active control for very flexible aircraft an ... · active control for very flexible aircraft an...

Active Control for Very Flexible AircraftAn Aeroservoelastic Modeling and Control Study

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Hugh H.T. Liu⇠”õ

University of TorontoInstitute for Aerospace Studies

Flight Systems and Control Laboratory

August 5, 2014

UTIAS

Introduction

Toronto ON Canada

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Introduction

University of Toronto⇢&⇢'f

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Introduction

U of T Facts

67,000 undergraduate students (10,000 international), 16,000graduate students (2,500 international)China: 6,047Faculty Members: 12,500Research Funds: $1.2B530,000 alumniInternational Institution Ranking (2013): 20 (Times Higher Ed), 28(SJTU), 17 (QS World), and 8 (NTU)


Introduction

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}Bi�Norman Bethune �;fZÎ�∫S;I⇧�˝E;I⇧Úy�John Charles Fields �pf∂�ÚyV�À⇧p+·Ú—�Jeffrey Skoll �eBayl¯ñ˚;¡'q�Mark Henry Rowswell �W�(NíSŒ⇢∫Xó◊≤�~�!

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Introduction

U of T Engineering

5,241 undergraduate students, 1,933 graduate students238 faculty membersresearch funds: $63.5MTimes Higher Education World University Rankings forEngineering and Technology: 22nd (2014)Academic Departments, Divisions and Institutions

Institute for Aerospace Studies (UTIAS)Institute of Biomaterials and Biomedical Engineering (IBBME)Department of Chemical Engineering and Applied Chemistry(ChemE)Department of Civil Engineering (CivE)Division of Engineering ScienceDivision of Environmental Engineering and Energy SystemsDepartment of Electrical and Computer EngineeringDepartment of Mechanical and Industrial EngineeringDepartment of Materials Science and Engineering


Introduction

UTIAS: Excellence in Research and Education

Recently awarded the Pioneer Award from the Canadian Air &Space Museum for its part in the rescue of Apollo 13Faculty have won numerous national and international awards,including three Members of the Order of CanadaGraduates play an important role in Canada’s aerospace sectorUniversity of Toronto Engineering Faculty ranked 18th in world in2013-14 (Times Higher Education: Engineering and Technology)UTIAS ranked among top 5 public aerospace departments inNorth America


Introduction

UTIAS: Academic Unit in Aerospace Science andEngineering

19 full-time faculty membersUndergraduate teachingAerospace Option of Engineering ScienceProgram ( 35 graduating students)Graduate program (2013-14)56 M.A.Sc., 66 Ph.D., 25 M.Eng. studentsAeronautical Engineering

aircraft flight systems, propulsion,aerodynamics, computational fluiddynamics, and structural mechanics,sustainable aviation

Space Systems Engineeringspacecraft dynamics and control, space(mobile) robotics and mechatronics,and microsatellite technology

first human-powered ornithoptor

MOST microsatellite (2003)


Introduction

Personal Experience

University EducationPh.D. (June 1998) University of TorontoM.A.Sc. (March 1994) Beihang University (BUAA)B.A.Sc. (July 1991) Shanghai Jiao Tong University

Experiences1998-2000, Systems Engineering/Lead, Honeywell Canada2000-05, Assistant Professor, UTIAS2006-12, Associate Professor, UTIAS2007, Visiting Professor, BUAA2006-07, Honeywell Engines and Systems2006-07, Visiting Scholar, Univ. of Michigan2006-07, Senior Research Fellow, City University of Hong Kong2008 - present, Associate Director, Graduate Studies, UTIAS2013 - present, Professor, UTIAS2009-15, Guest Professor (status only), BUAA2011-16, Guest Professor (status only), SJTU


Introduction

Research Interests

Systems Modelling, Simulation and Control

• Fail-Safe Flight Control• Flexible Wing, Active Control• Landing Gear Systems (MBD)• Systems Sim., Integration

Multi-Vehicle Systems Estimation and Control• Motion-Sync Formation Flight• Vision-based Tracking• Wild Fire Monitoring

Autonomous UVS Applications• Localization and Mapping• Soaring and Surveillance• Border Patrolling


Introduction

Today’s Talk: Outline

1 Introduction

2 Aircraft Flight Dynamics

3 Highly Flexible Wing with Active Control

4 Control Integrated MDO

5 Fault Tolerant Flight Control of Damaged Aircraft

6 Active Flutter Suppression

7 Integrated Flight Control and Simulation


Aircraft Flight Dynamics

2. Aircraft Flight Dynamics I

An inertial Earth Surface Frame is defined with its originsomewhere on the surface and near the vehicle as much aspossible (so that a flat surface might be assumed in simplifyingdynamics analysis); axis z

~E directed vertically down; x

~E points

north, and y

~E east.

The air-trajectory reference frame is also called the wind axessystem. It has origin fixed to the vehicle (usually at the center ofmass), and x

~W is directed along the velocity vector v

~of the vehicle

relative to the atmosphere. The axis z

~W lies in the plane of

symmetry of the vehicle. If the atmosphere is at rest, then OWwould trace out the trajectory of the vehicle relative to the Earth,and x

~W would be always tangent to it.



2. Aircraft Flight Dynamics II

Any set of axes fixed in a rigid-body is a body-fixed referenceframe (and the angular velocity of the frame is the same as that ofthe body). In flight vehicles, the origin is usually chosen at thecenter of mass, the axes x

~,z~

are chosen in the plane of vertical

symmetry, where z

~is directed downwards.



The Vehicle Euler Angles I

In flight dynamics, the Euler angles used are those which rotate theearth-surface frame FE into coincidence with the relevant axis system(frame), usually either FB or FW , denoted by ( , ✓,�) and (µ, �,�)respectively. Figure 1 shows the sequence of rotations.



The Vehicle Euler Angles II

Figure 1: Airplane Orientation



1 a rotation about z

~E = z

~1, is called the yaw angle: �⇡ < ⇡

or 0 < 2⇡,

{x

~E ,y~

E ,z~

E} = {x

~1,y~

1,z~

1} =) x

~2,y~

2,z~

2

2 a rotation ✓ about y

~2, ✓ is called the pitch angle: �⇡/2 ✓ ⇡/2,

{x

~2,y~

2,z~

2} =) {x

~3,y~

3,z~

3}

3 a rotation � about x

~3 = x

~, � is called the roll angle: �⇡ � < ⇡ or

0 � 2⇡,{x

~3,y~

3,z~

3} =) {x

~,y~,z~}



The rotation matrix from FE to FB is given by

F~ B

= C

¯BEF~ E

(1)

where C

¯BE = C

¯1(�)C

¯2(✓)C

¯3( )

2

4

cos ✓ cos cos ✓ sin � sin ✓sin� sin ✓ cos � cos� sin sin� sin ✓ sin + cos� cos sin� cos ✓cos� sin ✓ cos + sin� sin cos� sin ✓ sin � sin� cos cos� cos ✓

3

5

(2)and C

¯EB

2

4

cos ✓ cos sin� sin ✓ cos � cos� sin cos� sin ✓ cos + sin� sin cos ✓ sin sin� sin ✓ sin + cos� cos cos� sin ✓ sin � sin� cos

� sin ✓ sin� cos ✓ cos� cos ✓

3

5

(3)



Similarly, the rotation matrix from FE to FW is given by

F~W

= C

¯WEF

~ E(4)

where C

¯WE = C

¯1(µ)C

¯2(�)C

¯3(�).

