Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives...

Nonlinear Adaptive Dynamic Inversion Applied to

a Generic Hypersonic Vehicle

Elizabeth Rollins and John ValasekVehicle Systems & Control Laboratory, Texas A&M University

andJonathan A. Muse and Michael A. Bolender

U.S. Air Force Research Laboratory, Wright-Patterson Air Force Base

Approved for Public Release; Distribution Unlimited. Case Number88ABW-2013-3391.

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Outline

Introduction

Nonlinear Adaptive Dynamic Inversion Control Architecture

Analysis of the Nonlinear Adaptive Dynamic Inversion ControlArchitecture During Inlet Unstarts

Conclusions

Extensions

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Outline

IntroductionMotivationLiterature SurveyResearch IssuesObjectives



Conclusions

Extensions

3 / 50

Motivation

Control of Hypersonic Vehicles

• Wide range of flight conditions

• Highly integrated elastic vehicle

• Uncertainties

4 / 50

Motivation

Inlet Unstart

• Safety concern in hypersonic flight

• Three main causes of inlet unstarts:

1 A flow to the inlet that is slower than the required operatingMach number for the engine,

2 An altered flow that no longer passes through the throat of theengine, and

3 An increase in the back pressure in the engine that causes theshock wave to move ahead of the throat [1].

• Example - Boeing X-51A Waverider

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Literature Survey

• Use of linearized models of hypersonic vehicles to designcontrollers:

- Annaswamy, A. M., Jang, J., and Lavretsky, E. ”AdaptiveGain-Scheduled Controller in the Presence of ActuatorAnomalies.” 2008. [2]

- Gibson, T. E. and Annaswamy, A. M. ”Adaptive Control ofHypersonic Vehicles in the Presence of Thrust and ActuatorUncertainties.” 2008. [3]

- Groves, K. P., et al. ”Reference Command Tracking for aLinearized Model of an Air-breathing Hypersonic Vehicle.”2005. [4]

- Bolender, M. A., Staines, J. T., and Dolvin, D. J. ”HIFiRE 6:An Adaptive Flight Control Experiment.” 2012. [5]

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Literature Survey

• Use of nonlinear models of hypersonic vehicles to designcontrollers:

- Johnson, E. N., et al. ”Adaptive Guidance and Control forAutonomous Hypersonic Vehicles.” 2006. [6]

- Fiorentini, L., et al. ”Nonlinear Robust Adaptive Control ofFlexible Air-breathing Hypersonic Vehicles.” 2009. [7]

- Parker, J. T., et al. ”Approximate Feedback Linearization ofan Air-breathing Hypersonic Vehicle.” 2006. [8]

- Brocanelli, M., et al. ”Robust Control for Unstart Recovery inHypersonic Vehicles.” 2012. [9]

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Research Issues

• Control of hypersonic flight vehicles modeled as couplednonlinear equations with significant parametric uncertainty inthe aerodynamics

• Preventing inlet unstart due to exceeding limits on states

• Preventing inlet unstart due to control surface failures

• Maintaining reasonable tracking following an inlet unstart

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Objectives

Develop a nonlinear adaptive dynamic inversion controlarchitecture that:

1 Uses the complete coupled nonlinear dynamic equations for aninelastic, rigid body model of a hypersonic vehicle

2 Can enforce state constraints to prevent inlet unstarts thatoccur because of changes in angle-of-attack (α) and sideslipangle (β)

3 Is capable of preventing the loss of the vehicle following aninlet unstart

4 Is easily extensible to fault tolerant control methods

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Scope

Plant: The Generic Hypersonic Vehicle (GHV)

• Academic hypersonic vehicle model created at AFRL.

• Nonlinear, 6-DOF, inelastic, no rotors, CFD aero fromshock-expansion viscous corrected. [10]

• Four control surfaces - Two elevons, two ruddervators.

