Study of the Best Linear Approximation of Nonlinear Systems with Arbitrary Inputs

Student thesis: Doctoral Thesis


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 input-output 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 ofthen 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 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 discrete-time time-invariant 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.
Date of Award21 Jun 2013
Original languageEnglish
SupervisorJoannes Schoukens (Promotor), Steve Vanlanduit (Jury), Johan Deconinck (Jury), Gerd Vandersteen (Jury), Keith Richard Godfrey (Promotor), Nigel G. Stocks (Promotor), Tadeusz Dobrowiecki (Jury), Peter Jones (Jury) & J. Antoni (Jury)


  • Best Linear Approximation
  • Nonlinear Systems

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