1 a rotation � about z

~E = z

~1, � is called the heading angle,

{x

~E ,y~

E ,z~

E} = {x

~1,y~

1,z~

1} =) x

~2,y~

2,z~

2

2 a rotation � about y

~2, � is called the flight path angle,

{x

~2,y~

2,z~

2} =) {x

~3,y~

3,z~

3}

3 a rotation µ about x

~3 = x

~, µ is called the bank angle,

{x

~3,y~

3,z~

3} =) {x

~,y~,z~}



The rotation matrix from FE to FB is given by

F~W

= C

¯WEF

~ E(5)

where C

¯WE = C

¯1(µ)C

¯2(�)C

¯3(�)

2

4

cos � cos� cos � sin� � sin �sinµ sin � cos� � cosµ sin� sinµ sin � sin� + cosµ cos� sinµ cos �cosµ sin � cos� + sinµ sin� cosµ sin � sin� � sinµ cos� cosµ cos �

3

5

(6)and C

¯EW

2

4

cos � cos� sinµ sin � cos� � cosµ sin� cosµ sin � cos� + sinµ sin�cos � sin� sinµ sin ✓ sin� + cosµ cos� cosµ sin ✓ sin� � sinµ cos�

� sin ✓ sinµ cos ✓ cosµ cos ✓

3

5

(7)



The angular velocity of FB relative to the inertial frame FE is denotedby !

~B and is expressed by !

¯B = [p q r ]T . The angular velocity vector

following the Euler rotations are

!~

= z

~1 + ✓y

~2 + �x

~3 (8)

= [x~

3 y

~2 z

~1]

2

4

�

✓

3

5 (9)

Transform axes z

~1,y~

2,x~

3 to body frame axes x

~,y~,z~

, one obtains

x

~3 = x

~= [x~

y

~z

~]

2

4

100

3

5 (10)

y

~2 = y

~3 = cos�y

~� sin�z

~= [x~

y

~z

~]

2

4

0cos�

� sin�

3

5 (11)



z

~1 = z

~2 = � sin ✓x

~3 + cos ✓z

~3 (12)

= � sin ✓x~+ cos ✓(sin�y

~+ cos�z

~) (13)

= [x~

y

~z

~]

2

4

� sin ✓cos ✓ sin�cos ✓ cos�

3

5 (14)

Therefore !¯

B = S

¯B⇥¯

B

2

4

pqr

3

5 =

2

4

1 0 � sin ✓0 cos� sin� cos ✓0 � sin� cos� cos ✓

3

5

2

4

�

✓

3

5

�=S

¯B

2

4

�

✓

3

5 (15)



Similarly The angular velocity of FW relative to the inertial frame FE isdenoted by !

~and is expressed by !

¯W . The angular velocity vector

following the Euler rotations are

!~

= �z

~1 + �y

~2 + µx

~3 (16)

= [x~

3 y

~2 z

~1]

2

4

µ��

3

5 (17)

Transform axes z

~1,y~

2,x~

3 to body frame axes x

~,y~,z~

, one obtains

x

~3 = x

~= [x~

y

~z

~]

2

4

100

3

5 (18)

y

~2 = y

~3 = cosµy

~� sinµz

~= [x~

y

~z

~]

2

4

0cosµ

� sinµ

3

5 (19)



z

~1 = z

~2 = � sin �x

~3 + cos �z

~3 (20)

= � sin �x

~+ cos �(sinµy

~+ cosµz

~) (21)

= [x~

y

~z

~]

2

4

� sin �cos � sinµcos � cosµ

3

5 (22)

Therefore !¯

W = S

¯W⇥

¯W

!¯

W =

2

4

1 0 � sin �0 cosµ sinµ cos �0 � sinµ cosµ cos �

3

5

2

4

µ��

3

5

�=S

¯W

2

4

µ��

3

5 (23)



The Aerodynamic Angles I

The velocity of the vehicle relative to the atmosphere v

~can either be

given by its three coordinates (u, v ,w) in a body-fixed frame FB, oralternatively by the magnitude V and two suitable defined angles.These angles, which are of fundamental importance in determining theaerodynamic forces that act on the vehicle, are defined as the angle ofattack and sideslip angle respectively.Angle of Attack:

↵ = tan�1 wu

� ⇡ ↵ ⇡ (24)

Sideslip Angle:� = sin�1 v

V� ⇡ � ⇡ (25)



The Aerodynamic Angles II

It follows that the velocity components in the body-fixed frame arerepresented by

u = V cos� cos↵ (26)v = V sin� (27)w = V cos� sin↵ (28)

This relationship reveals the rotation matrix from FW to FB by thefollowing Euler rotation sequence: (��,↵, 0), i.e.

F~ B

= C

¯BWF

~W(29)



The Aerodynamic Angles III

where

C

¯BW = C

¯1(0)C

¯2(↵)C

¯3(��)

=

2

4

cos↵ 0 � sin↵0 1 0

sin↵ 0 cos↵x

3

5

2

4

cos� � sin� 0sin� cos� 0

0 0 1

3

5

=

2

4

cos↵ cos� � cos↵ sin� � sin↵sin� cos� 0

sin↵ cos� � sin↵ sin� cos↵

3

5 (30)

Obviously, the velocity of the vehicle relative to the atmosphere v

~expression under FW is v

¯W = [V 0 0]T , and the formulas 26 - 28 can

be obtained by v

¯B = C

¯BWv

¯W .



Force Equations in Trajectory Frame

Fromf

~= Wz

~E + Tx

~B � Dx

~W � Lz

~W (31)

one obtains

T � D � W sin � = mV (32)L cosµ � W cos � = mV � (33)

L sinµ = mV � cos � (34)



Equations in Body-Fixed Frame I

Given the expressions under body-fixed frame FB : {x

~,y~,z~}:

f

¯B = [X Y Z ]T (35)

!¯

B = [p q r ]T (36)

v

¯B = [u v w ]T (37)

one gets the following force equation expressions:

f

¯B = m(v

¯B + !

