Research Doctorate - Doctor of Philosophy (PhD) / This Thesis addresses the problem of robust experiment design, i.e., how to design an input signal to maximise the amount of information obtained from an experiment given limited prior knowledge of the true system. The majority of existing literature on experiment design specifically considers optimal experiment design, which, typically depends on the true system parameters, that is, the very thing that the experiment is intended to find. This obviously gives rise to a paradox. The results presented in this Thesis, on robust experiment design, are aimed at resolving this paradox. In the robust experiment design problem, we assume that the parameter vector is a-priori known to belong to a given compact set, and study the design of an input spectrum which maximises the worst case scenario over this set. We also analyse the problem from a different perspective where, given the same assumption on the parameter vector, we examine cost functions that give rise to an optimal input spectrum independent of the true system features. As a first approach to this problem we utilise an asymptotic (in model order) expression for the variance of the system transfer function estimator. To enable the extension of these results to finite order models, we digress from the main topic and develop several fundamental integral limitations on the variance of estimated parametric models. Based on these results, we then return to robust experiment design, where the input design problems are reformulated using the fundamental limitations as constraints. In this manner we establish that our previous results, obtained from asymptotic variance formulas, are valid also for finite order models. Robustness issues in experiment design also arise in the area of `identification for (robust) control'. In particular, a new paradigm has recently been developed to deal with experiment design for control, namely `least costly experiment design'. In the Thesis we analyse least costly experiment design and establish its equivalence with the standard formulation of experiment design problems. Next we examine a problem involving the cost of complexity in system identification. This problem consists of determining the minimum amount of input power required to estimate a given system with a prescribed degree of accuracy, measured as the maximum variance of its frequency response estimator over a given bandwidth. In particular, we study the dependence of this cost on the model order, the required accuracy, the noise variance and the size of the bandwidth of interest. Finally, we consider the practical problem of how to optimally generate an input signal given its spectrum. Our solution is centered around a Model Predictive Control (MPC) based algorithm, which is straightforward to implement and exhibits fast convergence that is empirically verified.

Research Doctorate - Doctor of Philosophy (PhD) / This Thesis addresses the problem of robust experiment design, i.e., how to design an input signal to maximise the amount of information obtained from an experiment given limited prior knowledge of the true system. The majority of existing literature on experiment design specifically considers optimal experiment design, which, typically depends on the true system parameters, that is, the very thing that the experiment is intended to find. This obviously gives rise to a paradox. The results presented in this Thesis, on robust experiment design, are aimed at resolving this paradox. In the robust experiment design problem, we assume that the parameter vector is a-priori known to belong to a given compact set, and study the design of an input spectrum which maximises the worst case scenario over this set. We also analyse the problem from a different perspective where, given the same assumption on the parameter vector, we examine cost functions that give rise to an optimal input spectrum independent of the true system features. As a first approach to this problem we utilise an asymptotic (in model order) expression for the variance of the system transfer function estimator. To enable the extension of these results to finite order models, we digress from the main topic and develop several fundamental integral limitations on the variance of estimated parametric models. Based on these results, we then return to robust experiment design, where the input design problems are reformulated using the fundamental limitations as constraints. In this manner we establish that our previous results, obtained from asymptotic variance formulas, are valid also for finite order models. Robustness issues in experiment design also arise in the area of `identification for (robust) control'. In particular, a new paradigm has recently been developed to deal with experiment design for control, namely `least costly experiment design'. In the Thesis we analyse least costly experiment design and establish its equivalence with the standard formulation of experiment design problems. Next we examine a problem involving the cost of complexity in system identification. This problem consists of determining the minimum amount of input power required to estimate a given system with a prescribed degree of accuracy, measured as the maximum variance of its frequency response estimator over a given bandwidth. In particular, we study the dependence of this cost on the model order, the required accuracy, the noise variance and the size of the bandwidth of interest. Finally, we consider the practical problem of how to optimally generate an input signal given its spectrum. Our solution is centered around a Model Predictive Control (MPC) based algorithm, which is straightforward to implement and exhibits fast convergence that is empirically verified.

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We discuss a formulation for active example selection for function learning problems. This formulation is obtained by adapting Fedorov's optimal experiment design to the learning problem. We specifically show how to analytically derive example selection algorithms for certain well defined function classes. We then explore the behavior and sample complexity of such active learning algorithms. Finally, we view object detection as a special case of function learning and show how our formulation reduces to a useful heuristic to choose examples to reduce the generalization error.

