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This document must be cited according to its fjnal version which is published in a conference as: C. Afri, V. Andrieu, L. Bako, P. Dufour, Identifjcation of linear systems with nonlinear Luenberger Observers, 2015 IEEE-IFAC American

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This document must be cited according to its fjnal version which is published in a conference as:

  • C. Afri, V. Andrieu, L. Bako, P. Dufour,

“Identifjcation of linear systems with nonlinear Luenberger Observers”, 2015 IEEE-IFAC American Control Conference (ACC), Chicago, IL, USA,

  • pp. 3373-3378, july 1-3, 2015. DOI :

10.1109/ACC.2015.7171853 You downloaded this document from the CNRS open archives server, on the webpages of Pascal Dufour: http://hal.archives-ouvertes.fr/DUFOUR-PASCAL-C-3926-2008

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1/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Identification of linear systems with nonlinear Luenberger observers

Chouaib Afri1 Vincent Andrieu2 Laurent Bako 3 Pascal Dufour4

LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago

1Ph.D. student France MENRT funding since October 2013 2Supervisor 3Supervisor 4Ph.D Director

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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2/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Outline

1

Problem description

2

Solution by Luenberger observers approach

3

Numerical illustration

4

Perspectives

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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3/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Outline

1

Problem description

2

Solution by Luenberger observers approach

3

Numerical illustration

4

Perspectives

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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4/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

We are looking for a mathematical model that describes the dynamic behaviour in order to better supervise, diagnose or control it. It may belong to a class in the continuous time domain: ˙ x(t) = f (x(t),u(t),t) y(t) = g(x(t),t)

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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5/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Different modelling approaches

Output measurements knowledge Input measurements knowledge → white box model Physical model structure Known parameters

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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5/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Different modelling approaches

Output measurements knowledge Input measurements knowledge → white box model Physical model structure Known parameters Output measurements knowledge Input measurements knowledge → gray box model Physical model structure Unknown parameters

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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5/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Different modelling approaches

Output measurements knowledge Input measurements knowledge → white box model Physical model structure Known parameters Output measurements knowledge Input measurements knowledge → gray box model Physical model structure Unknown parameters Output measurements knowledge Input measurements knowledge → black box model Assumed model structure Unknown parameters

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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6/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

LTI system

Assume that the process has a linear dynamic described by (Σ)

  • ˙

x(t) = A(θ)x(t)+B(θ)u(t) y(t) = C(θ)x(t), with: u ∈ R in L ∞

loc(R+).

y ∈ R. x ∈ Rn. θ ∈ Θ ⊂ Rq. A(·),B(·) and C(·) are sufficiently smooth and known.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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7/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Estimation problem

Goal

Estimate on-line the state and parameters (x(t),θ) by knowing the inputs and outputs.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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7/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Estimation problem

Goal

Estimate on-line the state and parameters (x(t),θ) by knowing the inputs and outputs.

Method

Construct and asymptotic observer for the augmented system.    ˙ x(t) = A(θ)x(t)+B(θ)u(t) ˙ θ = ← − added states y(t) = C(θ)x(t).

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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8/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Outline

1

Problem description

2

Solution by Luenberger observers approach

3

Numerical illustration

4

Perspectives

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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9/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Synthesis steps

Introduce an extended dynamic system controlled by the known process inputs and outputs. ˙ z = Λz +Ly , ˙ w = g(w,u) ,z ∈ Rr ,w ∈ Rr .

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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9/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Synthesis steps

Introduce an extended dynamic system controlled by the known process inputs and outputs. ˙ z = Λz +Ly , ˙ w = g(w,u) ,z ∈ Rr ,w ∈ Rr . Find a mapping (x,θ,w) → T(x,θ,w) in C 1 which satisfies the following ODE ˙ T(x,θ,w) = ΛT(x,θ,w)+Ly .

