3. Model

The search of a way to formalize maintenance management tasks and building models and algorithms for searching for optimizing strategies is useful to start with the formalization of the management process in an extremely general setting. The approach presented here is rather complicated, but it is useful for the development of further formalizations and construction of algorithms.

To do this, we will present the task of managing, using the following definitions. Let the considered technical object have in its arsenal several parameters, the variation of which affects the state of the mechanism, changing all the permissible modes of its operation. More precisely, the variation of the control parameters allows the mechanism to be switched from one operating mode to another physically acceptable mode. Next, consider some abstract mathematical space; often, these are certain subsets of a multidimensional space RN<sup>∗</sup> . Further constructions show that these subsets of space RN<sup>∗</sup> are topological manifolds with a complex topology.

Each point of such space determines the state of the mechanism at a fixed time; the sequence of states defines a trajectory in the state space. We also accept, as an empirically understandable assumption, that when the control parameters are varied, the continuous trajectories change in the same way without discontinuities. That is, a small perturbation of the parameters also causes a slight perturbation of the trajectory; in other words, for small perturbations the new trajectory is in some sense close to the original trajectory.

As a result, a continuous mapping from the parameter space to the state space is determined. Figure 1 demonstrates the mapping of the management loop Ω, consisting of two management parameters to the state space. It is assumed that all values of the parameters inside the circuit are physically realizable. When mapping the management interval I ¼ f g λ<sup>i</sup> ≝½ � 0; 1 , the path is formed from the initial to the final state in the state space. As a result, at the change of parameters and with changes in the state of the mechanism during operation, many paths are generated. The set of paths in the state space X defines a new space [9], designated as ΩX—the loop space of space X. In this case, the next parameter determines already the mapping of the management interval:

$$I \to \mathfrak{Q} \mathbf{X} \tag{1}$$

Ω2

Mapping the management loop X <sup>R</sup><sup>n</sup> � <sup>R</sup><sup>1</sup> in state space; <sup>λ</sup>1…λn; —Variable parameters, <sup>t</sup>-time.

Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies…

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

cisely to the k-fold loop space [9].

I

9

Figure 1.

Ω<sup>k</sup>X ≝ Ωk�<sup>1</sup>

Further presentation will require some information from algebraic topology, more precisely the homotopy theory. In view of the complexity of mathematical constructions, one must sacrifice a mathematically rigorous exposition in favor of simplification and clarity. Thus, the above arguments necessarily lead to an analysis of the set of paths ΩX in the state space X defined by the mapping Eq. (1) for one management parameter. With an increasing number of parameters, the path space is generated in the path space of the previous parameter Eq. (3), and so on with the growth of the number of control parameters. As a result, taking into account all control parameters leads to consideration of the k-fold space of paths, more pre-

Returning to the basic concepts of homotopy theory, it should be noted that the methods mentioned here were used in the 1970s in the physics of a condensed state for the analysis of singularities in condensed media, including superconductors (Abrikosov vortices), superfluid liquids, and liquid crystals. The methods of homotopy topology are effective not only for general analysis and classification of singularities of condensed media [10] but also transferred to the analysis of processes expressed in the form of multiple spaces of loops. This fact can be explained as follows. The management contour at the mapping to the state space defines a contour in the state space itself or on the corresponding loop space, the multiplicity of which is determined by the number of management parameters. The following

problem arises, solved by the homotopy theory methods. Can the image ∂I

on it the loop space be continued from the boundary of the set I

the mapping and defined by the contour in the state space or constructed Eq. (4, 5)

<sup>k</sup> in a continuous manner? Or such continuation is impossible, that means the presence of topological obstacles, expressed by the nontriviality of the topological (homotopy) type of the state space, the loop space. In the case of obstacles, any continuation will undergo a discontinuity in the corresponding topology of the loop space. In the case when the mapping F to the loop space is topologically nontrivial, that is, corresponds to a nontrivial element of the homotopy group of the state space or loop spaces, then a discontinuity will occur when the management parameters are varied. This means that it is not possible to continue the regularity from the

