Identification of Input/Output LPV Models V. Laurain, M. Gilson, R. - - PowerPoint PPT Presentation

Identification of Input/Output LPV Models

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

CRAN, Nancy-Universit´ e, DCSC, TU Delft

Delft Center for System s and Control Delft Center for System s and Control

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 1 / 32

Outline

1

Introduction

2

Problem Description LPV model description Scheduling dependency model

3

Prediction Error for LPV systems Usual linear regression The literature methods Reformulation of the problem RIV Algorithm

4

Results LPV-RIV performance

5

Application example

6

Conclusion

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 2 / 32

Introduction

Outline

1

Introduction

2

Problem Description LPV model description Scheduling dependency model

3

Prediction Error for LPV systems Usual linear regression The literature methods Reformulation of the problem RIV Algorithm

4

Results LPV-RIV performance

5

Application example

6

Conclusion

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 3 / 32

Introduction

What is an LPV model ?

Definition Linear relationship between the input and output The model parameters vary in time according to an external variable : The scheduling variable Identification of LPV models Control theory of LPV models well developed (Gain scheduling) LPV State-Space identification methods : Multiple models approaches, LMI’s based methods, Gradient methods Input/Output identification methods : Linear regression methods, Nonlinear

• ptimization

Advantages Tradeoff between LTI systems and NL systems Wide range of behavior representation capability Many practical applications (environment, automotive, wafer scanners...)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 4 / 32

Introduction

What is an LPV model ?

Definition Linear relationship between the input and output The model parameters vary in time according to an external variable : The scheduling variable Identification of LPV models Control theory of LPV models well developed (Gain scheduling) LPV State-Space identification methods : Multiple models approaches, LMI’s based methods, Gradient methods Input/Output identification methods : Linear regression methods, Nonlinear

• ptimization

Advantages Tradeoff between LTI systems and NL systems Wide range of behavior representation capability Many practical applications (environment, automotive, wafer scanners...)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 4 / 32

Introduction

What is an LPV model ?

Definition Linear relationship between the input and output The model parameters vary in time according to an external variable : The scheduling variable Identification of LPV models Control theory of LPV models well developed (Gain scheduling) LPV State-Space identification methods : Multiple models approaches, LMI’s based methods, Gradient methods Input/Output identification methods : Linear regression methods, Nonlinear

• ptimization

Advantages Tradeoff between LTI systems and NL systems Wide range of behavior representation capability Many practical applications (environment, automotive, wafer scanners...)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 4 / 32

Introduction

LPV modeling concept of a plane

Linear system for a given altitude The system parameters vary with the altitude

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 5 / 32

Introduction

LPV modeling concept of a bar under constraint

Consider a metal bar under a constraint F. F x The displacement is x For a constant temperature : x = kF with k the stiffness coefficient When considering the temperature θ, the LPV model can be written : x = k(θ)F

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 6 / 32

Problem Description

Outline

1

Introduction

2

Problem Description LPV model description Scheduling dependency model

3

Prediction Error for LPV systems Usual linear regression The literature methods Reformulation of the problem RIV Algorithm

4

Results LPV-RIV performance

5

Application example

6

Conclusion

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 7 / 32

Problem Description LPV model description

System

LPV IO model

LTI

M ( A(q−1)χ(tk) = B(q−1)u(tk) y(tk) = χ(tk) + v(tk) χ is the noise-free output u is the input v is the additive noise with bounded spectral density y is the noisy output of the system q is the time-shift operator, i.e. q−iu(tk) = u(tk−i)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 8 / 32

Problem Description LPV model description

System

LPV IO model

LPV

M ( A(pk, q−1)χ(tk) = B(pk, q−1)u(tk) y(tk) = χ(tk) + v(tk) χ is the noise-free output u is the input v is the additive noise with bounded spectral density y is the noisy output of the system q is the time-shift operator, i.e. q−iu(tk) = u(tk−i) p is the scheduling parameter

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 8 / 32

Problem Description LPV model description

System

LPV IO model

LTI

M ( A(q−1)χ(tk) = B(q−1)u(tk) y(tk) = χ(tk) + v(tk) A(q−1) and B(q−1) are polynomials in q−1 of degree na and nb respectively : A(q−1) = 1 +

na

X

i=1

aiq−i B(q−1) =

nb

X

j=0

bjq−i,

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 8 / 32

Problem Description LPV model description

System

LPV IO model

LPV

M ( A(pk, q−1)χ(tk) = B(pk, q−1)u(tk) y(tk) = χ(tk) + v(tk) A(pk, q−1) and B(pk, q−1) are polynomials in q−1 of degree na and nb respectively : A(pk, q−1) = 1 +

na

X

i=1

ai(pk)q−i B(pk, q−1) =

nb

X

j=0

bj(pk)q−i,

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 8 / 32

Problem Description Scheduling dependency model

Model

Process Model

Gρ : “ A(pk , q−1, ρ), B(pk , q−1, ρ) ” = ` Aρ, Bρ ´ where the p dependent polynomials A and B are parameterized as Aρ : A(pk , q−1, ρ) = 1 +

na

X

i=1

ai (pk )q−i , Bρ : B(pk , q−1, ρ) =

nb

X

j=0

bj (pk )q−i .

