System Identification and Models for Flight Control

Transcription

1 System Identification and Models for Flight Control -./, &! %! $! #! "! x α α k+1 A ERA x = 1 t α 1 α x C L (k t) = C ERA C Lα C L α α α ERA Model k + quasi-steady contribution k B ERA t α k input + D ERA α k!! " # $ % & ' ( )*+, Steve Brunton and Clancy Rowley Princeton University FAA/JUP Meeting October 7, 21

2 Major Applications 1. Autopilot 2. Flight Simulators a. small, agile UAV b. severe weather c. wake vorticity d. cheaper than full CFD to compute aerodynamic response of airframe e. necessary when using for onboard control

3 Goals 1. System Identification a. Stability derivatives b. Additional fast dynamics c. Markov parameters and ERA algorithm x α α k+1 A ERA x = 1 t α 1 α x C L (k t) = C ERA C Lα C L α α α ERA Model k + quasi-steady contribution k B ERA t α k input + D ERA α k 2. Flight Control a. Stability augmentation b. Control augmentation c. 6DOF inertial model d. Coupled aerodynamic model coupled model T L D M I yy q = M mv γ = L + T sin(α) mg sin(γ) m V = T cos(α) D mg sin(γ) α = q (L + T sin(α) mg cos(γ)) /mv /123)*+,#-. ẋ = Ax + Bu y = Cx + Du!"#$%&'()*+,#-. q γ V α e. Interesting control problem when inertial/aerodynamic timescales are close

4 NextGen ConOps V2.: UAVs Unmanned Aircraft Systems UAS operations are some of the most demanding operations in NextGen. UAS operations include scheduled and on-demand flights for a variety of civil, military, and state missions. Because of the range of operational uses, UAS operators may require access to all NextGen airspace Vertical Flight... Rotorcraft are also used for UAS applications for commercial, police, and security operations. These operations add to the density and complexity of operations, particularly in and around urban areas Integrated Environmental Operations UAS performing security functions and the airport perimeter security intrusion detection system may have the capability to assist with wildlife management programs Weather Information Enterprise Services! Enterprise Service 3: UASs Are Used for Weather Reconnaissance. [R-169] En route weather reconnaissance UASs are equipped to collect and report in-flight weather data. Specialized weather reconnaissance UASs are used to scout potential flight routes and trajectories to identify available weather-favorable airspace...

5 UAV Challenges UAVs and NextGen: May require access to all NextGen airspace Civil, military and state missions Mobile communications relays Security/Policing Shadow (Aerocam) Weather reconnaissance and much more... Safety Hazards: Extremely light, very difficult to control in high crosswinds No human failsafes Especially dangerous in takeoff and landing of the 195 Predators the Air Force has acquired since 1994 had been lost because of Class A mishaps % were attributed to human error. And 15% of the accidents occurred during landing Government Computer News (Oct. 9, 29) Predator (General Atomics)

6 Wind Disturbances Free air turbulence Wind rotors Wake vorticity Microburst wind shear ( Slides and history courtesy of Rob Stengel ) Safety Hazards: Landing and takeoff (congestion during storms, takeoff waiting lines) Especially problematic for lightweight UAVs

7 Flight Simulators Goal: Pilot flies real aircraft for 5-1 minutes, and reduced order aerodynamic model is automatically generated. Not specific to unsteady aerodynamics Physics based, generalizable to nonlinear affects, such as wake vorticity, turbulence, etc.

8 Flight Dynamic Model coupled model flight dynamics aerodynamics Our Research controllers observers gust disturbance rejection optimized flight paths evasive maneuvering early warning systems Goals

9 Theodorsen s Model C L = π a ḧ + α 2 2 α Added-Mass +2π α + ḣ + 12 α 12 a Circulatory C(k) Theodorsen, NACA-496, Leishman, 26. k = πfc U G(s) α 1/s α G QS (s) C L (α eff ) CL ν Quasi-Steady C(s) 1/s α C L + Added Mass C φ L

10 Wagner s Indicial Response Given y δ (t) for an impulse response u = δ(t), The response to an arbitrary input u(t) is given by linear superposition t τ 1 τ 2 τ 3 t!'#$()*+,*)#&")* y δ (t t ) y(t) =y δ (t)u() + t y δ (t τ)u(τ)dτ!"#$% u(t) y δ (t τ 1 ) τ 1 y δ (t τ 2 ) In particular, input is pitch rate, and output is lift coefficient: τ 2 y δ (t τ 3 ) u = α τ 3 y(t) =y δ u Wagner, Leishman, 26. y = C L &$%#$%

11 (Indicial) Step Response u A u u T Time u k u(t) y(t) y k PLANT

12 Eigensystem Realization Algorithm x(k + 1) y(k) = Ax(k)+ Bu(k) = Cx(k) Reduction x r(k + 1) = A r x r (k)+b r u(k) y(k) = C r x r (k) x R n x r R r n large r small 1. Gather outputs y(k) =CA k B from an impulse-response experiment, and arrange into Hankel matrices: H = CB CAB CA m c B CAB CA 2 B CA mc+1 B CA m o B CA mo+1 B CA m c+m o B H = CAB CA 2 B CA mc+1 B CA 2 B CA 3 B CA mc+2 B CA mo+1 B CA mo+2 B CA m c+m o +1 B 2. Compute the singular value decomposition of H : H = UΣV = U 1 U 2 Σ 1 V 1 V 2 = U 1 Σ 1 V 1 3. Let be the first block of and the first columns of so that the reduced order model A r,b r,c r is given by: Σ r r r Σ 1 U r,v r r U 1,V 1 Juang and Pappa, J. Guid. Contr. Dyn., 8:5, Ma, Z., Ahuja, S., and C. Rowley, Theor. Comput. Fluid. Dyn., to appear. A r = Σ 1/2 r Ur H V r Σ 1/2 r B r = first p columns of Σ 1/2 r C r = first q rows of U r Σ 1/2 r Recently shown to yield reduced order models equivalent to those obtained through Balanced Proper Orthogonal Decomposition V 1

