Performance Reliability Monitoring and Prediction

성능신뢰도 모니터링 및 예측

Yon Soo Kim*

김연수*

Abstract

This paper presents the conceptual development of performance reliability, which utilizes on-line performance information. When physical performance is a measure that represents some degree of system, subsystem, component or device success in a continuous sense, Performance reliability represents the probability that physical performance will remain satisfactory over a finite period of time or usage cycles in the future.  If applicable sensing and monitoring means exist, physical performance can be monitored and predicted over time, along with explanatory variables or covariables. An empirical physical performance function is constructed to incorporate explanatory variables (operating and environmental conditions) over a time or usage dimension. This function enables one to model device performance and the associated classical reliability measures simultaneously, in the performance domain. The process requires a performance critical limit (which represents an appropriate definition of failure in terms of acceptable performance level) set at a fixed level. The conditional performance reliability structure developed represents a tool to help maximize system performance over time or usage.

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* Department of Industrial Engineering, University of Inchon

+ 이 논문은 1997학년도 인천대학교 교내 연구비 지원에 의해 연구되었음

1. Introduction

Reliability is considered an  integral part of performance. Reliability is classically thought of in a binomial sense (success/failure) and performance is thought of in a continuous sense (power output, thrust, torque and so forth). The classical approach to the prediction of reliability is based on numerical values of recorded system/component operating times to failure. Hence, traditional reliability is posterial, because it can be determined only after failure has occurred [3]. Our present day abilities to monitor, measure and model allow us the opportunity to link reliability more closely to the total system performance concept. Therefore, one can quite naturally relate concepts of reliability and classes of degraded performance, where a failure occurs when performance degrades below (or above) a specified value (threshold) [4]. The performance reliability concept provides a linkage between physical performance and reliability, which will ultimately provide better engineering insights into the design, development, manufacture, deployment and support of components, subsystems and systems. Furthermore, the conditional performance reliability structure enables us to envisage reliability as a conditional indicator and make reliability predictions (conditionally) once product usage is under way. The use of such on-line performance reliability measures can lead to a notable increase in the safety and availability of the equipment to which they are connected through continuous performance monitoring [6].

This paper describes the conceptual development of performance reliability and demonstrates the concept through monitoring and its prediction within computerized manufacturing environment.

2. Performance Reliability

2.1 Notation

t   :  time or cycles

[eqn: {m}_{t}]  :  the performance measure of central tendancy at any point in time t

[eqn: { m}_{0 }]  :  the initial value of product performance at the be ginning

[eqn: { m}_{f }]  :  some lower (or upper) limiting value considered to be a failure

[eqn: {u}_{{m }_{t }}] :  the mean value of[eqn: { m}_{t }] at time t

[eqn: { G}_{m}](t) :   the rate of approach of [eqn: { m}_{t }] to the  performance plane and the rate of descent/ascent below/above the performance plane once[eqn: { m}_{t }] passes through the performance plane

A :   the total area under the curve of [eqn: { G}_{m }](t)

[eqn: { X}_{1 }],...,[eqn: { X}_{n-1 }] :  multiple load variables

[eqn: { beta }_{0 }], [eqn: { beta }_{1 }],...,[eqn: { beta }_{n }]: regression coefficients

P.C.L. :  the performance critical limit

O.C.P. :  a vector of operating conditions

E.C.P. :  a vector of environmental conditions

T :  a random variable t hat represents time to failure

2.2 Background

Usually, the physical performance of a component, subsystem, system, device, or equipment can be measured, to some degree, with regard to effect iveness of product. Moreover, these meas urements represent a relative degree of success that the component, subsystem, system, device, or equipment is experiencing. The term "performance" used in conjunction with performance reliability, or product reliability is defined as the physical output of a product which is functioning or operating according to its designed purpose, specification, and intended function from the beginning of its deployment (into service) until withdrawal or retirement from service, whether it is used as a component, subsystem, or system [1]. These performance measures, at any point in time, differ mainly depending on usage level, operating conditions, environmental conditions, and product variation and deterioration.

These measurements represent quantitative information which is readily available at any time during the life ti me of a product, if a sensing mechanism is available. Performance data monitored and collected over time represent a pe rformance history for a product. It is desirable to measure and record explanatory variables along with performance information. This historic profile can be used to structure inferences about various reliability measures of a component, product, device, equipment, subsystem, or system.

The concept of performance reliability is based on a continuous scale of performance, including concomitant covariates.  The concept can be stated through a probability statement as

Pr(Performance > y □ operating         condition, t > x)          (1)

The conceptual notion of performance reliability is very appealing and tied directly to physical performance.

