Modeling and Control Challenges for Airbreathing Hypersonic Vehicles

1
Modeling and Control Challenges for Airbreathing
Hypersonic Vehicles
SAE Aerospace Guidance Control Committee Meeting
2 Mar 2007 Dr. David B. Doman Control Design and
Analysis Branch Air Vehicles Directorate Air
Force Research Laboratory
2
Outline

• Background
• Modeling
• Aerodynamics
• Propulsion
• Structural
• Thermal
• Unsteady Aerodynamics
• Control Approaches
• Conclusions

3
Scramjet Modeling and Control Overview

• Highly coupled airframe/propulsion system with
  aeroelastic interactions.
• Unstable and non-minimum phase
• First-principles modeling approach
• Oblique Shock Theory
• Prandtl-Meyer Expansion Theory
• Quasi-1D flow with heat addition in combustor
• Spillage effects on inlet performance
• Aeroelastic modeling
• Plume shear layer modeling
• Thermal effects on structure
• Unsteady aerodynamic effects via nonlinear piston
  theory
• Advantages
• Flexibility to add/move control surfaces, change
  sensor locations
• A fundamental understanding of what is being
  controlled

2-D X-43A Profile Scaled to 100 ft
4
Animation
5
Outline

• Background
• Modeling
• Aerodynamics
• Propulsion
• Structural
• Thermal
• Unsteady Aerodynamics
• Control Approaches
• Conclusions

6
Aerodynamic Forces Moments

• Oblique Shock and Prandtl-Meyer Expansion Theory
  used to compute pressures over top, forebody and
  underbody of engine
• Approximation for plume shear layer location
  verified against panel solution
• Pressure over rear ramp computed from plume
  results
• Aerodynamic Forces and Moments computed by
  integrating pressures over each surface
  (including deformation due to bending)
• Inlet turning forces included

7
Aerodynamic Flow Regions
Region 5
M5, P5
Region 1
f
?2
?e
M1, P1
?1,U
Region 4
xb
c
?1,L
M4, P4
q
zb
Region 2
V
e
V
d
?
M2, P2
hi
g
M?, P?
Region 3
M3, P3
8
Outline

• Background
• Modeling
• Aerodynamics
• Propulsion
• Structural
• Thermal
• Unsteady Aerodynamics
• Control Approaches
• Conclusions

9
Scramjet Modeling

• Variable Geometry Diffuser
• Allows pressure at combustor inlet to be adjusted
• Assumed to be isentropic
• Constant area combustor with simple heat addition
• Heat addition/change in total temperature a
  function of equivalence ratio
• Fixed area ratio internal nozzle
• Assumed to be isentropic
• Very easy for the combustor to become thermally
  choked!

10
Aeroelastic Influence on Propulsion System

• Euler-Bernoulli Beam Theory
• Beam loading
• Normal component of surface pressures changes
  with beam deflection
• Variable point load due to elevon
• Heave and rotational effects included
• Deflection is a function of load
• Load is a function of deflection
• Modal analysis for beam dynamics
• On/Off-design conditions modeled
• Shock on lip (no spillage)
• Shock forward of lip (spillage)
• Shock inside inlet (no spillage)
• Quasi-1D flow with heat addition in combustor
  associated with fuel flow
• First dynamic model to include nonlinear on/off
  design effects
• Captures interaction between aero, structure and
  propulsion system

11
Outline

• Background
• Modeling
• Aerodynamics
• Propulsion
• Structural
• Thermal
• Unsteady Aerodynamics
• Control Approaches
• Conclusions

12
Refined Structural Model

• Model the effects of mass and temperature on the
  structural dynamics of a hypersonic aircraft
• Avoid FEM analysis of frequencies/mode shapes
• Approximation of fuselage first bending mode
• Assume ?n 18 rad/sec and ? 0.02 when fully
  fueled
• Use assumed modes method to estimate mode shapes
  and frequencies

13
HSV Mission Analysis
LH2 40 of GTOW
LOX 31 of GTOW

• Mission
• 8000 n.mi. Cruise at Mach 10, 120 kft
• Transition from rocket to scramjet at Mach 5
  during ascent
• Descent assumed to be unpowered

