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Dissertations / Theses on the topic 'Linear systems'
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1
Medina, Enrique A. "Linear Impulsive Control Systems: A Geometric Approach." Ohio : Ohio University, 2007. http://www.ohiolink.edu/etd/view.cgi?ohiou1187704023.
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Belayneh, Berhanu Bekele. "Time-varying linear systems." [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=98553530X.
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Mayo, Maldonado Jonathan. "Switched linear differential systems." Thesis, University of Southampton, 2015. https://eprints.soton.ac.uk/383678/.
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In this thesis we study systems with switching dynamics and we propose new mathematical tools to analyse them. We show that the postulation of a global state space structure in current frameworks is restrictive and lead to potential difficulties that limit its use for the analysis of new emerging applications. In order to overcome such shortcomings, we reformulate the foundations in the study of switched systems by developing a trajectory-based approach, where we allow the use of models that are most suitable for the analysis of a each system. These models can involve sets of higher-order differential equations whose state space does not necessarily coincide. Based on this new approach, we first study closed switched systems, and we provide sufficient conditions for stability based on LMIs using the concept of multiple higher order Lyapunov function. We also study the role of positive-realness in stability of bimodal systems and we introduce the concept of positive-real completion. Furthermore, we study open switched systems by developing a dissipativity theory. We give necessary and sufficient conditions for dissipativity in terms of LMIs constructed from the coefficient matrices of the differential equations describing the modes. The relationship between dissipativity and stability is also discussed. Finally, we study the dynamics of energy distribution networks. We develop parsimonious models that deal effectively with the variant complexity of the network and the inherent switching phenomena induced by power converters. We also present the solution to instability problems caused by devices with negative impedance characteristics such as constant power loads, using tools developed in our framework.
4
Markovsky, Ivan. "Exact and approximate modeling of linear systems : a behavioral approach /." Philadelphia, Pa. : Society for Industrial and Applied Mathematics, 2006. http://www.loc.gov/catdir/enhancements/fy0708/2005057537-d.html.
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Xu, Rui Hui. "Windowed linear canonical transform and its applications." Thesis, University of Macau, 2011. http://umaclib3.umac.mo/record=b2493220.
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Hopkins, Mark A. "Pseudo-linear identification: optimal joint parameter and state estimation of linear stochastic MIMO systems." Diss., Virginia Polytechnic Institute and State University, 1988. http://hdl.handle.net/10919/53941.
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This dissertation presents a new method of simultaneous parameter and state estimation for linear, stochastic, discrete—time, multiple-input, multiple-output (MIMO) (B systems. This new method is called pseudo·Iinear identification (PLID), and extends an earlier method to the more general case where system input and output measurements are corrupted by noise. PLID can be applied to completely observable, completely controllable systems with known structure (i.e., known observability indexes) and unknown parameters. No assumptions on pole and zero locations are required; and no assumptions on relative degree are required, except that the system transfer functions must be strictly proper.
Under standard gaussian assumptions on the various noises, for time-invariant systems in the class described above, it is proved that PLID is the optimal estimator (in the mean-square·error sense) of the states and the parameters, conditioned on the output measurements. It is also proved, under a reasonable assumption of persistent excitation, that the PLID parameter estimates converge a.e. to the true parameter values of the unknown system.
For deterministic systems, it is proved that PLID exactly identifies the states and parameters in the minimum possible time, so—called deadbeat identification. The proof brings out an interesting relation between the estimate error propagation and the observability matrix of the time-varying extended system (the extended system incorporates the unknown parameters into the state vector). This relation gives rise to an intuitively
appealing notion of persistent excitation.
Some results of system identification simulations are presented. Several different cases are simulated, including a two-input, two-output system with non-minimum-phase zeros, and an unstable system. A comparison of PLID with the widely used extended Kalman filter is presented for a single-input, single·output system with near cancellation of a pole-zero pair.
Results are also presented from simulations of the adaptive control of an unstable. two-input, two-output system In these simulations, PLID is used in a se1f—tuning regulator to identify the parameters needed to compute the feedback gain matrix, and (simultaneously) to estimate the system states, for the state feedback<br>Ph. D.
7
Newsham, Samantha. "Linear systems and determinants in integrable systems." Thesis, Lancaster University, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.663238.
