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This document comprises of an analysis undertaken by "hand", and as such is a scanned document, written in ink.The analysis considers the discretisation of the standard form of the continuous time PID controller, for digital implementation in Microcontrollers of FPGA driven circuitry as a discretised PI or PID controller.The discretisation is first undertaken for the design of a discretised PI controller using the backward-finite difference approximation of the first-order derivative and this is repeated with the forward-finite difference approximation of the first-order derivative. An alternative method is also detailed for obtaining both of the previous results. The trapezoidal approximation is considered for the integral component.The same discretisation is applied to the continuous time PID controller. Not only is the discrete time control law shown, U(t_k), known as the "positional algorithm" but they are further expanded to give the "incremental" or "velocity algorithms".
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