Overview of optimization methods in diagnostics of

pracujących w zakresie częstotliwości terahercowych. Nowoczesne terahercowe systemy antenowe zapewniają znacznie większe możliwości funkcjonalne niż pracujące w zakresie mikrofal czy w systemach optycznych. Ze względu na fakt, że w zakresie częstotliwości terahercowych znane techniki mikrofalowe lub optyczne nie sprawdzają się lub są bardzo drogie w realizacji, wymaga się zupełnie nowych metod – na przykład projektowania zintegrowanych struktur antenowych i szyków antenowych wielkiej skali, również dla technik MIMO, inteligentnych szyków antenowych, w tym o rekonfigurowalnej aperturze, zapewniających rozmaite scenariusze, anten wykorzystujących nowe materiały, w tym ferroelektryki i  grafen do opracowania przestrajalnych struktur antenowych. Zwiększenie funkcjonalności każdego systemu powoduje zwiększenie jego złożoności. Z tego powodu należy znaleźć kompromis między złożonością, ceną a możliwościami funkcjonalnymi. Wyniki ostatnich badań wskazują, że terahercowe systemy antenowe, ze względu na swoje możliwości, będą miały duży wpływ na charakterystyki przyszłych systemów bezprzewodowych. LITERATURA

[1] Yashchyshyn Y., Modelski J.: Radioelektronika terahercowa – oczekiwania, możliwości i ograniczenia. Przegląd telekomunikacyjny i Wiadomości Telekomunikacyjne, nr 4, 2015

[2] Ian F. Akyildz I.F, Josep Miquel Jornet J.M, Chong Han.: Terahertz band: Next frontier for wireless communications. Physical Communication, nr 12, 2014 [3] Wesołowski K.: Systemy bezprzewodowe piątej generacji – nowości i wyzwania. Przegląd Telekomunikacyjny i Wiadomości Telekomunikacyjne, nr 4, 2015 [4] Yashchyshyn Y., Marczewski J., Derzakowski K., Modelski J., Grabiec P.: Development and Investigation of an Antenna System With Reconfigurable Aperture, IEEE Trans. on Antennas and Propagation. vol. 57, no. 1, January 2009 [5] Wallace P. R. (1947): The band theory of graphite. Phys. Rev., 71 [6] Novoselov K. S., Geim A.K., Morozov S. V., Jiang D., Zhang Y., Dubonos S.V., Grigorieva I.V. and Firsov A.A.: Electric field effect in atomically thin carbon films. Science (2004) [7] Gusynin V. P., Sharapov S.G., Carbotte J. P.: Magneto-optical conductivity in graphene. Journal of Physics: Condensed Matter,19, 026222 (2007) [8] Attiya A.M.: Proposed Applications of Graphen for Millimeter-wave Passive Networks. In Proc. Of the 5th Intern. Conference on Nanotechnology: Fundamental and Applications. Prague, Czech republic, August 11–13, 2014 [9] Xu-Chen Wang, Wen-Sheng Zhao, Jun Hu, Tian Zhang: A Novel Tunable Antenna at THz Frequencies Using Graphen-Based Artifical Magnetic Conductor (AMC). Progress in Electromagnetics Research Letters, vol. 41, 2013

Adrian Bilski*, Piotr Bilski*, Jacek Wojciechowski* DOI: 10.15199/59.2015.6.5

Overview of optimization methods in diagnostics of analog systems Zastosowania metod optymalizacyjnych w diagnostyce systemów analogowych 1. Introduction applications of analog systems despite their unification with digital parts still remains important in various technical domains, such as military, acoustic or radio frequency (RF). The continuously growing the number of elements makes their testability (i.e. the ability to distinguish between particular fault sources) difficult to obtain. In case of high frequency and data acquisition systems, conducting the diagnostics of analog and digital parts separately is crucial in achieving satisfying work evaluation results. The testability of digital circuits have already been well defined, there are no such procedures prepared for the analog and mixed systems [1]. Identification of such systems is a key to decrease the production costs of modern electronics [2]. The analog systems, diagnostics is complicated by the tolerances of elements or noise (which must be treated during the signal processing operations prior to the fault detection or identification). The testability of mixed systems is included in the IEEE 1149.4 norm [3]. For analog systems, no such standards exist. The Artificial Intelligence (AI) approaches have been extensively used to monitor the state of analog systems during the last twenty years. Currently, sophisticated computer systems are * Instytut Radioelektroniki, Wydział Elektroniki i Technik Informacyjnych Politechniki Warszawskiej, e-mail: [email protected]

