Modelling of systems is focused on approaches adopted for the determination of the math- ematical relationships existing between the input and output of any given system whether linear or non-linear, single-input single-output, or multi-input multi-output systems. Such a model adequate for control purposes is usually a compromise arising from many factors, thus a perfect model of a system is only an idealisation as a result of practical exigencies. A system model is often regarded as the mathematical formulation which relates the system output to its past input(s), and/or past output(s). A good system model often makes it possible to infer peculiarities, nature and properties of the system it represents, and also to predict future output(s). Hence it is a pre-condition for system analysis and controller designs. System mod- elling is thus the process of determining the best mathematical structure representative of the system in question. This area of research which is a subset of the control theory has evolved over the second half of the last century and attracted much research attention [75], [76]. Linear systems modelling is therefore a mature research area with established techniques and algorithms unlike non-linear systems even though almost all real life scenarios are non-linear. In this section an attempt is made to discuss some of the general trends in the modelling of systems with a view to understanding their strengths and weaknesses and tools which can be modified and/or adopted in the present research. As such, modelling of systems and the
various approaches adopted in recent years are studied as they relate to friction. In system modelling one pertinent question that must be answered a priori is the purpose to which the model would be deployed. This is very important due to the fact that research indicates that there exists no ”perfect” model of any real system since most systems are inherently non-linear. Rather models which are useful for specific purposes can be obtained usually as a compromise among many variables where the increased accuracy in determination of one variable leads to increase in the uncertainty of another, [77]. Most modelling activities and dynamic model applications have been greatly motivated and influenced by some underlying pre-cursors, which are;
1. Satisfaction of scientific curiosity
2. Explanation of system behaviour
3. Simulation and validation (optimization)
4. Prediction and control
5. Fault detection and diagnosis
2.4.1
System identification techniques
Some approaches have been adopted over the years in the modelling of systems in general and some of these techniques are discussed next. Most of the techniques adopted in system modelling can be grouped into three main categories depending on the level of physical knowledge of the system taken into account. These techniques are;
1. The white-box,
2. The black-box, and
3. The grey-box.
The white box and the black box identification techniques to modelling are the two extremes adopted in system modelling while more practical models often are with shades of grey in between these extremes.
White-box approach
In white box method, a model of the system is derived from first principles considering the physical, biological, chemical parameters and components influencing the system and their
interactions. Because of the many considerations this phenomenological method often gives rise to very complex models which often are computationally impossible or expensive at the best. Sometimes adequate model structures of complex systems may even prove impossible to obtain in reasonable time frame owing to complexities in the systems, processes and their interactions. This may be feasible for some very simple systems and processes, however as the system increases in complexity it becomes nearly impossible to adopt this identification approach. The main appeal of this method is the fact that the entire model reflects the true nature of the system and the corresponding parameters are of real physical interpretation. As a result of this difficulty in modelling complex systems, the black box and grey box modelling are better alternatives for modelling complex systems, [77], [78]. An example of such modelling approach adopted for friction is illustrated in [24].
Black-box approach
Another approach is to deduce system models based on the input-output data set obtained by measurements without any attribution or emphasis on the exact nature and physics of the system itself. This makes the black box method empirical and reduces the modelling problem to that of identification of the existing relationship between input-output data from which future output could be inferred. This pre-supposes the possibility of obtaining input and output data through measurement procedures. The downside to this approach is in the fact that it entirely relies on the authenticity of the data used, thus the model can at best be as good as the data used for modelling. Relying on this data makes the method dependent on some factors like the type of devices used for data acquisition, and analysis, this process could thus become subjective. other demerits of this approach are; it yields opaque models often without mathematical interpretation, difficulties in model analysis and the isolation of cause and effect is not easy. This method without a priori knowledge of the system is termed the black box approach, [79], [80]. Typical examples of friction model structures obtain through this technique are the machine based models of section 2.2
Grey-box approach
Often-times a good or partial understanding of the system under study is of immense help in determining the structure of system’s model. A combination of this part knowledge of the system (phenomenological) and input-output identification (empirical) process often leads to models that are somewhat in-between these extremes of black and white. These models are therefore called Grey models, and different shades of grey depicts the extent of physical insight involved. They consider some properties and physical nature of the system
in deducing the model structure [77], [81]. A typical example is when a model structure has been deduced for a given system based on the white-box approach and its parameters are estimated through identification (black-box) method.