O método de identificação por minimização do erro de predição está relacionado com um problema de otimização não convexo. É comum utilizar algoritmos iterativos para resolver o problema de otimização. Contudo, os algoritmos iterativos podem ficar presos em mínimos locais da função custo ou convergir para a borda do domínio de busca. Uma análise da função custo e condições suficientes para garantir a convergência dos algoritmos iterativos para o mínimo global são apresentadas neste trabalho. Observa-se que estas condições dependem do espectro do sinal de entrada utilizado no experimento. Este trabalho apresenta ferramentas para melhorar a convergência dos algoritmos para o mínimo global, as quais são baseadas na manipulação do espectro do sinal de entrada. / The Prediction Error Method is related to a non-convex optimization problem. It is usual to apply iterative algorithms to solve this optimization problem. However, iterative algorithms can get stuck at a local minimum of the cost function or converge to the border of the searching space. An analysis of the cost function and sufficient conditions to ensure the convergence of the iterative algorithms to the global minimum are presented in this work. It is observed that this conditions depend on the spectrum of the input signal used in the experiment. This work presents tools to improve the convergence of the algorithms to the global minimum, which are based on the manipulation of the input spectrum.

O método de identificação por minimização do erro de predição está relacionado com um problema de otimização não convexo. É comum utilizar algoritmos iterativos para resolver o problema de otimização. Contudo, os algoritmos iterativos podem ficar presos em mínimos locais da função custo ou convergir para a borda do domínio de busca. Uma análise da função custo e condições suficientes para garantir a convergência dos algoritmos iterativos para o mínimo global são apresentadas neste trabalho. Observa-se que estas condições dependem do espectro do sinal de entrada utilizado no experimento. Este trabalho apresenta ferramentas para melhorar a convergência dos algoritmos para o mínimo global, as quais são baseadas na manipulação do espectro do sinal de entrada. / The Prediction Error Method is related to a non-convex optimization problem. It is usual to apply iterative algorithms to solve this optimization problem. However, iterative algorithms can get stuck at a local minimum of the cost function or converge to the border of the searching space. An analysis of the cost function and sufficient conditions to ensure the convergence of the iterative algorithms to the global minimum are presented in this work. It is observed that this conditions depend on the spectrum of the input signal used in the experiment. This work presents tools to improve the convergence of the algorithms to the global minimum, which are based on the manipulation of the input spectrum.

O método de identificação por minimização do erro de predição está relacionado com um problema de otimização não convexo. É comum utilizar algoritmos iterativos para resolver o problema de otimização. Contudo, os algoritmos iterativos podem ficar presos em mínimos locais da função custo ou convergir para a borda do domínio de busca. Uma análise da função custo e condições suficientes para garantir a convergência dos algoritmos iterativos para o mínimo global são apresentadas neste trabalho. Observa-se que estas condições dependem do espectro do sinal de entrada utilizado no experimento. Este trabalho apresenta ferramentas para melhorar a convergência dos algoritmos para o mínimo global, as quais são baseadas na manipulação do espectro do sinal de entrada. / The Prediction Error Method is related to a non-convex optimization problem. It is usual to apply iterative algorithms to solve this optimization problem. However, iterative algorithms can get stuck at a local minimum of the cost function or converge to the border of the searching space. An analysis of the cost function and sufficient conditions to ensure the convergence of the iterative algorithms to the global minimum are presented in this work. It is observed that this conditions depend on the spectrum of the input signal used in the experiment. This work presents tools to improve the convergence of the algorithms to the global minimum, which are based on the manipulation of the input spectrum.

