Abstract

The chapter describes a mathematical model of the early prognosis of the state of high-complexity mechanisms. Based on the model, systems of recognizing automata are constructed, which are a set of interacting modified Turing machines. The purposes of the recognizing automata system are to calculate the predictors of the sensor signals (such as vibration sensors) and predict the evolution of hidden predictors of dysfunction in the work of the mechanism, leading in the future to the development of faults of mechanism. Hidden predictors are determined from the analysis of the internal states of the recognizing automata obtained from wavelet decompositions of time series of sensor signals. The results obtained are the basis for optimizing the maintenance strategies. Such strategies are chosen from the classes of solutions to management problems. Models and algorithms for self-maintenance and self-recovery systems are discussed.

Keywords: turing machine, maintenance optimization, preventive maintenance, remaining useful life, remote calculating cluster

### 1. Introduction

This chapter describes a mathematical model that allows to unify the multiplicity of approaches to the creation of intelligent maintenance systems on the one hand and also allows more to accurately formalize and then algorithmize the optimization tasks of maintenance strategies.

Consideration of problems in the management of maintenance is useful to begin with the formalization of the basic tasks of PHM. The main trend of today is the development of prognostics and health management (PHM) in the sequence condition-based maintenance (CBM)-predictive maintenance (PdM) [1]. This concept may be called "CBM +" or "proactive management of materials degradation," and then on the horizon, there are new concepts, bearing a semantic load, in particular self-maintenance and self-recovery systems. However, all these concepts need further formalization (mathematical). It should also be noted that the effectiveness of any maintenance strategy depends on how reliably the PHM system is

**4**

*Maintenance Management*

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#### Maintenance Management

able to predict the state of technical objects of high complexity. Construction of the effective maintenance strategies is possible on the basis of a reliable prognosis.

We will assume that the system is equipped with all the necessary sensors, registering the vibration of the engine housing, the sensors of the angle of rotation of the shaft or crankshaft, pressure sensors in high-pressure fuel lines and other necessary sensors, most of which are included in the system of traditional onboard diagnostics or control systems. It also means the possibility of transmitting sensor signals (time

Remote Computing Cluster for the Optimization of Preventive Maintenance Strategies…

Further input to the computing cluster signals or time series is presented in the form of their wavelet coefficients. The fixation of all indices of the wavelet coefficients except for the quasi-period index, that is, the current number of the cycle of turbine engine, etc., determines the so-called cascades. The entire set of cascades is considered. Their number is equal to the number of wavelet coefficients of the decomposition of the time series multiplied by the number of sensors from which signals are received. A set of finite segments of fixed cascade defines a state vector in its sequence. The evolution of the state vector at successive change of segments determines the vector of trajectory. The multi-trajectory is determined by the vector of trajectories of all cascades, that is, a set of state vectors determines the

The first prognosis problem is reduced to the definition of the evolution equa-

Depending on the properties of the process, these equations are known in the sections of nonequilibrium thermodynamics called the "basic kinetic equation." The basic kinetic equation is reduced depending on the properties of the cascade (stationarity, ergodicity, nonstationarity, Markovity, non-Markovity, etc.) and reduces to equations such as the Fokker-Planck equation, the Schrodinger equation,

The prognosis task is formulated as a definition of the probability of a transition from the initial state vector to the final one and preassigned [6, 7], for example, preassigned on the boundary of the failure region, on the boundary of the region of the nucleation of failure predictor, or on the boundary of failure predictors. The development or evolution of predictors or hidden predictors is also described by evolutionary equations of the type listed above. Thus, there is a set of trajectories or multi-trajectories of sequences of states of the system. Further formalization requires the classification of trajectories in order to determine the trajectories leading to the boundaries of failure. The boundaries between classes of different trajectories may not be physically observable. However, these boundaries affect the trajectories, changing their characteristics. For example, in the case of interpreting

a trajectory as a random walk in a multidimensional lattice or its continual counterpart, the evolution equations themselves and, consequently, the RUL

Thus, there arises the problem of classifying a set of physically feasible trajectories or the task of representing trajectories in the form of a set of classes and the task of describing the boundaries between classes. Separation of the set of trajectories into classes is a rather ambiguous task, and often there are problems with changing the classification when changing the types of processes. However, it follows from the constructed model that the separation of trajectories into classes is related to the transformation of the topological characteristics of the state space and trajectory

In the case under consideration, each class is characterized by its own group of symmetries of the probability density of transitions between vectors and/or the group of symmetries of the generating functional. Factorization of the symmetry group by the isotropy subgroup, leaving the vector state in place, generates a homogeneous space [8]. It is in this space that the vector process wanders. In the process of operation of the mechanism and degradation of the material, the

series) to a remote computing cluster.