Usual assumption : static dependency on p

ai (pk ) = ai,0 +

nα

X

l=1

ai,l fl (pk ) i = 1, . . . , na bj (pk ) = bj,0 +

nβ

X

l=1

bj,l gl (pk ) j = 0, . . . , nb {fl }nα

l=1 and {gl } nβ l=1 are analytic functions of p known a priori (polynomials for example)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 9 / 32

Problem Description Scheduling dependency model

Model

Process Model

Gρ : “ A(pk , q−1, ρ), B(pk , q−1, ρ) ” = ` Aρ, Bρ ´ where the p dependent polynomials A and B are parameterized as Aρ : A(pk , q−1, ρ) = 1 +

na

X

i=1

ai (pk )q−i , Bρ : B(pk , q−1, ρ) =

nb

X

j=0

bj (pk )q−i .

Usual assumption : static dependency on p

ai (pk ) = ai,0 +

nα

X

l=1

ai,l fl (pk ) i = 1, . . . , na bj (pk ) = bj,0 +

nβ

X

l=1

bj,l gl (pk ) j = 0, . . . , nb

Parameter vector ρ = [a1,0 . . . ana,nα, b0,0 . . . bnb,nβ]⊤

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 9 / 32

Prediction Error for LPV systems

Outline

1

Introduction

2

Problem Description LPV model description Scheduling dependency model

3

Prediction Error for LPV systems Usual linear regression The literature methods Reformulation of the problem RIV Algorithm

4

Results LPV-RIV performance

5

Application example

6

Conclusion

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 10 / 32

Prediction Error for LPV systems Usual linear regression

Mθ ( A(pk, q−1, ρ)χ(tk)=B(pk, q−1, ρ)u(tk) y(tk)=χ(tk) + v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 11 / 32

Prediction Error for LPV systems Usual linear regression

Mθ ( A(pk, q−1, ρ)χ(tk)=B(pk, q−1, ρ)u(tk) A(pk, q−1, ρ)y(tk)=A(pk, q−1, ρ)χ(tk) + A(pk, q−1, ρ)v(tk) A(pk, q−1, ρ)y(tk)=B(pk, q−1, ρ)u(tk) + A(pk, q−1, ρ)v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 11 / 32

Prediction Error for LPV systems Usual linear regression

A(pk, q−1, ρ)y(tk)=B(pk, q−1, ρ)u(tk) + A(pk, q−1, ρ)v(tk) 1 +

na

X

i=1

ai(pk)q−i ! y(tk)= nb X

j=0

bj(pk)q−j ! u(tk) + A(pk, q−1, ρ)v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 11 / 32

Prediction Error for LPV systems Usual linear regression

A(pk, q−1, ρ)y(tk)=B(pk, q−1, ρ)u(tk) + A(pk, q−1, ρ)v(tk) 1 +

na

X

i=1

ai(pk)q−i ! y(tk)= nb X

j=0

bj(pk)q−j ! u(tk) + A(pk, q−1, ρ)v(tk) y(tk) +

na

X

i=1

ai(pk)y(tk−i)=

nb

X

j=0

bj(pk)u(tk−j) + A(pk, q−1, ρ)v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 11 / 32

Prediction Error for LPV systems Usual linear regression

A(pk, q−1, ρ)y(tk)=B(pk, q−1, ρ)u(tk) + A(pk, q−1, ρ)v(tk) 1 +

na

X

i=1

ai(pk)q−i ! y(tk)= nb X

j=0

bj(pk)q−j ! u(tk) + A(pk, q−1, ρ)v(tk) y(tk) +

na

X

i=1

ai(pk)y(tk−i)=

nb

X

j=0

bj(pk)u(tk−j) + A(pk, q−1, ρ)v(tk) y(tk) +

na

X

i=1

nα X

l=0

ai,lfl(pk) ! y(tk−i)=

nb

X

j=0

nβ X

l=0

bj,lgl(pk) ! u(tk−j) + A(pk, q−1, ρ)v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 11 / 32

Prediction Error for LPV systems Usual linear regression

A(pk, q−1, ρ)y(tk)=B(pk, q−1, ρ)u(tk) + A(pk, q−1, ρ)v(tk) 1 +

na

X

i=1

ai(pk)q−i ! y(tk)= nb X

j=0

bj(pk)q−j ! u(tk) + A(pk, q−1, ρ)v(tk) y(tk) +

na

X

i=1

ai(pk)y(tk−i)=

nb

X

j=0

bj(pk)u(tk−j) + A(pk, q−1, ρ)v(tk) y(tk) +

na

X

i=1 nα

X

l=0

ai,lfl(pk)y(tk−i)=

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j) + A(pk, q−1, ρ)v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 11 / 32

Prediction Error for LPV systems Usual linear regression

Usual regression

y(tk) +

na

X

i=1 nα

X

l=0

ai,lfl(pk)y(tk−i)=

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j) + A(pk, q−1, ρ)v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 12 / 32

Prediction Error for LPV systems Usual linear regression

Usual regression

y(tk) +

na

X

i=1 nα

X

l=0

ai,lfl(pk)y(tk−i)=

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j) + A(pk, q−1, ρ)v(tk) Usual Linear regression associated y(tk) = ϕ⊤(tk)ρ + A(pk, q−1, ρ)v(tk) ϕ(tk) = 2 6 6 6 6 6 6 6 6 4 −y(tk)f0(pk) . . . −y(tk)fnα(pk) u(tk)g0(pk) . . . u(tk)gnβ(pk) 3 7 7 7 7 7 7 7 7 5 ∈ Rnρ, with y(tk) = 2 6 4 y(tk−1) . . . y(tk−na) 3 7 5, u(tk) = 2 6 4 u(tk) . . . u(tk−nb) 3 7 5,

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 12 / 32

Prediction Error for LPV systems The literature methods

The ARX case y(tk) = ϕ⊤(tk)ρ + A(pk, q−1, ρ)v(tk) | {z }

ǫ(tk )

ǫ(tk) = e(tk) with e a white noise The least squares Least squares : minimize

N

X

k=1

ǫ(tk)2 =

N

X

k=1

||y(tk) − ϕ⊤(tk)ρ||2 Least squares solution minimizes the prediction error.