X ≝ Ω Ωð Þ X (2)

ð Þ ΩX (3)

k , ∂I k <sup>k</sup> under

, to its interior

thus defining a twofold loop space Ω<sup>2</sup> X in the space X. Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies… DOI: http://dx.doi.org/10.5772/intechopen.81996

Figure 1. Mapping the management loop X <sup>R</sup><sup>n</sup> � <sup>R</sup><sup>1</sup> in state space; <sup>λ</sup>1…λn; —Variable parameters, <sup>t</sup>-time.

$$\Omega^2 \mathbf{X} \overset{\text{def}}{=} \Omega(\Omega \mathbf{X}) \tag{2}$$

$$
\Omega^k \mathbf{X} \overset{\text{def}}{=} \Omega^{k-1} (\Omega \mathbf{X}) \tag{3}
$$

Further presentation will require some information from algebraic topology, more precisely the homotopy theory. In view of the complexity of mathematical constructions, one must sacrifice a mathematically rigorous exposition in favor of simplification and clarity. Thus, the above arguments necessarily lead to an analysis of the set of paths ΩX in the state space X defined by the mapping Eq. (1) for one management parameter. With an increasing number of parameters, the path space is generated in the path space of the previous parameter Eq. (3), and so on with the growth of the number of control parameters. As a result, taking into account all control parameters leads to consideration of the k-fold space of paths, more precisely to the k-fold loop space [9].

Returning to the basic concepts of homotopy theory, it should be noted that the methods mentioned here were used in the 1970s in the physics of a condensed state for the analysis of singularities in condensed media, including superconductors (Abrikosov vortices), superfluid liquids, and liquid crystals. The methods of homotopy topology are effective not only for general analysis and classification of singularities of condensed media [10] but also transferred to the analysis of processes expressed in the form of multiple spaces of loops. This fact can be explained as follows. The management contour at the mapping to the state space defines a contour in the state space itself or on the corresponding loop space, the multiplicity of which is determined by the number of management parameters. The following problem arises, solved by the homotopy theory methods. Can the image ∂I <sup>k</sup> under the mapping and defined by the contour in the state space or constructed Eq. (4, 5) on it the loop space be continued from the boundary of the set I k , ∂I k , to its interior I <sup>k</sup> in a continuous manner? Or such continuation is impossible, that means the presence of topological obstacles, expressed by the nontriviality of the topological (homotopy) type of the state space, the loop space. In the case of obstacles, any continuation will undergo a discontinuity in the corresponding topology of the loop space. In the case when the mapping F to the loop space is topologically nontrivial, that is, corresponds to a nontrivial element of the homotopy group of the state space or loop spaces, then a discontinuity will occur when the management parameters are varied. This means that it is not possible to continue the regularity from the

topology of the homogeneous space is changed; this change generates the boundary

Let us return to the tasks of management. In the concepts defined above, the task of management is formalized as follows in which variations of the controlled parameters preserve the trajectory of the states of the system in the given class for as long as possible. The following formulation concerns the estimates of RUL as an estimate of the time to reach the class boundary when the controlled parameters are varied. When crossing class boundaries, the task of evaluating the RUL and maximizing the time of stay in the class is solved again. In this case, the evolution

The model described above makes it possible to formalize the problem of finding optimal maintenance strategies as a task of determining control parameters or more precisely determining the range of admissible control parameters under which the

The search of a way to formalize maintenance management tasks and building models and algorithms for searching for optimizing strategies is useful to start with the formalization of the management process in an extremely general setting. The approach presented here is rather complicated, but it is useful for the development

To do this, we will present the task of managing, using the following definitions. Let the considered technical object have in its arsenal several parameters, the variation of which affects the state of the mechanism, changing all the permissible modes of its operation. More precisely, the variation of the control parameters allows the mechanism to be switched from one operating mode to another physically acceptable mode. Next, consider some abstract mathematical space; often, these are

