A Soft Computing Approach for Fault Prediction of Electronic Systems

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1 A Soft Computing Approach for Fault Prediction of Electronic Systems Ajith Abraham School of Computing & Information Technology Monash University (Gippsland Campus), Victoria 3842, Australia Abstract This paper presents a soft computing approach for intelligent online performance monitoring of electronic circuits and systems. Reliability modeling of electronic circuits can be best performed by the stressor susceptibility interaction model. A circuit or a system is deemed to be failed once the stressor has exceeded the susceptibility limits. For online prediction, validated stressor vectors may be obtained by direct measurements or sensors, which after preprocessing and standardization is fed into the neuro-fuzzy model. For comparison purpose, we also trained a artificial neural network using backpropagation learning and evaluated the comparative performance. The performance of the proposed method of prediction is evaluated by comparing the experimental results with the actual failure model values. Test results reveal that neuro-fuzzy models outperformed neural network in terms of performance time and error achieved. Keywords: Soft computing, hybrid system, neuro-fuzzy, artificial neural network, stressor, susceptibility, Monte Carlo analysis, failure prediction.. Introduction Real time monitoring of the healthiness of complex electronic systems/ circuits is a difficult challenge to both human operators and expert systems. When the electronic circuit or system is controlling a critical task fault prediction will be very important. Soft computing was first proposed by Zadeh [4] to construct new generation computationally intelligent hybrid systems consisting of neural networks, fuzzy inference system, approximate reasoning and derivative free optimization techniques. It is well known that the intelligent systems, which can provide human like expertise such as domain knowledge, uncertain reasoning, and adaptation to a noisy and time varying environment, are important in tackling practical computing problems. In contrast with conventional Artificial Intelligence (AI) techniques which only deal with precision, certainty and rigor the guiding principle of hybrid systems is to exploit the tolerance for imprecision, uncertainty, low solution cost, robustness, partial truth to achieve tractability, and better rapport with reality. The new technique of electronic system failure prediction using stressor- susceptibility interaction is briefly discussed in Section 2 [][2], [5]. Section 3 presents practical methods for obtaining stressor sets. The derivation of stressor sets using Monte Carlo Analysis is given in Section 4 followed by Section 5 wherein we had derived a stressor-susceptibility model for a circuit. Section 6 and 7 gives some theoretical background about neuro-fuzzy systems and artificial neural networks. In section 8 we have reported the experimentation results and finally conclusions are provided in Section Stressor-Susceptibility Interaction Stressor is a physical entity influencing the lifetime of a component or circuit. A stressor, indicating a physical entity x will be denoted as ψ x. Stressors can be broadly classified into three main groups. First group contains the electrical stressors, parameters related to the electrical behavior of the circuit. Second group of stressors is the mechanical stressors, which are related to the mechanical environment of the component. Third group of parameters influencing the lifetime of components is related to the thermal environment of the component. Susceptibility of a component to a certain failure mechanism is defined as the probability function indicating the probability that a component will not remain operational for a certain time under a given combination of stressors. The susceptibility related to the failure mechanism y is usually defined as S y (t, ψ p, ψ q,ψ r ). Failure probabilities require detailed analysis of both stressors and susceptibility. Most components tend to have more than one failure mechanism, resulting in more than one failure probability. It can be shown that there is a strong correlation between the various failure mechanisms existing within a component. Figure. Stressor-Susceptibility interaction for single failure mechanism. Figure illustrates the stressor - susceptibility interaction for a single failure mechanism. It is clear that the main source of problem is the overlap between stressor and susceptibility density. The first step is to calculate the failure probability for this stressor distribution on a failure mechanism with a single, one variable, time independent catastrophic susceptibility model. This results in the following probability f fail, y, Ψ ( ) = f y ( Ψ) d Ψ To calculate the failure probability as a function of more complex susceptibility model, it will be necessary to calculate the failure probability of a part of the susceptibility model, for a certain stressor interval, characterized by its mean value ψ o and the corresponding susceptibility density function at that point S y (ψ o ). Considering the probability that a part has failed at a lower susceptibility level, result in the possibility to predict the failure probability per time interval of a certain failure at stressor level ψ 0 using the following relation ψ 0 f fail, y, Ψ ( ) = ( S Y ( ) f y ( Ψ) dψ) f fail, y ( ψ ) dψ 0 The last term is introduced to subtract failures caused by stressors at a lower susceptibility level. As, most often, failure probabilities are very small, in many cases the previous expression will simplify to f fail, y, ψ ( ψ0) = ( Sy( ) f y( Ψ) dψ) Ψ 0