¯Bv

¯B) (38)



Equations in Body-Fixed Frame II

leading to

X = m(u + qw � rv) (39)Y = m(v + ru � pw) (40)Z = m(w + pv � qu) (41)

Given the expressions in Body-Fixed Frame FB:

⌧¯

cmB = [L M N]T (42)

I

¯B =

2

4

Ixx Ixy IxzIyx Iyy IyzIzx Izy Izz

3

5 (43)

According to⌧¯

B = !¯

BI

¯B!¯

B + I

¯B!¯

B (44)



Equations in Body-Fixed Frame III

L = Ixx p + Ixy (q � pr) + Ixz(r + pq) + Iyz(q2 � r2) � (Iyy � Izz)qrM = Iyy q + Iyz(r � pq) + Iyx(p + qr) + Izx(r2 � p2) � (Izz � Ixx)prN = Izz r + Izx(p � qr) + Izy (q + pr) + Ixy (p2 � q2) � (Ixx � Iyy )pq


Highly Flexible Wing with Active Control

3. Highly Flexible Wing with Active ControlHigh performance ) highly-flexible, high-aspect-ratio configurations

unified dynamics frameworkof rigid body along withstructural dynamics. 2

integrated design & controlfor maneuvrability andaeroelastic criteria. 3

increased endurance throughactive load alleviationsystem designed concurrentlywith aerodynamic shape andstructural sizing

2Sohrab Haghighat, J. Martins, and H. H. T. Liu, Journal of Aircraft, Vol. 49,No. 2 Mar-Apr. 2012, 432–443

3Sohrab Haghighat, H.H.T. Liu and J. Martins, Journal of Guidance,Control and Dynamics, Vol. 35, No. 6, 2012, 1751– 1766



Motivation

Highly flexible aircraft can deform in unexpected ways (eg.Helios 2003)

Wreckage of the HeliosAircraft became uncontrollable and broke up

Successful development of highly-flexible aircraft is dependent on,availability of appropriate design and analysis tools, specificallycontrol systems and time-domain analysis methods.performing concurrent design optimization to take advantage of thesynergy between different disciplines.



Aeroservoelastic formulation

Rc

R

r

d

dm

FB

FI

R = Rc + r + d,V = R = Vc + ! ⇥ (r + d) + v.

Structural displacement and velocity expressed using shape functions,d = �

r

⇠ ) d = �r

⌘Liu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 33 / 100


Nonlinear equations of motion

2

4

mI3⇥3 X T1 S1

⇤ J S2 + X2⇤ ⇤ Mee

3

5

8

<

:

Vc!⌘

9

=

;

=

�

2

4

m! !X1T 2!S1

!X1 !J + J + VcX1T ! (S2 + X2)

�ST1 !

T R

� ⇢�rT !r T

d� 2R

� ⇢�rT !�rd� + Cee

3

5

8

<

:

Vc!⌘

9

=

;

�

2

4

O3⇥3 O3⇥3 O3⇥nO3⇥3 O3⇥3 O3⇥nOn⇥3 On⇥3

R

� ⇢�rT !2�rd� + Kee

3

5

8

<

:

Rc✓⇠

9

=

;

+

2

4

mCbiX1CbiST

1 Cbi

3

5

�

g

+

8

<

:

FMfe

9

=

;

S1 =R

� ⇢�rd�, S2 =R

� ⇢r�rd�Constant terms that depend on the shape functions and the rigid-bodymass distribution.

X1 =R

� ⇢g�r⇠d�, X2 =

R

� ⇢g�r⇠�rd�

Time-dependent terms that vary as the aircraft structure deforms.

R

� ⇢�Tr !

2�rd�Geometric stiffening, accounts for the coupling between the transverseand the axial deformations and the rigid-body rotational velocity.

J =R

� ⇢

g�r⌘⇣

r + f�r⇠⌘T

+⇣

r + f�r⇠⌘

g�r⌘T�

d�



Nonlinear equations of motion

2

4

mI3⇥3 X T1 S1

⇤ J S2 + X2⇤ ⇤ Mee

3

5

8

<

:

Vc!⌘

9

=

;

=

�

2

4

m! !X1T 2!S1

!X1 !J + J + VcX1T ! (S2 + X2)

�ST1 !

T R

� ⇢�rT !r T

d� 2R

� ⇢�rT !�rd� + Cee

3

5

8

<

:

Vc!⌘

9

=

;

�

2

4

O3⇥3 O3⇥3 O3⇥nO3⇥3 O3⇥3 O3⇥nOn⇥3 On⇥3

R

� ⇢�rT !2�rd� + Kee

3

5

8

<

:

Rc✓⇠

9

=

;

+

2

4

mCbiX1CbiST

1 Cbi

3

5

�

g

+

8

<

:

FMfe

9

=

;



Challenges in controlling a highly flexible aircraft

Traditional aircraft control: rigid-body tracking is performedseparately from structural control, and the interaction is avoided byusing notch filters.

1 A unified controller needs to address both rigid-body motion andthe aircraft flexibility.

2 Structural flexibility results in a time delay between the controlinput deflections and the rigid-body motion of the aircraft.,! Non-minimum phase system

3 High frequency structural modes are very lightly damped whichmakes the design of a state observer challenging.,! Filter’s transient response can adversely affect theclosed-loop stability and performance.



Linearized State-Space Representation

The linearized equations of motion for a straight level flight are,

8

<

:

�Vc�!�⌘

9

=

;

= M�1

0

B

@

2

6

4

O3⇥3 m fVc0 O3⇥n

O3⇥3 X T10

Vc0 � fVc0X10 O3⇥n

O3⇥3 ST1 Vc0 �Cee

3

7

5

8

<

:

�Vc�!�⌘

9

=

;

+

2

4

O3⇥3 mG O3⇥nO3⇥3 X10G O3⇥nO3⇥3 ST

1 G �Kee

3

5

8

<

:

�Rc�✓�⇠

9

=

;

+

8

<

:

�F�M�fe

9

=

;

1

A

,

where M =

2

6

4

m X T10

S1X T

10J0 S2 + X20

ST1 ST

2 + X T20

Mee

3

7

5

and G is a matrix of projected

gravity accelerations.M includes off-diagonal terms that indicates the coupling between thetwo dynamics.



Model Comparison

The proposed unified dynamics framework allows for integrated designand control as it reveals strong inertial interactions, especially criticalfor very flexible aircraft. To further illustrate this point, a frequencyanalysis is given by comparing frequency characteristics of threemodels,

8

<

:

xTac =

�

V ! ⇠ ⌘ ✓ Rc xaero

xac = Axac + Bu + B!w

y = Cxac

1 a statically deformed aircraft model (deformed model);2 the statically deformed aircraft model augmented by a series of

elastic modes (augmented model); and3 the fully developed model under unified framework (unified model).