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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9/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Synthesis steps

Introduce an extended dynamic system controlled by the known process inputs and outputs. ˙ z = Λz +Ly , ˙ w = g(w,u) ,z ∈ Rr ,w ∈ Rr . Find a mapping (x,θ,w) → T(x,θ,w) in C 1 which satisfies the following ODE ˙ T(x,θ,w) = ΛT(x,θ,w)+Ly . Implies the following equation ˙ e(t) = Λe(t), where e(t) = z(t)−T(x(t),θ,w(t))

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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10/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Synthesis steps

If Λ is a Hurwitz matrix then z is an estimate of T lim

t→+∞|z(t)−T(x(t),θ,w(t))| = 0 .

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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10/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Synthesis steps

If Λ is a Hurwitz matrix then z is an estimate of T lim

t→+∞|z(t)−T(x(t),θ,w(t))| = 0 .

The nonlinear Luenberger observer is given as ˙ z = Λz +Ly ˙ w = g(w,u) (ˆ x, ˆ θ) = T ∗(z(t),w(t)) T ∗: is the left inverse of T.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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11/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Difficulties

1- Is there an explicit expression for T ?

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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11/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Difficulties

1- Is there an explicit expression for T ? 2- Is T injective and full rank ?

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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11/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Difficulties

1- Is there an explicit expression for T ? 2- Is T injective and full rank ? 3- Is there an explicit expression for T ∗ ?

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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12/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 1: Existence of an explicit expression for T

Theorem 1

For any r-uplet of real negative elements (λ1,...,λr) such that λi / ∈

  • θ∈Θ

σ{A(θ)}

  • i = 1,...,r

The mapping T(x,θ,w) =

  • T1(x,θ,w1)...Tr(x,θ,wr)

⊤ is a solution. with Ti : Rn ×Θ×R → R (x,θ,wi) → Ti(x,θ,wi) = C(θ)(A(θ)−λiIn)−1[x −B(θ)wi], and g : Rr ×R → Rr (w,u) → g(w,u) = Λw +Lu Λ = Diag{λ1,...,λr}; L = 1r.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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13/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 2: Is T injective and full rank ?

T(x,θ,w) =

  • T1(x,θ,w1)...Tr(x,θ,wr)

⊤

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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13/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 2: Is T injective and full rank ?

T(x,θ,w) =

  • T1(x,θ,w1)...Tr(x,θ,wr)

⊤ Ti : Rn ×Θ×R → R (x,θ,wi) → Ti(x,θ,wi) = C(θ)(A(θ)−λiIn)−1[x −B(θ)wi],

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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13/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 2: Is T injective and full rank ?

T(x,θ,w) =

  • T1(x,θ,w1)...Tr(x,θ,wr)

⊤ Ti : Rn ×Θ×R → R (x,θ,wi) → Ti(x,θ,wi) = C(θ)(A(θ)−λiIn)−1[x −B(θ)wi], Is it an injective and full rank function?

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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13/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 2: Is T injective and full rank ?

T(x,θ,w) =

  • T1(x,θ,w1)...Tr(x,θ,wr)

⊤ Ti : Rn ×Θ×R → R (x,θ,wi) → Ti(x,θ,wi) = C(θ)(A(θ)−λiIn)−1[x −B(θ)wi], Is it an injective and full rank function?

Answer

If the input makes the extended system observable, then by choosing the eigenvalues λi sufficiently large the function T is injective and full rank after a transient.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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14/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 2: Is T injective and full rank ?

Let the mapping Hr : Θ → Rr×n be defined as θ → Hr(θ) =      C(θ) C(θ)A(θ) . . . C(θ)A(θ)r−1     ,

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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14/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 2: Is T injective and full rank ?

Let the mapping Hr : Θ → Rr×n be defined as θ → Hr(θ) =      C(θ) C(θ)A(θ) . . . C(θ)A(θ)r−1     , and Gr : Θ → Rr×r be defined as

θ → Gr(θ) =         ··· ··· ··· C(θ)B(θ) ··· ··· ··· C(θ)A(θ)B(θ) C(θ)B(θ) ··· ··· ··· . . . . . . . . . ··· ··· ··· C(θ)A(θ)r−2B(θ) C(θ)A(θ)r−3B(θ) ··· ··· C(θ)B(θ)         ,

then

  • y(t)

˙ y(t) ··· y (r−1)(t)

  • = Hr(θ)x +Gr(θ)v
  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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15/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 2: Is T injective and full rank ?

