Flight Dynamics & Control

Eigenvalue-Eigenvector Assignment

Harry G. KwatnyDepartment of Mechanical Engineering & Mechanics

Drexel University

OutlineState Feedback

Pole placement revisitedEigenvector assignment

Example: F-16Example: L1011

State Space Design

Consider the usual linear plant

find an output feedback control that shapes the transient response ofThe feedback stabilization pro

the closed loop to meet pres

bl

cribed obj

em:

ectives. T

x Ax Buy Cx Du

= += +

&

( )( ) ( )

his will be done in three steps:1) Design a state feedback control,

ˆ2) Design a state estimator, that generates state estimates

from available information, i.e., , [0, )

3) Implement the comp

u Kxx t

u y tτ τ τ

=

∈

( ) ( )ˆosite controller, u t x t=

x Ax Buy Cx Du

= += +

&0r =

Pole Assignment Problem

( )

{ } { }1 1

Given a self conjugate set of scalars , , and vectors

, , completely controllable

, , find a real matrix such that the eigenvalues

, ran

and eige

Pole assignment

nvecto

prok

blem:

n nv

x Ax Bu A B B

v n

m

m Kλ λ… ×

= + =

K

&

{ }1

rs of ( ) are precisely the given sets.

The system is controllable if and only iffor every self-conjugate set of scalars , , there exists aTheorem (Wo

real

matrix such that

nham, 19

(

67):

n

A BK

m n K

λ λ

+

…

× { }1) has , , as its eigenvalues. nA BK λ λ+ …

Some Definitions

[ ]

Define the matrices : [ - | ]

: {columns form a basis for ker[ ]} , ,

Note:controllability dim kerrank rank

n k m k

S I A BN

R S N R M RM

S nB m N m

N N

λ

λλ λ λ λ

λ

λ

λ

λλ

λ

× ×

=

⎡ ⎤= = ∈ ∈⎢ ⎥

⎣ ⎦

⇒ =

= ⇒ =

=

Main Result on Pole Assignment

{ }

( )

{ }

Let , 1, , be a set of self-conjugate scalars. There exists a real matrix such that

, 1, ,if and only if

1)

Theorem (Moore

, 1, , are linearly independent

2) , whe

1976): i

i i i

i

i j

i nm n K

A BK v v i n

v i n

v v

λ

λ

=

×

+ = =

=

=

K

K

K

n

3) Im

Also, if exists and rank then is unique.i

i j

iv N

K B m Kλ

λ λ=

∈

=

Proof: Necessity

( )( )

[ ] 0

0 Im Imi i i

i i i

i i i

ii

i

i ii

i i

A BK v v

I A v BKv

vI A B

Kv

v vS R v N

Kv Kvλ λ λ

λ

λ

λ

+ =

⇒ − = −

⎡ ⎤⇒ − =⎢ ⎥−⎣ ⎦

⎡ ⎤ ⎡ ⎤= ⇒ ∈ ⇒ ∈⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

Proof: Sufficiency, 1{ }

( )( )

k

assume the set , 1, , satisifies 1), 2), 3)

3) there exists C such that

By definition 0

0

0

Suppose, can be chosen such that

Then, it wo

i

i i

i i

i i

i

i

i i i

i

i i i

i i

v i n

z v N z

S R

I A N BM

I A N z BM z

K M z Kv

λ

λ λ

λ λ

λ λ

λ

λ

λ

=

⇒ ∈ =

= ⇒

− + =

− + =

= −

K

( )

[ ]11 1

uld follow that 0.

Thus, real is to be chosen so that

n

i

n

i

n

I

K v v M z M

A BK v

K

zλ λ

λ

⎡ ⎤= −

− + =⎡ ⎤⎣

−

⎦

⎣ ⎦L L

Proof: Sufficiency, 2

[ ]1

11 1

Assumption 1) implies that this is always possible. If the 's are real, we simply compute

If some 's are complex, proceed as follows for each complex

conjugate pair. Supp

n

i

n n

i

K M z M z v vλ λ

λ

λ

−⎡ ⎤= − −⎣ ⎦L L

1 2 1 2 1 2ose so that from 2) .For simplicity suppose all other eigenvalues are real. Define

: ,

and use the notion, for any complex quantity . Thenii i

R I

v v z z

w M z

a a jaλ

λ λ= = ⇒ =

=

= +

Proof: Sufficiency 3[ ][ ]

[ ]

