4. Review of the solutions of the prognosis and management task from abstract models to implementation

The constructed model on the basis of the introduced assumptions using the concepts of homotopy topology gives a general classification of admissible trajectories, their evolution. The model demonstrates the complexity of the prognosis in view of the need to take into account symmetry breaking, in other words, the removal of degeneracy by one or more subgroups of process symmetries or cascade of wavelet coefficients. That is, the description of the topological transformation of the degeneracy spaces and the association with its spaces of k-fold paths allow one to look at the tasks of prognosis and management in a different interpretation. The model allows us to describe all admissible types of topological transformations, defines classes of admissible trajectories, determines all possible transformations of classes of trajectories and subsets, and characterizes subsets in the state space and

Part of the Hopf fibration is a mapping <sup>S</sup><sup>3</sup> ! <sup>S</sup><sup>2</sup> [13–15]. This mapping is the

This is under the assumption that the degeneracy space does not change. When the homotopy type of the degeneration space changes, the classification of trajectories also changes. The change in the topology of the degeneracy space is due to a change in the symmetry groups of the process. There is a violation of symmetry due to a change in the characteristic features of the process, as already noted, for example, with the appearance of trends. Predictor of the trend is, in fact, a change in the structure in the set of transition probabilities, as will be discussed below. Symmetry breaking or removal of degeneracy by isotropy subgroups can occur for various reasons. One such mechanism is associated with noise-induced transitions. In [16] examples of this kind are given. The reason for removing the degeneracy and, consequently, changing the topological type of the degeneracy space is the presence of multiplicative noise. As a result of the growth of the amplitude of such noise, a change occurs in the characteristics of the process, in particular, the

Returning to the tasks of management, the following should be noted. Thus, a space of degeneracy for the system and constructed on it k-fold loop space that is homotopy equivalent to the space of paths on the degeneracy space are defined sufficient roughly. The classification of paths is determined by the set of classes of homotopy equivalent paths. The transition from one class of paths to another class of paths is accompanied by symmetry breaking. Very conditionally the process of development of failure and dysfunctions of the mechanism can be shown in Figure 3. The colored concentric rings represent different types of homogeneous spaces on which it is necessary to keep the trajectory as long as possible. At the same

That is, the process of the development of faults as a result of operation passes from one class to another, reaching at the end of the failure field. In this case, the intersection of the conditional boundary is determined by a violation of the symmetry of the process. Further, the degeneration space itself and the character of the

The mathematical model described above allows us to make the first step in the formulation and formalization of optimization of the maintenance strategy. The optimization task is reduced to determining the number of management parameters and determining the image of the management loop in the state space or the k-fold space of paths that hold the trajectory of the process in a given homotopy class or in a given degeneracy space. If necessary, homotopic obstacles are overcome by

time, the time to reach the boundary of the RUL class is estimated.

generator of the homotopy group <sup>π</sup><sup>3</sup> <sup>S</sup><sup>2</sup> <sup>¼</sup> <sup>Z</sup>.

Maintenance Management

density of the distribution function changes.

transition from one class to another change.

Interpretation of optimization problem and maintenance strategies.

Figure 3.

12

trajectory spaces related to regions of failure. Further, the same methods describe the evolution of such regions, their interaction, and pair interaction on the basis of the group structure of homotopy classes. Involving physical models from the physics of failure makes it possible to determine the physical meaning of topological nontriviality and to connect topological obstacles and topological prohibitions with physical mechanisms that ensure topological transformations.

The next step to the construction of computational algorithms for prognosis and management is the transition from topological dynamics described above to the construction of the evolution equations of trajectories and states and finally to the construction of algorithms for prognosis and management. Moreover, the conclusions and results of the homotopy model must be taken into account with necessity.

To do this, it is necessary to determine the specific content of the above concepts such as the state space and path space. A detailed exposition of this construction is contained in the works [18–23]. The observed signal is represented as the coefficients of its wavelet transformation:

$$\left\{{}^{k}\_{Hist}W^{N}\_{i,j}\right\},N=1,2,3,...,N^{\*}\tag{17}$$

the nontriviality of the homotopy types of the degeneracy space. Evolution equations at the same time are complicated. And to obtain evolution equations, it is required to introduce three-point, four-point, etc. density of the distribution function for transition probabilities. In the present, brief review of approaches should be mentioned often; there are cases where the probability density function of the process or the transition probabilities are not Gaussian but have a so-called heavy

Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies…

P Rð Þ� ; <sup>L</sup> <sup>L</sup>

Thus, the approaches described above lead to the evolution equations with

This section is devoted to the description of a family of automata, analogs of Turing machines, allowing to formalize and, ultimately, take into account the above difficulties in the development of algorithms of reliable prognosis and numerical estimates of RUL, without which it is sometimes impossible to build and optimize the maintenance strategy and also implement the management task when choosing a class of trajectories that optimize the operation modes of technical objects. The transition from abstract models to the construction of recognizing and predictive automata occurs in two stages. By recognizing automata in this case, we mean a set of single-tape Turing machines. At the first stage, the state space described in the previous section and in more detail in [7, 22] is constructed. The second stage involves the construction of the symbolic space described in the work [18–21]. For this, the transition from the initial state to the final state on the state space is represented as the product of matrices of special form acting in the affine

1

0

n1 n2 : ni nk 1

0

BBBBBBBBBB@

n1 n2 :

1

CCCCCCCCCCA

(20)

ni þ 1 nk � 1

1

for ∀ ð Þ i; k (21)

CCCCCCCCCCA ¼

1

CCCCCCCCCA

BBBBBBBBBB@

: : 0 1

1 �1 0 1

The product of such matrices eventually transforms the initial state vector into

k k<sup>R</sup> <sup>α</sup>þ<sup>1</sup> : (19)

tail in its distribution, expressed as

DOI: http://dx.doi.org/10.5772/intechopen.81996

fractional derivative [25].

space:

Ωi,k

the final state:

15

n1 n2 : ni nk 1

0

BBBBBBBBB@

: : 0 0

: : 1 0

<sup>Ω</sup>N<sup>∗</sup> <sup>≝</sup> <sup>Y</sup> N∗

signal. The successive multiplication of matrices of elementary steps

1 Ωi, <sup>k</sup> !

The symbolic space in this case is a space P whose dimension is equal to the number of columns of the frequency histogram of the vectors of the initial or final. For definiteness, the dimension is increased by adding columns with zero values, thereby encompassing the physically permissible range of values of the observed

CCCCCCCCCCA ≝

0

BBBBBBBBBB@

1

5. Construction of automata

N—number of cycle; i, j—indices of wavelet decomposition.

Hist—duration of cycles (unevenness of stroke) of histogram column index. k—numbering of vectors from wavelet coefficients of dimension N∗:

This takes into account the fact that the mechanism under consideration is a reciprocating or rotational mechanism for all fixed indices except that N is determined by stochastic process with discrete time, N cascade. Further, fixing the limiting value N as N<sup>∗</sup> is determined by a set of vectors of dimension N<sup>∗</sup>, chosen from the consistent values of the process under consideration with discrete time. As a result, the space RN<sup>∗</sup> is determined, consisting of all possible finite segments of dimension размерности N<sup>∗</sup>.

The state space is defined as follows:

$$\{\mathbf{R}\_k\} \stackrel{\text{def}}{=} \left\{ {}^{k}\_{H \text{ist}} W^N\_{i,j} \colon k \ N^\* \le \mathbf{N} \le (k+1)N^\*, k = \mathbf{0}, 1, 2, 3... \right\}, \mathbf{R}\_k \in \mathbf{R}^{N^\*} \tag{18}$$

Thus, a vector space of dimension H is defined. The numerical value of H is not yet specified. It is determined in the process of preprocessing. The task of prognosis here reduces to determining the probability of transition from an initial vector to a finite vector in j steps. In this case, such task is solvable either by an explicit solution of the evolution equations or by calculating the moments, mainly of the dispersion, that is, second moment. Similar calculations are given in [6] and are reduced in most cases to the calculation of the Feynman integral along trajectories [6, 18–24] or to the solution of evolution equations such as the Fokker-Planck equation. In this case, the trajectory of states is represented as a walk along a multidimensional lattice or its continual analog, that is, RN space [6], in those cases when the observed process possesses certain properties, for example, the Chapman-Kolmogorov condition, the Markov property, stationarity and ergodicity are hold. The above properties of the process determine the evolution equations for the transition probability in the form of the Fokker-Planck equations already mentioned or Hamilton-Jacobi type equations, Schrödinger equations, and so on.

Topological prohibitions, implying the existence of such prohibition by physical mechanisms, determine other scenarios for the evolution of trajectories, that is, the probability of transition from one vector to another for a fixed number of steps. Moreover, in the interpretations of the process as a random walk on a lattice or continuum, processes are realized with allowance for the prohibitions imposed by

Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies… DOI: http://dx.doi.org/10.5772/intechopen.81996

the nontriviality of the homotopy types of the degeneracy space. Evolution equations at the same time are complicated. And to obtain evolution equations, it is required to introduce three-point, four-point, etc. density of the distribution function for transition probabilities. In the present, brief review of approaches should be mentioned often; there are cases where the probability density function of the process or the transition probabilities are not Gaussian but have a so-called heavy tail in its distribution, expressed as

$$P(R,L) \sim \frac{L}{||R||^{a+1}}.\tag{19}$$

Thus, the approaches described above lead to the evolution equations with fractional derivative [25].

