Lecture 2: Controllability of nonlinear systems · DISC Systems and Control Theory of Nonlinear Systems 2 Recall: Kinematic model of the unicycle x˙1 = u1 cosx3 x˙2 = u1 sinx3 x˙3

DISC Systems and Control Theory of Nonlinear Systems 1

Lecture 2:Controllability of nonlinear systems

Nonlinear Dynamical Control Systems, Chapter 3

See www.math.rug.nl/˜arjan (under teaching) for info on course

schedule and homework sets.

Take-Home Exam I on homepage on March 16.


Recall: Kinematic model of the unicycle

x1 = u1 cosx3

x2 = u1 sinx3

x3 = u2

written as a system with two input vector fields and zero drift

vector field

x =

cosx3

sin x3

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.

Interpretation of the Lie bracket:

Proposition 1 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 .


In fact, let the control strategy u = col(u1, u2) be defined by

u(t) =

(1, 0), t ∈ [0, ε), ε > 0

(0, 1), t ∈ [ε, 2ε)

(−1, 0), t ∈ [2ε, 3ε)

(0,−1), t ∈ [3ε, 4ε),

Then the motion of the system is described by

x(4ε) = x0 + ε2[g1, g2](x0) + O(ε3).

which indicates controllability, since [g1, g2] is everywhere

independent from g1, g2.

This formula holds in general.

This is enough for systems with two inputs and three state

variables, but what can we do if the dimension of the state is > 3?


Answer: consider higher-order Lie brackets.

I

M

yr

θ

ϕ

xr

Example 2 Consider the cart with fixed rear axis

d

dt

x1

x2

ϕ

θ

=

cos(ϕ + θ)

sin(ϕ + θ)

sin θ

0

u1 +

0

0

0

1

u2

with u1 the driving input, and u2 the steering input.


Define

g1(x) =

cos(x3 + x4)

sin(x3 + x4)

sin(x4)

0

︸︷︷︸

Drive

, g2(x) =

0

0

0

1

︸︷︷︸

Steer

.


Compute

[Steer, Drive] =∂g1

∂xg2 −

∂g2

∂xg1

=

0 0 − sin(x3 + x4) − sin(x3 + x4)

0 0 cos(x3 + x4) cos(x3 + x4)

0 0 0 cos(x4)

0 0 0 0

0

0

0

1

− 0

=

− sin(x3 + x4)

cos(x3 + x4)

cos(x4)

0

=: Wriggle.


Another independent direction is obtained by the third-order Lie

bracket

[Wriggle, Drive] =

− sin(x3)

cos(x3)

0

0

=: Slide.

This shows that you can manoeuver your car into any parking lot

by applying controls corresponding to the ‘Slide’ direction, i.e., by

applying the control sequence {Wriggle, Drive, −Wriggle, −Drive}.


What to do with the drift vector field ?

The system

x = f(x) + g(x)u

can be considered as a special case of

x = g1(x)u1 + g2(x)u2,

with u1 = 1. This means that care has to be taken with respect to

brackets involving f :

[f, g], [g, [f, g]], [f, [f, g]], . . .


Example 3 Consider the system on R2

x1 = x22

x2 = u.

Compute the Lie brackets of the vector fields

f(x) =

x2

2

0

, g(x) =

0

1

,

yielding

[f, g](x) =

−2x2

0

, [[f, g], g](x) =

2

0

.

Clearly, we have obtained two independent directions. However,

since x22 ≥ 0, the x1-coordinate is always non-decreasing. Hence,

the system is not really controllable.


A weaker form of controllability: local accessibility

Let V be a neighborhood of x0, then RV(x0, t1) denotes the

reachable set from x0 at time t1 ≥ 0, following the trajectories

which remain in the neighborhood V of x0 for t ≤ t1, i.e., all points

x1 for which there exists an input u(·) such that the evolution of

the system for x(0) = x0 satisfies x(t) ∈ V, 0 ≤ t ≤ t1, and x(t1) = x1.

Furthermore, let

RV

t1(x0) =

⋃

τ≤t1

RV(x0, τ).

Definition 4 (Local accessibility) A system is said to be locally

accessible from x0 if RVt1

(x0) contains a non-empty open subset of

X for all non-empty neighborhoods V of x0 and all t1 > 0. If the

latter holds for all x0 ∈ X then the system is called locally

accessible.


Definition 5 (Accessibility algebra) Consider the system

x = f(x) + g1(x)u1 + · · · + gm(x)um

The accessibility algebra C are the linear combinations of

repeated Lie brackets of the form

[Xk, [Xk−1, [· · · , [X2, X1] · · · ]]], k = 1, 2, . . . ,

where Xi, is a vector field in the set {f, g1, . . . , gm}.

This linear space is a Lie algebra under the Lie bracket.

Definition 6 The accessibility distribution C is the distribution

generated by the accessibility algebra C:

C(x) = span{X(x) | X vector field in C}, x ∈ X


Intermezzo: Distributions on manifolds

A distribution D on a manifold X is specified by a subspace

D(x) ⊂ TxX

for all x ∈ X .

