AFRL MACCCS Review of the Generic Hypersonic Vehicle PI: Active- Laboratory Department of Mechanical Engineering Massachusetts Institute of Technology September 19, 2012, MIT AACL 1/38
Our Team MIT Team PI:, Director, MIT Active Laboratory Daniel Wiese (MS Student) Spent the summer at AFRL as an intern Travis Gibson, Megumi Matsutani, Benjamin Jenkins, Heather Hussain, Max Qu (PhD students; not supported on this project) AFRL Team Dr. Mike Bolender (Team Lead) Dr. Jonathan Muse UM Team Prof. Jim Driscoll Sean Torrez, Derek Dalle Advisor Dr. Eugene Lavretsky, Senior Technical Fellow, Boeing Research & Technology Project started October 2011, biweekly telecons held with AFRL, MIT AACL 2/38
Advanced Architectures Goal Develop algorithms, methods, and models for the control of hypersonic vehicle configurations that include combinations of control surfaces, morphing and other configuration changes, response to uncertain or ambiguous environments, and recovery from subsystem failure., MIT AACL 3/38
Need for x-15 x-43 Limited wind tunnel data Harsh uncertain environments Poorly known physical models Actuator anomalies Largely varying operating conditions GHV From Doman s slides, MIT AACL 4/38
, MIT AACL 5/38
Execute desired maneuvers at various points on the flight envelope: Mach 4-8, q = 750-2000 psf High bank angle turns Large angle of attack climbs Nominal flight condition is Mach 6 at 80,000 ft altitude, MIT AACL 6/38
AFRL 6 DOF GHV (Generic Hypersonic Vehicle) Aircraft assumed to be rigid Earth centered, inertial reference frame Spherical, rotating Earth Aerodynamic forces and moments determined using look-up table Aerodynamic table values based on α, β, M, and control surface deflections Engine data tabulated as a function of M, α, ϕ, q Sideslip β not accounted for in thrust tables Data tables are interpolated over based on input for given flight condition, MIT AACL 7/38
State Space Representation The GHV has four aerodynamic control surfaces and throttle Left and right rudder Left and right elevon Input vector U accomplished by mixing effective elevator and aileron commands to generate right and left elevon deflections U = [ δ th δ elv δ ail δ rud ] T The GHV is maneuvered by commanding V T, φ, α Assuming small linearization error, the equations of motion when linearized about a trim point X eq and U eq are given by ẋ p = A p x p + B p u where x p and u are the perturbed state and input about trimmed flight condition, MIT AACL 8/38
Linear : Flight Modes An open-loop analysis of the linear equations about the nominal flight condition was performed, yielding the following modes of flight. Irregular Short Period (λ 1,3 ) - an unstable mode dominated by α and q. Fast, purely real poles, with λ 1,3 ±2. Rolling (λ 2 ) - a stable mode, dominated by β and p. Fast, purely real pole at λ 2 = 4.87 Dutch-Roll (λ 4,5 ) - an unstable mode, which is a combination of a rolling, pitching, and yawing motion in flight. Phugoid (λ 6,7 ) - a neutrally stable, slow phugoid mode. Velocity (λ 8 ) - neutrally stable. Spiral (λ 9 ) - a slow, but stable mode., MIT AACL 9/38
Linear : Open-Loop Poles Imaginary 0.8 0.6 0.4 0.2 0 0.2 Open Loop Poles Dutch Roll Phugoid Spiral 0.4 Short Period 0.6 Rolling 0.8 5 4 3 2 1 0 1 2 Real, MIT AACL 10/38
