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ROCOND06 T. Hilaire, Pole Sensitivity Stability Related Measure of P. Chevrel, J.P. Clauzel FWL Realizations with the Implicit State-Space Introduction Formalism A pole sensitivity stability related measure T. Hilaire 1 , 3 P.

slide-1
SLIDE 1

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 1/31

Pole Sensitivity Stability Related Measure of FWL Realizations with the Implicit State-Space Formalism

  • T. Hilaire1,3
  • P. Chevrel1,2

J.P. Clauzel3

1IRCCyN UMR CNRS 6597 NANTES FRANCE 2´

Ecole des Mines de Nantes NANTES FRANCE

3PSA Peugeot Citro¨

en LA GARENNE COLOMBES FRANCE

ROCOND’06 - 5-7 July 2006 - Toulouse France

slide-2
SLIDE 2

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 2/31

Context

Implementation of Linear Time Invariant controllers Finite Word Length context Motivation Evaluate the impact of the quantization of the embedded coefficients Compare various realizations and find an optimal one

slide-3
SLIDE 3

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 2/31

Context

Implementation of Linear Time Invariant controllers Finite Word Length context Motivation Evaluate the impact of the quantization of the embedded coefficients Compare various realizations and find an optimal one

slide-4
SLIDE 4

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 3/31

Outline

1

A pole sensitivity stability related measure

2

Macroscopic representation of algorithms through the implicit state-space framework

3

Extension of the pole-sensitivity stability related measure

4

Optimal realization

5

Conclusion and Perspectives

slide-5
SLIDE 5

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 4/31

Outline

1

A pole sensitivity stability related measure

2

Macroscopic representation of algorithms through the implicit state-space framework

3

Extension of the pole-sensitivity stability related measure

4

Optimal realization

5

Conclusion and Perspectives

slide-6
SLIDE 6

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 5/31

FWL degradation

Origin of the degradation The deterioration induced by the FWL implementation comes from : Quantization of the involved coefficients → parametric errors Roundoff noises in numerical computations → numerical noises Only the deterioration induced by the quantization of coefficients is considered here.

slide-7
SLIDE 7

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 5/31

FWL degradation

Origin of the degradation The deterioration induced by the FWL implementation comes from : Quantization of the involved coefficients → parametric errors Roundoff noises in numerical computations → numerical noises Only the deterioration induced by the quantization of coefficients is considered here.

slide-8
SLIDE 8

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 6/31

Problem setup

Let’s consider a discrete plant P P X p

k+1

= ApX p

k + Bp(Rk + Yk)

Uk = CpX p

k

and a LTI controller C C Xk+1 = AXk + BUk Yk = CXk + DUk P C +

Rk Yk Uk

The realizations of the form (T −1AT, T −1B, CT, D), with T a non-singular matrix, are all equivalent in infinite precision. They are no more in finite precision. The degradation of the realization depends on the realization.

slide-9
SLIDE 9

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 6/31

Problem setup

Let’s consider a discrete plant P P X p

k+1

= ApX p

k + Bp(Rk + Yk)

Uk = CpX p

k

and a LTI controller C C Xk+1 = AXk + BUk Yk = CXk + DUk P C +

Rk Yk Uk

The realizations of the form (T −1AT, T −1B, CT, D), with T a non-singular matrix, are all equivalent in infinite precision. They are no more in finite precision. The degradation of the realization depends on the realization.

slide-10
SLIDE 10

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 7/31

Problem setup

When quantized, the parameters X D C B A

  • are changed in

X + ∆X and the closed-loop system can became unstable. Let’s denote

  • λk(¯

A(X))

  • 1kl the eigenvalues of the

closed-loop system ¯ Xk+1 = ¯ A¯ Xk + ¯ BRk Uk = ¯ C ¯ Xk (1) with ¯ A Ap + BpDCp BpC BCp A

  • ¯

B Bp

  • ¯

C

  • Cp
  • (2)
slide-11
SLIDE 11

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 7/31

Problem setup

When quantized, the parameters X D C B A

  • are changed in

X + ∆X and the closed-loop system can became unstable. Let’s denote

  • λk(¯

A(X))

  • 1kl the eigenvalues of the

closed-loop system ¯ Xk+1 = ¯ A¯ Xk + ¯ BRk Uk = ¯ C ¯ Xk (1) with ¯ A Ap + BpDCp BpC BCp A

  • ¯

B Bp

  • ¯

C

  • Cp
  • (2)
slide-12
SLIDE 12

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 8/31

Pole-sensitivity stability related measure

Chen, Wu, Li,... (2000-2005) have proposed a pole-sensitivity measure defined by µ(X) min

1kr

1 −

  • λk(¯

A(X))

  • √NΨk

with N is the number of non-trivial elements in X (non-zero elements in ∆X) and Ψk is the pole sensitivity of the closed-loop with respect to the parameters : Ψk

  • i,j

(WX)i,j

  • ∂
  • λk(¯

A(X))

  • ∂Xi,j
  • 2

WX is the weighting matrix associated to the realization matrix X, defined by (WX)i,j =

  • if Xi,j is exactly implemented

1 if not

slide-13
SLIDE 13

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 8/31

Pole-sensitivity stability related measure

Chen, Wu, Li,... (2000-2005) have proposed a pole-sensitivity measure defined by µ(X) min

1kr

1 −

  • λk(¯

A(X))

  • √NΨk

with N is the number of non-trivial elements in X (non-zero elements in ∆X) and Ψk is the pole sensitivity of the closed-loop with respect to the parameters : Ψk

