The minimum degree algorithm is known as an effective scheme for identifying a fill reduced ordering for symmetric, positive definite, sparse linear systems, to be solved using a Cholesky factorization. Although the original algorithm has been enhanced to improve the efficiency of its implementation, ties between minimum degree elimination candidates are still arbitrarily broken. For many systems, the fill levels of orderings produced by the minimum degree algorithm are very sensitive to the precise manner in which these ties are resolved. This thesis introduces several tiebreaking enhancements of the minimum degree algorithm. Emphasis is placed upon a tiebreaking strategy based upon the deficiency of minium degree elimination candidates, and which can consistently identify low fill orderings for a wide spectrum of test problems. All tiebreaking strategies are fully integrated into implementations of the minimum degree algorithm based upon a quotient graph model, including indistinguishable sets represented by uneliminated supernodes. The resulting programs are tested on a wide variety of sparse systems in order to investigate the performance of the algorithm enhanced by the tiebreaking strategies and the quality of the orderings they produce. / Science, Faculty of / Computer Science, Department of / Graduate

In this thesis we study the model reduction problem in terms of the geometric concepts of linear system theory. By appropriate selection of reducing subspaces, useful lower-order system models can be achieved. The reducing subspaces can be chosen as parts of a system which are "most" and "least" controllable or observable; retaining, of course, the most controllable/observable subspace for model reduction. We review results showing how several measures of controllability and observability can provide this information. Balanced, Jordan canonical form, and dual GHR representations are shown to be state space realizations which naturally identify the reducing subspaces based on these measures. Several results unifying these methods are given. In another approach, we show that the reducing subspaces can be chosen such that after completing model reduction, a number of Markov parameters and time moments of the full system are retained by the reduced order model. We show how the dual GHR can be used as a tool which identifies these subspaces and state space realizations which naturally display them. Along these lines, a connection between model reduction in the state space and second-order systems is established, particularly the reduction of structures via the Lanczos algorithm. / Master of Science

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In today’s linear control systems, exact solutions can be obtained by the use of Laplace Transforms in the frequency domain. In dealing with nonlinear systems, exact solutions are not always achievable. For this reason, it is necessary to linearize the system and then apply frequency response methods. This paper shows the comparison of a nonlinear system with the linearized model of the same system. For both proportional and proportional-integral control, the response to a unit step change in the set point showed minimal difference between the linearized and nonlinear system.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy / In this thesis we used the following four types of optimal control problems: (i) Problems governed by systems of ordinary differential equations; (ii) Problems governed by systems of ordinary differential equations with time-delayed arguments appearing in both the state and the control variables; (iii) Problems governed by linear systems subject to sudden jumps in parameter values; (iv) A chemical reactor problem governed by a couple of nonlinear diffusion equations. • The aim of this thesis is to devise computational algorithms for solving the optimal control problems under consideration. However, our main emphasis are on the mathematical theory underlying the techniques, the convergence properties of the algorithms and the efficiency of the algorithms. Chapters II and III deal with problems of the first type, Chapters IV and V deal with problems of the second type, and Chapters VI and VII deal with problems of the third and fourth type respectively. A few numerical problems have been included in each of these Chapters to demonstrate the efficiency of the algorithms involved. The class of optimal control problems considered in Chapter II consists of a nonlinear system, a nonlinear cost functional, initial equality constraints, and terminal equality constraints. A Sequential Gradient-Restoration Algorithm is used to devise an iterative algorithm for solving this class of problems. 'I'he convergence properties of the algorithm are investigated. The class of optimal control problems considered in Chapter III consists of a nonlinear system, a nonlinear cost functional, and terminal as well as interior points equality constraints. The technique of control parameterization and Liapunov concepts are used to solve this class of problems, A computational algorithm for solving a class of optimal control problems involving terminal and continuous state constraints or inequality type was developed by Rei. 103 in 1989. In Chapter IV, we extend the results of Ref. 103 to a more general class of constrained time-delayed optimal control problems, which involves terminal state equality constraints, as well as terminal state inequality constraints and continuous state inequality constraints. In Ref. 104, a computational scheme using the technique of control parameterization was developed for solving a class of optimal control problems in which the cost functional includes the full variation of control. Chapter V is a straightforward extension of Ref. 104 to the time-delayed case. However the main contribution of this chapter is that many numerical examples have been solved. In Chapter VI, a class of linear systems subject to sudden jumps in parameter values is considered. To solve this class of stochastic control problem, we try to seek for the best feedback control law depending only on the measurable output. Based on this idea, we convert the original problem into an approximate constrained deterministic optimization problem, which can be easily solved by any existing nonlinear programming technique. In Chapter VII, a chemical reactor problem and its control to achieve a desired output temperature is considered. A finite element Galerkin method is used to convert the original distributed optimal control problem into a quadratic programming problem with linear constraints, which can he solved by any standard quadratic programming software . / Andrew Chakane 2018

