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DISC Systems and Control Theory of Nonlinear Systems 1
Lecture 1:
Mathematical preliminariesand 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 2
Very simple example of a nonlinear system: unicycle
x1 = u1 cosx3
x1 = u1 sinx3
x3 = u2
Example of a general nonlinear system
x = f(x, u), y = h(x, u)
DISC Systems and Control Theory of Nonlinear Systems 3
One approach to analysis and control of nonlinear systems:
linearization.
Let
0 = f(x, u)
Then linearized system is
z = Az +Bv
where
A =∂f
∂x(x, u), B =
∂f
∂u(x, u)
Approximation of the nonlinear system with z = x− x, v = u− u.
DISC Systems and Control Theory of Nonlinear Systems 4
Linearization of the unicycle at any point (x1, x2, x3) and
(u1, u2) = (0, 0) :
z1
z1
z3
=
cos x3 0
sin x3 0
0 1
u1
u2
Is never controllable ! Contrary to intuition.
How do we study nonlinear controllability ?
DISC Systems and Control Theory of Nonlinear Systems 5
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.
DISC Systems and Control Theory of Nonlinear Systems 6
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) · · · ∂f1
∂xn(x)
...
∂fm
∂x1
(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 7
Example 2 Every open subset V of Rn is a manifold of dimension
n (Take m = 0).
Example 3 The circle S1 is a manifold of dimension 1, since
S1 = {(x1, x2) ∈ R2|x2
1 + x22 − 1 = 0}.
Example 4 Consider the group O(n) of orthogonal (n, n)-matrices
(i.e. A ∈ O(n) satisfies ATA = In).
Consider the set g`(n) of all (n, n) matrices, identified with Rn2
.
Define the map f from g`(n) to the space of symmetric (n, n)
matrices (identified with R1
2n(n+1)) as
f(A) = ATA
Then O(n) = {A ∈ g`(n)|f(A) = In}. The rank of the Jacobian matrix
of f (seen as a map from Rn2
to R1
2n(n+1)) equals 1
2n(n+ 1) at every
point A ∈ O(n). Therefore O(n) is a smooth manifold of dimension
n2 −1
2n(n+ 1) =
1
2n(n− 1)
DISC Systems and Control Theory of Nonlinear Systems 8
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
· · · ∂f1
∂xm
...
∂fm
∂x1
· · · ∂fm
∂xm
is non-singular at xo. By the implicit function theorem there now
exists a neighborhood W1 ⊂ Rn of xo, a neighborhood W2 ⊂ R
n−m of
(xom+1, · · · , x
on), and a smooth map g : W2 → R
m 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 9
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.
DISC Systems and Control Theory of Nonlinear Systems 10
Example 5 Consider the circle S1 = {(x1, x2)|x21 + x2
2 − 1 = 0}. Take
any point xo = (xo1, x
o2) ∈ S1. If xo
1 6= 0 we have that∂
∂x1
(x21 + x2
2 − 1)|(xo1,xo
2) 6= 0, and thus we can solve for x1, i.e.
x1 = ±√
1 − x22 (with sign depending on the sign of xo
1). The
x2-coordinate may thus serve as coordinate function in both cases.
Alternatively, if xo2 6= 0 we solve for x2, i.e. x2 = ±
√
1 − x21, leading
to neighborhoods U1 and U2 which are respectively in the upper-
and the lower half-plane.
DISC Systems and Control Theory of Nonlinear Systems 11
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] , i = 1, · · · , k
are independent as functions of xm+1, · · · , xn.
DISC Systems and Control Theory of Nonlinear Systems 12
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
(= dimM) 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)).
DISC Systems and Control Theory of Nonlinear Systems 13
Let M be an (n−m)-dimensional manifold. Let xo ∈M , then the
tangent space Tx0M at x0 to the manifold M is given as the linear
space
Tx0M =
{
z ∈ Rn|∂f
∂x(x0)z = 0
}
= ker∂f
∂x(x0)
(Notice that because the rank of ∂f∂x
(x0) equals m the dimension of
Tx0M equals n−m, i.e. the dimension of the manifold M .)
Furthermore the tangent bundle TM is defined as the manifold
TM = {(x, z) ∈ V × Rn|f1(x) = · · · = fm(x) = 0,
∂f
∂x(x)z = 0}
and equals⋃
x∈M
TxM .