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Outline

Introduction

Nonlinear Adaptive Dynamic Inversion Control ArchitectureGeneral Adaptive Dynamic Inversion EquationsSimulation ResultsRobustness Analysis


Conclusions

Extensions

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Nonlinear Adaptive Dynamic Inversion Control

Architecture for the GHV

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Case 1: Equal Number of Controls and Controlled

Variables

Given a general nonlinear equation of a system in the form

x = f(x) + g(x)u (1)

the dynamic equations for α, β, and µ can be written in the sameform as Equation (1) as

βαµ

=

1

mVT

(

(Ys + FTy)Cβ + mgSµCγ − FTx

CαSβ − FTzSαSβ

)

1

mVTCβ

(

−Ls + mgCµCγ − FTxSα + FTz

Cα)

1

mVT

(

Ls(Tβ + TγSµ) + (Ys + FTy)TγCµCβ − mgCγCµTβ

+ (FTxSα − FTz

Cα)(TγSµ + Tβ) − (FTxCα + FTz

Sα)TγCµSβ

)

+

Sα 0 −Cα

−TβCα 1 −TβSα

Cα/Cβ 0 Sα/Cβ

pqr

.

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Variables

Suppose that a desired reference model for the system is chosen tobe

xm = Axm +Br. (2)

The equation for the error between the reference model and theactual system is

e = xm − x. (3)

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Variables

Taking the time derivative of Equation (3) results in

e = xm − x = xm − f(x)− g(x)u. (4)

The control u is chosen to be

u = [g(x)]−1[xm − f(x) +Ke− ν]. (5)

Substituting Equation (5) into Equation (4) and defining∆ = f(x)− f(x) produces the error dynamics

e = −Ke+∆+ ν. (6)

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Variables

Assume that ∆ can be represented in the form ∆ = W Tβ(x; d),where d is a vector of bounded continuous exogenous inputs.Choose ν to be ν = −W Tβ(x; d). Then, Equation (6) can bewritten as

e = −Ke− W Tβ(x; d) (7)

where W = W −W , which is the weight estimation error.

Finally, the adaptive law to ensure Lyapunov stability is defined as

˙W = ΓW Proj(W , β(x; d)eT ) (8)

where Proj represents the projection operator [11], which is usedto maintain specified bounds on the weights

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Case 2: Number of Controls > Number of Controlled

Variables

Given a general nonlinear equation of a system in the form

x = f(x) + g(x)Λu (9)

the dynamic equations for p, q, and r can be written in the sameform as

ω = I−1(ω × Iω) + I−1(MT +MA(δ = 0))︸︷︷︸f(x)

+ I−1Mδδ︸︷︷︸g(x)Λu

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Variables

ω = I−1(ω × Iω) + I−1(MT +MA(δ = 0))︸︷︷︸f(x)

+ I−1Mδδ︸︷︷︸g(x)Λu

where(ω × Iω) = −

−Jxzpq + (Jz − Jy)qr

(Jx − Jz)pr + Jxz(p2− r2)

Jxzqr + (Jy − Jx)pq

MA(δ = 0) =

qSb

(

C`,baseline + b2VT

(C`pp)

)

qSc

(

Cm,baseline +c

2VT

(Cmq q + Cmαα) + 2Cmδf,r

(δf,r = 0) + 2Cmδt,r(δt,r = 0)

)

− 2CNδf,r(δf,r = 0)qSxcg − 2CNδt,r

(δt,r = 0)qSxcg

qSb

(

Cr,baseline + b2VT

(Cnr r)

)

− 2CYδf,r(δf,r = 0)qSxcg − 2CYδt,r

(δt,r = 0)qSxcg

Mδ =

b∂C`∂δf,r

b∂C`∂δf,l

b∂C`∂δt,r

b∂C`∂δt,l

(

c ∂Cm∂δf,r

− xcg∂CN∂δf,r

) (

c ∂Cm∂δf,l

− xcg∂CN∂δf,l

) (

c ∂Cm∂δt,r

− xcg∂CN∂δt,r

) (

c ∂Cm∂δt,l

− xcg∂CN∂δt,l

)

