Chapter 2 - Methods and models of failure prediction

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2.1. Failure Detection Isolation Prediction (FDIP)

FDIP in technical systems has become more important with the advances in technology. The technical advances have increased system complexity, while reducing the success of maintenance crews. To combat this trend, requirements for new systems have had ever increasing Built In Test (BIT) requirements. 

Great strides have been made in the areas of self test and fault tolerance, while very little has been done in the area of prediction. 

BIT improvements have been made by adding hardware to the circuit designs which verify the integrity of the LRUs and interfaces. Examples of these are output-wrap monitors, parity checkers, and static tests of output vs feedback. 

Fault tolerance improvements have been made by use of redundancy. Several companies are also involved in research to take advantage of the cross-effects of multi-input-multi-output systems. For example, an aircraft can be turned without the use of rudders by using the ailerons and elevators. 

Parameter estimation is in use for controller design. IR&D efforts are currently investigating the use of estimation in fault detection applications. 

2.2 Methods of trend analysis and prediction

Trend analysis has been used extensively in such applications as the stock market, and weather prediction. However, the only method of trend analysis applied to control systems seems to be Kalman filtering. Box-Jenkins filters are very similar to the Kalman filter in that the prediction is based on the minimization of the error squared. The difference is that the Kalman filter uses the state transition matrix to project the estimates into future time, while the Box-Jenkins filter uses an iterating recursive approach. 

The approach used in this research is to use a RLSM estimator and then iterate to the desired lead using the Box-Jenkins concept. The reason for selection of the Box-Jenkins approach over the Kalman filter is the complexity of calculating the state transition matrix 15 for a system of an arbitrary order with drifting constants in the difference equation. 

2.3 Problem Definition

Analysis of the publications indicates that predictive diagnostics is an important issue for flight critical servo systems. Since it is a relatively new area, an opportunity exists to pioneer in this area. 

Redundancy and self test are used in flight control systems to increase flight safety. Since redundancy is very expensive, the goal of this thesis is to investigate the feasibility of an on-line fault prediction/detection applied to the servo actuator systems interfaced with a flight control system. After such a prediction system is installed, it can be tested to determine it's effectiveness. If the effectiveness is good enough to reduce the redundancy levels required, then future system cost reductions are possible. 

Interfaces are presumed known. Thus, isolation is assumed known with the fault detection. While it may be possible to isolate to subcomponents of an actuator3, the primary goal is to detect or predict failure of the servo system. Further isolation to components within the servo system is very difficult due to the multitude of parameters which can cause the same effect on system performance. 

To ease system modeling, the whole study has been done on a mathematic model. Since second order transfer functions are used most often to characterize servo performance, a second order model has been used for the plant. A third order servo system and the second order tendency principal are used to demonstrate this approach. 

The on-line test procedure is capable of tracking parameters to provide information for diagnostics. A diagnostic routine interprets the results using statistical trend analysis for predicting failures, status and trends. 

2.3.1 General System Architecture 

A computer controlled servomechanism is required for the predictive diagnostic system. 

Figure 2 shows the general system architecture assumed for the servo system where predictive diagnostics are to be applied. 

Figure 2, General System Architecture

The Electro-mechanical actuator or Control-Plant can be any type of electrical to mechanical converter. 

In order to perform the estimation of plant parameters, a system model must be defined. Trade-offs must be made between the order of the model and the extent to which isolation must be performed. Also, the model must be of sufficient order to approximate the plant performance. 

For most systems with inherent LRU isolation, a second order model is sufficient based on the second order tendency principal17

To demonstrate the validity of the second order tendency principal, a typical control-plant order system is compared to it's second order approximation. 


2.3.2 Typical Control-Plant System

Consider a dc-motor and gear train driven by a control computer, with a position feedback as a typical control plant. Figure 3 shows the block diagram of such a system18

Figure 3, Typical Control-Plant System

In Figure 3 the parameters are: 

G1 = Open Loop Gain, Kol,
G2 = Armature Circuit 1/(Ra(1+s*ta)), 

Ra = Armature Resistance, 
ta = Armature Constant = La/Ra, 

G3 = Armature Torque const Kt, 
G4 = Torque-Speed conv. 1/(F(1+s*tm)), 

F = viscous friction of motor and load, 
tm = J/F, 
J = inertia of load and armature Jm+Jl, 

G5 = Gear Ratio Kg=N:1, 
G6 = speed to position conversion 1/s, 
G7 = speed feedback constant, Kw, and 
G8 = position feedback, Kc. 

