DISC Systems and Control Theory of Nonlinear Systems, 2010 1 Stabilization and Passivity-Based Control Lecture 8 Nonlinear Dynamical Control Systems, Chapter 10, plus handout from R. Sepulchre, Constructive Nonlinear Control (pp.25–70, pp.229–239) available at www.montefiore.ulg.ac.be/˜sepulch
tutorial on passivity-based control for dynamic systems.
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DISC Systems and Control Theory of Nonlinear Systems, 2010 1
Stabilization and Passivity-Based Control
Lecture 8
Nonlinear Dynamical Control Systems, Chapter 10,
plus handout from
R. Sepulchre, Constructive Nonlinear Control (pp.25–70, pp.229–239)
available at www.montefiore.ulg.ac.be/˜sepulch
DISC Systems and Control Theory of Nonlinear Systems, 2010 2
Standard definitions on the stability of an autonomous system
x = f(x)
where x = (x1, . . . , xn) are local coordinates for X .
Let x0 be an equilibrium point, i.e.
f(x0) = 0 .
The equilibrium point x0 is stable if for any neighborhood V of x0
there exists a neighborhood V of x0 such that if x ∈ V , then the
solution x(t, 0, x) belongs to V for all t ≥ 0.
The equilibrium x0 is unstable if it is not stable.
The equilibrium x0 is (locally) asymptotically stable if x0 is stable
and there exists a neighborhood V0 of x0 such that all solutions
x(t, 0, x) with x ∈ V0, converge to x0 as t → ∞.
The equilibrium x0 is globally asymptotically stable if V0 = X
DISC Systems and Control Theory of Nonlinear Systems, 2010 3
In Lyapunov’s first method the local stability of x0 is related to
the stability of the linearization around the equilibrium point
˙x = Ax ,
with
A =∂f
∂x(x0) .
Theorem 1 (First method of Lyapunov) The equilibrium x0 is
asymptotically stable if the matrix A is asymptotically stable, i.e.,
the matrix A has all its eigenvalues in the open left half plane. The
equilibrium point x0 is unstable if at least one of the eigenvalues of
the matrix A has a positive real part.
DISC Systems and Control Theory of Nonlinear Systems, 2010 4
Consider the control system
x = f(x, u) ,
where x = (x1, . . . , xn) are local coordinates for a smooth manifold
X , u = (u1, . . . , um) ∈ U ⊂ Rm, the input space, and f(·, u) a smooth
vector field for each u ∈ U .
We assume U to be an open part of Rm and that f depends
smoothly on the controls u.
Let (x0, u0) an equilibrium point:
f(x0, u0) = 0 .
(N.B.: usually u0 = 0.)
DISC Systems and Control Theory of Nonlinear Systems, 2010 5
Problem 2 Under which conditions does there exist a smooth
strict static state feedback u = α(x), α : X → U , with α(x0) = u0,
such that the closed loop system
x = f(x, α(x))
has x0 as an asymptotically stable equilibrium?
DISC Systems and Control Theory of Nonlinear Systems, 2010 6
Consider the linearization of the system around the point (x0, u0)
˙x = Ax + Bu ,
where
A =∂f
∂x(x0, u0) , B =
∂f
∂u(x0, u0) .
Define R as the reachable subspace of the linearized system
R = im[
B AB · · · An−1B]
Clearly R is invariant under A, i.e., AR ⊂ R, so after a linear
change of coordinates
d
dt
x1
x2
=
A11 A12
0 A22
x1
x2
+
B1
0
u
where the vectors (x1, 0)T correspond to vectors in R.
DISC Systems and Control Theory of Nonlinear Systems, 2010 7
Theorem 3 The feedback stabilization problem admits a local
solution around x0 if all eigenvalues of the matrix A22 are in C−,
the open left half plane of C. Moreover if one of the eigenvalues of
A22 has a positive real part, then there does not exist a solution to
the local feedback stabilization problem.