Figures 1, 2 and 3 depict conceptual performance curves. Figure 1 shows a simple performance "curve," actually a volume, between two curves defined by the worst and best performa nces observed across the time and environment dimensions. Figure 2 shows a single slice (taken out at one environm

ental level) out of the performance volume of Figure 1. This slice shows individual performance curves for each of 8 devices (e.g., batteries) tested or monitored. These curves show performance dropping off as a function of time for a given environment and at a given load (the slice taken).

A performance plane is drawn at a low performance level in Figure 2. The points at which these performance curves pass through this plane are equivalent to failure times (life times) that would be recorded in a classical reliability study. The classical practice has been to take these failure times and fit them to parametric models, such as a Weibull model, and then use this model as a reliability predictor. From a conceptual standpoint, the use of conte

mporary covariate models can extend t

his plane back across the environmental dimension. Thus, one can think of the current practice of reliability assessment and analysis as a subset of the concept

s shown in figures 1 and 2.

In order to add further detail to this hypothetical illustration, one can think about passing a plane in a vertical direction through a point on the time axis. This intersection is shown in Figure 3. In Figure 3, the vertical plane represents a censoring plane. Placement of the censoring plane at a preselected point in time would be known as a Type I or time censoring concept in classical reliability work. If this same plane were passed through the jth ordered failure time, it could be referred to as a Type II or failure censoring concept. In the physical performance domain, we are concerned with developing the conditional probability (reliability), as stated by equation (1), that the performance will be above the performance plane for the amount of time stated, under a given set of operating and environmental conditions.

2.3 Assumptions

We are interested in establishing models and data analysis techniques for tracking and assessing product performance over time under known load conditions and under known environmental conditions. Conceptually, this is done by extrapolating the performance parameter to predict the time to reach a critical performance plane (failure level). Such performance analysis will allow life projections and conditional reliability predictions to be calculated. These projections and predictions can serve as valuable information for decision making, as to future action, such as component replacement. In order to develop these mathematical or probabilistic relationsh-

ips, assumptions are necessary.  As sumptions underlying the performance models developed in this paper are described below [4]:

1. Performance measures may drift up or down, but must ultimate ly decrease or increase. That is per formance degradation is, in the long run, not reversible. As shown in Figure 4, typical performance curves gradually decrease or   increase depending on the quality characteristic. For example, battery  performance in terms of voltage moves downward over time when batteries     are discharged under a given load.

2. A model typically applies to a single failure process (mechanism or failure mode), where failure is defined by some threshold performance level which produces unsatisfactory results (a performance plane).

3. Degradation of product performance before service starts is negligible.

4. Performance can be measured with a limited amount of random error (the performance signal is detectable, even though noise in the measurement exists).

2.4 Performance Models and Reliability

In the physical performance domain, empirical mathematical relationships representing location as well as dispersion measures can be fit using regression or response surface modeling techniques [5]. Field or laboratory perfo

rmance data can be fit to performance curves using existing modeling technolo

gy and tools. Hence, a first-order gener

al performance model would be

[eqn: { m}_{t }]=[eqn: { beta }_{0 }]+[eqn: { beta }_{1 }][eqn: { X}_{1 }]+...+

[eqn: { beta }_{n-1 }][eqn: { X}_{n-1 }]+[eqn: { beta }_{n }t]          (2)

Provided physical performance can be measured, three basic possibilities exist: (1) higher is better, (2) lower is better, or (3) a nominal value is best. Figure 4 (a), (b), and (c) shows some typical cases of physical performance along with critical performance (failure) limits over time or usage. Figure 5 shows a conceptual relationship between a performance and a failure distribution. For practical purposes, the range of the possible failure time covers from the point where the upper 3 sigma band (assuming 3 sigma limits are used) meets the performance plane to the point where the lower 3 sigma band meets the performance plane. Here, sigma is an  estimate of the true standard deviation of the performance measure. The shaded area of the individual performance distribution at t in Figure 5 depicts the probability of failing at time t. The shaded area in Figure 5 at t is the probability of failure at or before t. The conditional probability of failure is changing as performance changes. A one-to-one mapping between performan-

ce and time exists.

It can be shown that reliability measures can be extracted from performance data. Let the event F represent a failure. Let P(F,t) denote the probability that the failure may occur at any point in time, t. Let f([eqn: {m}_{t }]) represent the probability distribution function of the performance measure,[eqn: { m}_{t }], at any point in time t. Then, for a higher is better case, P(F,t) is given by:

P(F,t)=Pr([eqn: { m}_{t }][eqn: <=][eqn: { m}_{f }])  (3)

where [eqn: { m}_{t }][eqn: >=] 0, 0 [eqn: <=][eqn: { m}_{f }],

and  [eqn: { m}_{t }]→ 0 as [eqn: t]→∞

Another measure associated with P(F,t) is the intensity (density) that the failure occurs at an instant of time t, defined as [eqn: { G}_{m }](t), where [eqn: { G}_{m }](t) is the rate of approach of [eqn: { m}_{t }] to the performance plane and the rate of descent below the performance plane once [eqn: { m}_{t }] passes through the performance plane:

