David A. Castanon, Marcel Coderch, Alan S. Willsky. / Caption title. "Presented at the 21st IEEE Conference on Decision and Control, Dec. 1982." / Bibliography: leaf [1] / Air Force Office of Scientific Research Grant AFOSR-82-0258

Shankar Sastry. / Caption title. / Bibliography: leaf [6] / Air Force Office of Scientific Research Grant AFOSR 82-0258

Systems of large sparse linear equations frequently arise in engineering and science. Therefore, there is a great need for methods that can solve these systems. In this thesis we will present three of the earliest and simplest iterative methods and also look at two more sophisticated methods. We will study their rate of convergence and illustrate them with examples.

Modern neural network (NN) based control schemes have surmounted many of the limitations found in the traditional control approaches. Nevertheless, these modern control techniques have only recently been introduced for use on high-specification Programmable Logic Controllers (PLCs) and usually at a very high cost in terms of the required software and hardware. This ‗intelligent‘ control in the sector of industrial automation, specifically on standard PLCs thus remains an area of study that is open to further research and development. The research documented in this thesis examined the effectiveness of linear traditional control schemes such as Proportional Integral Derivative (PID), Lead and Lead-Lag control, in comparison to non-linear NN based control schemes when applied on a strongly non-linear platform. To this end, a mechatronic-type balancing system, namely, the Ball-on-Wheel (BOW) system was designed, constructed and modelled. Thereafter various traditional and intelligent controllers were implemented in order to control the system. The BOW platform may be taken to represent any single-input, single-output (SISO) non-linear system in use in the real world. The system makes use of current industrial technology including a standard PLC as the digital computational platform, a servo drive and wireless access for remote control. The results gathered from the research revealed that NN based control schemes (i.e. Pure NN and NN-PID), although comparatively slower in response, have greater advantages over traditional controllers in that they are able to adapt to external system changes as well as system non-linearity through a process of learning. These controllers also reduce the guess work that is usually involved with the traditional control approaches where cumbersome modelling, linearization or manual tuning is required. Furthermore, the research showed that online-learning adaptive traditional controllers such as the NN-PID controller which maintains the best of both the intelligent and traditional controllers may be implemented easily and with minimum expense on standard PLCs.

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 mainly study preconditioning techniques for all-at-once linear systems arising from discretization of three types of time-dependent advection-diffusion equation: linear diffusion equation, constant-coefficients advection-diffusion equation, time-fractional sub-diffusion equation. The proposed preconditioners are used with Krylov subspace solvers. The preconditioner developed for linear diffusion equation is based on -circulant ap- proximation of temporal discretization. Diagonalizability, clustering of spectrum and identity-plus-low-rank decomposition are derived for the preconditioned matrix. We also show that generalized minimal residual (GMRES) solver for the preconditioned system has a linear convergence rate independent of matrix-size. The preconditioner for constant-coefficients advection-diffusion equation is based on approximating the discretization of advection term with a matrix diagonalizable by sine transform. Eigenvalues of the preconditioned matrix are proven to be lower and upper bounded by positive constants independent of discretization parameters. Moreover, as the preconditioner is based on spatial approximation, it is also applicable to steady-state problem. We show that GMRES for the preconditioned steady-state problem has a linear convergence rate independent of matrix size. The preconditioner for time-fractional sub-diffusion equation is based on approximat- ing the discretization of diffusion term with a matrix diagonalizable by sine transform. We show that the condition number of the preconditioned matrix is bounded by a constant independent of discretization parameters so that the normalized conjugate gradient (NCG) solver for the preconditioned system has a linear convergence rate independent of discretization parameters and matrix size. Fast implementations based on fast Fourier transform (FFT), fast sine transform (FST) or multigrid approximation are proposed for the developed preconditioners. Numerical results are reported to show the performance of the developed preconditioners

In this thesis, we mainly study preconditioning techniques for all-at-once linear systems arising from discretization of three types of time-dependent advection-diffusion equation: linear diffusion equation, constant-coefficients advection-diffusion equation, time-fractional sub-diffusion equation. The proposed preconditioners are used with Krylov subspace solvers. The preconditioner developed for linear diffusion equation is based on -circulant ap- proximation of temporal discretization. Diagonalizability, clustering of spectrum and identity-plus-low-rank decomposition are derived for the preconditioned matrix. We also show that generalized minimal residual (GMRES) solver for the preconditioned system has a linear convergence rate independent of matrix-size. The preconditioner for constant-coefficients advection-diffusion equation is based on approximating the discretization of advection term with a matrix diagonalizable by sine transform. Eigenvalues of the preconditioned matrix are proven to be lower and upper bounded by positive constants independent of discretization parameters. Moreover, as the preconditioner is based on spatial approximation, it is also applicable to steady-state problem. We show that GMRES for the preconditioned steady-state problem has a linear convergence rate independent of matrix size. The preconditioner for time-fractional sub-diffusion equation is based on approximat- ing the discretization of diffusion term with a matrix diagonalizable by sine transform. We show that the condition number of the preconditioned matrix is bounded by a constant independent of discretization parameters so that the normalized conjugate gradient (NCG) solver for the preconditioned system has a linear convergence rate independent of discretization parameters and matrix size. Fast implementations based on fast Fourier transform (FFT), fast sine transform (FST) or multigrid approximation are proposed for the developed preconditioners. Numerical results are reported to show the performance of the developed preconditioners

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.