### 5. Construction of automata

trajectory spaces related to regions of failure. Further, the same methods describe the evolution of such regions, their interaction, and pair interaction on the basis of the group structure of homotopy classes. Involving physical models from the physics of failure makes it possible to determine the physical meaning of topological nontriviality and to connect topological obstacles and topological prohibitions with

The next step to the construction of computational algorithms for prognosis and management is the transition from topological dynamics described above to the construction of the evolution equations of trajectories and states and finally to the construction of algorithms for prognosis and management. Moreover, the conclusions and results of the homotopy model must be taken into account with necessity. To do this, it is necessary to determine the specific content of the above concepts such as the state space and path space. A detailed exposition of this construction is contained in the works [18–23]. The observed signal is represented as the coeffi-

, N <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, …, N<sup>∗</sup> (17)

,Rk ∈ RN<sup>∗</sup>

(18)

physical mechanisms that ensure topological transformations.

k HistW<sup>N</sup> i,j n o

N—number of cycle; i, j—indices of wavelet decomposition.

Hist—duration of cycles (unevenness of stroke) of histogram column index. k—numbering of vectors from wavelet coefficients of dimension N∗:

This takes into account the fact that the mechanism under consideration is a reciprocating or rotational mechanism for all fixed indices except that N is determined by stochastic process with discrete time, N cascade. Further, fixing the limiting value N as N<sup>∗</sup> is determined by a set of vectors of dimension N<sup>∗</sup>, chosen from the consistent values of the process under consideration with discrete time. As a

> : k N<sup>∗</sup> <sup>≤</sup> <sup>N</sup> <sup>≤</sup>ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>N</sup>∗; <sup>k</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>; <sup>3</sup>… n o

Thus, a vector space of dimension H is defined. The numerical value of H is not yet specified. It is determined in the process of preprocessing. The task of prognosis here reduces to determining the probability of transition from an initial vector to a finite vector in j steps. In this case, such task is solvable either by an explicit solution of the evolution equations or by calculating the moments, mainly of the dispersion, that is, second moment. Similar calculations are given in [6] and are reduced in most cases to the calculation of the Feynman integral along trajectories [6, 18–24] or to the solution of evolution equations such as the Fokker-Planck equation. In this case, the trajectory of states is represented as a walk along a multidimensional lattice or its continual analog, that is, RN space [6], in those cases when the observed process possesses certain properties, for example, the Chapman-Kolmogorov condition, the Markov property, stationarity and ergodicity are hold. The above properties of the process determine the evolution equations for the transition probability in the form of the Fokker-Planck equations already mentioned or Hamilton-Jacobi

Topological prohibitions, implying the existence of such prohibition by physical mechanisms, determine other scenarios for the evolution of trajectories, that is, the probability of transition from one vector to another for a fixed number of steps. Moreover, in the interpretations of the process as a random walk on a lattice or continuum, processes are realized with allowance for the prohibitions imposed by

is determined, consisting of all possible finite segments of

cients of its wavelet transformation:

Maintenance Management

result, the space RN<sup>∗</sup>

f g Rk <sup>≝</sup> <sup>k</sup>

14

dimension размерности N<sup>∗</sup>.

The state space is defined as follows:

type equations, Schrödinger equations, and so on.

HistW<sup>N</sup> i,j

This section is devoted to the description of a family of automata, analogs of Turing machines, allowing to formalize and, ultimately, take into account the above difficulties in the development of algorithms of reliable prognosis and numerical estimates of RUL, without which it is sometimes impossible to build and optimize the maintenance strategy and also implement the management task when choosing a class of trajectories that optimize the operation modes of technical objects.

The transition from abstract models to the construction of recognizing and predictive automata occurs in two stages. By recognizing automata in this case, we mean a set of single-tape Turing machines. At the first stage, the state space described in the previous section and in more detail in [7, 22] is constructed. The second stage involves the construction of the symbolic space described in the work [18–21]. For this, the transition from the initial state to the final state on the state space is represented as the product of matrices of special form acting in the affine space:

$$
\Omega\_{i,k} \begin{pmatrix} n\_1 \\ n\_2 \\ \cdot \\ \cdot \\ n\_i \\ n\_k \\ \cdot \\ 1 \end{pmatrix} \overset{\scriptstyle{\underline{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\cdots}}}}}}}}}}}}}}}}}}}} \right)}} \end}} \right)}} \right)}} \right)}} \right)}}}}
$$

The product of such matrices eventually transforms the initial state vector into the final state:

$$\mathfrak{Q}\_{N^r} \stackrel{\text{def}}{=} \left( \prod\_{\mathbf{1}}^{N\*} \mathfrak{Q}\_{i,k} \right) \text{ for } \forall \, (i,k) \tag{21}$$

The symbolic space in this case is a space P whose dimension is equal to the number of columns of the frequency histogram of the vectors of the initial or final. For definiteness, the dimension is increased by adding columns with zero values, thereby encompassing the physically permissible range of values of the observed signal. The successive multiplication of matrices of elementary steps

(coordinatewise) with subsequent renormalization determines transition probabilities in the form of a matrix <sup>w</sup><sup>b</sup> k,m defined by the operator <sup>Ω</sup><sup>N</sup> .

