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

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Nonlinear Adaptive Dynamic Inversion Applied to a Generic Hypersonic Vehicle Elizabeth Rollins and John Valasek Vehicle Systems & Control Laboratory, Texas A&M University and Jonathan A. Muse and Michael A. Bolender U.S. Air Force Research Laboratory, Wright-Patterson Air Force Base Approved for Public Release; Distribution Unlimited. Case Number 88ABW-2013-3391. 1 / 50

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

Page 1: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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.

1 / 50

Page 2: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Outline

Introduction

Nonlinear Adaptive Dynamic Inversion Control Architecture

Analysis of the Nonlinear Adaptive Dynamic Inversion ControlArchitecture During Inlet Unstarts

Conclusions

Extensions

2 / 50

Page 3: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Outline

IntroductionMotivationLiterature SurveyResearch IssuesObjectives

Nonlinear Adaptive Dynamic Inversion Control Architecture

Analysis of the Nonlinear Adaptive Dynamic Inversion ControlArchitecture During Inlet Unstarts

Conclusions

Extensions

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Page 4: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Motivation

Control of Hypersonic Vehicles

• Wide range of flight conditions

• Highly integrated elastic vehicle

• Uncertainties

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Page 5: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 6: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 7: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 8: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 9: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 10: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 11: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Outline

Introduction

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

Analysis of the Nonlinear Adaptive Dynamic Inversion ControlArchitecture During Inlet Unstarts

Conclusions

Extensions

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Page 12: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Nonlinear Adaptive Dynamic Inversion Control

Architecture for the GHV

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Page 13: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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

.

13 / 50

Page 14: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 1: Equal Number of Controls and Controlled

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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Page 15: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 1: Equal Number of Controls and Controlled

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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Page 16: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 1: Equal Number of Controls and Controlled

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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Page 17: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 18: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 2: Number of Controls > Number of Controlled

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

)

18 / 50

Page 19: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 2: Number of Controls > Number of Controlled

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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Page 20: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 2: Number of Controls > Number of Controlled

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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Page 21: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 2: Number of Controls > Number of Controlled

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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Page 22: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 2: Number of Controls > Number of Controlled

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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Page 23: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 2: Number of Controls > Number of Controlled

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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Page 24: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 2: Number of Controls > Number of Controlled

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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Page 25: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 2: Number of Controls > Number of Controlled

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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Page 26: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Case 2: Number of Controls > Number of Controlled

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.

26 / 50

Page 27: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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.

27 / 50

Page 28: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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

28 / 50

Page 29: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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

29 / 50

Page 30: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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)

30 / 50

Page 31: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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

31 / 50

Page 32: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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

32 / 50

Page 33: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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

33 / 50

Page 34: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Outline

Introduction

Nonlinear Adaptive Dynamic Inversion Control Architecture

Analysis of the Nonlinear Adaptive Dynamic Inversion ControlArchitecture During Inlet Unstarts

Modeling an Inlet UnstartFlight Path Angle Reference Trajectory GenerationSimulation Results

Conclusions

Extensions

34 / 50

Page 35: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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)

35 / 50

Page 36: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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

Page 37: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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.

37 / 50

Page 38: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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

38 / 50

Page 39: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 40: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 41: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 42: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 43: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 44: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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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Page 45: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

Questions?

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Page 46: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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.

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Page 47: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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.

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Page 48: Vehicle Systems & Control Laboratory - Nonlinear Adaptive … · 2018. 12. 5. · Objectives Develop a nonlinear adaptive dynamic inversion control architecture that: 1 Uses the complete

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.

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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.

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References V

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