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Correcting Sensor Drift and Intermittency Faults With Data Fusion and Automated Learning Kai Goebel and Weizhong Yan, Member, IEEE
Abstract—Many fault detection algorithms deal with fault signatures that manifest themselves as step changes. While detection of these step changes can be difficult due to noise and other complicating factors, detecting slowly developing faults is usually even more complicated. Tradeoffs between early detection and false positive avoidance are more difficult to establish. Often times, slow drift faults go completely undetected because the monitoring systems assume that they are ordinary system changes and some monitoring schemes may adapt to the changes. Where redundant sensors are used, a drifting sensor may cause the logic to latch on to the “bad” sensor. Another problem may be intermittent sensors faults where the detection logic is too sluggish to recognize a problem before the sensor has returned to seemingly normal behavior. To address these classes of problems, we introduce here a set of algorithms that learns to avoid the bad sensor, thus indirectly recognizing the aberrant sensor. We combine advanced sensor validation techniques with learning. The sensor validation is inspired by fuzzy principles. The parameters of this algorithm are learned using competing optimization approaches. We compare the results from a particle swarm optimization approach with those obtained from genetic algorithms. Results are shown for an application in the transportation industry. Index Terms—Data fusion, drift fault, fuzzy fusion, intermittency, intermittent fault, sensor validation, soft fault.
I. INTRODUCTION
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ENSORS play a critical role in process control and modern system health monitoring. Even with the most advanced design of instruments, sensor failures, defined as “a non-permitted deviation from a characteristic property” [1], are almost unavoidable, in particular when sensors are located in a harsh environment (high temperature and/or subject to high vibration). Hence, sensor validation, consisting of the tasks of sensor failure detection, isolation, and accommodation (DIA), is essential in ensuring that sensors reliably perform their desired functions. Broadly speaking, there are two types of sensor failures/faults: those with abrupt changes that are manifested as step changes and slowly developing failures that are characterized as drift. The former type of faults is often referred to as a “hard fault” and the latter is called the “soft fault.” In real world applications, detecting sensor failures (hard or soft) can be a challenging task since faults are often masked by Manuscript received May 2, 2007; revised April 8, 2008. K. Goebel is with the NASA Ames Research Center, MS 269-4, Moffett Field, CA 94035 USA (e-mail:
[email protected]). W. Yan is with the GE Global Research, Niskayuna, NY 12309 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSYST.2008.925262
sensor noise, deterioration, system dynamics, and changing environmental conditions (among other things). While there may be some opportunity to detect hard failures (depending on the characteristics of the faults), soft failures are even more difficult to detect because the detection scheme needs to be able to distinguish between transient system states changes in operating conditions or environmental conditions and the soft fault. Over the years, numerous sensor validation methods have been proposed for various applications. Some sensor validation methods produce its own sensor health information with just information from that sensor alone, while other sensor validation methods are based on information from multiple sensors. Among the former, limit checking, or more generally limit filtering [2], is one of the classical methods of finding sensor faults. In this setting, sensor readings are compared to a preestablished nominal value and a faulty sensor is declared whenever a threshold value is exceeded. Recently, fuzzy limits (in lieu of hard limits) have also been investigated [3], for more reliable sensor validation and outlier detection. More sophistical individual-based methods include algorithmic sensor validation (ASV) filter [4], Kalman filter [5], [6], and wavelet transform [7], among many other methods. The redundancy-based methods are further categorized into analytical redundancy and physical redundancy. Analytical redundancy utilizes functional relationship among the sensors that are of different types and/or located at different locations. Hence, analytical redundancy approach involves building the model capturing relationships among the measured parameters, thus is often called model-based validation methods. The models can be analytical equations derived from first principles or those empirically derived mathematical relations estimated using data-based system identification techniques. With the model established, a residual error between the present sensor reading and the model predicted value can be calculated and analyzed. Among the model-based methods, auto-associative neural networks (AANNs) [6], [8]–[10] are the most popular ones. Others include the use of principal component analysis (PCA) [11], Bayesian network [12], and Nadaraya-Watson statistical estimator [13]. More recently, particle filtering has been proposed by Wei et al. [14] as a model-based sensor validation technique. A survey paper by Frank [15] provides further references on model-based sensor validation. Physical redundancy (also called hardware redundancy), on the other hand, involves utilizing redundant sensors to measure the same parameter of the system. This is a fairly common approach in application with safety-critical systems such as aviation or space. In this paper, we discuss detection and accommodation of intermittent sensor faults and soft sensor faults using physical