• L. Giarr´

e, D. Bauso, P. Falugi, and B. Bamieh. LPV model identification for gain scheduling control : An application to rotating stall and surge control problem. Control Engineering Practice, 14 :351-361, 2006 Problems LPV-ARX structure is very unrealistic LS are biased if the model structure is not correct

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 13 / 32

Prediction Error for LPV systems The literature methods

The ARX case y(tk) = ϕ⊤(tk)ρ + A(pk, q−1, ρ)v(tk) | {z }

ǫ(tk )

ǫ(tk) = e(tk) with e a white noise The least squares Least squares : minimize

N

X

k=1

ǫ(tk)2 =

N

X

k=1

||y(tk) − ϕ⊤(tk)ρ||2 Least squares solution minimizes the prediction error.

• L. Giarr´

e, D. Bauso, P. Falugi, and B. Bamieh. LPV model identification for gain scheduling control : An application to rotating stall and surge control problem. Control Engineering Practice, 14 :351-361, 2006 Problems LPV-ARX structure is very unrealistic LS are biased if the model structure is not correct

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 13 / 32

Prediction Error for LPV systems The literature methods

The ARX case y(tk) = ϕ⊤(tk)ρ + A(pk, q−1, ρ)v(tk) | {z }

ǫ(tk )

ǫ(tk) = e(tk) with e a white noise The least squares Least squares : minimize

N

X

k=1

ǫ(tk)2 =

N

X

k=1

||y(tk) − ϕ⊤(tk)ρ||2 Least squares solution minimizes the prediction error.

• L. Giarr´

e, D. Bauso, P. Falugi, and B. Bamieh. LPV model identification for gain scheduling control : An application to rotating stall and surge control problem. Control Engineering Practice, 14 :351-361, 2006 Problems LPV-ARX structure is very unrealistic LS are biased if the model structure is not correct

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 13 / 32

Prediction Error for LPV systems The literature methods

The OE or BJ case y(tk) = ϕ⊤(tk)ρ + A(pk, q−1, ρ)v(tk) | {z }

ǫ(tk )

v(tk) = C(q−1, η) D(q−1, η)e(tk) with e a white noise

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 14 / 32

Prediction Error for LPV systems The literature methods

The OE or BJ case y(tk) = ϕ⊤(tk)ρ + A(pk, q−1, ρ)v(tk) | {z }

ǫ(tk )

v(tk) = C(q−1, η) D(q−1, η)e(tk) with e a white noise Problems Least squares : minimize

N

X

k=1

ǫ(tk)2 =

N

X

k=1

||y(tk) − ϕ⊤(tk)ρ||2 Therefore ǫ(tk) = A(pk, q−1, ρ)v(tk) = e(tk) Least squares solution is biased

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 14 / 32

Prediction Error for LPV systems The literature methods

The OE or BJ case y(tk) = ϕ⊤(tk)ρ + A(pk, q−1, ρ)v(tk) | {z }

ǫ(tk )

v(tk) = C(q−1, η) D(q−1, η)e(tk) with e a white noise Possible solution Possible solution : find a filter such that ǫf(tk) = e(tk) Optimal filter : D(q−1, η) C(q−1, η A†(pk, q−1, ρ)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 14 / 32

Prediction Error for LPV systems The literature methods

The OE or BJ case y(tk) = ϕ⊤(tk)ρ + A(pk, q−1, ρ)v(tk) | {z }

ǫ(tk )

v(tk) = C(q−1, η) D(q−1, η)e(tk) with e a white noise Problems This filter is a priori unknown Exact inverse of an LPV filter is tedious (impossible in practice) LPV filters ARE NOT COMMUTATIVE In practice it is not possible to minimize the prediction error

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 14 / 32

Prediction Error for LPV systems The literature methods

The OE or BJ case y(tk) = ϕ⊤(tk)ρ + A(pk, q−1, ρ)v(tk) | {z }

ǫ(tk )

v(tk) = C(q−1, η) D(q−1, η)e(tk) with e a white noise Solution offered in the literature for OE models

• H. Abbas and H. Werner. An instrumental variable technique for open-loop and

closed-loop identification of input-ouput LPV models. In Proceedings of the European Control Conference 2009, pages 2646-2651, Budapest, Hungary, 23-26 August 2009.

• M. Butcher, A. Karimi, and R. Longchamp. On the consistency of certain

identification methods for linear parameter varying systems. In Proceedings of the 17th IFAC World Congress, pages 4018-4023, Seoul, Korea, July 2008. Not aiming at minimizing the prediction error Conclusion : Display a large variance in comparison to least squares

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 14 / 32

Prediction Error for LPV systems Reformulation of the problem

LTI MISO interpretation of the model Mθ ( A(pk, q−1, ρ)χ(tk)=B(pk, q−1, ρ)u(tk) y(tk)=χ(tk) + v(tk) Mθ 8 > > < > > : χ(tk) +

na

X

i=1 nα

X

l=0

ai,lfl(pk)χ(tk−i)=

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j) y(tk)=χ(tk) + v(tk) Mθ 8 > > > > < > > > > : χ(tk) +

na

X

i=1

ai,0χ(tk−i) | {z }

F(q−1)χ(tk )

+

na

X

i=1 nα

X

l=1

ai,lfl(pk)χ(tk−i) | {z }

χi,l (tk )

