Advanced Algorithms for Diagnostics of Nonlinear Systems

There are two main approaches for fault diagnostics of dynamical systems: data-driven based and model-based approach. In data-driven approach, features are extracted from the collected data and then mapped to predefined classes of fault scenarios. Data-driven methods are based on the priori knowledge of the relationship between signal symptoms and faults and are therefore, heavily dependent on data. Because there is no physics of the system involved, this approach cannot accommodate the changes easily. For instance, this method has a poor performance in diagnosing defects with different geometry, size and location if they are not already included in the training data. In addition, newer phenomena can appear in the behavior of the system which is difficult to predict using these methods. Model-based methods on the other hand, use physics-based model of the system to identify any defect or abnormality in the performance of the system. Since the understanding of the physics of the system is involved in this approach, the diagnostics algorithm can still be effective in different domains of the system response with even unknown phenomena and faults. The main drawback of this approach however, is the difficulty of derivation of accurate models for complex systems.

Our main thesis is that the diagnostics process can be better performed with techniques that combine both model-based techniques and data-based approaches. This is an interesting trend of research on developing new algorithms that combine information from physics-based models and data-based models (time series). Since the mathematical model derived from the physics-based model is in general an approximation or incomplete (due to the presence of many unknowns), data-based techniques can be used to extract information in order to update the mathematical model and thus train a learning system for diagnostics. The overview of the diagnostics process is illustrated in the diagram below showing what we will call here a hybrid approach for diagnostics. According to this diagram, the main objective of the proposed work is to develop a model-based framework for diagnostics of nonlinear system in which the mathematical model is updated and perfected using the measured data.

Defects in mathematical point of view

From the mathematical point of view, a defective system would in general have parameters or models different from the perfect system. The decreased stiffness of a structure due to a crack, or changes in the parameters of a degraded electronic motor are examples of parametric defects. Most of defects in practical systems can be modeled as parametric defects. On the other hand, structural defects which are due to the alteration of the intrinsic structure of the system, change the structure of the mathematical model of the system. A broken coupling in a mechanical system or a broken capacitor in an electrical system which change the dimension of the mathematical model are two examples of the structural defects.

Diagnostics as an inverse problem

Given the mathematical model of a system along with its parameter values, one can determine the nonlinear response of the system. This would be the “forward problem” in the context of system dynamic analysis. As mentioned earlier, in general, a defective system would have parameters different from a perfect system (at its simplest representation), or could have a model that is substantially different from the model of the healthy system. In contrast to the “forward problem”, the diagnostics problem is the “inverse problem,” where we would like to predict the changes in the system model and its parameters given its nonlinear response.

Our approach for diagnostics

There are two aspects of the work overlooked in almost all previous efforts for model-based diagnostics of system. First of all, the efforts to solve the "inverse problem" are only restricted to the estimation of system parameters; whereas, structural defects which can change the structure of the model are an important class of defects as well which need to be addressed. There is almost no research report to the best of our knowledge that has looked at the diagnostics problem from this point of view.

Furthermore, the unavoidable nonlinear nature of real systems makes this approach quite complicated. This is true especially in applications where prominence of nonlinear phenomena such as multi-resonance, chaos, quasi-periodicity, bursting oscillations, etc. is possible. Many studies have reported the emergence of these complex nonlinear phenomena in machinery originating from defects or even due to their nonlinear nature in healthy conditions. The prevailing estimation methods which are mostly based on optimization algorithms show poor performance coping with such complexities. In many cases, the models are linearized to simplify the estimation problem or the behavior of the system in such complex domains is simply ignored. This is the second aspect of the work that we aim to address in this research and try to make a contribution into its solution.

There are several techniques that we use to extract features and information from the nonlinear response of the system. “Recurrence Quantification Analysis (RQA)” and “Phase Space Density" are two of the techniques we have successfully applied to our problems. The following provides a brief description to two problems in which we have used these methods.

Sample Case Studies

Dynamic characterization and diagnostics of a MDOF nonlinear oscillator using the method of phase space topology (PST)

A phase space is a space in which all states of a system are represented and a phase portrait is a visual representation of the trajectory of this space. For the two-dimensional case, the phase space will turn into a phase plane. The phase plane trajectory consists of a closed single loop for a periodic response and multiple loops for a multi-periodic behavior. The topology of the phase space trajectory provides valuable information regarding the behavior of a system in a qualitative fashion. The method of Phase Space Topology (PST) which was developed by our team at Villanova University is a technique for characterizing this topology in the periodic and multi-periodic domain with quantitative measures. PST quantifies the topology of these closed curves by computing the density of points along each axis of the phase portrait. For simplicity of illustration, the examples here are presented in the two dimensional space; however, it can be extended to higher dimensions. For dimensions higher than three, even though the visualization of the phase space trajectory is not possible, the method is still applicable. In fact, the computations are performed individually and independently for each state of the system.