(

b ∂Cn∂δf,r

− xcg∂CY∂δf,r

) (

b ∂Cn∂δf,l

− xcg∂CY∂δf,l

) (

b ∂Cn∂δt,r

− xcg∂CY∂δt,r

) (

b ∂Cn∂δt,l

− xcg∂CY∂δt,l

)

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Variables

Suppose that the desired dynamics of the closed loop system aregiven by

xm = Axm +Br. (10)

The equation for the error between the reference model and theactual system is

e = xm − x. (11)

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Variables

Taking the time derivative of Equation (11) results in

e = xm − x = xm − f(x)− g(x)Λu. (12)

The desired final form for e is

e = −Ke− W Tβ(x; d) + g(x)Λu (13)

which is the same as the final form for e in Case 1, except for thefinal term g(x)Λu.

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Variables

In order to express e in the final desired form of Equation (13), theterm g(x)Λu is added and subtracted from Equation (12) toproduce

e = xm − f(x)− g(x)Λu+ g(x)Λu. (14)

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Variables

In order to determine a specific control law for the system, aconstrained optimization problem is solved in which the costfunction

J = uTQu (15)

where Q = QT > 0, will be minimized, subject to the constraintg(x)Λu = `, which must be satisfied at all times. The term ` isbased on the control from Case 1 and is expressed as

` = xm − f(x) +Ke− ν. (16)

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Variables

To derive the control law, first the constraint must be included inthe cost function to form the augmented cost function

J = uTQu+ λT (g(x)Λu− `) (17)

where λ ∈ Rn is a Lagrange multiplier. The first order necessary

conditions for minimizing J are given by

∂J

∂λ= g(x)Λu− ` = 0 (18)

∂J

∂u= 2Qu+ ΛT gT (x)λ = 0. (19)

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Variables

From the first order necessary conditions, the control law isdetermined to be

u = Q−1ΛT gT (x)(g(x)ΛQ−1ΛT gT (x))−1`. (20)

where` = xm − f(x) +Ke− ν (21)

Note that for the case where n = m, Equation (20) simplifies tothe control law in Case 1, which is

u = [g(x)]−1[xm − f(x) +Ke− ν]. (22)

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Variables

Continuing with the derivation of e, let ∆ = f(x)− f(x).Substituting Equation (20) and Equation (16) into Equation (14)produces the equation

e = −Ke+∆+ ν + g(x)Λu. (23)

Again, assume that ∆ can be represented in the form∆ = W Tβ(x; d), and choose ν to be ν = −W Tβ(x; d), where d isa vector of bounded continuous exogenous inputs. Then, Equation(23) can be written as

e = −Ke− W Tβ(x; d) + g(x)Λu (24)

where W = W −W , which is the weight estimation error.

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Variables

Finally, the adaptive laws to ensure Lyapunov stability are definedas

˙W = ΓW Proj(W , β(x; d)eT ) (25)

˙Λ = ΓΛ Proj(Λ,−ueT g(x)) (26)

where Proj represents the projection operator, which is used to

maintain specified bounds on W and Λ.

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Simulation Results

Objective: Evaluate tracking of a specified reference trajectory,6/80K.

Commands to α, β, and µ are given as ramp signals from 0degrees to a desired angle in a fixed amount of time.

Current simulation includes:

• Velocity PID Controller

• Second-order actuator dynamics with ζ = 0.7 and ωn = 25 Hz

• Time delay of 0.03 s (Note: time delays of 0.04 s can betolerated as well)

α, β, µ inversion controller: β(x; d) =[c α β µ M

]T, where

c is a constant bias term.

P, Q, R inversion controller: β(x; d) =[c p q r α β M

]T,

where c is a constant bias term.

