Differential Geometric Approach
• Introduction
Vector field : A mapping f : D  R n where D  R n is a domain,
is said to be a vector field. (n - dim column ve ctor)
Covector field : A transpo se of a vector field is said to be a covector field.
(n - dim row vector)
n
Inner Product :  w, f  w( x) f ( x)   wi ( x) fi ( x)
i 1
where w is a covector field and f is a vector field.
Differential(gradient) : Let h : D  R. The differenti al of h is a covector
field, i.e., dh  hx  [ xh1 
(h)
h
x n
]
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Lie Derivative
Lie derivative: Let h : D  R and f : D  R n . The Lie derivative of h with
respect f or along f , is defined by L f h( x)  hx f ( x)
the directional derivative of
h along the direction of f
Let : L0f h( x)  h( x)
Lif h( x)  L f ( Lif1h( x))  ( Lif1h( x)) f ( x)
Ex:
x  f ( x)  g ( x)u
y  h( x )
 scalar function
If Lg Lif h( x)  0 for 0  i  r  2
r 1
g f
L L h( x )  0
x  D0
10-28
Lie Derivative (Continued)
then the system has relative degree r in D0
y  hx  h( f  gu )  L f h  Lg hu

y r  ( Lrf1h) x  ( Lrf1h)( f  gu )  Lrf h  Lg Lrf1hu
0
Lie bracket (Lie product) : Let f and g be two vector fields on D  R n .
Then Lie bracket of f and g , [ f , g ] is a vector field
g
f
defined by [ f , g ]( x)  x f ( x)  x g ( x)
 gf  fg  L f g  Lg f
Let ad g ( x)  g ( x)
0
f
where
g f
x x
,
are Jacobian matrix
ad f g ( x)  [ f , g ]( x)
ad kf g ( x)  [ f , ad kf 1 g ]( x)
10-29
Example
Ex:
x2


0
Let f ( x)  
,
g


x 
 sin x1  x2 
 1
x2
1  0 
  0
0 0  
Then [ f , g ]  

 
 

1 0  sin x1  x2   cos x1  1  x1 
  x1 

  ad f g
x

x
 1 2
ad 2f g  [ f , ad f g ]
x2
1    x1 
  0
  1 0 


 



 1 1  sin x1  x2   cos x1  1  x1  x2 
 x1  2 x2




 x1  x2  sin x1  x1 cos x1 
 If f and g are constant [ f , g ]  0
10-30
Lemma
Lemma: Lie brackets have the following properties
(i) Bilinearity : Let f1 , f 2 , g1 , g 2 be vector fields and  1 ,  2 be real numbers. Then
[ 1 f1   2 f 2. , g1 ]   1[ f1 , g1 ]   2 [ f 2 , g1 ]
[ f1 ,  1 g1   2 g 2 ]   1[ f1 , g1 ]   2 [ f1 , g 2 ]
(ii) Skew commutativity : [ f , g ]  [ g , f ]
(iii) Jacobi identity : If f and g are vector fields and h is a real valued
function. Then
L[ f , g ]h( x)  L f Lg H ( x)  Lg L f H ( x)
i.e., h [ f , g ]  ( Lg h) f  ( L f h) g
Proof: See Chapter 6.2 in Applied Nonlinear Control.
10-31
Diffeomorphism
Diffeomorphism : A mapping T : D  R n is a diffeomorp hism if it is invertible
on D, i.e., T 1 ( x) such that T 1 (T ( x))  x for all x  D, and
T ( x), T 1 ( x) are continuous ly differenti able.
Lemma:
If the Jacobian matrix
T x  is
nonsingula r at a point x0  D, then
T ( x) defines a local diffeomorp hism in a subregion of D.
T is a global diffeomorp hism if it is a diffeomorp hism on R n and
T (Rn )  Rn.
10-32
Example
Lemma:
T is a global diffeomorp hism iff
(i)
Tx  is
nonsingula r for all x  R n
(ii) lim T ( x)  
x 
Ex:
 e x1 
Let f ( x)    x 
1
 x2e 
Then
f
x
 e x1
  x
1
 e x2
0 
 f 