The Bode diagrams are plotted representing transfer functions relatingvertical gust excitation to the load factor



Model Comparison: Bode Plots

Bode diagrams for three aircraft models



Improved Model Predictive Control I

A quadratic performance function is used.

V (xk ,U,Nu,Np) =

Np�1X

i=1

⇣

�

xk+i � xref,k+i

�T Q�

xk+i � xref,k+i

�

⌘

+Nu�1X

i=0

⇣

uTk+iRuk+i

⌘

+�

xk+Np � xref,k+Np

�T Qf�

xk+Np � xref,k+Np

�

Corrected future prediction can be expressed as,

Xp = HU + Gxk + GoLek



Improved Model Predictive Control II

Plantref

xk

uk

x

k+1

= Ad

x

k

+Bd

uk

+ L (xk

� x

p

k

)+

X = HU+Gx

k

+Go

Lek

min V (x0

,U, Nu

, Np

)w.r.t U

s.t. F.U c

yk

Linear/Nonlinear

Future state prediction

x

pk

Future inputoptimization

V (xk ,U,Nu,Np) = UT⇣

HT QH + R⌘

U

+ 2⇣

xTk GT + eT

k LT GTo � X

ref

⌘

QHU.

The developed controller is well suited to perform gust loadalleviation, an integral part of the aeroservoelastic optimization.



Compact form matrices

H =

2

6

6

6

6

6

6

6

6

6

6

6

4

CdBdCdAdBd CdBd

... . . .CdANu�1

d Bd CdANu�2d Bd . . . CdBd

CdANud Bd CdANu�1

d Bd . . . CdAdBd... . . . ...

CdANp�1d Bd CdANp�2

d Bd . . .PNp�Nu

i=0 CdAidBd

3

7

7

7

7

7

7

7

7

7

7

7

5

, G =

2

6

6

6

4

AdA2

d...

ANpd

3

7

7

7

5

GTo =

h

I ATd Ad

2T · · ·i

F =

2

6

6

4

INu⇥m�INu⇥m

M�M

3

7

7

5

, c =

2

6

6

4

Umax�Umin

�Umax + b��Umax � b

3

7

7

5

, with M =

2

6

6

6

4

I�I I

. . . . . .�I I

3

7

7

7

5

and

bT =�

uk�1T 0 · · ·



Gust Profile

For discrete gust evaluation, a (1 - cos) gust profile of the followingform is used:

wg =wg

2

✓

1 � cos2⇡Vt

Lg

◆

where wg is the maximum gust velocity, and Lg is the gust length.For continuous gust modelling, the Dryden model is used to createstochastic gust excitations. A gust filter is used to generate thenumerical values of gust,

Gw (s) = wg

r

L⇡V

1 +p

3Lv s

�

1 + LV s

�2 , Gq (s) =sV

�

1 +� 4b⇡V

�

s�Gw (s)

The parameter L represents the gust scale and V represents theaircraft flying speed.In this work, wg is set to 5 m/s for both continuous and discrete profiles.



Stress-level reduction for the different controllersLg (s) 1.0 0.5 0.25MPC with PE 45.4% 43.9% 42.9%MPC 41.6% 37.7% 29.3%LQR 44.5% 37.9% 26.7%

Load factor and rigid-body parameters (Lg = 0.25s)Liu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 44 / 100


Structural modes and wing root and mid stresses (Lg = 0.25s)



Average RMS and peak values in response to Dryden gustLoad factor �

root

(MPa) �e (deg) �a (deg)

Aver

age

RM

S MPC with PE 0.4162 2.5867 2.00002 0.5254MPC 0.5724 5.0891 1.2768 0.9227LQR 0.5658 4.9946 1.2852 0.9128Without GLA 0.4394 4.7404 2.6167 –

Aver

age

peak MPC with PE 1.0426 5.1686 4.3849 1.36003

MPC 1.5687 1.0166 2.9316 2.3634LQR 1.5317 9.9287 2.9467 2.3456Without GLA 1.1521 1.0802 5.9604 –



Load factor and rigid-body parameters for the crippled aircraft(Lg = 0.25s)



Structural modes and wing root and mid stresses for the crippledaircraft (Lg = 0.25s)



Average RMS and peak values in response to Dryden gust for crippledaircraft

Load factor �root

(MPa) �e (deg) �a (deg)Av

erag

eR

MS

MPC with P.E. 0.7128 4.9537 3.4353 1.4580LQR 0.9821 8.9246 2.5772 3.3027

Aver

age

peak

MPC with P.E. 1.9363 1.0473 7.6642 3.6368LQR 2.6036 2.0225 6.1842 7.6825

Haghighat et al., Journal of Guidance, Control, and Dynamics (2012)



MPC Stability

Stabilizability and detectability + standard LQ infinite horizonoptimal control ) optimal stabilizing controller.A linear finite horizon model predictive controller with a quadraticperformance index is not asymptotically stabilizing in general.An LQ model predictive controller is stable if and only if the pair(A,B) is stabilizable, the pair

�

A,Q1/2� is observable, and the costmonotonicity condition, V (xk ,U⇤,N + 1) V (xk ,U⇤,N), issatisfied.

Terminal equality constraintsTerminal set constraintsTerminal penalty function

The first two approaches are very restrictive and reduce thefeasibility region.The terminal penalty approach is used,Qf � Q + K T RK + (Ad � BdK )T Qf (Ad � BdK )for some K 2 Rm⇥n.



MP Controller Applied to a Non-minimum PhaseSystem

One of the difficulties of controlling a non-minimum phase systemis the internal instability problem.When a high performance level is of interest, the controller isforced to contain an implicit approximate inverse of the systemmodel.For non-minimum phase systems, this leads to the cancellation ofunstable poles and zeros causing internal instability.If no control and state constraints are considered, the optimalcontrol sequence can be calculated as follows,

U =⇣

HT QH + R⌘�1

HT Q (Gxk � Xref

)

If the control effort is not penalized (R = O), and the predictionand the control horizon are equal (H is a square matrix),U⇤ = H�1 (Gxk � X

ref

)Substituting the optimal control sequence into compact form )cancellation of unstable poles with unstable zeros that causes theinternal instability.Penalizing the control effort and also the application of a longerprediction horizon than the control horizon (tall H matrix) havebeen reported to avoid the internal instability.