able to perform fault detection, location and identification in the real time. The advanced concepts of the AI include the classifier fusion [7] or the combination of supervised and unsupervised learning systems [23]. Contemporary approaches are heuristic and their parameters must be optimized to maximize the accuracy. Therefore one of the most pressing issues in the fault detection and location domain is the investigation and applications of optimization methods. The paper presents the overview of optimization algorithms and their applications to particular problems. The most widely used approaches are briefly described and their diagnostic implementations considered. The computational example demonstrates the usefulness of optimization in the selected application. The outline of the paper is as follows. In Section 2 the important definition symptoms and concepts for the diagnostics are introduced. Section 3 reviews challenges of the modern diagnostics. Overview and taxonomy of the optimization algorithms are in Section 4, while the categorization of their applications in diagnostics are provided by Section 5. The experimental example regarding the automated testing node selection is in Section 6. Section 7 contains conclusions and future prospects of these approaches.

2. Basic definitions The aim of the diagnostics is determining if the System Under Test (SUT) is working correctly. Otherwise the auxiliary procedures

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recognize the source of the incorrect SUT behavior (fault location) and its actual value (fault identification). These aims require applying AI methods to solve classification or regression tasks (i.e. finding the discrete information about the SUT state, or determining real values of its parameters). The information about the system’s behavior is acquired from its output signals (at accessible or partially accessible nodes). Therefore three problems must be solved for every system separately: selection of the optimal set of accessible nodes, determining symptoms which maximize the accuracy of the diagnostic system and selection of the excitation signal. To make a decision about the SUT state, it is analyzed in the time, frequency, DC or mixed domain [24]. It was proven [25] that the correct selection of characteristic signal features (symptoms) has the greatest impact on the testability. The computer methods of analog systems diagnostics have been developed for the last 40 years. The advancement in technology allowed for implementation of various numerical data analysis algorithms. Such methods are more sophisticated, designed to work with complex environmental conditions, such as noise. The generic architecture of the intelligent diagnostic system is presented in Fig. 1, where the algorithm is run on the microprocessor (often single-board computer) directly connected to the accessible nodes of the SUT. If the algorithm is to be working in the Real-Time, the duration of the conducted operations is important.

„„Fig. 1. Architecture of the on-line diagnostic system „„Rys. 1. Architektura systemu diagnostyki on-line

Diagnostic methods are divided into multiple groups, where the important feature is the knowledge representation, based on which the decision about the SUT state is made. The presented framework follows the Simulation Before Test (SBT) principle. In this approach (also described as the model-based diagnostics) the system is simulated before the actual diagnostic process takes place. This way data sets are obtained, which can further be processed by decision making methods to extract knowledge applied later for the on-line fault detection and location. The alternative approach (also called model-based diagnostics) focuses on the model simulations in parallel with the measurements of the actual system [4]. Optimization approaches are applicable to both methodologies. The subjects of the analysis in the presented research are linear time-invariant systems. Their analysis domains have the following characteristics: In the time domain the parameters are acquired from periodic or quasi-periodic waveforms. The symptoms may be the maximum or minimum signal values or the points of zero-crossings (Fig. 2). In the frequency domain the spectral analysis is performed first. The most popular methods acquiring frequency components are the Fast Fourier Transform (FFT) or the Total Harmonic Distortion (THD) [6]. In the mixed domain both time and frequency components are considered. The methods include the Short Time Fourier Transform (STFT) or the Wavelet Transform (WT) [7]. The analyzed system may be affected by catastrophic and parametric faults. The former change the structure of the SUT and include short or open circuits in the system. Although they are more dangerous for the system’s operation (as may lead