Anti-control of chaos has attracted a lot of attention recently due to its potential applications in science and engineering. How to generate useful chaos that is also practically implementable and useful is a current focus of research. This research aims at developing new chaos generation schemes which demonstrate complex dynamical behaviours using simple linear systems with hysteresis function series. A continuous-time linear unstable second-order system with a feedback of hysteresis function is first proposed for generating chaos. The design for chaos generation is studied theoretically. A Poincaré map is used to demonstrate the dynamical behaviour of the system. The existence and the analytic solution of the limit cycle that bounds the basin of attraction of the chaotic attractor are derived. Conditions for the existence of chaotic attractors are studied. A hysteresis based system with a maximum chaotic stability margin is designed. Second, systematic methods for generating 1D n-scroll chaotic attractors in the directions of the state variables and 2D nxm-grid scroll chaotic attractors in the phase plane via continuous-time linear unstable second-order systems with a feedback of hysteresis function series are proposed. Furthermore, systematic methods for generating 1D n-scroll, 2D nxm-grid scroll and 3D nxmxl-space scroll chaotic attractors via continuous-time linear unstable third-order systems using hysteresis function series feedback are also presented in this thesis. Simulation results are presented to demonstrate effectiveness of the schemes. It is shown that the multi-scroll chaos generation systems can be represented in Lur'e form, and as a result it may be used within synchronization schemes for secure communication. Third, the limit cycle that bounds the basin of attraction in the multi-scroll chaos generation with second-order systems case is studied. The relationship of the size of the basin of attraction with the numbers of hysteresis function series is studied. The multi-scroll chaos generation mechanism is then further explored by analyzing the system trajectories; the switching boundaries, switching rules and the trajectories on each subspace. The chaotic behaviours are confirmed theoretically and it is proved that a non-ordinary attractor exists in the multi-scroll chaotic attractor of the second-order systems case. The abundant dynamical behaviour of the multi-scroll chaos generation systems using different hysteresis feedback are demonstrated. A double-hysteresis function, which is the superimposition of two basic hysteresis functions, is proposed for the implementation of the hysteresis based chaotic system. In this design, the double-hysteresis block and its series are constructed via a systematic method. The ideal hysteresis function series can be implemented easily with the proposed double-hysteresis function. The number of scroll attractors can be designed arbitrarily, and the multi-scroll chaotic attractors can be located anywhere and cover any chosen area of the phase plane. The circuitry implementation for generating 1D n-scroll, 2D nxm-grid scroll chaotic attractors with linear second-order systems and hysteresis function series is given. And the oscilloscope illustrated waveforms which included as many as 9x9=81 scrolls chaotic attractor are presented. The experimental results confirmed the theoretical analysis very well and validated the effectiveness as well as the feasibility of the proposed multi-scroll chaos generation schemes. This research may find potential engineering applications in areas such as digital coding and image processing, etc.

Systems of linear equations are central to many science and engineering application domains. Given the abundance of low-cost parallel processing fabrics, the study of fast and accurate parallel algorithms for solving such systems is receiving attention. Fast linear solvers generally use a form of LU factorization. These methods face challenges with workload distribution and communication overhead that hinder their application in a true broadcast communication environment. Presented is an efficient framework for solving large-scale linear systems by means of a novel utilization of Cramer's rule. While the latter is often perceived to be impractical when considered for large systems, it is shown that the algorithm proposed has an order N^3 complexity with pragmatic forward and backward stability. To the best of our knowledge, this is the first time that Cramer's rule has been demonstrated to be an order N^3 process. Empirical results are provided to substantiate the stated accuracy and computational complexity, clearly demonstrating the efficacy of the approach taken. The unique utilization of Cramer's rule and matrix condensation techniques yield an elegant process that can be applied to parallel computing architectures that support a broadcast communication infrastructure. The regularity of the communication patterns, and send-ahead ability, yields a viable framework for solving linear equations using conventional computing platforms. In addition, this dissertation demonstrates the algorithm's potential for solving large-scale sparse linear systems.

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics