112 B.H. Aghdam et al. The concept of energy in specific frequency bands has been used by many researchers. A specific method of this kind which considers the energy contribution of each single vibration mode has been applied by Yao and Fang to multidimensional force and acceleration signals [7]. Modal energy component in frequency domain was introduced as an effective wear sensitive feature by Roth and Pandit [8] for estimation of tool wear in milling. Sick [9], performed an extensive review on online indirect tool wear estimation in turning with artificial neural network (ANN) methods and outlined possible directions for future studies. He concluded that certain features in time or frequency domain can be used for online tool wear estimation Features in time domain depend significantly on the values of cutting conditions; whereas, frequency domain features depend on fundamental resonant frequencies of the tool-workpiece system, chip lamination frequencies, chatter, etc. In another review paper, Rehorn et al. [10] organized the results according to the type of the machining operation. They stated that time-frequency methods can be used to identify wear and breakage in machining operations but very few researches have been carried out on these items. Authors concluded that even though ANNs and similar methods can provide highly accurate results, simple force models can as well provide similar accuracies. Another important aspect of tool wear estimation is the extraction of wear sensitive features from the recorded signals, which is independent from cutting conditions. Especially, Features in frequency domain have the property of being independent from cutting conditions. Some statistical features of tool bending and longitudinal vibration obtained by Singular Spectrum Analysis (SSA) in certain frequency bands has been employed for wear estimation [11, 12]. ANNs and its variants are among the more commonly used methods in tool wear estimation [13–15]. In many of the papers, different methods has been applied to train the networks to achieve higher performance in wear estimation [16–19]. Many researchers tried to exploit multiple sensors and sensor fusion techniques to enhance the accuracy and robustness of the wear estimation algorithm [20–25], however in some cases it did not guarantee the enhancement of the estimations. Simultaneous use of Back-propagation neural networks (BPNs) and adaptive neuro-fuzzy inference system (ANFIS) was investigated by Liu et al. [26] and it was shown that the error is lower when ANFIS method is used. In a more recent review paper [27], based on a comprehensive investigation of research works it has been concluded that the use of frequency or time-frequency features is more advantageous. According to the review papers [9, 27] it can be said that very often researchers employed ANNs and its variants for modelling tool wear and features relation for TCM researches and have proposed different procedures for neural network training. These methods have important disadvantages such as sensitivity to cutting conditions and the need for too many training samples. Time invariant (TI) systems under stationary input produce stationary vibration signals with TI statistical characteristics, while time varying (TV) systems have non-stationary responses with time varying statistical characteristics [28]. However, TI structures which are subjected to non-stationary input can also produce non-stationary response, for instance bridges and tool-holder set of a lathe can be considered as TI structures that have non-stationary response due to ambient excitations and cutting forces, respectively. ANN’s that are capable of modelling non-stationary signals has been extensively used for estimation of tool acceleration signals. However, ANN’s do not have a parametric structure and system dynamic parameters such as damping ratios and natural frequencies cannot be extracted using them. In certain dynamic systems, the input to the system cannot be measured or it is unobservable. In such cases, the identification methods that only rely on system output are attractive. These methods that are known as output-only identification problems in the focus of this study and they are used for modelling the recorded tool vibration. In estimation of tool/holder system non-stationary response measured by an accelerometer, output-only FS-TARMA method can be used. Although it is a very effective method, it has never been used in identification of tool-holder dynamics and tool wear estimation. For the modelling and analysis of non-stationary signals, two types of methods can be used, non-parametric methods and parametric methods. Parametric methods are mainly consisting of parameterized versions of Time-Dependent AutoRegressive Moving Average (TARMA). The difference between these versions and ARMA counterparts is that in the former models, the parameters are time-dependent. Parametric methods have some advantages over the non-parametric counterparts such as representation parsimony, improved accuracy, improved frequency resolution, improved tracking of time-dependent dynamics, and flexibility in analysis [28]. Parametric methods can be classified as unstructured and structured, stochastic and deterministic parameter evolution model considering the “structure” that is imposed upon time evolution of the model parameters [28]. Deterministic parameter evolution methods impose deterministic “structure” upon the evolution of TV parameters. Functional Series TAR and TARMA (FS-TAR and FS-TARMA) types are of the main version of these methods and employ
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