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redundancy. Prior work on soft fault detection has focused mainly on electrical systems where soft faults are often described as a drift of parameters away from their tolerance range. Tadeusiewicz et al. [16] model the circuit with the node equation and use a linear programming approach to solve it. Catelani and Fort [17] compare two techniques for soft fault classification in analog electronic circuits. Both techniques are based on a priori collected signatures that are used to train two different classifiers: a fuzzy rule base and a radial basis function network. The fuzzy approach provides better results due to more flexible mechanism in rule modification that allows better segmentation in ambiguous regions. Hamida and Kaminska [18] use bond graphs and sensitivity computation to detect soft (and catastrophic) faults. In [10], Najafi et al. describe the use of an auto-associative network to address sensor drift. This paper introduces a set of algorithms customized to respond to problems of one of two redundant sensors by avoiding the bad sensor, thus indirectly recognizing the aberrant sensor. To that end, we combine advanced sensor validation techniques with learning. The sensor validation is inspired by fuzzy principles and the algorithm parameters are learned using optimization approaches.
Fig. 1. Response of baseline algorithms.
II. BACKGROUND In the application that motivated this work, abnormal system behavior was caused by sensor faults. The sensors were measuring characteristics of equipment that aids in the distribution of a substance used for energy conversion. Proper functioning of this equipment is critical for the overall safe operation of the system. The sensor validation algorithm that was used historically was found to be unsuitable for a certain class of sensor faults. However, because these sensor faults, including drift faults, do not happen very often, the problem did not arise for quite a while during which the system was in operation. The algorithm used was basically calculating the mean of the two sensor values unless the difference between the two sensor values was surpassing a threshold value. In that case, the algorithm would then choose the sensor value that was closer to the mean. That worked well for hard faults, the only fault considered. However, for some faults, that logic resulted in the undesired effect of latching on to the wrong sensor about half of the time. Because the sensor information was an input to a controller that was responding to this input as designed, it did produce in undesired output. The original algorithm output (here labeled “base”) is exemplified by Fig. 1 which shows the two sensors already diverged . One can see how the baseline to some degree prior to algorithm chooses the mean of the two sensor values. Then, as the values diverge more (here clearly shown by a fairly drastic ), the algorithm abandons the mean change around value strategy and chooses to go with the bad sensor. Clearly, this demonstrates poor algorithm performance. There are several potential ways to deal with this problem. A fairly easy fix might be to implement outlier detection. However, the issue gets a little more complicated when the sensor values diverge slowly. In such a case, outlier detection is not that straightforward.
Fig. 2. Response of baseline algorithms in the presence of soft faults.
Such a scenario is shown in Fig. 2 (the time window of the example is chosen wider because of the nature of the slow drift). Here, the sensor values slowly diverge and the base algorithm output jumps initially back and forth between the mean and one , the base algorithm output jumps of the sensors. Around between the mean of the two sensors and the bad sensor. Around , the base algorithm jumps from the mean to the faulted sensor and stays with it for almost 90 time units. That behavior . Outlier detection would not help in deis repeated at tecting a malfunction in any one of these sensors. There are a number of algorithms that have been proposed in the literature to detect abnormal conditions of time series. These include statistical approaches such as cumulative sum [19], sequential probability ratio test [20], and generalized likelihood ratio test [21], etc. Other algorithms for abnormal condition detection include signal processing techniques (wavelet [22]), regression (autoregressive process [23]), and computational intelligence techniques (neural networks [24], [25] and fuzzy logic [26], etc.). Typical steps in change detection include data validation, features extraction, classification, and perhaps post-processing. Moreover, fusion approaches have been suggested [27]
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that are designed to further boost the performance of these detection algorithms. However, none of these attempt to solve slow drift problems.