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„„Fig. 2. Example of a system’s response to the sine excitation. The characteristic points are denoted with circles „„Rys. 2. Przykład odpowiedzi systemu na pobudzenie sinusoidalne. Punkty charakterystyczne zaznaczono kółkami

to its complete destruction), they are relatively easy to detect. The latter group (considered in this paper) is more difficult to analyze and covers gradual changes in parameters’ values. The SUT performance changes with time because of wearing out the elements. This is the common situation related to the long term SUT application. The parametric fault detection is justified as the tool to predict the systems’ behavior in the future. Also, in some systems (such as audio amplifiers) it is economically practical to exchange the faulty element with its healthy counterparts. The parametric fault detection is supplemented with the performance-driven diagnostics, i.e. constant checking if the SUT responses are within the tolerance margins or not. The SUT operates within the margins: ymin (t) ≤ y(t) ≤ ymax (t),

(1)

even if its parameters are out of tolerances.

3. Modern diagnostic problems Although diagnostics is a well-developed branch of science, some problems still remain unsolved. They must be considered in the course of implementing novel fault detection and classification approaches. Element tolerances. Values of parameters in actual systems are different from their nominal counterparts due to the manufacturing process itself or the changes over time. If the parameter deviations from its nominal value are smaller than the tolerance values, it is assumed that the SUT behaves according to the design. Tolerances modify the output signals’ shape and decreas the accuracy of the characteristic points’ acquisition. Thus tolerances must be considered by the intelligent diagnostic module. The moment of leaving the tolerance region should be predicted and detected. Ambiguity groups. The existence of ambiguity groups depends on the amount of available information about the systems behavior. They are the sets of parameters indistinguishable from each other, based on the performed measurements. Two types of ambiguity groups exist. In the first one, changes in two or more parameters are visible in the output signal, but do not allow for distinguishing between them (e.g. parallel resistors). In this case fault detection is possible, but the identification and location abilities are limited [8]. In the second category changes of parameter values compensate each other. They are undetectable on the output, which makes the fault detection impossible.

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Presence of noise. The noise influences the waveforms measured at the accessible nodes and makes symptoms extraction difficult. The de-noising procedures should be applied before the features are acquired from signals. Various noise elimination approaches are used [3], including lowpass filters or the wavelet transform, facilitating signal observation in a frequency domain. Multiple faults. Another categorization refers to the number of simultaneously existing faults. A single faulty element changes the behavior of SUT and influences other elements, initiating a sequence of changes. It is crucial for the diagnostic process to quickly identify the fault source. Different faults may influence identical symptoms, which makes their detection difficult. There is no effective method to detect multiple parametric faults yet, but successful attempts to diagnose multiple catastrophic faults were made [9]. Only single faults are considered here. Complex systems analysis. The increasing complexity of analyzed systems makes the proper diagnostics more difficult: the occurrence of ambiguity groups is in this case more probable. The worst-case scenario is when the input-output nodes of the SUT are the only ones available for analysis. To solve such a diagnostic problem the utilization of subsequent nodes is necessary. Systems with multiple parameters can also be decomposed into a group of smaller subsystems that can be diagnosed independently. They are simpler for analysis, as the output node of one subsystem becomes the input of the next.

4. Optimization methods characteristics Optimization methods have multiple applications in diagnostics. The most important include searching for the optimal parameters of the heuristic methods (to maximize the classification or regression accuracy), finding the best subset of accessible or partially accessible nodes (at which signals are measured and features extracted), and determining the parameters of the excitation signals (such as the frequency of the sine or the slope of signal). Another is the group of optimization methods used in the SAT approach, where the model is optimized online to produce output signal as close to the actual system as possible. As each problem has individual characteristics (including discrete, continuous or global optimization), different approaches to are applied. The most widely used algorithms are in Fig. 3, including both traditional, numerical approaches (such as bisection or gradient methods) and modern heuristic algorithms (such as Particle Swarm Optimization – PSO or evolutionary methods).

