Στοχαστική αναγνώριση μηχανικών συστημάτων με υστέρηση

Abstract

Hysteresis is a nonlinear phenomenon which is common in various branches of science and technology, including physics, mechanics, biology, civil and mechanical engineering. It is well known that many applications stimulate and provide lively support to mathematical construction. For instance, several well - known models which named after the physicists and engineers who proposed them (Rayleigh, Duhem, Weiss, Prandtly, Preisach, Bouc and so on) was entangled with applications before being viewed as models of hysteresis. The study of hysteresis phenomena as well as their mathematical analysis have been systematically researched [1,2] and several mathematical modelling tools have been proposed. Nevertheless the problem of inverse modelling (that is identifying from available experimental data a model that provides the optimum representation of the system under study) of a system with hysteresis have not been investigated thoroughly. In the most of the cases the hysteresis identification is often limited to practical issues such that postulating a hysteresis model and applying an estimation technique (i.e. nonlinear optimization) for obtaining its parameters. However the problem is much deeper and a more theoretical analysis, employing principles and tools from the System Theory, should be applied in order the optimum results to be obtained. The dissertation aims at proposing a methodology for detecting nonlinearities (such as hysteresis) utilizing experimentally obtained excitation - response data only. The key feature for doing so is the Coherence Function, which is expected to be decreased with the appearance of nonlinear behavior [3]. Following this a complete methodology for identifying systems with hysteresis based upon the Maxwell Slip model [4] is established. This problem is addressed from the ``System Identification point of view, and therefore the critical issues primarily addressed here are:  The a priori identifiability (or uniqueness of representation) of the Maxwell Slip model structure.  The excitation design for achieving ``rich enough data for the Maxwell Slip model identification purpose.  The selection and numerical implementation of an estimator  The a posteriori identifiability (whether the model can be identified from a actual experiment under real conditions) of the Maxwell Slip model.  The estimator asymptotic statistical properties, such that consistency and asymptotic normality  The model structure selection  The model validation Based upon these questions, answers are provided and detailed guidelines are established. In order the established theoretical findings to be solidify and their practicality to be revealed, their application to an actual system with hysteresis is required. From the huge number of the available systems with hysteresis, a real (experimental) mechanical system with friction is selected. This kind of systems are not only quite common in mechanical engineering applications (friction perhaps is the most common source of nonlinearities in mechanical systems), but also extremely challenging due to its complicated hysteresis nature (different types of hysteresis appear according to the operational regime). In order the latter to be more clear consider that whenever a system with friction operates within presliding regime (no macroscopic movement between the two surface in contact - the relative displacement is approximately 2 – 5 μm for steel materials [5], then there is a (almost) rate-independent hysteresis with nonlocal memory between the displacement and friction force [6,7,4]. As the relative displacement increases then more and more junctions break and finally there is a macroscopic relative motion and therefore the sliding regime begins. In this case the previous type of hysteresis disappears, and a new rate-dependent hysteresis between the displacement and friction force appears [8,9,4]. However, the system transits from the one regime to the other, (i.e at velocity reversals) several time during its typical operation, thus the type of hysteresis appears depends on the operational regime and thus to the system motion. Unavoidably this makes the identification problem quite challenging. Firstly the system under study is considered within the presliding regime only. Experiments were carried out and actual presliding displacement - friction force signals were obtained. The application of the proposed methodology yields almost excellent results, indicating its ability of capturing the underlying presliding frictional dynamics. The obvious next step is to consider the most common case, which is operation within both presliding - sliding dynamics. Experiments were implemented and displacement - friction force signals were collected. In this case the identification results based upon the Maxwell Slip model, though not so good as before, appear to be very promising. For that reason a proper extension of the basic Maxwell Slip model structure is proposed and implemented. This modification yields to significant improvement, and excellent results are achieved. In order the potentiality of the proposed extended Maxwell Slip model to be demonstrated better, a simple feedforward friction compensation scheme, based upon the extended model, is implemented. The results demonstrate excellent friction compensation, yielding extremely low tracking error, not only in each operational regime but also during regime transitions.