DISC Systems and Control Theory of Nonlinear Systems
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Lecture 1:
Mathematical preliminaries
and introduction to nonlinear
controllability
Nonlinear Dynamical Control Systems, Chapters 1, 2 + handout
See www.math.rug.nl/˜arjan (under teaching) for info on course
schedule and homework sets.
DISC Systems and Control Theory of Nonlinear Systems
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Very simple example of a nonlinear system: unicycle
ẋ1
= u1 cos x3
ẋ1
= u1 sin x3
ẋ3
= u2
Example of a general nonlinear system
ẋ = f (x, u),
y = h(x, u)
DISC Systems and Control Theory of Nonlinear Systems
One approach to analysis and control of nonlinear systems:
linearization.
Let
0 = f (x̄, ū)
Then linearized system is
ż = Az + Bv
where
∂f
∂f
(x̄, ū), B =
(x̄, ū)
∂x
∂u
Approximation of the nonlinear system with z = x − x̄, v = u − ū.
A=
3
DISC Systems and Control Theory of Nonlinear Systems
Linearization of the unicycle at any point (x̄1 , x̄2 , x̄3 ) and
(ū1 , ū2 ) = (0, 0) :
  

ż1
cos x̄3 0  
  
 u
ż1  =  sin x̄3 0  1 
  

u2
ż3
0
1
Is never controllable ! Contrary to intuition.
How do we study nonlinear controllability ?
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DISC Systems and Control Theory of Nonlinear Systems
First observation:
Nonlinearity shows up in
• Nonlinearity of differential equations for the state evolution or
nonlinear output map.
• Nonlinearity also shows up in the structure of the state space,
which is in general not anymore Rn .
We will start by defining nonlinear state spaces; or in
mathematical terminology, (smooth) manifolds.
Analogy:
Subspaces of Rn of dimension n − m are defined by m independent
linear equations.
Manifolds of dimension n − m are subsets of Rn , which are defined
by m independent nonlinear equations.
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Definition 1 Let f1 , · · · , fm , m ≤ n, be smooth functions on an
open part V of Rn . Define the set
M = {x ∈ V |f1 (x) = · · · = fm (x) = 0}
Suppose that the rank of the Jacobian matrix of f = (f1 , · · · , fm )T





∂f1
∂x1 (x)
···
∂fm
∂x1 (x)
···
..
.
∂f1
∂xn (x)
∂fm
∂xn (x)


∂f

=:
(x)

∂x

is m at each x ∈ M . Then M is a manifold of dimension n − m (if
M is non-empty).
DISC Systems and Control Theory of Nonlinear Systems
Example 2 Every open subset V of Rn is a manifold of dimension
n (Take m = 0).
Example 3 The circle S 1 is a manifold of dimension 1, since
S 1 = {(x1 , x2 ) ∈ R2 |x21 + x22 − 1 = 0}.
Example 4 Consider the group O(n) of orthogonal (n, n)-matrices
(i.e. A ∈ O(n) satisfies AT A = In ).
n2
Consider the set g`(n) of all (n, n) matrices, identified with R .
Define the map f from g`(n) to the space of symmetric (n, n)
1
matrices (identified with R 2 n(n+1) ) as
f (A) = AT A
Then O(n) = {A ∈ g`(n)|f (A) = In }. The rank of the Jacobian matrix
2
1
of f (seen as a map from Rn to R 2 n(n+1) ) equals 21 n(n + 1) at every
point A ∈ O(n). Therefore O(n) is a smooth manifold of dimension
1
1
n − n(n + 1) = n(n − 1)
2
2
2
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The basic feature of a manifold M of dimension n − m is that it is
locally Rn−m in the following sense.
Let xo ∈ M . By permuting the coordinates x1 , · · · , xn for Rn we may
assume that the (m, m) matrix