14
Assumed Modes Method

• Based on Lagranges Equations

• The transverse displacement along the structure
  is given by a separation of the time and spatial
  coordinates

• The shape functions ?i(x) are the assumed modes.
• ?i(x) satisfy the geometric boundary conditions
  and possess all required derivatives

15
Choice of Assumed Modes

• Basis functions are uniform free-free beam mode
  shapes
• Satisfies geometric boundary conditions
• Expected to be close to real mode shapes
• Requires fewer assumed mode shapes for
  convergence
• Alternatively assessing viability of Chebyshev
  Polynomials as basis functions

16
Assumed Modes MethodMass Effects
17
Outline

• Background
• Modeling
• Aerodynamics
• Propulsion
• Structural
• Thermal
• Unsteady Aerodynamics
• Control Approaches
• Conclusions

18
Plant CharacterisiticsHeating Effects

• Modulus of Elasticity a function of Temperature
• ?n f(E1/2)
• Expect ?n to vary with T1/2
• 125 deg ?T predicts 3 change in frequency
• 2 change in first mode freq after 2 hrs
• No effect on aircraft controllability!

From Vosteen, NACA TR 4348
19
Assumed Modes MethodTemperature Effects

• Uniform Temperature Distribution along beam (1500
  deg F)
• Vehicle structure modeled as a Ti beam
• Only freqs affected by a uniform ?T
• Spatially varying temperature will affect mode
  shapes
• Currently under study

20
Aerothermal Modeling

• Sustained Flight at high Mach number requires TPS
  
• Either passive or active
• Capture effects of heat transfer on the vehicle
  structural dynamics
• Controllability considerations
• Structural Dynamics estimated using the Assumed
  Modes Method
• Frequencies and Mode Shape dependency on vehicle
  mass and temperature can be modelled
• Modulus of Elasticity a function of temperature
• Thermal Protection System Architecture assumed

21
Aerothermal ModelingProblem Formulation

• Three layer model assumed
• PM2000 honeycomb
• SiO2 Insulation
• Titanium Load Bearing Structure
• Explicit, finite difference method used to
  calculate unsteady heat transfer during cruise
• 1-D heat transfer only
• Discrete Points along Structure Considered
• Boundary Conditions
• Convective Heating
• Radiative Cooling
• nth node insulated

PM2000 Honeycomb
Insulation
Ti Beam
1st node
22
Problem FormulationSolution Method

• Once all the difference equations are obtained,
  they can be written in the form

• A is a tri-diagonal matrix
• Convergence requires eigenvalues of A to lie in
  the unit circle
• Max allowable ?t given ?y material properties
• b is a non-linear forcing term
• Without loss of generality, lump both radiative
  cooling and convective heating into this term as
  well

23
Aerothermal ModelingResults
PM2000

• Case 1 Temp profile due to constant heating
• Point 50 ft behind the nose
• dq/dt 15 BTU/(ft2 s)
• Emissivity 0.6
• Radiation to a perfect black body at T0 deg R
• Steady state reached for outer TPS
• Large ?T across insulation layer
• Minimal ?T in structure
• Model matches ABACUS predictions

SiO2
Ti Beam
24
Aerothermal ModelingResults
Structure
Outer Layer
25
Plant CharacterisiticsHeating Effects

• Temperature Along Structure Determined

Trailing Edge
Leading Edge
26
Outline

• Background
• Modeling
• Aerodynamics
• Propulsion
• Structural
• Thermal
• Unsteady Aerodynamics
• Control Approaches
• Conclusions

27
Piston Theory

• Piston Theory
• Method for Calculating Aerodynamic Loads
• Local Pressure from Bodys Motion Is Related to
  Local Normal Component of Fluid Velocity
• Same Way As Pressure and Velocity Are Related At
  Face of Piston Moving in a 1-D Channel
• Provides Expression for Time Varying Pressure as
  a Function of Local Surface Velocity
• Valid for M 4
• Accurate Calculation
• Of Pressures