The equation describing the evolution of certain states of the Turing machine in this way is a known balance equation [5]:

$$\frac{\partial p\_m}{\partial t} = \sum\_k \left[ \hat{w}\_{k,m} p\_k - \hat{w}\_{m,k} p\_m \right] \tag{22}$$

paths are in the same homotopy class of k-fold loops of the degeneracy space. The scenario described is valid for Gaussian processes with zero correlation length over the time variable. As noted in this case and taking into account the symmetry with respect to time reversal, the degeneracy space is an m-dimensional projective space. When time correlations of nonzero length appear, the degeneracy of the group Z<sup>2</sup> occurred. When a trend appears in the signal, degeneration by subgroups of the permutation group is removed. The admissible class of paths, on which the transition from one state vector to another takes place, is narrowed. Further dynamics of the internal states of the automaton is associated with the creation of new filling cells and the destruction of some old ones. At the same time, new ways of transition from one state to another are determined. And with the appearance of new ways, the estimate of the time to reach the class boundary or the field of the failure is changed. A complete reconstruction of the whole class of paths occurs when the homotopy type of the degeneracy space is reconstructed, for example, under noiseinduced transitions reminding the second-order phase transitions in condensed

Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies…

DOI: http://dx.doi.org/10.5772/intechopen.81996

7. Reproduction and birth of automata with increasing complexity

product of multipoint functions of lower orders.

Additional properties of the family of automata:

1. Automata must be pairwise interacting.

automata with increasing complexity.

automata should possess.

17

However, the constructed set of automata is still not enough for an effective prognosis. This deficiency is closely related to topological dynamics, in particular, to the already mentioned effects of excluded volume. In other words, taking into account the homotopy classes of paths, that is, in those important cases, when the processes under consideration are not Markovian, the Chapman-Kolmogorov identity is not satisfied. In the analytic approach, three-particle and then many-particle distribution functions are considered in such cases. Here, the system of equations for these functions can have infinite dimension. Another analog of such equations is the transition from evolution equations in PDF to equations in moments, where an infinite-dimensional system of equations also appears. Most often, such equations are solved by truncating in dimension, assuming that the finite-dimensional part of the system of differential equations approximates an infinite chain of equations in some sense. In the case when multiparticle distribution functions are introduced to obtain solutions, it is assumed that, beginning with a certain number, many-particle functions are assumed to be approximately Markovian, that is, are represented as a

The examples given represent some analogies for completing the construction of a set of automata. The family of constructed automata with the necessity for complete account of topological dynamics must be supplemented by some additional properties. The analogies described above demonstrate what properties a family of

2. In a number of cases, automata must analyze the described situations, that is, construct automata by merging one-tape automata, thereby passing to

A pair interaction between automata can be introduced in different ways, depending on the language describing these automata in terms of evolution equations or in constructing the Feynman path integral. In this case, the simple way of constructing interacting automata is to introduce interaction through a statistical

media.

or the basic kinetic equation [4].

The renormalization, which determines the transition from the frequency representation to the probabilistic one, also allows us to interpret the change of states as a walk in an N<sup>∗</sup>—dimensional simplex ΣN<sup>∗</sup> . Since each one-tap automaton processes only one cascade of wavelet coefficients, then for the entire set of cascades, the number of automata is equal to the number of wavelet coefficients of the signal in one period. Since in the general case there is a scatter of periods over lengths, the set of recognizing automata is determined for each interval in the histogram of the lengths of the periods, depending on the values of the lengths of the elementary intervals and the probable transition to packet wavelet decomposition. At a signal sampling frequency of 2 kHz, the number of automata is estimated from below by a number equal to 40 � <sup>10</sup><sup>3</sup> . If vibration signals are analyzed in a wide frequency range, then the number of automata is estimated by orders 10<sup>6</sup> .

### 6. Accounting for topological dynamics and how the automata work

Thus, the operation of an automaton reduces to change in its internal state when it shifts by one step in the input cascade of wavelet coefficients of the observed signal. It is assumed here that the equations describing the changes in the state of the automaton are independent of time, that is, the internal states of the automaton are stationary. In practice, small deviations from stationarity in the Levy metric are allowed [26]. The value of the permissible deviations is determined on the basis of the chronological database. Thus, the quasi-stationary nature is verified by checking the approximate fulfillment of the stationarity conditions of the basic kinetic equation Eq. (22):

$$
\hat{\boldsymbol{w}}\_{k,m} = \sum\_{m} \hat{\boldsymbol{w}}\_{m,k} \tag{23}
$$