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Simulation Results

For α = ±2 deg, β = 0 deg, and µ = 70 deg,

0 5 10 15 20 25 30−10

0

10

20

30

40

time (s)

angu

lar

rate

(de

g/s)

P, Q, R

p

q

rp

c

qc

rc

0 5 10 15 20 25 30

−2

−1

0

1

2

time (s)

angl

e (d

eg)

Angle−of−Attack and Sideslip Angle

αβα

c

βc

0 5 10 15 20 25 30−20

0

20

40

60

80

time (s)

angl

e (d

eg)

Bank Angle

µµ

c

0 5 10 15 20 25 30−1000

0

1000

2000

3000

4000

5000

6000

7000

time (s)

velo

city

(ft/

s)

U, V, W

u

v

w

0 5 10 15 20 25 30

5865.3

5865.4

5865.5

5865.6

5865.7

5865.8

5865.9

5866

5866.1

5866.2

time (s)

tota

l vel

ocity

(ft/

s)

Total Velocity

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Simulation Results

For α = ±2 deg, β = 0 deg, and µ = 70 deg,

0 5 10 15 20 25 30−0.2

−0.1

0

0.1

0.2

time (s)

adap

tive

wei

ght

Adaptive Weights forP, Q, R Inversion

0 5 10 15 20 25 30−2

−1

0

1

2

time (s)

adap

tive

Λ w

eigh

t

Adaptive Λ Weightsfor P, Q, R Inversion

0 5 10 15 20 25 30−1

−0.5

0

0.5

1x 10

−3

time (s)

adap

tive

wei

ght

Adaptive Weights forα, β, µ Inversion

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

time (s)

equi

vale

nce

ratio

Equivalence Ratio

0 5 10 15 20 25 30−30

−20

−10

0

10

20

30

time (s)

cont

rol s

urfa

ce d

efle

ctio

n (d

eg)

Control Surfaces

δf,r

δf,l

δt,r

δt,l

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Robustness Analysis

Uncertainties in the plant examined in the analysis include theadditive uncertainties ∆Cmα , ∆Cnβ

, and ∆Cm and multiplicativegains D on the control surface deflections, given in terms ofequations as

Cm = Cmbaseline+∆Cmαα (27)

Cn = Cnbaseline+∆Cnβ

β (28)

Cm = Cmbaseline+∆Cm (29)

Cδ = DCδo . (30)

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Robustness Analysis

Table 1: Additive uncertainty ∆Cmαover a 30 s period with 0.03 s time

delay.

α (deg) β (deg) µ (deg) max ∆Cmα min ∆Cmα

(deg−1) (deg−1)

5 0 0 0.0005 -0.0013

5 1 20 0.0003 -0.0011

Table 2: Additive uncertainty ∆Cnβover a 30 s period with 0.03 s time

delay.

α (deg) β (deg) µ (deg) max ∆Cnβmin ∆Cnβ

(deg−1) (deg−1)

0 1 0 0.007 -0.003

5 0 20 0.01 -0.004

5 1 20 0.006 -0.003

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Robustness Analysis

Table 3: Additive uncertainty ∆Cm over a 30 s period with 0.03 s timedelay.

α (deg) β (deg) µ (deg) max ∆Cm min ∆Cm

5 0 0 0.0005 -0.003

5 1 20 0.0005 -0.002

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Robustness Analysis

Table 4: Multiplicative gains D on control surface deflection terms over a30 s period with 0.03 s time delay.

α (deg) β (deg) µ (deg) Dδf,r Dδf,l Dδt,r Dδt,l

5 0 0 1 0.14 1 1

5 0 0 1 1 1 0.01

5 0 0 0.15 0.15 1 1

5 0 0 1 1 0.15 0.15

5 0 20 1 0.31 1 1

5 0 20 1 1 1 0.01

5 0 20 0.21 0.21 1 1

5 0 20 1 1 0.30 0.30

5 1 20 1 0.42 1 1

5 1 20 1 1 1 0.05

5 1 20 0.38 0.38 1 1

5 1 20 1 1 0.38 0.38

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Outline

Introduction



Modeling an Inlet UnstartFlight Path Angle Reference Trajectory GenerationSimulation Results

Conclusions

Extensions

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Modeling an Inlet Unstart

For this simplified model, an inlet unstart is triggered at a specifiedtime, and the following changes occur in the GHV plant:

