Stabilization and Passivity-based Control

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


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


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.


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.)


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?


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.


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.


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.


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.


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−.


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 .


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.


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.


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.


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.)


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.


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 ,


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 .


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}


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 .


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

at ω = 0. Computing LfL yields

LfL(ω) = I1I23ω1ω2ω3 + I2I31ω1ω2ω3 + I3I12ω1ω2ω3 = 0 ,

which shows that ω = 0 is a stable equilibrium point when u = 0.


Define the smooth feedback

u = −LcL(ω) = −(c1I1ω1 + c2I2ω2 + c3I3ω3) .

Assume that

c1c2c3 6= 0 .

(So the control axis is not perpendicular to any of the principal

axes of the rigid body.)

Then the above feedback asymptotically stabilizes ω = 0.


Passive systems theory

A system with outputs

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

y = h(x)

is called passive if there exists a function S(x) with S(x) ≥ 0 for all

x if and only ifd

dtS ≤ yT u

In physical situations S is often the ’stored energy’ in the system,

while yT u is the supplied power. Formally, S(x) is called the

storage function and s(y, u) = yT u the supply rate.


ddt

S ≤ yT u is equivalent to

∂T S

∂x(x)[f(x) + g(x)u] ≤ hT (x)u

for all x and u, or equivalently

LfS(x) = ∂T S∂x

(x)f(x) ≤ 0

h(x) = gT (x)∂S∂x

(x)

Hence, passivity can be regarded as an extension of the notion of

Lyapunov function.


Furthermore, the above feedback u = α(x) can be seen to be equal

to

u = −gT (x)∂S

∂x(x) = −y

leading tod

dtS ≤ yT u = −||y||2

and asymptotic stability of a minimum of S is guaranteed if the

largest invariant subset contained in the set where the passive

output y is equal to zero is equal to this minimum.


Two fundamental properties of passive systems

1) In general a passive system has uniform relative degree 1.

Indeed, since

yj = LgjS

the input-output decoupling matrix is generally given as

[LgiLgj

S]i,j=1,··· ,p

which has full rank if the Hessian matrix of the storage function S

has full rank.

2) The zero-dynamics of a passive system is stable. Indeed, by

substituting y = 0 in ddt

S ≤ yT u it follows that the zero-dynamics

satisfiesd

dtS ≤ 0


Both properties are if and only if conditions for ’passifiability’.

Rewrite the system satisfying both conditions as

z = q(z, ξ)

ξ = a(z, ξ) + b(z, ξ)u

y = ξ

with b(z, ξ) full rank. Furthermore, factorize

q(z, ξ) = q(z, 0) + p(z, ξ)ξ

Since q(z, 0) is asymptotically stable there exists a Lyapunov

function W (z) for z = q(z, 0). Then

S(z, ξ) := W (z) +1

2ξT ξ

is a Lyapunov function for the system after applying the feedback

u = b−1(z, ξ)(−a(z, ξ) − Lp(z,ξ)W + v)


Passivity interpretation of ’backstepping’ procedure for

stabilization

Suppose we want to stabilize the system

x1 = f1(x1, x2)

x2 = f2(x1, x2, x3)

x3 = f3(x1, x2, x3, u)

Construct a ’passive output’

y = x3 − α2(x1, x2)

such that the zero-dynamics

x1 = f1(x1, x2)

x2 = f2(x1, x2, α2(x1, x2))

is asymptotically stable.


How to find α(x1, x2) ? This is a stabilization problem for a

reduced-order dynamics. Can be solved by constructing a ’passive’

output

y = x2 − α1(x1)

such that the zero dynamics

x1 = f1(x1, α1(x1))

is asymptotically stable.

This procedure can be generalized to arbitrary systems that are

feedback linearizable.


Dual interpretation

Stabilize

x1 = f1(x1, x2)

x2 = f2(x1, x2, x3)

x3 = f3(x1, x2, x3, u)

by working ’top-down’: first consider the system

x1 = f1(x1, v)

and construct a stabilizing feedback v = β1(x1). Define a new

coordinate

z2 = x2 − β(x1)


and rewrite the system as

x1 = f1(x1, β(x1) + z2)

z2 = f2(x1, z2, x3)

x3 = f3(x1, z2, x3, u)

and stabilize the first two lines by a feedback x3 = β2(x1, z2), etc.


Passivity is a compositional property

Consider k passive subsystems with input-output pairs

ui, yi, i = 1, · · · , k and storage functions Si(xi), interconnected in

such a way that

yT1 u1 + yT

2 u2 + · · · + yTk uk = yT u

Then the overall system is again passive with storage function

S1(x1) + S2(x2) + · · · + Sk(xk)


From a network modeling point of view, passive systems arise as

port-Hamiltonian systems:

Port-based network modeling leads to a representation of a

physical system as a graph, where each edge is decorated with a

(vector) pair of flow variables f ∈ Rm, and effort variables

e ∈ Rm, i.e., a bond graph

H1 fH1

eH1

0

R1

1

IC : f = 0

T H2

H3 0fR2

eR2

R2

Figure 1: Port-based network modeling


and each vertex corresponds to one of the following ideal elements:

• Energy-storing elements H:

x = −fH

eH = ∂H∂x

(x), H(x1, · · · , xm) ∈ R energy

• Power-dissipating elements R:

R(fR, eR) = 0, eTRfR ≤ 0

• Power-conserving elements: transformers T, gyrators GY, ideal

constraints IC.

• 0- and 1-junctions:

e1 = e2 = · · · = ek, f1 + f2 + · · · + fk = 0

f1 = f2 = · · · = fk, e1 + e2 + · · · + ek = 0


• 0- and 1-junctions are the basic conservation laws of the

system, and are also power-conserving:

e1f1 + e2f2 + · · · + ekfk = 0

• Transformers, gyrators are energy-routing devices, and may

correspond to exchange between different types of energy.

All power-conserving elements have the following properties in

common. They are described by linear equations:

Ff + Ee = 0, f, e ∈ Rl

whose solutions f, e satisfy

eT f = e1f1 + e2f2 + · · · + elfl = 0,

rank[

F E]

= l

All power-conserving elements taken together define a Dirac

structure.


A geometric definition of port-based network models

Take all power-conserving elements

(T, G, IC, 0- and 1-junctions)

together in a single power-conserving interconnection structure:

H1fH1

eH1

0

R1

1

IC : f = 0

T H2

H3 0fR2

eR2

R2

D

Figure 2: Power-conserving interconnection structure


Input-state-output port-Hamiltonian systems:

Particular case is a Dirac structure D(x) ⊂ TxX × T ∗

xX × F × F∗

given as the graph of the skew-symmetric map

fx

eP

=

−J(x) −g(x)

gT (x) 0

ex

fP

,

leading (fx = −x, ex = ∂H∂x

(x)) to a port-Hamiltonian system as

before

x = J(x)∂H∂x

(x) + g(x)fP , x ∈ X , fP ∈ Rm

eP = gT (x)∂H∂x

(x), eP ∈ Rm


Power-dissipation is included by terminating some of the ports by

static resistive elements

fR = −F (eR), where eTRF (eR) ≥ 0, for all eR.

d

dtH ≤ eT

P fP

This leads, e.g. for linear damping, to input-state-output

port-Hamiltonian systems in the form

x = [J(x) − R(x)]∂H∂x

(x) + g(x)fP

eP = gT (x)∂H∂x

(x)

where J(x) = −JT (x), R(x) = RT (x) ≥ 0 are the interconnection and

damping matrices, respectively.

Stabilization and Passivity-based Control

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