faults, although rare, are also possible and should be investigated in the future. (2) The classification or regression method is trained and tested on them, so the decision making module quality can be measured. The optimized object (the heuristic method, configuration of nodes or the parameters of the excitation signal) is represented by the vector of attributes α (further called solution), which values must be adjusted. The aim of the optimization procedure is then to find the best vector α* minimizing the value of the evaluation function fe (where U is the domain containing all solutions):

α*

α

(3) The important aspect of the optimization procedure is the definition of this function, i.e. the tool for determining the quality of the solution. In the considered problems, its generic form is as follows: α

α

ß

(4) Here Es is the sample error, i.e. the percentage of incorrectly classified examples from V (5) in the classification task or the distance between the real values, expected c(ei) and produced by the diagnostic module h(ei) (6). Coefficient the α is the vector of optimized values (depending on the particular problem) and ß is the optional evaluation criterion, depending on the solved problem. For instance, in the optimal node subset selection, α is the binary vector indicating, which nodes should be analyzed, while α is the number of nodes included in the analysis. Both factors should be minimized. Weights w1 and w2 determine the influence of both on the overall value of the evaluation function. (5) (6) The optimization procedure consists in the repeated execution of the diagnostic module for various α vectors in the hope of the gradual improvement of evaluation function (3). Such “best" values of α are preferred, which provide the smallest possible classification error. From the of the usefulness perspective of each algorithm to the particular diagnostic problem is related to the following features: the number of solutions processed at the same time, the character of the optimization methodology parameters, the type of the optimization problem applicable for the method. The Numerical Methods, Simulated Annealing (SA) and Particle Swarm Optimization (PSO) algorithms can be utilized to solve continuous problems. On the other hand, Tabu Search (TS), Evolutionary Algorithm (EA) or Ant Colony (AC) are generally proper tools for solving discrete problems, like the accessible nodes configuration selection. What’s important, each method from Fig. 3 is heuristic; therefore its parameters may be optimized as well. The following are of the most important heuristic optimization algorithms. Simulated Annealing is the stochastic method applicable mainly for continuous problems. In every step it processes only two α vectors simultaneously: the actual αa and the new one αn, which is selected randomly from the neighborhood of αa. Depending on which solution provides better diagnostic quality, it is selected for the next iteration with the probability determined by the temperature and the difference between the quality of

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„„Fig. 3. Classification of AI optimization methods in analog systems diagnostics „„Rys. 3. Klasyfikacja metod optymalizacji z użyciem sztucznej inteligencji w diagnostyce analogowej

The latter are more popular in practical applications, therefore will be presented in detail. To adjust such methods to the specific diagnostic problem, the training L and validating V datasets are required. They include n examples representing the behavior of the SUT for the particular configuration of parameters. Each example ei contains m symptoms, supplemented with the information about the fault source ci (discrete for classification and continuous for regression). In both sets only single faults are considered, where one parameter is out of the tolerance region, while all other are considered nominal. This is the most frequent situation in the actual systems. Multiple