Let X1, X2, . . . , Xk be vector fields on X . Then

D(x) = span(X1(x), X2(x), . . . , Xk(x)).

defines a distribution.

A distribution D is called involutive if, whenever f, g ∈ D, also

[f, g] ∈ D.

The distribution D is called constant-dimensional whenever the

dimension of D(x) is constant.


Example 7 Let X = R3 and D = span(f1, f2), where

f1(x) =

2x2

1

0

, f2(x) =

1

0

x2

.

Since f1 and f2 are linearly independent, we have that

dim(D(x)) = 2, for all x. Furthermore, we have

[f1, f2](x) =∂f2

∂x(x)f1(x) −

∂f1

∂x(x)f2(x) =

1

0

0

.


[f1, f2] ∈ D if and only if rank(f1(x), f2(x), [f1, f2](x)) = 2, for all x.

However,

rank(f1(x), f2(x), [f1, f2](x)) = rank

2x2 1 0

1 0 0

0 x2 1

= 3,

for all x. Hence, D is not involutive.


Let D be a nonsingular distribution on X , generated by the

independent vector fields f1, . . . , fr. Then D is said to be integrable

if for each x0 ∈ X , there exists a neighborhood N of x0 and n − r

real-valued independent functions h1(x), . . . , hn−r(x) defined on N ,

such that h1(x), . . . , hn−r(x) satisfy the partial differential equations

∂hj

∂x(x)fi(x) = 0, (1)

for all indices i = 1, . . . , r, j = 1, . . . , n − r.

Frobenius’ theorem

A constant-dimensional distribution is integrable if and only if it is

involutive.

The necessity of involutivity for complete integrability is easily

seen. Indeed, suppose that (1) is satisfied. This is the same as

Lfihj = 0


It follows that

L[fi,fk]hj = LfiLfk

hj − LfkLfi

hj = 0

Since the functions h1(x), . . . , hn−r(x) are independent, this implies

that the Lie brackets [fi, fk] are (pointwise) linear combinations of

the vector fields f1, . . . , fr, and are thus contained in the

distribution D.

A geometric description of Frobenius’ theorem is as follows. Let

the independent functions h1(x), . . . , hn−r(x) satisfy (1). Then their

level sets, i.e., all sets of the form

{x | h1(x) = c1, . . . , hn−r(x) = cn−r}

for arbitrary constants c1, . . . , cn−r, are well-defined r-dimensional

submanifolds of X , to which all the vector fields f1, . . . , fr are

tangent, and, as a consequence, also all their Lie brackets are

tangent.


Example 8 Consider the following set of partial differential

equations

0 = x1∂φ

∂x1+ x2

∂φ

∂x2+ x3

∂φ

∂x3

0 =∂φ

∂x3

Define the vector fields

f1(x) =

x1

x2

x3

, f2(x) =

0

0

1

.

It is checked that D := span(f1, f2) has constant dimension = 2 on

the set X = {x ∈ R3 | x2

1 + x22 6= 0} (that is, R

3 excluding the x3-axis),

and is involutive. Thus, by Frobenius’ theorem, D is integrable.


Consequently, for each x0 ∈ X , there exists a neighborhood N of x0

and a real-valued function φ(x) with dφ(x) 6= 0 that satisfies the

given set of partial differential equations. In fact, φ(x) = lnx1 − lnx2

is a (global) solution.

Note that the solution is not unique. In particular, φ(x) = tan−1 x2

x1

is

also a global solution.


By construction, the accessibility distribution C is involutive.

Theorem 9 (Local accessibility) A sufficient condition for the

system to be locally accessible from x ∈ X is

dimC(x) = n (2)

If this holds for all x ∈ X then the system is locally accessible.

Conversely, if the system is locally accessible then (2) holds for all

x in an open and dense subset of X .

We call (2) the accessibility rank condition at x.


Key idea of the proof

Consider the system

x = f(x) + g1(x)u1 + . . . gm(x)um

and its generated system vector fields

F = {X | ∃u1, . . . , um such that X(x) = f(x) + g1(x)u1 + . . . gm(x)um}

Then for every k ≤ n there exists a submanifold Nk around x0 of

dimension k given as

Nk = {x | x = Xtk

k ◦ Xtk−1

k−1 ◦ · · · ◦ Xt11 (x0), 0 ≤ σi < ti < τi}

with Xi ∈ F. Indeed, suppose for a certain k < n we cannot

construct Nk+1. This means that all system vector fields X ∈ F are

tangent to Nk, and hence all vector fields f, g1, . . . , gm. This also

means that all Lie brackets of these vector fields are tangent to

Nk, and thus dim C(x) ≤ k, which is a contradiction.


If there is no drift vector field then we obtain real controllability:

Theorem 10 Consider

x = g1(x)u1 + g2(x)u2 + . . . + gm(x)um

If dimC(x) = n for all x ∈ X then the system is controllable.

