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Grey-Box Modelling for Nonlinear Systems

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Grey-Box Modelling for Nonlinear Systems i i i i Dissertation im Fachbereich Mathematik der Technischen Universität Kaiserslautern Grey-Box Modelling for Nonlinear Systems Jan Hauth Dezember 2008 1. Gutachter: Prof. Dr. Dieter Prätzel-Wolters, Technische Universität Kaiserslautern 2. Gutachter: Prof. Dr. Jürgen Franke, Technische Universität Kaiserslautern Datum der Disputation: 5. Juni 2008 Vom Fachbereich Mathematik der Universität Kaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften (Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation D 386 i i i i i i i i i i i i i i i i To my parents Irma born Scholtes and Kurt Hauth In memoriam Prof. Walter Blankenheim i i i i i i i i i i i i i i i i Abstract Grey-box modelling deals with models which are able to integrate the following two kinds of information: qualitative (expert) knowledge and quantitative (data) knowledge, with equal importance. The doctoral thesis has two aims: the improvement of an existing neuro-fuzzy ap- proach (LOLIMOT algorithm), and the development of a new model class with corresponding identification algorithm, based on multiresolution analysis (wavelets) and statistical methods. The identification algorithm is able to identify both hidden differential dynamics and hysteretic components. After the presentation of some improvements of the LOLIMOT algorithm based on readily normalized weight functions derived from decision trees, we investigate several mathematical theories, i.e. the theory of nonlinear dynamical systems and hysteresis, statistical decision the- ory, and approximation theory, in view of their applicability for grey-box modelling. These theories show us directly the way onto a new model class and its identification algorithm. The new model class will be derived from the local model networks through the following modifications: Inclusion of non-Gaussian noise sources; allowance of internal nonlinear dif- ferential dynamics represented by multi-dimensional real functions; introduction of internal hysteresis models through two-dimensional “primitive functions”; replacement respectively approximation of the weight functions and of the mentioned multi-dimensional functions by wavelets; usage of the sparseness of the matrix of the wavelet coefficients; and identification of the wavelet coefficients with Sequential Monte Carlo methods. We also apply this modelling scheme to the identification of a shock absorber. v i i i i i i i i vi i i i i i i i i Abstrakt Grey-Box-Modellierung beschäftigt sich mit Modellen, die in der Lage sind folgende zwei Arten von Information über ein reales System gleichbedeutend einzubeziehen: qualitatives (Experten-)Wissen, und quantitatives (Daten-)Wissen. Die Dissertation hat zwei Ziele: die Verbesserung eines existierenden Neuro-Fuzzy-Ansatzes (LOLIMOT-Algorithmus); und die Entwicklung einer neuen Modellklasse mit zugehörigem Identifikations-Algorithmus, basie- rend auf Multiskalenanalyse (Wavelets) und statistischen Methoden. Der resultierende Iden- tifikationsalgorithmus ist in der Lage, sowohl verborgene Differentialdynamik als auch hyste- retische Komponenten zu identifizieren. Nach der Vorstellung einiger Verbesserungen des LOLIMOT-Algorithmus basierend auf von vorneherein normalisierten Gewichtsfunktionen, die auf einer Konstruktion mit Entschei- dungsbäumen beruhen, untersuchen wir einige mathematische Theorien, das sind die Theorie nichtlinearer Systeme und Hysterese, statistische Entscheidungstheorie and Approximations- theorie, im Hinblick auf deren Anwendbarkeit für Grey-Box-Modellierung. Diese Theorien führen dann auf direktem Wege zu einer neuen Modellklasse und deren Identifikationsalgo- rithmus. Die neue Modellklasse wird von Lokalmodellnetzwerken durch folgende Modifika- tionen abgeleitet: Einbeziehung von nicht-Gaußschen Rauschquellen; Zulassung von inter- ner nichtlinearer Differentialdynamik repräsentiert durch mehrdimensionale reelle Funktio- nen; Einführung interner Hysterese-Modelle mittels zweidimensionaler „Stammfunktionen“; Ersetzung bzw. Approximation der Gewichtsfunktionen und der erwähnten mehrdimensiona- len Funktionen durch Wavelet-Koeffizienten; Ausnutzung der Dünnbesetztheit der Wavelet- Koeffizienten-Matrix; und Identifikation der Wavelet-Koeffizienten mit Sequentiellen Monte Carlo-Methoden. Wir wenden dieses Modellierungsschema dann auf die Identifikation eines Stoßdämpfers an. vii i i i i i i i i viii i i i i i i i i Contents Thanks xv Overview xvii Notations xxv 1 Introduction: Grey-box models and the LOLIMOT algorithm 1 1.1 Systemsandmodels............................... 2 1.1.1 Nonlineardynamicalsystemsand modelschemes . ..... 2 1.1.2 Propertiesofsystemsandmodels . 6 1.1.3 Separationofdynamics. .. .. .. .. .. .. .. 