[PDF] - This document must be cited according to its fjnal version which is PDF Document

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

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

"Adaptive identifjcation of continuous time MIMO state-space models", 54rd IEEE Conference on Decision and Control (CDC), Osaka, Japan, pp. 5677-5682, december 15-18, 2015 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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1/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Adaptive identification of continuous-time MIMO state-space models

C. Afri*
L. Bako
V. Andrieu
P. Dufour

Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015

Acknowledgments: * Ph.D. student, funded by the French Ministry of Research.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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2/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Outline

1

Identification problem

2

Estimation method

3

Exponential convergence

4

Simulation

5

Conclusion and future work

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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3/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Outline

1

Identification problem

2

Estimation method

3

Exponential convergence

4

Simulation

5

Conclusion and future work

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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4/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

General schema

Linear systems identification Identification

f the Input/Output models

Identification

f the state models
nline
ffline

Adaptive observers Extended observers requires a parameterization of model matrices Subspaces method based on SVD decomposition

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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5/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Studied approach

Linear systems identification Identification

f the Input/Output models

Identification

f the state models
nline
ffline

Adaptive observers Extended observers requires a parameterization of model matrices Subspaces method based on SVD decomposition

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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6/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

MIMO (Multiple-Input Multiple-Output) systems

Consider a linear system described by a state-space model of the form S :

˙

x(t) = Ax(t)+Bu(t) y(t) = Cx(t)+Du(t),

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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6/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

MIMO (Multiple-Input Multiple-Output) systems

Consider a linear system described by a state-space model of the form S :

˙

x(t) = Ax(t)+Bu(t) y(t) = Cx(t)+Du(t), u ∈ Rnu measured inputs signals y ∈ Rny measured outputs signals x ∈ Rn, unknown state A ∈ Rn×n ,B ∈ Rn×nu ,C ∈ Rny ×n and D ∈ Rny ×nu are unknown parameters

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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7/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Systems assumptions & estimation problem

System assumptions

A1. The system S is stable → A is Hurwitz.
A2. (A,B,C) is minimal → (A,B) is controllable and (A,C) is
bservable.
A3. C is full row rank.
C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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7/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Systems assumptions & estimation problem

System assumptions

A1. The system S is stable → A is Hurwitz.
A2. (A,B,C) is minimal → (A,B) is controllable and (A,C) is
bservable.
A3. C is full row rank.

Estimation problem

From the measurements of inputs u and outputs y → Estimate both the state x and the parameters (A,B,C,D) in an arbitrary state space basis.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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8/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Outline

1

Identification problem

2

Estimation method

3

Exponential convergence

4

Simulation

5

Conclusion and future work

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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9/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Main idea

Search for a linear expression M(A,B,C,D)ϕ(u,y) = f (u,y)

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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9/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Main idea

Search for a linear expression M(A,B,C,D)ϕ(u,y) = f (u,y) Use the Recursive Least Squares (RLS) method

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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9/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Main idea

Search for a linear expression M(A,B,C,D)ϕ(u,y) = f (u,y) Use the Recursive Least Squares (RLS) method

Search for a criterion Vt(M) such that

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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9/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Main idea

Search for a linear expression M(A,B,C,D)ϕ(u,y) = f (u,y) Use the Recursive Least Squares (RLS) method

Search for a criterion Vt(M) such that

M(t) = argminM Vt(M) converges to M0 the solution of the linear expression d dt ∇Vt(M) = 0 → a continuous-time recursive estimator.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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9/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Main idea

Search for a linear expression M(A,B,C,D)ϕ(u,y) = f (u,y) Use the Recursive Least Squares (RLS) method

Search for a criterion Vt(M) such that

M(t) = argminM Vt(M) converges to M0 the solution of the linear expression d dt ∇Vt(M) = 0 → a continuous-time recursive estimator.

Extract A, B, C and D from M.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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10/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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10/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

We can get yf = Of x +Tf uf , f > n with yf =

y⊤

˙ y⊤ ··· y (f −1)⊤⊤ ∈ Rny n uf =

u⊤

˙ u⊤ ··· u(f −1)⊤⊤ ∈ Rnun Of =      C CA . . . CAf −1      ∈ Rny n×n ,Tf =      D ··· CB D ··· . . . . . . ... . . . CAf −2B CAf −3B ··· D      ∈ Rny n×nun.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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11/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

yf =

y⊤

˙ y⊤ ··· y (f −1)⊤⊤ uf =

u⊤

˙ u⊤ ··· u(f −1)⊤⊤

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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11/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

yf =

y⊤

˙ y⊤ ··· y (f −1)⊤⊤ uf =

u⊤

˙ u⊤ ··· u(f −1)⊤⊤

Problem (noisy measurements)

The time derivatives of u and y are not measured. The only available signals are the inputs and outputs .

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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11/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

yf =

y⊤

˙ y⊤ ··· y (f −1)⊤⊤ uf =

u⊤

˙ u⊤ ··· u(f −1)⊤⊤

Problem (noisy measurements)

The time derivatives of u and y are not measured. The only available signals are the inputs and outputs .

Solution (Linear stable filter)

Avoid numerical computation of derivatives → work with filtered data.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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12/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

yf = Of x +Tf uf F(s) =

1 sf +αf −1sf −1+···+α0

p = Of ¯ x +Tf q

filter

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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12/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

yf = Of x +Tf uf F(s) =

1 sf +αf −1sf −1+···+α0

p = Of ¯ x +Tf q

filter

with ˙ p =

Λc ⊗Iny
p +
l ⊗Iny
y

˙ q = (Λc ⊗Inu)q +(l ⊗Inu)u , Λc =    

1 ··· 1 ··· . . . . . . ··· ... . . . ··· ··· 1 −α0 −α1 ··· ··· −αf −1

   , l =    

. . . 1

   . and ¯ x is the filtered state of x.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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12/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

yf = Of x +Tf uf F(s) =

1 sf +αf −1sf −1+···+α0

p = Of ¯ x +Tf q

filter

with ˙ p =

Λc ⊗Iny
p +
l ⊗Iny
y

˙ q = (Λc ⊗Inu)q +(l ⊗Inu)u , Λc =    

1 ··· 1 ··· . . . . . . ··· ... . . . ··· ··· 1 −α0 −α1 ··· ··· −αf −1

   , l =    

. . . 1

   . and ¯ x is the filtered state of x.

Problem

We are still facing the difficulty that ¯ x is unknown.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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13/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

Solution

By a change of state coordinates → We define a new state ˜ x as ˜ x = T ¯ x , with T = HOf ∈ Rn×n is nonsingular

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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13/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

Solution

By a change of state coordinates → We define a new state ˜ x as ˜ x = T ¯ x , with T = HOf ∈ Rn×n is nonsingular

Remark (choice of H)

(A,C) is observable ⇒ Of is of full rank column ⇒ the probability to get a nonsingular T is one if H is randomly generated.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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13/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

Solution

By a change of state coordinates → We define a new state ˜ x as ˜ x = T ¯ x , with T = HOf ∈ Rn×n is nonsingular

Remark (choice of H)

(A,C) is observable ⇒ Of is of full rank column ⇒ the probability to get a nonsingular T is one if H is randomly generated. For multiple inputs single output systems H = [In 0n×(f −n)].

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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13/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Is there a linear relation M(A,B,C,D)ϕ(y,u) = f (y,u)?

Solution

By a change of state coordinates → We define a new state ˜ x as ˜ x = T ¯ x , with T = HOf ∈ Rn×n is nonsingular

Remark (choice of H)

(A,C) is observable ⇒ Of is of full rank column ⇒ the probability to get a nonsingular T is one if H is randomly generated. For multiple inputs single output systems H = [In 0n×(f −n)]. For MIMO systems, if a set of Of rows indexed by [i1,...,in] is linearly independent, then H = [ei1 ··· ein]⊤ with ei ∈ Rfny the i-th canonical basis vector.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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14/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

There is a linear relation M( ˜ A, ˜ B, ˜ C, ˜ D)ϕ(y,u) = f (y,u)

The system realization becomes ˜ S :

