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lecture_9

# lecture_9 - 16.333 Lecture 9 Basic Longitudinal Control...

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16.333 Lecture # 9 Basic Longitudinal Control Basic aircraft control concepts Basic control approaches

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Fall 2004 16.333 8–1 Basic Longitudinal Control Goal: analyze aircraft longitudinal dynamics to determine if the be- havior is acceptable, and if not, then modify it using feedback control. Note that we could (and will) work with the full dynamics model, but for now, let’s focus on the short period approximate model from lecture 7–5. x ˙ sp = A sp x sp + B sp δ e where δ e is the elevator input, and w Z w /m U 0 x sp = q , A sp = I 1 ( M w + M w ˙ Z w /m ) I 1 ( M q + M w ˙ U 0 ) yy yy Z δ e /m B sp = I 1 ( M δ e + M w ˙ Z δ e /m ) yy Add that θ ˙ = q , so = q Take the output as θ , input is δ e , then form the transfer function θ ( s ) 1 q ( s ) = = 0 1 ( sI A sp ) 1 B sp δ e ( s ) e ( s ) As shown in the code, for the 747 (40Kft, M = 0 . 8 ) this reduces to: θ ( s ) 1 . 1569 s + 0 . 3435 = δ e ( s ) s ( s 2 + 0 . 7410 s + 0 . 9272) G θδ e ( s ) so that the dominant roots have a frequency of approximately 1 rad/sec and damping of about 0.4
Fall 2004 16.333 8–2 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0.09 0.95 0.09 0.2 0.3 0.42 0.54 0.68 0.84 1 0.95 0.2 0.3 0.42 0.54 0.68 0.84 0.2 0.4 0.6 0.8 0.8 0.2 0.4 0.6 Pole-Zero Map Real Axis Imaginary Axis Figure 1: Pole-zero map for G e Basic problem is that there are vast quantities of empirical data to show that pilots do not like the ﬂying qualities of an aircraft with this combination of frequency and damping What do they prefer? Acceptable 0.1 1 0 2 3 4 5 6 7 0.2 0.4 2 4 s s Unacceptable Poor Satisfactory 0.6 0.8 1 Undamped natural frequency rad/sec Damping ratio Figure 2: “Thumb Print” criterion This criterion has been around since the 1950’s, but it is still valid. Good target: frequency 3 rad/sec and damping of about 0 . 6 Problem is that the short period dynamics are no where near these numbers, so we must modify them.

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Fall 2004 16.333 8–3 Could do it by redesigning the aircraft, but it is a bit late for that... -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 -3 -2 -1 0 1 2 3 0.84 5 0.7 0.56 0.1 0.2 0.32 0.44 0.7 0.44 0.32 0.95 0.1 0.2 2.5 0.84 0.56 3 0.95 0.5 1 1.5 2 Pole-Zero Map Real Axis Imaginary Axis Figure 3: Pole-zero map and target pole locations Of course there are plenty of other things that we will consider when we design the controllers Small steady state error to commands δ e within limits No oscillations Speed control
Fall 2004 16.333 8–4 First Short Period Autopilot First attempt to control the vehicle response: measure θ and feed it back to the elevator command δ e . Unfortunately the actuator is slow, so there is an apparent lag in the response that we must model δ c e / 4 s + 4 δ a e / G θδ e ( s ) θ / o k θ O θ c o Dynamics: δ a is the actual elevator deﬂection, δ e c is the actuator e command created by our controller 4 θ = G θδ e ( s ) δ e a ; δ a = H ( s ) δ e c ; H ( s ) = e s + 4 The control is just basic proportional feedback δ c = k θ ( θ θ c ) e Which gives that θ = G θδ e ( s ) H ( s ) k θ ( θ θ c ) or that θ ( s ) G θδ e ( s ) H ( s ) k θ = θ c ( s ) 1 + G θδ e ( s ) H ( s ) k θ Looks good, but how do we analyze what is going on?

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