2 ψ 0 when ( f fail, y ( ψ ) dψ ) =. 0 Since the susceptibility is defined as the probability that a component will not remain operational during a certain time, it is therefore possible to calculate the failure probability during a certain observation time t obs. f fail, y, ψ ( ψ 0 ) = t obs * * ( S y ( ) f y ( Ψ) dψ) The important requirement for using of the above equation is that the observation time t obs must be larger than the total elapsed sampling time to obtain an ergodic description of the associated stressors t total sample (t obs > t total sample ); f ail, y, ψ, (t, ψ) is assumed to be constant during the time interval t obs. From the previous equation it is possible to calculate the failure probability of a part per fail mechanism per time interval using the relation f fail, y = f fail, y, ψ ( t, ψ ) dψ 0 The above equation can now be used to calculate the part failure probability per time interval f = n fail f fail, i i = Using the previous assumptions it is also possible to calculate the probability that a component survives from time t to t+dt. The following expression can be used to calculate the failure probability for one single failure mechanism within one single device k Devices operational at time ( t + t) R( t... t+ t) = i = n Devices operational at ( t) i = = R( t) F( t) = R( t) tf ( t) As for large series of components, the physical structures of the individual components will be different for every component, the survival probability of such a series of components will also show individual differences. The stress on a component may vary with time due to circuit behavior and circuit use. The circuit behavior will differ amongst a series of circuits due to physical differences in the individual circuit components, the physical structure of a circuit, the use of a circuit and the environment (electrical, thermal, etc.) of the circuit. To summarize the variety of effects it is useful to describe stressors as stochastic signals with properties depending on the influencing factors mentioned above. These assumptions make it possible to derive the failure probability and reliability of a component using a Markov approach. For Markov approach the following requirements should be fulfilled Susceptibility of all failure mechanisms in a component is known and is constant in the time interval (t, t+ t). All stressors ψ a (t), ψ b (t), are known as stochastic signals for the time interval (t, t+ t). The failure probability (or reliability) is known at a certain (initial) time t. Using these properties it is possible to calculate the reliability and failure probability for components, derived from internal failure mechanisms for time t+ t. For this purpose the following relationships are used P(t+ t) = P(t) ( t) where P(t) is the state probability vector of a component. This state probability vector is defined as n P j (t) = j = P (t) = P operational (t) = R(t) P 2.n (t) = P fail, 2 n (t) = F fail, 2. n (t) P x y = P (y (t+ t) x (t) ) = f y (t) t P x (t) It is possible to replicate this calculation process for a whole batch of circuits. In this case for every circuit the individual stressor/ susceptibility interaction is calculated thus simulating batch behavior. Using this method it is possible to derive the failure probability for many parts in many practical situations, also in cases where considerable differences (in stressors and susceptibility) exist within a batch. 3. Modeling Stressor Sets and Susceptibility There are two different categories of failure mechanisms applicable to electronic components [3][0][][2]]. First, the failure mechanisms that are related to the electrical stress in a circuit. Second there are failure mechanisms related to the intrinsic aspects of a component. [4][5] There are two possible ways to obtain stressor sets for practical circuits. The First possibility involves usage of computer simulation models to derive all circuit signals using one single simulation. Second possibility is to derive stressor sets from practical measurements. In those cases where sufficient systems are available it is possible to do a statistical evaluation of the individual stressor functions existing in individual systems. As the stressor sets are dependent on the conditions of use and the operation modes of a system it is important that the measured stressor is based on all the possible

3 operation modes of a circuit and all the possible transitions between the various operation modes. This can become a quite tedious job as the entire operation is to be repeated for a number of systems to obtain an accurate statistical mean stressor model. Accurate description of a stressor set needs a sampling frequency of at least twice the highest frequency in the stressor frequency spectrum. Accurate description of a stressor set will require a number of samples sufficient to cover all the different states of the system. As a signal has often more than one quasi-stationary states, each characterized by their stressor set, it is possible to derive the overall stressor set function from the individual state stressor sets using: f n Ti str, y( x) = Σ f i = str, y, i( x) T total f str,y (x) is the stressor probability density function of quasistationary state i. T i / T total is the fraction of time that the stressor is in quasi-stationary state i. 4. Monte Carlo Analysis for Deriving Stressor Sets In a Monte Carlo analysis, a logical model of the system being analyzed is repeatedly evaluated, each run using different values of the distributed parameters []. The selection of parameter values is made randomly, but with probabilities governed by the relevant distribution functions. Statistical exploration covers the tolerance space by means of the generation of sets of random parameters within this tolerance space. Each set of random parameters represents one circuit. Multiple circuit simulations, each with a new set of random parameters, explore the tolerance space. Statistically the distribution of all random selections of one parameter represents the parameter distribution. Figure 2. Monte Carlo Analysis Although the number