=

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j | {z })

uj,l (tk )

y(tk)=χ(tk) + v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 15 / 32

Prediction Error for LPV systems Reformulation of the problem

LTI MISO interpretation of the model Mθ ( A(pk, q−1, ρ)χ(tk)=B(pk, q−1, ρ)u(tk) y(tk)=χ(tk) + v(tk) Mθ 8 > > < > > : χ(tk) +

na

X

i=1 nα

X

l=0

ai,lfl(pk)χ(tk−i)=

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j) y(tk)=χ(tk) + v(tk) Mθ 8 > > > > < > > > > : χ(tk) +

na

X

i=1

ai,0χ(tk−i) | {z }

F(q−1)χ(tk )

+

na

X

i=1 nα

X

l=1

ai,lfl(pk)χ(tk−i) | {z }

χi,l (tk )

=

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j | {z })

uj,l (tk )

y(tk)=χ(tk) + v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 15 / 32

Prediction Error for LPV systems Reformulation of the problem

LTI MISO interpretation of the model Mθ ( A(pk, q−1, ρ)χ(tk)=B(pk, q−1, ρ)u(tk) y(tk)=χ(tk) + v(tk) Mθ 8 > > < > > : χ(tk) +

na

X

i=1 nα

X

l=0

ai,lfl(pk)χ(tk−i)=

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j) y(tk)=χ(tk) + v(tk) Mθ 8 > > > > < > > > > : χ(tk) +

na

X

i=1

ai,0χ(tk−i) | {z }

F(q−1)χ(tk )

+

na

X

i=1 nα

X

l=1

ai,lfl(pk)χ(tk−i) | {z }

χi,l (tk )

=

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j | {z })

uj,l (tk )

y(tk)=χ(tk) + v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 15 / 32

Prediction Error for LPV systems Reformulation of the problem

LTI MISO interpretation of the model Mθ ( A(pk, q−1, ρ)χ(tk)=B(pk, q−1, ρ)u(tk) y(tk)=χ(tk) + v(tk) Mθ 8 > > < > > : χ(tk) +

na

X

i=1 nα

X

l=0

ai,lfl(pk)χ(tk−i)=

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j) y(tk)=χ(tk) + v(tk) Mθ 8 > > > < > > > : F(q−1, ρ)χ(tk) =−

na

X

i=1 nα

X

l=1

ai,lfl(pk)χ(tk−i) | {z }

χi,l (tk )

+

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j | {z })

uj,l (tk )

y(tk)=χ(tk) + v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 15 / 32

Prediction Error for LPV systems Reformulation of the problem

An example

LTI MISO model interpretation Mθ 8 > > > < > > > : F(q−1, ρ)χ(tk) =−

na

X

i=1 nα

X

l=1

ai,lfl(pk)χ(tk−i) | {z }

χi,l (tk )

+

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j | {z })

uj,l (tk )

y(tk)=χ(tk) + v(tk) S 8 > > < > > : “ 1 + ` 0.5 + 0.3p(tk) ´ | {z }

a1(tk )

q−1” χ(tk) = “ 1 + 0.5p(tk) | {z }

b0(tk )

” u(tk) y(tk) = χ(tk) + v(tk) S 8 > > < > > : “ 1 + 0.5q−1 | {z }

F(q−1)

” χ(tk) = −0.3p(tk)q−1χ(tk) | {z }

χ1,1(tk )

+ u(tk) | {z }

u0,0(tk )

+ 0.5p(tk)u(tk) | {z }

u0,1(tk )

y(tk) = χ(tk) + v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 16 / 32

Prediction Error for LPV systems Reformulation of the problem

An example

LTI MISO model interpretation Mθ 8 > > > < > > > : F(q−1, ρ)χ(tk) =−

na

X

i=1 nα

X

l=1

ai,lfl(pk)χ(tk−i) | {z }

χi,l (tk )

+

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j | {z })

uj,l (tk )

y(tk)=χ(tk) + v(tk) S 8 > > < > > : “ 1 + ` 0.5 + 0.3p(tk) ´ | {z }

a1(tk )

q−1” χ(tk) = “ 1 + 0.5p(tk) | {z }

b0(tk )

” u(tk) y(tk) = χ(tk) + v(tk) S 8 > > < > > : “ 1 + 0.5q−1 | {z }

F(q−1)

” χ(tk) = −0.3p(tk)q−1χ(tk) | {z }

χ1,1(tk )

+ u(tk) | {z }

u0,0(tk )

+ 0.5p(tk)u(tk) | {z }

u0,1(tk )

y(tk) = χ(tk) + v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 16 / 32

Prediction Error for LPV systems Reformulation of the problem

An example

LTI MISO model interpretation Mθ 8 > > > < > > > : F(q−1, ρ)χ(tk) =−

na

X

i=1 nα

X

l=1

ai,lfl(pk)χ(tk−i) | {z }

χi,l (tk )

+

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j | {z })

uj,l (tk )

y(tk)=χ(tk) + v(tk) S 8 > > < > > : “ 1 + ` 0.5 + 0.3p(tk) ´ | {z }

a1(tk )

q−1” χ(tk) = “ 1 + 0.5p(tk) | {z }

b0(tk )

” u(tk) y(tk) = χ(tk) + v(tk) S 8 > > < > > : “ 1 + 0.5q−1 | {z }

F(q−1)

” χ(tk) = −0.3p(tk)q−1χ(tk) | {z }

χ1,1(tk )

+ u(tk) | {z }

u0,0(tk )

+ 0.5p(tk)u(tk) | {z }

u0,1(tk )

y(tk) = χ(tk) + v(tk)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 16 / 32

Prediction Error for LPV systems Reformulation of the problem

LTI MISO model interpretation Mθ 8 > > > < > > > : F(q−1, ρ)χ(tk) =−

na

X

i=1 nα

X

l=1

ai,lfl(pk)χ(tk−i) | {z }

χi,l (tk )

+

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j | {z })

uj,l (tk )

y(tk)=χ(tk) + v(tk) Associated linear regression y(tk) = ϕ⊤(tk)ρ + F(q−1, ρ)v(tk) where, ϕ(tk)= ˆ −y(tk−1) . . . −y(tk−na) | −χ1,1(tk) . . . −χna,nα(tk) u0,0(tk) . . . unb,nβ(tk) ˜

(usually ϕ(tk )= h −y(tk−1) . . . −y(tk−na ) −y1,1(tk ) . . . −yna,nα (tk ) | u0,0(tk ) . . . unb,nβ (tk ) i⊤ )

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 17 / 32

Prediction Error for LPV systems Reformulation of the problem

LTI MISO model interpretation Mθ 8 > > > < > > > : F(q−1, ρ)χ(tk) =−

na

X

i=1 nα

X

l=1

ai,lfl(pk)χ(tk−i) | {z }

χi,l (tk )

+

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j | {z })

uj,l (tk )

y(tk)=χ(tk) + v(tk) Associated linear regression y(tk) = ϕ⊤(tk)ρ + F(q−1, ρ)v(tk) Real LTI configuration Filtering commutativity Theoretical possibility to minimize the prediction error in practice

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 17 / 32

Prediction Error for LPV systems Reformulation of the problem

LTI MISO model interpretation Mθ 8 > > > < > > > : F(q−1, ρ)χ(tk) =−

na

X

i=1 nα

X

l=1

ai,lfl(pk)χ(tk−i) | {z }

χi,l (tk )

+

nb

X

j=0 nβ

X

l=0

bj,lgl(pk)u(tk−j | {z })

uj,l (tk )

y(tk)=χ(tk) + v(tk) Associated linear regression y(tk) = ϕ⊤(tk)ρ + F(q−1, ρ)v(tk) Optimal filter a priori unknown Deterministic output a priori unknown In the Box-Jenkins case the LS method is biased Which method from the LTI framework could be used ?

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 17 / 32

Prediction Error for LPV systems RIV Algorithm

Instrumental variable method

Consider the LTI Box-Jenkins case y(tk) = ϕ⊤(tk)ρ+F(q−1, ρ) C(q−1, η) D(q−1, η)e(tk)

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• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 18 / 32

Prediction Error for LPV systems RIV Algorithm

Instrumental variable method

Consider the LTI Box-Jenkins case yf(tk) = ϕ⊤

f (tk)ρ+e(tk)

Using the Optimal filter : D(q−1, η) C(q−1, η)F(q−1, ρ)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 18 / 32

Prediction Error for LPV systems RIV Algorithm

Instrumental variable method

Consider the LTI Box-Jenkins case yf(tk) = ϕ⊤

f (tk)ρ+e(tk)

Using the Optimal filter : D(q−1, η) C(q−1, η)F(q−1, ρ) Least Square Solution ρLS(N)= " N X

k=1

ϕf(tk)ϕ⊤

f (tk)

#−1 N X

k=1

ϕf(tk)yf(tk) Instrumental Variable Solution ρIV(N)= " N X

k=1

ζf(tk)ϕ⊤

f (tk)

#−1 N X

k=1

ζf(tk)yf(tk) ζf(tk) is the so called instrument

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 18 / 32

Prediction Error for LPV systems RIV Algorithm

Instrumental variable method

Consider the LTI Box-Jenkins case yf(tk) = ϕ⊤

f (tk)ρ+e(tk)

Using the Optimal filter : D(q−1, η) C(q−1, η)F(q−1, ρ) Properties ρIV is consistent if :

Eζf(tk)ϕ⊤

f (tk) is non-singular (correlation to the data)

Eζf(tk)ǫf(tk) = 0 (no correlation to the noise)

For example delayed version of the regressor (S¨

• derstr¨
• m 1983)

Optimal choice (minimum variance) : the instrument is the noise-free version of the regressor Why an IV method over a LS method ?

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 18 / 32

Prediction Error for LPV systems RIV Algorithm

Instrumental variable method

Consider the LTI Box-Jenkins case yf(tk) = ϕ⊤

f (tk)ρ+e(tk)

Using the Optimal filter : D(q−1, η) C(q−1, η)F(q−1, ρ) Advantages of IV methods In practice the noise model is unknown Therefore the optimal filter is unknown LS method is biased Advantage of IV methods : consistent even if wrong noise model assumption

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 18 / 32

Prediction Error for LPV systems RIV Algorithm

Initialise Step 1 : Assume that as an initialization, an LPV-ARX estimate of Mθ is available by the LPV-LS approach Simulate Step 2 : Compute an estimate of the deterministic output χ(tk) based on the estimated parameters of the previous iteration. Deduce the output terms {ˆ χi,l(tk)}

na,nα i=1,l=0.

Filter Step 3 : Compute the estimated filter based on previous iteration ˆ Q(q−1, ˆ θ(τ)) = D(q−1, ˆ η(τ)) C(q−1, ˆ η(τ))F(q−1, ˆ ρ(τ)) and the associated filtered signals {uf

j,l(tk)} nb,nβ j=0,l=0, yf(tk) and {χf i,l(tk)}na,nα i=1,l=0.

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 19 / 32

Prediction Error for LPV systems RIV Algorithm

Initialise Step 1 : Assume that as an initialization, an LPV-ARX estimate of Mθ is available by the LPV-LS approach Simulate Step 2 : Compute an estimate of the deterministic output χ(tk) based on the estimated parameters of the previous iteration. Deduce the output terms {ˆ χi,l(tk)}

na,nα i=1,l=0.

Filter Step 3 : Compute the estimated filter based on previous iteration ˆ Q(q−1, ˆ θ(τ)) = D(q−1, ˆ η(τ)) C(q−1, ˆ η(τ))F(q−1, ˆ ρ(τ)) and the associated filtered signals {uf

j,l(tk)} nb,nβ j=0,l=0, yf(tk) and {χf i,l(tk)}na,nα i=1,l=0.

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 19 / 32

Prediction Error for LPV systems RIV Algorithm

Initialise Step 1 : Assume that as an initialization, an LPV-ARX estimate of Mθ is available by the LPV-LS approach Simulate Step 2 : Compute an estimate of the deterministic output χ(tk) based on the estimated parameters of the previous iteration. Deduce the output terms {ˆ χi,l(tk)}

na,nα i=1,l=0.

Filter Step 3 : Compute the estimated filter based on previous iteration ˆ Q(q−1, ˆ θ(τ)) = D(q−1, ˆ η(τ)) C(q−1, ˆ η(τ))F(q−1, ˆ ρ(τ)) and the associated filtered signals {uf

j,l(tk)} nb,nβ j=0,l=0, yf(tk) and {χf i,l(tk)}na,nα i=1,l=0.

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 19 / 32

Prediction Error for LPV systems RIV Algorithm

Build regressor and Instrument Step 4 : Build the filtered estimated regressor ˆ ϕf(tk) : ˆ ϕf(tk)= ˆ −yf(tk−1) . . . −yf(tk−na) −ˆ χf

1,1(tk) . . . −ˆ

χf

na,nα(tk) uf 0,0(tk) . . . uf nb,nβ(tk) ˜⊤

Compute the filtered estimated instrument (which is optimal if the instrument equals the noise-free regressor) : ˆ ζf(tk)= ˆ −ˆ χf(tk−1) . . . −ˆ χf(tk−na) −ˆ χf

1,1(tk) . . . −ˆ

χf

na,nα(tk) uf 0,0(tk) . . . uf nb,nβ(tk) ˜⊤

Compute the IV solution Step 5 : Compute the IV solution for the process model ˆ ρ(τ+1)(N) at iteration τ + 1 based on the prefiltered input/output data.

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 20 / 32

Prediction Error for LPV systems RIV Algorithm

Build regressor and Instrument Step 4 : Build the filtered estimated regressor ˆ ϕf(tk) : ˆ ϕf(tk)= ˆ −yf(tk−1) . . . −yf(tk−na) −ˆ χf

1,1(tk) . . . −ˆ

χf

na,nα(tk) uf 0,0(tk) . . . uf nb,nβ(tk) ˜⊤

Compute the filtered estimated instrument (which is optimal if the instrument equals the noise-free regressor) : ˆ ζf(tk)= ˆ −ˆ χf(tk−1) . . . −ˆ χf(tk−na) −ˆ χf

1,1(tk) . . . −ˆ

χf

na,nα(tk) uf 0,0(tk) . . . uf nb,nβ(tk) ˜⊤

Compute the IV solution Step 5 : Compute the IV solution for the process model ˆ ρ(τ+1)(N) at iteration τ + 1 based on the prefiltered input/output data.

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 20 / 32

Prediction Error for LPV systems RIV Algorithm

Compute noise model Step 6 : An estimate of the noise signal v is obtained as ˆ v(tk) = y(tk) − ˆ χ(tk, ˆ ρ(τ)) Based on ˆ v, the estimation of the noise model parameter vector ˆ η(τ+1) follows, using in this case the ARMA estimation algorithm of the MATLAB identification toolbox (an IV approach can also be used for this purpose ). Iterate Iterate until convergence

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 21 / 32

Prediction Error for LPV systems RIV Algorithm

Compute noise model Step 6 : An estimate of the noise signal v is obtained as ˆ v(tk) = y(tk) − ˆ χ(tk, ˆ ρ(τ)) Based on ˆ v, the estimation of the noise model parameter vector ˆ η(τ+1) follows, using in this case the ARMA estimation algorithm of the MATLAB identification toolbox (an IV approach can also be used for this purpose ). Iterate Iterate until convergence

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 21 / 32

Results

Outline

1

Introduction

2

Problem Description LPV model description Scheduling dependency model

3

Prediction Error for LPV systems Usual linear regression The literature methods Reformulation of the problem RIV Algorithm

4

Results LPV-RIV performance

5

Application example

6

Conclusion

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 22 / 32

Results LPV-RIV performance

Example The system taken into consideration is mathematically described as follows : So 8 > > < > > : Ao(q, pk) = 1 + ao

1(pk)q−1 + ao 2(pk)q−2

Bo(q, pk) = bo

0(pk)q−1 + bo 1(pk)q−2

Ho(q) = 1 1 − q−1 + 0.2q−2 where ao

1(pk) = 1 − 0.5pk − 0.1p2 k

ao

2(pk) = 0.5 − 0.7pk − 0.1p2 k

bo

0(pk) = 0.5 − 0.4pk + 0.01p2 k

bo

1(pk) = 0.2 − 0.3pk − 0.02p2 k

The scheduling signal p is considered as a periodic function of time :pk = 0.5 sin(0.35πk) + 0.5 u(tk) is taken as a white noise with a uniform distribution U(−1, 1) and with length N = 4000 Method compared LS : assumes an ARX model (Bamieh, Giarr´ e) OSIV : uses the ARX model to build an instrument (Butcher, Abbas) LPV-SRIV : RIV technique for OE models. Assumes H(q−1) = 1 (Laurain et. al) LPV-RIV : RIV technique for BJ models. (Laurain et. al)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 23 / 32

