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.
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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
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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.
DISC Systems and Control Theory of Nonlinear Systems 2
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
DISC Systems and Control Theory of Nonlinear Systems 3
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 .
DISC Systems and Control Theory of Nonlinear Systems 4
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?
DISC Systems and Control Theory of Nonlinear Systems 5
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.
DISC Systems and Control Theory of Nonlinear Systems 6
Define
g1(x) =
cos(x3 + x4)
sin(x3 + x4)
sin(x4)
0
︸ ︷︷ ︸
Drive
, g2(x) =
0
0
0
1
︸ ︷︷ ︸
Steer
.
DISC Systems and Control Theory of Nonlinear Systems 7
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.
DISC Systems and Control Theory of Nonlinear Systems 8
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}.
DISC Systems and Control Theory of Nonlinear Systems 9
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]], . . .
DISC Systems and Control Theory of Nonlinear Systems 10
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.
DISC Systems and Control Theory of Nonlinear Systems 11
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.
DISC Systems and Control Theory of Nonlinear Systems 12
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