of simulations required for MCA is quite large, this analysis method is useful, especially because the number of parameters in the failure prediction of circuits is often too large to allow the use of other techniques. Figure 2 illustrates the MCA. With MCA it is possible to simulate the behavior of a large batch of circuits and derive stressor sets. The next phase will be the combination of the derived stressor sets with the component susceptibilities in order to decide whether a component will fail or not. As for the failure prediction, the most important aspect is to prevent failures; susceptibility will be expressed using the susceptibility limit. To distinguish circuits where failures are possible any circuit in the MCA causing to exceed a susceptibility limit are marked as fail. Circuits where no stressors exceed susceptibility limit are marked as pass. 5. Experimental Model - Stressor Sets and Susceptibility The analysis was carried out on a power circuit and the main cause of the failure of the circuit was a Schottky diode. Analysis shows that the main failure mechanisms are leakage current and excess crystal temperature. Using the procedure described earlier, it was possible to derive a complete individual stressor set for the failure mechanism of this diode. Figure 3 shows the joint stressor susceptibility interaction model in terms of voltage and current. The susceptibility limit for leakage current is set at -.5A. Figure 3. Stressor - susceptibility interaction model 6. Neuro Fuzzy Systems A Fuzzy Inference System (FIS) [9] can utilize human expertise by storing its essential components in rule base and database, and perform fuzzy reasoning to infer the overall output value. The derivation of if-then rules and corresponding membership functions depends heavily on the a priori knowledge about the system under consideration. However there is no systematic way to transform experiences of knowledge of human experts to the knowledge base of a FIS. There is also a need for adaptability or some learning algorithms to produce outputs within the required error rate. On the other hand, ANN learning mechanism does not rely on human expertise. Due to the homogenous structure of ANN, it is hard to extract structured knowledge from either the weights or the configuration of the ANN. The weights of the ANN represent the coefficients of the hyper-plane that partition the input space into two regions with different output values. If we can visualize this hyper-plane structure from the training data then the subsequent learning procedures in an ANN can be reduced. However, in reality, the a priori knowledge is usually obtained from human experts and it is most appropriate to express the knowledge as a set of fuzzy if-then rules and it is not possible to encode into an ANN. Table 3 summarizes the comparison of FIS and ANN. Black box Table. Complementary features of ANN and FIS ANN Learning from scratch Interpretable FIS Making use of linguistic knowledge To a large extent, the drawbacks pertaining to these two approaches seem complementary. Therefore it is natural to consider building an integrated system combining the concepts of FIS and ANN modeling. A common way to apply a learning algorithm to a FIS is to represent it in a special ANN like architecture. However the conventional ANN learning algorithms (gradient descent) cannot be applied directly to such a system as the functions used in the inference process are usually non

4 differentiable. This problem can be tackled by using differentiable functions in the inference system or by not using the standard neural learning algorithm. In our simulation, we used Adaptive Network Based Fuzzy Inference System (ANFIS) [7] as we found this to be the best neuro-fuzzy system available for function approximation problems [3]. ANFIS implements a Takagi Sugeno Kang (TSK) fuzzy inference system [8] in which the conclusion of a fuzzy rule is constituted by a weighted linear combination of the crisp inputs rather than a fuzzy set. For a first order TSK model, a common rule set with two fuzzy ifthen rules is represented as follows: Rule : If x is A and y is B, then f = p x + q y + r Rule 2: If x is A 2 and y is B 2, then f 2 = p 2 x + q 2 y + r 2 where x and y are linguistic variables and A, A 2,, B,, B 2 are corresponding fuzzy sets and p, q, r and p 2, q 2, r 2, are linear parameters. Figure 4. TSK type fuzzy inference system Figure 4 illustrates the TSK fuzzy inference system when 2 membership functions each are assigned to the 2 inputs (x and y). TSK fuzzy controller usually needs a smaller number of rules, because their output is already a linear function of the inputs rather than a constant fuzzy set. Figure 5. Architecture of the ANFIS Figure 5 depicts the 5 layered architecture of ANFIS and the functionality of each layer is as follows: Layer- Every node in this layer has a node function Oi = µ Ai ( x), for i =, 2 or Oi = µ ( y), for i=3,4,. B i 2 O i is the membership grade of a fuzzy set A ( = A, A 2, B or B 2 ) and it specifies the degree to which the given input x (or y) satisfies the quantifier A. Usually the node function can be any parameterized function.. A gaussian membership function is specified by two parameters c (membership function center) and σ.(membership function width) ( ) 2 x c σ guassian (x, c, σ) = e 2 Parameters in this layer are referred to premise parameters. Layer-2 Every node in this layer multiplies the incoming signals and sends the product out. Each node output represents the firing strength of a rule. 2 Oi = wi = µ A i ( x) µ B i ( y), i =,2.... In general any T-norm operators that perform fuzzy AND can be used as the node function in this layer. Layer-3 Every i-th node in this layer calculates the ratio of the i-th rule s firing strength to the sum of all rules firing strength. 