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compared solutions. SA was used for the parameter optimization of the SVM classification and regression in the analog filters and induction machines diagnostics [26]. The efficiency of such algorithm is comparable to GA and PSO. Its parameters include the initial and terminal temperature and the annealing scheme (i.e. the method of decreasing the temperature). Tabu Search is the discrete optimization method selecting the best new solution from the vicinity of the actual one αe. The admissibility is verified by checking if the solution was created using the operation not present in the list of forbidden moves (tabu list). This prevents the algorithms from going in circles and using the same solutions repeatedly. Parameters of the algorithm include the length of the short-term and long-term tabu list and the aspiration criterion. The method was used to select the best set of accessible nodes for the analysis of complex circuits (Section 6). It was also applied in data optimization of training set for the neuro-fuzzy inference systems, optimal node set selection in diagnostics of complex SUT, and multiple fault detection in distribution networks. Particle Swarm Optimization processes multiple solutions simultaneously (their number is the parameter of the algorithm) to find the best αa vector, based on its position and velocity. This method was extensively used for the parameter optimization of Artificial Neural Networks: Multilayered Perceptrons (MLP), Radial Basis Function (RBF) networks or Support Vector Machines (SVM) in the diagnostics of electrical machines [10] or hydraulic systems. Its computational efficiency is greater than SA, with lower chances of ending in the local optimum. Evolutionary Algorithms processes multiple sets of solutions (their number is the parameter of the method) called populations at the same time, which makes it computationally demanding. The method consists in generating new populations from the previous one. The structure of the new population depends on the quality of constituent solutions (the better ones have greater chances to be copied to the new population). The generation of new solutions is possible through the genetic operations, such as cross-over and mutation. They ensure high variability in population: EA is insensitive to ending in the local minima. This method was used in parameter selection of the excitation signal of the analyzed SUT [11]. Ant Colony Optimization is based on constructing a pseudograph which vertices are discretized variable vectors, values. Each agent (ant) constructs its path by a random move from the variable value to another. Initially, each ant k randomly chooses a path and forms a directed graph while randomly leaving fraction of the pheromone at the visited graph edges. In each iteration the path giving the minimum value of the objective function sees its rate increase. ACO was successfully used for the optimization of digital circuits [12] and its application to the analog design field was recently proposed [13]. In [14] the ACO technique was applied for the optimal sizing of two CMOS analog circuits: a differential pair current conveyor and an inverted current conveyor.

5. Applications of optimization methods in diagnostics This section presents the most popular applications of the optimization approaches to diagnostic problems.

The search for optimal parameters of the classification or regression module

Because of the heuristic nature of most diagnostic algorithms, their particular parameters need to be adjusted to get the best fault classification/identification efficiency. The optimization depends on the particular diagnostic method. The procedure requires the single generation of the training and validating sets, which are repeatedly used to train and verify efficiency of the classifier or

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regression module. The scheme is illustrated in Fig. 4. The evaluation function has the form (4) and no coefficient ß is required. The solution vector α is the set of real values (usually one or two). Because most of parameters are real-valued (such as of the SVM kernel or the ANN activation function), the continuous optimization approaches are used here. The most widely used is the SVM classifier, for which SA, PSO and EA were applied [27].

„„Fig. 4. Heuristic optimization scheme in diagnostics „„Rys. 4. Schemat optymalizacji heurystycznej w diagnostyce

The search for optimal excitation signal parameters The optimal selection of excitation signal (e.g. sinusoid, step function or sinc) is of tremendous importance. It justifies the usefulness of symptoms extracted from signals. For instance, the step function was used to determine dynamic features of servomechanisms [28]. On the other hand, the sinc function may be used universally, as its spectrum is uniform in the whole operating frequency range. Each signal can be parameterized. For instance the sinusoid frequency is crucial for determining the change in the filter passband. The abruptness of moving from the one signal level to an other may also be of importance (e.g. in ramp signal) [29]. The optimization of the excitation signal for the analog system requires constructing training and validating data sets for each particular excitation. The evaluation function is the simplified version of (4), where the coefficient ß=0, as there are no additional requirements for the optimized parameters. The excitation signal selection was also considered in multiple diagnostic applications. Starting from the systematic selection of the center frequency of the sinc signal [30], the use of heuristic optimization is performed more often. In [31] the EA was used to adjust parameters of the PWL signal. Other applications include the usage of the SA to increase the accuracy of analysis of nonlinear circuits. The closed-loop systems also require the optimized excitation signal (such as the step function), which may also be adjusted using the information channel concept.

