Fa2004 16.333105 E Without Washout With Washout 0.5 15 Time Without Washout 15 Time Figure 6: Impulse response of closed loop system with and without the Washout filter(T= 4.2). Commanded Tc=0, and both have(8)ss =0, but without the filter, Tss=0, whereas with it, Tss #0 For direct comparison with and without the filter, applied impulse as re to both closed-loop systems and then calculated r and dr Bottom plot shows that control signal quickly converges to zero in both cases, i.e., no more control effort is being applied to correct the motion But only the one with the washout filter produces a zero control input even though the there is a steady turn=, the controller will not try to fight a commanded steady turn
� Fall 2004 16.333 10–5 0 5 10 15 20 25 30 −0.5 0 0.5 1 Time Response r Without Washout With Washout 0 5 10 15 20 25 30 −0.5 0 0.5 1 Time Control δ r Without Washout With Washout Figure 6: Impulse response of closed loop system with and without the Washout filter (τ = 4.2). Commanded rc = 0, and both have (δr)ss = 0, but without the filter, rss = 0, whereas with it, rss = 0. • For direct comparison with and without the filter, applied impulse as rc to both closedloop systems and then calculated r and δr. • Bottom plot shows that control signal quickly converges to zero in both cases, i.e., no more control effort is being applied to correct the motion. • But only the one with the washout filter produces a zero control input even though the there is a steady turn ⇒ the controller will not try to fight a commanded steady turn
Fa2004 16.33310-6 Heading Autopilot Design So now have the yaw damper added correctly and want to control the heading yl Need to bank the aircraft to accomplish this Result is a "coordinated turn"with angular rate y FRoA◆HW R PATH oF U AlC R Aircraft banked to angle so that vector sum of mg and mvo l is along the body z-aXIs ng in the body y-axis direction gives muo? cos o= mg sin g tan Since typically 1, we have gives the desired bank angle for a specified turn rate
Fall 2004 16.333 10–6 Heading Autopilot Design • So now have the yaw damper added correctly and want to control the heading ψ. – Need to bank the aircraft to accomplish this. – Result is a “coordinated turn” with angular rate ψ˙ Aircraft banked to angle φ so that vector sum of mg and mU0ψ˙ • is along the body zaxis – Summing in the body yaxis direction gives mu0ψ˙ cos φ = mg sin φ U0ψ˙ tan φ = g • Since typically φ � 1, we have U0ψ˙ φ ≈ g gives the desired bank angle for a specified turn rate
Fa2004 16.33310-7 Problem now is that yb tends to be a noisy signal to base out bank angle on, so we generate a smoother signal by filtering it Assume that the desired heading is known vd and we want y to follow v,d relatively slowly Choose dynamics T1 +yb=yd yd T1s+ with T1=15-20sec depending on the vehicle and the goals A low pass filter that eliminates the higher frequency noise Filtered heading angle satisfies which we can use to create the desired bank angle 719
Fall 2004 16.333 10–7 Problem now is that ψ˙ • tends to be a noisy signal to base out bank angle on, so we generate a smoother signal by filtering it. – Assume that the desired heading is known ψd and we want ψ to follow ψd relatively slowly – Choose dynamics τ1ψ˙ + ψ = ψd ψ 1 ⇒ = ψd τ1s + 1 with τ1=1520sec depending on the vehicle and the goals. – A low pass filter that eliminates the higher frequency noise. • Filtered heading angle satisfies 1 ψ˙ = (ψd − ψ) τ1 which we can use to create the desired bank angle: U0 φd = ψ˙ = U0 (ψd − ψ) g τ1g
Fa2004 16.333108 Roll Control e Given this desired bank angle we need a roll controller to ensure that the vehicle tracks it accurately Aileron is best actuator to use: Ba= ho(od -o)-kpp To design hio and kp, can just use the approximation of the roll mode cp= Lpp o=p which gives L For the design, add the aileron servo dynamics Ha(s) 0.15s+1 8a= ha(s8 ● PD controller ko(sy +1)+kood, adds zero at s=-1/ I-Pick y=2/3 0.120.06 s Axe igure 7: Root Locus for roll loop -closed Loop poles for Kp=-20, Ko=-30
−7 −6 −5 −4 −3 −2 −1 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 0.88 0.68 0.52 0.38 0.28 0.2 0.12 0.06 0.68 0.38 0.28 0.2 0.12 0.06 8 0.88 10 0.52 6 2 4 6 8 10 2 4 Root Locus Real Axis Imaginary Axis � Fall 2004 16.333 10–8 Roll Control • Given this desired bank angle, we need a roll controller to ensure that the vehicle tracks it accurately. – Aileron is best actuator to use: δa = kφ(φd − φ) − kpp • To design kφ and kp, can just use the approximation of the roll mode I� I� ¨ ˙ xxp˙ = Lpp + Lδaδa xxφ − Lpφ˙ = Lδaδa φ = p which gives φ Lδa = δa s(I� xxs − Lp) • For the design, add the aileron servo dynamics 1 Ha(s) = , δa = Ha(s)δc a 0.15s + 1 a • PD controller δc a = −kφ(sγ + 1) + kφφd, adds zero at s = −1/γ – Pick γ = 2/3 Figure 7: Root Locus for roll loop – closed Loop poles for Kp = −20, Kφ = −30