8 1.1.4 Linear combinations of basis functions and networks . ........ 11 1.2 Localmodelnetworks.............................. 14 1.3 TheLOLIMOTalgorithm............................ 19 1.4 Problemsandpossibleimprovements. ..... 25 1.4.1 Decisiontreesundweightfunctions . ... 30 1.4.2 Gradientbasedoptimization . 44 1.4.3 Applications of the gradient based optimization to the improvement oftheLOLIMOTalgorithm . 57 2 Dealing with time: Dynamics 63 2.1 Deterministicmodelsfordynamicalsystems . ........ 64 2.2 Preisachhysteresis .............................. 74 2.2.1 Definitionandproperties . 75 2.2.2 Implementation............................. 93 2.2.3 Identification .............................. 109 2.3 Conclusions................................... 113 3 Stochastic decision theory: Bridge between theory and reality 117 3.1 Modelsforreality................................ 118 3.2 Bayesianstatistics. .. .. .. .. .. .. .. .. 123 3.2.1 Bayes’theorem............................. 124 3.2.2 Foundationsofdecisiontheory. 126 3.2.3 JustificationsforBayesianinference . ..... 135 3.3 Priors ...................................... 140 3.3.1 Strategiesforpriordetermination . .... 140 3.3.2 HierarchicalBayes .. .. .. .. .. .. .. .. 147 ix i i i i i i i i Contents 3.4 StochasticmodelsandBayesianestimation . ....... 150 3.4.1 Staticnormalmodels . 150 3.4.2 Dynamicmodels ............................ 153 3.4.3 Markovchains ............................. 158 3.4.4 Graphicalmodels............................ 168 3.5 Computationalissues . .. .. .. .. .. .. .. .. 170 3.5.1 Bayesiancalculations. 170 3.6 Statespacesystemsandrecursivecomputations . ......... 183 3.6.1 Generalstatespacemodels. 183 3.6.2 Filteringandsmoothing . 186 3.6.3 Exactalgorithmsforfilteringandsmoothing . ..... 192 3.6.4 Approximations ............................ 194 4 Signal processing, representation and approximation: Wavelets 209 4.1 Wavelets..................................... 211 4.1.1 Signalanalysis ............................. 211 4.1.2 Time-scalewavelets . 215 4.1.3 Thecontinuouswavelettransform . 217 4.1.4 Thediscretewavelettransform. 219 4.1.5 Multiresolution analysis and Fast Wavelet Transform (FWT) . 221 4.1.6 Waveletpackets............................. 231 4.2 Nonlinearapproximation . 234 4.2.1 Approximationtheory . 236 4.2.2 Approximationandwavelets . 248 4.2.3 Highlynonlinearapproximation . 254 4.3 WaveletsandBayesiantechniques: Denoising . ........ 262 4.4 Waveletsanddynamicalsystems . 275 4.4.1 Nonparametricestimation . 275 4.4.2 Linearsystemsandframes . 275 5 Putting things together: Implementation and application 277 5.1 Summary .................................... 278 5.2 Model,algorithmandimplementation . ..... 281 5.2.1 Model.................................. 282 5.2.2 Algorithm................................ 285 5.2.3 Implementation............................. 292 5.3 Examples .................................... 292 5.3.1 First example: Linear mass-spring-damper system . ....... 293 5.3.2 Second example: Nonlinear mass-spring-damper system ....... 298 5.3.3 Thirdexample:Preisachhysteresis . 300 5.4 Identificationofrealdata . 303 5.4.1 Thedata................................. 303 5.5 Conclusionandfuturework. 312 5.5.1 Résumé:Usageofidentificationmethods . 312 x i i i i i i i i Contents 5.5.2 Futurework............................... 317 5.5.3 Conclusion ............................... 317 Appendix: Basic notions 319 Bibliography 327 Index of definitions 347 xi i i i i i i i i Contents xii i i i i i i i i Schritt — Atemzug — Besenstrich Beppo Straßenkehrer in Michael Ende’s Momo xiii i i i i i i i i xiv i i i i i i i i Thanks My thanks go to the Fraunhofer Institut für Techno- und Wirtschaftsmathematik (ITWM; In- stitute for Industrial Mathematics) in Kaiserslautern, Germany, for the provision of room and means, and in particular to the head of this institute Prof. Dr. Dieter Prätzel-Wolters, my Dok- torvater, and to Dr. Patrick Lang, head of the department of Adaptive Systems of the same Institute, both for supervising me during my doctoral transition steps. My thank goes also to the Graduiertenkolleg Mathematik und Praxis (Graduate Research Training Programme Mathematics and Practice) which provided a scholarship to me during the years 2003-2006, and especially to its head Prof. Dr. Jürgen Franke for his valuable hints. I also want to give thanks to my industry partner, the company LMS Deutschland GmbH, Kaiserslautern, and especially to Dr. Manfred Bäcker, who always was very interested in the results of this thesis and always willing in providing measurement data to test the models. Thanks go also to the department MDF of the ITWM, especially to Michael Speckert who prepared simulation data to test the models (which I regrettably could not manage to include into my thesis before finishing it). Thanks go also to my
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