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

multistate or state of the entire system.

estimates change.

spaces.

7

tions describing the evolution of the state vector.

the balance equation, to single-step processes, etc. [4, 5].

Let us dwell in more detail on this question. Any prognostic system is based on the statistical processing of the signals received by sensors mounted on the technical objects. This can be a variety of vibration sensors, sensors of pressure, and measurement of currents and voltages. In this chapter, we will appeal to the examples of prognosis of the technical state of rotating machinery and the reciprocating action mechanisms, demonstrating the commonality of models.

Also, the chapter will pay attention to medical applications, in particular, to remote cardiac monitoring systems. Here, the task of the prognosis consists of three sub-tasks: a prognosis of the state of the heart on the basis of wearable or implanted in the body-miniaturized ECG recorders; construction of the management model of the heart state with the help of variable parameters of implantable devices in the human body, such as ICD and CRT devices; and prognosis and estimation of the remaining useful life (RUL) of the implanted devices [2, 3]. It should be noted that not only statistical methods are the basis of the prognosis, but also, more importantly, in this basis, physical models of the monitored object and its subsystems should be contained. Ultimately, we are talking about digital counterparts, that is, accounting for all components and processes occurring in a working device is necessary.

Thus, there is a task of prognosis of the technical state of the object and a time estimation under the general name remaining useful life (RUL). The prognosis of the technical condition and RUL estimation are the bases for constructing costeffective maintenance strategies.

It is on the basis of the prognosis and RUL estimates is possible formalization of the task of determining the cost-effective maintenance strategies, taking into account the conditions of goal setting. Goal setting involves taking into account the requirements for the technology and determining the ultimate goal of its operation. For example, in monitoring vehicles, extending the time of operation with a minimum change in operating parameters, for example, the engine and its subsystems, is a natural requirement that determines the strategy for calculating optimal operating conditions. Obviously, the calculated strategy is unsatisfactory for military applications, since it will explicitly prohibit operation with violation of optimal speed regimes and all kinds of extreme exploitation. When calculating maintenance strategies for military systems, the goal setting conditions change, where the determining strategy is the delivery of one's own weapons to a given point of space at any cost, taking into account the impact of enemy-striking factors.

The noted condition directly points to the fact that a maintenance strategy with necessity must be determined in a number of cases in real-time conditions, while the goal setting itself will change during operation. The transition to earlier prediction methods that can be called the diagnosis or prognosis of the root causes, hidden predictors of prognosis, etc. creates the conditions to search for more effective maintenance strategies. And, finally, the creation of self-maintenance and self-recovery systems requires the presence of a physical model of processes, within which functional dependencies between the parameters of management of process and its state are determined.

#### 2. Basis

For precise algorithmized formulations of optimization tasks for maintenance strategies, mathematically rigorous formalizations of the basic concepts of prediction tasks and tasks of management of the state of technical object are necessary.

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

We will assume that the system is equipped with all the necessary sensors, registering the vibration of the engine housing, the sensors of the angle of rotation of the shaft or crankshaft, pressure sensors in high-pressure fuel lines and other necessary sensors, most of which are included in the system of traditional onboard diagnostics or control systems. It also means the possibility of transmitting sensor signals (time series) to a remote computing cluster.

Further input to the computing cluster signals or time series is presented in the form of their wavelet coefficients. The fixation of all indices of the wavelet coefficients except for the quasi-period index, that is, the current number of the cycle of turbine engine, etc., determines the so-called cascades. The entire set of cascades is considered. Their number is equal to the number of wavelet coefficients of the decomposition of the time series multiplied by the number of sensors from which signals are received. A set of finite segments of fixed cascade defines a state vector in its sequence. The evolution of the state vector at successive change of segments determines the vector of trajectory. The multi-trajectory is determined by the vector of trajectories of all cascades, that is, a set of state vectors determines the multistate or state of the entire system.

The first prognosis problem is reduced to the definition of the evolution equations describing the evolution of the state vector.

Depending on the properties of the process, these equations are known in the sections of nonequilibrium thermodynamics called the "basic kinetic equation." The basic kinetic equation is reduced depending on the properties of the cascade (stationarity, ergodicity, nonstationarity, Markovity, non-Markovity, etc.) and reduces to equations such as the Fokker-Planck equation, the Schrodinger equation, the balance equation, to single-step processes, etc. [4, 5].

