Title:

Study of the best linear approximation of nonlinear systems with arbitrary inputs

System identification is the art of modelling of a process (physical, biological, etc.) or to predict its behaviour or output when the environment condition or parameter changes. One is modelling the inputoutput relationship of a system, for example, linking temperature of a greenhouse (output) to the sunlight intensity (input), power of a car engine (output) with fuel injection rate (input). In linear systems, changing an input parameter will result in a proportional increase in the system output. This is not the case in a nonlinear system. Linear system identification has been extensively studied, more so than nonlinear system identification. Since most systems are nonlinear to some extent, there is significant interest in this topic as industrial processes become more and more complex. In a linear dynamical system, knowing the impulse response function of a system will allow one to predict the output given any input. For nonlinear systems this is not the case. If advanced theory is not available, it is possible to approximate a nonlinear system by a linear one. One tool is the Best Linear Approximation (Bla), which is an impulse response function of a linear system that minimises the output differences between its nonlinear counterparts for a given class of input. The Bla is often the starting point for modelling a nonlinear system. There is extensive literature on the Bla obtained from input signals with a Gaussian probability density function (p.d.f.), but there has been very little for other kinds of inputs. A Bla estimated from Gaussian inputs is useful in decoupling the linear dynamics from the nonlinearity, and in initialisation of parameterised models. As Gaussian inputs are not always practical to be introduced as excitations, it is important to investigate the dependence of the Bla on the amplitude distribution in more detail. This thesis studies the behaviour of the Bla with regards to other types of signals, and in particular, binary sequences where a signal takes only two levels. Such an input is valuable in many practical situations, for example where the input actuator is a switch or a valve and hence can only be turned either on or off. While it is known in the literature that the Bla depends on the amplitude distribution of the input, as far as the author is aware, there is a lack of comprehensive theoretical study on this topic. In this thesis, the Blas of discretetime timeinvariant nonlinear systems are studied theoretically for white inputs with an arbitrary amplitude distribution, including Gaussian and binary sequences. In doing so, the thesis offers answers to fundamental questions of interest to system engineers, for example: 1) How the amplitude distribution of the input and the system dynamics affect the Bla? 2) How does one quantify the difference between the Bla obtained from a Gaussian input and that obtained from an arbitrary input? 3) Is the difference (if any) negligible? 4) What can be done in terms of experiment design to minimise such difference? To answer these questions, the theoretical expressions for the Bla have been developed for both WienerHammerstein (Wh) systems and the more general Volterra systems. The theory for the Wh case has been verified by simulation and physical experiments in Chapter 3 and Chapter 6 respectively. It is shown in Chapter 3 that the difference between the Gaussian and nonGaussian Bla’s depends on the system memory as well as the higher order moments of the nonGaussian input. To quantify this difference, a measure called the Discrepancy Factor—a measure of relative error, was developed. It has been shown that when the system memory is short, the discrepancy can be as high as 44.4%, which is not negligible. This justifies the need for a method to decrease such discrepancy. One method is to design a random multilevel sequence for Gaussianity with respect to its higher order moments, and this is discussed in Chapter 5. When estimating the Bla even in the absence of environment and measurement noise, the nonlinearity inevitably introduces nonlinear distortions—deviations from the Bla specific to the realisation of input used. This also explains why more than one realisation of input and averaging is required to obtain a good estimate of the Bla. It is observed that with a specific class of pseudorandom binary sequence (Prbs), called the maximum length binary sequence (Mlbs or the msequence), the nonlinear distortions appear structured in the time domain. Chapter 4 illustrates a simple and computationally inexpensive method to take advantage this structure to obtain better estimates of the Bla—by replacing mean averaging by median averaging. Lastly, Chapters 7 and 8 document two independent benchmark studies separate from the main theoretical work of the thesis. The benchmark in Chapter 7 is concerned with the modelling of an electrical Wh system proposed in a special session of the 15th International Federation of Automatic Control (Ifac) Symposium on System Identification (Sysid) 2009 (Schoukens, Suykens & Ljung, 2009). Chapter 8 is concerned with the modelling of a ‘hyperfast’ Peltier cooling system first proposed in the U.K. Automatic Control Council (Ukacc) International Conference on Control, 2010 (Control 2010).