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have to be chosen separately for the left where parameters and right curves. The curves then need to be scaled from 0 and 1
III. METHOD The output of advanced sensor validation should not just report that a sensor works (or does not work). Instead, it should produce a correction of the raw sensor measurement when needed to ensure that machine controllers do not work incorrectly. This paper uses a fusion algorithm and we will therefore also refer to the corrected sensor value as the “fused” value. The fusion algorithm is based on the FUSVAF algorithm [28], [29]. As explained in more detail in the following, the FUSVAF algorithm has several tunable parameters that directly affect the algorithm’s performance. It has been found that manual tuning of the parameters is rather burdensome and does not guarantee optimal solutions. Therefore, this paper explores learning of key parameters using an optimization approach to explore the potential performance improvement.
Fig. 3. Validation gate for the assignment of confidence values.
between the validation border and the predicted value . The resulting assignment for confidence values is
A. Fuzzy Sensor Fusion Fuzzy sensor fusion operates by building kernel-like validation curves around local estimates. These curves can be dynamically adjusted based on domain knowledge and subjective confidence measures. Fuzzy validation (in contrast to its probabilistic counterpart) does not make any assumptions (Gaussian distribution, etc.) about the data. The confidence values obtained for each sensor reading from validation curves are aggregated using weighted average. With increasing distance from the predicted value, readings are discounted through a nonlinear validation function. Incoming sensor readings are validated using the validation gate and the old fused value. This fused value is then used to assess the state of the system represented by an exponential moving average predictor. It is also used for prediction which in turn is necessary to perform the validation of the next time step. The confidence value that is assigned to all sensor measurements depends on the specific sensor characteristics, the predicted value, and the physical limitations of the sensor value. The assignment takes place in a validation gate that is bounded by the physically possible changes the system can undergo in one time step. If the sensor readings show a change beyond that limit, it cannot be a correct value. These limits are the boundaries of the validation gate. Sensor readings outside the validation gate are assigned a confidence value of since they do not make physical sense. Within the region, a maximum value will be assigned to readings that coincide with the of predicted value. The curve between the maximum and the two minima is dependent on the sensor behavior. Generally, this is a non-symmetric curve that is wider around the maximum value if the sensor is known to have little variance and narrower if the sensor exhibits noisy behavior. for a particular situation A choice for validation curves could be a piece-wise bell curve of the form (1)
(2)
where and and
is the confidence value for a particular sensor; are the right and left validation gate borders, respectively; are the parameters for the left and right validation curve; is the sensor reading; is the predicted value.
A validation gate is shown in Fig. 3 where the validation gate is bounded by the validation gate borders and . Shown are three validation curves for three sensors. Measure, and correspond to the three sensors, respecments tively. They in turn are associated with the three confidence , and . Displayed is also the fused value at the values and the predicted value . previous time step Here sensor measurements; sensor confidence values; predicted value; old value at previous time step. The operative equation in the fuzzy fusion algorithm is (3)
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where fused value; measurements; confidence values; adaptive parameter representing the system state; constant scaling factor; predicted value; # of sensor measurements. Note that if all sensors lie on one side of the predicted value, the fused value will also be pulled to the same side. This ensures that evidence from the sensors is closely followed yet discounted the further it gets away from the predicted value. The parameter is used for an exponential weighted moving average predictor that is used to model the next expected value. In the baseline approach (i.e., the manually optimized setting) is . Both and are values which determine set to the shape of the left and right side of the validation gate. In the baseline setting, these two parameters are set to 0.95 and 1, respectively. In addition, the width of the validation gate is set to 3.24 which corresponds to the maximum possible application specific change of the device considered here within one time step (15 ms). The curves change dynamically with the operating conditions which allows to capture the change in behavior of the sensor over its operating span. An example is to modify the confidence in sensor performance expressed by the sensors’ variance. That is, if the sensor readings are all over the place, then the sensor is probably bad. In that case, the width of the validation curve and for this specific sensor should be modified. Such a relation can be captured by (4) (5) is the difference of two consecutive readings for a where particular sensor and is a parameter that scales the impact of the difference between the readings. It is one of the parameters that will later be subject to tuning via the optimization routines. The algorithm will adjust the shape of the validation curve based on the observation that the fault mode exhibits an increase in noise. To get fastest response, the nominal change from one is observed for each device and then time step to the next translated into the parameter of the curves , and . Next, an evaluation takes place which checks whether the observations are within the validation gate. Is this not the case, the confidence for that particular sensor value is zero. Otherwise, depending on whether the sensor reading is on the left (smaller) or right (larger) side of the predicted value, the confidence value is computed using the generalized bell curve. The same procedure is then performed for source B. A final check makes sure that both confidence values are greater than zeros. Should they be both zero, the algorithm is reinitialized using the average of both measurements and the predicted value. If at least one confidence value is greater than
Fig. 4. GA-based parameter optimization.