3 w O = w = i i i, i =,2.... w + w2 Layer-4 Every node i in this layer is with a node function O 4 = wi fi = wi ( pix + qi y + ri ), where wi is the output of layer3, and { p i, q i, r i } is the parameter set. Parameters in this layer will be referred to as consequent parameters. Layer-5 The single node in this layer computes the overall output as the summation of all incoming signals: = = = iwi fi O 5 Overall output wi fi. i iwi ANFIS makes use of a mixture of back propagation to learn the premise parameters and least mean square estimation to determine the consequent parameters. A step in the learning procedure has two parts: In the first part the input patterns are propagated, and the optimal conclusion parameters are estimated by an iterative least mean square procedure, while the antecedent parameters (membership functions) are assumed to be fixed for the current cycle through the training set. In the second part the patterns are propagated again, and in this epoch, back propagation is used to modify the antecedent parameters, while the conclusion parameters remain fixed. This procedure is then iterated. 7. Artificial Neural Network Model Artificial Neural Networks (ANNs) [6] have been developed as generalizations of mathematical models of biological nervous systems. A neural network is characterized by the network architecture, the connection strength between pairs of neurons (weights), node properties, and updating rules. The updating or learning rules control weights and/or states of the processing elements (neurons). Normally, an objective function is defined that represents the complete status of the network, and its set of minima corresponds to different stable states of the network. It can learn by adapting its weights to changes in the surrounding environment, can handle imprecise information, and generalise from known tasks to unknown ones. Each neuron is an elementary processor with primitive operations, like summing the weighted inputs coming to it and then amplifying or thresholding the sum. There are three broad paradigms of learning: supervised, unsupervised (or self-organized) and reinforcement (a special case of supervised learning). We used a feedforward neural network using supervised backpropagation learning.

5 An important factor in the design of an ANN is that the designer has to specify the number of neurons, their distribution over several layers and interconnection between them. Besides a good choice of learning parameters and initial weights are required for training success and speed of the ANN. The following guideline will be of help as a step-by-step methodology for training a network. Choosing the number of neurons The number of hidden neurons affects how well the network is able to separate the data. A large number of hidden neurons will ensure the correct learning and the network is able to correctly predict the data it has been trained on, but its performance on new data, its ability to generalise, is compromised. With too few a hidden neurons, the network may be unable to learn the relationships amongst the data and the error will fail to fall below an acceptable level. Thus, selection of the number of hidden neurons is a crucial decision. Often a trial and error approach is taken starting with a modest number of hidden neurons and gradually increasing the number if the network fails to reduce its error. Choosing the initial weights The learning algorithm uses a steepest descent technique, which rolls straight downhill in weight space until the first valley is reached. This valley may not correspond to a zero error for the resulting network. This makes the choice of initial starting point in the multidimensional weight space critical. However, there are no recommended rules for this selection except trying several different starting weight values to see if the network results are improved. Choosing the learning rate Learning rate effectively controls the size of the step that is taken in multidimensional weight space when each weight is modified. If the selected learning rate is too large then the local minimum may be overstepped constantly, resulting in oscillations and slow convergence to the lower error state. If the learning rate is too low, the number of iterations required may be too large, resulting in slow performance. If the network fails to reduce its error, then reducing the learning rate may help improve convergence to a local minimum of the error surface. Figure 6. Test Result - Predicted Temperature using ANFIS Figure 7. Test Result - Predicted Temperature using ANN Figure 8. Test Result - Leakage current prediction using ANFIS Choosing the activation function The learning signal is a function of the error multiplied by the df gradient of the activation function. The larger the value of d(net) the gradient, the more weight the learning will receive. For training, the activation function must be monotonically increasing from left to right, differentiable and smooth. 8. Experimentation Results The experimental system consists of two stages: Network training and performance evaluation. From our circuit analysis, we modeled the stressor- susceptibility model as shown in Figure 3. The failure of the system occurs either due to rising crystal temperature or high leakage current. We attempted to predict the component temperature and leakage current for a given voltage and current. Data was generated by simulating circuit failure. All the training data were standardized before training. The input parameters considered are the Voltage (V) and Current (I). Predicted outputs are the junction temperature and leakage current. Figure 9. Test Result - Leakage current prediction using ANN ANFIS training In the ANFIS network, we used 4 Gaussian membership functions for each input parameter variable. Sixteen rules were learned based on the training data. ANFIS was trained using 80% of the data and the remaining 20% data was used for testing and validation. The training was terminated after 20 epochs.