DISC Systems and Control Theory of Nonlinear Systems 14
Let (z1, z2, . . . , zn) be a coordinate system (local on U) on the
n-dimensional manifold M . This defines on every tangent space
TpM , with p ∈ U , a basis for this linear space, denoted as
∂
∂z1
∣
∣
∣
∣
p
, · · · ,∂
∂zn
∣
∣
∣
∣
p
Indeed, every tangent vector Xp ∈ TpM, p ∈M can be associated
with a derivation. Define
c : (−ε, ε) −→M, ε > 0, c(0) = p
such that c′(0) = dcdt
(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
dth(c(t))|t=0
DISC Systems and Control Theory of Nonlinear Systems 15
The derivation corresponding to ∂∂zi
|p is defined as
∂
∂zi
|p h =∂h
∂zi
(p)
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
F := ψ ◦ F ◦ ϕ−1 : ϕ(U) ⊂ Rn1 → ψ(V ) ⊂ R
n2
where ϕ = (ϕ1, · · · , ϕn1)T , ψ = (ψ1, · · · , ψn2
)T , are smooth maps.
DISC Systems and Control Theory of Nonlinear Systems 16
F 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
h := 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 : TpM → TF (p)N
as follows. Let Xp ∈ TpM . For any f ∈ C∞(F (p)) set
F∗pXp(f) = Xp(f ◦ F )
where Xp ∈ TpM is identified with the corresponding derivation at
p ∈M .
DISC Systems and Control Theory of Nonlinear Systems 17
Definition 7 A (smooth) vector field X on a manifold M is
defined as a smooth mapping
X : M −→ TM
satisfying π(X(p)) = p, ∀p ∈M , where π : TM →M is the canonical
projection mapping (p,Xp) ∈ TM to p ∈M .
Thus a vector field X on M assigns to every point p ∈M an
element of TpM :
X(p) ∈ TpM
Let now (U,ϕ1, · · · , ϕn) = (U, x1, · · · , xn) be a coordinate system for
M . For every p ∈ U this yields a basis
{
∂∂x1
∣
∣
∣p, · · · ,∂
∂xn
∣
∣
∣
p
}
for TpM .
DISC Systems and Control Theory of Nonlinear Systems 18
It thus follows that locally on U the vector field X can be expressed
by a column-vector
X(x) =
X1(x1, · · · , xn)...
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
x1 = X1(x1, · · · , xn)...
xn = Xn(x1, · · · , xn)
DISC Systems and Control Theory of Nonlinear Systems 19
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 x = f(x) then z = S(x) should satisfy
z =∂S
∂x(S−1(z))f(S−1(z))
where ∂S∂x
(x) denotes the Jacobian of the coordinate transformation
S.
It follows that f(x) transforms under z = S(x) to
f(z) := ∂S∂x
(S−1(z))f(S−1(z)). Here, f denotes the same vector field,
but now expressed in the new coordinates. As an example, any
linear set of differential equations
x = Ax
transforms under a linear coordinate transformation z = Sx (with S
an invertible matrix) to
z = SAS−1z
DISC Systems and Control Theory of Nonlinear Systems 20
Using this machinery we are now able to give a coordinate-free
definition of a nonlinear state space system
x = f(x) + g(x)u , u ∈ Rm, x ∈ X ,
y = h(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 21
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
DISC Systems and Control Theory of Nonlinear Systems 22
belongs to the space of derivations at p. Indeed
[X,Y ]p(fg) = Xp(Y (fg)) − Yp(X(fg)) =
= 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) (1)
Thus [X,Y ]p can be uniquely identified with an element in the
tangent space TpM , and [X,Y ] defines a new vectorfield on M .
DISC Systems and Control Theory of Nonlinear Systems 23
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
[X,Y ](x) =∂Y
∂x(x)X(x) −
∂X
∂x(x)Y (x)
with ∂Y∂x, ∂X
∂xdenoting the Jacobian matrices.
DISC Systems and Control Theory of Nonlinear Systems 24
Proof
Compute for any j = 1, · · · , n
[X,Y ]p(xj) = Xp(Y (xj)) − Yp(X(xj)) =
= Xp(Yj) − Yp(Xj) =n∑
i=1
[
∂Yj
∂xiXi −
∂Xj
∂xiYi
]
(x(p))
Since [X,Y ]p(xj) is the j-th component of [X,Y ]p in these
coordinates the result follows. �
DISC Systems and Control Theory of Nonlinear Systems 25
It readily follows that the Lie bracket satisfies the following
properties
(a) [fX, gY ] = fg[X,Y ] + f · LXg · Y − g · LY f ·X f, g ∈ C∞(M)
(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
DISC Systems and Control Theory of Nonlinear Systems 26
Towards nonlinear controllability
Consider the unicycle example
x =
cosx3
sinx3
0
u1 +
0
0
1
u2
The Lie bracket of the two input vector fields is given as
−
0 0 − sinx3
0 0 cosx3
0 0 0
0
0
1
=
sinx3
− cosx3
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.
DISC Systems and Control Theory of Nonlinear Systems 27
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 .
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