noise xcmd 100 Hz Controller 600 Hz τ delay Actuator Dynamics Equations of Motion Simulation block diagram Assume full state accessible for feedback control Sensors measuring at 600 Hz with sample and hold Low-pass first order discrete sensor output filter Gaussian white noise injected into each sensor channel Controller operating at 100 Hz using zero order hold Input delay can be added to control input signal, MIT AACL 11/38
Actuators and Sensors Actuator Model (Control Surface) Second order actuators drive each aerodynamic control surface Saturation included on deflection angle and rate 30 deg deflection limit 100 deg/s rate limit δ cmd deflection saturation Σ ω n 2 δ Σ 1 s δ 2ζω n rate saturation 2nd order actuator model: dynamics and saturation 1 s δ deflection saturation Actuator Model (Throttle) Engine modeled as first order system with cutoff τ = 10 Sensor Model First order output filters defined in continuous time and implemented discrete time Velocity output filter with τ = 20 All other output filters τ = 150, MIT AACL 12/38 δ act
X = [ V T α q θ h β p r φ ] T U = [ δ th δ elv δ ail δ rud ] T Baseline design consists of three linear controllers Modal analysis allows the velocity, longitudinal, and lateral subsystems to be considered independently for linear control design about nominal flight condition Timescale separation between phugoid and irregular short period allows feedback of fast states only An LQR-PI control was implemented on each of the three subsystems V T,cmd α cmd φ cmd β cmd = 0 Velocity Controller Longitudinal Controller Lateral Controller δ th V T δ e α, q δ a, δ r β, p, r, φ GHV, MIT AACL 13/38
r 0 Σ e K(s) d n u o u i y o y i Σ G(s) Σ Frequency domain analysis block diagram The LQR-PI control architecture can be represented using the block diagram above Bode and singular value plots of the transfer function matrix were used to guide selection of state and input weighting matrices Q and R, changing the feedback gains Quadratic cost function dictates feedback gains J = [x(t) T Qx(t) + u(t) T Ru(t)]dt 0, MIT AACL 14/38
: Velocity and Longitudinal Subsystems Magnitude [db] 150 100 50 0 Velocity Controller Bode Plot GM:Inf db PM:65.56 Gain Grossover:0.31 Hz Magnitude [db] 100 50 0 50 100 150 Longitudinal Controller Bode Plot GM:-10.97 db PM:49.24 Gain Grossover:1.98 Hz Phase [deg] 50 10 4 10 3 10 2 10 1 10 0 10 1 10 2 90 Frequency ω [rad/s] Phase 180 10 4 10 3 10 2 10 1 10 0 10 1 10 2 Frequency ω [rad/s] Phase [deg] 200 10 1 10 0 10 1 10 2 10 3 10 4 270 180 90 0 90 180 Frequency ω [rad/s] Phase 10 1 10 0 10 1 10 2 10 3 10 4 Frequency ω [rad/s] allowed for sufficient margins while maintaining desirable crossover frequencies, MIT AACL 15/38
: Lateral Controller Magnitude [db] Magnitude [db] Magnitude [db] 200 100 0 100 Control Loop: σ(lu) 200 10 2 10 0 10 2 10 4 Frequency ω [rad/s] Stability Robustness: σ(1+l 1 u ) at Plant Input 200 150 100 50 0 50 10 2 10 0 10 2 10 4 Frequency ω [rad/s] Complementary Output Sensitivity: σ(t(s)): r to y 40 20 0 20 40 60 10 1 10 0 10 1 10 2 Frequency ω [rad/s] Magnitude [db] Magnitude [db] Magnitude [db] Return Difference: σ(1+lu) at Plant Input 200 150 100 50 0 50 50 0 50 100 150 200 100 80 60 40 20 10 2 10 0 10 2 Frequency ω [rad/s] Output Sensitivity: σ(s(s)): n to y 10 2 10 0 10 2 10 4 Frequency ω [rad/s] Noise to Control: σ(k) 0 10 1 10 0 10 1 10 2 10 3 Frequency ω [rad/s] Lateral subsystem singular value plots, MIT AACL 16/38
