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Modelling and Control Methods with Applications to Mechanical Waves

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Modelling and Control Methods with Applications to Mechanical Waves Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1174 Modelling and Control Methods with Applications to Mechanical Waves HANS NORLANDER ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-9023-2 UPPSALA urn:nbn:se:uu:diva-229793 2014 Dissertation presented at Uppsala University to be publicly examined in room 2446, ITC, Lägerhyddsvägen 2, Uppsala, Friday, 17 October 2014 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Prof. Jonas Sjöberg (Chalmers). Abstract Norlander, H. 2014. Modelling and Control Methods with Applications to Mechanical Waves. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1174. 72 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9023-2. Models, modelling and control design play important parts in automatic control. The contributions in this thesis concern topics in all three of these concepts. The poles are of fundamental importance when analyzing the behaviour of a system, and pole placement is an intuitive and natural approach for control design. A novel parameterization for state feedback gains for pole placement in the linear multiple input case is presented and analyzed. It is shown that when the open and closed loop poles are disjunct, every state feedback gain can be parameterized. Other properties are also investigated. Hammerstein models have a static non-linearity on the input. A method for exact compensation of such non-linearities, combined with introduction of integral action, is presented. Instead of inversion of the non-linearity the method utilizes differentiation, which in many cases is simpler. A partial differential equation (PDE) can be regarded as an infinite order model. Many model based control design techniques, like linear quadratic Gaussian control (LQG), require finite order models. Active damping of vibrations in a viscoelastic beam, modelled as a PDE, is considered. The beam is actuated by piezoelectric elements and its movements are measured by strain gauges. LQG design is used, for which different finite order models, approximating the PDE model, are constructed. The so obtained controllers are evaluated on the original PDE model. Minimization of the measured strain yields a satisfactory performance, but minimization of transversal deflection does not. The effect of the model accuracy of the finite order model approximations is also investigated. It turns out that a model with higher accuracy in a specified frequency interval gives controllers with better performance. The wave equation is another PDE. A PDE model, with one spatial dimension, is established. It describes wave propagation in a tube perforated with helical slots. The model describes waves of both extensional and torsional type, as well as the coupling between the two wave types. Experimental data are used for estimation of model parameters, and for assessment of the proposed model in two different cases. The model is found adequate when certain geometrical assumptions are valid. Keywords: automatic control, pole placement, linearization, vibration control, distributed parameter systems, wave propagation Hans Norlander, Department of Information Technology, Division of Systems and Control, Box 337, Uppsala University, SE-75105 Uppsala, Sweden. Department of Information Technology, Automatic control, Box 337, Uppsala University, SE-75105 Uppsala, Sweden. © Hans Norlander 2014 ISSN 1651-6214 ISBN 978-91-554-9023-2 urn:nbn:se:uu:diva-229793 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-229793) To Malin, Simon and M˚ans List of contributions This thesis is based on the following contributions, which are referred to in the text by their Roman numerals. I H. Norlander, “Parameterization of state feedback gains for pole assign- ment”, Licentiate thesis, Department of Information Technology, Uppsala University, 2001. II P. Samuelsson, H. Norlander, and B. Carlsson, “An integrating lineariza- tion method for Hammerstein models”, Automatica, vol. 41, no. 10, pp. 1825–1828, 2005 III P. Naucl´er, H. Norlander, A. Jansson, and T. S¨oderstr¨om, “Modeling and control of a viscoelastic piezolaminated beam”, in Proceeding of the 16th IFAC World Congress, Prague, Czech Republic, July 2005. IV H. Norlander, “On the impact of model accuracy for active damping of a viscoelastic beam”, Department of Information Technology, Uppsala Uni- versity, Tech. Rep. 2011-013, 2011. V H. Norlander, U. Valdek, B. Lundberg, and T. S¨oderstr¨om, “Parameter estimation from wave propagation tests on a tube perforated by helical slots”, Mechanical Systems and Signal Processing, vol. 40, no. 1, pp. 385– 399, 2013. Reprints were made with permission from the publishers. Contents 1 Introduction 9 1.1Thesisoutline............................ 10 2 Models of dynamical systems 13 2.1Systemsandmodels........................ 13 2.2Statespacemodels......................... 14 2.2.1 Linear time-invariant state space models . 15 2.3Transferfunctions......................... 17 2.4Frequencyresponse......................... 18 2.5 Infinite-dimensional models . 18 3 Modelling 21 3.1 Mathematical modelling . 21 3.2 Empirical modelling . 21 3.2.1 Black box modelling . 22 3.2.2 Grey box modelling . 23 3.2.3 Modelvalidation...................... 23 3.3Modelreduction.......................... 23 4 Control design 25 4.1Feedbackcontrol.......................... 25 4.2Modelbasedcontroldesign.................... 27 4.3Observerbasedstatefeedbackcontrol.............. 30 4.3.1 Poleplacement....................... 32 4.3.2 LQG............................. 34 5 The contributions in perspective 37 5.1 Pole placement for multiple input systems . 37 5.2 Integrating linearization of static non-linearities on the input . 41 5.3Activedampingofaviscoelasticbeam.............. 44 5.3.1 Modelling and control of a viscoelastic beam . 47 5.3.2 Theeffectofmodelaccuracy............... 50 5.3.3 Further comments and open questions . 54 7 8CONTENTS 5.4 1D modelling and parameter estimation of couplings between extensional and torsional waves in a helical structured tube . 54 Summary in Swedish 61 Acknowledgments 67 Bibliography 68 Chapter 1 Introduction utomatic control is a quite broad discipline. In very general terms it A can be described as the study of dynamical systems and their control. As a technology it has been used frequently for several hundred years. As a recognized research area, in its own right, automatic control emerged during the middle of the 20th century from the insight that dynamical systems from radically different applications, but with equivalent behaviour, can be described by the same theory. A nice recent overview of the historical development of automatic control is given in [42]. Today control systems are ubiquitous in the technical systems we rely on in our everyday life. Typically the control systems are essential for the function of these technical systems. Still most people are not aware of the presence of the control systems, or of their role. One could argue whether this is the cause for, or an effect of, the fact that many people lack an intuitive understanding for what automatic control is about. Either way, as a reflection of this fact automatic control has been referred to as the “hidden technology” ([41]). So what is automatic control about? Two key concepts are models and feedback. In order to perform any form of analysis of a dynamical system some sort of quantitative description is required, and this is typically obtained as a mathematical model. One subfield within automatic control concerns models; different types of them and their properties. Another subfield concerns the procedure of producing models. Feedback is an essential principle for control, and is in general an integrated part in control design. Control design in itself is yet another subfield within automatic control. While this admittedly is a rather meager description of automatic control (there are plenty more subfields, and subdivisions within these), models, modelling and control design together still constitute a core of the subject. There are a few quite different topics treated in the thesis. What the contri- butions have in common is that they are all strongly related to this core. This is also reflected in the title of the thesis. Some of the contributions concern more general situations, while for the others there are some special cases that 9 10 1. Introduction are considered. Common for the special cases is that they can be described as mechanical wave propagation applications. In order to provide a proper background to the contributions a brief presen- tation of some important concepts concerning models, modelling and control is given; see below. This is by no means a complete description, but is rather aimed at giving a comprehensive picture, covering the different aspects consid- ered in the contributions. 1.1 Thesis outline The thesis is composed of two parts. The first part, comprising Chapters 1– 5 here, is a comprehensive summary of the thesis work. Chapters 2–4 are providing the necessary background for the topics covered in the contributions, and in Chapter 5 summaries of the contributions are presented. The second part contains the contributions that the thesis is based on. Chapter 2: Models of dynamical systems Here different mathematical models of dynamical systems are presented. A few features, that define the model type, are discussed, and some model properties are also described.
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