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Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

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Appendix I / Utilities / Digital Filters<br />

_ _<br />

22.8 Digital Filters<br />

The author would like to acknowledge the work of Dr. R. Stirling in the<br />

conversion of continuous filters in the Laplace domain to the digital domain<br />

using bi-linear transforms.<br />

Digital filters functional forms are used in the text that operate on a scalar<br />

argument (ϕXYZ), or a vectorised input (ϕXYZ). If a scalar argument is given<br />

in a vector function it applies to all the elements of the vector input.<br />

When using the functional form it is assumed that the update rate is<br />

provided in-situ. Digital filters should be used with an update frequency at<br />

least 10x the bandwidth of their continuous counterpart so as to maintain<br />

accuracy.<br />

22.8.1 Digital First Order Lag Filters<br />

D_LAG propagates up to 150 digital lags over time interval (∆t), with time<br />

varying bandwidth (ωC).<br />

( X , ∆t<br />

, t , N , I ) ≡ ϕ ( X , 1 )<br />

y : = D_LAG<br />

ω<br />

I<br />

22.8-1<br />

C<br />

D1L<br />

I<br />

C<br />

Equation 22.8-1<br />

These filters are the discrete equivalent to the continuous transfer function:<br />

χ<br />

k<br />

: =<br />

χ<br />

χ<br />

y<br />

k − 1<br />

0<br />

( s )<br />

: =<br />

⎛ X<br />

+<br />

⎜<br />

⎝<br />

I<br />

− 1<br />

XI<br />

1 + s ⋅ t<br />

( k −1<br />

)<br />

C<br />

− 2 ⋅ χ<br />

∆t<br />

+ 2 ⋅ t<br />

−1<br />

C<br />

: = χ : = 2 ⋅ X<br />

yk : = χk<br />

+ χk<br />

−1<br />

k −1<br />

I<br />

⎞<br />

⎟ ⋅ ∆t<br />

⎠<br />

Equation 22.8-2<br />

Equation 22.8-3<br />

Equation 22.8-4<br />

Equation 22.8-5<br />

If (I := 1), and on first use, the filter is re-initialised without integration<br />

using input (XI). Each filter is identified by a unique number (N).

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