DISC Systems and Control Theory of Nonlinear Systems, 2010 8
Consider the linearized dynamics around (x0, u0) and assume all
eigenvalues of A22 belong to C−. Then by linear control theory
there is a linear state feedback u = F x which asymptotically
stabilizes the linearized system. (We may actually take u = F1x1.)
Then the affine feedback u = u0 + F (x− x0) for the nonlinear system
yields the closed loop system
x = f(x, u0 + F (x − x0)) ,
of which the linearization around x0 equals
˙x = (A + BF )x .
By construction this linear dynamics is asymptotically stable and so
by Lyapunov’s first method x0 is an asymptotically stable
equilibrium point.
DISC Systems and Control Theory of Nonlinear Systems, 2010 9
Next suppose that at least one of the eigenvalues of the matrix A22
has a positive real part. Let u = α(x) be an arbitrary smooth
feedback with α(x0) = u0. Linearizing the dynamics around x0 yields
˙x = (A + B∂α
∂x(x0))x ,
which still has the same unstable eigenvalue of the matrix A22, and
thus x0 is an unstable equilibrium point of the closed loop system.
DISC Systems and Control Theory of Nonlinear Systems, 2010 10
One step further by using center manifold theory
Suppose the set of eigenvalues of A, σ(A), can be written as the
disjoint union
σ(A) = σ− ∪ σ0 ,
where the eigenvalues in σ− lie in the open left half plane and those
in σ0 lie on the imaginary axis. Let l be the number of eigenvalues
(counted with their multiplicity) contained in σ−, so that there are
n − l eigenvalues (counted with their multiplicity) in σ0 .
Then there exists a linear coordinate transformation T such that
TAT−1 =
A0 0
0 A−
where the (n − l, n − l)-matrix A0 and the (l, l)-matrix A− have as
eigenvalues σ(A0) = σ0, respectively σ(A−) = σ−.
DISC Systems and Control Theory of Nonlinear Systems, 2010 11
In the transformed coordinates z = Tx − x0 the nonlinear system
takes the form
z1 = A0z1 + f0(z1, z2)
z2 = A−z2 + f−(z1, z2)
where
f0(0, 0) = 0 ,
f−(0, 0) = 0 ,
df0(0, 0) = 0 ,
df−(0, 0) = 0 .
DISC Systems and Control Theory of Nonlinear Systems, 2010 12
Theorem 4 (Center Manifold Theorem) For each k = 2, 3, . . .
there exists a δk > 0 and a Ck-mapping
φ : {z1 ∈ Rn−l | ||z1|| < δk} → Rl with φ(0) = 0 and dφ(0) = 0, such that
the submanifold (the center manifold)
z2 = φ(z1) , ||z1|| < δk ,
is invariant under the nonlinear dynamics.
Remark 5 (i) In general the nonlinear dynamics does not possess
a unique center manifold, but may have an infinite number of such
invariant manifolds.
(ii) The smooth dynamics has a Ck center manifold for each
(finite) k = 2, 3, . . .. However the size of the center manifold (δk
depends on k and may shrink with increasing k. Even in case the
system is analytic, there does not necessarily exist an analytic
center manifold.
DISC Systems and Control Theory of Nonlinear Systems, 2010 13
Theorem 6 The dynamics on the center manifold are given as
z1 = A0z1 + f0(z1, φ(z1)) .
If z1 = 0 is asymptotically stable, stable, or unstable, respectively,
for this center manifold dynamics then (z1, z2) = 0 is asymptotically
stable, stable or unstable for the full-order system.
The linearized system resulting from applying the linear feedback
u = u0 + F1(x1 − x1
0) + F2(x2 − x2
0)
A11 + B1F1 A12 + B1F2
0 A22
Although the matrix F2 does not affect the eigenvalues, it does
affect the orientation of the imaginary eigenspace, and thus the
dynamics on the center manifold.
DISC Systems and Control Theory of Nonlinear Systems, 2010 14
Second or direct method of Lyapunov
A smooth function L defined on some neighborhood V of x0 is
positive definite if L(x0) = 0 and L(x) > 0 for all x 6= x0.