If the quasi-stationary conditions are violated, for example, a trend appeared in one of the columns of the histogram or in several columns, the prognosis of the evolution of the internal states of the automaton is determined already by solving the nonstationary basic kinetic equation. Meanwhile, under the quasi-stationary conditions, the change in the internal states of the automaton is possible. As an example, we can mention the noise-induced transitions [16]. In this case, the change in the type of internal states is connected with a slow evolution of the coefficients of polynomial approximation of stationary solutions of the basic equation [3, 20]. Thus, in the transition from the initial state vector in the state space or in the symbol space introduced above, there are many transition paths in the formalism of birth-death process. However, all paths under quasi-stationary conditions reduce to permutations in the commutative subgroup of matrices of elementary transitions Ωi,k. That is, the transition from one vector to another takes place under the condition that the form of the symbolic histogram is stationary or that small deviations in the Levy metric are assumed. Moreover, the set of transition

Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies… DOI: http://dx.doi.org/10.5772/intechopen.81996

paths are in the same homotopy class of k-fold loops of the degeneracy space. The scenario described is valid for Gaussian processes with zero correlation length over the time variable. As noted in this case and taking into account the symmetry with respect to time reversal, the degeneracy space is an m-dimensional projective space. When time correlations of nonzero length appear, the degeneracy of the group Z<sup>2</sup> occurred. When a trend appears in the signal, degeneration by subgroups of the permutation group is removed. The admissible class of paths, on which the transition from one state vector to another takes place, is narrowed. Further dynamics of the internal states of the automaton is associated with the creation of new filling cells and the destruction of some old ones. At the same time, new ways of transition from one state to another are determined. And with the appearance of new ways, the estimate of the time to reach the class boundary or the field of the failure is changed. A complete reconstruction of the whole class of paths occurs when the homotopy type of the degeneracy space is reconstructed, for example, under noiseinduced transitions reminding the second-order phase transitions in condensed media.

### 7. Reproduction and birth of automata with increasing complexity

However, the constructed set of automata is still not enough for an effective prognosis. This deficiency is closely related to topological dynamics, in particular, to the already mentioned effects of excluded volume. In other words, taking into account the homotopy classes of paths, that is, in those important cases, when the processes under consideration are not Markovian, the Chapman-Kolmogorov identity is not satisfied. In the analytic approach, three-particle and then many-particle distribution functions are considered in such cases. Here, the system of equations for these functions can have infinite dimension. Another analog of such equations is the transition from evolution equations in PDF to equations in moments, where an infinite-dimensional system of equations also appears. Most often, such equations are solved by truncating in dimension, assuming that the finite-dimensional part of the system of differential equations approximates an infinite chain of equations in some sense. In the case when multiparticle distribution functions are introduced to obtain solutions, it is assumed that, beginning with a certain number, many-particle functions are assumed to be approximately Markovian, that is, are represented as a product of multipoint functions of lower orders.

The examples given represent some analogies for completing the construction of a set of automata. The family of constructed automata with the necessity for complete account of topological dynamics must be supplemented by some additional properties. The analogies described above demonstrate what properties a family of automata should possess.

Additional properties of the family of automata:


A pair interaction between automata can be introduced in different ways, depending on the language describing these automata in terms of evolution equations or in constructing the Feynman path integral. In this case, the simple way of constructing interacting automata is to introduce interaction through a statistical

(coordinatewise) with subsequent renormalization determines transition probabili-

The equation describing the evolution of certain states of the Turing machine in

The renormalization, which determines the transition from the frequency representation to the probabilistic one, also allows us to interpret the change of states as

only one cascade of wavelet coefficients, then for the entire set of cascades, the number of automata is equal to the number of wavelet coefficients of the signal in one period. Since in the general case there is a scatter of periods over lengths, the set of recognizing automata is determined for each interval in the histogram of the lengths of the periods, depending on the values of the lengths of the elementary intervals and the probable transition to packet wavelet decomposition. At a signal sampling frequency of 2 kHz, the number of automata is estimated from below by a

6. Accounting for topological dynamics and how the automata work

<sup>w</sup><sup>b</sup> k,m <sup>¼</sup> <sup>∑</sup> m

If the quasi-stationary conditions are violated, for example, a trend appeared in one of the columns of the histogram or in several columns, the prognosis of the evolution of the internal states of the automaton is determined already by solving the nonstationary basic kinetic equation. Meanwhile, under the quasi-stationary conditions, the change in the internal states of the automaton is possible. As an example, we can mention the noise-induced transitions [16]. In this case, the change in the type of internal states is connected with a slow evolution of the coefficients of polynomial approximation of stationary solutions of the basic equation [3, 20]. Thus, in the transition from the initial state vector in the state space or in the symbol space introduced above, there are many transition paths in the formalism of birth-death process. However, all paths under quasi-stationary conditions reduce to permutations in the commutative subgroup of matrices of elementary transitions Ωi,k. That is, the transition from one vector to another takes place under the condition that the form of the symbolic histogram is stationary or that small deviations in the Levy metric are assumed. Moreover, the set of transition

Thus, the operation of an automaton reduces to change in its internal state when it shifts by one step in the input cascade of wavelet coefficients of the observed signal. It is assumed here that the equations describing the changes in the state of the automaton are independent of time, that is, the internal states of the automaton are stationary. In practice, small deviations from stationarity in the Levy metric are allowed [26]. The value of the permissible deviations is determined on the basis of the chronological database. Thus, the quasi-stationary nature is verified by checking the approximate fulfillment of the stationarity conditions of the basic kinetic equa-

<sup>w</sup><sup>b</sup> k,mpk � <sup>w</sup><sup>b</sup> m,kpm

� � (22)

. If vibration signals are analyzed in a wide frequency

. Since each one-tap automaton processes

.