∂f1
∂x1
···
..
.
∂fm
∂x1
···
∂f1
∂xm
∂fm
∂xm





is non-singular at xo . By the implicit function theorem there now
exists a neighborhood W1 ⊂ Rn of xo , a neighborhood W2 ⊂ Rn−m of
(xom+1 , · · · , xon ), and a smooth map g : W2 → Rm such that M ∩ W1
equals
{[g1 (xm+1 , · · · , xn ), · · · , gm (xm+1 , · · · , xn ), xm+1 , · · · , xn ] |(xm+1 , · · · , xn ) ∈ W2 }
DISC Systems and Control Theory of Nonlinear Systems
Then on U := M ∩ W1 we define coordinate functions
ϕi , i = 1, · · · , n − m, by
ϕi [g1 (xm+1 , · · · , xn ), · · · , gm (xm+1 , · · · , xn ), xm+1 , · · · , xn ] = xm+i
U is called a coordinate neighborhood of xo . In this way the
neighborhood U of xo becomes identified with an open part of
Rn−m .
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DISC Systems and Control Theory of Nonlinear Systems
Example 5 Consider the circle S 1 = {(x1 , x2 )|x21 + x22 − 1 = 0}. Take
any point xo = (xo1 , xo2 ) ∈ S 1 . If xo1 6= 0 we have that
∂
2
2
o o
∂x1 (x1 + x2 − 1)|(x1 ,x2 ) 6= 0, and thus we can solve for x1 , i.e.
p
x1 = ± 1 − x22 (with sign depending on the sign of xo1 ). The
x2 -coordinate may thus serve as coordinate function in both cases.
p
o
Alternatively, if x2 6= 0 we solve for x2 , i.e. x2 = ± 1 − x21 , leading
to neighborhoods Ũ1 and Ũ2 which are respectively in the upperand the lower half-plane.
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From now on we look at manifolds as objects on their own.
Let h : M → R be a function on M . Let U be a coordinate
neighborhood of xo ∈ M as above. Then h is smooth on U if the
function
h [g1 (xm+1 , · · · , xn ), · · · , gm (xm+1 , · · · , xn ), xm+1 , · · · , xn ]
depends smoothly on its arguments xm+1 , · · · , xn .
The function h is smooth on M if it is smooth on a covering set
of coordinate neighborhoods of M .
Let h1 , · · · , hk be smooth functions on M . Then h1 , · · · , hk are
called independent on U if the functions
hi [g1 (xm+1 , · · · , xn ), · · · , gm (xm+1 , · · · , xn ), xm+1 , · · · , xn ] ,
are independent as functions of xm+1 , · · · , xn .
i = 1, · · · , k
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With the aid of the above definition the notion of a coordinate
neighborhood and of coordinate functions defined on it can be
immediately generalized. Indeed, any open subset V of M with n
(= dim M ) independent smooth functions (ϕ1 , · · · , ϕn ) defined on it
defines a coordinate neighborhood and coordinate functions for M ,
or, briefly, a coordinate system
(V, (ϕ1 , · · · , ϕn ))
Definition 6 Let M now be a manifold of dimension n. A subset
P ⊂ M is called a submanifold of dimension k < n if for each p ∈ P
there exists a coordinate system (V, ϕ1 , · · · , ϕn ) for M about p such
that
P ∩ V = {q ∈ V |ϕi (q) = ϕi (p), i = k + 1, · · · , n}
Notice that a submanifold P of a manifold M is a manifold in its
own right, with coordinate system (P ∩ V, (ϕ1 , · · · , ϕk )).
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Let M be an (n − m)-dimensional manifold. Let xo ∈ M , then the
