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Analysis and Control of NonAffine, NonStandard, Singularly Perturbed SystemsNarang, Anshu 14 March 2013 (has links)
This dissertation addresses the control problem for the general class of control nonaffine, nonstandard singularly perturbed continuoustime systems. The problem of control for nonlinear multiple time scale systems is addressed here for the first time in a systematic manner. Toward this end, this dissertation develops the theory of feedback passivation for nonaffine systems. This is done by generalizing the KalmanYakubovichPopov lemma for nonaffine systems. This generalization is used to identify conditions under which nonaffine systems can be rendered passive. Asymptotic stabilization for nonaffine systems is guaranteed by using these conditions along with wellknown passivitybased control methods. Unlike previous nonaffine control approaches, the constructive static compensation technique derived here does not make any assumptions regarding the control influence on the nonlinear dynamical model. Along with these control laws, this dissertation presents novel hierarchical control design procedures to address the two major difficulties in control of multiple time scale systems: lack of an explicit small parameter that models the time scale separation and the complexity of constructing the slow manifold. These research issues are addressed by using insights from geometric singular perturbation theory and control laws are designed without making any assumptions regarding the construction of the slow manifold. The control schemes synthesized accomplish asymptotic slow state tracking for multiple time scale systems and simultaneous slow and fast state trajectory tracking for two time scale systems. The control laws are independent of the scalar perturbation parameter and an upper bound for it is determined such that closedloop system stability is guaranteed.
Performance of these methods is validated in simulation for several problems from science and engineering including the continuously stirred tank reactor, magnetic levitation, six degreesoffreedom F18/A Hornet model, nonminimum phase helicopter and conventional takeoff and landing aircraft models. Results show that the proposed technique applies both to standard and nonstandard forms of singularly perturbed systems and provides asymptotic tracking irrespective of the reference trajectory. This dissertation also shows that some benchmark nonminimum phase aerospace control problems can be posed as slow state tracking for multiple time scale systems and techniques developed here provide an alternate method for exact output tracking.

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Normally elliptic singular perturbation problems: local invariant manifolds and applicationsLu, Nan 18 May 2011 (has links)
In this thesis, we study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be nonautonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small
singular parameters. We also use our results on local invariant manifolds to study
the persistence of homoclinic solutions under weakly dissipative and conservative per
turbations. We apply Semigroup Theory and LyapunovPerron Integral Equations with some
careful estimates to handle the O(1) driving force in the system so that we can approximate the full system through some simpler limiting system. In the investigation of homoclinics, a diagonalization procedure and some normal form transformation should be first carried out. Such diagonalization procedure is not trivial at all. We discuss this issue in the appendix. We use Melnikov type analysis to study the weakly
dissipative case, while the conservative case is based on some energy methods. As a concrete example, we have shown rigrously the persistence of homoclinic solutions of an elastic pendulum model which may be affected by damping, external
forcing and other potential fields.

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The robustness of the hierarchical a posteriori error estimator for reactiondiffusion equation on anisotropic meshesGrosman, Serguei 01 September 2006 (has links) (PDF)
Singularly perturbed reactiondiffusion problems
exhibit in general solutions with anisotropic
features, e.g. strong boundary and/or interior
layers. This anisotropy is reflected in the
discretization by using meshes with anisotropic
elements. The quality of the numerical solution
rests on the robustness of the a posteriori error
estimator with respect to both the perturbation
parameters of the problem and the anisotropy of the
mesh. The simplest local error estimator from the
implementation point of view is the socalled
hierarchical error estimator. The reliability
proof is usually based on two prerequisites:
the saturation assumption and the strengthened
CauchySchwarz inequality. The proofs of these
facts are extended in the present work for the
case of the singularly perturbed reactiondiffusion
equation and of the meshes with anisotropic elements.
It is shown that the constants in the corresponding
estimates do neither depend on the aspect ratio
of the elements, nor on the perturbation parameters.
Utilizing the above arguments the concluding
reliability proof is provided as well as the
efficiency proof of the estimator, both
independent of the aspect ratio and perturbation
parameters.

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Nonlinear oscillation and control in the BZ chemical reaction.Li, Yongfeng 25 August 2008 (has links)
In this thesis, a reversible LotkaVolterra model was proposed to study the nonlinear oscillation of the BelousovZhabotinsky(BZ) reaction in a closed isothermal chemical system. The reaction zone can be divided into two zones, oscillation zone and transition zone, where the oscillation time and the transition time and the number of the complete oscillations are estimated. By applying the geometric singular perturbation method, it was proved that there exist an oscillation axis in the oscillation zone, a strongly stable twodimensional invariant manifold in transition zone, on which there is also a onedimensional stable invariant
manifold, which is the part of the central axis. There is no oscillation in the vicinity of the equilibrium, as indicated by Onsager's reciprocal symmetry relation. Furthermore, the damped oscillation is studied in terms of the actionactionangle variables. In the end, the model reference control technique is employed to control the oscillation amplitude in the
BZ reaction.