• Instantaneous loss of thrust,

• Slight increase in the coefficient of the axial force (CA),

• Slight decrease in the coefficient of the normal force (CN ),and

• Inclusion of additive variations in Cmα and Cnβthrough the

equationsCm = Cmbaseline

+∆Cmαα (31)

Cn = Cnbaseline+∆Cnβ

β. (32)

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Flight Path Angle Reference Trajectory Generation

To allow the GHV simulation to track a flight path angle trajectorygenerated using a nonzero setpoint (NZSP) controller, a methodfrom Reference [12] was applied in which the equation for h, whereh represents the altitude of the aircraft, is written in the form

h =[b0V + b1β + b2α

]+

[a0 a1 a2

]p

q

r

(33)

wherea0 = b4

a1 = b3Cφ + b4SφTθ

a2 = b4CφTθ − b3Sφ

andb0 = CβCαSθ − SβSφCθ − CβSαCφCθ

b1 = V (−SβCαSθ − CβSφCθ + SβSαCφCθ)

b2 = V (−CβSαSθ − CβCαCφCθ)

b3 = V (CβCαCθ + SβSφSθ + CβSαCφSθ)

b4 = V (−SβCφCθ + CβSαSφCθ) .36 / 50

Flight Path Angle Reference Trajectory Generation

The original reference trajectory that is generated for γ using theNZSP controller can be converted to h using the relation fromaircraft kinematics that h = V Sγ . The h, β, µ inversion controllerreplaces the original α, β, µ inversion controller in the GHVsimulation, and now desired trajectories for γ can be tracked.

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Simulation Results

Track flight path angle trajectory during inlet unstart at 10 seconds,6/80K.

0 10 20 30 40 50 60−5

0

5

time (s)

angu

lar

rate

(de

g/s)

P, Q, R

p

q

rp

c

qc

rc

0 10 20 30 40 50 60

−200

0

200

400

600

time (s)

altit

ude

rate

(ft/

s)

Altitude Rate

h

h c

0 10 20 30 40 50 60−1

−0.5

0

0.5

1

time (s)

angl

e (d

eg)

Bank Angle and Sideslip Angle

µβµ

c

βc

0 10 20 30 40 50 60−1

0

1

2

3

4

5

6

time (s)

angl

e (d

eg)

Flight Path Angle

γγp

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Simulation Results

For the generated flight path angle trajectory during an inlet unstart at10 seconds

0 10 20 30 40 50 60−0.5

0

0.5

time (s)

adap

tive

wei

ght

Adaptive Weights forP, Q, R Inversion

0 10 20 30 40 50 60−2

−1

0

1

2

time (s)

adap

tive

Λ w

eigh

t

Adaptive Λ Weightsfor P, Q, R Inversion

0 10 20 30 40 50 60−5

0

5

time (s)

adap

tive

wei

ght

Adaptive Weights forα, β, µ Inversion

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

time (s)

equi

vale

nce

ratio

Equivalence Ratio

0 10 20 30 40 50 60−30

−20

−10

0

10

20

30

time (s)

cont

rol s

urfa

ce d

efle

ctio

n (d

eg)

Control Surfaces

δf,r

δf,l

δt,r

δt,l

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Outline

Introduction



Conclusions

Extensions

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Conclusions

• The approach of nonlinear adaptive dynamic inversion controlworks well as a candidate control architecture for hypersonicvehicles because

- the control architecture is able to maintain trackingperformance without excessive control effort while beingrobust to

• decreases in control surface effectiveness,• changes in system parameters, and• time delays of 0.04 seconds or less, and

- the control architecture can tolerate an inlet unstart andmaintain nominal tracking of a specified flight path angletrajectory.

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Conclusions

• The nonlinear adaptive dynamic inversion control architecturecan maintain reference trajectory tracking with only a slightdegradation in tracking performance following an inlet unstart.

