Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing

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1 Ouline Moivaion Sochasic formulaion Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing Aircraf and gus modeling Resuls Deparmen of Mechanical and Aerospace Engineering Rugers, The Sae Universiy of New Jersey IEEE American Conrol Conference Milwaukee, WI 06/27/2018 Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 1 / 20

2 Ouline Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Moivaion Sochasic Formulaion Aircraf and Gus Modeling Resuls and Fuure Work Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 2 / 20

3 Moivaion Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Aircraf carrier landing challenges Amospheric urbulence Carrier airwakes Carrier moion Requiremen: Real-ime opimal feedback conrol Previous research: l 1 adapive conrol (Ramesh and Subbarao, 2016), nominal (Ngo and Sulan, 2015), dynamic inversion (Denison, 2007) Sochasic naure of guss and airwakes sochasic opimal conrol Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 3 / 20

4 Sochasic Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Opimizaion based conrol for offse recovery due o gus N 1 minimize E[ (x T k Qx k + u T k Ru k) + x T N Q Nx N ] subjec o k=0 x k+1 = Ā d x k + B d u k + Ē d η k x k X u k U Hard polyopic sae and conrol consrains relaxed o individual chance consrains Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 4 / 20

5 Sochasic Ouline Moivaion Sochasic formulaion Aircraf and gus modeling In compac form x = Ax 0 + Bu + Eη Opimal conrol problem wih probabilisic consrains Resuls minimize subjec o E[x T Qx + u T Ru] P[x X] 1 α P[u Ū] 1 β Adjus α, β for rade-off beween conservaism and performance. Inracable wih non-convex probabilisic consrains Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 5 / 20

6 Sochasic Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Assume full sae feedback, reconsruc pas noise from sae and conrol inpu Affine disurbance feedback policy Compac form k 1 u k = G k,i η k + s k i=0 u = Gη + s Subopimal bu racable; Origin is ISS w.r. disurbance inpu under mild assumpions (Goular & Kerrigan, 2008) Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 6 / 20

7 Sochasic Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Infinie dimensional problem Finie dimensional η N (0, Σ), individual chance consrains second order cone consrains Φ 1 (1 α i ) H xi G + E 2 p i H xi (AX 0 + Bs) Φ 1 (1 β j ) H uj G 2 l j H uj s Consrain se X = {H x x p} wih H x = diag(h x,...h x ) U = {H u u l} wih H u = diag(h u,...h u ) l = [l T,..., l T ] T, p = [p T,..., p T ] T Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 7 / 20

8 Sochasic Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Second order cone program formulaion of S minimize b T s + r(m 2 GΣ + G T M 1 GΣ) + s T M 1 s subjec o Φ 1 (1 α i ) H xi G + E 2 k 1 Φ 1 (1 β j ) H uj G 2 k 2 where k 1 = p i H xi (AX 0 + Bs) k 2 = l j H uj s b T = 2(Ax 0 ) T QB, M 1 = B T QB + R and M 2 = 2E T QB Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 8 / 20

9 Aircraf moion Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Linear longiudinal dynamics wih gus u X u X w u 0 sin θ 0 g cos θ 0 u ẇ q = Z u Z w u 0 cos θ 0 g sin θ 0 w M u M w M q 0 q θ θ X δ X δt [ ] X u X w 0 + Z d Z δt δe M δ M δt + Z u Z w 0 u g δ T M u M w M q w g q g Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 9 / 20

10 Aircraf moion Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Aerodynamic coefficiens based on he F/A-18 High angle of aack (HARV) model. Landing configuraion wih nominal speed 134 knos and sea level aliude Aerodynamic model Leading and railing edge flaps compleely down o 17.6 degrees and 45 degrees Boh lef and righ ailerons down o 42 deg Longiudinal aerodynamics acuaor dependency only on elevaor deflecion Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 10 / 20

11 Aircraf moion Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Assuming seady-sae descen fligh u rim = f/s w rim = 28.4 f/s q rim = 0 deg/s θ rim = 3.72 deg Corresponds o a rim AOA of 7.26 deg and 3.5 deg glideslope Trimmed conrols δ e = deg δ T = 0.29 Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 11 / 20

12 Gus modeling Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Only coninuous guss sudied Spaially varying sochasic processes wih Gaussian disribuion Dryden form given as Φ ug (Ω) = σu 2 L u 1 π 1 + (L u Ω) 2 Φ wg (Ω) = σw 2 L w 1 + 3(L w Ω) 2 2 π (1 + (L w Ω) 2 ) Φ qg (Ω) = Ω ( 4bΩ Φ π )2 w g (Ω) Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 12 / 20

13 Gus modeling Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls For low aliude ( 200 f) h L w = 100 f L u = f ( h) 1.2 σ w σ w = 0.1W 20 f/s σ u = f/s ( h) 0.4 Specral facorizaion ransfer funcion linear filer driven by whie noise ξ w = A w ξ w + E w η d = C w ξ w Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 13 / 20

14 Gus modeling Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls πb Significance of roary gus q g if 16L w C mq > C mα Augmening linearized aircraf model wih wind dynamics [ ] ẋl ẋ = = ξ Āx + Bu + Ēη w Discreized version x k+1 = Ādx k + B d u k + Ēdη k, k N 0 Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 14 / 20

15 Gus modeling Ouline Moivaion Wind gus a low, moderae, and high urbulence Sochasic formulaion Aircraf and gus modeling Resuls Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 15 / 20

16 Simulaion resuls Ouline Moivaion Sochasic formulaion Perurbed fligh wih iniial sae x = [ ] T. Predicion horizon N p = 10 s, Toal ime 20 s. Aircraf and gus modeling Resuls Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 16 / 20

17 Simulaion resuls Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 17 / 20

18 Ouline Simulaion Resuls Randomized iniial condiions Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Noise/wind reconsrucion Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 18 / 20

19 Resuls Ouline Comparison wih cerainy equivalen Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Cos comparison Mehod Cos AD-S CE Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 19 / 20

20 and Fuure Work Ouline Moivaion Sochasic formulaion Aircraf and gus modeling Resuls Summary Sochasic for aircraf glideslope recovery in gus Chance consrained affine-disurbance feedback formulaion Tracable, cos efficien soluion compared o cerainy equivalen Fuure direcions Incomplee sae informaion and measuremen noise Inclusion of carrier burble componens Sochasic Model Predicive Conrol for Gus Alleviaion during Aircraf Carrier Landing 20 / 20