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The thesis concerns linear systems and scattering theory. In particular, it presents lineal' systems for some integrable systems and finds discrete analogues for many well known results for continuous variables. It introduces some new tools from linear systems and applies them to standard integrable systems. We begin by expressing the first Painleve equation as the compatibility condition of a certain Lax pair and introduce the Korteweg-de Vries partial differential equation. We introduce the spectral curve for algebraic families and the Toda lattice. The Fredholm determinant of a trace class Hankel integral operator gives rise to a tau function. Dyson used the tau function to solve an inverse spectral problem for Schrodinger operators. When a plane wave is subject to Schrodinger's equation and scattered by a potential u, the output is described at great distances by a scattering function. The spectral problem is to find the spectrum of Schrodinger's operator in L2 and hence the scattering function. The inverse spectral problem is to find the potential given the scattering function. The scattering and inverse scattering problems are linked by the Gelfand- Levitan equation. In this thesis, for a discrete linear system, we introduce a scattering function and Hankel matrix and a version of the Gelfand-Levitan equation for discrete linear systems. We introduce the discrete operator ∑∞/k=n AkBCAk and use it to solve the Gelfand-Levitan equation and compute Fredholm determinants of Hankel operators. We produce a discrete analogue of a calculation of Poppe giving a solution to the Korteweg-de Vries equation and via the methods of linear systems find an analogous solution in terms of Hankel matrices. We then produce a discrete analogue of the Miura transform. Thus the main new contributions of this thesis are the discrete analogues of the R operator, the Gelfand- Levitan equation, the Lyapunov equation and the Miura transform.
8
Marinosson, Sigurdur Freyr. "Stability Analysis of Nonlinear Systems with Linear Programming - Stabilitätsanalyse nicht-linearer Systeme mit linearer Optimierung." Gerhard-Mercator-Universitaet Duisburg, 2002. http://www.ub.uni-duisburg.de/ETD-db/theses/available/duett-02152002-111745/.
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In this thesis the stability and the region of attraction of nonlinear dynamical systems' equilibrium points are considered. Methods from linear programming are combined with theorems from the Lyapunov theory of dynamical systems to develop numerical algorithms. These algorithms deliver non-trivial information about the stability-behaviour of an equilibrium of a continuous, autonomous, nonlinear system. Two linear programs, LP1 and LP2, are developed. LP1 depends on a simply connected open neighborhood N of the equilibrium at the origin and two constants, a and m. The construction of LP1 implies that if it does not possess a feasible solution, then the corresponding system is not a,m-exponentially stable on N. LP2 has the property that every feasible solution of the linear program defines a piecewise-affine (piecewise-linear) Lyapunov function or a Lyapunov-like function V for the system.
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Enqvist, Martin. "Linear Models of Nonlinear Systems." Doctoral thesis, Linköping : Linköpings universitet, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-5330.
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Haddleton, Steven W. "Steady-state performance of discrete linear time-invariant systems /." Online version of thesis, 1994. http://hdl.handle.net/1850/11795.
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Tse, Wilfred See Foon. "Linear equivalents of nonlinear systems." Thesis, University of British Columbia, 1987. http://hdl.handle.net/2429/26652.
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Consider the following nonlinear system
[Formula Omitted]
where ϰ ∈ Rⁿ, f, ℊ₁,…,ℊｍ are C∞ function in Rⁿ and ℎ is a C∞ function in R⍴, all defined on a neighborhood of 0. The problem of finding a necessary and sufficient condition such that system (1) can be transformed to a linear controllable system by a state coordinate change and feedback has been studied
quite well. In this thesis, we first discuss a few different approaches to this problem and eventually we will show that the slightly different versions of the necessary and sufficient condition discovered are equivalent. Next we consider
system (1) with all սi,= 0 together with system (2), and study the dual problem of transforming it to a linear observable system by a state and output coordinate change. Finally, we consider briefly system (l) and (2) with nonzero սi and study the problem of transforming it to a linear system that is both completely controllable and observable. Examples are given and applications to local stabilization and estimation are discussed.<br>Science, Faculty of<br>Mathematics, Department of<br>Graduate
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He, C., and V. Mehrmann. "Stabilization of large linear systems." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800595.
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We discuss numerical methods for the
stabilization of large linear multi-input
control systems of the form x=Ax + Bu via a
feedback of the form u=Fx. The method
discussed in this paper is a stabilization
algorithm that is based on subspace splitting.
This splitting is done via the matrix
sign-function method. Then a projection into
the unstable subspace is performed followed by
a stabilization technique via the solution of
an appropriate algebraic Riccati equation.
There are several possibilities to deal with the
freedom in the choice of the feedback as well
as in the cost functional used in the Riccati
equation. We discuss several optimality criteria
and show that in special cases the feedback
matrix F of minimal spectral norm is obtained
via the Riccati equation with the zero constant term.
A theoretical analysis about the distance to
instability of the closed loop system is given
and furthermore numerical examples are presented
that support the practical experience with
this method.
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Miri, Seyed Ali. "Modeling of multidimensional linear systems." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ30631.pdf.
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Chen, Shyh-Huei. "Solving Systems of Linear Inequalities." NCSU, 2001. http://www.lib.ncsu.edu/theses/available/etd-20010502-233936.
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<p>The problem of finding a feasible solution to a system of linear inequalities arises in numerous contexts. In this dissertation, we consider solving a system of linear inequalities in view of unconstrained convex programming problems. Solution methods for solving systems with either finitely or infinitely many linear inequalities are proposed. Convergence properties and implementation issues are discussed. Some computational results are also included. <P>
15
Woolf, Matthew Jacob. "Relative Jacobians of Linear Systems." Thesis, Harvard University, 2014. http://dissertations.umi.com/gsas.harvard:11522.
