The Library
Study of the best linear approximation of nonlinear systems with arbitrary inputs
Tools
Wong, Hin Kwan Roland (2013) Study of the best linear approximation of nonlinear systems with arbitrary inputs. PhD thesis, University of Warwick.

Text
WRAP_THESIS_Wong_2013.pdf  Submitted Version Download (3063Kb)  Preview 
Official URL: http://webcat.warwick.ac.uk/record=b2690389~S1
Abstract
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).
Item Type:  Thesis or Dissertation (PhD)  

Subjects:  Q Science > QA Mathematics T Technology > TA Engineering (General). Civil engineering (General) 

Library of Congress Subject Headings (LCSH):  Nonlinear systems, System analysis, Approximation theory  
Official Date:  January 2013  
Dates: 


Institution:  University of Warwick  
Theses Department:  School of Engineering  
Thesis Type:  PhD  
Publication Status:  Unpublished  
Supervisor(s)/Advisor:  Stocks, Nigel G.; Godfrey, Keith; Schoukens, J. (Johan)  
Description:  Completed in conjunction with Vrije Universiteit Brussel. Faculty of Engineering 

Extent:  xxviii, 190 leaves : illustrations  
Language:  eng 
Request changes or add full text files to a record
Repository staff actions (login required)
View Item 
Downloads
Downloads per month over past year