<sup>w</sup><sup>b</sup> m,k (23)

ties in the form of a matrix <sup>w</sup><sup>b</sup> k,m defined by the operator <sup>Ω</sup><sup>N</sup> .

<sup>∂</sup>pm <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∑</sup> k

range, then the number of automata is estimated by orders 10<sup>6</sup>

this way is a known balance equation [5]:

Maintenance Management

or the basic kinetic equation [4].

a walk in an N<sup>∗</sup>—dimensional simplex ΣN<sup>∗</sup>

number equal to 40 � <sup>10</sup><sup>3</sup>

tion Eq. (22):

16

interaction, taking into account this interaction from the first principles, when the measure of interaction is d P; P<sup>0</sup> ð Þ [11, 26]:

$$d(P, P') = \left\{ \int\_{\Omega} \left| \psi(\alpha) - \psi'(\alpha) \right|^2 dQ(\alpha) \right\}^{1/2} \tag{24}$$

where P, P<sup>0</sup> are two probability measures:

$$Q = \frac{1}{2}(P + P') \tag{25}$$

of the failure physics confirm this fact. Turning to concrete implementations of the family of prognostic automata, it should be noted that, in spite of the complexity of topological dynamics, specific algorithms prescribe fully realizable requirements for the costs of supporting their work in parallel architectures. In this case, as practice

In addition, in the optimization process, there are opportunities that allow some automata to be transferred to the computing power of onboard computers when it comes to, for example, monitoring of mobile objects, in particular, monitoring of all kind of transport. At the same time, onboard automata perform not only signaling functions but also are able to manage remote computing cores, thereby optimizing the computational processes on the remote computing cluster. And to the contrary, the automata of computing cluster can change the structure and functionality of peripheral automata, located on onboard computers or other computational capacities inherent in microcontrollers of onboard electronics. Returning to the complexity of topological dynamics, it should be noted that the automata for prognosis that support this complexity allow us to formalize and algorithmize the models of the root cause prognosis and, in the end, algorithmize the tasks for the intellectual self-

From the model constructed above, a certain hierarchy of prognostic problems also follows, since the set of physically acceptable trajectories is divided into the classes of homotopic equivalence. In turn, the classes themselves are changed during transformations of the space of degeneracy; other admissible trajectories and their classes appear that differ from the previous ones and have their own predictors and time estimates of the RUL. This means that with each transformation of the space of degeneration there is a change in the prognosis and changes in the RUL estimates. Thus, with RUL estimates, it is necessary to take into account not only the time to reach the class boundary but also the time to reach the moment when the transformation of degeneration space begins, for example, the time to reach the bifurcation set in the bifurcation tasks of the stationary solution of the Fokker-Planck equation. Another example is the time to reach a certain critical value of the amplitude of multiplicative noise. And each time the task of determining the RUL is updated. A good example of such an update is the calculation of the probability density of the transition from the original value to the preassigned one when determining the RUL in the representation of the probability density of the transition for a fixed time in the form of the Feynman path integral. At the same time, a change in the class of admissible transition trajectories, when the degeneracy by one of the isotropy groups or by its subgroups is removed, changes the evolution equation itself. In some cases, with the emergence of new topological obstacles, the Smoluchowski-Chapman-Kolmogorov identity does not hold the system and becomes non-Markov etc.; finally, an integro-differential equation appears as the Fokker-Planck evolution equation. In this case, all the previous predictors are changed, as well as all the time estimates. And so it happens with each new trans-

The family of interacting automata presented here changes traditional approaches to the learning of automata in recognizing early predictors of failure, in other words, in identifying the characteristics of the trajectories, the movement along which leads to the boundaries of the failure regions. In the examples described above and from the general model, it should be that learning is reduced to a set of segments of wavelet coefficient cascades as long as the automaton output to the quasi-stationary regime is not going to happen. For simple automata, the segment length is estimated at about 1000 full cycles of the engine operation or the number of revolutions of the turbine shaft. Further, the algorithm during monitoring verifies compliance with the conditions of quasi-stationarity. At the birth of more

has shown in the operation of automata, their structures are optimized.

Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies…

maintenance and self-recovery systems.

DOI: http://dx.doi.org/10.5772/intechopen.81996

formation of the space of degeneration.

19

$$
\psi = \left(\frac{dP}{dQ}\right)^{\mathbb{1}\_2} \psi = \left(\frac{dP'}{dQ}\right)^{\mathbb{1}\_2} \tag{26}
$$

is the Radon-Nicodym density.

