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

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 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 1 / 20

tugraz Outline Outline Motivation Stochastic MPC formulation Motivation Aircraft and gust modeling Stochastic MPC Formulation Numerical Aircraft and Gust Modeling Results Conclusions Numerical Results Conclusions and Future Work Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 2 / 20

tugraz Motivation Outline Motivation Aircraft carrier landing challenges Stochastic Atmospheric turbulence MPC formulation Carrier airwakes Aircraft and Carrier motion gust modeling Numerical Requirement: Real-time optimal feedback control Results Previous research: ℓ 1 adaptive control (Ramesh and Conclusions Subbarao, 2016), nominal MPC (Ngo and Sultan, 2015), dynamic inversion (Denison, 2007) Stochastic nature of gusts and airwakes → stochastic optimal control Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 3 / 20

tugraz Stochastic MPC Outline Optimization based control for offset recovery due to Motivation Stochastic gust MPC formulation N − 1 Aircraft and ( x T k Qx k + u T k Ru k ) + x T � gust modeling minimize E [ N Q N x N ] Numerical k =0 Results x k +1 = ¯ A d x k + ¯ B d u k + ¯ subject to E d η k Conclusions x k ∈ X u k ∈ U Hard polytopic state and control constraints relaxed to individual chance constraints Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 4 / 20

tugraz Stochastic MPC Outline In compact form Motivation Stochastic x = A x 0 + Bu + E η η η MPC formulation Optimal control problem with probabilistic constraints Aircraft and gust modeling Numerical E [ x T Qx + u T Ru ] minimize Results Conclusions P [ x ∈ ¯ subject to X ] ≥ 1 − α P [ u ∈ ¯ U ] ≥ 1 − β Adjust α , β for trade-off between conservatism and performance. Intractable with non-convex probabilistic constraints Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 5 / 20

tugraz Stochastic MPC Assume full state feedback, reconstruct past noise Outline Motivation from state and control input Stochastic Affine disturbance feedback policy MPC formulation Aircraft and k − 1 gust modeling � u k = G k , i η k + s k Numerical Results i =0 Conclusions Compact form u = G η η η + s Suboptimal but tractable; Origin is ISS w.r.t disturbance input under mild assumptions (Goulart & Kerrigan, 2008) Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 6 / 20

tugraz Stochastic MPC Outline Motivation Infinite dimensional problem → Finite dimensional Stochastic η ∼ N (0 , Σ), individual chance constraints → second MPC formulation order cone constraints Aircraft and gust modeling Φ − 1 (1 − α i ) � ¯ 2 ≤ p i − ¯ � � H x i G + E H x i ( A X 0 + Bs ) Numerical � Results Φ − 1 (1 − β j ) � ¯ 2 ≤ l j − ¯ � � H u j G H u j s Conclusions � Constraint set X = { H x x ≤ p } with H x = diag( H x , ... H x ) U = { H u u ≤ l } with H u = diag( H u , ... H u ) l = [ l T , ..., l T]T, p = [ p T , ..., p T]T Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 7 / 20

tugraz Stochastic MPC Outline Second order cone program formulation of SMPC Motivation Stochastic MPC b T s + tr ( M 2 G Σ Σ + G T M 1 G Σ Σ) + s T M 1 s Σ Σ minimize formulation Aircraft and � ¯ Φ − 1 (1 − α i ) � � gust modeling subject to H x i G + E 2 ≤ k 1 � Numerical � ¯ Φ − 1 (1 − β j ) � � H u j G 2 ≤ k 2 Results � Conclusions where k 1 = p i − ¯ H x i ( A X 0 + Bs ) k 2 = l j − ¯ H u j s b T = 2( A x 0 )T QB , M 1 = B T QB + R and M 2 = 2 E T QB Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 8 / 20

