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Project: Numerical Solutions to Ordinary Differential Equations in Hardware. Joseph Schneider EE 800 March 30, 2010. Ordinary Differential Equations. Function with one independent variable and derivatives of the dependent variable Eg . y’ = 1 + y/x

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Project numerical solutions to ordinary differential equations in hardware

Project: Numerical Solutions to Ordinary Differential Equations in Hardware

Joseph Schneider

EE 800

March 30, 2010


Ordinary differential equations
Ordinary Differential Equations Equations in Hardware

  • Function with one independent variable and derivatives of the dependent variable

  • Eg. y’ = 1 + y/x

  • Requires some initial condition in order to be solved

  • Eg. y(1) = 2


Ordinary differential equations1
Ordinary Differential Equations Equations in Hardware

  • Found in many areas of engineering

  • Radioactive decay, heat equation, motion…

  • In electrical engineering, charge, flux, voltage, current, all intertwined by differential equations


Ordinary differential equations2
Ordinary Differential Equations Equations in Hardware

  • In some cases, original equation can be derived with relative ease; Exact solutions are then available

  • In other cases, we will only have the ODE to work with

  • Numerical solutions have been developed to deal with these cases


Ordinary differential equations3
Ordinary Differential Equations Equations in Hardware

  • Most basic case: Euler’s Method

  • Selecting a step size h, iterate from initial value to desired value using the derivative function


Ordinary differential equation
Ordinary Differential Equation Equations in Hardware

  • Euler’s Method most basic case – Simple, but inaccurate

  • Variety of other methods that have been developed


Ordinary differential equations4
Ordinary Differential Equations Equations in Hardware


Ordinary differential equations5
Ordinary Differential Equations Equations in Hardware


Ordinary differential equations6
Ordinary Differential Equations Equations in Hardware

  • Error directly linked with step size

  • As step size decreases, error decreases; However, takes longer for process to complete

  • Implemented in software (eg. Matlab), more accurate methods take several seconds to complete for smaller scale cases; Several minutes for larger cases


Ordinary differential equations7
Ordinary Differential Equations Equations in Hardware

  • Original project goal: Implement the Runge-Kutta 4th order method in hardware for improved speed


Project
Project Equations in Hardware

  • Euler’s method also implemented for comparisons on area, timing, and accuracy.

  • Current implementation: Uses fixed-point number representation, state machine

  • Further steps on this implementation

    • Variable-point number representation to improve accuracy

    • Modify for parallelism to examine impacts on area, timing


Project1
Project Equations in Hardware

  • Second implementation: Error-controlled design of Runge-Kutta method

  • Error is specified at beginning of process, step size is varied to ensure final result meets error specifications


Matlab implementations
Matlab Equations in Hardware Implementations


Project2
Project Equations in Hardware

  • For second implementation, desired to use a hardware design to improve on bottleneck of software design

  • Comparisons to software on time vs. error threshold


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