13 ERA Model x α α k+1 = C L (k t) = ERA Model A ERA x 1 t α 1 α C ERA C Lα C L α k + x α α k B ERA t α k input + D ERA α k quasi-steady plus added-mass contribution additional fast dynamics

14 Canonical Pitch-ramp Maneuver 4 Coefficient of Lift Wagner! cos( ) ERA! cos( ) (16 mode) DNS (Re=3) Eldredge (Re=5k) Time Canonical Maneuver M. Ol, J. Eldredge et al, 48th AIAA ASM, 21. Developed to compare models, simulations and experiments Qualitatively similar for range of Reynolds numbers from 3-4k Pitch-up to Hold at Pitch-down to Leading-edge pitch-ramp maneuver Large added-mass forces appear as spikes Reduced order ERA model captures unsteady lift

15 Canonical Pitch-ramp Maneuver Canonical Maneuver &! M. Ol, J. Eldredge et al, 48th AIAA ASM, /, %! $! #! "!!! " # $ % & ' ( )*+, Developed to compare models, simulations and experiments Qualitatively similar for range of Reynolds numbers from 3-4k 1. Pitch-up to Hold at Pitch-down to Leading-edge pitch-ramp maneuver Large added-mass forces appear as spikes Reduced order ERA model captures unsteady lift

16 Canonical Pitch-ramp Maneuver 1 8 Angle of Attack DNS ERA QS+AM (CLa) CL Time

17 Combined Pitch/Plunge Maneuver Vertical Position Angle of Attack DNS ERA ERA plunge ERA pitch CL Time

18 Pitching at Quarter Chord 6 4 Magnitude (db) Phase (deg) 5 1 ERA, r=6 Wagner Theodorsen DNS Frequency (rad U/c)

19 Vertical Plunging 6 5 Magnitude (db) Phase (deg) ERA, r=7 Wagner Theodorsen DNS Frequency (rad U/c)

20 Summary 1. Reduced Order Model for Wagner a. Stability derivatives b. Additional fast dynamics c. Markov parameters and ERA algorithm x α α k+1 A ERA x = 1 t α 1 α x C L (k t) = C ERA C Lα C L α α α ERA Model k + quasi-steady contribution k B ERA t α k input + D ERA α k 2. Advantages a. More accurate than Quasi-steady b. More accurate than Theodorsen c. Efficient d. ODE framework ideal for control e. Fits naturally into fight dynamic framework coupled model T L D M I yy q = M mv γ = L + T sin(α) mg sin(γ) m V = T cos(α) D mg sin(γ) α = q (L + T sin(α) mg cos(γ)) /mv /123)*+,#-. ẋ = Ax + Bu y = Cx + Du!"#$%&'()*+,#-. q γ V α

21 Flight Dynamic Model gust disturbance rejection coupled model T L D M I yy q = M mv γ = L + T sin(α) mg sin(γ) m V = T cos(α) D mg sin(γ) α = q (L + T sin(α) mg cos(γ)) /mv /123)*+,#-. ẋ = Ax + Bu y = Cx + Du!"#$%&'()*+,#-. q γ V α controllers observers optimized flight paths evasive maneuvering Our Research early warning systems Goals

22 Flight Dynamic Model coupled model T L D M!"#$%&'()*+,#-. I yy q = M mv γ = L + T sin(α) mg sin(γ) m V = T cos(α) D mg sin(γ) α = q (L + T sin(α) mg cos(γ)) /mv q γ V α /123)*+,#-. ẋ = Ax + Bu y = Cx + Du Interesting control scenario when time-scales of flight dynamics are close to time-scales of aerodynamics

23 Flight Dynamic Model reference trajectory wind disturbances coupled model flight dynamics deviation from desired path thrust, elevator, aileron aerodynamics position aerodynamic state controller estimator

24 Next Step DISTURBANCE: Gust Field INPUT: Flaperon INPUT: Elevator

25 Moving Base Flow Base flow velocity: Vorticity: u(x, y, t) =U cos(α + α ) ẋ α(y y C ) v(x, y, t) =U sin(α + α ) ẏ + α(x x C ) (u, v) =v x u y = α + α =2 α Moving Base Flow where Faster simulations (Cholesky decomposition) allows more aggressive maneuvers and gusts subject of current research (x C,y C ) is the center of mass. Immersed Boundary Method T. Colonius and K. Taira, 28 A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions.

26 Flight Control Kornfeld, Hansman, and Deyst, ICAT-99-5, Reference Trajectory Controller Aircraft SAS / CAS Flight Control SAS: Stability Augmentation System CAS: Control Augmentation System Guidance Figure 2.1: Classical Flight Control Loops

27 Flight Control Kornfeld, Hansman, and Deyst, ICAT-99-5, Controller Aircraft SAS / CAS {Body Accelerations a x a y a z } {Body Rates p q r} Accelerometers Gyros Flight Control Pseudo- Attitude ~! " Pseudo- Attitude Synthesis d dt Guidance Velocity Position Single- Antenna GPS Figure 2.7: Single-Antenna GPS-Based Instrumentation Architecture