1

3

3

1 1

1 1 1 1 3 1 1 1

1 1

1 3

1 1 3 1 1 3

post multiply by the nonsingular matrix1/ 2 1/ 2

01/ 2 1/ 2

0to obtain

n

n

n

R

n n

R I R I n R I R I n

R I n R I n

K w w

K v v M z M z

K v jv v jv v v w jw w jw M z M z

jj

I

K v v v v w w M z M z

λ λ

λ λ

λ λ

⎡ ⎤= − − ⇒⎣ ⎦⎡ ⎤+ − = + − − −⎣ ⎦

−⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤= − −⎣ ⎦

=

L L

L L

L L

[ ]( )

3

13 1 1 3

Finally, since a fixed eigenstructure uniquely defines , it canbe proved that is unique when rank .

nI n R I nM z M z v v v

A BKK B m

vλ λ−⎡ ⎤

+

− −⎣ ⎦

=

L L

Geometry

{ }

{ }

1 2

1 2 1 1 2

1

A subset of the linear space (over field ) is a linear subspace of if , and

, ,If ( 1, , ), then span , , is a

subspace of . then

,

&

x

i k

x xc c c x c x

x i k x x

r s r s

x x

•∀ ∈

∀ ∈ + ∈

• ∈ =

• ⊂

+ = + ∈ ∈

∩ = ∈

S XX SS

XX

R,S X

R S R S

R S R

F

FK K

{ }Two subspaces are independent if 0

x ∈

• ∩ =

S

R,S R S

Geometry 2

1

1

If , 1, , are independent subspaces, then the sum

is called an indirect sum and may be written

The symbol presuposes independence.Let = . For each , there are unique ,

i

k

k

i k

x r

• == + +

= ⊕ ⊕⊕

• ⊕ ∈ ∈

RR R R

R R R

X R S X R

K

L

L

so that . This implies a unique function calledthe projection on along .

sx r s x r

∈= +

S

R Sa

Geometry 3

( )( )

( ) 2

2

The projection is a linear map : , such that Im and ker , and

Note that is the projection on along . Thus,

0

Conversly, for any map : such that Im ker

i.e.

QQ Q

Q I Q

I Q

Q I Q Q Q

Q Q QQ Q

• →= =

= ⊕ −

• −

− = ⇔ =

• → == ⊕

X XR S

X X X

S R

X X X, is the projection on Im along ker .Q Q Q

Geometry 4

( )( ) ( ) ( )

( )( )( )

1

1 1 12

1

1

Im Im ker ker Im ker

is the projection on Im along ker

is the projection on ker along Im

Q B CB C

Q B CB CB CB C B CB C QQ B Q C X B C

B CB C B C

I B CB C C B

−

− − −

−

−

=

= = =

= = = ⊕

−

XIm B

kerC

Example: F-16 landing approach

[ ]

0.507 3.861 0 32.17 00.00117 0.5164 1 0 0.07170.000129 1.4168 0.4932 0 1.645

0 0 1 0 0

0 0 1 0

E

u u

q q

u

yq

α αδ

θ θ

α

θ

− − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

&

&

&&

0.999978 00.000484 0.0002676

phugoid: 0.0438167 0.2064610.001343 0.00022640.000272 0.0064497

0.9942870.063373

short period: 1.7036, 0.7309370.0740730.043481

j h j

h

λ

λ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= − ± = ±⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

−⎡ ⎤⎢ ⎥−⎢ ⎥= − =⎢⎢−⎣ ⎦

0.9995080.014171

,0.0165070.022584

⎡ ⎤⎢ ⎥−⎢ ⎥

⎥ ⎢ ⎥−⎥ ⎢ ⎥−⎣ ⎦

Example: F-16 state feedback

1 1

1,2

3,4

Desired poles -short period: 1.25 2.16506phugoid: 0.01 0.0994987

0.942209 00.049274 0.0482930.088225 0.135264Re , Im0.029211 0.057615

0.260982 0.095482

jj

R Rλ λ

λ

λ

= − ±

= − ±

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−= =⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦

[ ]

3 3

0.999992 00.001453 0.0001170.000320 0.000077Re , Im

0.001090 0.0031070.001428 0.000104

0.004076 3.87578 0.718424 0.095189

R R

K

λ λ

⎥

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −= =⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

=

Example: F-16 Rynaski “robust observer”

1 2 3

"place observer poles at LHP plant zeros, remainder are placed arbitrarily"

0, 0.04231, 0.5865,0 0.0009340 0.0067330 0.000293, ,

0.707107 0.7105150.707107 0.703649

1

R R R

λ = − −

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

−

[ ]