Assumption (Uniform differential observability)

There exist two compact subsets Cθ ∈ Θ, Cx ∈ Rn, an integer r and Ur a bounded subset of Rr−1 such that the mapping Hr is injective and foll rank Hr(x,θ,v) = Hr(θ)x +Gr(θ)v , for all (x,θ) and (x∗,θ ∗) both in Cl(Cθ)×Cl(Cx) and all v in Ur. Or, there exist L > 0 such that |Hr(x∗,θ ∗,v)−Hr(x,θ,v)| ≥ L

  • x −x∗

θ −θ ∗

  • .
  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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16/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 2: Is T injective and full rank ?

Theorem 2

Assume the assumption holds. Let u(·) be a bounded C r−2([0,+∞]) function with bounded r −2 first derivatives ¯ u(r−2)(·). For all (˜ λ1,...,˜ λr), for all positive time τ and all t1 ≥ τ, if ¯ u(t1) is in Ur then for all (x,θ) and (x∗,θ ∗) in Cx ×Cθ, T is injective and full rank. Or, there exist two positive real numbers k∗ and LT such that for all k > k∗ |T(x,θ,w(t1))−T(x∗,θ ∗,w(t1))| ≥ LT kr

  • x −x∗

θ −θ ∗

  • .

With the mapping T is defined by taking λi = k˜ λi.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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17/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 3: Existence of an explicit expression for T ∗

The general form of T ∗ is the optimization of the following criteria (ˆ x(t), ˆ θ(t)) = argmin

x,θ

T(x(t),θ,w(t))−z(t)2

2

T(x,θ,w) is nonlinear ⇒ iterative optimization methods.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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17/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 3: Existence of an explicit expression for T ∗

The general form of T ∗ is the optimization of the following criteria (ˆ x(t), ˆ θ(t)) = argmin

x,θ

T(x(t),θ,w(t))−z(t)2

2

T(x,θ,w) is nonlinear ⇒ iterative optimization methods.

We are looking for an explicit expression of T ∗.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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18/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 3: Existence of an explicit expression for T ∗

Let the canonical observable structure A(θ) = A(θa) =        −θa1 1 ·· −θa2 1 ·· : : : ... : : : : 1 −θan ··        ∈ Rn×n, C(θ) = C =

  • 1

···

  • ∈ R1×n,

B(θ) = B(θb) =

  • θb1

θb2 ··· θbn ⊤ ∈ Rn×1, All strictly proper linear SISO systems can be written in this form.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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19/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Question 3: Existence of an explicit expression for T ∗

For all i = 1,...,r, we have zi(t) = Ti(ˆ x(t), ˆ θ,wi(t)) = C(A(ˆ θ)−λiIn)−1 ˆ x(t)−B(ˆ θ)wi(t)

  • By using the Kronecker algebra, we can get the following expression

zi =

  • ziV T

i

V T

i

−(wT

i ⊗V T i )

  • Pi (zi,wi )

  ˆ θa ˆ x ˆ θb   The solution is given by (ˆ θa(t), ˆ x(t), ˆ θb(t))T = (P(z(t),w(t))⊤P(z(t),w(t)))−1P(z(t),w(t))⊤z(t) where: P(z,w) = [P1(z1,w1)⊤,··· ,Pr(zr,wr)⊤]⊤ ∈ Rr×(2n+1) and Vi =

  • 1

λi

...

1 λ n

i

⊤ ∈ Rn , r ≥ 4n −1

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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Question 3: Existence of an explicit expression for T ∗

Question ?

Under which conditions on T(x,θ,w) and λi is the matrix P full rank column?

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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Question 3: Existence of an explicit expression for T ∗

Question ?