Inner loop flight Controller Subsystem Order Subsystem Order Integrators Augmented Order Velocity 1 V T 2 Longitudinal 2 α 3 Lateral 4 φ, β 6 Loop transfer function crossover frequencies Subsystem Crossover [rad] Crossover [Hz] Delay Margin [ms] Velocity 6.2 0.98 588 Longitudinal 12.5 1.99 69 Lateral 11.1-17.2 1.77-2.73 44 V T,cmd Velocity Controller δ th V T α cmd Longitudinal Controller δ e α, q GHV φ cmd β cmd = 0 Lateral Controller δ a, δ r β, p, r, φ, MIT AACL 17/38
(a) Control surface effectiveness (b) CG movement (c) Stability derivatives (d) Actuator anomalies Failure Saturation (e) Time delay Control surface failure CG shift Saturation Time delay, MIT AACL 18/38
How do lers Work? GHV, MIT AACL 19/38
How do lers Work? GHV, MIT AACL 19/38
How do lers Work? Controller GHV θ Adaptive System, MIT AACL 19/38
How do lers Work? Controller GHV e Adaptive System θ, MIT AACL 19/38
How do lers Work? Controller GHV Adaptive System Adapt θ so that e(t) 0 θ e (error), MIT AACL 19/38
How do lers Work? Controller GHV Adaptive System Adapt θ so that e(t) 0 θ Construct a suitable e Find an adaptive law for θ e (error), MIT AACL 19/38
Choice of Adaptive Law Makes a Difference! 1 x-15 MH-96 AFCS x-15 Provably Correct AFCS 1 Z.T. Dydek, A.M., and E. Lavretsky, and the NASA X-15 Program: A Concise History, Lessons Learned, and a Provably Correct," IEEE Control Systems Magazine, June 2010. (Best Paper Award Winner), MIT AACL 20/38
A Closer Look MH-96 AFCS, MIT AACL 21/38
A Closer Look Provably Correct AFCS MH-96 AFCS, MIT AACL 21/38
A Closer Look Provably Correct AFCS MH-96 AFCS, MIT AACL 21/38
A Closer Look 24 Adaptive Parameters 3 Adaptive Parameters, MIT AACL 21/38
A Closer Look 8 Adaptive Parameters (Coupling Removed), MIT AACL 21/38
A Closer Look 8 Adaptive Parameters (Coupling Removed), MIT AACL 21/38
A Closer Look 8 Adaptive Parameters (Coupling Removed), MIT AACL 21/38
A Closer Look 3 Adaptive Parameters (α, β integral states removed; e u replaced with e), MIT AACL 21/38
A Closer Look 3 Adaptive Parameters (α, β integral states removed; e u replaced with e), MIT AACL 21/38
A Closer Look 3 Adaptive Parameters (α, β integral states removed; e u replaced with e), MIT AACL 21/38
A Closer Look 3 Adaptive Parameters (PC Adaptive law replaced with MH-96 logic), MIT AACL 21/38
A Closer Look 3 Adaptive Parameters (PC Adaptive law replaced with MH-96 logic) PC Adaptive law is key!, MIT AACL 21/38
x cmd Baseline Controller Adaptive Controller u nom u ad Σ u Plant : (a) Control Effectiveness (b) CG movement (c) Stability derivatives (d) Saturation x Adaptive controller added around baseline controller (a-d), after linearization, lead to ẋ p = A pλ x p + B p Λu The matrix Λ is diagonal with entries of known sign With an integral state, the underlying plant dynamics become ẋ = A λ x + B 1 Λu + B 2 x cmd, MIT AACL 22/38
Reference Model Linearize Ẋ = f (X, U) about equilibrium: f (X,U) A pλ = B pλ = X eq f (X,U) U Augment the plant matrices with x e = x x cmd [ ] [ ][ ] [ ] [ d xp Apλ 0 xp Bpλ 0 = + u + x dt x e H 0 x e 0 I] cmd Nominal plant dynamics: set Λ = I; A λ = A; B λ = B 1 Λ = B 1 Baseline control law ẋ = Ax + B 1 u + B 2 x cmd u = K T x Reference model: nominal plant + baseline controller ẋ m = A m x m + B m x cmd eq A m = A + B 1 K T and B m = B 2, MIT AACL 23/38