For the system of figure 3, the transfer function is third order. The transfer function is 

C(s) / CMD(s) = Kt*Kol*Kg/(tm*ta*Ra*F)  {*divided by:*}

s3 + (tm*ta / (tm*ta))s2 + (Ra*F+Kt*Kw /(tm*ta*Ra*F))*s + Kc*Kt*Kg*Kol/(tm*ta*Ra*F) 

A group of undergraduate engineering students at the State University of New York at Binghamton performed measurements on a Hitachi permanent magnet DC motor to determine it's characteristics3. The tests performed were 

1. Blocked Rotor Test - Ra,La,ta, 
2. Static Torque Test - Kt, 
3. No Load Test - F,Jm, 
4. Open Ckt Speed-V - Kw. 

The nominal values of the motor's parameters are 

Ra = 5.5 Ohms, 
La = .025 H, 
Kt = .25 V-Sec/Rad or N-m/Amp, 
Jm = 1.19E-4 Kg-m^2, 
F = 1.05E-3 N-m-Sec, 
Kw = 100 N-m/Amp / H, 
ta = 4.54E-3 H/Ohms, 
tm = .1133 Kg-m^2 / N-m-Sec 

Assume for the purpose of this exercise that the gear reduction ratio is at Kg = 1/100, and thus the load torque, and load inertia (TL(s), and Jl), are negligible. Also assume that the feedback gain Kc=1 is set to make the overall system gain unity. Also assume that the system performance requires a 1 second settling. Thus to design the servo robustly, Kol=5000, making tset = .78 sec. Therefore, the servo transfer function is 

C(s)/CMD(s) = 425E5/(s3 + 231*s2 + 85E5*s + 425E5). 

2.3.3 Second Order Tendency  

The procedure to establish the second order model of a higher order servo system is 

1. Determine the poles (eigenvalues), 

2. Define a second order transfer function with the same gain and the two lowest value poles. 

The eigenvalues of the servo are 

1 = -112 + j910 

2 = -112 - j910 

3 = -5. 

To form a second order approximation of this system notice that the dominant pole is 3, and that the other poles are a complex pair. Thus, the second order approximation is 

C(s)/CMD(s) = 5*112/((s+5)*(s+112)). 

Figure 4 shows the comparison of the step response between the real system and the second order system approximation. 

Figure 4, 3rd Order vs 2nd Order Step Responses

It is important to note here that by using a second order approximation, all hope of converting estimated parameter values into real plant parameters is lost. Since the control-plant is typically a Line Replaceable Unit (LRU) on an aircraft, the lack of isolation to internal servo components is typically not a problem. 
On systems where isolation to components within the servo is important, a higher than second order LSM and a pattern recognition technique is possible3

2.3.4 Simulation of Failure Development 

Failure development in a technical system can occur in many different ways. Components of the system can break, wear out, or have intermittent problems. 

The LSM estimation procedure can detect when the system breaks, but without a trend no prediction can be made. Similarly, intermittent faults can be detected by LSM parameter estimation but again no prediction is possible due to the lack of a trend in the parameters. 

Wear out failures can occur very slowly, or sometimes all at once. A wear out failure which occurs suddenly is considered the same as when a component breaks because the trend toward failure is too short to analyze. Components of the system which wear out and have a slow deterioration effect on system performance do exhibit a trend. Components such as bushings, hydraulic seals, bearings, and motors are all subject to wear-out. Of these, some may be more sudden than others. 

The method used to simulate failure development here is to use an input data generator routine to generate input X(1), and output Y data according to a second order transfer function equation which contains a dominant pole drifting with time. 

In order to apply predictive diagnostics to the servo system developed in section 2.3.2, a performance criteria is needed. Since the system was designed to have a step response with <1 second settling time based on the system requirements, the dominant pole can be monitored to insure tset remains below1

For the second order transfer function developed in section 2.3.3, the step response is 

y(t) = 1 - 1.046*e-5t + 0.046*e-112t

From this equation, the settling time is 

tset = ln(.02) / (-5) = .782. 

By rearranging the same equation to solve for the minimum pole value, the minimum pole value is 

p1min = ln(.02) / (tset=1) = 3.91. 

Therefore, to detect a failure in the modeled system, the parameters of the second order transfer functions are estimated by the LSM estimation routine. Then the value of the lower pole is calculated from parameter estimates by the LSM routine). Fault detection is achieved if the estimated pole goes below p1min minus the confidence interval. Likewise, fault prediction is achieved if the predicted minus the confidence interval pole goes below the same threshold. 

To see how the motor parameters effect the dominant pole, a routine has been used to plot the drift of the dominant pole vs the value of the servo constant. 

The torque constant should decrease as the field magnates get weaker with age. Figure 5 shows the relationship between the dominant pole and the torque constant Kt. This graph demonstrates that the magnets would have to loose nearly all of their magnetism to adversely effect the servo performance. Therefore, this seems to be an unlikely failure. 

The friction constant should increase as the armature bushings wear out, or the gear train is not properly lubricated. Figure 6 shows the relationship between the dominant pole and the motor friction constant F. This graph demonstrates that the relationship is nearly linear in region where the dominant pole would be declared failed. 

The armature circuit resistance should increase when the motor overheats, when the brushes wear out, or when the commutator becomes dirty (varnish build-up). Figure 7 shows the relationship between the dominant pole and the armature resistance Ra. This graph demonstrates a linear relationship between Ra and the dominant pole. 

Figure 5, Dominant Pole vs Torque Constant Kt

Figure 6, Dominant Pole vs Friction Constant F

Figure 7, Dominant Pole vs Armature Resistance, Ra


Chapter 3 - On-Line Parameter Estimation 

Abstract - Fault Prediction With Regression Models