- Through a robustness analysis on the GHV, it wasdetermined that the maximum additive variations inCmα and Cnβ

that the control architecture could toleratewere ∆Cmα = 0.001 deg−1 and ∆Cnβ

= −0.001 deg−1.

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Outline

Introduction



Conclusions

Extensions

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Extensions

• Enforcing state constraints with a projection operator, andincluding the projection operator in the control laws.

• Combine controllers that can enforce state constraints withfault-tolerant controllers.

• Development of control logic associated with the inlet unstartenvelope for a hypersonic vehicle to determine the appropriatecourse of action to ensure its preservation.

• Development and testing of controllers using an elastic modelof a hypersonic vehicle.

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Questions?

45 / 50

References I

[1] Heiser, W. H. and Pratt, D. T., Hypersonic Airbreathing

Propulsion, AIAA Education Series, American Institute ofAeronautics and Astronautics, Washington D. C., 1994.

[2] Annaswamy, A. M., Jang, J., and Lavretsky, E., “Adaptivegain-scheduled controller in the presence of actuatoranomalies,” AIAA Guidance, Navigation and Control

Conference and Exhibit, Honolulu, Hawaii, August 2008.

[3] Gibson, T. E. and Annaswamy, A. M., “Adaptive Control ofHypersonic Vehicles in the Presence of Thrust and ActuatorUncertainties,” AIAA Guidance, Navigation and Control

Conference and Exhibit, Honolulu, Hawaii, August 2008.

46 / 50

References II

[4] Groves, K. P., Sigthorsson, D. O., Serrani, A., Yurkovich, S.,Bolender, M. A., and Doman, D. B., “Reference CommandTracking for a Linearized Model of an Air-breathingHypersonic Vehicle,” AIAA Guidance, Navigation and Control

Conference and Exhibit, San Francisco, California, August2005.

[5] Bolender, M. A., Staines, J. T., and Dolvin, D. J., “HIFiRE 6:An Adaptive Flight Control Experiment,” 50th AIAA

Aerospace Sciences Meeting including the New Horizons

Forum and Aerospace Exposition, Nashville, TN, January2012.

47 / 50

References III

[6] Johnson, E. N., Calise, A. J., Curry, M. D., Mease, K. D., andCorban, J. E., “Adaptive Guidance and Control forAutonomous Hypersonic Vehicles,” Journal of Guidance,

Control, and Dynamics, Vol. 29, No. 3, May - June 2006,pp. 725–737.

[7] Fiorentini, L., Serrani, A., Bolender, M. A., and Doman,D. B., “Nonlinear Robust Adaptive Control of FlexibleAir-breathing Hypersonic Vehicles,” Journal of Guidance,

Control, and Dynamics, Vol. 32, No. 2, Mar.-Apr. 2009,pp. 401–416.

[8] Parker, J. T., Serrani, A., Yurkovich, S., Bolender, M. A., andDoman, D. B., “Approximate Feedback Linearization of anAir-breathing Hypersonic Vehicle,” AIAA Guidance,

Navigation and Control Conference and Exhibit, Keystone,Colorado, August 2006.

48 / 50

References IV

[9] Brocanelli, M., Gunbatar, Y., Serrani, A., and Bolender,M. A., “Robust Control for Unstart Recovery in HypersonicVehicles,” AIAA Guidance, Navigation and Control

Conference, Minneapolis, Minnesota, August 2012.

[10] Vick, T. J., “Documentation for Generic Hypersonic VehicleModel,” Tech. rep., U.S. Air Force Research Laboratory,Wright-Patterson AFB.

[11] Pomet, J.-B. and Praly, L., “Adaptive Nonlinear Regulation:Estimation from the Lyapunov Equation,” IEEE Transactions

on Automatic Control , Vol. 37, No. 6, June 1992,pp. 729–740.

[12] Menon, P., Badgett, M., Walker, R., and Duke, E.,“Nonlinear Flight Test Trajectory Controllers for Aircraft,”Journal of Guidance, Control, and Dynamics, Vol. 10, No. 1,Jan.-Feb. 1987, pp. 67–72.

49 / 50

References V

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