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Let X be a smooth projective variety. Given any basepoint-free linear system, |D|, there is a dense open subset parametrizing smooth divisors, and over that subset, we can consider the relative Picard variety of the universal divisor, which parametrizes pairs of a smooth divisor in the linear system and a line bundle on that divisor. In the case where X is a surface, there is a natural compactification of the relative Picard variety, given by taking the moduli space of pure one-dimensional Gieseker-semistable sheaves with respect to some polarization. In the case of the projective plane, this is an irreducible projective variety of Picard number 2. We study the nef and effective cones of these moduli spaces, and talk about the relation with variation of Bridgeland stability conditions.<br>Mathematics
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Hadad, Zarif M. "Structural properties of linear systems." Thesis, City University London, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.332557.
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MacCaig, Marie. "Integrality in max-linear systems." Thesis, University of Birmingham, 2015. http://etheses.bham.ac.uk//id/eprint/6024/.
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This thesis deals with the existence and description of integer solutions to max-linear systems. It begins with the one-sided systems and the subeigenproblem. The description of all integer solutions to each of these systems can be achieved in strongly polynomial time. The main max-linear systems that we consider include the eigenproblem, and the problem of determining whether a matrix has an integer vector in its column space. Also the two-sided systems, as well as max-linear programming problems. For each of these problems we construct algorithms which either find an integer solution, or determine that none exist. If the input matrix is finite, then the algorithms are proven to run in pseudopolynomial time. Additionally, we introduce special classes of input matrices for each of these problems for which we can determine existence of an integer solution in strongly polynomial time, as well as a complete description of all integer solutions. Moreover we perform a detailed investigation into the complexity of the problem of finding an integer vector in the column space. We describe a number of equivalent problems, each of which has a polynomially solvable subcase. Further we prove NP-hardness of related problems obtained by introducing extra conditions on the solution set.
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Pechev, Alexandre Nikolov. "Robust linear and non-linear control of magnetically levitated systems." Thesis, Cardiff University, 2004. http://orca.cf.ac.uk/55944/.
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The two most advanced applications of contactless magnetic levitation are high-speed magnetic bearings and magnetically levitated vehicles (Maglev) for ground transportation using superconducting magnets and controlled d.c. electromagnets. The repulsion force from superconducting magnets provide stable levitation with low damping, while the suspension force generated by electromagnets is inherently unstable. This instability, due to the in verse force-distance relationship, requires the addition of feedback controllers to sustain stable suspension. The problem of controlling magnetically levitated systems using d.c. electromagnets under different operating conditions has been studied in this thesis with a design process primarily driven by experimental results from a representative single-magnet test rig and a multi-magnet vehicle. The controller-design stages are presented in detail and close relationships have been constructed between selection of performance criteria for the derivation process and desired suspension characteristics. Both linear and nonlinear stabilising compensators have been developed. Simulation and experimental results have been studied in parallel to assess operational stability and the main emphasis has been given to assessing performance under different operational conditions. For the experimental work, a new digital signal processor-based hardware platform has been designed, built with interface to Matlab/Simulink. The controller design methods and algorithmic work presented in this thesis can be divided into: non-adaptive, adaptive, optimal linear and nonlinear. Adaptive algorithms based on model reference control have been developed to improve the performance of the suspension system in the presence of considerable variations in external payload and force disturbances. New design methods for Maglev suspension have been developed using robust control theory (%oo and fi synthesis). Single- and multi-magnet control problems have been treated using the same framework. A solution to the Hoo controller-optimisation problem has been derived and applied to Maglev control. The sensitivity to robustness has been discussed and tools for assessing the robustness of the closed-loop system in terms of sustaining stability and performance in the presence of uncertainties in the suspension model have been presented. Multivariable controllers based on %00 and /i synthesis have been developed for a laboratory scale experimental vehicle weighing 88 kg with four suspension magnets, and experimental results have been derived to show superiority of the proposed design methods in terms of ability to deal with external disturbances. The concept of Hoo control has been extended to the nonlinear setting using the concepts of energy and dissipativity, and nonlinear state-feedback and out put-feed back controllers for Maglev have been developed and reported. Simulation and experimental results have been presented to show the improved performance of these controllers to attenuate guideway-induced disturbances while maintaining acceptable suspension qualities and larger operational bandwidth.
19
Zheng, Gan. "Optimization in linear multiuser MIMO systems." Click to view the E-thesis via HKUTO, 2007. http://sunzi.lib.hku.hk/HKUTO/record/B39557923.
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Zheng, Gan, and 鄭淦. "Optimization in linear multiuser MIMO systems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2007. http://hub.hku.hk/bib/B39557923.