Under which conditions on T(x,θ,w) and λi is the matrix P full rank column?

Proposition

If the matrices A(θ), B(θ) and C(θ) have the observable form, if the λi’s are different from A(θ) eigenvalues and if the dimension of T r ≥ 4n −1, then for any (z,x,θ,w) such that z = T(x,θ,w) and rank

  • ∂T

∂(x,θ)(x,θ,w)

  • = r

the matrix P is a full rank column.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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Question 3: Existence of an explicit expression for T ∗

Result

When z(t) is in Im(T), the Luenberger observer    ˙ z(t) = Λz(t)+Ly(t) ˙ w(t) = Λw(t)+Lu(t) (ˆ x(t), ˆ θ(t)) =

  • PT(w(t),z(t))P(w(t),z(t))

−1 PTz(t), is well defined.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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Question 3: Existence of an explicit expression for T ∗

Result

When z(t) is in Im(T), the Luenberger observer    ˙ z(t) = Λz(t)+Ly(t) ˙ w(t) = Λw(t)+Lu(t) (ˆ x(t), ˆ θ(t)) =

  • PT(w(t),z(t))P(w(t),z(t))

−1 PTz(t), is well defined. In the transient phase we can not guarantee that P is of full rank

  • column. If it is numerically not the case, we keep the old values of

the unknown parameters.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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Outline

1

Problem description

2

Solution by Luenberger observers approach

3

Numerical illustration

4

Perspectives

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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23/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Black box model

˙ x(t) =

  • −θa1

1 −θa2

  • x(t)+
  • θb1

θb2

  • u(t)

y(t) =

  • 1
  • x(t)

We want to estimate θa1, θa2, θb1, θb2 and state x2 from knowledge of signals u(t) and y(t). (z(0) , w(0)) = (0 , 0) (x1(0),x2(0),θa1(0),θa2(0),θb1(0),θb2(0)) = (0,0.5,2,3,1,−1) (ˆ x1(0), ˆ x2(0), ˆ θa1(0), ˆ θa2(0), ˆ θb1(0), ˆ θb2(0)) = (0,0,0,0,0,0)

Table: Initial configuration of the system and the observer states.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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Simulation without noise effect

Matrix A(θ) parameters ˆ θa1 Time (s) ˆ θa2 θa1 θa2

Figure: Estimation of parameters θa1 and θa2.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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25/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Simulation without noise effect

Matrix B(θ) parameters Time (s) ˆ θb1 ˆ θb2 θb1 θb2

Figure: Estimation of parameters θb1 and θb2.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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26/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Simulation without noise effect

Output y(t) vs ˆ y(t) State ˆ x2(t) Time (s) Estim Real Figure: Comparison between estimated output ˆ y(t) and real output y(t).

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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27/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Simulation with output added noise of 5%

Matrix A(θ) parameters Time (s) ˆ θa1 ˆ θa2 θa1 θa2

Figure: Estimation of parameters θa1 and θa2.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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28/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Simulation with output added noise of 5%

ˆ θb1 ˆ θb2 θb1 θb2 Time (s) Matrix B(θ) parameters

Figure: Estimation of parameters θb1 and θb2 .

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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29/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Simulation with output added noise of 5%

Estim Real Output y(t) vs ˆ y(t) State ˆ x2(t) Estim Time (s) Figure: Comparison between estimated output ˆ y(t) and real output y(t).

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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30/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Outline

1

Problem description

2

Solution by Luenberger observers approach

3

Numerical illustration

4

Perspectives

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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31/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

Perspectives Numerical comparison with other approaches in literature. Study of the observer robustness with respect to noise. Study of the persistency excitation of input and observer order. Application of this approach on a real system.

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers

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32/32 Problem description Solution by Luenberger observers approach Numerical illustration Perspectives

THANK YOU

  • C. Afri:

afri@lagep.univ-lyon1.fr LAGEP – AMP` ERE LAGEP, UMR 5007, UCBL1-CNRS, ACC 2015 Chicago Identification of linear systems with nonlinear Luenberger observers