4

0.001445 0.0008630.665243 0.731830.028975 0.247431,

0.079296 0.0279340.741837 0.634367

0.168343 1.02106 0.56851 1T

R

L

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= − − −

Example: F-16

( ) ( )( )( )( )( )

( ) ( )( )( )( )( )

2

0.0423101 0.5865431.645

0.730937 1.7036 0.0876334 0.044546

2.45962 0.0148335 0.1475084.46035

0.423102 0.586577 2.45962

p

c

s s sG s

s s s s

s s jG s

s s s s

+ +=

− + + +

+ + ±=

+ + +

0.01 0.05 0.1 0.5 1 5 10radêsec

0

10203040

50

dB

0.01 0.05 0.1 0.5 1 5 10MAGNITUDE

Example F-16

-3 -2 -1 0 1Re@sD

-3

-2

-1

0

1

2

3

sD

0

1

-0.4 -0.2 0 0.2 0.4Re@sD

-0.2

-0.1

0

0.1

0.2

0

1

F-16 CCVThe first YF-16 (72-1567) was rebuilt in December 1975 to become the USAF Flight Dynamics Laboratory's Control Configured Vehicle (CCV). CCV aircraft have independent or "decoupled" flight control surfaces, which make it possible to maneuver in one plane without movement in another--for example, turning without having to bank. The CCV YF-16 was fitted with twin vertical canards added underneath the air intake, and flight controls were modified to permit use of wing trailing edge flaperons acting in combination with the all moving stabilator. The YF-16/CCV flew for the first time on March 16, 1976, piloted by David J. Thigpen. On June 24, 1976, it was seriously damaged in a crash landing after its engine failed during a landing approach. The aircraft was repaired and its flight test program was resumed. The last flight of the YF-16/CCV was on June 31, 1977, after 87 sorties and 125 air hours had been logged.

F-16 AFTIThe Flight Dynamics Laboratory of the Air Force Systems Command sponsored an Advanced Fighter Technology Integration (AFTI) program. In 1979, General Dynamics was awarded a contract to convert the fifth FSD F-16A (75-0750) into an AFTI aircraft. It capitalized on the experience gained with the CCV (Control Configured Vehicle) F-16 (72-1567). The AFTI F-16 was fitted with twin canard surfaces mounted below the air intake, these surfaces having been taken from the CCV/F-16. It had a full-authority triplex Digital Flight Control System (DFCS) and an Automated Maneuvering Attack System (AMAS). This system provides six independent degrees of freedom. It was designed to be fault tolerant, so that no single failure should affect correct operation. In the event of a second fault, the system reverts to a standby condition which will permit safe flight to continue. The system incorporates an analog backup flight-control system. The AFTI first took to the air July 10. Phase I testing involved the demonstration of direct translational maneuvering capability. Phase II testing (1984-87) involved F-16C-standard avionics with AMAS. The AMAS enabled the AFTI/F-16 to translate in all three axes at a constant angle of attack and to be pointed up to six degrees off the flight vector. In recent years, the AFTI/F-16 became associated with close air support (CAS) studies, some of them conducted by NASA. These studies began in 1991.

Multimode, High Maneuverability Flight Control

Sobel & Shapiro, 1985Longitudinal

Pitch pointing/ vertical translation - command the pitch angle without a change in flight path angleDirect lift – command normal acceleration (or flight path angle rate) without affecting angle of attack

LateralYaw pointing/ lateral translation– decouple the lateral directional response from roll (bank) angle and rate and yaw rateDirect sideforce – command lateral acceleration without a change in sideslip angle

Example: F-16 CCV

F-16 CCV – pitch pointing

Objectives: command the pitch angle while maintaining the flight path angleStabilize short period mode, ζ=0.8, ω=7 rad/s

Measured variables: pitch rate, normal acceleration (at pilot station), flight path angle, surface deflections

F-16 CCV – pitch pointingReplace θ by γ+α, so that θ equation is replaced by θ equation.Choose eigenvectors in an attempt to decouple pitch rate and flight path angle.

F-16 CCV pitch poining

Example L-1011, Shapiro & Chung, 1983

.117 .000295 .996 .0386 .02 05.2 1.0 .249 0 .337 1.12

1.54 .0042 .154 0 .744 .0320 1 0 0 0 0

r

a

p pdr rdt

β βδδ

φ φ

− − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − − ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= + ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − − − ⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Eigenvalue Desired Eigenvalue

Dutch Roll -.0882 ± i 1.2695 -1.5 ± i 1.5Roll Subsidence -1.0855 -2.0 ± i 1.5Spiral -.0092

1 0 00 0 1

Dutch roll: roll:1 0 00 0 1

xx

xx

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