2-DOF control architecture

Kff

Kfb

Wg

!g

A/C

Ms

w

y

e

s

u

Ws

refu

�

�

y�

u�

Me

Ws

Wref

�

Wu

err

Wu, Wref

, We, and Ws are the frequency-domain weighting parameters



Control synthesis

The control system can be represented in linear fractionaltransformation (LFT) form

Pw =

⇢

ref!g

�

u

v =

(

�y

ref

)

y

�

K

z

flexible

=n

s u

oT

�u

�

z

rigid

=n

e u

oT

The feed-back and feed-forward are combined as K =⇥

Kfb Kff⇤



xp =Apxp + Buu + B!w + B�u�

zrigid

=Cr xp + Druuz

flexible

=Cf xp + Dfuuy� =C�x + D�uu + D�!w

y =Cyxp

A mixed-norm minimization is of interest in this work:

min kTy�u�k1 +P

i ⇢ikTzi wk↵i

w.r.t. K

LMI techniques and convex optimization methods are employed tomake the problem of finding the controller gains more tractable



LMI representative of H1, Hg and H2 problem

The resulting LMI representation for the H1 problem is

0

@

(AX + BNCv ) + (AX + BuNCy )T B� (C�X + D�uNCy )

T

⇤ ��I O⇤ ⇤ ��I

1

A < 0,

X > 0,MCv = Cv X .

N and M are two auxiliary matrices that used in the transformation



LMI representative of H1, Hg and H2 problem

The LMI formulation of the Hg problem is

✓

(AX + BNCy ) + (AX + BNCy )T Bw

⇤ �I

◆

< 0,✓

X (Cf X + DfuNCv )T

⇤ ↵I

◆

> 0,

MCy = CyX .

And finally the LMI formulation of the H2 problem is

✓

(AX + BNCy ) + (AX + BNCy )T Bw

⇤ �I

◆

< 0,✓

X (Cr X + DruNCv )T

⇤ Q

◆

> 0,

Tr (Q) < ↵,

MCv = Cv X .Liu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 56 / 100


Optimal control problem

The control design problem can be casted as follows,

min. � +P

i ⇢i↵iw.r.t. X ,Ns.t. LMIs

The X and N matrices are found by solving the above problem,the M matrix can be found by solving MCy = CyXThe controller gain has the following form,

K = NM�1

The above LMI problems are casted in semi-definite programmingform and solved using CVXOPT



Uncertainty modelling

A = Ao + A1�v + A2�2v where �V = V�Vo

0.2Voand �V 2 [�1, 1]

The actual plant dynamics:

xac =⇣

A +P2

i=1 �iv Ai

⌘

xac +⇣

B +P2

i=1 �iv Bi

⌘

u +⇣

B!

+P2

i=1 �iv B

! i

⌘

w

y = Cxac

Using singular value decomposition (SVD)

xac = Axac + Bu + B!w

+⇥

B�1 B�2⇤

u�z }| {

�vIr1

�2vIr2

�

C�1 D�u1 D�w1

C�2 D�u2 D�w2

�

8

<

:

xacuw

9

=

;

| {z }

y�

where ri = rank

�⇥

Ai Bi B! i⇤�

.Liu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 58 / 100


Altitude maneuver results (V = 80 m/s)

0 5 10 15 20 25 30 35 40�0.8

0.0

0.8

1.6

2.4

3.2Lo

adfa

ctor

(g)

H2

H2/H1

0 5 10 15 20 25 30 35 40Time (s)

�30

0

30

60

90

120

Alti

tude

(m)




0 5 10 15 20 25 30 35 40Time (s)

�6000000

�3000000

0

3000000

6000000

Mro

ot(N

m)

0 14(b/2) 1

2(b/2) 34(b/2) b/2

Span position

0

100

200

300

400

500

Max

.Stre

ss(M

Pa)

H2

H2/H1



Gust excitation results (V = 80 m/s, Lg = 20 m)

0 2 4 6 8 10�3.0

�1.5

0.0

1.5

3.0

4.5Lo

adfa

ctor

(g)

H2

H2/H1

0 2 4 6 8 10Time (s)

0.0

0.2

0.4

0.6

Alti

tude

(m)




0 2 4 6 8 10Time (s)

�600000

�300000

0

300000

600000

900000

Mro

ot(N

m)

0 14(b/2) 1

2(b/2) 34(b/2) b/2

Span position

0

50

100

150

200

250

Max

.Stre

ss(M

Pa)

H2

H2/H1




0 5 10 15 20 25 30 35 40�1

0

1

2

3Lo

adfa

ctor

(g)

H2

H2/H1

0 5 10 15 20 25 30 35 40Time (s)

�30

0

30

60

90

120

Alti

tude

(m)




0 2 4 6 8 10Time (s)

�4000000

�2000000

0

2000000

4000000

Mro

ot(N

m)

0 14(b/2) 1

2(b/2) 34(b/2) b/2

Span position

0

80

160

240

320

400

Max

.Stre

ss(M

Pa)

H2

H2/H1

3AIAA GNC 2011Liu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 64 / 100

Control Integrated MDO

4. Control Integrated MDOBaseline geometry

X

0

10

20Y

-30

-20

-10

0

10

20

Z

0

2

High aspect ratio UAV

Property ValueFuselage length 26.4 mWing span 58.6 mWing area 196 m

2

Wing taper ratio 0.48HT span 18 mHT area 53.5 m

2

HT taper ratio 0.7VT span 4 mVT area 8.9 m

2

VT taper ratio 0.81A NACA 4415 airfoil is used.Chord-wise location of the spar is set to 45% of the chordThe optimization is performed for a cruise speed of 80 m/s



Multidisciplinary Design Optimization of a HighlyFlexible Aeroservoelastic Wing

x

0,7!1:Optimization

6 : x 2 : Swing

, AR, �i

3 : Swing

, AR, ti

5 : ⇢rigid

, ⇢elastic

7:Endurance,C

l,max

, �i

6:Function

evaluations

1,4!2:MDA

2 : ⇠target

L/D⇤ 6 : L/D

2:

3 : fe

5 : ↵o

, �e

o

W ⇤ 6 : W 4 : ⇠

3:

e1

e2

e3

t1

t2

t3

u1

u2

u3

X2

X1

d2

d1

5 : ⇠o

K⇤ 6 : A,B,C,D,K

5:

Kff

Kfb

Wg

!g

A/C

Ms

w

y

e

s

u

Ws

refu

�

�

y�

u�

Me

Ws

Wref

�

Wu

err



Aerodynamic modelling

A three-dimensional unsteady aerodynamic panel code is used toobtain aerodynamic loads.The lifting surfaces are discretized using source and doubletpanels and the wake elements are represented by doublet panels.The combination of the Volterra theory with an eigensystemrealization algorithm is used to construct a ROM.Fidkowski et al. showed that, for a transport aircraft, gustencounters at typical flight conditions can be reasonablymodelled using quasi-steady aerodynamics assumption.