28
Piston Theory - Basics
Basic Result From Linear Piston Theory

• Isentropic Simple Wave Expression for Pressure
  on Surface of Moving Piston

• P Pressure
• ? Ratio of Specific Heats

• Binomial Expansion Gives

• Use Perfect Gas Law (P?RT) and Speed of Sound
  (a2?RT) Yields

Basic Result

• Vn Velocity of Surface Normal to Flow
• Subscript Refers To Flow Conditions Past
  Surface, i.e., Behind Shock or Expansion Fan or
  Freestream
• ? density
• a speed of sound

29
Piston Theory - Basics

• Infinitesimal Force Due To Pressure ? dF -P dA
  n
• F force (normal/axial)
• dA surface element (unit depth into page)
• n outward pointing normal vector

• From Basic Result, Need Vn

• Subscript
• U (upper surface)
• Lcd (lower surface cd)
• Lgh (lower surface gh)
• Lef (lower surface ef)

• Infinitesimal Force Becomes

• Find dA, n, and V

30
Stability Derivative Approximations for Unsteady
Aerodynamic Effects

• Unsteady effects captured in the form of
  stability deriviatives
• CZa, CXa, CMa, CMQ, CZQ

Structural mode stability derivatives are also
included
31
Results

• Poles of Linearized Model - Mach 8, H85,000

Without Unsteady Effects
With Unsteady Effects
-2.458 2.314 -.00048 -4.93e-5 - j0.0335
More Unstable
-1.508 1.444 -.0004759 -5.15e-5 - j0.0335
32
Effects of Unsteady Aerodynamics on Airbreathing
Hypersonic Vehicle Dynamics

• Continued development of first principles model
  of scramjet vehicle
• Aero-thermo-servo-elasticity effects captured in
  multidisciplinary model suitable for control
  studies
• Unsteady Aero Modeling via Piston Theory
• Accounts for Fluid-Structure Interaction as
  Vehicle Vibrates
• Used to Compute Damping and Flex-body stability
  derivatives
• Steady and Unsteady Aerodynamics in Model
• Significant shifts in pole-zero locations
• Heat transfer and thermal effects on structure
  modeled

Unsteady Aero Terms Move Unstable Zero
Pole to Right in S-Plane Affect Stability and
Closed-Loop Bandwidth
Unsteady
Steady
33
Outline

• Background
• Modeling
• Aerodynamics
• Propulsion
• Structural
• Thermal
• Unsteady Aerodynamics
• Control Approaches
• Conclusions

34
Flexible Aircraft Equations-of-Motion

• Derived using Lagranges Equations
  
  

• Kinetic Energy is the sum of rigid elastic
  kinetic energies

• Potential Energy is a superposition of the strain
  energy of the structure and the altitude
  potential energy

• Generalized forces
• Proportional Damping on structure (2 ? ?n)
• Aerodynamic forces/moments and Thrust
• Modal forces

35
Flexible Aircraft DynamicsNon-linear
Equations-of-Motion

• Non-linear, longitudinal equations-of-motion

• L, D, M, Fn are functions of ?in ?(x) ?I

• Variable mass and frequency effects are captured

36
Plant Characteristics

• Unstable Rigid-body modes s-4.47, 4.41
• Very Slow (albeit stable) Altitude mode s-0.001
• Unstable, very lightly damped phugoid (time to
  double 3.6 hrs)
• Non-minimum phase transmission zeros for typical
  input/output combinations
• Controllable/Observable
• Structural modes are the most Controllable
  Observable

yV ?T u?e ?T
37
Configuration Changes for NMP Mitigation

• Goal Elimination of Low Frequency Nonminimum
  Phase Behavior
• Large elevon contributes substantially to total
  lift
• Increasing forebody AOA requires short-term loss
  of lift at the elevon to create pitching moment
• Long-term effect-Increased Lift
• Short-term effect-Decreased Lift
• Canard added with interconnnect gain to elevator
  greatly improves control law potential.
• Moves center of rotation from off-vehicle to near
  CG.