There are more complex forms of interaction by introducing a potential or some vector field, as described in the work [6]. With the pair interaction between automata taken into account, it becomes possible to construct a graph whose automata are placed in 0-dimensional vertices. Since we are talking about automata on the wavelet cascades, the resulting graph allows us to analyze the situation with a root cause prognostics if the interaction is defined for automata with different time indices. Or, if automata interact with different scaling indices, then an analysis of multiscale processes or processes that occur at different scale levels and are interconnected becomes available.

### 8. Conclusions

The result of the completed constructions is the family of predictive automata. The set of automata is large, and depending on the sampling frequencies of incoming signals from sensors installed on the mechanisms, it is estimated from 10<sup>6</sup> to 10<sup>9</sup> . The automata themselves represent some analog of the Turing machine. In this case, the set of automata interacts in pairs. The interaction leads to the construction of more complicated automata and is an analogy of transitions to many-particle distribution functions or their densities. Thus, the family of interacting automata generates the next generation of automata with increased complexity. Many features of the operation of automata and methods for constructing state spaces and degeneration spaces remain outside the scope of this chapter. It is only necessary to note that as the state space it is considered a sequence nested with respect to the dimension of spaces, as is the sequence of path spaces whose multiplicity can tend to infinity. In such limiting cases, infinite-dimensional symmetry groups appear. And when implementing limit transitions, there are mathematical models that allow us to algorithmize the problem in some sense.

The complexity of family of automata is determined by the complexity of topological dynamics. If we are talking about the observed signals, then the account of symmetry groups, the appearance, and the removal of degeneracy in different subgroups are determined by the complexity of the signals themselves, reflecting in turn the complexity of physicochemical processes occurring in complex mechanical and electronic systems at various scale levels. Symmetry groups appear in all existing time series. Most often the groups of symmetries of incoming signals are caused by the concrete physical processes of the failure physics occurring in complex mechanisms in the presence of friction, gas hydrodynamics, physical and chemical processes, etc. In the overwhelming number of cases, the physical models

#### Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies… DOI: http://dx.doi.org/10.5772/intechopen.81996

of the failure physics confirm this fact. Turning to concrete implementations of the family of prognostic automata, it should be noted that, in spite of the complexity of topological dynamics, specific algorithms prescribe fully realizable requirements for the costs of supporting their work in parallel architectures. In this case, as practice has shown in the operation of automata, their structures are optimized.

In addition, in the optimization process, there are opportunities that allow some automata to be transferred to the computing power of onboard computers when it comes to, for example, monitoring of mobile objects, in particular, monitoring of all kind of transport. At the same time, onboard automata perform not only signaling functions but also are able to manage remote computing cores, thereby optimizing the computational processes on the remote computing cluster. And to the contrary, the automata of computing cluster can change the structure and functionality of peripheral automata, located on onboard computers or other computational capacities inherent in microcontrollers of onboard electronics. Returning to the complexity of topological dynamics, it should be noted that the automata for prognosis that support this complexity allow us to formalize and algorithmize the models of the root cause prognosis and, in the end, algorithmize the tasks for the intellectual selfmaintenance and self-recovery systems.

From the model constructed above, a certain hierarchy of prognostic problems also follows, since the set of physically acceptable trajectories is divided into the classes of homotopic equivalence. In turn, the classes themselves are changed during transformations of the space of degeneracy; other admissible trajectories and their classes appear that differ from the previous ones and have their own predictors and time estimates of the RUL. This means that with each transformation of the space of degeneration there is a change in the prognosis and changes in the RUL estimates. Thus, with RUL estimates, it is necessary to take into account not only the time to reach the class boundary but also the time to reach the moment when the transformation of degeneration space begins, for example, the time to reach the bifurcation set in the bifurcation tasks of the stationary solution of the Fokker-Planck equation. Another example is the time to reach a certain critical value of the amplitude of multiplicative noise. And each time the task of determining the RUL is updated. A good example of such an update is the calculation of the probability density of the transition from the original value to the preassigned one when determining the RUL in the representation of the probability density of the transition for a fixed time in the form of the Feynman path integral. At the same time, a change in the class of admissible transition trajectories, when the degeneracy by one of the isotropy groups or by its subgroups is removed, changes the evolution equation itself. In some cases, with the emergence of new topological obstacles, the Smoluchowski-Chapman-Kolmogorov identity does not hold the system and becomes non-Markov etc.; finally, an integro-differential equation appears as the Fokker-Planck evolution equation. In this case, all the previous predictors are changed, as well as all the time estimates. And so it happens with each new transformation of the space of degeneration.