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Babaali, Mohamed. "Switched Linear Systems: Observability and Observers." Diss., Available online, Georgia Institute of Technology, 2004:, 2004. http://etd.gatech.edu/theses/available/etd-04122004-073020/unrestricted/babaali%5Fmohamed%5F200405%5Fphd.pdf.
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Thesis (Ph. D.)--Electrical and Computer Engineering, Georgia Institute of Technology, 2004.<br>Verriest, Erik, Committee Member ; Wardi, Yorai, Committee Member ; Yezzi, Anthony, Committee Member ; Wang, Yang, Committee Member ; Egerstedt, Magnus, Committee Chair. Vita. Includes bibliographical references (leaves 80-85).
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Xiong, Dapeng. "Stability analysis and controller synthesis of linear parameter varying systems /." Digital version accessible at:, 1998. http://wwwlib.umi.com/cr/utexas/main.
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Chadwick, Mark Anthony. "Identification and analysis of linear and non-linear fast-sampled systems." Thesis, University of Sheffield, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.489362.
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This thesis presents an investigation into the identification and analysis of and non-linear fast-sampled systems. Traditionally, discrete-time sampled-data systems are represented using shift-operator parameterisations utilising the d-operator. An alternative parameterisation using the q-operator is explored and shown to maintain a close correspondence to the continuous-time. This approach offers the ability to unify discrete and continuous-time theory as each can be considered a special case of the d-operator approach. In addition, these parameterisations possess numerical advantages when compared to the shift-operator representations.
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Landschoot, Timothy P. "Suppression of the transient response in linear time-invariant systems /." Online version of thesis, 1994. http://hdl.handle.net/1850/11794.
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Jackson, Billy Davis John M. "A general linear systems theory on time scales transforms, stability, and control /." Waco, Tex. : Baylor University, 2007. http://hdl.handle.net/2104/5066.
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Chan, Ka Hou. "Bayesian methods for solving linear systems." Thesis, University of Macau, 2011. http://umaclib3.umac.mo/record=b2493250.
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Stoll, Martin. "Solving linear systems using the adjoint." Thesis, University of Oxford, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.504598.
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Plischke, Elmar. "Transient effects of linear dynamical systems." [S.l.] : [s.n.], 2005. http://elib.suub.uni-bremen.de/diss/docs/00010211.pdf.
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Nader, Babak. "Parallel solution of sparse linear systems." Full text open access at:, 1987. http://content.ohsu.edu/u?/etd,138.
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Lin, M. "Multidimensional linear systems : factorisation and stabilisation." Thesis, University of Cambridge, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.233963.
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This thesis is concerned with various problems associated with the factorisation and feedback stabilisation of multidimensional linear discrete systems, which may be represented by rational matrices in several variables. Some factorisation techniques for polynomial and rational matrices in several variables have been explored and applied to the study of feedback stabilisation of multidimensional linear systems. The work presented here may be divided into two parts. The first part (Chapters 2 and 3) is concerned with two-dimensional systems, while the second part (Chapters 4 and 5) deals with three- and higher-dimensional systems. The emphasis of the first part is placed on the development of constructive algorithms for several kinds of factorisation of polynomial and rational matrices in two variables. In Chapter 2, an algorithm for obtaining primitive factorisation of polynomial matrices in two variables is developed, which is then followed by an algorithm for the decomposition of a rational matrix in two variables into factor coprime matrix fraction descriptions. Chapter 3 presents a procedure for the analysis and compensator design of two-dimensional feedback systems. A constructive algorithm for solving a Diophantine-type equation in two variables is derived. A necessary and sufficient condition for the feedback stabilisability of two-dimensional systems is obtained. The complete set of stabilising compensators for a given two-dimensional plant is then characterised. The role played by the matrix fraction description approach in the study of three- and higher dimensional systems, particularly with respect to the feedback stabilisation of these systems, is then investigated in detail in the second part. Chapter 4 deals with various kinds of factorisations for polynomial and rational matrices in three or more variables. For example, a criterion for the existence of primitive factorisation of a class of polynomial matrices in three or more variables is derived. By introducing a new concept: <i>generating polynomials</i>, it is shown that a direct generalisation of several existing results in two-dimensional systems theory to their higher-dimensional counterparts is not possible. In chapter 5, applying the generating polynomials, we obtain a stability test and a necessary and sufficient condition for feedback stabilisability of three- and higher-dimensional systems.
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Chou, Chun Tung. "Geometry of linear systems and identification." Thesis, University of Cambridge, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.320010.
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Keller, Carsten. "Constraint preconditioning for indefinite linear systems." Thesis, University of Oxford, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.343007.
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Sheen, Nicholas I. "Mossbauer spectroscopy of linear chain systems." Thesis, University of Canterbury. Physics, 1994. http://hdl.handle.net/10092/8074.