˙

z(t) = ˜ Az(t)+ ˜ Bu(t) y(t) = ˜ Cz(t)+ ˜ Du(t), with ˜ A = TAT −1, ˜ B = TB, ˜ C = CT −1, ˜ D = D, z = Tx and ˜ x is the filtered state of z.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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14/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

There is a linear relation M( ˜ A, ˜ B, ˜ C, ˜ D)ϕ(y,u) = f (y,u)

The system realization becomes ˜ S :

˙

z(t) = ˜ Az(t)+ ˜ Bu(t) y(t) = ˜ Cz(t)+ ˜ Du(t), with ˜ A = TAT −1, ˜ B = TB, ˜ C = CT −1, ˜ D = D, z = Tx and ˜ x is the filtered state of z.

Result

This change of coordinates leads to ˜ Of ˜ x = p − ˜ Tf q and ˜ x = Hp −HTf q ⇒ Mϕ = p M = ˜ Of ˜ Tf − ˜ Of HTf

∈ Rfny ×(n+fnu) , ϕ =
Hp

q

∈ Rn+fnu.
C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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15/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

RLS method

Vt Criterion

Vt(M) = 1 2e−αttr

(M −M0)P−1

0 (M −M0)⊤

+ 1 2

t

e−α(t−τ) r(τ)2 p(τ)−Mϕ(τ)2

2

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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15/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

RLS method

Vt Criterion

Vt(M) = 1 2e−αttr

(M −M0)P−1

0 (M −M0)⊤

+ 1 2

t

e−α(t−τ) r(τ)2 p(τ)−Mϕ(τ)2

2

M(t) = argmin

M

Vt(M) ⇒ M(t) is the solution of ∇Vt(M) = 0

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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15/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

RLS method

Vt Criterion

Vt(M) = 1 2e−αttr

(M −M0)P−1

0 (M −M0)⊤

+ 1 2

t

e−α(t−τ) r(τ)2 p(τ)−Mϕ(τ)2

2

M(t) = argmin

M

Vt(M) ⇒ M(t) is the solution of ∇Vt(M) = 0 P0 > 0 → P0 is a symmetric positive-definite matrix. α > 0 is a design parameter. r(τ)2 ≥ 1 is a normalizing upper-bounded factor. M0 = M(0) is the initial value of M(t).

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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16/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Parameters ˜ A and ˜ C adaptive identifier

Result

Setting d dt ∇Vt(M) = 0 yields the continuous-time RLS estimator: ˙ M = (p −Mϕ)ϕ⊤P r(t)2 , M(0) = M0 ˙ P = αP − Pϕϕ⊤P r(t)2 , P(0) = P0 ≻ 0 r(t)2 = 1+ϕ⊤Pϕ with M =   M11 M12 M21 M22 M31 M32   ⇒ ˜ C = M11 , ˜ A =

M11

M21 † M21 M31

.
C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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17/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Parameters ˜ B and ˜ D adaptive identifier

The system output can be expressed on a linear regression form y(t) = ˜ Ce

˜ At ˜

x0 +Φ(t)θ, with θ =

vec( ˜

B)⊤ vec( ˜ D)⊤⊤ ;Φ(t) =

Ψ(t)

u(t)⊤ ⊗Iny

,

and Ψ(t) obeys the following differential equation ˙ Ψ(t) = Ψ(t)

Inu ⊗ ˜

A

+u(t)⊤ ⊗ ˜

C ,Φ(0) = 0

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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18/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Parameters ˜ B and ˜ D adaptive identifier

Result

we can derive the RLS estimator of θ ˙ θ = QΦ⊤ (y −Φθ) ρ(t)2 , θ(0) = θ0 ˙ Q = βQ − QΦ⊤ΦQ ρ(t)2 , Q(0) = Q0 ≻ 0 where ρ(t)2 = 1+tr

Φ(t)Q(t)Φ(t)⊤

.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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19/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Estimation of the state z = Tx

z can be generated by constructing a Luenberger observer based on the identified realization ( ˜ A, ˜ B, ˜ C, ˜ D). ˙ ˆ z = ˜ Aˆ z + ˜ Bu +L(y −( ˜ C ˆ z + ˜ Du)) such that ( ˜ A−L ˜ C) is a Hurwitz matrix.

