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lecture_13 - 16.333 Lecture 13 Aircraft Longitudinal...

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Unformatted text preview: 16.333: Lecture # 13 Aircraft Longitudinal Autopilots Altitude Hold and Landing Fall 2004 16.333 11–1 Altitude Controller • In linearized form, we know from 1–5 that the change of altitude h can be written as the flight path angle times the velocity, so that h ˙ ≈ U sin γ = U ( θ − α ) = U θ − U w = U θ − w U – For fixed U , h ˙ determined by variables in short period model • Use short period model augmented with θ state ⎡ ⎤ ⎧ w ⎨ x ˙ = A ˜ sp x + ˜ B sp δ e x = ⎣ q ⎦ θ ⇒ ⎩ h ˙ = − 1 0 U x where ˜ A ˜ sp = A sp , B sp = B sp [ 0 1 0 ] • In transfer function form, we get h K ( s + 4)( s − 3 . 6) = δ e s 2 ( s 2 + 2 ζ sp ω sp s + ω 2 sp )-5-4-3-2-1 1 2 3 4 5-3-2-1 1 2 3 0.945 0.89 0.3 0.5 0.68 0.81 5 0.81 0.976 0.994 0.3 0.5 0.68 0.994 0.89 0.945 0.976 4 1 2 3 Pole-Zero Map Real Axis Imaginary Axis Figure 1: Altitude root locus #1 Fall 2004 16.333 11–2-5-4-3-2-1 1 2 3 4-4-3-2-1 1 2 3 4 0.68 0.54 0.38 0.18 5 0.18 0.986 0.95 0.89 0.8 0.8 0.54 0.38 3 0.68 0.89 4 2 0.95 0.986 1 Altitude Gain Root Locus: k>0 Real Axis Imaginary Axis-5-4-3-2-1 1 2 3 4-4-3-2-1 1 2 3 4 0.68 0.54 0.38 0.18 5 0.18 0.986 0.95 0.89 0.8 0.8 0.54 0.38 3 0.68 0.89 4 2 0.95 0.986 1 Altitude Gain Root Locus: k<0 Real Axis Imaginary Axis Figure 2: Altitude root locus #2 • Root locus versus h feedback clearly NOT going to work! • Would be better off designing an inner loop first. Start with short period model augmented with the θ state δ e = − k w w − k q q − k θ θ + δ e c = k w k q k θ x + δ c = − K IL x + δ e c e − – Target pole locations s = − 1 . 8 ± 2 . 4 i , s = − . 25 – Gains: K IL = − . 0017 − 2 . 6791 − 6 . 5498-5-4-3-2-1 1 2 3 4-4-3-2-1 1 2 3 4 0.8 0.68 0.54 0.38 0.18 0.54 0.18 0.986 0.95 0.89 4 5 0.38 0.89 0.68 2 0.8 3 0.95 0.986 1 Inner loop target poles Real Axis Imaginary Axis Figure 3: Inner loop target pole locations – won’t get there with only a gain. Fall 2004 16.333 11–3 • Giving the closed-loop dynamics x ˙ = A sp x + ˜ ˜ B sp ( − K IL x + δ e c ) δ c = ( A ˜ sp − ˜ B sp e B sp K IL ) x + ˜ h ˙ = − 1 0 U x In transfer function form • ˜ h K ( s + 4)( s − 3 . 6) = δ c s ( s + . 25)( s 2 + 3 . 6 s + 9) e with ζ d and ω d being the result of the inner loop control.-6-5-4-3-2-1 1-4-3-2-1 1 2 3 4 6 0.58 0.44 0.3 0.14 0.3 0.98 0.92 0.84 0.72 4 0.44 5 0.14 0.84 0.58 2 0.72 3 0.92 0.98 1 CLP zeros pole locations with inner loop Altitude Gain Root Locus: with inner loop Real Axis Imaginary Axis-0.5-0.4-0.3-0.2-0.1 0.1 0.2-0.5-0.4-0.3-0.2-0.1 0.1 0.2 0.3 0.4 0.5 0.4 0.66 0.52 0.4 0.26 0.12 0.52 0.26 0.12 0.97 0.9 0.8 0.2 0.66 0.3 0.4 0.97 0.8 0.4 0.9 0.1 0.1 0.2 0.3 CLP zeros pole locations with inner loop Altitude Gain Root Locus: with inner loop Real Axis Imaginary Axis Figure 4: Root loci versus altitude gain K h < with inner loop added (zoomed on right). Much better than without inner loop, but gain must be small ( K h ≈ − . 01 )....
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lecture_13 - 16.333 Lecture 13 Aircraft Longitudinal...

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