Consider the map

(t1, . . . , tn) → Xtn

n ◦ Xtn−1

n−1 ◦ · · · ◦ Xt11 (x0), 0 ≤ σi < ti < τi

having image Nn, which is an n-dimensional open part of X .

Now let s1, . . . , sn be such that 0 ≤ σi < si < τi. Then the map

(t1, . . . , tn) → (−X1)s1◦(−X2)

s2◦· · ·◦(−Xn)sn◦Xtn

n ◦Xtn−1

n−1 ◦· · ·◦Xt11 (x0), 0 ≤ σi < ti < τi

has an image which is an open neighborhood of x0. Thus the

reachable set R(x0) from x0 contains an open neighborhood of x0.


Suppose now that the reachable set is smaller than X . Then take

any point on the boundary of the reachable set R(x0). Then the set

of points reachable from this point is again open. Contradiction.

Thus the unicycle and the cart are indeed controllable.

Note that the actual construction of the input functions which

steers the system from x0 to x1 has not been addressed.


Sometimes local accessibility is heavily depending on the flow of

the drift vector field; consider for example the system

x1 = 1

x2 = u

This system is locally accessible, but of course very far from

controllability. In order to improve the situation we look at a

stronger form of accessibility: local strong accessibility

A system is locally strongly accessible from x0 if for any

neighborhood V of x0 the set RV(x0, t1) contains a non-empty set

for any t1 > 0 sufficiently small. If the latter holds for all x ∈ X then

the system is called locally strongly accessible.

(The example given above is not locally strongly accessible.)


Define C0 as the smallest algebra which contains g1, . . . , gm and

satisfies [f, w] ∈ C0 for all w ∈ C0. Define the corresponding

involutive distribution

C0(x) := span{X(x) | X vector field in C0}.

We refer to C0 and C0 as the strong accessibility algebra and the

strong accessibility distribution, respectively.


Notice that the strong accessibility algebra C0 does not contain the

drift vector field f).

Theorem 11 (Strong accessibility) A sufficient condition for the

system to be locally strongly accessible from x is

dimC0(x) = n

Furthermore, the system is locally strongly accessible if this holds

for all x. Conversely, if the system is locally strongly accessible

then it holds for all x in an open and dense subset of X .

The system given before:

x1 = x22

x2 = u.

is not only locally accessible, but also locally strongly accessible,

since g(x) and [[f, g], g](x) are everywhere independent.


Let us apply the theory developed above to a linear system

x = Ax +

m∑

i=1

biui, x ∈ Rn,

where b1, . . . , bm are the columns of the input matrix B.

Clearly, the Lie brackets of the constant input vector fields given by

the input vectors b1, . . . , bm are all zero, i.e.,

[bi, bj ] = 0, for all i, j = 1, . . . , m.

Furthermore, the Lie bracket of the linear drift vector field Ax with

an input vector field bi yields the constant vector field

[Ax, bi] = −Abi.

The Lie brackets of Abi with Abj or bj are again all zero, while

[Ax,−Abi] = A2bi.


Hence we conclude that C is spanned by all constant vector fields

bi, Abi, A2bi, . . . , i ∈ m, together with the linear drift vector field Ax,

i.e.,

C = {Ax, bi, Abi, A2bi . . . , An−1bi, i = 1, . . . , m}.

while

C0 = columns of (B, AB, A2B . . . , An−1B)

We see that for linear systems the rank condition for strong

accessibility coincides with the Kalman rank condition for

controllability. Hence, if we would not have known anything special

about linear systems, then at least a linear system which satisfies

the Kalman rank condition is locally strongly accessible.


Example 12 (Actuated rotating rigid body)

Consider

ω1

ω2

ω3

=

A1ω2ω3

A2ω3ω1

A3ω1ω2

+

α1

0

0

u1 +

0

α2

0

u2

with α1 6= 0, α2 6= 0. Here the constants A1, A2, A3 are determined by

the moments of inertia a1, a2, a3. Compute

[g1, f ](ω) =

0

α1A2ω3

α1A3ω2

[g2, f ](ω) =

α2A1ω3

0

α2A3ω1


On the other hand

[g2, [g1, f ]] =

0

0

α1α2A3

Thus the system is locally strongly accessible if

A3 6= 0

which is equivalent to a1 6= a2.

In fact, this is the if and only if condition. Indeed, if A3 = 0 then

ω3 = 0, showing that the system is not locally strongly accessible.

Remark 13 Due to the specific properties of the drift vector field,

i.c. Poisson stability, it can be shown that the system is in fact

controllable if and only if the two first moments of inertia a1 and

a2 are different.)

Documents

Lecture 2: Controllability of nonlinear systems · DISC Systems and Control Theory of Nonlinear Systems 2 Recall: Kinematic model of the unicycle x˙1 = u1 cosx3 x˙2 = u1 sinx3 x˙3