F

aero

=X

j

pj nj�Aj ,

M

aero

=X

j

r�

pj nj�

�Aj ,

f ei =X

j

�i |j pj nj�AjLiu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 67 / 100


ERA Algorithm

The recorded pulse responses, are the Markov parameters(Yk = CAk�1B) that are required by the ERA technique toreconstruct the state matrices.The algorithm is based on forming the r · p ⇥ s · m generalizedHankel matrix as,

Hrs (k � 1) =

2

6

6

6

4

Yk Yk+t1 . . . Yk+ts�1

Yk+j1 Yk+j1+t1 . . . Yk+j1+ts�1...

...Yk+jr�1 Yk+jr�1+t1 . . . Yk+jr�1+ts�1

3

7

7

7

5

where

ji (i : 1 ! r � 1) and ti (i : 1 ! s � 1) are arbitrary integers.Using singular value decomposition (SVD), Hrs (0) can bedecomposed into the following form, Hrs (0) = U⌃V T ,The size of the reduced order model can be determined using therank of Hrs (0) (number of singular values, ⌃, that are greater thanthe desired accuracy).A minimum realization,⇥

⌃�1/2UT Hrs (1)V⌃�1/2, ⌃1/2VEm, ETp U⌃1/2⇤ where

ETm =

⇥

Im Om · · · Om⇤

and ETp =

⇥

Ip Op · · · Op⇤

.



Structural modelling

In the co-rotational formulation :A local coordinate system that translates and rotates with theelement to represent rigid-body motion.The local nodal variables represent the element’s strainingdeformations with respect to the local frame .This approach the geometric nonlinearities are captured throughthe rigid-body motion of the local frame and are shown in form oftransformation matrices.

X

12

14

16

Y

0

5

10

15

20

25

30

Z

0

2

4



Co-rotational axes systems

e

1

e

2

e

3

t

1

t

2

t

3

u

1

u

2

u

3

X

2

X

1

d

2

d

1

E =⇥

e1 e2 e3⇤

is the local coordinate system .e1, is placed along the line that connects the first node to thesecond node.T =

⇥

t1 t2 t3⇤

and U =⇥

u1 u2 u3⇤

are the nodalcoordinates connected to the elements two end-nodes androtate as the element deforms.



The local and global nodal variables

The local nodal variables for a 2-node 3-dimensional beam,

pTl =

�

ul ✓Tl1 ✓T

l2

ul represents the element elongation and is defined as ul =ln2�lo2

ln+lo✓

l1 and ✓l2 are pseudo-vectors that represent the rotation

transformation from the element coordinate, E , to the nodalcoordinates T and U

The global nodal variables are,

pTl =

�

dT1 ↵T dT

2 �T

↵T and �T are pseudo-vectors that represent the rotationtransformation from the inertial frame to the nodal coordinates Tand U



Stiffness matrix derivation

pl = f (p,E)

Through differentiation of the above equation the variation of thelocal and global nodal variables can be related as follows:�pl = F�p where F T =

⇥

f1 f2 · · · f7⇤

Virtual work is equal in both frames,qi = F T qil

The local internal force is related to the local displacementthrough element stiffness matrix�qil = Kl�pl

Differentiating the force equation,�qi = F T �qil + �F T qil = (Ktl + Kt�) �p

where Ktl = F tKlF and Kt��p =P

j qil (j) �fj



Aerostructural trim calculation

All derivatives and ! are set to zero, yielding,

Faero

+ Fprop

+ mCbig = 0M

aero

+ Mprop

+ X1Cbig = 0

�Kee (⇠) ⇠ + (S1)T Cbig + fe = 0

↵o

, �e

o

, ⇠o

Ltarget

MDA

1

6!22

2 : ⇠target

↵trim

, �e

trim

Aerodynamics(Trim calculator)

2

33 : f

e

⇠trim

6 : ⇠

NonlinearStructures

(Co-rotationalframework)

6

3

4

5! 4

4 : �(ST

1

Cbi

g + fe

)

5 : ⇠n+1

Internal solver(updates

displacement)

4

5

1

Aerostructural iterative solverLiu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 73 / 100


Optimization results

Optimization results with and without load alleviation system.Load alleviation Off OnSref (m2) 219.18 191.47 14.5% smallerAR 13.98 14.03L/D 34.29 34.37qelastic 1499.95 1499.88qrigid 90.63 75.71Wing mass (kg) 13,378 7,817 41.5% lighterEndurance factor 31.90 38.83 21.7% higher



Thickness and twist distribution

0 14(b/2) 1

2(b/2) 34(b/2) b/2

0.02

0.04

0.06

0.08

0.10

t spa

r(m

)

Load Alleviation: ONLoad Alleviation: OFF

0 14(b/2) 1

2(b/2) 34(b/2) b/2

Span position

�8

�6

�4

�2

0

2

�(d

eg)



Stress and maneuver constraints

0 1 2 3 4 5Time (s)

�2

�1

0

1

2

3

Load

fact

or(g

)

Load Alleviation: ONLoad Alleviation: OFF

0 14(b/2) 1

2(b/2) 34(b/2) b/2

Span position

0.0

0.6

1.2

1.8

2.4

3.0

Gus

t�m

ax(P

a)

⇥108

0 5 10 15 20 25 30 35 40Time (s)

1800

2100

2400

2700

3000

Alti

tude

(m)

0 14(b/2) 1

2(b/2) 34(b/2) b/2

Span position

0.0

0.6

1.2

1.8

2.4

3.0

Man

euve

r�m

ax(P

a) ⇥108



Contributions

Developed a unified dynamics framework for modeling andsimulating highly flexible aircraft that has the following capabilities,

Independent of the structural representation.The framework allows for integrated control synthesis.

Developed two control techniques adept at controlling highlyflexible aircraft control.Applied the framework to two design cases: 1) control systemsimply worked towards achieving or maintaining a target altitude,2) control system was also performing load alleviation.

For a HALE UAV, the use of the active load alleviation systemresulted in a 21.7% improvement in the endurance.Load alleviation system in flying wing design optimization reducedpower by 2.4%, and the structural weight is lowered by17.4%.


Fault Tolerant Flight Control of Damaged Aircraft

5. Fault Tolerant Flight Control of Damaged Aircraft

Fault Detectionsensor failuresactuator (control surface)failuresstructural damage

Fault Tolerant Control 4

Range of Tolerance 51-2) DHL (A300), 2003; 3) Japan Air (B747),

1985; 4) UA (DC-10), 1989

4Yoshi Hitachi and H. H. Liu, Canadian Aeronautics and Space Journal,vol. 56, no. 2, pp. 53–65, 2010.

5Xiaobo Li and H. H. T. Liu, ASME Journal of Dynamic Systems,Measurement, and Control, 2012 (in print)



5. Fault Tolerant Flight Control of Damaged Aircraft

changes of mass, inertia.changes of stability andcontrol derivatives.shift of Center of gravity.



