ControllabilityHarry G. Kwatny

Department of Mechanical Engineering & MechanicsDrexel University

Outline Affine Systems Controllability Controllability Distributions Controllability Rank Condition

Examples

Affine Systems

1( ) ( ) ( ) ( )

( ), ,

m

i ii

n p m

x f x G x u f x g x u

y h xx R y R u R

=

= + = +

=

∈ ∈ ∈

∑

Controllability xf is U-reachable from x0 if given a neighborhood U of x0 containing xf,

there exists tf>0 and u(t) on [0,tf] such that x0 goes to xf along a trajectory contained entirely in U.

The system is locally reachable from x0 if for each neighborhood U of x0 the set of states U-reachable from x0 contains a neighborhood of x0. If the reachable set contains merely an open set the system is locally weakly reachable from x0 .

The system is locally (weakly) controllable if it is locally (weakly) reachable from every initial state.

R. Hermann and A. J. Krener, "Nonlinear Controllability and Observability," IEEE Transactions on Automatic Control, vol. 22, pp. 728-740, 1977

0x

0x

neighborhood

Openset

Controllability Distributions{ }

{ }1 1

1 1

, , , span , , ,

, , , span , ,O

C m m

C m m

f g g f g g

f g g g g

∆ =

∆ =

{ }{ } { }

{ }

, satisfy

span

a regular point of span ( ) span ( ) ( )

if and span are of constant dim, then

dim dim 1

O

O

O O

O O

O

C C

C C

C C C

C C

C C

f

x f x f x x

f

∆ ∆

− ∆ + ⊆ ∆

− ∆ + ⇒ ∆ + = ∆

− ∆ ∆ +

∆ − ∆ ≤

Controllability Rank Condition

Proposition:A necessary and sufficient condition for the system to be locally weakly controllable is

A necessary and sufficient condition for the system to be locally controllable is

( )0 0dim , nC x n x R∆ = ∀ ∈

( )0 0 0dim , n

C x n x R∆ = ∀ ∈

Example: Linear System Controllability

[ ]

[ ]

( ) , ( ) , 1, ,

,

, 0 ,

,

0

,

,

n m

i i

ii i i

i j

f x Ax g x b i mb AxAx b Ax

x Ax Bu x R u

b Abx x

b b A

R

x Ax

= = =∂ ∂

= − =

= + ∈

∂ ∂ = =

∈

Example: Linear System Continued{ }

{ }

{ }{ }0

0

1

1

1

span

span

span

span

kk

nC

B

B AB

B AB A B

CH Thm B AB A B

−

−

∆ =

∆ =

∆ =

− ⇒ ∆ =

{ }

{ }

0

1

span ,

span , , , nC

Ax B

Ax B AB A B−

∆ =

∆ =

[ ]0

1 11

1

,

stop when

qk k i ki

k k

τ− −=

−

∆ = ∆

∆ = ∆ + ∆

∆ = ∆∑

Example: Wheeled Robot

cos 0sin 0

0 1

x

xvd y

dt

θθ

ωθ

=

driverotate

θ

x

y

body fixed frame

space frame (x,y)

xv

caster wheel

independent drive wheels

d

dlF

rF

Wheeled Robot 3

{ }0

2

23

1 0span drive,rotate tan , 0

1

1 0 0tan , 0 sec

1 0

0span drive,rotate, sec

0I

θθ

θ θθ

θ

∆ = =

=

=

sideslip

1

2

3

45

6

6

Implementing Lie Bracket

1v

2v

Add

Example: Extended Wheeled RobotThe extended system includes dynamics and , :

0 0cos 0 0sin 0 0

0 00 0 10 1 0

x

x

x

z F z u

x vy v udv z Tdt

z

θ ωθθ

ω

= =

= +

wheeledvehicle

1s

u F

T

xyθ

Example: Extended Wheeled Robot - 2

0

0 0

1 0 00 1 0

span , , , , rank 6 controllable at generic state

0 0 1

1 0 0 00 1 00 tan 0

span , , ,0 0 10 0 00 0 0

C

C x

θ=

∆ = ⇒

∆ =

00 00 0

, , rank 5 uncontrollable at 00 01 00 1

x

⇒ =

Actually, this is true not only at the origin,but at any point with and both zero.For generic points with only 0,the system is controllable.