A set W in M is an invariant set if for all x ∈ W the solutions
x(t, 0, x) belong to W for all t.
Theorem 7 (Second method of Lyapunov) Consider the
dynamics x = f(x) around the equilibrium point x0. Let L be a
positive definite function on some neighborhood V0 of x0. Then x0
is stable if
LfL(x) ≤ 0 , ∀x ∈ V0
The function L is called a Lyapunov function.
DISC Systems and Control Theory of Nonlinear Systems, 2010 15
Furthermore, x0 is asymptotically stable if
LfL(x) < 0 , ∀x 6= x0
or more generally if the largest invariant set contained in the set
W = {x ∈ V0 |LfL(x) = 0}
equals {x0}; i.e. the only solution x(t, 0, x) starting in x ∈ W which
remains in W for all t ≥ 0, coincides with x0.
(This called LaSalle’s Invariance principle.)
DISC Systems and Control Theory of Nonlinear Systems, 2010 16
Consider the system
x = f(x) +
m∑
i=1
gi(x)ui
with
f(x0) = 0 .
Suppose there exists a Lyapunov function L for the dynamics with
u ≡ 0
LfL(x) ≤ 0 , ∀x ∈ V0 .
so x0 is already stable for the system with u = 0.
DISC Systems and Control Theory of Nonlinear Systems, 2010 17
Consider the smooth feedback u = α(x) defined as
αi(x) = −LgiL(x) , i = 1, . . . , m , x ∈ V0
yielding the closed loop behavior
x = f(x) +m
∑
i=1
gi(x)αi(x) .
satisfying
LfL(x) +m
∑
i=1
LαigiL(x) = LfL(x) −
m∑
i=1
(LgiL(x))2 ≤ 0 ,
DISC Systems and Control Theory of Nonlinear Systems, 2010 18
In order to study the asymptotic stability of x0 we introduce the set
W = {x ∈ V0 |LfL(x) −∑m
i=1(LgiL(x))2 = 0}
= {x ∈ V0 |LfL(x) = 0, LgiL(x) = 0, i ∈ m}.
Let W0 be the largest invariant subset of W under the dynamics.
Observe that any trajectory xα(t, 0, x) in W0 is a trajectory of the
original dynamics; this because the feedback u = α(x) is identically
zero for each point in W .
DISC Systems and Control Theory of Nonlinear Systems, 2010 19
Consider the distribution
D(x) = span{f(x), adkfgi(x), i ∈ m, k ≥ 0} , x ∈ V0 .
Lemma 8 Suppose there exists a Lyapunov-function L for x = f(x)
on a neighborhood V0 of the equilibrium point x0. Suppose that
dimD(x0) = n ,
which implies that on some neighborhood V0 ⊂ V0 of x0
dimD(x) = n .
Then x0 is asymptotically stable for the closed loop system. The
same holds if
dimD(x) = n , for all x ∈ V0 \ {x0}
DISC Systems and Control Theory of Nonlinear Systems, 2010 20
Example 9 Consider the equations for the angular velocities of a
rigid body with one external torque
Iω = S(ω)Iω + bu
with ω = (ω1, ω2, ω3),
S(ω) =
0 ω3 −ω2
−ω3 0 ω1
ω2 −ω1 0
I =
I1 0 0
0 I2 0
0 0 I3
I3 > I2 > I1 > 0 denote the principal moments of inertia. Let
I23 = (I2 − I3)/I1 ,
I31 = (I3 − I1)/I2 ,
I12 = (I1 − I2)/I3 .
DISC Systems and Control Theory of Nonlinear Systems, 2010 21
Then the dynamics may be written as
ω1 = I23ω2ω3 + c1u
ω2 = I31ω3ω1 + c2u
ω3 = I12ω1ω2 + c3u
with c = (c1, c2, c3)T = I−1b. An obvious choice for a Lyapunov
function for the drift vector field is the kinetic energy of the rigid
body, i.e.
L(ω) = 12 (I1ω
21 + I2ω
22 + I3ω
23) .
L is a smooth positive definite function having a unique minimum