Damaged Control/Stabiliity Derivatives

2

6

6

4

�Cy� �Cn� �Cl��Cyp �Cnp �Clp�Cyr �Cnr �Clr�Cy�r �Cn�r �Cl�r

3

7

7

5

= µ

2

6

6

6

4

�Cmaxy� �Cmax

n��Cmax

l��Cmax

yp �Cmaxnp �Cmax

lp�Cmax

yr �Cmaxnr �Cmax

lr�Cmax

y�r�Cmax

n�r�Cmax

l�r

3

7

7

7

5

where µ is the parameter representing the so-called damage degree.�Yv = 1

2⇢u0S�Cmaxy� �Yp = 1

4⇢u0bS�Cmaxyp

�Yr =14⇢u0bS�Cmax

yr �Y�r =12⇢u2

0S�Cmaxy�r

�Lv = 12⇢u0bS�Cmax

l� �Lp = 14⇢u0b2S�Cmax

lp�Lr =

14⇢u0b2S�Cmax

lr �L�r =12⇢u2

0bS�Cmaxl�r

�Nv = 12⇢u0bS�Cmax

n��Np = 1

4⇢u0b2S�Cmaxnp

�Nr =14⇢u0b2S�Cmax

nr �N�r =12⇢u2

0bS�Cmaxnr



Modeling of Damaged Aircraft

Linear parameter dependent model:Define x(t) =

⇥

u w q ✓ v p r �⇤0 and

u(t) =⇥

�e �f �a �r⇤0

Healthy Aircraft:x(t) = Ax(t) + Bu(t)

Damaged aircraft:

x(t) = (A � µA)x(t) + (B � µB)u(t), µ 2 [0, 1]

where µ is the damage degree, which is related to the percentage oftail loss.

µ = 0 represents un-damaged caseµ = 1 represents the complete loss of vertical tail0 < µ < 1 represents partial loss of vertical tail

and look for the maximum tolerable damage degreeLiu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 81 / 100


Quadratic stabilizationPerformance requirement: maintain stabilityIn theory, a linear parameter-dependent system can be written as a polytopic system.Polytopic description:

x(t) = Ai x(t) + Bi u(t) 8 i = 1, 2, · · · ,N.

The N systems (45) can be quadratically stabilized, if there exists a positive definite matrix Q(Q > 0) and Y such that the following LMIs are satisfied(Boyd1994),

Ai Q + QA0i + Bi Y + Y 0B0

i < 0 8 i = 1, 2, · · · ,N.

controller isK = YQ�1

The aircraft system of our study can be treated in an N = 2 polytopic format to present healthyand faulty dynamics respectively,

A1 = A, A2 = A � µA, B1 = B, B2 = B � µB.

In order to set limits to control surfaces, the following LMI conditions with respect to Q and Y(Boyd1994) are also included

Q � W (45)

Q Y 0

Y ⇢2I

�� 0 (46)

where W = x(0)x(0)0 reflects the upper bound of the initial states, and ⇢ represents the actuatorlimits. Liu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 82 / 100


Bisection algorithm

Note: feasibility of LMI () desired performance is satisfied.

Figure 2: Bisection Algorithm



Case Study: Boeing 747

Altitude 20,000 ft CL 0.40 Cl�

-0.16

Air density 0.001268 slug/ft3 CD 0.025 Clp -0.34Speed 673 fps CT 0.025 Clr 0.13

Wing area 5500 ft2 Cmu 0.013 Cl�a 0.013

Wing span 196 ft Cm↵

-1.00 Cl�r 0.008

Wing mean chord 27.3 ft Cm↵

-4.00 Cn�

0.16Weight 636,636 lbs Cmq -20.5 Cnp -0.026

Ixx 182⇥106 slug/ft2 CLu 0.13 Cnr -0.28Iyy 33.1 ⇥ 106 slug/ft2 CL

↵

4.4 Cn�a 0.0018

Izz 49.7 ⇥ 106 slug/ft2 CL↵

7.0 Cn�r -0.100

Ixz 0.97 ⇥ 106 slug/ft2 CLq 6.6 Cy�

-0.90Air velocity 673 fps CD

↵

0.20 Cyp 0Thrust 9870 lb CDU

0.0 Cyr 0Air density 0.001268 slug/ft3 CL

�e 0.32 Cy�a 0

Pressure ratio 0.4695 - CD�e 0.0 Cy

�r 0.12Cm

�e -1.3

�Cmaxy�

-0.480 �Cmaxl�

-0.0577 �Cmaxn�

0.3029

�Cmaxyp -0.1154 �Cmax

lp-0.0194 �Cmax

np 0.0729�Cmax

yr 0.06058 �Cmaxlr

0.0887 �Cmaxnr -0.3826

�Cmaxy�r

0.12 �Cmaxl�r

0.008 �Cmaxn�r

-0.100



Comparison of the maximum tolerance

Control strategy Critical damage degreeLQR 0.101

Quadratic stabilization 0.205Guaranteed cost control 0.158

Guaranteed cost control with robust pole placement 0.106



Simulation Results - LQR

0 50 100 150 200672

674

676

678

680

682

684

Time (sec)

u (

fps)

µ=0

µ=0.05

µ=0.10

(a) u vs t

0 0.5 1 1.5 2 2.5 3 3.5 4−15

−10

−5

0

5

10

15

Time (sec)

v (

fps)

µ=0.0

µ=0.05

µ=0.10

µ=0.15

(b) v vs t

0 50 100 150 200−4

−3

−2

−1

0

1

2

3x 10

16

Time (sec)

w (

fps)

µ=0

µ=0.05

µ=0.10

µ=0.15

(c) w vs t

0 20 40 60 80 100−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

29

Time (sec)

! (

rad

)

µ=0

µ=0.05

µ=0.10

µ=0.15

(d) � vs tLiu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 86 / 100


Simulation Results - QS

0 50 100 150 200664

666

668

670

672

674

676

678

680

682

684

Time (sec)

u (

fps)

µ=0

µ=0.05

µ=0.10

µ=0.15

(e) u vs t

0 0.5 1 1.5 2 2.5 3 3.5 4−3

−2

−1

0

1

2

3

4

Time (sec)

v (

fps)

µ=0.0

µ=0.05

µ=0.10

µ=0.15

(f) v vs t

0 50 100 150 200−1.5

−1

−0.5

0

0.5

1

Time (sec)

w (

fps)

µ=0

µ=0.05

µ=0.10

µ=0.15

(g) w vs t

0 20 40 60 80 100−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

Time (sec)

! (

rad

)

µ=0

µ=0.05

µ=0.10

µ=0.15

(h) � vs tLiu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 87 / 100

Active Flutter Suppression

6. Active Flutter Suppression

flutter is an aeroelastic instability phenomenon caused by thecoupling between aerodynamics, structural dynamics and inertialforces as the speed of aircraft increases, which can lead to failureon aircraft wings.traditional preventive technique which adds weight penalty on thestructure designactive control techniques provide a promising alternative to ensurethe safety of aircraft and expand the flutter boundary.a preliminary work6

active flutter suppression design using linear parameter-varying(LPV) controlfor a nonlinear aeroservoelastic model that captures wing-sectionstall flutterThe loss-of-effectiveness fault is considered for the control surface.

