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−8 −6 −4 −2 0 2 4 6 Time (s) Temp erature ( °C) Simulated True

Figure 8.7: Intermediate signal forSubsystem 2 for simulation and experiment data

that the contribution of Subsystem 1 to the overall system output is considerably smaller than that ofSubsystem 2. This means that the model developed provided a good fit to the overall system output for both the training data and the validation data, even though the modelling of Subsystem 1was not particularly accurate.

8.4

Comparison with other approaches

Results from other approaches on the same benchmark exercise are given in Table

8.1. It can be seen that the current ‘hybrid empirical and mechanistic’ approach—a mix between blackbox and whitebox modelling, has resulted in good performance while retaining the physical interpretations of the system structure.

Table 8.1: Comparison between several models

Source Model Mae Mse

(×10 ) (×10 ) Cham, Tan and Ramar

(2010)

Variable time delay 59.1 5.71 Larkowski and Burnham

(2010)

Box-Jenkins 56.5 5.05 Marconato et al. (2010) Parametric Best Linear

Approximation(Bla)

73.2 8.40

Taylor and Young (2010) Data-based mechanistic 81 10 Wong and Godfrey

(2010)

Hybrid empirical and mechanistic

8.5. Conclusions

8.5

Conclusions

This chapter is concerned with the the ‘hyperfast’ switching Peltier cooling system benchmark, first proposed in the U.K. Automatic Control Council (Ukacc) Inter- national Conference on Control, 2010 (Control 2010). The chapter has illustrated how a simple modelling exercise using Matlab Simulink can provide valuable in- sights into the structure of a system such as the time delay of a signal path, when some physical information is known beforehand. The models forSubsystems 2 and

3 were linear. The principal nonlinearities in the actual system were the stepper motor and the steady-state gain characteristic between angle and flow rate inSub- system 1 as stated in Cham, Tan and Tan (2010), which could be modelled readily in Simulink, and this meant that the other parts of modelling could be reduced to a simple linear fitting exercise.

It was necessary to make small changes to the gains 𝐾 and 𝐾 of the paths from Subsystems 1 and 2 and to the temperature offset 𝑇_o in the model between the training data and the validation data. This is not unreasonable, since the two sets of data were collected at different times, and in all probability, the ambient conditions were not quite the same between the measurements.

C

9

Conclusions and Future Work

All systems are nonlinear to some extent, and while nonlinear modelling techniques exist, it is sometimes preferable to apply linearisation techniques as linear models, especially when applied to control theory, are well understood. The Best Linear Approximation (Bla) is one linearisation technique that achieves the least total sum squared error when approximating a time-invariant nonlinear system.

It is known that the Best Linear Approximation (Bla) is dependent upon the power spectrum and the amplitude distribution of the input. TheBlaobtained from input signals or sequences having a Gaussian amplitude distribution, called the Gaussian Bla, has certain desirable properties, some of which are well known. For example, in the frequency domain, theBlais composed of the combined linear dynamics of all the linearities in a block-structured system, so that it is possible to separate them in terms of their poles and zeros (see Chapter 7 for a modelling exercise utilising this property). However, this is not the case for a Bla estimated by non-Gaussian inputs.

The dependence on the amplitude distribution has not previously been stud- ied in detail in the literature. In Chapter3of this thesis, the theoretical closed-form expressions were derived for the Blas for both discrete-timeWiener-Hammerstein

(Wh) systems and discrete-time Volterra systems for inputs with an arbitrary amp- litude distribution. The theory is valid for binary and ternary sequences for example, but they must have white power spectra and zero mean. In practice, the zero-mean requirement is not of concern, as having non-zero mean only introduces a constant bias in the operating point about which the Bla is linearised, and does not affect the estimated dynamics. The developed theory has been verified by both simulation experiments in Chapter 3 as well as a set of physical experiments performed on an

9.1. Research impact

electronic Wiener system in Chapter 6. The theory has shown that the Gaussian and non-GaussianBlaindeed differ, and the discrepancy is due to the higher order moment terms of the input. A relative measure to quantify the discrepancy, called the Discrepancy Factor (Df), was proposed in Section 3.5. It was found that in general, theDf decreases with the memory of the linear dynamics of the nonlinear system. In addition, a method was proposed to design discrete multilevel sequences for Gaussianity, in order to minimise the Df. This was shown in Chapter 5 to be viable and effective; more so than the commonly used uniformly distributed se- quences. This has obvious implications in experiment design if the objective is to extract the GaussianBla, especially when the principal degrees of the nonlinearities are known.

Lastly, Chapters 7and 8document benchmark studies which provided valu- able training opportunities for the author in system identification and as general contributions to the academic discussion in the conferences where the work was presented.

9.1

Research impact

There are systems where applying continuous level signals, for example, Gaussian inputs, is either impractical or impossible. For instance, if the actuator of a system is a valve that can only be opened or closed, only a binary input may be applied. The thesis has provided theoretical insights to the identification of nonlinear sys- tems with respect to the amplitude distribution of the input. The discrete-time theory of theBladeveloped in Chapter3is applicable to a large variety of systems, thanks to the generalisation of the closed-form expressions to the Volterra theory. In addition, the amplitude distribution of the inputs is arbitrary, allowing the the- oreticalBlato be studied for any given input. This piece of theoretical work sheds light on the behaviour of the Bla when the amplitude distribution (or the higher order moments) of the input are non-Gaussian. This, coupled with the method of designing multilevel sequences for Gaussianity in Chapter 5, allow the Gaussian Bla to be estimated more accurately with discrete-level signals. The research can benefit potential applications where discrete or digital input signals predominate, for instance, identification of digital communication channels, characterisation of delta-sigma digital-to-analogue (Dac) converters, chemical processes involving valve control or heating elements, and micro-fluidics in micro-electromechanical (Mem) systems.

9.2. Future work