The search for optimal accessible nodes configuration in complex analog systems

Each SUT consists of at least two accessible nodes (which is the maximum number in medium or large circuits); it is too costly to test each node response to conduct the faulty system diagnostics. Not every node can be made accessible. It is then imperative to determine the optimal set of nodes, required to increase the efficiency of the diagnostic classifier. Solution vectors are binary with the length equal to the number of nodes in the SUT: value 1 is assigned to the node selected for testing. The globally optimal set of nodes is guaranteed by the exhaustive search, i.e. checking of all combinations of nodes. It is an NP-hard problem [15]. Therefore, the testing time of the diagnosed SUT should be significantly shorter, which is provided by such heuristics as GA, TS or ACO. Before the optimization procedure, the L and V sets are generated once, for the selected excitation signal and all SUT nodes accessible. The subsequent solution vectors are generated by selecting symptoms only from the particular nodes. The example of the solution α for the 5-node problem (where the first, the third

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and the fifth node are analyzed) is as follows: {1,0,1,0,1}. The evaluation function has the form of (4), where ß is the minimized number of nodes in α. The test-point selection has already been the subject of some studies. In [16] the heuristic method based on the performance indices was proposed. In [17] the same aim was realized by developing logical rules. In [18] two heuristic methods for testpoints selection were described based on the criteria proposed by Hochwald and Bastian. In [19] the QR factorization for the circuit sensitivity matrix was introduced. The decomposition of the systems sensitivity matrix to select the test-point set was presented in [20]. Other approaches include the method of finding test-points by computing the information content of all the candidate testpoints or the entropy to select the minimal test-point set [15]. In [21] the GA was used to assess the optimal set of characteristic points. These studies utilized the extended fault dictionary in the process of test-points selection. Finally, the fuzzy logic and the ant colony algorithm were used to select the optimal test-points sets in analog circuits’ diagnostics.

SUT description

The object of presented studies is the 52-element fourth-order elliptic active low-pass filter with 19 nodes (Fig. 5), modeled using PSpice. The nominal values of elements are presented in Fig. 5 as well. This object consists of 8 operational amplifiers modeled by the controlled source (VCCS) and the input and output resistances. The faults simulated in the amplifiers provide the information about which VCCS is corrupted. The system has been excited by a 10 mA sinusoidal current source and of the 10 kHz frequency.

Model-based diagnostics

Most diagnostic algorithms rely on models of the analyzed object to acquire knowledge about its behavior. Some systems like power plants need to be constantly monitored to avoid dangerous consequences of their possible faults [22]. The proper means to analyze them is through the Model-Based Diagnostics (MBD). The model of the system is simulated in parallel with the actual object and the diagnostic decision is made based on the residual signals (i.e. differences between the responses produced by the system and its model). The core idea relies on comparison of the output signals provided by the actual system with model’s responses. The most similar one (with known parameters ensuring the minimal residual signals) is selected as the model identifying the state of the actual system. The optimization procedure consists in the repeated simulation of the model with known excitation and adjusting its parameters until the residual signals are small enough. The main problem of the SAT approach is that the simulation must be done in RT leaving small amount of time to perform all simulations. Therefore the effective optimization must be performed here. The popular approach in the SAT methodology is the Recurrent Neural Network (RNN), applied to optimize the system’s model [32], while the modified architectures (wavelet RNN or diagonal RNN) were also proposed for this purpose. Other methods described earlier are applied as well. For instance, faults in power transformers and rotating machinery were detected using the genetic programming.

6. Experimental example This section presents the numerical example presenting the application of the selected optimization method (briefly discussed in Section 4) to the problem of the optimal node subset selection in the complex analog circuit. It is the discrete optimization problem with the evaluation function having the form as (4), where ß is the diagnostic time in seconds (depending on the number of analyzed nodes). The classification accuracy is calculated as in (5). The single SVM classifier is used to perform the fault detection. Apart from the node configuration selection, the parameters of the kernel are optimized separately. The following kernels were verified in the experiment: polynomial, RBF, ERBF, sigmoidal, bspline, ANOVA and linear (the latter does not contain any parameters and is used as the reference function, usually giving poor results). Both the classification and optimization methods were implemented in the Matlab environment. In the following subsections the analyzed object is introduced. Next, the modelling and classification assumptions are provided. Finally, results of the simulations are presented and discussed.