tugraz Aircraft motion Outline Motivation Linear longitudinal dynamics with gust Stochastic MPC formulation  ∆ ˙ u   − u 0 sin θ 0 − g cos θ 0   ∆ u  X u X w Aircraft and ∆ ˙ w Z u Z w u 0 cos θ 0 − g sin θ 0 ∆ w gust modeling        =       ∆ ˙ q M u M w M q 0 ∆ q Numerical      Results ∆ ˙ 0 0 1 0 ∆ θ θ Conclusions  X δ X δ T   − X u − X w 0    u g � ∆ δ e � − Z u − Z w 0 Z d Z δ T     + + w g       M δ M δ T ∆ δ T − M u − M w − M q     q g 0 0 0 0 0 Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 9 / 20

tugraz Aircraft motion Outline Motivation Aerodynamic coefficients based on the F/A-18 High Stochastic MPC angle of attack (HARV) model. formulation Landing configuration with nominal speed 134 knots Aircraft and gust modeling and sea level altitude Numerical Results Aerodynamic model Conclusions 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 on elevator deflection Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 10 / 20

tugraz Aircraft motion Outline Motivation Assuming steady-state descent flight Stochastic MPC u trim = 223 . 1 ft/s formulation w trim = 28 . 4 ft/s Aircraft and q trim = 0 deg/s gust modeling θ trim = 3 . 72 deg Numerical Results Corresponds to a trim AOA of 7 . 26 deg and − 3 . 5 deg Conclusions glideslope Trimmed controls δ e = 11 . 36 deg δ T = 0 . 29 Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 11 / 20

tugraz Gust modeling Outline Only continuous gusts studied Motivation Stochastic Spatially varying stochastic processes with Gaussian MPC distribution formulation Aircraft and Dryden form given as gust modeling Numerical L u 1 Results Φ u g (Ω) = σ 2 u Conclusions 1 + ( L u Ω) 2 π 2 1 + 3( L w Ω) 2 L w Φ w g (Ω) = σ 2 w (1 + ( L w Ω) 2 ) π Ω 2 Φ q g (Ω) = π ) 2 Φ w g (Ω) 1 + ( 4 b Ω Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 12 / 20

tugraz Gust modeling Outline For low altitude ( ∼ 200 ft) Motivation Stochastic MPC h formulation L w = 100 ft L u = (0 . 177 + 0 . 000823 h ) 1 . 2 ft Aircraft and gust modeling σ w σ w = 0 . 1 W 20 ft/s σ u = (0 . 177 + 0 . 000823 h ) 0 . 4 ft/s Numerical Results Conclusions Spectral factorization → transfer function → linear filter driven by white noise ˙ ξ w = A w ξ w + E w η d = C w ξ w Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 13 / 20

tugraz Gust modeling Outline Motivation � π b Significance of rotary gust q g if 16 L w C m q > C m α Stochastic MPC formulation Augmenting linearized aircraft model with wind Aircraft and dynamics gust modeling � ˙ Numerical � x l Results = ¯ Ax + ¯ Bu + ¯ x = ˙ E η ˙ ξ w Conclusions Discretized version x k +1 = ¯ A d x k + ¯ B d u k + ¯ E d η k , k ∈ N 0 Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 14 / 20

tugraz Gust modeling Outline Wind gust at low, moderate, and high turbulence Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 15 / 20

tugraz Simulation results Outline Perturbed flight with initial state Motivation � T . � x = 15 − 10 0 0 . 1 Stochastic MPC Prediction horizon N p = 10 s, Total time 20 s. formulation Aircraft and gust modeling Numerical Results Conclusions Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 16 / 20

tugraz Simulation results Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Conclusions Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 17 / 20

tugraz Simulation Results Randomized initial conditions Outline Motivation Stochastic MPC formulation Aircraft and gust modeling Numerical Results Noise/wind reconstruction Conclusions Gaurav Misra and Xiaoli Bai Stochastic Model Predictive Control for Gust Alleviation during Aircraft Carrier Landing 18 / 20