16.333: Lecture #13 Aircraft Longitudinal Autopilots Altitude Hold and Landing
16.333: Lecture # 13 Aircraft Longitudinal Autopilots Altitude Hold and Landing 0
Fa2004 16.33311-1 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 sIny=00(0 (6-a)=U06-U U06- For fixed Uo, h determined by variables in short period model Use short period model augmented with g state 汇=Agnx+B q h=|-10U where 0 B In transfer function form, we get hK(s+4)(s-36) × Figure 1: Altitude root locus #1
� � � 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˙ ≈ U0 sin γ = U0 (θ − α) = U0 θ − U0 w = U0 θ − w U0 – For fixed U0, h˙ determined by variables in short period model • Use short period model augmented with θ state ⎡ ⎤ ⎧ w ⎨ x˙ = A˜spx + B˜spδe x = ⎣ q ⎦ θ ⇒ ⎩ h˙ = � −1 0 U0 x where � � � � 0 A˜ ˜ sp = Asp , Bsp = Bsp [ 0 1 0 ] 0 • In transfer function form, we get h K(s + 4)(s − 3.6) = δe s2(s2 + 2ζspωsps + ω2 sp) −5 −4 −3 −2 −1 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 0.945 0.89 0.81 0.68 0.5 0.3 5 0.81 0.976 0.994 0.68 0.5 0.3 0.994 0.89 0.945 0.976 4 3 2 1 Pole−Zero Map Real Axis Imaginary Axis Figure 1: Altitude root locus #1
Fa2004 16.33311-2 Altitude Gain Root Locus: k>0 Altitude gain Root locus: k<0 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 0 state ww-kq9-h00+8c L hw kg ke x+de=-KIlz+de Target pole locations s=-1.8+2.41, s=-0 25 gains:K=[-0.0017-26791-6.5498 Inner loop target poles Figure 3: Inner loop target pole locations - t get there with only a gain
� � � � Fall 2004 16.333 11–2 −5 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 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 0 1 2 3 4 −4 −3 −2 −1 0 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 δe = −kww−kqq−kθθ+δe c = kw kq kθ x+δc = −KILx+δe c − e – Target pole locations s = −1.8 ± 2.4i, s = −0.25 – Gains: KIL = −0.0017 −2.6791 −6.5498 −5 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 0.8 0.68 0.54 0.38 0.18 0.54 0.18 0.986 0.95 0.89 5 4 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
Fa2004 16.33311-3 Giving the closed-loop dynamics i= Asp C+ Bsp(KIL +de (Asp-Bspkil)r+Bs 10U In transfer function form K(s+4)(s-36) 6es(s+0.25)2+3.6s+9) with Sd and wd being the result of the inner loop control Altitude gain Root Altitude gain Root locations with inner lo Figure 4: Root loci versus altitude gain Kh <0 with inner loop added(zoomed on right). Much better than without inner loop, but gain must be small(Kh x-0.01) Final step then is to select the feedback gain on the altitude kh and implement(hc is the commanded altitude. de=kn(he-h Design inner loop to damp the short period poles and move one of the poles near the origin -Then select Kh to move the 2 poles near the origin
� Fall 2004 16.333 11–3 • Giving the closedloop dynamics x˙ = A˜spx + B˜sp (−KILx + δe c ) δc = (A˜sp − B˜spK Bsp e � IL)x + ˜ h˙ = −1 0 U0 x • In transfer function form h K˜ (s + 4)(s − 3.6) = δc s(s + 0.25)(s2 + 3.6s + 9) e with ζd and ωd being the result of the inner loop control. −6 −5 −4 −3 −2 −1 0 1 −4 −3 −2 −1 0 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 0.84 0.58 0.14 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 0.1 0.2 −0.5 −0.4 −0.3 −0.2 −0.1 0 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 Kh < 0 with inner loop added (zoomed on right). Much better than without inner loop, but gain must be small (Kh ≈ −0.01). • Final step then is to select the feedback gain on the altitude Kh and implement (hc is the commanded altitude.) δc = −Kh(hc − h) e – Design inner loop to damp the short period poles and move one of the poles near the origin. – Then select Kh to move the 2 poles near the origin
Fa2004 16.33311-4 Poles near origin dominate response s=-01056+0.281li →n=0.3,=0.35 Rules of thumb for 2nd order response 10-90% rise time t=1+1.1+1.42 Settling time(5%) Time to peak amplitude tp Peak overshoot Figure 5: Time response for the altitude controller Predictions: tr=5.2sec, ts=284sec, tp=11.2sec, Mp=0. 3 Now with h feedback, things are a little bit better, but not much Real design is complicated by the location of the zeros Any actuator lag is going to hinder the performance also Typically must tweak inner loop design for altitude controller to work well
Fall 2004 16.333 11–4 • Poles near origin dominate response s = −0.1056 ± 0.2811i ⇒ ωn = 0.3, ζ = 0.35 – Rules of thumb for 2nd order response: 1 + 1.1ζ + 1.4ζ2 1090% rise time tr = ωn 3 Settling time (5%) ts = ζωn Time to peak amplitude tp = � π ωn 1 − ζ2 Peak overshoot Mp = e−ζωntp 0 10 20 30 40 50 60 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Altitude controller height time Figure 5: Time response for the altitude controller. • Predictions: tr = 5.2sec, ts = 28.4sec, tp = 11.2sec, Mp = 0.3 • Now with h feedback, things are a little bit better, but not much. – Real design is complicated by the location of the zeros – Any actuator lag is going to hinder the performance also. ⇒ Typically must tweak inner loop design for altitude controller to work well