„„Fig. 5. 52-element low-pass active filter with 19 accessible nodes „„Rys. 5. 52-elementowy filtr dolnoprzepustowy o 19 dostępnych węzłach

The output signal was analyzed in the steady state. The symptoms described earlier were extracted to provide the information required for diagnostics. The parametric sweep was commenced to determine the SUT behavior for various configurations of parameters. After changing the value of the single parameter, the SUT was excited and symptoms recorded from the response. This allowed for creating the learning and testing sets for the classifier. Each element has been assigned a range of values, from significantly smaller to larger than the nominal values. For instance: R2 was assigned values from 182 to 1000 Ohms, C3 was assigned values from 0.001 to 0.05 mF, R28 was assigned values from 500 to 1000 Ohms, R31 was assigned values from 9000 to 10 000 Ohms, The gain K of the first VCCS from 103 to 0.1 S. The fault-free state was also simulated and included in the datasets. The example in Fig. 2 depicts how the response of the 52-element lowpass filter to the sine excitation changes with the increase of the R17 value in the range of 500 – 2000 Ohms. The classification accuracy of the method depends on the measurement accuracy and differences at the accessible nodes between the faulty and nominal values. If R8 decreases to 1k Ohms or C10 increases from 0.05 to 0.06 mF, the relative changes in node voltages may not be sufficient to locate or even detect a faulty element. This is because sensitivity of these elements is too small.

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Assumptions for the experiment

The learning set of the fault classifier contains examples describing the system’s behavior in various faulty scenarios (Tab. 1). After changing the value of the selected parameter, the response of the SUT is recorded. The procedure is repeated for every parameter separately (the remaining parameters are at nominal values). Faults of each element are simulated by eight examples with different fault intensity (up to 80 percent of the nominal values of subsequent parameters). The tolerance margins assumed for all parameters were 10 percent of the nominal values. The time and frequency analysis was conducted, during which the following symptoms were acquired:

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„„Table 1. The dataset fragment „„Tabela 1. Fragment przykładowego zbioru danych

are selected to form the training and validating subsets, subsequently used to train and test the classifier. Element Element P1 T1 P2 T2 Fault code The optimal configuration of nodes (that is code parameter value the one that provides better results than input1.46e-001 3.23e-006 -1.46e-001 3.73e-006 3.00e+0 2.50e+001 0 output analysis in the shortest time possible) 2.22e-002 3.23e-006 -2.24e-002 3.73e-006 1.00e+0 7.00e+002 1 is in Tab. 2. 3.92e-002 4.22e-006 -3.90e-002 4.72e-006 1.00e+0 5.00e+002 1 In the node selection process some nodes 5.44e-002 4.22e-006 -5.19e-002 4.70e-006 4.00e+0 1.00e+002 1 can be replaced by other without any loss on the quality of the fault detection. The subsets {2, 3}, {4, 5}, {6, 7}, {9, 10}, {13, 14}, {15, 16} and {18, 19} can time domain analysis – the value of the first two maxima and minima values in the output signal and the time instants reqube measured interchangeably. ired to reach these values (see Fig. 2). Zero-crossing coordinaTab. 3 presents the classification accuracy for the 52-element tes were also considered. low-pass filter in the optimal node configuration. The kernel paramfrequency domain analysis – the 3db-frequency is extracted eter values were acquired by using the quasi-discrete method from the frequency spectrum curve. which is based on repeating the fault classification for each kernel While making the decision which nodes need to be consi- ten thousand times. Then the set of kernel parameter values was divided into a hundred subsets, from which the values providing dered, intermediate solutions are required. Two heuristic reprethe best classification results were selected. The fault detection sentatives, i.e. TS and GA were selected for the task. The first one is simple, but powerful, while the second is used as the last chance using the polynomial kernel requires function of the higher degree method, when all others fail. Both approaches work assuming that (at least 33). This indicates a high level of complexity of the data transformation from the original feature space. The high order of all nodes in the analyzed system can be accessible. The optimal solution contains nodes that should be excited and measured to the polynomial needed in complex analog systems diagnostic is caused by the complexity of the analyzed system and requires get better results than just the input-output analysis, working in the shortest possible time. Such information can then be used by the more specialized functions. Smaller systems were successfully designer to make the diagnostics easier and cheaper. diagnosed in the past using polynomial kernels of the lower order. Solution vectors are binary with the length equal to the number The classification results for the RBF and ERBF kernels are identiof nodes in the SUT. The successive solutions are created from the cal, but require different values of parameters. For the Gaussian latter ones by choosing a different tabu element, thus including or and polynomial kernels a significant improvement in the fault excluding a node from the newly generated solution. diagnostics can be observed when the nodes of the lower index (such as 3, 7 or 8) are included in the diagnostics. Results and discussion The node selection using TS achieved similar results to GA, but The sizes of the data sets were: |L|=416, |V|=208 exam- did it much faster (Tab. 4). Being less computationally demanding, ples. All contained the complete set of symptoms collected from it is the better approach for the task of node selection in complex all nodes. According to the node configuration, the symptoms analog systems.