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Since the prediction by Villain in 1975 of moving magnetic domain walls (or solitons) in 1-dimensional Ising-like antiferromagnets, there has been interest in compounds with these properties. Mössbauer spectra of CsCo₀.₉₉Fe₀.₀₁Cl₃ in the magnetically ordered phases below TN₁ = 21.2 K were analysed by Ward et al (1987) assuming that only the two lowest energy electronic states of the ⁵⁷Fe⁺ ion were significantly occupied (the 2-level relaxation model). This assumption is unreliable above approximately 18 K because it was estimated that the third lowest-lying electronic state of Fe⁺ was significantly occupied at those temperatures. Nevertheless, it was found that two relaxation processes were present, one attributed to moving domain walls, and the other to transitions between the low-lying electronic states of the Fe⁺ ion.
In this work the 2-level relaxation model was extended to include the third lowest-lying electronic state of the Fe⁺ ion. A further extension, the combined relaxation model, enables Mössbauer spectra to be fitted when both 3-level electronic relaxation and relaxation due to moving domain walls occur with similar rates at the same ⁵⁷Fe⁺ site. The quasi 1-dimensional Ising-like antiferromagnetic salt NH₄CₒCl₃ doped with less than 1 atomic % ⁵⁷Fe⁺ was synthesised, and Mössbauer spectra were taken at temperatures between 1.3 and 250 K. The Mössbauer spectra of both CₛC₀₁₋ₓFeₓCl₃ and NHâ‚„Coâ‚ -â‚“Feâ‚“Cl3 in the magnetically ordered phases were analysed using the 2-level, 3-level and combined relaxation models. Although good fits to the Mössbauer spectra of CₛC₀₁₋ₓFeₓCl₃ and NH₄FeₓCl₃ were obtained using all the relaxation models, the combined relaxation model was the most satisfactory since the assumptions of this model were not invalidated by the parameters obtained from the fits. The soliton relaxation rates obtained were much slower than those predicted theoretically and found from other experiments on CsCoCl3.
One reason for this which is examined in this thesis is that the determined rates are unduly affected by the approximations made in the relaxation models. Another possible explanation of the discrepancy is that the presence of iron in the cobalt chains changes the soliton dynamics.
In order to study the effect of doping NNH₄FeₓCl₃with Fe⁺, the isomorphous crystal NH₄FeₓCl₃ was grown. The magnetic structure of NH₄FeₓCl₃ is expected to be governed by ferromagnetic intra-chain interactions and weaker antiferromagnetic inter-chain interactions. The linewidth broadening of the Mössbauer spectra of NH₄FeₓCl₃ at temperatures up to 10 K is evidence for the existence of magnetic correlations above the Neel temperature (1.7 K). The 4.2 and 1.3 K spectra show that a distribution of magnetic hyperfine fields B are present, possibly due to incommensurate magnetic ordering. At 4.2 K the main components of the Mössbauer spectrum are approximately 79 % with B = 0 and 21 % with B = 5.32 T. At 1.3 K the main components of the Mössbauer spectrum are approximately 72 % with B = 5.2 T and 28 %with B = 0. The non-magnetic subspectrum at 1.3 K may be caused by cancellation between the different components of the magnetic hyperfine field.
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Galanis, Georgios. "Dynamic polynomial combinants and linear systems." Thesis, City University London, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.527471.
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Eccles, Mark Richard. "Linear Jahn-Teller systems with troughs." Thesis, University of Nottingham, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.311773.
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Mhana, Khalid Jalal. "Optimal control of non-linear systems." Thesis, University of Sheffield, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.412720.
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Nankoo, Daniel. "Linear systems and control structure selection." Thesis, City University London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.397927.
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Kalogeropoulos, G. E. "Matrix pencils and linear systems theory." Thesis, City University London, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.355580.
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Benbow, Steven James. "Iterative methods for augmented linear systems." Thesis, University of Bath, 1997. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.760703.
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Saif, Abdul-Wahid. "Simultaneous stabilization of multivariable linear systems." Thesis, University of Leicester, 1995. http://hdl.handle.net/2381/34823.
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The simultaneous stabilization of a collection of systems has received considerable attention over a number of years. The practical motivation for a solution to the simultaneous stabilization problem (SSP) stems from the stability requirements of multimode systems in practical engineering. For example, a real plant may be subjected to several modes due to the failure of sensors and nonlinear systems are often represented by a set of linear models for design purposes. To examine these problems, it is necessary to establish a simultaneous stabilization theory. This dissertation considers the problem of simultaneously stabilizing a set of linear multivariable time-invariant systems. Three methodologies are presented. The first method is based on finding new approaches to solving the strong stabilization problem (i.e. stabilization by a stable controller) which can then be used in the SSP of two plants. New sufficient conditions and algorithms are derived for the solution to this problem. The second method utilizes robust stability theory applied to a "central" plant obtained from a given set of plants. A generalized two-block L-optimization problem is formulated and solved to find the central plant. The third method utilizes the parametrization of all stabilizing controllers. Sufficient conditions for the existence of a solution are derived and in the case of two plants a formula is derived for finding a simultaneously stabilizing controller. The work advances the theory of the SSP (and the Strong Stabilization Problem) by introducing and investigating several new approaches, and deriving new sufficient conditions. The work is less successful in deriving practical algorithms for the SSP except in the second method where a reliable algorithm is given for finding a central plant on which existing robust stabilization methods can be applied. This method is illustrated by its application to helicopter control.