zero, is computed as the weighted average of the sensor readings and confidence values. The fused reading is then passed on. As a last step, the predicted value is computed using the exponential weighted moving average (EWMA) predictor (6) where is the predicted value of the previous time step and is the smoothing factor that is also subject to the optimization. Also, subject to the tuning is the width of the validation gate, i.e., (7) In summary, the five parameters that will be tuned are: , and . The tuning methods are detailed in the following two subsections. B. Particle Swarm Optimization The particle swarm optimization algorithm, originally introduced in terms of social and cognitive behavior [30], has come to be widely used as a problem solving method in engineering and computer science. The technique is fairly simple and comprehensible as it derives its simulation form social behavior of individuals. The individuals, called particles, are flown through the multidimensional search space, with each particle representing a possible solution to the multidimensional problem. The movement of the particles is influenced by two factors: as a result of the first factor, each particle stores in its memory the best po, and experiences a pull towards sition visited by it so far, this position as it traverses through the search space. As a result of the second factor, the particle interacts with all the neighbors and stores in its memory the best position visited by any particle in the search space and experiences a pull towards this position . The first and second factors are called cognitive and soand cial components, respectively. After each iteration the are updated if a more dominating solution (in terms of fitness) is found, by the particle and population, respectively. This process is continued iteratively until either the desired result is achieved or the computational power is exhausted. The PSO formulae define each particle in the D-dimensional , where the subscript “ ” repspace as resents the particle number and the second subscript is the dimension. The memory of the previous best position is repreand a velocity along each sented as dimension as . After each iteration, the
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Fig. 5. Twelve sensor measurement trajectories.
velocity term is updated and the particle is pulled in the direcand the global best position, tion of its own best position, , found so far. This is apparent in the velocity update equation [30]
(8) (9)
For more details of the particle swarm optimization algorithm, the reader is referred to [31]. To use PSO to optimize the five FUSVAF algorithm parameters concerned in this paper, each particle represents a possible solution, which is a point in a 5-D solution space. The fitness function in this case is the summed error over all example cases (described in detail in Section IV). C. Genetic Algorithms The second optimization algorithm used for optimizing the parameters of our fusion algorithm is the Genetic Algorithm
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TABLE I OPTIMIZATION RESULTS
TABLE II LEAVE-ONE-OUT RESULTS FROM GA
Fig. 6. Convergence of parameters to specific values in PSO optimization. TABLE III LEAVE-ONE-OUT RESULTS FROM PSO
(GA). Like PSO, GA is also a derivative-free, population-based, and stochastic optimization method. The GA was first introduced by John Holland in 1975 [32]. GA is an optimization method that is based loosely on the concepts of natural selection and evolutionary processes. Since its introduction, GA has become popular in both academia and industry. The following characteristics of GA contribute to its popularity: • it is conceptually simple and easy to implement; • it is applicable to a wide range of problems (linear or nonlinear objective and constrain functions; continuous or discrete variable); • it is a parallel-search procedure that is suitable for distributed computation for complex optimization. In GA-based parameter optimization, a solution is represented as a chromosome. In each iteration (generation) of the algorithm, a fixed number (population) of possible solutions is generated in a stochastic fashion. Each of the possible solutions is evaluated and modified following the defined genetic operators. Fig. 4 illustrates the concept of GA-based parameter optimization. GA employs the following three operators in the optimization process, namely evaluation, reproduction, and modification. The evaluation operator calculates the fitness value of each member in the population based on the objective function. The reproduction operator determines which parents participate in producing offspring for the next generation, which mimics the principle of “survival of the fittest” in natural selection. The modification operator consists of two functions: crossover and mutation. While crossover is similar to that of mating in the natural evolutionary process by combining elements of the two meting members of the initial population, mutation intends to promote the diversity of the population by randomly changing bits in the representation of the solution, This is meant to prevent the optimization process from getting trapped in local minima. There are various GA codes available. In this paper, we use the MATLAB-based Genetic Algorithm Optimization Toolbox