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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19/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Estimation of the state z = Tx

z can be generated by constructing a Luenberger observer based on the identified realization ( ˜ A, ˜ B, ˜ C, ˜ D). ˙ ˆ z = ˜ Aˆ z + ˜ Bu +L(y −( ˜ C ˆ z + ˜ Du)) such that ( ˜ A−L ˜ C) is a Hurwitz matrix.

Remarks

The state ˜ x is a filtered version of z If the filter F(s) is properly chosen, then ˜ x contains less noise than z

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

SLIDE 39

20/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Outline

1

Identification problem

2

Estimation method

3

Exponential convergence

4

Simulation

5

Conclusion and future work

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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21/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Convergence of the ( ˜ A, ˜ C)-estimates

Assume that M0 is the real value of M, then we get the error eM = (M −M0) dynamic equation as ˙ eM = −eM ϕϕ⊤ r 2 P

C. Afri:

afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models

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21/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Convergence of the ( ˜ A, ˜ C)-estimates

Assume that M0 is the real value of M, then we get the error eM = (M −M0) dynamic equation as ˙ eM = −eM ϕϕ⊤ r 2 P

Definition (persistence of excitation)

A locally integrable signal ϕ : R+ → Rd is said to be Persistently Exciting (PE) if there exist constant numbers α1,α2,T > 0 and t0 ≥ 0 such that α1I

t+T

t

ϕ(τ)ϕ(τ)⊤dτ α2I ∀t ≥ t0. Here the notation S R is used to mean that R −S is positive semi-definite.

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22/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Convergence of the ( ˜ A, ˜ C)-estimates

Definition (sufficiency of richness)

A smooth signal u : R+ → Rnu is said to be Sufficiently Rich (SR) of

rder k if
u(i)(0) = 0 for i = 0,...,k −1 and
for any γ > 0, the signal ¯

uk ∈ Rknu defined by ¯ uk = 1 (s +γ)k

u⊤

˙ u⊤ ··· (u(k−1))⊤⊤ is PE.

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Convergence of the ( ˜ A, ˜ C)-estimates

Proposition

Assume that

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23/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Convergence of the ( ˜ A, ˜ C)-estimates

Proposition

Assume that The matrices ˜ A and Λc (the state matrix of filter F(s)) are Hurwitz

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Convergence of the ( ˜ A, ˜ C)-estimates

Proposition

Assume that The matrices ˜ A and Λc (the state matrix of filter F(s)) are Hurwitz The pair ( ˜ A, ˜ B) is controllable

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Convergence of the ( ˜ A, ˜ C)-estimates

Proposition

Assume that The matrices ˜ A and Λc (the state matrix of filter F(s)) are Hurwitz The pair ( ˜ A, ˜ B) is controllable The input u is SR of order f +m, where m = deg(m ˜

A(λ))

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23/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Convergence of the ( ˜ A, ˜ C)-estimates

Proposition

Assume that The matrices ˜ A and Λc (the state matrix of filter F(s)) are Hurwitz The pair ( ˜ A, ˜ B) is controllable The input u is SR of order f +m, where m = deg(m ˜

A(λ))

then the signal ϕ defined as ϕ(t) =

Hp

q

.

is PE.

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24/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Convergence of the ( ˜ A, ˜ C)-estimates

Theorem

Assume that The system ˜ S is stable → ˜ A is Hurwitz. ( ˜ A, ˜ B, ˜ C) is minimal → ( ˜ A, ˜ B) is controllable and ( ˜ A, ˜ C) is

bservable.

The input u is SR of order f +m then M converges exponentially to Mo.

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25/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Convergence of the ( ˜ B, ˜ D)-estimates

Assume that θ 0 is the real value of θ, then we get the error eθ = (θ −θ 0) dynamic equation as ˙ eθ = −QΦ⊤Φ ρ2 eθ +v with v = QΦ⊤ ρ2

˜

C oe

˜ Aot ˜

x0 +(Φo −Φ)θ o .