6W. Fan, HHTL, R.Kwong, ASME DSCC 2014Liu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 88 / 100


Wing-Section Model

• the plunge motion h, the pitch motion✓ and the trailing-edge flap deflection�, which is used as the control sur-face.

• two linear spring constants are de-noted by kh and k✓ respectively.

• the points P, C, and Q denote theelastic center, the center of mass,and the aerodynamic center, respec-tively.

• the dimensionless parameters e anda represent the respective locationsof the points C and P from the beamreference line.



Equations of Motion

m mbx✓mbx✓ Ip

�

h✓

�

+

kh 00 k✓

�

h✓

�

=

�LT0 � L�

MT0 + M�

�

where m is the airfoil mass, Ip represents the moment of inertia aboutthe elastic axis, x✓ = e � a is the dimensionless distance between thecenter of mass and elastic axis. To account for the dynamic stall, theONERA model is used. The corrected unsteady lift and momentbecome

LT0 = ⇢U20 bCl0 + L0 + ⇢UT�L � CdUT (h + ab✓ � �0)

MT0 = 2⇢U20 b2Cm0 + M0 + 2⇢UT b�M

Aerodynamic lift and moment generated by the deflection of thetrailing-edge control surface � are

L� = 2⇡⇢bc1U2

M� = (12+ a)bL� + 2⇡⇢b2c2U2�



LPV Control

H2 gain-scheduled state-feedback control is designed for the LPVmodel subject to dynamic disturbances and control surfaceeffectiveness loss. Consider the following LPV system

x = A(U)x + B(✏)u + B1d

y = Cx

where d is exogenous disturbance signal, B1 = I , C = I . Let X (U, ✏)denote a parameter-dependent Lyapunov matrix. The H2 norm of theclosed-loop system is less than a prescribed value � if there existX = X

T and W = W

T satisfying

X (U, ✏) + AT

clX (U, ✏) + X (U, ✏)Acl X (U, ✏)BclBT

clX (U, ✏) �I

�

< 0

X (U, ✏) CT

clCcl W

�

> 0

Trace(W ) < �2



Simulation Results

NACA 0012 airfoil is chosen for study in this paper. The first LCOsare observed to emerge at U = 10.6m/s, which is lower than thelinearly estimated flutter point U = 11m/s.An LPV controller is synthesized over a grid region of schedulingparameters. The grid points are taken at speeds 10, 11, 12, 13, 14and 15m/s and control surface effectiveness factors 1, 0.8 and0.5. The bound of speed variation rate ⌫ = 1m/s

2. The LMIs aresolved for these grid points simultaneously. The resulting H2performance index is � = 8.207.Control input constraint is considered with the limit of controlsurface deflection set at ±15�.



0 2 4 6 8 10 12 14 16 18 20−0.5

0

0.5

1

Plun

ge (m

)

Closed−loop Open−loop

0 2 4 6 8 10 12 14 16 18 20−0.5

0

0.5

1

Pitc

h (ra

d)

Closed−loop Open−loop

0 2 4 6 8 10 12 14 16 18 20−10

0

10

Time (s)Flap

Def

lect

ion

(deg

)

Plunge and pitch responses at U=10.8 m/s when ✏ = 1.0



0 2 4 6 8 10 12 14 16 18 20−0.2

00.20.4

Plun

ge (m

)

Designed controller LQR controller

0 2 4 6 8 10 12 14 16 18 20−0.2

00.20.4

Pitc

h (ra

d)


0 2 4 6 8 10 12 14 16 18 20−5

0

5

Time (s)Flap

Def

lect

ion

(deg

)

Plunge and pitch responses at U=12.6 m/s when ✏ = 0.5



0 2 4 6 8 10 12 14 16 18 20−0.5

00.5

11.5

Plun

ge (m

)


0 2 4 6 8 10 12 14 16 18 20

0

0.5

1

Pitc

h (ra

d)


0 2 4 6 8 10 12 14 16 18 20

−10

0

10

Time (s)Flap

Def

lect

ion

(deg

)

Plunge and pitch responses at U=15 m/s when ✏ = 0.5Liu (UTIAS.FSC) www.flight.utias.utoronto.ca 05-Aug-2014 95 / 100

Integrated Flight Control and Simulation

7. Integrated Flight Control and SimulationA CRJ200 Example

The 50-seat Canadair Regional Jet 200 (CRJ200) series wasannounced by Bombardier in 1995 and introduced into service in 1996.The design of the CRJ200, which evolved from the Challenger 604business jet, is for regional airline operations. The aircraft is equippedwith high efficiency engines giving a range up to 3,050km. The wingsare transonic and are fitted with winglets for efficient high-speed flight7.In this project, we provide a CRJ200 type aircraft model that is alsoimplemented in a flight training device (FTD). The developmentconsists of the following steps.

to develop an altitude-hold autopilot to command a steady climbfrom 33,000ft to 35,000ft, andto develop a heading-hold autopilot to command a steady 90degree turnrequirements: 1) overshoot 10%; 2) steady-state error 5%

• full nonlinear flight dynamics• stability derivatives estimation• linearization and control de-

sign• nonlinear simulation• flight training, cockpit checklist• flight simulation7www.aerospace-technology.com/projects/crj200/



Simulation Block



Flight Simulation Results


Acknowledgement

Acknowledgement

Dr. Sohrab Haghighat (PhD 2012)Research Staff, MRI, Boston

Dr. Xiaobo Li (PDF 2011)

Dr. Ruben Perez (PhD 2007)Assist. Professor, RMC

Dr. Anton de Ruiter (PhD 2005)Assist. Professor, Ryerson U.

Ms. Wen Fan (PhD Candiate 2013 - )

Ms. Yan Zhang s (BUAA, Visiting 2009)

Dr. Peng Zhang O (HIT, Visiting 2009-11)


Acknowledgement

Thank You

Prof. Hugh H.T. LiuUniversity of TorontoInstitute for Aerospace Studies4925 Dufferin StreetToronto, OntarioCANADA M3H 5T6Tel: 416-667-7928Email: [email protected]


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