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Pinto, Joao Moreira de Sousa. "Decidability boundaries in linear dynamical systems." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:9dde757f-6de1-47a6-a628-73f46e4bdf70.
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The object of this thesis is the study of the decidability properties of linear dynamical systems, which have fundamental ties to theoretical computer science, software verification, linear hybrid systems, and control theory. In particular, we describe a method for deciding the termination of simple linear loops, partly solving a 10-year-old open problem of Tiwari (2004) and Braverman (2006). We also study the membership problem for semigroups of matrix exponentials, which we show to be undecidable in general by reduction from Hilbert's Tenth Problem, and decidable for all instances where the matrices defining the semigroup commute. In turn, this entails the undecidability of the generalised versions of the Continuous Orbit and Skolem Problems to a multi-matrix setting. We also study point-to-point controllability for linear time-invariant systems, which is a central problem in control theory. For discrete-time systems, we show that this problem is undecidable when the set of controls is non-convex, and at least as hard as the Skolem Problem even when it is a convex polytope; for continuous-time systems, we show that this problem reduces to the Continuous Orbit Problem when the set of controls is a linear subspace, which entails decidability. Finally, we show how to decide whether all solutions of a given linear ordinary differential equation starting in a given convex polytope eventually leave it; this problem, which we call the "Polytope Escape Problem'', relates to the liveness of states in linear hybrid automata. Our results rely on a number of theorems from number theory, logic, and algebra, which we introduce in a self-contained way in the preamble to this thesis, together with a few new mathematical results of independent interest.
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Chonev, Ventsislav. "Reachability problems for linear dynamical systems." Thesis, University of Oxford, 2015. https://ora.ox.ac.uk/objects/uuid:e73d1a5b-edce-4e1d-a593-fd8df7e2a817.
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The object of principal interest in this thesis is linear dynamical systems: deterministic systems which evolve under a linear operator. They are specified by an initial state set I, contained in ℝ<sup>m</sup>, and a real m-by-m evolution matrix A. We distinguish two varieties of linear dynamical systems: discrete-time and continuous-time. In the discrete-time setting, the state x(n) of the system at time n for natural n is governed by the difference equation x(n)=Ax(n-1). Similarly, in the continuous case, the state x(t) at real, non-negative times t is determined by a system of first-order linear differential equations: x'(t) = Ax(t). In both cases, x(0) lies in I. Throughout this thesis, we will be interested in the Reachability Problem for linear dynamical systems, which may be formulated in a general way as follows: given a target set T contained in ℝ<sup>m</sup> and a (discrete- or continuous-time) linear dynamical system specified by the evolution matrix <b>A</b> and the set of initial states I, determine whether for all x(0) in I, starting from x(0), the system will eventually be in a state which lies in T. In order to make the decision problem well-defined, one must first fix an admissible class of initial sets and, similarly, a class of target sets of interest. For the purposes of expressing the problem instance, it is also necessary to restrict the domain of the input data to a subset of the reals which may be represented effectively, such as the rational numbers or the algebraic numbers. As we vary the choice of domain, the types of initial and target sets under consideration and the discreteness of time, a rich landscape of decision problems emerges. The goal of the present thesis is to explore pointwise reachability problems, that is, reachability from a single initial state. Under the assumption that I consists of a single point in ℝ<sup>m</sup> provided as part of the input data, we will study reachability to polyhedral targets, in the context of both discrete- and continuous-time linear dynamical systems. We prove both upper complexity bounds and hardness results, employing in the process a wide-ranging arsenal of techniques and mathematical tools. We rely on powerful number-theoretic results, such as Baker's Theorem on inhomogeneous linear forms of logarithms of algebraic numbers, Schanuel's Conjecture on the transcendence degree of certain field extensions of the rationals, and Kronecker's Theorem on simultaneous inhomogeneous Diophantine approximation. We draw interesting connections with the study of linear recurrence sequences and exponential polynomials, and relate pointwise reachability to open problems concerning the approximability by rationals of algebraic numbers and logarithms of algebraic numbers. Albeit a simple model, linear dynamical systems are of profound interest, both from a theoretical and a practical standpoint. Reachability problems for linear dynamical systems have recently elicited considerable attention, due to their frequent occurrence in practice and their deep and wide-ranging connections with other fascinating areas of study, such as problems on Markov chains (Akshay et al., 2015), quantum automata (Derksen et al., 2005), Lindenmayer systems (Salomaa and Soittola, 1978), linear loops (Braverman, 2006), linear recurrence sequences (Everest et al., 2003) and exponential polynomials (Bell et al., 2010).