6 ANN training We used a feedforward neural network with 2 hidden layers in parallel, 2 input neurons corresponding to the input variables and 3 output neurons. Each of the input neurons was directly connected to each of the neurons in the hidden layers as well as to each of the neurons in the output layer. The network was trained using 80% of the data and the remaining 30% data was used for testing and validation. Initial weights, learning rate and momentum used were 0.3, 0. and 0., respectively. The training was terminated after 3500 epochs. Performance and results achieved Prediction of reactive power demand: Figures 6 and 7 illustrate the test results for junction temperature prediction using ANFIS and ANN. Similarly, Figures 8 and 9 depicts the test results for leakage current prediction using ANFIS and ANN. Table 2 summarizes the comparative performance of ANFIS and ANN. Table 2. ANFIS and ANN comparative performance ANFIS ANN Learning epochs Training time 0 Sec 58 Sec Training Error (RMSE) Junction Temperature Leakage Current Testing Error (RMSE) Junction Temperature Leakage Current For network training, ANN took more time than ANFIS to achieve the error stated in Table 2. Surprisingly, the starting error after the first epoch in ANFIS was much lower than the final error of ANN. From the experimental results, it is clear that ANFIS outperformed ANN in terms of performance time and achieving lower error on test data. 9. Conclusions In this paper we attempted to predict the failures of electronic circuits and systems using neuro-fuzzy systems and artificial neural networks. The proposed neuro-fuzzy model outperformed artificial neural network in terms of performance time and prediction error. Compared to neural network, an important advantage of neuro-fuzzy system is its reasoning ability of any particular state. Moreover, the considered connectionist models are very robust, capable of handling the noisy and approximate data that are typical in circuits, and therefore should be more reliable during worst conditions. The problem modeling using stressor susceptibility interaction method can be widely applied to a wide range of electronic circuits or systems. However, it requires intense knowledge on the circuit behavior to model the various dependent input parameters to predict the results accurately. References [] Abraham A, Reliability Modeling and Optimization of Electronic circuits by Stressor Susceptibility Analysis, Master of Science (Computer Control & Automation) Thesis, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, July 998. [2] Abraham A and Nath B, Failure Prediction of Critical Electronic Systems in Power Plants Using Artificial Neural Networks, First International Power and Energy Conference, INTPEC99, Australia, November 999. [3] Anthony Chan H, A formulation of environmental stress testing and screening, Proceedings of the Annual Reliability and Maintainability Symposium (IEEE Reliability Society), pp 99-04, January 994. [4] Arsenault J E and J A Roberts, Reliability and maintainability of electronic systems, Computer Science press, Maryland, 980. [5] Brombacher A C, Reliability by Design, Wiley, 995 [6] Zurada J M, Introduction to Artificial Neural Systems, PWS Pub Co, 992. [7] Jang J S R, ANFIS: Adaptive Network based Fuzzy Inference Systems, IEEE Transactions Systems, Man & Cybernetics, 99. [8] Sugeno M, Industrial Applications of Fuzzy Control, Elsevier Science Pub Co., 985. [9] Cherkassky V, Fuzzy Inference Systems: A Critical Review, Computational Intelligence: Soft Computing and Fuzzy- Neuro Integration with Applications, Kayak O, Zadeh LA et al (Eds.), Springer, pp.77-97, 998. [0] Jenson F, Electronic Component Reliability, Wiley, 995. [] Klion J, Practical Electronic Reliability Engineering, VNR Edition, 992. [2] Fuqua N B, Reliability Engineering for Electronic Design, Marcel Dekker, 987. [3] Abraham A and Nath B, Optimal Design of Neuro-Fuzzy Systems for Intelligent Control, The Sixth International Conference on Control, Automation, Robotics and Vision, (ICARCV 2000), December 2000,Singapore. [4] Zadeh LA, Roles of Soft Computing and Fuzzy Logic in the Conception, Design and Deployment of Information/Intelligent Systems, Computational Intelligence: Soft Computing and Fuzzy-Neuro Integration with Applications, O Kaynak, LA Zadeh, B Turksen, IJ Rudas (Eds.), pp-9, 998.