identification there may be used bi-orthogonal wavelets [14], wavelet-frames
[15], or even wavelet- networks [16].
There exist many different ways of applying wavelets for linear system identifica-
tion. Preisig [17] studied the identification of systems with a specific input/output struc-
ture, in which the parameters are identified via spline-wavelets and their derivatives. In
the paper of [18], an extended by use of an orthonormal transformation least squares
method is presented in order to revealing useful information from data.
In the present paper the analysis of possibilities of joint applying the wavelet theory
is presented.
10 Conditions of the associative model stability in the aspect of the
analysis of the spectrum of multi-scale wavelet expansion
Let a predicting associative model of a non-linear time-varying plant meet equation
(3). For the selected detail level for the current input vector , we obtain the multi-
scale expansion [9]:
(10)
148 N. N. Bakhtadze, V. E. Pyatetsky and E. A. Sakrutina
where: is the depth of the multi-scale expansion ( , where
and is the power of the set of states of the system in the System
Dynamics Knowledge Base); – are scaling functions; are the wavelet
functions that are obtained from the mother wavelets by the tension/combustion and shift
(as the mother wavelets, in the present case we consider the Haar wavelets); is the
level of data detailing; are the scaling coefficients, are the detailing coefficients.
The coefficients are calculated by use of the Mallat algorithm [10].
Let us expand equation (3) over wavelets:
(11)
In papers [8, 9] it was shown that a sufficient condition of the stability of plant (3)
(and, hence, also (11)) is as follows: for meeting the inequalities is to be pro-
vided:
if , then the condition for the detailing coefficients:
for the approximating coefficients:
if , then the condition for the detailing coefficients:
for the approximating coefficients:
if , then the condition of the stability for the detailing
coefficients:
Predictive Model’s Development Based on the Wavelet Analysis Technique 149
for the approximating coefficients:
if , then the condition of the stability for the detailing
coefficients:
for the approximating coefficients:
The investigation of the stability by use of the wavelet analysis may be demonstrated
by use of the example when model has the input and output memory depth
For this model, criteria of the stability for the approximating and detailing coefficients
will have the form:
,
In Fig. 2, 3, meeting the stability criterion is displayed for the approximating and de-
tailing coefficients of the sample for the prediction based on the associative search,
where is the depth of the expansion ( ), and is the quantity of vectors
selected to derive the approximating model by use of the associative search. The simula-
tion results show that for a number of vectors selected the stability criterion is not met
for the prediction. Thus, in these instants a non-stationarity is present that requires addi-
tional studying.
Investigating expansion coefficients in accordance to the wavelet stability criterion,
in entity, provides not only solving the inverse problem of the spectral analysis for the
non-linear operator , but also permits to select such control algorithm that it will keep
the system stability. Just in such a manner one should form the operator under solving
the problem of the synthesis of the variable structure systems.
11 Conclusions
For non-linear and time-varying dynamic systems, a methodology of the variable struc-
ture systems synthesis is proposed, based on deriving operator feed-backs and virtual
models of the plant dynamics. The predicting virtual models are based on the intelligent
data analysis of the dynamic dossier of the system status. On the basis of approach pro-
posed, it looks possible to synthesize systems meeting set dynamic properties. It is pre-
150 N. N. Bakhtadze, V. E. Pyatetsky and E. A. Sakrutina
dicted approaching to the stability bounds on the basis of investigating the dynamics of
the coefficients of the multi-scale wavelet analysis.
Fig. 2. Stability criterion for the approximating coefficients
Fig. 3. Stability criterion for the detailing coefficients
Predictive Model’s Development Based on the Wavelet Analysis Technique 151
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