x cmd Baseline Controller u nom Σ u Reference Model Plant x Σ e Adaptive Controller u ad Plant: ẋ = A λ x + B 1 Λu + B 2 x cmd Reference model: ẋ m = (A + B 1 K T )x m + B 2 x cmd Error: e = x x m Control law: u = (θ + K) T x Adaptive gain update law: θ = Proj Γ (θ, Γxe T PB 1 sign(λ)), MIT AACL 24/38
Adaptive Update Law A Projection algorithm is used to ensure parameter boundedness. θ = Proj Γ (θ, Γxe T PB 1 sign(λ)) Proj(θ,y) y f (θ) m = θ θ b θ θ b Ω 0 Ω A {θ f (θ) = 0} {θ f (θ) = 1} Projection operator in θ space, MIT AACL 25/38
Stability and Convergence θ : True control parameter Assumption: θ satisfies θ T = Λ 1 (I Λ)K T W T A λ + B 1 Λ(θ + K) T = A m Global stability and convergence ( θ = θ θ ) V = e T Pe + tr ( θ T Γ 1 θ Λ ) V 0 lim e(t) = 0 t x cmd Baseline Controller θ Adaptive Controller u nom u ad Σ u Plant x, MIT AACL 26/38
noise xcmd 100 Hz Baseline 600 Hz τ delay Adaptive Actuator Dynamics Equations of Motion Simulation block diagram The following simulation results demonstrate adaptive controller performance for two commanded tasks, M = 6, h = 80, 000 ft 2 : 3 deg angle of attack doublet : 80 deg roll step Command tasks performed in the presence of uncertainties (a) Control ineffectiveness (b) CG shift (c) Stability derivative uncertainty The time delay is set to zero 2 GNC Paper in Progress of the Generic Hypersonic Vehicle in Presence of Actuator, CG, and Aerodynamic, MIT AACL 27/38
with Uncertainty (a) α [deg] 5 0 Angle of Attack Command Baseline Reference Adaptive az [ft/s 2 ] δe [deg] 5 0 5 10 15 time [s] 20 0 20 200 0 Elevator Deflection Angle 0 5 10 15 time [s] Normal Acceleration 200 0 5 10 15 time [s] Reduced control surface effectiveness: 50% on all surfaces For this uncertainty and command, the adaptive controller can tolerate a time delay of up to τ delay = 28 ms, MIT AACL 28/38
with Uncertainty (b) α [deg] 5 0 Angle of Attack Command Baseline Reference Adaptive az [ft/s 2 ] δe [deg] 5 0 5 10 15 time [s] 20 0 20 200 0 Elevator Deflection Angle 0 5 10 15 time [s] Normal Acceleration 200 0 5 10 15 time [s] Longitudinal CG shift: -0.7 ft rearward CG shift of -0.7 ft is 5% of the vehicle length For this uncertainty and command, the adaptive controller can tolerate a time delay of up to τ delay = 20 ms, MIT AACL 29/38
with Uncertainty (c) α [deg] 5 0 Angle of Attack Command Baseline Reference Adaptive az [ft/s 2 ] δe [deg] 5 0 5 10 15 time [s] 20 0 20 200 0 Elevator Deflection Angle 0 5 10 15 time [s] Normal Acceleration 200 0 5 10 15 time [s] Pitching moment coefficient scaled 350% For this uncertainty and command, the adaptive controller can tolerate a time delay of up to τ delay = 18 ms, MIT AACL 30/38
with Uncertainty (a) φ [deg] δa [deg] δr [deg] 100 50 0 20 0 20 20 0 20 Roll Angle Command Baseline Reference Adaptive 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time [s] Aileron Deflection Angle 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time [s] Rudder Deflection Angle 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time [s] Reduced control surface effectiveness: 50% on all surfaces Tolerable time delay τ delay = 56 ms, MIT AACL 31/38