tangent space Tx0 M at x0 to the manifold M is given as the linear
space
∂f
∂f
Tx0 M = z ∈ Rn | (x0 )z = 0 = ker
(x0 )
∂x
∂x
(Notice that because the rank of ∂f
∂x (x0 ) equals m the dimension of
Tx0 M equals n − m, i.e. the dimension of the manifold M .)
Furthermore the tangent bundle T M is defined as the manifold
T M = {(x, z) ∈ V × Rn |f1 (x) = · · · = fm (x) = 0,
and equals
S
x∈M
Tx M .
∂f
(x)z = 0}
∂x
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Let (z1 , z2 , . . . , zn ) be a coordinate system (local on U ) on the
n-dimensional manifold M . This defines on every tangent space
Tp M , with p ∈ U , a basis for this linear space, denoted as
∂
∂z1
,··· ,
p
∂
∂zn
p
Indeed, every tangent vector Xp ∈ Tp M, p ∈ M can be associated
with a derivation. Define
c : (−, ) −→ M,
> 0,
c(0) = p
such that c0 (0) = dc
dt (0) = Xp . For any function h : M → R define the
derivative of h in the direction Xp at the point p ∈ M as
Xp (h) :=
d
h(c(t))|t=0
dt
DISC Systems and Control Theory of Nonlinear Systems
The derivation corresponding to
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∂
∂zi |p
is defined as
∂
∂h
|p h =
(p)
∂zi
∂zi
We have defined what we mean by a smooth function on a
manifold M . Similarly we define what we mean by a smooth
mapping
F : M1 → M2
with M1 and M2 manifolds. Indeed, let M1 and M2 be manifolds of
dimension n1 and n2 , respectively. Then for any p ∈ M1 there exist
local coordinate systems (U, (ϕ1 , · · · , ϕn1 )) for p and (V, (ψ1 , · · · , ψn2 ))
for F (p) ∈ M2 . We now require that the maps
F̂ := ψ ◦ F ◦ ϕ−1 : ϕ(U ) ⊂ Rn1 → ψ(V ) ⊂ Rn2
where ϕ = (ϕ1 , · · · , ϕn1 )T , ψ = (ψ1 , · · · , ψn2 )T , are smooth maps.
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F̂ is nothing else than the local coordinate expression of the map
F : M → N . Similarly we may rephrase the definition of a smooth
function h : M → R by requiring that the functions
ĥ := h ◦ ϕ−1 : ϕ(U ) ⊂ Rn → R
are smooth, where (U, (ϕ1 , · · · , ϕn )) is a local coordinate system for
M.
Let now F : M → N be a smooth map. Define a linear map (called
the tangent map of F at p ∈ M )
F∗p : Tp M → TF (p) N
as follows. Let Xp ∈ Tp M . For any f ∈ C ∞ (F (p)) set
F∗p Xp (f ) = Xp (f ◦ F )
where Xp ∈ Tp M is identified with the corresponding derivation at
p ∈ M.
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Definition 7 A (smooth) vector field X on a manifold M is
defined as a smooth mapping
X : M −→ T M
satisfying π(X(p)) = p, ∀p ∈ M , where π : T M → M is the canonical
projection mapping (p, Xp ) ∈ T M to p ∈ M .
Thus a vector field X on M assigns to every point p ∈ M an
element of Tp M :
X(p) ∈ Tp M
Let now (U, ϕ1 , · · · , ϕn ) = (U, x1 , · · · , xn )be a coordinate system for
∂
∂
M . For every p ∈ U this yields a basis ∂x
,
·
·
·
,
for Tp M .
p
∂xn
1
p
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It thus follows that locally on U the vector field X can be expressed
by a column-vector