[33] as the GA engine. For the FUSVAF algorithm parameters optimization concerned in this paper, a possible solution is represented by a chromosome, a five-element vector, and each of the five elements represents one of the five FUSVAF parameters , and ). The fitness function used (i.e., in GA is the summed error over all example cases (described in detail in Section IV). IV. RESULTS AND DISCUSSION The performance of the algorithm was evaluated on data sets from a real-world application in the transportation industry as explained previously in more detail. A total of 12 trajectories, each one with a unique observed sensor fault were used for training. Fig. 5 shows the 12 trajectories for two sensor measurements. It can be seen that the type of fault actually varies between the cases. Case 1 is perhaps the most compelling because system dynamics fluctuate greatly while the two sensors, sensor A and sensor B, slowly deviate from each other. In contrast, case 2 shows sensor faults of bigger magnitude which occurs intermittently. Cases 3 and 4 are a superposition of the first 2 cases with intermittent behavior and increasing bias. Similar behavior occurs for the rest of the cases, although the scale prevents one to see the characteristics well for some cases. Because of the limited number of training cases, we employed a jackknife process, i.e., a leave-one-out cross-validation where all but one trajectories are used for training and the remaining case is used for validation. The test case is then rotated while the training process is repeated with the other eleven cases. The performance metric is the summed error across all test cases. While it cannot be guaranteed that the algorithm will perform well for drastically different trajectories, it should be expected that the algorithm works reasonably well for the types of faults represented
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Fig. 7. (a) Baseline fusion. (b) GA optimized FUSVAF. (c) PSO optimized FUSVAF.
in this set of trajectories. This may not be satisfactory in all situations, but it served to solve a real problem in this instance. The error was computed as the difference between the fused value and the measurements of the unfaulted sensor. For both GA and PSO optimization schemes, the population size of 50 and number of generations of 25 are used. Other parameters are set to default values. Both optimization schemes improve over the previously established baseline performance that uses the same fusion scheme but with manually tuned parameters. The results are summarized in Table I. Table II shows the results from the GA-based runs. The summed error, as shown in the last row, is different from the one seen in Table I because the results of the individual optimizations were added up, as opposed to the overall optimization with all 12 runs shown in Table I. The individual results display considerable variability, which is due to the nature of the different runs as seen in Fig. 5. Fig. 6 illustrates the convergence of two of the five parameand , in the PSO optimization process. ters, namely, Table III shows the results of the PSO optimization. They are qualitatively similar to the GA-based optimization results although one can observe a generally higher error. It is also noteworthy that the parameters are all considerably smaller. The GA-based optimization has overall best results in this particular application. This may be due to a very flat optimiza-
tion landscape where locally optimal solutions are more easily missed in the PSO approach. It has to be noted that the PSO approach did come up with considerably different results when different start conditions were used, further supporting this argument. To appreciate the performance difference among the three tuning methods, we show the residuals between the ground truth and the fused vales in Fig. 7. Smaller residuals means better sensor fusion. To make the figures more readable, we only show five trajectories as the representatives of the original 12 trajectories. Fig. 7(a) shows the baseline results, that is, the residuals between ground truth and baseline fusion outputs. Fig. 7(b) and (c) shows the residuals between the ground truth and the FUSVAF with GA and PSO optimizations, respectively. Comparing Fig. 7(a)–(c), one can see that both GA and PSO optimizations of the FUSVAF parameters result in a much smaller residuals for all trajectories, indicating their effectiveness in accommodating soft and intermittent sensor faults. Between GA and PSO optimizations [see Fig. 7(b) and (c)], the difference is qualitatively not significant in terms of effectiveness of sensor fusion. To further appreciate the difference between the three optimization schemes, we show in Fig. 8 the residuals from the three optimization schemes for one trajectory. is alThe results displayed in Tables I–III indicate that . This is explained by taking a closer ways smaller than
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Fig. 8. Comparison of different optimization schemes.