x

x

vvω

=

Example: Parking

x b

y b

φ

(x,y)

v

θ

x

y

steering & drive wheel

body fixed frame

space frame 1

2

cos( ) 0sin( ) 0

sin 00 1

xuydudt

φ θφ θ

φ θθ

+ + =

drive

steer

1 2,u v u ω= =

Parking, Continued

{ }

{ }

( )( )( )

( )( )( )

1 1

1 1

, , , span , , ,

, , , span , ,

0 0cos cos0 0sin sin

, span ,0 0sin sin1 10 0

O

O

C m m

C m m

C C

f g g f g g

f g g g g

φ θ φ θφ θ φ θθ θ

∆ =

∆ =

+ + + + ∆ = ∆ =

Example: Parking, new directions from Lie bracket

[ ]

[ ]

sin( )cos( )

,cos

0

sincos

,00

wriggle steer drive

slide wriggle drive

θ φθ φθ

φφ

− + + = =

− = =

Parking, Continued( )( )( )

0 sin( ) sincos0 cos( ) cossin

span , , ,0 cos 0sin1 0 00

1 0 0 00 1 0 0

, , ,0 0 1 00 0 0 1

θ φ φφ θθ φ φφ θθθ

− + − + ++

Recall Lie Bracket Interpretation as Commutator of Flows

Let us consider the Lie bracket as a commutator of flows. Beginning at point xin M follow the flow generated by v for an infinitesimal time which we take asfor convenience. This takes us to point

Then follow w for the same length of time, then -v, then -w. This brings us to apoint ψ given by

ε

xy )exp( vε=

( , )x e e e e xε ε ε εψ ε − −= w v w v

Example: Parking, implementation

(x,y)

θ

x b

y b

φ

v

wriggle=steer+drive-steer-drive

slide=wriggle+drive-wriggle-drive

More Controllability Distributions

{ }{ }0

0 1

span , ,1 ,0 1

span ,

( ) , ,

1 ,

(

0 1

)

k

k kv v v

L f i

kL f i

f ad g i m k n

ad g i m k

ad w ad v ad w

n

w − = =

∆ = ≤ ≤ ≤ ≤ −

∆ = ≤ ≤ ≤ −

≤

0 0

C L

C L

weak local controllability n n

local controllability n n

⇔ ∆ = ⇐ ∆ =⇑ ⇑ ⇑

⇔ ∆ = ⇐ ∆ =

What is missing in these earlier attemptsto obtain a controllability determinationis any Lie Brackets between fields , . They are sufficient but not necessaryconditions for controllability.

i jg g

Example: Linear Systems Revisited

{ }{ }0 0

1

1

( ) , ( )

span , , , ,

span , , ,

nC L

nC L

f x Ax G x B

Ax B AB A B

B AB A B

−

−

= =

∆ = ∆ =

∆ = ∆ =

Controllability Hierarchy

( ) ( )

( ) ( )0 0

0 0

0 0

1

dim dim

dim dim

dim

C L

C L

n

weak local controllability x n x n

local controllability x n x n

linear controllability B AB A B n−

⇐ ∆ = ⇐ ∆ =⇑ ⇑ ⇑

⇐ ∆ = ⇐ ∆ =⇑

⇔ =

{ }

{ }

{ }{ }0

1 1

1 1

, , , span , , ,

, , , span , ,

span , ,1 ,0 1

span ,1 ,0 1

O

C m m

C m m

kL f i

kL f i

f g g f g g

f g g g g

f ad g i m k n

ad g i m k n

∆ =

∆ =

∆ = ≤ ≤ ≤ ≤ −

∆ = ≤ ≤ ≤ ≤ −