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A new renormalization method for the asymptotic solution of multiple scale singular perturbation problems /Mudavanhu, Blessing. January 2002 (has links)
Thesis (Ph. D.)University of Washington, 2002. / Vita. Includes bibliographical references (p. 97104).

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Robust local problem error estimation for a singularly perturbed reactiondiffusion problem on anisotropic finite element meshesGrosman, Serguei 05 April 2006 (has links) (PDF)
Singularly perturbed reactiondiffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. An estimator that has shown to be one of the most reliable for reactiondiffusion problem is the <i>equilibrated residual method</i> and its modification done by Ainsworth and Babuška for singularly perturbed problem. However, even the modified method is not robust in the case of anisotropic meshes. The present work modifies the equilibrated residual method for anisotropic meshes. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. A numerical example confirms the theory.

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A posteriori error estimation for a finite volume discretization on anisotropic meshesKunert, Gerd, Mghazli, Zoubida, Nicaise, Serge 31 August 2006 (has links) (PDF)
A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertexcentered finite volume method. The anisotropy of the solution is reflected by using \emph{anisotropic meshes} which can improve the accuracy of the discretization considerably. The main focus is on \emph{a posteriori} error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient \emph{a posteriori} error estimation is achieved for the finite volume method on anisotropic meshes.

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A posteriori error estimation for a finite volume discretization on anisotropic meshesKunert, Gerd, Mghazli, Zoubida, Nicaise, Serge 31 August 2006 (has links)
A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertexcentered finite volume method. The anisotropy of the solution is reflected by using \emph{anisotropic meshes} which can improve the accuracy of the discretization considerably. The main focus is on \emph{a posteriori} error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient \emph{a posteriori} error estimation is achieved for the finite volume method on anisotropic meshes.

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Initialvalue Technique For Singularly Perturbed Two Point Boundary Value Problems Via Cubic SplineNegron, Luis G. 01 January 2010 (has links)
A recent method for solving singular perturbation problems is examined. It is designed for the applied mathematician or engineer who needs a convenient, useful tool that requires little preparation and can be readily implemented using little more than an industrystandard software package for spreadsheets. In this paper, we shall examine singularly perturbed two point boundary value problems with the boundary layer at one end point. An initialvalue technique is used for its solution by replacing the problem with an asymptotically equivalent first order problem, which is, in turn, solved as an initial value problem by using cubic splines. Numerical examples are provided to show that the method presented provides a fine approximation of the exact solution. The first chapter provides some background material to the cubic spline and boundary value problems. The works of several authors and a comparison of different solution methods are also discussed. Finally, some background into the specific singularly perturbed boundary value problems is introduced. The second chapter contains calculations and derivations necessary for the cubic spline and the initial value technique which are used in the solutions to the boundary value problems. The third chapter contains some worked numerical examples and the numerical data obtained along with most of the tables and figures that describe the solutions. The thesis concludes with some reflections on the results obtained and some discussion of the error bounds on the calculated approximations to the exact solutions for the numeric examples discussed

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Energy management of threedimensional minimumtime interceptVisser, Hendrikus January 1985 (has links)
A realtime computer algorithm to control and optimize aircraft flight profiles is described and applied to a threedimensional minimumtime intercept mission.
The proposed scheme has roots in two wellknown techniques: singular perturbations and neighboringoptimal guidance. Use of singularperturbation ideas is made in terms of the assumed trajectoryfamily structure. A heading/energy family of prestored pointmassmodel stateEuler solutions is used as the baseline in this scheme. The next step is to generate a nearoptimal guidance law that will transfer the aircraft to the vicinity of this reference family. The control commands fed to the autopilot consist of the reference controls plus correction terms which are linear combinations of the altitude and pathangle deviations from reference values, weighted by a set of precalculated gains. In this respect the proposed scheme resembles neighboringoptimal guidance. However, in contrast to the neighboringoptimal guidance scheme, the reference control and state variables as well as the feedback gains are stored as functions of energy and heading in the present approach.
A detailed description of the feedback laws and of some of the mathematical tools used to construct the controller is presented. The construction of the feedback laws requires a substantial preflight computational effort, but the computation times for onboard execution of the feedback laws are very modest. Other issues relating to practical implementation are addressed as well.
Numerical examples, comparing openloop optimal and approximate feedback solutions for a sample highperformance fighter, illustrate the attractiveness of the guidance scheme. Optimal threedimensional flight in the presence of a terrain limit is studied in some detail. / Ph. D. / incomplete_metadata

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