det
 1  0,
 x1 



x
e 
 
x  R 2
However f ( R 2 )  R 2 (Note that f1  e x1  0, x1 )
10-33
Distribution
Distribution : Let f1 ,, f k be vector fields on D  R n . At any fixed point
x  D, f1 ( x), f 2 ( x),, f k ( x) are vectors in R n and
( x)  span{ f1 ( x), f 2 ( x),, f k ( x)}
is a subspace of R n . To each point x  R n , we assign a subspace
( x). We refer to this assignment by
  span{ f1 , f 2 ,, f k }
which we call a distributi on. In other wor ds,  is the collection
of all vector spaces ( x) for x  D.
Note that dim( ( x))  rank { f1 ( x), f 2 ( x),, f k ( x)} may vary with x.
If   span{ f1 ,, f r } where { f1 ,, f r } are linearly independen t for
all x  D, then dim( ( x))  r for all x  D. Then every g   can
be expressed as
r
g ( x)   ci ( x) f i ( x), ci ( x) : smooth function
i 1
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Involutive distribution
Involutive distribution : A distributi on  is involutive if g1   and g 2  
 [ g1 , g 2 ]  
Ex: Let D  R 3 ,   span{ f1 , f 2 } where
2 x2 
1
f1   1 , f 2   0 
 
 
 0 
 x2 
 dim ( ( x))  2
0 
f
f
[ f1 , f 2 ]  2 f1  1 f 2  0
 
x
x
1
[ f1 , f 2 ]   iff rank{ f1 ( x), f 2 ( x), [ f1 , f 2 ]( x)}  2, x  D.
2 x2
However rank{ f1 , f 2 ,[ f1 , f 2 ]}  rank  1

 0
Hence  is not involutive
1
0
x2
0
0  3, x  D.

1
10-35
Codistribution & Complete Integrability
Codistribution :  ( x)  {w  ( R n )* :  w, v  0, v  ( x)}
where ( R n )* is the n - dimensiona l space of row vectors.
Complete Integrability : Let  be a nonsingula r distributi on on D, generated
by f1 , , f r . Then  is said to be completely integrable
if for each x0  D, there exists a neighborho od N of x0
and n  r real valued smooth functions h1 ( x), , hn  r ( x)
such that h1 ( x), , hn  r ( x) satisfy t he PDE
h j
f i ( x)  0,  1  i  r , 1  j  n  r
x
and the covector fields h j ( x) are linearly independen t
for all x  D, i.e.,
  span{h1 , , hn  r }
A key result from differenti al geometry is Frobenius theorem which states
that a nonsingula r distributi on is completell y integrable iff it is involutive .
10-36
Input-state linearization
• Input-state linearization
Consider the SI system
x  f ( x)  g ( x)u
1
where f , g are smooth ve ctor fields
Note that 1 are said to be linear in control.
Definition: A single input nonlinear system in the form above, with f ( x), g ( x)
being smooth ve ctor fields in R n , is said to be input state linearizab le
if there exists a region  in R n , a diffeomorp hism T :   R n and
a nonlinear control law u   ( x)   ( x)v
2
such that the new state variable s z  T ( x) and the new input v satisfy
a linear ti me invariant relation
z  Az  bv
10-37
Input-state linearization (Continued)
0
0

where A   

0
0
The new state
1 0  0
0 
0 
0 1 0 0

 
    , b    

 
0 0  1
0 
1
0 0  0
z is called the linearizin g state, 2
the linearizin g control law.
is called
Question: Can all nonlinear state eq. of 1 be input - state linearizab le?
If not, when do such linearizat ions exist?
Theorem:
The nonlinear system 1 with f ( x), g ( x) being smooth ve ctor
fields, is input - state linearizab le iff there exists a region 
such that the following conditions hold :
10-38
Input-state linearization (Continued)
 the vector fields {g , ad f g , , ad nf 1 g} are linearly independent in .
 the distribution D  span{g , ad f g ,, ad nf  2 g} is involutive in .
P roof: See Ch.13.3
 T he first condition can be interprete
d as a controllability condition
for nonlinear system. For linear system [ g , ad f g , , ad nf 1 g ] becomes
[b, Ab, , An 1b] .
 It is easy to show that if a system's linear approximation in a closed
connected region  in R n are all controllable, then under mild smoothness
assumption, the system can be driven from any point in  to any point
in . However, a nonlinear system can be controllable while its linear
approximation is not.
 T he involutivity condition is trivially satisfied for linear systems which
have constant vector fields.
10-39