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# topic2 - Topic#2 16.31 Feedback Control Systems Basic Root...

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Topic #2 16.31 Feedback Control Systems Basic Root Locus Basic aircraft control concepts Basic control approaches Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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Fall 2007 16.31 2–1 Aircraft Longitudinal Control Consider the short period approximate model of an 747 aircraft. ˙ x sp = A sp x sp + B sp δ e where δ e is the elevator input, and x sp = w q , A sp = Z w /m U 0 I 1 yy ( M w + M ˙ w Z w /m ) I 1 yy ( M q + M ˙ w U 0 ) B sp = Z δ e /m I 1 yy ( M δ e + M ˙ w Z δ e /m ) Add that ˙ θ = q , so = q Take the output as θ , input is δ e , then form the transfer function θ ( s ) δ e ( s ) = 1 s q ( s ) δ e ( s ) = 1 s 0 1 ( sI A sp ) 1 B sp For the 747 (40Kft, M = 0 . 8 ) this reduces to: θ ( s ) δ e ( s ) = 1 . 1569 s + 0 . 3435 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 -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 0.2 0.3 0.42 0.54 0.68 0.84 0.95 0.09 0.2 0.3 0.42 0.54 0.68 0.84 0.95 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 Pole-Zero Map Real Axis Imaginary Axis Figure 1: Pole-zero map for G e September 2, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 2007 16.31 2–2 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 is preferred? Figure 2: “Thumb Print” criterion This criterion was developed in 1950’s, and more recent data is provided in MILSPEC8785C Based on this plot, a 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. Could do it by redesigning the aircraft, but it is a bit late for that... September 2, 2007 6 7 4 5 2 3 0 0.1 0.2 0.4 0.6 0.8 1 2 4 1 POOR ACCEPTABLE UNACCEPTABLE SATISFACTORY Damping ratio ς s Undamped natural frequency ω s rad/sec Figure by MIT OpenCourseWare. Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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Fall 2007 16.31 2–3 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 k θ θ c Dynamics: δ a e is the actual elevator deﬂection, δ c e is the actuator command created by our controller θ = G θδ e ( s ) δ a e ; δ a e = H ( s ) δ c e ; H ( s ) = 4 s + 4 The control is just basic proportional feedback δ c e = k θ ( θ θ c ) which gives that θ = G θδ e ( s ) H ( s ) k θ ( θ θ c ) or that θ ( s ) θ c ( s ) = G θδ e ( s ) H ( s ) k θ 1 + G θδ e ( s ) H ( s ) k θ Looks good, but how do we analyze what is going on?
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topic2 - Topic#2 16.31 Feedback Control Systems Basic Root...

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