constructive invariant manifolds to stabilize pendulum-like ...jaar/datos/talks/ifac08_web.pdf'...

© 2008

Constructive Invariant Manifolds toStabilize Pendulum–like systems Via

Immersion and InvariancePlease, let us be constructive !

J. A. Acosta♦, R. Ortega♠, A. Astolfi♣ and I. Sarras♠

[email protected], [email protected], [email protected],

[email protected]

♦University of Sevilla (Spain)♠Supelec (France)

♣Imperial College (UK) & University of Rome (Italy)

J.A. Acosta et al. - Please, let us be constructive ! - IFAC World Congress, July 6-11, 2008, Seoul, Korea – p.1/15

© 2008

OutlineMotivation.

Framework. I&I for stabilization.

The pendulum on a cart:

Assumptions.

Stability results.

Simulations and Experiments.

Extensions:

Solution for pendulum-like systems.

General UMS.

Conclusions and future research.


© 2008

MotivationNonlinear methodologies:

Forwarding.Backstepping.Energy Shaping.Feedback Linearization.Immersion & Invariance (I&I).Sliding modes....

Which of them are constructive ?Cascade systems: Feedback & forward.Energy Shaping and I&I: Classes ofUnderactuated Mechanical Systems (UMS).


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Framework - Immersion & Invariance (I&I)x = f(x) + g(x)u, x ∈ R

n, u ∈ Rm and x∗ ∈ R

n.

p < n: α(·) : Rp → R

p, π(·) : Rp → R

n, c(·) : Rp → R

m,

φ(·) : Rn → R

n−p, ψ(·, ·) : Rn×(n−p) → R

m

(H1) Target system: ξ = α(ξ), with state ξ ∈ Rp and

ξ∗ ∈ Rp is an AS equilibrium, with x∗ = π(ξ∗).

(H2) Immersion condition: For all ξ ∈ Rp

f(π(ξ)) + g(π(ξ))c(π(ξ)) =∂π

∂ξα(ξ).

(H3) Implicit manifold:

{x ∈ Rn | φ(x) = 0} = {x ∈ R

n | x = π(ξ) for some ξ ∈ Rp}.

(H4) Manifold attractivity and trajectory boundedness:

z =∂φ

∂x[f(x) + g(x)ψ(x, z)]

(

z = φ(x), limt→∞

z(t) = 0

)

x = f(x) + g(x)ψ(x, z)

Then x∗ is an AS equilibrium of the CL x = f(x) + g(x)ψ(x, φ(x)).

PSfrag replacements

φ(x) = 0


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Pendulum on a Cart (PoC) - Key Idea !

PSfrag replacements

x1

x1

x2

x3

x3

ξ1

ξ2

ξ1

π

Σ :

x1 = x2

x2 = a sinx1 − u b cosx1

x3 = u

ΣT :

ξ1 = ξ2

ξ2 = −V ′(ξ1) −R(ξ1, ξ2)ξ2


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PoC - H1. Target system

Single pendulum:Energy: H(ξ1, ξ2) = 1

2ξ22

+ V (ξ1)

Dynamics:

ΣT :

{

ξ1 = ξ2,

ξ2 = −V ′(ξ1) − R(ξ1, ξ2)ξ2,

Equilibrium ξ∗ = (ξ1, ξ2) = (0, 0).

Assumptions to render ξ? AS:Potential V (ξ1): V ′(0) = 0 and V ′′(0) > 0.Damping R(ξ): R(ξ∗) > 0.

PSfrag replacements

ξ1


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PoC - H2. Immersion condition

Mapping: π(ξ) =

ξ1

ξ2

π3(ξ1, ξ2)

PDE:(

b cos ξ1∂π3

∂ξ1−R(ξ1, ξ2)∆(ξ)

)

ξ2 = a sin ξ1 + ∆(ξ)V ′(ξ1),

with ∆(ξ) , 1 + ∂π3

∂ξ2

b cos ξ1.

Assumptions to solve the PDE:∂π3

∂ξ2

= Φ1(ξ1) ⇒ ∆ 6= Φ2(ξ2).

|∆(0)| =∣

∣

∣1 + b∂π3

∂ξ2

(0)∣

∣

∣≥ ε > 0, with ε > 0.

The solution reads:V ′(ξ1) = −

a sin ξ1∆(ξ)

, R(ξ1, ξ2) =b cos ξ1∆(ξ)

∂π3

∂ξ1.

PSfrag replacements

x1

x1

x2

x3

x3

ξ1

ξ2

ξ1

π


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PoC - H3 & H4H3. Implicit manifold: M = {x ∈ R

3 | φ(x) = 0} with

z := φ(x) = x3 − π3(x1, x2) (Off − the − manifold).