Results LPV-RIV performance

Example The system taken into consideration is mathematically described as follows : So 8 > > < > > : Ao(q, pk) = 1 + ao

1(pk)q−1 + ao 2(pk)q−2

Bo(q, pk) = bo

0(pk)q−1 + bo 1(pk)q−2

Ho(q) = 1 1 − q−1 + 0.2q−2 where ao

1(pk) = 1 − 0.5pk − 0.1p2 k

ao

2(pk) = 0.5 − 0.7pk − 0.1p2 k

bo

0(pk) = 0.5 − 0.4pk + 0.01p2 k

bo

1(pk) = 0.2 − 0.3pk − 0.02p2 k

The scheduling signal p is considered as a periodic function of time :pk = 0.5 sin(0.35πk) + 0.5 u(tk) is taken as a white noise with a uniform distribution U(−1, 1) and with length N = 4000 Method compared LS : assumes an ARX model (Bamieh, Giarr´ e) OSIV : uses the ARX model to build an instrument (Butcher, Abbas) LPV-SRIV : RIV technique for OE models. Assumes H(q−1) = 1 (Laurain et. al) LPV-RIV : RIV technique for BJ models. (Laurain et. al)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 23 / 32

Results LPV-RIV performance

Monte-Carlo simulation of 100 runs SNR from 15dB to 0dB The bias and variance on the estimated parameter vector is computed

10 20 30 40 50 60 70 80 90 100 −0.5 0.5 1

time

• uput

noise−free output

• utput

10 20 30 40 50 60 70 80 90 100 −0.5 0.5

input time

Input and Output signals example at 0dB

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 24 / 32

Results LPV-RIV performance Method 15dB 10dB 5dB 0dB LS Bias norm 2.9107 3.2897 3.0007 2.8050 Variance norm 0.0074 0.0151 0.0215 0.0326 OSIV Bias norm 0.1961 1.8265 6.9337 10.8586 Variance norm 1.3353 179.4287 590.7869 11782 LPV-SRIV Bias norm 0.0072 0.0426 0.1775 0.2988 Variance norm 0.0149 0.0537 0.4425 0.4781 mean iteration number 22 22 25 30 LPV-RIV Bias norm 0.0068 0.0184 0.0408 0.1649 Variance norm 0.0063 0.0219 0.0696 0.2214 mean iteration number 31 30 30 32

LS estimates are biased OSIV delivers unbiased estimates but unusable under 15dB due to large variance LPV SRIV is consistent and delivers low variance in estimates even if the noise model is false LPV RIV estimates are unbiased and present variance similar to LS method

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 25 / 32

Results LPV-RIV performance Method 15dB 10dB 5dB 0dB LS Bias norm 2.9107 3.2897 3.0007 2.8050 Variance norm 0.0074 0.0151 0.0215 0.0326 OSIV Bias norm 0.1961 1.8265 6.9337 10.8586 Variance norm 1.3353 179.4287 590.7869 11782 LPV-SRIV Bias norm 0.0072 0.0426 0.1775 0.2988 Variance norm 0.0149 0.0537 0.4425 0.4781 mean iteration number 22 22 25 30 LPV-RIV Bias norm 0.0068 0.0184 0.0408 0.1649 Variance norm 0.0063 0.0219 0.0696 0.2214 mean iteration number 31 30 30 32

LS estimates are biased OSIV delivers unbiased estimates but unusable under 15dB due to large variance LPV SRIV is consistent and delivers low variance in estimates even if the noise model is false LPV RIV estimates are unbiased and present variance similar to LS method

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 25 / 32

Results LPV-RIV performance Method 15dB 10dB 5dB 0dB LS Bias norm 2.9107 3.2897 3.0007 2.8050 Variance norm 0.0074 0.0151 0.0215 0.0326 OSIV Bias norm 0.1961 1.8265 6.9337 10.8586 Variance norm 1.3353 179.4287 590.7869 11782 LPV-SRIV Bias norm 0.0072 0.0426 0.1775 0.2988 Variance norm 0.0149 0.0537 0.4425 0.4781 mean iteration number 22 22 25 30 LPV-RIV Bias norm 0.0068 0.0184 0.0408 0.1649 Variance norm 0.0063 0.0219 0.0696 0.2214 mean iteration number 31 30 30 32

LS estimates are biased OSIV delivers unbiased estimates but unusable under 15dB due to large variance LPV SRIV is consistent and delivers low variance in estimates even if the noise model is false LPV RIV estimates are unbiased and present variance similar to LS method

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 25 / 32

Results LPV-RIV performance Method 15dB 10dB 5dB 0dB LS Bias norm 2.9107 3.2897 3.0007 2.8050 Variance norm 0.0074 0.0151 0.0215 0.0326 OSIV Bias norm 0.1961 1.8265 6.9337 10.8586 Variance norm 1.3353 179.4287 590.7869 11782 LPV-SRIV Bias norm 0.0072 0.0426 0.1775 0.2988 Variance norm 0.0149 0.0537 0.4425 0.4781 mean iteration number 22 22 25 30 LPV-RIV Bias norm 0.0068 0.0184 0.0408 0.1649 Variance norm 0.0063 0.0219 0.0696 0.2214 mean iteration number 31 30 30 32