Closed-loop system identification and validation are important components in dynamic system modelling. In this dissertation, a comprehensive literature survey is compiled on system identification with a specific focus on closed-loop system identification and issues of identification experiment design and model validation. This is followed by simulated experiments on known linear and non-linear systems and experiments on a pilot scale distillation column. The aim of these experiments is to study several sensitivities between identification experiment variables and the consequent accuracy of identified models and discrimination capacity of validation sets given open and closed-loop conditions. The identified model structure was limited to an ARX structure and the parameter estimation method to the prediction error method. The identification and validation experiments provided the following findings regarding the effects of different feedback conditions: <ul> <li>Models obtained from open-loop experiments produced the most accurate responses when approximating the linear system. When approximating the non-linear system, models obtained from closed-loop experiments were found to produce the most accurate responses.</li> <li>Validation sets obtained from open-loop experiments were found to be most effective in discriminating between models approximating the linear system while the same may be said of validation sets obtained from closed-loop experiments for the nonlinear system.</li> </ul> These finding were mostly attributed to the condition that open-loop experiments produce more informative data than closed-loop experiments given no constraints are imposed on system outputs. In the case that system output constraints are imposed, closed-loop experiments produce the more informative data of the two. In identifying the non-linear system and the distillation column it was established that defining a clear output range, and consequently a region of dynamics to be identified, is very important when identifying linear approximations of non-linear systems. Thus, since closed-loop experiments produce more informative data given output constraints, the closed-loop experiments were more effective on the non-liner systems. Assessment into other identification experiment variables revealed the following: <ul> <li>Pseudo-random binary signals were the most persistently exciting signals as they were most consistent in producing models with accurate responses.</li> <li>Dither signals with frequency characteristics based on the system’s dominant dynamics produced models with more accurate responses.</li> <li>Setpoint changes were found to be very important in maximising the generation of informative data for closed-loop experiments</li></ul> Studying the literature surveyed and the results obtained from the identification and validation experiments it is recommended that, when identifying linear models approximating a linear system and validating such models, open-loop experiments should be used to produce data for identification and cross-validation. When identifying linear approximations of a non-linear system, defining a clear output range and region of dynamics is essential and should be coupled with closed-loop experiments to generate data for identification and cross-validation. / Dissertation (MEng)--University of Pretoria, 2009. / Chemical Engineering / unrestricted

System identification concerns how to construct mathematical models of dynamic systems based on experimental data. A very important application of system identification is in model-based control design. In such applications it is often possible to externally excite the system during the data collection experiment. The properties of the exciting input signal influence the quality of the identified model, and well-designed input signals can reduce both the experimental time and effort. The objective of this thesis is to develop algorithms and theory for minimum cost experiment design for system identification while guaranteeing that the estimated model results in an acceptable control performance. We will use the framework of application oriented Optimal Input Design (OID). First, we study how to find a convex approximation of the set of models that results in acceptable control performance. The main contribution is analytical methods to determine application sets for controllers with no explicit control law, for instance Model Predictive Control (MPC). The application oriented OID problem is then formulated in time domain to enable the handling of signals constraints, which often comes from the physical limitations on the plant and actuators. The framework is the extended to closed-loopsystems. Here two different cases are considered. The first case assumes that the plant is controlled by a general (either linear or non-linear) but known controller. The main contribution here is a method to design an external stationary signal via graph theory such that the identification requirements and signal constraints are satisfied. In the second case application oriented OID problem is studied for MPC. The proposed approach here is a modification of a results where the experiment design requirements are integrated to the MPC as a constraint. The main idea is to back off from the identification requirements when the control requirements are violating from their acceptable bounds. We evaluate the effectiveness of all the proposed algorithms by several simulation examples. / <p>QC 20150126</p>

<p>Nanofluids are engineered colloidal dispersions of nanoparticles in a liquid. The field of nanofluids has seen much interest due to reported heat transfer enhancements over the corresponding pure fluids at low particle concentrations. Particularly, a large increase in critical heat flux (CHF) has been widely reported along with modification of the boiling interface. Inconsistencies in reported impact on nucleate boiling heat transfer and the degree of CHF enhancement illustrate the need for further study.</p> <p>A pool boiling experiment has been designed and constructed at McMaster University to allow for the study the boiling of water-based nanofluids. The facility has been commissioned with saturated distilled water tests at atmospheric pressure, heat flux levels up to 1200 kW·m^-2, and at wall superheat levels up to 19.5^oC. Wall superheat and heat flux uncertainties were estimated to be ±0.6^oC and ±20 kW∙m^-2, respectively. For the installed test section, heat flux is limited to 2.62 ± 0.06 MW·m^-2. A high speed video system for the analysis of bubble dynamics was tested and used for qualitative comparisons between experimental runs. This system was tested at 2500 FPS and an imaging resolution of 39 pixels per mm, but is capable of up to 10 000 FPS at the same spatial resolution. Heat flux versus wall superheat data was compared to the Rohsenow correlation and found to qualitatively agree using surface factor <em>C_sf</em> = 0.011. Results were found to have a high degree of repeatability at heat flux levels higher than 600 kW·m^-2.</p> <p>The new facility will be used to conduct studies into the pool boiling of saturated water-based nanofluids at atmospheric pressure. Additional work will involve the control and characterization of heater surface conditions before and after boiling. Quantitative analysis of bubble dynamics will be possible using high speed video and particle image velocimetry.</p> / Master of Applied Science (MASc)