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Fretwell, Paul. "Equivalence transformations in linear systems theory." Thesis, Loughborough University, 1986. https://dspace.lboro.ac.uk/2134/33259.
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There is growing interest in infinite frequency structure of linear systems, and transformations preserving this type of structure. Most work has been centred around Generalised State Space (GSS) systems. Two constant equivalence transformations for such systems are Rosenbrock's Restricted System Equivalence (RSE) and Verghese's Strong Equivalence (str.eq.). Both preserve finite and infinite frequency system structure. RSE is over restrictive in that it is constrained to act between systems of the same dimension. While overcoming this basic difficulty str.eq. on the other hand has no closed form description. In this work all these difficulties have been overcome. A constant pencil transformation termed Complete Equivalence (CE) is proposed, this preserves finite elementary divisors and non-unity infinite elementary divisors. Applied to GSS systems CE yields Complete System Equivalence (CSE) which is shown to be a closed form description of str.eq. and is more general than RSE as it relates systems of different dimensions. Equivalence can be described in terms of mappings of the solution sets of the describing differential equations together with mappings of the constrained initial conditions. This provides a conceptually pleasing definition of equivalence. The new equivalence is termed Fundamental Equivalence (FE) and CSE is shown to be a matrix characterisation of it. A polynomial system matrix transformation termed Full Equivalence (fll.e.) is proposed. This relates general matrix polynomials of different dimensions while preserving finite and infinite frequency structure. A definition of infinite zeros is also proposed along with a generalisation of the concept of infinite elementary divisors (IED) from matrix pencils to general polynomial matrices. The IED provide an additional method of dealing with infinite zeros.
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Shearer, J. M. "Interval methods for non-linear systems." Thesis, University of St Andrews, 1986. http://hdl.handle.net/10023/13779.
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In numerical mathematics, there is a need for methods which provide a user with the solution to his problem without requiring him to understand the mathematics underlying the method of solution. Such a method involves computable tests to determine whether or not a solution exists in a given region, and whether, if it exists, such a solution may be found by using the given method. Two valuable tools for the implementation of such methods are interval mathematics and symbolic computation. In. practice all computers have memories of finite size and cannot perform exact arithmetic. Therefore, in addition to the error which is inherent in a given numerical method, namely truncation error, there is also the error due to rounding. Using interval arithmetic, computable tests which guarantee the existence of a solution to a given problem in a given region, and the convergence of a particular iterative method to this solution, become practically realizable. This is not possible using real arithmetic due to the accumulation of rounding error on a computer. The advent of packages which allow symbolic computations to be carried out on a given computer is an important advance for computational numerical mathematics. In particular, the ability to compute derivatives automatically removes the need for a user to supply them, thus eliminating a major source of error in the use of methods requiring first or higher derivatives. In this thesis some methods which use interval arithmetic and symbolic computation for the solution of systems of nonlinear algebraic equations are presented. Some algorithms based on the symmetric single-step algorithm are described. These methods however do not possess computable existence, uniqueness, and convergence tests. Algorithms which do possess such tests, based on the Krawczyk-Moore algorithm are also presented. A simple package which allows symbolic computations to be carried out is described. Several applications for such a package are given. In particular, an interval form of Brown's method is presented.
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Malakorn, Tanit. "Multidimensional Linear Systems and Robust Control." Diss., Virginia Tech, 2003. http://hdl.handle.net/10919/26845.
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This dissertation contains two parts: Commutative and Noncommutative Multidimensional ($d$-D) Linear Systems Theory. The first part focuses on the development of the interpolation theory to solve the $H^{\infty}$ control problem for $d$-D linear systems. We first review the classical discrete-time 1D linear system in the operator theoretical viewpoint followed by the formulations of the so-called Givone-Roesser and Fornasini-Marchesini models. Application of the $d$-variable $Z$-transform to the system of equations yields the transfer function which is a rational function of several complex variables, say $\mathbf{z} = (z_{1}, \dots, z_{d})$.
We then consider the output feedback stabilization problem for a plant $P(\mathbf{z})$. By assuming that $P(\mathbf{z})$ admits a double coprime factorization, then a set of stabilizing controllers $K(\mathbf{z})$ can be parametrized by the Youla parameter $Q(\mathbf{z})$. By doing so, one can convert such a problem to the model matching problem with performance index $F(\mathbf{z})$, affine in $Q(\mathbf{z})$. Then, with $F(\mathbf{z})$ as the design parameter rather than $Q(\mathbf{z})$, one has an interpolation problem for $F(\mathbf{z})$. Incorporation of a tolerance level on $F(\mathbf{z})$ then leads to an interpolation problem of multivariable Nevanlinna-Pick type. We also give an operator-theoretic formulation of the model matching problem which lends itself to a solution via the commutant lifting theorem on the polydisk.