38
Control of Unstable/NMP VehicleOutput
Modification

• New controlled variable flight path angle
  measured ahead of cg
• ?p azp/(U0 s)
• Accelerometer placed ahead of Instantaneous
  Center-of-Rotation
• Real Zeros of N??(s) bifurcate to complex
  conjugate pair in LHP
• Improved Tracking Performance since NMP behavior
  is eliminated

39
Control Approaches
Modified Dynamic Inversion Results

• Control Performance Limitations
• Feedback control required (O.L unstable)
• Low frequency non-minimum phase behavior
  prevalent
• Limited speed of response
• Dynamic Inversion with NMP coupling strategically
  ignored
• LQR Classical Loop Shaping
• Modified Dynamic Inversion
• Place poles at mirror image of NMP zeros
• Result Decoupled bank of integrators all-pass
  filters

40
Outline

• Background
• Modeling
• Aerodynamics
• Propulsion
• Structural
• Thermal
• Unsteady Aerodynamics
• Control Approaches
• Conclusions

41
Conclusions

• First principles based model that captures
  coupling of aerodynamic, structural, and
  propulsion systems
• Mass flow spillage effects on thrust added
• Control of flexible hypersonic aircraft is a
  difficult problem, extreme coupling,
  nonlinearities, NMP behavior, unstable
• Low-frequency flexible mode lt actuator bandwidth
  (20 rad/sec)
• Possible interactions between structural dynamics
  and control system
• Non-minimum phase transmission zero due to
  location of center-of-rotation limits controller
  bandwidth
• Unsteady aero effects captured using piston
  theory in conjunction with quasi-steady gas
  dynamics.
• Unsteady aero appears to significantly affect
  pole-zero locations

42
Conclusions

• Model validation comparisons to X-43 flight test
  data show aero-results are reasonable.
• Thermal and mass effects included in structural
  model
• Frequencies Increase and Mode Shapes change with
  fuel burn
• Structural frequencies change due to aerodynamic
  heating (lowered by 2-3)
• Future work
• Improve scramjet model by integrating cycle code
• Propulsion mode transition and low speed
  modeling
• Viscous effects

43
Backup Slides
44
Assumed Modes MethodKinetic Energy

• The kinetic energy is

• Which can be written in vector-matrix form

• Where

45
Assumed Modes MethodPotential Energy

• Similarly, the potential energy is

• In vector-matrix form becomes

• Where

46
Problem FormulationSurface Node

• At the surface, the energy balance is

Which, after discretizing, becomes
Where
Node index
pth Time step
47
Problem FormulationInterior Point Nodes

• For the interior points, the energy balance gives

When discretized, this becomes
48
Problem FormulationMaterial Intefaces

• Again compute the energy balance at the ith node

Applying the following
Gives
49
Aeroelastic Influence on Propulsion System

• Euler-Bernoulli Beam Theory
• Beam loading
• Normal component of surface pressures changes
  with beam deflection
• Variable point load due to elevon
• Heave and rotational effects included
• Deflection is a function of load
• Load is a function of deflection
• Modal analysis for beam dynamics
• On/Off-design conditions modeled
• Shock on lip (no spillage)
• Shock forward of lip (spillage)
• Shock inside inlet (no spillage)
• Quasi-1D flow with heat addition in combustor
  associated with fuel flow
• First dynamic model to include nonlinear on/off
  design effects
• Captures interaction between aero, structure and
  propulsion system

50
Nonlinear Model and Linearization Results
Nonlinear Longitudinal Equations of Motion

• Dynamic model derived using Lagrangian method
• Captured both heave and rotational effects on
  structure
• Captured on/off design effects propulsion (use
  gas dynamics versus Newtonian flow)
• Include effects of uncommanded control rotation
  due to flexible structure
• Linearized Model reveals
• Short Period Phugoid modes
• Height mode
• Fore/aft first bending modes
• Open-loop unstable
• Non-minimum phase
• Off-vehicle center of rotation

51
Control Approaches
Modified Dynamic Inversion Results

• Control Performance Limitations
• Feedback control required (O.L unstable)
• Low frequency non-minimum phase behavior
  prevalent
• Limited speed of response
• Dynamic Inversion with NMP coupling strategically
  ignored
• LQR Classical Loop Shaping
• Modified Dynamic Inversion
• Place poles at mirror image of NMP zeros
• Result Decoupled bank of integrators all-pass
  filters

52
Current Status

• Work-in-Progress
• Include thermal effects on structure
• Include unsteady aerodynamic effects via
  nonlinear piston theory
• Include higher order structural modes
• Lateral directional dynamics
• Advanced control methodologies as required