2.4.2
Non-linear systems structure
Many variations of non-linear systems such as friction have been studied and reported as indicated by the volume of publications in this regard. However, for the present research interest we categorise them into four main groups.
The Volterra series model representation; In this representation a functional called the Volterra functional comprising a convolution integral and Taylor series is adopted for the model structure. The convolution integral represents the linear part of the system while the static non-linearity is captured by the Taylor series. Parameters estimation are often computationally involving because of the exponential relation between the number of pa- rameter and the degree of the kernel. This and the fact that it cannot be used to model some non-linearities such as saturation, backlash, dead zone and hysteresis are the major drawbacks of this model which resulted in the Wiener series model. A tensor based approach has been proposed to reduce the effect of complexities associated with parameter estimation such as the Volterra-Parafac model and the Volterra-Laguerre model, [82], [83].
Block oriented model representation; It has been possible to formulate models of systems in block formation relating the inputs and outputs, these blocks can be combinations of linear dynamic and non-linear static blocks interconnected in cascade or in parallel. Typical examples of this formulation are the Hammerstein and Wiener block models. In the former the linear block is after the static non-linear block in a series connection while the reverse is the case for the later. The Wiener formulation is better able to handle non-linearities resulting from input amplitude induced dynamics. The main appeal of the block diagram approach is its simplicity, however, the requirement of some level of a priori system knowledge and restriction on the nature of excitation used place great limitations to the structure, [84], [81]. Neural Network model representation; Neural Networks have drawn a lot of research attention in system modelling mainly due to their excellent approximation property. The radial basis function (RBF) and the multi-layer perceptions (MLP) are the most widely used feed-forward Neural Networks for modelling many industrial processes [79], [85].
NARMAX model representation; In Non-linear Auto-Regressive Moving Average with eX- ogenous input (NARMAX) model representation, an extension of the ARMAX model for linear systems is a form of system representation with non-linear difference equations or polynomials. This form has generated much interest as a result of these qualities;
1. No restrictions to the nature of signal inputs unlike the others previously studied
2. Its model parameters are linear thus rendering analysis simpler
3. Physical interpretations of the models are possible under certain conditions
4. It allows for integration of other formulations like the block oriented, and Volterra series model structures
The NARMAX model structure has successfully been applied to systems modelling and identification problems as illustrated in the following.
Piece-wise linear models; These are representations of approximations of the system non- linearities localized at various regions of operation. This technique has also be used often where it is nearly impossible to obtain a single model representation of the system capturing most of the system characteristics. One major issue with this model is the determination of the regions of model validity.
2.4.3
The system identification process
The basic motivation in system identification is; given a model structure of a system to obtain the estimation of model parameters adequate for replication of the system from the input-output data set. The models obtained could therefore be;
1. Linear or non-linear
2. Static or dynamic
3. Continuous-time or discrete-time
4. Deterministic or stochastic
5. Parametric (lumped or distributed) or non-parametric
6. Phenomenological or representative
Insight and prior knowledge of a system are often helpful in the determination of an adequate model structure for the system and estimation of its parameters to fit the data to a given model structure. Many a time identification problems reduce to curve fitting optimisation problems, minimizing the difference between the measured output data and the model structure often in a square-error-sense. Minimization of a cost function subject to some constraints can often take any of the many forms in the previous section depending on the purpose and the level of physical insight employed.
In chapter 4, a carefully designed set of experiments would be performed for the purpose of using the input-output data from such experiments to identify the parameters of the proposed model structure and some other model of friction. In the next section a review of friction compensation and control schemes adopted for frictional systems is undertaken since one of the research objectives is to identify models adequate for the purposes of control.