published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

Families of operators that are triangularizable must necessarily satisfy a number of spectral mapping properties. These necessary conditions are often sufficient as well. This thesis investigates such properties in finite dimensional and infinite dimensional Banach spaces. In addition, we investigate whether approximate spectral mapping conditions (being "close" in some sense) is similarly a sufficient condition.

Families of operators that are triangularizable must necessarily satisfy a number of spectral mapping properties. These necessary conditions are often sufficient as well. This thesis investigates such properties in finite dimensional and infinite dimensional Banach spaces. In addition, we investigate whether approximate spectral mapping conditions (being "close" in some sense) is similarly a sufficient condition.

The Multiplicative Ergodic Theorem is a powerful tool for studying certain types of dynamical systems, involving real matrix cocycles. It gives a block diagonalization of these cocycles, according to the Lyapunov exponents. We ask if it is always possible to refine the diagonalization to a block upper-triangularization, and if not over the real numbers, then over the complex numbers. After building up to the posing of the question, we prove that there are counterexamples to this statement, and give concrete examples of matrix cocycles which cannot be block upper-triangularized. / Graduate / 0405 / jahoran@uvic.ca

abstract: This thesis report aims at introducing the background of QR decomposition and its application. QR decomposition using Givens rotations is a efficient method to prevent directly matrix inverse in solving least square minimization problem, which is a typical approach for weight calculation in adaptive beamforming. Furthermore, this thesis introduces Givens rotations algorithm and two general VLSI (very large scale integrated circuit) architectures namely triangular systolic array and linear systolic array for numerically QR decomposition. To fulfill the goal, a 4 input channels triangular systolic array with 16 bits fixed-point format and a 5 input channels linear systolic array are implemented on FPGA (Field programmable gate array). The final result shows that the estimated clock frequencies of 65 MHz and 135 MHz on post-place and route static timing report could be achieved using Xilinx Virtex 6 xc6vlx240t chip. Meanwhile, this report proposes a new method to test the dynamic range of QR-D. The dynamic range of the both architectures can be achieved around 110dB. / Dissertation/Thesis / M.S. Electrical Engineering 2014

The main objective of this research is the development of a framework for the automatic generation of systems of kinematic and dynamic equations that are suitable for real-time applications. In particular, the efficient simulation of constrained multibody systems is addressed. When modelled with ideal joints, many mechanical systems of practical interest contain closed kinematic chains, or kinematic loops, and are most conveniently modelled using a set of generalized coordinates of cardinality exceeding the degrees-of-freedom of the system. Dependent generalized coordinates add nonlinear algebraic constraint equations to the ordinary differential equations of motion, thereby producing a set of differential-algebraic equations that may be difficult to solve in an efficient yet precise manner. Several methods have been proposed for simulating such systems in real time, including index reduction, model simplification, and constraint stabilization techniques. In this work, the equations of motion are formulated symbolically using linear graph theory. The embedding technique is applied to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. The theory of Gröbner bases is then used to triangularize the kinematic constraint equations, thereby producing recursively solvable systems for calculating the dependent generalized coordinates given values of the independent coordinates. For systems that can be fully triangularized, the kinematic constraints are always satisfied exactly and in a fixed amount of time. Where full triangularization is not possible, a block-triangular form can be obtained that still results in more efficient simulations than existing iterative and constraint stabilization techniques. The proposed approach is applied to the kinematic and dynamic simulation of several mechanical systems, including six-bar mechanisms, parallel robots, and two vehicle suspensions: a five-link and a double-wishbone. The efficient kinematic solution generated for the latter is used in the real-time simulation of a vehicle with double-wishbone suspensions on both axles, which is implemented in a hardware- and operator-in-the-loop driving simulator. The Gröbner basis approach is particularly suitable for situations requiring very efficient simulations of multibody systems whose parameters are constant, such as the plant models in model-predictive control strategies and the vehicle models in driving simulators.

The main objective of this research is the development of a framework for the automatic generation of systems of kinematic and dynamic equations that are suitable for real-time applications. In particular, the efficient simulation of constrained multibody systems is addressed. When modelled with ideal joints, many mechanical systems of practical interest contain closed kinematic chains, or kinematic loops, and are most conveniently modelled using a set of generalized coordinates of cardinality exceeding the degrees-of-freedom of the system. Dependent generalized coordinates add nonlinear algebraic constraint equations to the ordinary differential equations of motion, thereby producing a set of differential-algebraic equations that may be difficult to solve in an efficient yet precise manner. Several methods have been proposed for simulating such systems in real time, including index reduction, model simplification, and constraint stabilization techniques. In this work, the equations of motion are formulated symbolically using linear graph theory. The embedding technique is applied to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. The theory of Gröbner bases is then used to triangularize the kinematic constraint equations, thereby producing recursively solvable systems for calculating the dependent generalized coordinates given values of the independent coordinates. For systems that can be fully triangularized, the kinematic constraints are always satisfied exactly and in a fixed amount of time. Where full triangularization is not possible, a block-triangular form can be obtained that still results in more efficient simulations than existing iterative and constraint stabilization techniques. The proposed approach is applied to the kinematic and dynamic simulation of several mechanical systems, including six-bar mechanisms, parallel robots, and two vehicle suspensions: a five-link and a double-wishbone. The efficient kinematic solution generated for the latter is used in the real-time simulation of a vehicle with double-wishbone suspensions on both axles, which is implemented in a hardware- and operator-in-the-loop driving simulator. The Gröbner basis approach is particularly suitable for situations requiring very efficient simulations of multibody systems whose parameters are constant, such as the plant models in model-predictive control strategies and the vehicle models in driving simulators.

We present several methods that allow the exact computation of the exponential matrix etA. Methods that include computation of eigenvectors or Laplace transform are very well-known, and they are mentioned herefor completeness. We also present other methods, not well-known inthe literature, that do not need the computation of eigenvectors, and are easy to introduce in a classroom, thus providing us with general formulas that can be applied to any matrix. / Presentamos varios métodos que permiten el calculo exacto de la matriz exponencial etA. Los métodos que incluyen el calculo de autovectores y la transformada de Laplace son bien conocidos, y son mencionados aquí por completitud. Se mencionan otros métodos, no tan conocidos en la literatura, que no incluyen el calculo de autovectores, y que proveen de fórmulas genéricas aplicables a cualquier matriz.