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چكيده انگليسي :

A Characterization of Normal Form for control System Majid Darabzadeh m darabzadeh@math iut ac ir June 16 2018 Master of Sience Thesis Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Majid Gazor mgazor@cc iut ac irAdvisor Dr Mohsen Mojiri Foroshani mohsen mojiri@cc iut ac ir2000 MSC 34A34 93C10 93C15Keywords Nonlinear control system Inner product normal forms Pull up Push down Abstract The study of the behavior of solutions of ODEs often benefits from deciding on a convenient choice of coordinates This choice of coordinates may be used to simplify the functional expressions that appear in the vector field inorder that the essential features of the flow of the ODE near a critical point become more evident In the case of theanalysis of an ordinary differential equation in the neighborhood of an equilibrium point this naturally leads to theconsideration of the possibility to remove the maximum number of terms in the Taylor expansion of the vector fieldup to a given order This idea was introduced by H Poincar e and the simplified system is called normal form There have been severalapplications of the method of normal forms particularly in the context of bifurcation theory where one combinesbetween the method of normal forms and the center manifold theorem in order to classify bifurcations Numerouspapers were published during the last decade on the normal forms of nonlinear control systems with applications inbifurcation and its control It is well known that there are several normal forms for a linear control system If thesystem is controllable then the system can be transformed into controllable or controller normal form If the systemhas a linear output map and is observable then it can be transformed into observable or observer form If a nonlinearcontrol system admits a controller normal form it can be transformed into a linear system by a change of coordinatesand feedback Therefore the design of a locally stabilizing state feedback control law is a straightforward task On theother hand most nonlinear systems do not admit a controller normal form under change of coordinates and invertiblestate feedback In this thesis we introduce a normal form characterization for normal forms of ordinary differential systems andnonlinear control systems The normal form characterization here follows an important normal form style that iscalled inner product normal form style A normal form characterization for normal forms can be obtained by derivingand solving an associated linear partial differential equation We recall the method of characteristics for solvingthe linear PDE systems In this direction the relation between the characteristic curves and first integrals of thecorresponding differential system of equations are illustrated The first part relies on the results related to ODEs and is based on the results developed by elphick et al while thecharacterization for control systems follows the generalization of the results on the ODE systems for nonlinear con trol systems The latter is implemented and described by Hamzi et al The result are applied on a general linearlycontrollable system and the characterization of normal forms for these systems are presented The control systems inthe vicinity of an uncontrollable equilibrium is also considered The introduced method is applied on two examplesin order to show the method can be applied to linearly uncontrollable cases The second part of this thesis focuses on a normalization method for control system that is called pull up and push down methods Simultaneous application of pull up and push down methods on a control system provides a completenormalization process These methods are applied an a variety of examples to illustrate their applicability

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