The second part details a system whose time-axis is described by a free semigroup $\mathcal{F}_{d}$. Such a system can be represented by the so-called noncommutative Givone-Roesser, or noncommutative Fornasini-Marchesini models which are analogous to those in the first part. Application of a noncommutative $d$-variable $Z$-transform to the system of equations yields the transfer function expressed by a formal power series in several noncommuting indeterminants, say $T(z) = \sum_{v \in \mathcal{F}_{d}}T_{v}z^{v}$ where $z^{v} = z_{i_{n}} \dotsm z_{i_{1}}$ if $v = g_{i_{n}} \dotsm g_{i_{1}} \in \mathcal{F}_{d}$ and $z_{i}z_{j} \neq z_{j}z_{i}$ unless $i = j$. The concepts of reachability, controllability, observability, similarity, and stability are introduced by means of the state-space interpretation. Minimal realization problems for noncommutative Givone-Roesser or Fornasini-Marchesini systems are solved directly by a shift-realization procedure constructed from appropriate noncommutative Hankel matrices. This procedure adapts the ideas of Schützenberger and Fliess originally developed for "recognizable series" to our systems.<br>Ph. D.
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Mu, Huiying. "Predictive control of linear uncertain systems." Thesis, Imperial College London, 2007. http://hdl.handle.net/10044/1/8515.
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Predictive control is a very useful tool in controlling constrained systems, since the constraints can be satisfied explicitly by the optimisations. Sets, namely, reachable sets, controllable sets, invariant sets, etc, play fundamental roles in designing predictive control strategies for uncertain systems. Meanwhile, in addition to the commonly assumed boundedness of the uncertainty, the explicit use of its stochastic properties can lead to improvement in system response. This thesis is concerned with robust set theories, mainly for reachable sets, with applications to time-optimal control; and the use of stochastic properties of the uncertainty to achieve less conservative controls. In the first part of this thesis, we focus on LTI systems subject to, additional to the usual constraints, a constraint on the control change between sample times. One key ingredient in controlling such constrained systems is the initial control value, which, via analyses and simulations, is shown to be a useful extra degree of freedom. Reachable sets that incorporate this influential initial control value are derived and analyzed, with theoretical as well as computational algorithms developed for both nominal and uncertain systems under different types of feedback policy. Following this, the reachable set is discussed in connection with time-optimal control to obtain desired control laws. In addition, controllable sets, stabilisable sets and invariant sets for such constrained uncertain systems are studied. In the second part, the uncertainties are assumed to have stochastic properties. They are exploited in three different ways: the expected worst-case is used instead of the worst-case to achieve less conservative control even when the uncertainty is relatively large; the stochastic invariant set is proposed to provide alternative methods for approximating disturbance invariant sets; the relaxed set difference is developed to obtain less restrictive controls and/or replacing probabilistic constraint or slack variables.
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Leccese, Andrew J. "Stability of parametrically forced linear systems /." Online version of thesis, 1994. http://hdl.handle.net/1850/11789.
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Wilson, Jonathan P. "Non-linear dynamics and power systems." Thesis, University of Bath, 2000. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.341136.
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Lo, Hong Rui. "System characterisation and identification of non-linear systems (with particular reference to hysteretic systems)." Thesis, University of Southampton, 1988. https://eprints.soton.ac.uk/52277/.
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System identification is the process of building mathematical models of dynamical systems based on observed data. Many effective techniques have been developed for linear systems. For non-linear systems, some progress has been achieved, but techniques for practical use and which can deal with a large class of systems are limited. In particular few identification techniques have been found in the literature which can be applied to hysteretic systems. This thesis is devoted to the development of a system identification technique which can be applied to a relatively large class of non-linear systems, including hysteretic systems. The key to this technique is to select an appropriate subset of the state vector describing the system and generate a non-linear surface in this subspace which characterises the non-linearity. For non-hysteretic systems, this space is the normal state space. For hysteretic systems, the selection of the appropriate space usually needs some prior knowledge about the system. The procedure involves estimating the non-linear component as a function of time. This is approached via a deconvolution method, and a section of this thesis shows how an optimal deconvolution method may be used. The method of creating the surface is described, and identification is then conducted by analysing and fitting the surface. The success of identification is obviously affected by the quality of the surface, which is, in turn, affected by factors such as the type and the level of the excitation, the frequency range and the magnitude of the spectrum of the process, and errors in the signal processing. These problems are discussed in the application of this technique to several simulated non-linear systems (including both non-hysteretic and hysteretic types) and also to the practical case of a cable type vibration isolator.
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Rath, W. "Canonical forms for linear descriptor systems with variable coefficients." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800708.
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We study linear descriptor systems with rectangular variable coefficient matrices.
Using local and global equivalence transformations we introduce normal and
condensed forms and get sets of characteristic quantities. These quantities allow us to
decide whether a linear descriptor system with variable coefficients is regularizable
by derivative and/or proportional state feedback or not. Regularizable by feedback
means for us that their exist a feedback which makes the closed loop system uniquely
solvable for every consistent initial vector.