look at Fig. 5. The optimization routines have effectively learned the fault mode. Specifically, when a drift fault occurs, the deviation of the faulty sensor is always in negative direction. Assuming that this fault mode is the dominant one, this is not a problem. However, it also illustrates a limitation of machine learning approaches: They can only learn (and will inevitably learn) what is dictated by the data and warrants a sanity check with engineering understanding. V. CONCLUSION We present a method to correct for certain types of sensor faults, including slow drift sensor faults. To that end, we utilize a fusion scheme that takes advantage of different information sources with different noise characteristics. The method leverages a simplistic model that calculates an expected next value. Dynamically adjustable subjective confidence measures are interpreted as fuzzy memberships and used to compute a corrected value. Results from real world data demonstrate that the method can successfully remediate soft faults for those examples. REFERENCES [1] R. Isermann, “Process fault detection based on modeling and estimation methods,” Automatica, vol. 20, no. 4, pp. 387–404, 1984. [2] T. S. Sowers, L. M. Santi, and R. L. Bickford, “Performance evaluation of a data validation system,” presented at the 41st Joint Propulsion Conf., Tucson, AZ, Jul. 2005. [3] J. Nasi, A. Sorsa, and K. Leiviska, “Sensor validation and outlier detection using fuzzy limits,” presented at the 44th IEEE Conf. Dec. Control, Seville, Spain, Dec. 2005. [4] Y. J. Kim, W. H. Wood, and A. M. Agogino, “Signal validation for expert system development,” in Proc. 2nd Int. Forum Expert Syst. Comput. Simulations Energy Eng., Erlangen, Germany, Mar. 1992, pp. 9-5-1–9-5-6. [5] D. Gobbo, M. Napolitano, P. Famouri, and M. Innocenti, “Experimental application of extended Kalman filtering for sensor validation,” IEEE Trans. Control Syst. Technol., vol. 9, no. 2, pp. 376–380, Mar. 2001. [6] M. Napolitano, D. Windon, J. Casanova, M. Innocenti, and G. Silvestri, “Kalman filters and neural network schemes for sensor validation in flight control systems,” IEEE Trans. Control Syst. Technol., vol. 6, no. 5, pp. 596–611, Sep. 1998. [7] J. Ma, J. Q. Zhang, and Y. Yan, “Wavelet transform based sensor validation,” IEE Colloquium on Intelligent and Self-Validation Sensors, pp. 10/1–10/4, 1999.
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Kai Goebel received the Ph.D. degree from the University of California at Berkeley, Berkeley, in 1996. He is currently a Senior Scientist with NASA Ames Research Center, Moffett Field, CA, where he is leading the Diagnostics and Prognostics Group and is coordinating the Prognostics Center of Excellence. From 1997 to 2006, he worked at Global Research Center, General Electric, Niskayuna, NY. His research interests include advancing artificial intelligence, soft computing, and information fusion for real time monitoring, diagnostics, and prognostics. He holds 9 patents and has published more than 90 papers.
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Weizhong Yan (M’06) has been a Research Engineer with the Industrial Artificial Intelligence Lab, GE Global Research Center, Niskayuna, NY, since 2000. His research interests include machine learning, pattern recognition/classification, data analysis and modeling, soft computing, and information fusion. His specialties include application of soft computing technologies to monitoring and diagnosis of gas turbine engines and other mechanical systems. He has authored over 50 publications and filed over 20 U.S. patents. He is an Adjunct Professor of the Mechanical Engineering Department, Rensselaer Polytechnic Institute, Troy, NY, where he teaches control and modeling since 2004.