Each point of such space determines the state of the mechanism at a fixed time; the sequence of states defines a trajectory in the state space. We also accept, as an empirically understandable assumption, that when the control parameters are varied, the continuous trajectories change in the same way without discontinuities. That is, a small perturbation of the parameters also causes a slight perturbation of the trajectory; in other words, for small perturbations the new trajectory is in some

As a result, a continuous mapping from the parameter space to the state space is

determined. Figure 1 demonstrates the mapping of the management loop Ω, consisting of two management parameters to the state space. It is assumed that all values of the parameters inside the circuit are physically realizable. When mapping the management interval I ¼ f g λ<sup>i</sup> ≝½ � 0; 1 , the path is formed from the initial to the final state in the state space. As a result, at the change of parameters and with changes in the state of the mechanism during operation, many paths are generated. The set of paths in the state space X defines a new space [9], designated as ΩX—the loop space of space X. In this case, the next parameter determines already the

. Further constructions show that

I ! ΩX (1)

X in the space X.

are topological manifolds with a complex topology.

trajectory is kept as long as possible in a given class.

of further formalizations and construction of algorithms.

certain subsets of a multidimensional space RN<sup>∗</sup>

these subsets of space RN<sup>∗</sup>

sense close to the original trajectory.

mapping of the management interval:

8

thus defining a twofold loop space Ω<sup>2</sup>

between classes.

Maintenance Management

equations change.

3. Model

boundary of the management loop to its interior without discontinuities. Physically, with small variations in the management parameters, a transition from the initial process to the final process will take place abruptly. This, depending on the specific physical content of the management model, leads to dramatic changes in the state of the mechanism that is accompanied by a sharp change in the operating conditions and extreme loads, leading to accelerated degradation of the material: the nucleation and growth of microcracks, the development of abnormal wear in the corresponding mechanical junction and other troubles, the precursor of creation of avalanche changes in the material, and so on.

$$F: \partial I^k \to X \tag{4}$$

Taking into account other symmetries existing in the observed process changes the degeneracy space. For example, the vector processes under consideration can in some cases have symmetry with respect to time reversal, and then the vector field becomes a field of directors as in a nematic liquid crystal. In this case, the degeneracy space is transformed from a sphere into a projective space of dimension:

Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies…

The presence of trends or dynamic predictors removes such degeneracy, and the degeneracy space again becomes a sphere. The permutation group acting on the components of the vectors, that is, changes their places, turns the sphere into an even more complex homogeneous space, where the gluing takes place in the discrete orbits of the group of permutations during factorization, generating a space

homotopically equivalent to a bouquet of spheres of different dimensions

DS <sup>¼</sup> <sup>S</sup><sup>N</sup>�<sup>1</sup>

The transition to space trajectories (spaces of k-fold loops) determines in the final analysis ultimately a classification of trajectories, representing each class from the set of admissible trajectories as a set of homotopy equivalent trajectories. The set of homotopy classes of such spaces is denoted as in [9]. This set has a structure of

Useful relations for computing homotopy groups of homogeneous spaces and loop spaces are given below, along with examples of homotopy groups of spheres

(Figure 2):

Figure 2.

11

group as follows from the given examples.

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

ΣW—cited superstructure over W;

A bouquet of two-dimensional and two one-dimensional spheres.

and other homogeneous spaces:

SO Nð Þ=SO Nð Þ� �<sup>1</sup> <sup>Z</sup><sup>2</sup> ffi PRN�<sup>1</sup> (8)

<sup>⋁</sup>f g<sup>i</sup> <sup>S</sup><sup>1</sup> (9)

½ �\$ W; ΩX ½ � ΣW;X (10) πið Þffi ΩX π<sup>i</sup>þ<sup>1</sup>ð Þ X (11)

<sup>π</sup><sup>n</sup>þ<sup>15</sup> Sn ð Þ¼ <sup>Z</sup><sup>480</sup> <sup>⊕</sup> <sup>Z</sup><sup>2</sup> (13) <sup>π</sup><sup>1</sup> SOn ð Þ¼ <sup>Z</sup><sup>2</sup> (14)

