Corrections to "A Global High-Gain Finite-Time Observer"

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Corrections to ”A Global High-Gain Finite-Time Observer”

Tomas Menard, Emmanuel Moulay, Wilfrid Perruquetti

To cite this version:

Tomas Menard, Emmanuel Moulay, Wilfrid Perruquetti. Corrections to ”A Global High-Gain Finite-

Time Observer”. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics

Engineers, 2017, 62 (1), pp.509 - 510. �10.1109/TAC.2016.2518742�. �hal-01497178�

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1

Corrections to "A Global High-Gain Finite-Time Observer"

Tomas Ménard, Emmanuel Moulay and Wilfrid Perruquetti Abstract—This note fix the proof of Theorem 2 in the article [2].

Equations from the original paper will be denoted with a star (for example (1

^∗

)) whereas equations of the present corrected paper will be denoted without a star (for example (1)).

I. T

HE ERROR

The function V ˜

_α

used in the proof of Theorem 2 in [2], and derived from Theorem 10 in [3], is not C

¹

with respect to (e, α). Indeed, one has

∂

de

k

c

^αk−1¹ ^q

∂e

_k

= 1

α

_k

q de

_k

c

^αkq¹ ⁻¹

. (1) Hence, when α → 1,

_α¹

kq

→ 1 and when one of the component of e goes to zero, the limit lim

_{(α,e)→(1,e}₀₎_α¹

kq

de

k

c

^αkq¹ ⁻¹

does not exist. Thus the function V ˜

α

cannot be used as a candidate Lyapunov function.

II. T

HE FIX

Let us first recall Theorem 2 from [2].

Theorem 1. Let us consider system (3

^∗

) with a bounded input u. Then there exists θ

^∗

≥ 1 such that for all θ > θ

^∗

there exists > 0 such that system (3

^∗

) admits the following global finite-time observer:



 



 



˙ˆ

x

₁

= ˆ x

₂

+ k

₁

(de

1

c

^α¹

+ ρe

₁

) + P

m

j=1

g

_1,j

(ˆ x

₁

)u

_j

.. .

˙ˆ

x

n

= k

n

(de

1

c

^αⁿ

+ ρe

1

) + ϕ(ˆ x) + P

m

j=1

g

n,j

(ˆ x)u

j

for all α ∈]1 − , 1[, where e

1

= x

1

− x ˆ

1

, the powers α

i

are defined by (5

^∗

), the gains k

i

by (6

^∗

) and ρ =

n²θ²3S1+1 2

where S

₁

is defined by (8

^∗

).

In addition, the settling time of the error dynamics is bounded by T

₁

(e

₀

) + T

₂

(e

₀

) (with e

₀

= x

₀

− x ˆ

₀

), where T

₁

, T

₂

are respectively given by (18

^∗

) and (6).

The statement of Theorem 2 in [2] remains correct, except for the settling time which has to be corrected.

We can define the function V

1

(e) = e

^T

S

_∞

(1)e, for e ∈ R

ⁿ

, where S

_∞

(1) is the solution of (7

^∗

) for θ = 1. This choice

GREYC, UMR-CNRS 6072, Université de Caen, 6 Bd du Maréchal Juin, BP 5186-14032 Caen Cedex, France. (e-mail:

[email protected])

Xlim, UMR-CNRS 7252, Université de Poitiers, 11 bd Marie et Pierre Curie, 86962 Futuroscope Chasseneuil Cedex, France. (e-mail:

[email protected])

NON-A, INRIA Lille Nord Europe and CRIStAL UMR-CNRS 9189, Ecole Centrale de Lille, BP 48, 59651 Villeneuve D’Ascq, France. (e-mail:

[email protected])

corresponds to the linear case, that is α = 1. Proceeding as in [4], [5], one can construct a candidate Lyapunov function with properties stated next.

Proposition 1. Let a ∈ C

^∞

( R , R ) be such that a =

( 0 on (−∞, 1]

1 on [2, +∞) and a

⁰

≥ 0 on R . (2) There exists > 0 such that for all α ∈]1 − , 1 + [, the function V ¯

α

defined as

V ¯

_α

(e) = Z

+∞

0

1 t

³

(a ◦ V

₁

)(t

^r¹^(α)

e

₁

, . . . , t

^rⁿ^(α)

e

_n

)dt (3) if e ∈ R

ⁿ

\{0} and V ¯

_α

(0) = 0 is well defined, radially unbounded, of class C

¹

( R

ⁿ

, R ), and satisfies

a) V ¯

α

(δ

^r(α)_λ

e) = λ

²

V ¯

α

(e), for all e ∈ R

ⁿ

and λ > 0.

b) h∇ V ¯

α

(e), Ae−F(S

_∞⁻¹

(1)C

^T

, e)i ≤ −γ( ¯ V

α

(e))

^1+α²

, for all e ∈ R

ⁿ

, where γ > 0.

where F ,C and δ

^r(α)_λ

are defined in [2].

Proof of proposition 1. Let, α ∈]1 −

_n¹

, +∞[, proceeding as in [4], one directly shows that V ¯

_α

is well defined, radially unbounded, C

¹

on R

ⁿ

, and homogeneous of degree 2 with respect to the weights r(α). Then, only point b) remains to prove.