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„„Table 2. The best node selection for 52-element low-pass filter (the GA and Tabu search) „„Tabela 2. Konfiguracja optymalnych konfiguracji węzłów w 52-elementowym filtrze dolnoprzepustowym (algorytm genetyczny i przeszukiwanie z Tabu) Node number/ selector

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

GA

1

1

0

1

0

0

0

1

1

0

1

0

1

0

1

0

0

1

0

Tabu

1

0

1

1

0

0

1

0

1

0

0

1

0

1

0

1

0

1

0

„„Table 3. The classification accuracy for 52-element low-pass filter for the optimal node configuration (algorithm A – with node selection, algorithm B – input-output analysis only) „„Tabela 3. Dokładność klasyfikacji 52-elementowego filtru dolnoprzepustowego dla optymalnej konfiguracji węzłów (algorytm A – z wybranymi węzłami wewnętrznymi, algorytm B – analiza wejściowowyjściowa) Type of kernel

Algorithm A

Algorithm B

Parameter value

Rbf

79%

69%

0.000039-0.000096

Erbf

79%

69%

0.00108-0.00207

Poly

61%

55%

33-42

„„Table 4. The target function values for optimal node selection „„Tabela 4. Wartości funkcji celu dla optymalnych konfiguracji węzłów Kernel type

Target (GA)

Target (tabu)

Rbf

3.931e-4

2.65

616

Erbf

3,931e-4

2.65

poly

3,528e-4

2.38

7. Conclusions Optimization methods contemporary play crucial role in the diagnostics of analog systems. Although most of their implementation is auxiliary to the classification and regression approaches, they help to improve both testing conditions and expand abilities of heuristic approaches. Applications presented in the paper show the complexity of most problems that must be solved in the reasonable time and limited computer resources. The presented work demonstrated the usefulness of heuristic optimization to increase the parametric fault classification accuracy in complex analog systems using the minimum set of analyzed nodes. During the simulation, the desirable fault classification accuracy could be achieved within around 23-26 percent of the total number of nodes. The types of nodes to be included in the diagnostic process are those of high degree, which allow the expert system to acquire the information from a substantial number of elements connected to them. The future applications of the presented algorithms require testing their abilities with different systems and incorporating additional optimization algorithms into the framework.

PRZEGLĄD TELEKOMUNIKACYJNY  ROCZNIK LXXXVIII  WIADOMOŚCI TELEKOMUNIKACYJNE  ROCZNIK LXXxIV  nr 6/2015

The problems of optimization get signifficantly complex with increasing size of the considered problem (for instance, the number of heuristic classifier’s parameters). This forces the designer to develop diagnostic methods based on multiple algorithms from different fields of mathematical and statistical studies. As heuristic methods are the most flexible tools for that purpose, the maximization of diagnostic accuracy may be achieved with relatively small cost. As the presented approaches show different characteristics, they should be extensively studied and compared in the future, using systems from various domains. Bibliography

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Zapraszamy do prenumeraty Przeglądu Telekomunikacyjnego i Wiadomości Telekomunikacyjnych PRZEGLĄD TELEKOMUNIKACYJNY  ROCZNIK LXXXVIII  WIADOMOŚCI TELEKOMUNIKACYJNE  ROCZNIK LXXxIV  nr 6/2015

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