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
Fa2004 16.33311-5 Altitude controller Figure 6: Altitude controller time response for more complicated input The gain Kh must be kept very small since the low frequency poles are heading into the rhP Must take a different control approach to design a controller for the full dynamics Must also use the throttle to keep tighter control of the speed Example in Etkin and Reid, page 277. Simulink and Matlab code for this is on the Web page
Fall 2004 16.333 11–5 0 50 100 150 200 250 300 −20 0 20 40 60 80 100 120 Altitude controller height time h c h Figure 6: Altitude controller time response for more complicated input. • The gain Kh must be kept very small since the low frequency poles are heading into the RHP. ⇒ Must take a different control approach to design a controller for the full dynamics. – Must also use the throttle to keep tighter control of the speed. – Example in Etkin and Reid, page 277. Simulink and Matlab code for this is on the Web page
Fa2004 16.33311-6 Figure 7: Simulink block diagram for implementing the more advanced version of the altitude hold controller. Follows and extends the example in Etkin and Reid page 277
Fall 2004 16.333 11–6 Altitude Controller Completes Figure 8.17 in Etkin and Reid use with altit_simul.m T=alt(:,1);u=alt(:,2);w=alt(:,3); q=alt(:,4);theta=alt(:,5);h=alt(:,6); dele=alt(:,7);delt=alt(:,8);href=alt(:,9); K_h*Khn(s) Khd(s) height Lead alt To Workspace Sum4 Signal 2 Signal Builder Signal Generator Scope1 Scope Saturation Sat Ramp Mux Mux Mux Dele Delt u w q theta h all Longitudinal Model -KK_u K_th K_q Kun(s) Kud(s) Engine Lead JNt(s) JDt(s) Engine Dynamics JNe(s) JDe(s) Elevator Lag 0 Constant1 0 Clock u u_ref Figure 7: Simulink block diagram for implementing the more advanced version of the altitude hold controller. Follows and extends the example in Etkin and Reid, page 277
with q and theta FB to 8 Fa2004 16.33311-7 0.94 -3 Figure 8: Root loci associated for simulation(a, 6 to de) wth u FB to Figure 9: Root loci associated with simulation (u to dt). There is more authority in the linear control analysis but this saturates the nonlinear simulation igure 10: Root loci associated with simulation(h to de). Lead controller
Fall 2004 −3 −2.5 −2 −1.5 −1 −0.5 0 −3 −2 −1 0 1 2 3 0.82 0.66 0.52 0.4 0.28 0.18 0.09 3 0.52 0.4 0.28 0.18 0.09 0.94 0.5 0.82 0.66 2 2.5 0.94 1.5 1 1 2 1.5 0.5 2.5 3 with q and theta FB to δ e Real Axis Imaginary Axis 16.333 11–7 Figure 8: Root loci associated for simulation (q, θ to δe). −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0.94 0.82 0.66 0.52 0.4 0.28 0.18 0.09 0.94 0.66 0.52 0.4 0.28 0.18 0.09 1.75 0.25 2 0.82 1 0.5 1.25 0.75 1.25 0.75 1.5 1.5 1 0.5 1.75 2 0.25 with u FB to δ t Real Axis Imaginary Axis Figure 9: Root loci associated with simulation (u to δt). There is more authority in the linear control analysis, but this saturates the nonlinear simulation. −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0.82 0.6 0.42 0.31 0.22 0.16 0.1 0.045 0.6 0.31 0.22 0.16 0.1 0.045 1.75 0.82 2 0.42 0.75 0.25 1.25 0.5 1 0.75 1.5 1.25 0.25 0.5 1 2 1.5 1.75 with h FB to δ e command Real Axis Imaginary Axis Figure 10: Root loci associated with simulation (h to δe). Lead controller
Fa2004 16.333118 Initial Condition response to 90m altitude error. E-FB 5*U Initial Condition response to 90m altitude error 208, input Figure 11: Initial condition response using elevator controller. Note the initial ele- vator effort and that the throttle is scaled back as the speed picks up
Fall 2004 16.333 11–8 0 2 4 6 8 10 12 14 16 18 −20 0 20 40 60 80 100 Initial Condition response to 90m altitude error. E−FB H 5*U θ deg 0 2 4 6 8 10 12 14 16 18 −20 −15 −10 −5 0 5 10 15 20 Commands (degs) Initial Condition response to 90m altitude error δ e input 20*δ t input Figure 11: Initial condition response using elevator controller. Note the initial elevator effort and that the throttle is scaled back as the speed picks up
Fa2004 16.33311-9 DO 100 150 250 Figure 12: Command following ramps up and down. offset, but smooth transitions deg 250 200 Figure 13: Elevator controller -Overall response and inputs
Fall 2004 16.333 11–9 0 50 100 150 200 250 300 0 200 400 600 800 1000 1200 1400 1600 time height Altitude controller: elevator FB h c h Figure 12: Command following ramps up and down. Offset, but smooth transitions. 0 50 100 150 200 250 300 −5 0 5 Linear Response − elevator−FB U α deg θ deg 0 50 100 150 200 250 300 −3 −2 −1 0 1 2 3 δ e 10*δ t Figure 13: Elevator controller – Overall response and inputs