Following the same lines as in [4], there exists l, L > 0 such that for all e ∈

e ∈ R

ⁿ

| V ¯

α

(e) = 1 , one has h∇ V ¯

α

(e), Ae−F (S

_∞⁻¹

(1)C

^T

, e)i =

Z

L l

1 t

^α+2

a

⁰

V

1

δ

_t^r(α)

e

× D ∇V

1

δ

_t^r(α)

e

, Aδ

^r(α)_t

e − F

S

_∞⁻¹

(1)C

^T

, δ

^r(α)_t

e E dt

Consider the function g(e, t, α) =

D ∇V

1

δ

_t^r(α)

e

, Aδ

_t^r(α)

e − F

S

_∞⁻¹

(1)C

^T

, δ

_t^r(α)

e E

, where (e, t, α) ∈ {e ∈ R

ⁿ

, V ¯

_α

(e) = 1} × {t ∈ [l, L]}×]1 −

_n¹

, +∞[.

The function g is continuous, (e, t) belongs to a compact set and there exists γ

1

> 0 such that the image of g is included in ] − ∞, −γ

1

[ for (e, t) ∈ {e ∈ R

ⁿ

, V ¯

α

(e) = 1} × {t ∈ [l, L]}

and α = 1 (since it corresponds to the linear case).

We can then apply Lemma 26.8 in [1] (tube lemma) which gives the existence of > 0 such that for all (e, t, α) ∈ {e ∈ R

ⁿ

, V ¯

α

(e) = 1} × {t ∈ [l, L]}×]1 − , 1 + [, g(e, t, α) ≤ −γ

1

.

Then we have

h∇ V ¯

_α

(e), Ae − F (S

_∞⁻¹

(1)C

^T

, e)i

≤ −γ

1

Z

L l

1 t

^α+2

a

⁰

V

1

δ

^r(α)_t

e

dt ≤ −γ V ¯

α

(e)

^2+α−1₂

(4)

where γ > 0 is a lower bound of

γ

1

R

L l

1 t^α+2

a

⁰

V

1

δ

^r(α)_t

e

dt for (e, α) ∈ {e ∈ R

ⁿ

, V ¯

α

(e) = 1}×]1 − , 1 + [. Since V ¯

α

is homogeneous of degree 2 with respect to the weights r(α), inequality (4) is valid for all e ∈ R

ⁿ

.

Now that a new candidate Lyapunov function has been

defined, we explain how it will be used to correct the proof

(3)

2

of Theorem 2 in [2]. Please note that part 1 of the proof is correct, then it has already been proved that every trajectory starting from e

0

∈ R

ⁿ

enter the ball B

_k.k_S∞(θ)

(1) after time T

1

(e

0

) = log(1/V (e

0

))/κ(θ) (see equation (18

^∗

)).

Denote e ¯ = ∆

θ

e, where ∆

θ

= diag

1

¹_θ

. . .

_θn−1¹

, in the remaining, we will show that for every θ ≥ θ

₂

=

⁴ ²_γ

(M

₁

+ 2), there exists > 0 such that the following inequality

V ˙¯

α

(¯ e) ≤ − γ 2 θ − 1

V ¯

α

(¯ e)

^2+α−1₂

+ M

1

V ¯

α

(¯ e) (5) holds for every e ¯ ∈ B

_k.k_S

∞(1)

(1), α ∈]1 − , 1[ , where M

1

> 0 is a constant independent of θ. This inequality replaces inequality (19

^∗

). Inequality (5) directly implies that the error system (11*) is finite time stable on B

_k.k_S∞(θ)

(1).

Thus, after time T

₁

(e

₀

), the error enters B

_k.k_S∞(θ)

(1) and after time T

₁

(e

₀

) + T

₂

(e

₀

) the error reaches the origin, where the settling time T

₂

(e

₀

) is bounded as follows

T

2

(e

0

) ≤ ln

1 −

γ^M¹

2θ−1

V ¯

_α

(e

₀

)

¹⁻^2+α−1²

M

1

(

^2+α−1₂

− 1) . (6) The remaining of the corrected proof is very similar to the original one. The dynamics of e ¯ is given by

˙¯

e = θ A¯ e − F S

_∞⁻¹

(1)C

^T

, e ¯

− ρS

_∞⁻¹

(1)C

^T

C¯ e +∆

_θ

D(x, x, u). ˆ

One has

V ˙¯

α

(¯ e) =

⁴

θ W ¯

1

+ ¯ W

2

(7) with W ¯

1

= h∇ V ¯

α

(¯ e), A¯ e−F(S

_∞⁻¹

(1)C

^T

, e)−ρS ¯

_∞⁻¹

(1)C

^T

C¯ ei and W ¯

2

= h∇ V ¯

α

(¯ e), ∆

θ

D(x, x, u)i. ˆ

Following the same lines as in [2], one can show that there exists θ

₂

≥ 0 such that for every θ ≥ θ

₂

, there exists > 0 such that for all α ∈]1 − , 1[ inequality (5) holds true.

R

EFERENCES

[1] J. Munkres. Topology. Prentice Hall, 2 edition, 1999.

[2] T. Ménard, E. Moulay, and W. Perruquetti. A global high-gain finite-time observer. IEEE Transactions on Automatic Control, 55(6):1500–1506, 2010.

[3] W. Perruquetti, T. Floquet, and E. Moulay. Finite-time observers:

application to secure communication. IEEE Transactions on Automatic Control, 53(1):356–360, 2008.

[4] L. Rosier. Homogeneous lyapunov function for homogeneous continuous vector field.Systems & Control Letters, 19(6):467–473, 1992.

[5] H. Yu, Y. Shen, and X. Xia. Adaptive finite-time consensus in multi-agent networks.Systems & Control Letters, 62(10):880 – 889, 2013.