To illustrate the method, consider the three sample phase portraits (position x1 vs. velocity x2 of the first mass) shown below along with the corresponding estimated density of x1 and x2 time series. The data has been obtained from a MDOF nonlinear mass-spring system. Let us now see how the density of x1 and its peaks properties change based on the topology of the phase portraits. In Fig. a, we have a double-loop phase portrait which is a characteristics of a bi-periodic motion. The edges of the loops at the returning points in x1 direction have produced four sharp peaks in the density plot of x1. The two sharp peaks in the middle are higher than the other two due the lower radius of the phase trajectory in those areas. In addition, the depression of the phase trajectory in the middle has produced a smoother peak in the density plot. In Fig. b, we have a similar topology; however, the depressed part has moved rightward. It can be seen from the corresponding density plot that the smooth peak has shifted rightward to the middle of the two sharp peaks as well. In Fig. c another loop has evolved in the phase plane trajectory; representing a response with three frequencies. As a result, two more sharp peaks have emerged in the density plot of x1. The density plots of x2 can also be explained in a similar way.

This density function can be computed and plotted for any state of the system. According to the figures, the shape of the phase space trajectory which is a closed curve for periodic and multi-periodic motions is in a direct relationship with the properties of the peaks in the density plots. Each peak in the density plots can be characterized with its location, height, and sharpness values. These values can be used as effective features in order to train a neural network and estimate the defective parameters of the system.

Recurrence analysis of experimental time series of a rotor response with bearing outer race faults

Rolling element bearings are among the most common components used in expensive, high precision and critical machines such as gas turbines, rolling mills and gyroscopes. They can be subjected to various defects which could lead to catastrophic results. This includes inner and outer race defects and hence it is of interest to analyze the system response under such defects. A better understanding of the system performance under such defects can be beneficial when performing system diagnostics and system design. In this study we focused on the outer race defect and performed a comparative nonlinear time series analysis of a healthy system and a defective system. We considered various levels of outer race defects. The analysis is based on the recurrence properties of the system in its reconstructed state space. After determining the appropriate time lags through the average mutual information technique and the corresponding embedding dimensions through the false neighbor technique, we performed a sequential analysis of the system by subdividing the time series into bins and investigating the system response through recurrence quantification analysis parameters along with the entropy. This contributes to the enhancement of the science of diagnostics of outer race defects by analyzing the signature of various recurrence quantification analysis parameters as the system goes from a healthy state to a severely defective state.

Sample Publications
Book Chapter
  1. C. Kwuimy, M. Samadani, K Kappaganthu, and C. Nataraj (2015), “Sequential recurrence analysis of experimental time series of a rotor response with bearing outer race faults”, Vibration Engineering and Technology of Machinery, Pages 683-696, Springer.
Journal Publications
  1. M. Samadani, C. Kwuimy and C. Nataraj (2015), “Model-based fault diagnostics of nonlinear systems using the features of the phase plane response”, Communications in Nonlinear Science and Numerical Simulation, Volume 20, Issue 2, Pages 583-593.
  2. C. Kwuimy, M. Samadani and C. Nataraj (2013), “Bifurcation analysis of a nonlinear pendulum using recurrence and statistical methods: Applications to fault diagnostics", Nonlinear Dynamics, Volume 76, Issue 4, Pages 1963-1975.
Conference Publications
  1. M. Samadani, C. Kwuimy and C. Nataraj, ” Fault detection and severity analysis of servo valves using recurrence quantification analysis” Annual Conference of the Prognostics and Health Management Society, Sep-Oct. 2014, Fort Worth, TX, USA.
  2. C. Kwuimy, M. Samadani and C. Nataraj, “Recurrence analysis of experimental time series of a rotor response with bearing outer race faults”. 10th International Conference on Vibration Engineering and Technology of Machinery VETOMAC-X 2014, Sep. 2014, Manchester, UK.
  3. C. Kwuimy, M. Samadani and C. Nataraj, “Preliminary diagnostics of nonlinear dynamical systems using embedded dimension analysis: A case study”, Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2014, August 2014, Buffalo, NY, USA. to predict the oc-currence and severity of Periventricular Leukomalacia, Dynamic Systems and Controls Conference (DSCC), 2012.
  4. M. Samadani, C. Kwuimy and C. Nataraj, “Diagnostics of a nonlinear pendulum using computational intelligence", Proceedings of the 6th Annual Dynamic Systems and Control Conference, Oct. 2013, Palo Alto, California, USA.