Stochastic Model Predictive Control for Outline Gust Alleviation - - PowerPoint PPT Presentation

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Stochastic Model Predictive Control for Outline Gust Alleviation - - PowerPoint PPT Presentation

Apr 23, 2023 132 likes •341 views

tugraz Stochastic Model Predictive Control for Outline Gust Alleviation during Aircraft Carrier Motivation Stochastic Landing MPC formulation Aircraft and gust modeling Numerical Gaurav Misra and Xiaoli Bai Results Conclusions

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing

Gaurav Misra and Xiaoli Bai

Department of Mechanical and Aerospace Engineering Rutgers, The State University of New Jersey

IEEE American Control Conference Milwaukee, WI 06/27/2018

Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Outline

Motivation Stochastic MPC Formulation Aircraft and Gust Modeling Numerical Results Conclusions and Future Work

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Motivation

Aircraft carrier landing challenges

Atmospheric turbulence Carrier airwakes Carrier motion

Requirement: Real-time optimal feedback control Previous research: ℓ1 adaptive control (Ramesh and Subbarao, 2016), nominal MPC (Ngo and Sultan, 2015), dynamic inversion (Denison, 2007) Stochastic nature of gusts and airwakes → stochastic

  • ptimal control

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Stochastic MPC

Optimization based control for offset recovery due to gust minimize E[

N−1

  • k=0

(xT

k Qxk + uT k Ruk) + xT N QNxN]

subject to xk+1 = ¯ Adxk + ¯ Bduk + ¯ Edηk xk ∈ X uk ∈ U Hard polytopic state and control constraints relaxed to individual chance constraints

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Stochastic MPC

In compact form x = Ax0 + Bu + Eη η η Optimal control problem with probabilistic constraints minimize E[xTQx + uTRu] subject to P[x ∈ ¯ X] ≥ 1 − α P[u ∈ ¯ U] ≥ 1 − β Adjust α, β for trade-off between conservatism and performance. Intractable with non-convex probabilistic constraints

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Stochastic MPC

Assume full state feedback, reconstruct past noise from state and control input Affine disturbance feedback policy uk =

k−1

  • i=0

Gk,iηk + sk Compact form u = Gη η η + s Suboptimal but tractable; Origin is ISS w.r.t disturbance input under mild assumptions (Goulart & Kerrigan, 2008)

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Stochastic MPC

Infinite dimensional problem → Finite dimensional η ∼ N(0, Σ), individual chance constraints → second

  • rder cone constraints

Φ−1(1 − αi)

  • ¯

HxiG + E

  • 2 ≤ pi − ¯

Hxi(AX0 + Bs) Φ−1(1 − βj)

  • ¯

HujG

  • 2 ≤ lj − ¯

Hujs Constraint set

X = {Hxx ≤ p} with Hx = diag(Hx, ...Hx) U = {Huu ≤ l} with Hu = diag(Hu, ...Hu) l = [lT, ..., lT]T, p = [pT, ..., pT]T

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Stochastic MPC

Second order cone program formulation of SMPC minimize bTs + tr(M2GΣ Σ Σ + GTM1GΣ Σ Σ) + sTM1s subject to Φ−1(1 − αi)

  • ¯

HxiG + E

  • 2 ≤ k1

Φ−1(1 − βj)

  • ¯

HujG

  • 2 ≤ k2

where

k1 = pi − ¯ Hxi(AX0 + Bs) k2 = lj − ¯ Hujs bT = 2(Ax0)TQB, M1 = BTQB + R and M2 = 2ETQB

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Aircraft motion

Linear longitudinal dynamics with gust     ∆ ˙ u ∆ ˙ w ∆ ˙ q ∆ ˙ θ     =     Xu Xw −u0 sin θ0 −g cos θ0 Zu Zw u0 cos θ0 −g sin θ0 Mu Mw Mq 1         ∆u ∆w ∆q ∆θ     +     Xδ XδT Zd ZδT Mδ MδT     ∆δe ∆δT

  • +

    −Xu −Xw −Zu −Zw −Mu −Mw −Mq       ug wg qg  

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Aircraft motion

Aerodynamic coefficients based on the F/A-18 High angle of attack (HARV) model. Landing configuration with nominal speed 134 knots and sea level altitude Aerodynamic model

Leading and trailing edge flaps completely down to 17.6 degrees and 45 degrees Both left and right ailerons down to 42 deg Longitudinal aerodynamics actuator dependency only

  • n elevator deflection

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Aircraft motion

Assuming steady-state descent flight

utrim = 223.1 ft/s wtrim = 28.4 ft/s qtrim = 0 deg/s θtrim = 3.72 deg

Corresponds to a trim AOA of 7.26 deg and −3.5 deg glideslope Trimmed controls

δe = 11.36 deg δT = 0.29

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Gust modeling

Only continuous gusts studied Spatially varying stochastic processes with Gaussian distribution Dryden form given as Φug (Ω) = σ2

u

Lu π 1 1 + (LuΩ)2 Φwg (Ω) = σ2

w

Lw π 1 + 3(LwΩ)2 (1 + (LwΩ)2)

2

Φqg (Ω) = Ω2 1 + ( 4bΩ

π )2 Φwg (Ω)

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Gust modeling

For low altitude (∼ 200 ft) Lw = 100 ft Lu = h (0.177 + 0.000823h)1.2 ft σw = 0.1W20 ft/s σu = σw (0.177 + 0.000823h)0.4 ft/s Spectral factorization → transfer function → linear filter driven by white noise ˙ ξw = Awξw + Ewη d = Cwξw

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Gust modeling

Significance of rotary gust qg if

  • πb

16Lw Cmq > Cmα

Augmenting linearized aircraft model with wind dynamics ˙ x = ˙ xl ˙ ξw

  • = ¯

Ax + ¯ Bu + ¯ Eη Discretized version xk+1 = ¯ Adxk + ¯ Bduk + ¯ Edηk, k ∈ N0

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Gust modeling

Wind gust at low, moderate, and high turbulence

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Simulation results

Perturbed flight with initial state x =

  • 15

−10 0.1 T. Prediction horizon Np = 10 s, Total time 20 s.

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Simulation results

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Simulation Results

Randomized initial conditions Noise/wind reconstruction

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Numerical Results

Comparison with certainty equivalent MPC Cost comparison Method Cost AD-SMPC 3.23 × 106 CE-MPC 3.47 × 106

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Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions

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Conclusions and Future Work

Summary Stochastic MPC for aircraft glideslope recovery in gust Chance constrained affine-disturbance feedback MPC formulation Tractable, cost efficient solution compared to certainty equivalent MPC Future directions Incomplete state information and measurement noise Inclusion of carrier burble components

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