ΣW ≝ ð Þ ð Þ W � I =ðð Þ W � 0 ∪ ð Þ w\_0 � I ∪ ð Þ W � 1 (12)

$$I^k = \{\lambda\_i : i = 1, \ldots k\} \stackrel{\text{def}}{=} \prod\_k [0, 1] \tag{5}$$

The above results need more precise definitions of the state space X, the space of trajectories, and the identification of physical causes for the appearance of homotopically nontrivial state space:It is appropriate here again to use the analogy with topological defects of condensed media. It has already been noted above that when using analogies of this kind, it is only necessary to redefine the notion of a degeneracy space. The redefined degeneracy space in this case and thanks to the work [9] is nothing more than a k-fold loop space.

Topological singularities in condensed media are provided by the homotopy nontriviality of the so-called degeneracy space of the free energy functional of a condensed medium. The presence of the degeneracy group of the free energy and its further factorization with respect to the isotropy subgroup gives the required degeneration space, in mathematics called the homogeneous space [10]. In the task under consideration, the analog of the construction of the degeneracy space is in the most general case the characteristic functional of the stochastic process. The symmetry groups of such a functional are considered in [11]. To understand the methods of constructing degeneration spaces, one can consider the density of function of the distribution of the process. If we return to the cascades of the wavelet coefficients of the observed signal and then to the vector processes, then we consider the vector process or segments of length N or the set of such segments or vectors under certain assumptions about the properties of the observed process, for example, if the process reduces to a random walk in a multidimensional lattice or on a continuum. In the example under consideration, the group of probability density function (PDF) of the process has a Gaussian distribution, and hence the symmetry group of such a process is the group SO Nð Þ:

For example, in the problem of walk of R<sup>N</sup> [6, 12], the Gaussian function for the density of probability of falling into a point R∈R<sup>N</sup> after traversing the path of length L is the following:

$$G(R;L) = \left(\frac{N}{2\pi lL}\right)^{\frac{N}{2}} \exp\left(-\frac{N||R||^2}{2lL}\right). \tag{6}$$

The subgroup of isotropy is in this case the subgroup of rotations of the vector R about its axis, that is, SO Nð Þ � 1 . The result of the factorization SO Nð Þ of the group with respect to the subgroup SO Nð Þ � 1 is the N-1-dimensional sphere:

$$
\langle SO(N) \rangle\_{SO(N-1)} \cong \mathbb{S}^{N-1} \tag{7}
$$

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

Taking into account other symmetries existing in the observed process changes the degeneracy space. For example, the vector processes under consideration can in some cases have symmetry with respect to time reversal, and then the vector field becomes a field of directors as in a nematic liquid crystal. In this case, the degeneracy space is transformed from a sphere into a projective space of dimension:

$$\langle \text{SO}(N) \rangle\_{\text{SO}(N-1)\times Z\_2} \cong P \mathbf{R}^{N-1} \tag{8}$$

The presence of trends or dynamic predictors removes such degeneracy, and the degeneracy space again becomes a sphere. The permutation group acting on the components of the vectors, that is, changes their places, turns the sphere into an even more complex homogeneous space, where the gluing takes place in the discrete orbits of the group of permutations during factorization, generating a space homotopically equivalent to a bouquet of spheres of different dimensions (Figure 2):

$$\mathbf{DS} = \mathbf{S}^{\aleph - 1} \bigvee\_{\{\mathbf{i}\}} \mathbf{S}^{\mathbf{i}} \tag{9}$$

The transition to space trajectories (spaces of k-fold loops) determines in the final analysis ultimately a classification of trajectories, representing each class from the set of admissible trajectories as a set of homotopy equivalent trajectories. The set of homotopy classes of such spaces is denoted as in [9]. This set has a structure of group as follows from the given examples.