Mode Shapes
Hot
Cold
Aerothermoelasticity
Unsteady Aero. Piston Theory
53
Thermal Modeling and Implications for HSV Control
Aerodynamic Heating at cruise

• Problem Capture heat transfer effects on
  structural dynamics
• Structural Frequencies depend upon
• Vehicle mass
• Frequencies increase with decreasing mass
• Temperature of the underlying structure
• 10 reduction in frequencies for a hot
  structure
• Temperature effects calculated using unsteady,
  heat transfer code
• Explicit Finite Difference Method
• Ti Multiwall TPS Architecture w/ Aerodynamic
  heating function of flight condition
• Integrated with vehicle model
• On-going research effort to quantify heat
  transfer effects on vehicle controllability
  required robustness of controllers

54
Vehicle Configuration
L100
47Lf
20Ln
33La
Xcs
f
14.4?2
?e
Zcs
?1,U3
xb
c
?1,L6
q
zb
V
V
e
?
d
s
3.5hi
g
h
55
Outline

• Piston Theory
• Forces
• Flow Analysis
• Stability Derivatives
• Engine
• Simulation
• Conclusions

56
Assumed Modes MethodEigenproblem

• From Lagranges Equation

• Assuming simple harmonic motion for the
  free-vibration of the structure, set

• The unknown ? is found from the eigenproblem

• The eigenvectors ui determines the mode shape of
  the structure from

57
Aerothermal ModelingTPS Model

• TPS consists of PM2000 Honeycomb Outer wall and
  SiO2 Insulation
• Structure (aircraft) idealized as a Titanium beam
  with varying mass and stiffness

58
Plant CharacteristicsA Word On Zeros

• Origin of NMP zero
• Due to loss of lift on aircraft when elevator
  deflected to pitch aircraft up
• True for any tail controlled aircraft
• Position of zero a function of instantaneous
  center-of-rotation (for a rigid aircraft)

Field, E., Armor, J., Rossitto, K., and Mitchell,
D., Effects of Instantaneous Center of Rotation
Location on Flying Qualities, Proceedings of
the AIAA Atmospheric Flight Mechanics Conference
CD-ROM, Aug. 5-8 2002, AIAA 2002-4799.
59
Plant CharacteristicsA Word On Zeros

• Modification of ICR will move the transmission
  zeros
• Ideally, would like to apply a couple to the
  aircraft in order to minimize the reduction in
  lift that is experienced when deflecting the
  elevator
• How do we accomplish this?
• Augment HSV model with a canard
• Choose the mean aero. chord
• Interconnect with elevator using a gain, k
• Deflect canard in opposite direction of elevator
• Redefine the ICR
• We can pick k, M? c, and Z? c to place the ICR
  where we want and thus move the offending zero to
  a better location
• Optimal k exists that moves ICR to origin zero
  to 1

60
Plant CharacteristicsA Word On Zeros
61
Plant CharacteristicsA Word On Zeros

• Effect of flexibility is to move zero back to the
  j? axis
• Incomplete cancellation of lift
• Exact details being worked

• Small perturbations in Z? c can move zeros
  significantly

62
Surface Velocities

• Consider Small Perturbations From Steady Flight
  Condition at M? in u,w, and q

• Velocity of Point on Upper Surface Due To
  Perturbations

Unit Vectors in X and Z directions
Angular Rate Vector
Position Vector of Point on Upper Surface

• Integrate Differential Force

63
Incremental Forces
Upper Surface
Lower Surfaces

• Control Surface (Elevator)
• Modeled As Flat Plate Hinged at Midpt.
• Le Length of Elevator
• ?e Elevator Deflection ( trailing edge down)
• xcs, zcs x z Position of Elevator Relative to
  CG

64
Afterbody
f

• Flow On Afterbody Bounded By Vehicle Surface
  Shear Layer
• Pef Pressure on Afterbody
• Pe Pressure at Engine Exit
• P? Freestream Pressure

?2
?e
?e
?1,U
c
?1,L
e
V
V
d
s
hi
?
g
h
F. Chavez and D. Schmidt, Analytical
Aeropropulsive /Aeroelastic Hypersonic Vehicle
Model with Dynamic Analysis, JGCD, Vol. 17, No.
6, 1994.
65
Afterbody, cont.