P(F,t)=

[eqn: { G}_{m }](t)=  [eqn: INT _{ 0}^{ { m}_{f }}]f([eqn: { m}_{t }]) [eqn: d { m}_{t }]

if [eqn: { m}_{t }] > [eqn: { m}_{f }]   (4)

P(F,t)=

[eqn: { G}_{m }](t)= [eqn: INT _{{ m}_{f }}^{ ∞}]f([eqn: { m}_{t }]) [eqn: d { m}_{t }]

if [eqn: { m}_{t }][eqn: <=] [eqn: { m}_{f }]   (5)

Equation (4) represents the rate of approach to the performance plane when the mean performance of the device popu

lation has not yet reached the performance plane. Equation (5) represents the rate of ascent away from the performance plane after the mean performance passes through the critical plane ([eqn: { m}_{f }] ).

The total area under the curve of [eqn: {G}_{m }(t)] is given by:

A=[eqn: INT _{ 0}^{∞ }][eqn: { G}_{m }][eqn: (t)][eqn: dt]          (6)

After normalizing the area under the [eqn: {G}_{m }(t)] curve, a probability function is obtained. The failure time distribution can be developed as:

[eqn: f(t)]=[eqn: { { G}_{m }(t) } over { INT _{ 0}^{∞ } { G}_{m }(t)dt}]       (7)

Under the circumstance of monitoring at given time intervals, ti, f(t) is approximated by [eqn: f( {t }_{i } )]:

[eqn: { ∧} atop { f}(t)]=[eqn: f( {t }_{i } )]

=[eqn: { G}_{m } ( {t }_{i })]/[eqn: SUM from { { i}=1}to n][eqn: { G}_{m } ( {t }_{i })]              [eqn: i=1,2,...,n]         (8)

The cumulative distribution of failure time, F(t), is given by:

F(t)=[eqn: INT _{ 0}^{t } f(z)dz]

F(t)=[eqn: INT _{ 0}^{t }][eqn: { { G}_{m }(z) } over { INT _{ 0}^{∞ } { G}_{m }(z) dz}dz] (9)

or

F([eqn: { t}_{i }])=

[eqn: SUM from { { j}=1} to i]{[eqn: { G}_{m } ( {t }_{i })]/[eqn: SUM from { { j}=1} to n][eqn: { G}_{m }( {t }_{i } )]} (10)

Finally, the performance reliability is obtained by:

R(t) = 1 - F(t)

R(t)=1-[eqn: INT _{ 0}^{t }][eqn: { { G}_{m }(z) } over { INT _{ 0}^{∞ } { G}_{m }(z) dz}dz]

[eqn: i=1,2,...,n]      (11)

3. Performace Reliability         Monitoring and Prediction

Performance reliability utilizes perfor-

mance information which is very product specific. When an appropriate definition of failure, in terms of performance, is given, performance reliability is defined as "the conditional probability that the performance measures of a component, device, equipment, product, subsystem, or system is less (or greater) than a performance critical limit (which represents an appropriate definition of failure in terms of physical performance), given operating and environmental conditions, for a specified period of time or cycles." It can be summarized as  P(Performance>(<)P.C.L.□O.C.P=

([eqn: { O}_{1 },{ O}_{2 } ,..., { O}_{k }]),E.C.P=               ([eqn: { E}_{1 },{ E}_{2 },...,{ E}_{k }]),T>t)       (12)

Once an empirical performance functi-

on is fit from past or current performance data, the fitted performance function traces the central tendency measure of performance over time. The location information from the performance function represents the best available estimate for performance until significant new evidence is presented (from real-time monitoring) that suggests that performance is not on track. Dispersion may be assumed to be constant, [eqn: { sigma }^{ 2}], over time, but may also vary over time. If the dispersion is changing over time and/or with explanatory variables, a relationship function must be developed that can give the best available estimate of dispersion over time and explanatory variables.

As we can see in Figure 6, the conceptual performance function [eqn: { m}_{t }](t;  [eqn: { X}_{1 },][eqn: { X}_{2 } ,..., { X}_{k }]), which may contain explanatory variables, traces the mean performance at time t. At every instant of time, t, we assume that some distribution of the performance measure about the [eqn: { m}_{t }] value exists (see Figure 5). We expect that device performance will follow this distribution, until evidence to the contrary is presented. Furthermore from simple statistical process control, if the performance of an individual device is within [eqn: { u }_{ { m}_{t } }]±b[eqn: sigma], where b is a constant (usually 3), then the performance may be assumed to be on track. Otherwise, we may assume that the performance process has shifted and devise a new performance function [eqn: { m}`_{t } ^{' }] using the last few points as a trend projection for future expected performance.