Useful relations for computing homotopy groups of homogeneous spaces and loop spaces are given below, along with examples of homotopy groups of spheres and other homogeneous spaces:

$$[W, \Omega X] \hookrightarrow [\Sigma W, X] \tag{10}$$

$$
\pi\_i(\Omega X) \cong \pi\_{i+1}(X) \tag{11}
$$

$$
\Sigma W \stackrel{\text{def}}{=}((W \times I)) / ((W \times \mathbf{0}) \cup (w \\_ \mathbf{0} \times I) \cup (W \times \mathbf{1}) \tag{12}
$$

ΣW—cited superstructure over W;

$$
\pi\_{n+15}(\mathbb{S}^n) = Z\_{480} \oplus Z\_2 \tag{13}
$$

$$
\pi\_1(\text{SO}^n) = Z\_2 \tag{14}
$$

Figure 2. A bouquet of two-dimensional and two one-dimensional spheres.

boundary of the management loop to its interior without discontinuities. Physically, with small variations in the management parameters, a transition from the initial process to the final process will take place abruptly. This, depending on the specific physical content of the management model, leads to dramatic changes in the state of the mechanism that is accompanied by a sharp change in the operating conditions and extreme loads, leading to accelerated degradation of the material: the nucleation and growth of microcracks, the development of abnormal wear in the

corresponding mechanical junction and other troubles, the precursor of creation of

<sup>k</sup> ! <sup>X</sup> (4)

½ � 0; 1 (5)

F : ∂I

trajectories, and the identification of physical causes for the appearance of

<sup>k</sup> <sup>¼</sup> <sup>λ</sup><sup>i</sup> f g : <sup>i</sup> <sup>¼</sup> <sup>1</sup>; …<sup>k</sup> <sup>≝</sup> <sup>Y</sup>

The above results need more precise definitions of the state space X, the space of

homotopically nontrivial state space:It is appropriate here again to use the analogy with topological defects of condensed media. It has already been noted above that when using analogies of this kind, it is only necessary to redefine the notion of a degeneracy space. The redefined degeneracy space in this case and thanks to the

Topological singularities in condensed media are provided by the homotopy nontriviality of the so-called degeneracy space of the free energy functional of a condensed medium. The presence of the degeneracy group of the free energy and its further factorization with respect to the isotropy subgroup gives the required degeneration space, in mathematics called the homogeneous space [10]. In the task under consideration, the analog of the construction of the degeneracy space is in the most general case the characteristic functional of the stochastic process. The symmetry groups of such a functional are considered in [11]. To understand the

methods of constructing degeneration spaces, one can consider the density of function of the distribution of the process. If we return to the cascades of the wavelet coefficients of the observed signal and then to the vector processes, then we consider the vector process or segments of length N or the set of such segments or vectors under certain assumptions about the properties of the observed process, for example, if the process reduces to a random walk in a multidimensional lattice or on a continuum. In the example under consideration, the group of probability density function (PDF) of the process has a Gaussian distribution, and hence the symmetry

For example, in the problem of walk of R<sup>N</sup> [6, 12], the Gaussian function for the

The subgroup of isotropy is in this case the subgroup of rotations of the vector R about its axis, that is, SO Nð Þ � 1 . The result of the factorization SO Nð Þ of the group with respect to the subgroup SO Nð Þ � 1 is the N-1-dimensional sphere:

exp � N Rk k<sup>2</sup>

2lL !

SO Nð Þ=SO Nð Þ �<sup>1</sup> ffi <sup>S</sup><sup>N</sup>�<sup>1</sup> (7)

: (6)

density of probability of falling into a point R∈R<sup>N</sup> after traversing the path of

2πlL � �<sup>N</sup> 2

G Rð Þ¼ ; <sup>L</sup> <sup>N</sup>

k

avalanche changes in the material, and so on.