• For Stability Derivatives Need Force on Rear Ramp
  Due To Perturbations

Unsteady Portion Used To Compute Contributions to
Stab. Der.
66
Flow Analysis, cont.
Let Bowshock angle be denoted by
Shock on lip or inside engine inlet
Shock forward of pt. g
Expansion Fan
a lt 0 Expansion Fan
Flow Behind Oblique Shock from cd Used As
I.C. Wedge Angle
a gt 0 Shock
Freestream I.C
Wedge Angle a
67
Rigid Body Moments
Sign Convention

-

-
68
Outline

• Piston Theory
• Forces
• Flow Analysis
• Stability Derivatives
• Engine
• Simulation
• Conclusions

69
Scramjet Engine

• Working Fluid perfect gas, constant specific
  heats
• Two Controls Diffuser Area Ratio, Ad and
  Temperature Addition in Combustor, ?To

70
Engine

• Engine Inlet Conditions Flow Turned Parallel to
  Surface cd

• Force Acts At

• Turning Force Moment

• Engine Thrust

• Engine Moment

71
Engine Thrust and Moment

• Total Forces and Moments

72
Mass Flow/Capture Area
f
?2
?e
c
?1,U
xb
?1,L
q
A0
zb
d
e
hi
A1
g
h
As
V
V
Shock
?
73
Outline

• Piston Theory
• Forces
• Flow Analysis
• Stability Derivatives
• Engine
• Simulation
• Conclusions

74
Stability Derivatives Rear Ramp

• Stab. Der. Functions of density and speed of
  sound (Not Constant on Rear Ramp)

• Let Temperature on Rear Ramp Be Similar To
  Pressure, i.e.,

No Closed Form Solution

• Then

• Instead, Let

75
Plant CharacterisiticsHeating Effects
Convective heating

• Two hour cruise _at_ 85000 ft, 2000 psf (Mach 8)
• Compression due to oblique shock over a 3 deg
  ramp
• Mass changes neglected
• Convective heating due to hypersonic flow
• Eckerts reference Temp. Method
• Turbulent boundary layer assumed
• Radiative cooling occurs (same as before)
• Net Heating determined at discrete points along
  structure

Recovery Factor
Reference Temperature
The local Heat Transfer Coefficient is found from
76
HSV Geometry
Aircraft has Unit depth into paper
Reflected Shock
Spill Flow

• Capture area function of Mach and ?
• Mass flow spillage effects included in thrust
  computation
• Motivation for using oblique shock and expansion
  theory
• Elevator modeled as a flat plate hinged at
  mid-chord
• all-moving elevator
• Inlet turning force included in total aero force
  calculation

77
Piston Theory Conclusions

• Hypersonic Vehicles
• Highly Flexible Long and Slender
• Integrated Airframe/Propulsion
• Airframe Provides External Compression/Expansion

• Piston Theory
• Used to Calculate Aerodynamic Loads
• Local Pressure from Bodys Motion Is Related to
  Local Normal Component of Fluid Velocity
• Same Way As Pressure and Velocity Are Related At
  Face of Piston Moving in a 1-D Channel
• Provides Expression for Time Varying Pressure as
  a Function of Local Surface Velocity
• Unsteady Effects Appear Significant

78
Flow Analysis
Determine Properties of Flow
Engine Constraint
Freestream
Upper Surface
Expansion Fan
Compression (Shock)
Lower Surface cd
Compression (Shock)
Lower Surface gh Find angle at which shock
exactly impinges on point g of engine nacelle
79
Rigid Body Forces and Moments
FUa
FUf
FeU
f
?2
?e
xb
?1,U
FeL
c
?1,L
q
zb
V
V
Fa
e
?
d
hi
Flcd
g
h
Flgh
mg
80
Control Approaches

• Approximate Feedback Linearization
• Team Lead Prof Andrea Serrani
• Controls Oriented Model Developed from AFRL Model
• Analytically intractable otherwise
• Neglect elevator contribution to lift in the
  inversion (a la Sastry)
• Otherwise zero dynamics are unstable
• Ongoing research by CCCS Team to apply robust
  non-linear and adaptive control methods to the
  HSV model