H4. Manifold attractivity and trajectory boundedness:

z = x3 − π3(x1, x2)

= ψ(x, z) −∂π3

∂x1x2 −

∂π3

∂x2(a sinx1 − b cosx1ψ(x, z))

= −∂π3

∂x1x2 −

∂π3

∂x2a sinx1 + ∆(x1, x2) ψ(x, z) =: −γz, γ > 0.

The controller:

ψ(x, z)∣

∣

∣

z=φ(x)=

1

∆(x1, x2)

(

−γz +∂π3

∂x1x2 +

∂π3

∂x2a sinx1

)

∣

∣

∣

z=φ(x)

PSfrag replacements

φ(x) = 0


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PoC - Stability resultProposition. Any function π3 verifying the Assumptions,and with the functions V and R defined, renders the zeroequilibrium of the cart–pendulum system LAS.Proof. Boundedness: H(x1, x2) ⇒ x1, x2; (x1, x2) ⇒ π3

and π3 + z ⇒ x3. �Some selections

π3(x1, x2) ∆(x1) V (ξ1)

−k1x1 − k2x2 1 − k2b cosx1 − ak2b

(ln |1 − k2b cos ξ1| − ln |1 − k2b|)

a√k2b

(

− tanh−1(

k2b√k2b

)

−k1x1 − k2x2 cosx1 1 − k2b cos2 x1

+tanh−1(

k2b√k2b

cos ξ1))

−k1x1 − k2x2

cos x11 − k2b − a

k2b−1(−1 + cos ξ1)

Constructive ! What else ... ?


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PoC - Simulations

π3(x1, x2) , −k1x1 − k2x2

0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

1.5

t (sec)

Pen

dulu

m a

ngle

x1 (r

ad)

0 5 10 15 20−4

−3

−2

−1

0

1

2

t (sec)

Pen

dulu

m a

ngul

ar v

eloc

ity x

2 (rad

/sec

)

0 5 10 15 20−5

0

5

10

15

20

t (sec)

Car

t vel

ocity

x3 (m

/sec

)

0 5 10 15 20−10

−5

0

5

10

15

20

t (sec)

Con

trol i

nput

u

Scope. γ = 1, k1 = 3 and k2 = 4.

−6

−4

−2

0

2

−2

−1

0

1

2−1

0

1

2

3

4

5

Pendulum angular velocity x2

Pendulum angle x1

Dis

tanc

e z

from

inva

riant

man

ifold

φ (x

)=0

γ1 = 1

γ2 = 10

Manifold z=0

Attractivity.


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PoC - Experiments

Laboratoire d’Automatique, Supelec (France).

Control system. Set-up.

0 5 10 15−0.1

0

0.1

0.2

0.3

0.4

Time (sec.)

x 1 (rad

)

0 5 10 15−5

0

5

10

Time (sec.)

x 2 (rad

/sec

)

0 5 10 15−3

−2

−1

0

1

2

Time (sec.)

x 3 (rad

/sec

)0 5 10 15

−10

−5

0

5

10

Time (sec.)

u (v

olt)

State/time. k1 = 8, k2 = 3 and γ = 8.


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Extensions - Class of UMS (Automatica’08)Σ : x1, x2 ∈ R

p and x3, u ∈ Rm and g2(0) 6= 0, and

ΣT : 2p–dimensional pendulum with ξ1, ξ2 ∈ Rp

Σ :

x1 = x2

x2 = f2(x1) + g2(x1)u

x3 = f3(x1) + u

ΣT :

ξ1 = ξ2

ξ2 = −V ′(ξ1) −R(ξ1, ξ2)ξ2

Assumptions:∂V∂ξ1

(0) = 0 and ∂2V∂ξ2

1

(0) > 0.

R(0, 0) > 0.∂π3

∂ξ2

6= Φ1(ξ2) & det ∆(0) 6= 0(

∆ := Ip − g2(ξ1)[∂π3

∂ξ2

(ξ1)]>

)

.

Solution of PDEs:∂V

∂ξ1= ∆−1(g2f3 − f2), R = −∆−1g2(

∂π3

∂ξ1)>.

Off-the-manifold: det ∆(0) 6= 0(

∆(ξ1) := Im − [∂π3

∂ξ2

(ξ1)]>g2(ξ1)

)

.


© 2008

Conclusions & future researchB Conclusions:

Constructive solution of a class of pendulum-like systems.

No PDEs need to be solved !

No restrictions of underactuation degree.

B Current and future research:

Further extensions to a general UMS.

Target dynamics:

ΣT :

ξ1 = ξ2

ξ2 = −V ′(ξ1) −R(ξ1, ξ2)ξ2 + J(ξ1, ξ2)ξ2

J(ξ1, ξ2) possibly J(·, 0) = 0.

PSfrag replacementsx1

x2

x3


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