The family of interacting automata presented here changes traditional approaches to the learning of automata in recognizing early predictors of failure, in other words, in identifying the characteristics of the trajectories, the movement along which leads to the boundaries of the failure regions. In the examples described above and from the general model, it should be that learning is reduced to a set of segments of wavelet coefficient cascades as long as the automaton output to the quasi-stationary regime is not going to happen. For simple automata, the segment length is estimated at about 1000 full cycles of the engine operation or the number of revolutions of the turbine shaft. Further, the algorithm during monitoring verifies compliance with the conditions of quasi-stationarity. At the birth of more

interaction, taking into account this interaction from the first principles, when the

ψ ωð Þ� <sup>ψ</sup><sup>0</sup> j j ð Þ <sup>ω</sup> <sup>2</sup>

� �1=<sup>2</sup>

<sup>ψ</sup> <sup>¼</sup> dP<sup>0</sup> dQ � �<sup>1</sup>=<sup>2</sup>

There are more complex forms of interaction by introducing a potential or some

The result of the completed constructions is the family of predictive automata. The set of automata is large, and depending on the sampling frequencies of incoming signals from sensors installed on the mechanisms, it is estimated from 10<sup>6</sup> to

. The automata themselves represent some analog of the Turing machine. In this case, the set of automata interacts in pairs. The interaction leads to the construction of more complicated automata and is an analogy of transitions to many-particle distribution functions or their densities. Thus, the family of interacting automata generates the next generation of automata with increased complexity. Many features of the operation of automata and methods for constructing state spaces and degeneration spaces remain outside the scope of this chapter. It is only necessary to note that as the state space it is considered a sequence nested with respect to the dimension of spaces, as is the sequence of path spaces whose multiplicity can tend to infinity. In such limiting cases, infinite-dimensional symmetry groups appear. And when implementing limit transitions, there are mathematical models that allow

The complexity of family of automata is determined by the complexity of topological dynamics. If we are talking about the observed signals, then the account of symmetry groups, the appearance, and the removal of degeneracy in different subgroups are determined by the complexity of the signals themselves, reflecting in turn the complexity of physicochemical processes occurring in complex mechanical and electronic systems at various scale levels. Symmetry groups appear in all existing time series. Most often the groups of symmetries of incoming signals are caused by the concrete physical processes of the failure physics occurring in complex mechanisms in the presence of friction, gas hydrodynamics, physical and chemical processes, etc. In the overwhelming number of cases, the physical models

dQð Þ ω

P þ P<sup>0</sup> ð Þ (25)

(24)

(26)

ð: Ω

<sup>ψ</sup> <sup>¼</sup> dP dQ � �1=<sup>2</sup>

<sup>Q</sup> <sup>¼</sup> <sup>1</sup> 2

vector field, as described in the work [6]. With the pair interaction between automata taken into account, it becomes possible to construct a graph whose automata are placed in 0-dimensional vertices. Since we are talking about automata on the wavelet cascades, the resulting graph allows us to analyze the situation with a root cause prognostics if the interaction is defined for automata with different time indices. Or, if automata interact with different scaling indices, then an analysis of multiscale processes or processes that occur at different scale levels and

measure of interaction is d P; P<sup>0</sup> ð Þ [11, 26]:

Maintenance Management

is the Radon-Nicodym density.

are interconnected becomes available.

us to algorithmize the problem in some sense.

8. Conclusions

10<sup>9</sup>

18

d P; P<sup>0</sup> ð Þ¼

where P, P<sup>0</sup> are two probability measures:

complex automata with an increase in their dimension, each added dimension increases the length of the segments.

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And in conclusion, it is necessary to say a few words about the set of early predictors of failure. In accordance with the hierarchical construction of the prognostic model, when removing the next degeneration by one of the isotropy subgroups in each new class of trajectories, their predictors are determined. That is, the inheritance of predictors in the transition of the trajectory from one class to another is not necessary. And this non-obligation is connected with the mapping of the gopotopic groups of the previous space of degeneration into the space of degeneration after its transformation. Part of the predictors may persist, another part may disappear, and new predictors may appear. The listed mutations of the set of predictors are determined by the specific structure of the degeneration space, that is, by the set of symmetry groups and isotropy groups. In this case, the trajectory itself or its characteristics, for example, configurational entropy, can act as a predictor, along with other types of the Kullback type of entropy.

Another type of predictor exists, and here again the analogy between the topological singularities of condensed media and the singularities of multiple loop spaces is appropriate. We are talking about the structure of the singularity core. Conducting the noted analogy, if the trajectory passes through the core of the singularity, then the effect of changing the permissible number of trajectories gives rise to changes, for example, of pointwise holder regularity of the trajectory.

In terms of the evolution of the internal states of a set of interacting automata, the above conclusions are expressed as additional conditions imposed on the densities of the distribution functions and transition probabilities for automata in any dimension.

This chapter is mainly devoted to the presentation of theoretical prognostic models and the basic ideas of constructing predictive automata. Demonstration of examples of the work of predictive automata and more detailed description of the predictors of the early prognosis will be continued in the next edition of IntechOpen book Prognostics edited by Prof. Fausto Pedro García Márquez.