The probability that the device will fail at time t, given the operating conditions, is represented by the shaded areas (relative to the [eqn: { m}_{t }] curve) in Figure 5. Quantitatively, this area is described by equations (4) and (5). Assuming that performance location and dispersion dimensions can be fitted and modeled, some parametric form for the f([eqn: { m}_{t }]) is useful in order to quantitatively develop failure probabilities. At this point in the development of the performance reliability model, we will as sume that the performance distribution function, f([eqn: { m}_{t }]), can be approximated by the normal distribution. Of course, this assumption would need to be verified empirically in any specific application. However, the previously developed concepts and relationships will hold regardless of the distributional form chosen.

Therefore,

Pr([eqn: { m}_{t }] [eqn: <=] [eqn: { m}_{f }]) = P(F,t)

=□{([eqn: { m}_{f }] -[eqn: { u}_{ { m}_{t }}])/[eqn: sigma])}

[eqn: t >=0]     (11)

The performance reliability function is given by:

R(t) = 1 - F(t)

=1-[eqn: INT _{ 0}^{t }][eqn: { { G}_{m }(t) } over { INT _{ 0}^{∞ } { G}_{m }(t)dt}dz] (12)

where

[eqn: { G}_{m }(t)] = □{([eqn: { m}_{f }] -[eqn: { u}_{ { m}_{t } }])/[eqn: sigma])}                 [eqn: { m}_{t }] > [eqn: { m}_{f }]

[eqn: { G}_{m }(t)] = 0.5

[eqn: { m}_{t }] = [eqn: { m}_{f }]

[eqn: { G}_{m }(t)] = □{([eqn: {- m}_{f }] -[eqn: { u}_{ { m}_{t }}])/[eqn: sigma])}

[eqn: { m}_{t }] < [eqn: { m}_{f }]

Figure 6 shows the empirical performance C.D.F. and reliability function for a performance function. The conditional reliability for a future □t period, given survival to time t, R(t+□t|t) is given by:

R(t+□t□t)=[eqn: { R(t+ TRIANGLE t □t)} over {R(t) }]

[eqn: { R(t+ TRIANGLE t □t)} over {R(t) }]=                              [eqn: {1- INT _{0}^{ t+□t} { Gm(z)} over { INT _{ 0}^{∞ }Gm(u) du} dz} over {1- INT _{0}^{ t} { Gm(z)} over { INT _{ 0}^{∞ }Gm(u) du} dz }]

where

[eqn: { G}_{m }(t)] = □{([eqn: { m}_{f }] -[eqn: { u}_{ { m}_{t } }])/[eqn: sigma])}

[eqn: { m}_{t }] > [eqn: { m}_{f }]

[eqn: { G}_{m }(t)] = 0.5

[eqn: { m}_{t }] = [eqn: { m}_{f }]

[eqn: { G}_{m }(t)] = □{([eqn: {- m}_{f }] -[eqn: { u}_{ { m}_{t }}])/[eqn: sigma])}

[eqn: { m}_{t }] < [eqn: { m}_{f }]

4. Conclusion

In the performance domain, if physical performance can be monitered over time, along with explanatory variables or covariables, it can provide some a measure that represents some degree of system, subsystem, component or device success in a continuous sense, as opposed to a classical binomial sense (success or failure). An empirical physical performance prediction function is constructed to incorporate the explanatory variables, operating, and environmental conditions over a time or usage dimension. This function enables one to model deviceperforman-

ce and the associated classical reliability measures simultaneously, in the performance domain, when a performance critical limit (which represents an appropriate definition of failure in terms of performance) is set at a fixed level, based on application requirements. With this kind of scheme, we could  predict and assess the probability that performance will remain satisfactory over a finite period of time or usage cycles in the future.

In the case of inadequate knowledge of the failure mechanics, this empirical modeling concept along with performance degradation knowledge can serve as an important analysis tool in reliability work in product and process improvement. This research provides an innovative linkage between actuarial and physical based reliability work. Results are expressed in a non parametric form, with minimal assumptions. The parametric case is demonstrated using the normal distribution to represent performance measurem rents. Traditional regression analysis and response surface techniques can be used to develop performance function models. The resulting failure density, cumulative failure density, reliability and failure hazard functions are empirical in nature and follow the time or usage dimension of the performance data.

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Figure 1: Performance Reliability Volume

Figure 2: Individual Performance Curves    for a Specified Environment, with a      Performance Plane Superimposed

Figure 3: Individual Performance Curves for a Specified Environment, with a Performance Plane and a Censoring Plane Superimposed

Figure 4: Typical Performance Curves

Figure5 : Relationship between Perform-

ance and FailureDistribution

Figure 6:        Performance Function, C.D.F., and Reliability Function