with Uncertainty (b) φ [deg] δa [deg] δr [deg] 100 50 0 20 0 20 20 0 20 Roll Angle Command Baseline Reference Adaptive 0 5 10 15 time [s] Aileron Deflection Angle 0 5 10 15 time [s] Rudder Deflection Angle 0 5 10 15 time [s] Longitudinal CG shift: -1.7 ft rearward CG shift of -1.7 ft is 12% of the vehicle length Tolerable time delay τ delay = 35 ms, MIT AACL 32/38
Adaptive controller implemented on 6-DOF nonlinear sim, operating at 100 Hz, with sensor input delay and output noise Adaptive controller performance evaluated when performing two tasks : 3 deg angle of attack doublet : 80 deg roll step in the presence of uncertainties (a) Control ineffectiveness up to 50% (b) CG shift -0.7 to -1.7 ft (5-12% vehicle length) (c) Stability derivative uncertainty up to 350% Adaptive MRAC architecture with projection showed increased tracking performance and stability over baseline controller for each task with uncertainty Input time delay of 18-56 ms tolerated, MIT AACL 33/38
Collaboration and Tasks : Develop adaptive controllers for high performance maneuvers through unstart Simple model proposed by AFRL to capture essential effect of unstart, a function of: α, β, M When unstart occurs on the GHV All thrust forces and moments become zero Drag increases by 20% Lift decreases by 20% Pitching and yawing moment coefficient curves shifted and scaled: C M (α) = C Mnom (α) + 0.003 + 0.01α C N (β) = C Nnom (β) + 0.0003 + 0.002β continued, MIT AACL 34/38
Collaboration and Tasks : (continued) : Same adaptive controller as in earlier slides was implemented Preliminary results showed successful maneuver through unstart for a 5 deg doublet in α Collaboration with Prof. Driscoll to include more detailed use of engine data and unstart model to guide control design Incorporation of unstart features in the nonlinear GHV model M α ϕ q Nonlinear GHV model unstart Simulation block diagram, MIT AACL 35/38
Collaboration and Tasks Task 3 Adaptive control with improved transient response Investigate the use of methods in 3 to overcome unstart Task 4 Improved transient response using CRM-adaptive 2 Quantify delay margins for the adaptive controller using 4,5 3 Gibson, T.E.,, A.M. and Lavretsky, E. Adaptive Systems with Closed-loop Reference Models, Part I: Transient Performance, ACC13 4 Matsutani, M.,, A.M. and Lavretsky, E. Guaranteed Delay Margins for of Scalar Plants, CDC12 5 Matsutani, M.,, A.M. and Lavretsky, E. Guaranteed Delay Margins for Adaptive Systems with State Variables Accessible, ACC13, MIT AACL 36/38
Collaboration and Tasks Task 5 Task 6 Task 7 Investigate the use of rate-saturation for improving the adaptive control performance 6 Investigate the use of state constraints for improving the adaptive control performance 7,8 Develop adaptive control based on output feedback Reliable incidence measurements not available on hypersonic vehicles Instead of angle of attack and sideslip, measure accelerations A x A y A z Control using outputs C and D C = A z + k q q and D = A y + k p p 6 Matsutani, M.,, A.M. and Crespo, L. in the Presence of Rate Saturation with Application to GTM, GNC10 7 Muse, J. A Method For Enforcing State Constraints in 8 Lavretsky, E., and Gradient, R. Robust Adaptive for Aerial Vehicles with State-Limiting Constraints, JGCD10, MIT AACL 37/38
Long Term Goals x cmd Baseline Controller u nom Σ u x Adaptive Controller u ad Unstart model Develop adaptive architectures that accommodate uncertainty effects due to unstart, MIT AACL 38/38