LS estimates are biased OSIV delivers unbiased estimates but unusable under 15dB due to large variance LPV SRIV is consistent and delivers low variance in estimates even if the noise model is false LPV RIV estimates are unbiased and present variance similar to LS method

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 25 / 32

Results LPV-RIV performance

Detailed results at 10dB

,0 ,1 ,2

a1 a2 b0 b1

1 1.2 1.4 1.6 1.8 −1 1 2 1 1.2 1.4 1.6 1.8 −1 1 2 1 1.2 1.4 1.6 1.8 −2 −1 1 1 1.2 1.4 1.6 1.8 −0.5 0.5 1 1 1.2 1.4 1.6 1.8 −1.5 −1 −0.5 1 1.2 1.4 1.6 1.8 −1 −0.5 0.5 1 1.2 1.4 1.6 1.8 0.48 0.5 0.52 0.54 1 1.2 1.4 1.6 1.8 −0.5 −0.4 −0.3 1 1.2 1.4 1.6 1.8 −0.1 0.1 1 1.2 1.4 1.6 1.8 −0.5 0.5 1 1.2 1.4 1.6 1.8 −0.5 0.5 1 1 1.2 1.4 1.6 1.8 −1 −0.5 0.5

Detailed results at SNR=10dB for methods LS (black), SRIV(blue), RIV(green) with respect to the true value (blue line)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 26 / 32

Application example

Outline

1

Introduction

2

Problem Description LPV model description Scheduling dependency model

3

Prediction Error for LPV systems Usual linear regression The literature methods Reformulation of the problem RIV Algorithm

4

Results LPV-RIV performance

5

Application example

6

Conclusion

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 27 / 32

Application example

Rainfall/runoff modelling

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 28 / 32

Application example

Rainfall/runoff modelling

Problems Nonlinearity of the model (depending on the temperature, plant absorption, soil structure, surface) Conceptual models at the catchment scale are tedious (too many parameters) Noise cannot be expressed in a stochastic way Why an LPV model ? The response of the catchment depends on the moisture of the field (like a sponge) Simple parsimonious model Possibility of interpretation as time-varying pierced tank (physical meaning)

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 28 / 32

Application example

Rainfall/runoff modelling

100 200 300 400 500 2 4 6 8 10 12 14 16 18 time (min) flow (l/s) Measured flow Linear Simulation LPV Simulation

(a) LPV and linear model results for

a light rainfall event

20 40 60 80 100 120 20 40 60 80 100 120 140 160 180 time (min) flow (l/s) Measured flow Linear Simulation LPV Simulation

(b) LPV and linear model results for

a strong rainfall event Fig.: Comparison of linear and LPV OE models for light and strong rainfall events

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 28 / 32

Application example

Rainfall/runoff modelling

100 200 300 400 500 600 2 4 6 8 time (min) flow (l/s) Measured Flow LPV−ARX Simulation

(a) LPV-ARX model results for a

light rainfall event

100 200 300 400 500 600 2 4 6 8 10 time (min) flow (l/s) Measured Flow LPV−OE Simulation

(b) LPV-OE model results for a light

rainfall event

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

() LPV-RIV S´ eminaire LOUVAIN 28 / 32

Application example

Rainfall/runoff modelling

5 10 15 20 20 40 60 80 100 120 140 160 180 time (min) flow (l/s) Measured Flow LPV−ARX Simulation

(c) LPV-ARX model results for a

strong rainfall event

5 10 15 20 20 40 60 80 100 120 140 160 180 time (min) flow (l/s) Measured Flow LPV−OE Simulation

(d) LPV-OE

model results for a strong rainfall event

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

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Conclusion

Outline

1

Introduction

2

Problem Description LPV model description Scheduling dependency model

3

Prediction Error for LPV systems Usual linear regression The literature methods Reformulation of the problem RIV Algorithm

4

Results LPV-RIV performance

5

Application example

6

Conclusion

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• th, H. Garnier

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Conclusion

Conclusion

Many Open issues Full LPV models (LPV noise model) Conditions of excitation Convergence of RIV methods Conclusion Usual literature regressor can not lead to optimal estimates in practice By writing the LPV system under a MISO form makes possible the extension of LTI methods for solving LPV identification problem LPV-SRIV method outperforms existing methods even for noise model errors LPV-RIV method approaches (reaches ?) statistically optimal estimates Ongoing work Proposition of one of the very first methods for CT LPV models CT method soon available in the MATLAB toolbox

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• th, H. Garnier

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Conclusion

References

Book about LPV

• R. T´
• th. Modeling and Identification of Linear Parameter-Varying Systems.

Springer-Germany, may 2010. About this work

• V. Laurain, M. Gilson, R. T´
• th and H. Garnier. Refined Instrumental Variable Methods

for Identification of LPV Box-Jenkins Models. Automatica, Vol. 46, Issue 6 : 959–967, June 2010.

• V. Laurain, R. T´
• th, M. Gilson and H. Garnier. Direct Identification of Continuous-time

LPV Input/Ouput Models. Submitted to special issue “Continuous-time model Identification”, IET Control Theory & Applications, April 2010.

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

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Thank You

Thank you for your attention Identification of Input/Output LPV Models

• V. Laurain, M. Gilson, R. T´
• th, H. Garnier

CRAN, Nancy Universit´ e, DCSC, TU Delft

Delft Center for System s and Control Delft Center for System s and Control

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• th, H. Garnier

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