Maintenance Management

I

work [9] is nothing more than a k-fold loop space.

group of such a process is the group SO Nð Þ:

length L is the following:

10

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

increasing the dimension of the management loop, that is, by increasing the number of management parameters. The operation of the restructuring of the degeneracy

Restructuring degeneracy space is associated with symmetry breaking. To explain all of the above, we can use a very simplified example. Consider the degeneration group SO(3) and the isotropy group SО(2) \* Z2. Moreover, the degeneracy space is a projective space RP2. Removing degeneracy by Z2 is associated with the violation of time-reversal symmetry. Such loss of symmetry is possible with the appearance of the trends of the initial cascade of wavelet coefficients of the

observed time series. In this case, the degeneracy space is transformed into a sphere. Homotopy groups of projective space and spheres differ and are represented by

The homotopy groups of k-fold loop spaces are represented by Eqs. (10) and (11). The nontriviality of homotopy groups generates a classification of paths, that is, their division into classes of homotopically nonequivalent paths in a management

When removing degeneracy and the transformation of the degeneration space itself into the space of another homotopy type, respectively, the homotopy classes of paths also change. In this case, new hidden predictors of failure will appear. An example is the predictors of turbine surging, described in [17]. It is noted that the early predictor of surging is destroyed by the mixing of wavelet coefficients. In this approach this means that in the observed process (the observed signal) from the pressure sensor the violation symmetry with respect to the permutation group

The above general model is based only on two statements that need concrete implementation for the further use of such model in prognosis and management tasks. Two said assumptions are as follows: there exists a space of states realized as a vector multidimensional space then taking into account the symmetry groups the set of states of the system, and the set of trajectories was represented in the form of degeneracy space and multiple path spaces that take into account the management

4. Review of the solutions of the prognosis and management task from

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

task with two or three management parameters.

abstract models to implementation

<sup>π</sup><sup>1</sup> <sup>S</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> (15) <sup>π</sup><sup>1</sup> RP<sup>2</sup> <sup>¼</sup> <sup>Z</sup><sup>2</sup> (16)

space is inevitable, therefore with each new restructuring task literally

Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies…

reformulated.

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

Eqs. (15) and (16):

occurred.

parameters.

13

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 density of the distribution function changes.

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

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 transition from one class to another change.

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

Figure 3. Interpretation of optimization problem and maintenance strategies.

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

increasing the dimension of the management loop, that is, by increasing the number of management parameters. The operation of the restructuring of the degeneracy space is inevitable, therefore with each new restructuring task literally reformulated.

Restructuring degeneracy space is associated with symmetry breaking. To explain all of the above, we can use a very simplified example. Consider the degeneration group SO(3) and the isotropy group SО(2) \* Z2. Moreover, the degeneracy space is a projective space RP2. Removing degeneracy by Z2 is associated with the violation of time-reversal symmetry. Such loss of symmetry is possible with the appearance of the trends of the initial cascade of wavelet coefficients of the observed time series. In this case, the degeneracy space is transformed into a sphere. Homotopy groups of projective space and spheres differ and are represented by Eqs. (15) and (16):

$$
\pi\_1(\mathbb{S}^2) = \mathbf{0} \tag{15}
$$

$$
\pi\_1(RP^2) = Z\_2 \tag{16}
$$

The homotopy groups of k-fold loop spaces are represented by Eqs. (10) and (11). The nontriviality of homotopy groups generates a classification of paths, that is, their division into classes of homotopically nonequivalent paths in a management task with two or three management parameters.

When removing degeneracy and the transformation of the degeneration space itself into the space of another homotopy type, respectively, the homotopy classes of paths also change. In this case, new hidden predictors of failure will appear. An example is the predictors of turbine surging, described in [17]. It is noted that the early predictor of surging is destroyed by the mixing of wavelet coefficients. In this approach this means that in the observed process (the observed signal) from the pressure sensor the violation symmetry with respect to the permutation group occurred.

The above general model is based only on two statements that need concrete implementation for the further use of such model in prognosis and management tasks. Two said assumptions are as follows: there exists a space of states realized as a vector multidimensional space then taking into account the symmetry groups the set of states of the system, and the set of trajectories was represented in the form of degeneracy space and multiple path spaces that take into account the management parameters.
