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.

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.

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.