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25/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Convergence of the ( ˜ B, ˜ D)-estimates

Assume that θ 0 is the real value of θ, then we get the error eθ = (θ −θ 0) dynamic equation as ˙ eθ = −QΦ⊤Φ ρ2 eθ +v with v = QΦ⊤ ρ2

˜

C oe

˜ Aot ˜

x0 +(Φo −Φ)θ o .

Remark

If we can prove that v(t) vanishes as t → ∞. the homogeneous part is exponentially stable. then, we can conclude on the convergence of the estimation error eθ → 0.

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26/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Convergence of the ( ˜ B, ˜ D)-estimates

Question:

v(t) vanishes as t → ∞ ?

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26/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Convergence of the ( ˜ B, ˜ D)-estimates

Question:

v(t) vanishes as t → ∞ ?

Answer:

v(t) = QΦ⊤ ρ2

˜

C oe

˜ Aot ˜

x0 +(Φo −Φ)θ o . When ( ˜ A, ˜ C) converge to the true values ( ˜ Ao, ˜ C o) ⇒ limt→∞ ˜ C oe ˜

Aot ˜

x0 = 0 and limt→∞ Φ = Φo ⇒ limt→∞ v(t) = 0.

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Convergence of the ( ˜ B, ˜ D)-estimates

Lemma

Assume that The system ˜ S is stable → ˜ A is Hurwitz ( ˜ A, ˜ B, ˜ C) is minimal u is SR of order at least f +m the rate of convergence defined as the largest c > 0 such that sup

k∈N

˜

C(t) ˜ A(t)k − ˜ C o( ˜ Ao)k

2 ≤ λe−ct,

then Φ = Φ(t, ˜ A(t), ˜ C(t)) is PE. If Φ is PE, then the homogeneous part of ˙ eθ = −QΦ⊤Φ ρ2 eθ +v is exponentially stable.

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28/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Outline

1

Identification problem

2

Estimation method

3

Exponential convergence

4

Simulation

5

Conclusion and future work

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29/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Fourth order MIMO system

Consider the MIMO linear system with matrices: A =     −1.4 0.3 −0.6 −0.3 0.3 −0.9 −0.9 0.1 −0.6 −0.9 −2.6 −1.0 −0.3 0.1 −1.0 −1.8    ,B =     −0.4 1.3 0.6 0.2 0.6    , C =

0.9

1.3 −1.1 −0.8 +0.9 +0.4

, D =
0.8

−1.5

.

We estimate a system ( ˜ A, ˜ B, ˜ C, ˜ D) similar to the system (A,B,C,D).

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30/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Time performances

10 20 30 40 50 60 70 80 0.2 0.4 0.6 0.8 1 speed direction Time Error norm

Figure: The relative error εr =

˜

O0

f ˜

B0 − ˜ Of ˜ B

F
˜

O0

f ˜

B0

F

time evolution

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31/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Frequency performances

15
10
5

5 10 -3 10 -2 10 -1 10 0 10 1 45 90 135 180 C

n

v e r g e n c e d i r e c t i

n

C

n

v e r g e n c e d i r e c t i

n

system target initial system Bode diagram Magnitude (dB) Phase (deg) Frequency (rad/s)

Figure: Bode diagram of output1 / input 1 transfer function convergence

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32/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Outline

1

Identification problem

2

Estimation method

3

Exponential convergence

4

Simulation

5

Conclusion and future work

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33/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Conclusion

The main challenges with such a problem are Estimate time derivatives of input-output signals. Provide a cheap update process for the system matrices while the state is unknown. Analyse convergence of the whole algorithm.

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33/34 Identification problem Estimation method Exponential convergence Simulation Conclusion and future work

Conclusion

The main challenges with such a problem are Estimate time derivatives of input-output signals. Provide a cheap update process for the system matrices while the state is unknown. Analyse convergence of the whole algorithm.

Future work

Study of the algorithm robustness to noise. Search for the relation between choice of H and robustness. Improvement concerning the filtering of the input-output signals.

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THANK YOU

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afri@lagep.univ-lyon1.fr Universit´ e de Lyon, France Laboratories : AMPERE, LAGEP CDC, Osaka, 15-18 December, 2015 Adaptive identification of continuous-time MIMO state-space models