X1 (x1 , · · · , xn )


..


X(x) = 

.


Xn (x1 , · · · , xn )
It follows that in local coordinates x1 , · · · , xn a vector field X
corresponds to the n-dimensional set of first-order differential
equations
ẋ1
= X1 (x1 , · · · , xn )
..
.
ẋn
= Xn (x1 , · · · , xn )
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This implies that a vector field f transforms in a special way under
any coordinate transformation z = S(x). Indeed, if x satisfies the
differential equation ẋ = f (x) then z = S(x) should satisfy
ż =
where
S.
∂S
∂x (x)
∂S −1
(S (z))f (S −1 (z))
∂x
denotes the Jacobian of the coordinate transformation
It follows that f (x) transforms under z = S(x) to
−1
f˜(z) := ∂S
(z))f (S −1 (z)). Here, f˜ denotes the same vector field,
∂x (S
but now expressed in the new coordinates. As an example, any
linear set of differential equations
ẋ = Ax
transforms under a linear coordinate transformation z = Sx (with S
an invertible matrix) to
ż = SAS −1 z
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Using this machinery we are now able to give a coordinate-free
definition of a nonlinear state space system
ẋ = f (x) + g(x)u
y
= h(x)
, u ∈ Rm ,
x ∈ X,
, y ∈ Rp ,
living on a state space X that is a manifold. Indeed, f (x) is the
local coordinate expression of a vector field on X (called the drift
vector field), and also the columns of g(x) are local coordinate
expressions of vector fields on X (the input-vector fields), while h is
a smooth mapping from X to Rp .
DISC Systems and Control Theory of Nonlinear Systems
Lie brackets of vector fields
For X and Y any two vectorfields on M we define the Lie bracket
of X and Y , denoted [X, Y ], by setting
[X, Y ]p (f ) = Xp (Y (f )) − Yp (X(f ))
for every function f : M → R. It can be checked that [X, Y ]p
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belongs to the space of derivations at p. Indeed
[X, Y ]p (f g)
= Xp (Y (f g)) − Yp (X(f g)) =
= Xp {Y (f ) · g + f · Y (g)} − Yp {X(f ) · g + f · X(g)} =
= Xp [Y (f )]g(p) + Yp (f )Xp (g) + Xp (f )Yp (g) + f (p)Xp (Y (g))
− Yp (X(f ))g(p) − Xp (f )Yp (g) − Yp (f )Xp (g) − f (p)Yp (X(g))
= [X, Y ]p (f ) · g(p) + f (p) · [X, Y ]p (g)
Thus [X, Y ]p can be uniquely identified with an element in the
tangent space Tp M , and [X, Y ] defines a new vectorfield on M .
(1)
DISC Systems and Control Theory of Nonlinear Systems
In local coordinates the Lie bracket takes the following form:
Proposition 8 Let X and Y be vectorfields on M , given in local
coordinates (x1 , · · · , xn ) as X(x) = (X1 (x), · · · , Xn (x))T and
Y (x) = (Y1 (x), · · · , Yn (x))T . Then the local coordinate expression of
[X, Y ] is given as
∂Y
∂X
[X, Y ](x) =
(x)X(x) −
(x)Y (x)
∂x
∂x
with
∂Y
∂x
, ∂X
∂x denoting the Jacobian matrices.
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Proof
Compute for any j = 1, · · · , n
[X, Y ]p (xj ) = Xp (Y (xj )) − Yp (X(xj )) =
= Xp (Yj ) − Yp (Xj ) =
n h
P
∂Yj
i=1
∂xi Xi
−
∂Xj
∂xi Yi
i
(x(p))
Since [X, Y ]p (xj ) is the j -th component of [X, Y ]p in these
coordinates the result follows.
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It readily follows that the Lie bracket satisfies the following
properties
(a)
[f X, gY ]
= f g[X, Y ] + f · LX g · Y − g · LY f · X
(b)
[X, Y ]
= −[Y, X]
(c)
[X, Y1 + Y2 ] = [X, Y1 ] + [X, Y2 ]
Furthermore, the following property can be checked
[[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0
f, g ∈ C ∞ (M )
DISC Systems and Control Theory of Nonlinear Systems
Towards nonlinear controllability
Consider the unicycle example

 

0
cos x3

 




ẋ =  sin x3  u1 + 
0 u2
0
1
The Lie bracket of the two input vector fields is given as

  

0 0 − sin x3
0
sin x3

  






− 0 0 cos x3  0 = − cos x3 

0 0
0
1
0
which is a vector field that is independent from the two input
vector fields.
Claim: This new independent direction guarantees controllability
of the unicycle system.
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Interpretation of the Lie bracket
Proposition 9 Let X, Y be two vector fields such that
[X, Y ] = 0
Then the solution flows of the vector fields are commuting.
In fact, we may find local coordinates x1 , . . . , xn such that
∂
X=
,
∂x1
∂
Y =
∂x2
Thus, the Lie bracket [X, Y ] characterizes the amount of
non-commutativity of the vector fields X, Y .