The prognosis task is formulated as a definition of the probability of a transition from the initial state vector to the final one and preassigned [6, 7], for example, preassigned on the boundary of the failure region, on the boundary of the region of the nucleation of failure predictor, or on the boundary of failure predictors. The development or evolution of predictors or hidden predictors is also described by evolutionary equations of the type listed above. Thus, there is a set of trajectories or multi-trajectories of sequences of states of the system. Further formalization requires the classification of trajectories in order to determine the trajectories leading to the boundaries of failure. The boundaries between classes of different trajectories may not be physically observable. However, these boundaries affect the trajectories, changing their characteristics. For example, in the case of interpreting a trajectory as a random walk in a multidimensional lattice or its continual counterpart, the evolution equations themselves and, consequently, the RUL estimates change.

Thus, there arises the problem of classifying a set of physically feasible trajectories or the task of representing trajectories in the form of a set of classes and the task of describing the boundaries between classes. Separation of the set of trajectories into classes is a rather ambiguous task, and often there are problems with changing the classification when changing the types of processes. However, it follows from the constructed model that the separation of trajectories into classes is related to the transformation of the topological characteristics of the state space and trajectory spaces.

In the case under consideration, each class is characterized by its own group of symmetries of the probability density of transitions between vectors and/or the group of symmetries of the generating functional. Factorization of the symmetry group by the isotropy subgroup, leaving the vector state in place, generates a homogeneous space [8]. It is in this space that the vector process wanders. In the process of operation of the mechanism and degradation of the material, the

able to predict the state of technical objects of high complexity. Construction of the effective maintenance strategies is possible on the basis of a reliable prognosis. Let us dwell in more detail on this question. Any prognostic system is based on the statistical processing of the signals received by sensors mounted on the technical

Also, the chapter will pay attention to medical applications, in particular, to remote cardiac monitoring systems. Here, the task of the prognosis consists of three sub-tasks: a prognosis of the state of the heart on the basis of wearable or implanted in the body-miniaturized ECG recorders; construction of the management model of the heart state with the help of variable parameters of implantable devices in the human body, such as ICD and CRT devices; and prognosis and estimation of the remaining useful life (RUL) of the implanted devices [2, 3]. It should be noted that not only statistical methods are the basis of the prognosis, but also, more importantly, in this basis, physical models of the monitored object and its

subsystems should be contained. Ultimately, we are talking about digital counterparts, that is, accounting for all components and processes occurring in a working

Thus, there is a task of prognosis of the technical state of the object and a time estimation under the general name remaining useful life (RUL). The prognosis of the technical condition and RUL estimation are the bases for constructing cost-

It is on the basis of the prognosis and RUL estimates is possible formalization of

For precise algorithmized formulations of optimization tasks for maintenance strategies, mathematically rigorous formalizations of the basic concepts of prediction tasks and tasks of management of the state of technical object are necessary.

the task of determining the cost-effective maintenance strategies, taking into account the conditions of goal setting. Goal setting involves taking into account the requirements for the technology and determining the ultimate goal of its operation. For example, in monitoring vehicles, extending the time of operation with a minimum change in operating parameters, for example, the engine and its subsystems, is a natural requirement that determines the strategy for calculating optimal operating conditions. Obviously, the calculated strategy is unsatisfactory for military applications, since it will explicitly prohibit operation with violation of optimal speed regimes and all kinds of extreme exploitation. When calculating maintenance strategies for military systems, the goal setting conditions change, where the determining strategy is the delivery of one's own weapons to a given point of space at any cost, taking into account the impact of enemy-striking factors. The noted condition directly points to the fact that a maintenance strategy with necessity must be determined in a number of cases in real-time conditions, while the goal setting itself will change during operation. The transition to earlier prediction methods that can be called the diagnosis or prognosis of the root causes, hidden predictors of prognosis, etc. creates the conditions to search for more effective maintenance strategies. And, finally, the creation of self-maintenance and self-recovery systems requires the presence of a physical model of processes, within which functional dependencies between the parameters of management of

objects. This can be a variety of vibration sensors, sensors of pressure, and measurement of currents and voltages. In this chapter, we will appeal to the examples of prognosis of the technical state of rotating machinery and the recipro-

cating action mechanisms, demonstrating the commonality of models.

device is necessary.

Maintenance Management

effective maintenance strategies.

process and its state are determined.

2. Basis

6

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

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 equations change.

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