  • i,j

(WX)i,j

  • ∂
  • λk(¯

A(X))

  • ∂Xi,j
  • 2

WX is the weighting matrix associated to the realization matrix X, defined by (WX)i,j =

  • if Xi,j is exactly implemented

1 if not

slide-14
SLIDE 14

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 8/31

Pole-sensitivity stability related measure

Chen, Wu, Li,... (2000-2005) have proposed a pole-sensitivity measure defined by µ(X) min

1kr

1 −

  • λk(¯

A(X))

  • √NΨk

with N is the number of non-trivial elements in X (non-zero elements in ∆X) and Ψk is the pole sensitivity of the closed-loop with respect to the parameters : Ψk

  • i,j

(WX)i,j

  • ∂
  • λk(¯

A(X))

  • ∂Xi,j
  • 2

WX is the weighting matrix associated to the realization matrix X, defined by (WX)i,j =

  • if Xi,j is exactly implemented

1 if not

slide-15
SLIDE 15

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 9/31

Pole-sensitivity stability related measure

the measure is such that ∆Xmax µ(X) ⇒ ¯ A(X + ∆X) is stable it considers how close the eigenvalues are to 1 and how sensitive they are w.r.t the controller parameters ; this measure is directly linked an estimation of the smallest word-length bit needed to guarantee the closed-loop stability the optimal design problem associated consists in finding an equivalent realization (T −1AT, T −1B, CT, D) that maximizes this measure.

slide-16
SLIDE 16

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 9/31

Pole-sensitivity stability related measure

the measure is such that ∆Xmax µ(X) ⇒ ¯ A(X + ∆X) is stable it considers how close the eigenvalues are to 1 and how sensitive they are w.r.t the controller parameters ; this measure is directly linked an estimation of the smallest word-length bit needed to guarantee the closed-loop stability the optimal design problem associated consists in finding an equivalent realization (T −1AT, T −1B, CT, D) that maximizes this measure.

slide-17
SLIDE 17

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 9/31

Pole-sensitivity stability related measure

the measure is such that ∆Xmax µ(X) ⇒ ¯ A(X + ∆X) is stable it considers how close the eigenvalues are to 1 and how sensitive they are w.r.t the controller parameters ; this measure is directly linked an estimation of the smallest word-length bit needed to guarantee the closed-loop stability the optimal design problem associated consists in finding an equivalent realization (T −1AT, T −1B, CT, D) that maximizes this measure.

slide-18
SLIDE 18

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 9/31

Pole-sensitivity stability related measure

the measure is such that ∆Xmax µ(X) ⇒ ¯ A(X + ∆X) is stable it considers how close the eigenvalues are to 1 and how sensitive they are w.r.t the controller parameters ; this measure is directly linked an estimation of the smallest word-length bit needed to guarantee the closed-loop stability the optimal design problem associated consists in finding an equivalent realization (T −1AT, T −1B, CT, D) that maximizes this measure.

slide-19
SLIDE 19

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 10/31

Outline

1

A pole sensitivity stability related measure

2

Macroscopic representation of algorithms through the implicit state-space framework

3

Extension of the pole-sensitivity stability related measure

4

Optimal realization

5

Conclusion and Perspectives

slide-20
SLIDE 20

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 11/31

The need of a unifying framework

Various implementation forms have to be taken into consideration shift-realizations δ-realizations

  • bserver-state-feedback

direct form I or II cascade or parallel realizations etc...

slide-21
SLIDE 21

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 12/31

The need of a unifying framework

In order to encompass all these implementations, we have proposed a specialized implicit state-space realization to be used as a unifying framework : Interests macroscopic description of a FWL implementation more general than previous realizations more realistic with regard to the parameterization directly linked to the in-line computations to be performed

slide-22
SLIDE 22

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 12/31

The need of a unifying framework

In order to encompass all these implementations, we have proposed a specialized implicit state-space realization to be used as a unifying framework : Interests macroscopic description of a FWL implementation more general than previous realizations more realistic with regard to the parameterization directly linked to the in-line computations to be performed

slide-23
SLIDE 23

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 12/31

The need of a unifying framework

In order to encompass all these implementations, we have proposed a specialized implicit state-space realization to be used as a unifying framework : Interests macroscopic description of a FWL implementation more general than previous realizations more realistic with regard to the parameterization directly linked to the in-line computations to be performed

slide-24
SLIDE 24

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 12/31

The need of a unifying framework

In order to encompass all these implementations, we have proposed a specialized implicit state-space realization to be used as a unifying framework : Interests macroscopic description of a FWL implementation more general than previous realizations more realistic with regard to the parameterization directly linked to the in-line computations to be performed

slide-25
SLIDE 25

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 12/31

The need of a unifying framework

In order to encompass all these implementations, we have proposed a specialized implicit state-space realization to be used as a unifying framework : Interests macroscopic description of a FWL implementation more general than previous realizations more realistic with regard to the parameterization directly linked to the in-line computations to be performed

slide-26
SLIDE 26

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 13/31

Implicit State-Space Framework

The control algorithm is described with

1 J.Tk+1 = M.Xk + N.Uk 2 Xk+1 = K.Tk+1 + P.Xk + Q.Uk 3 Yk = L.Tk+1 + R.Xk + S.Uk

Intermediate variables computation Implicit State-Space Framework   J −K I −L I     Tk+1 Xk+1 Yk   =   M N P Q R S     Tk Xk Uk  

slide-27
SLIDE 27

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 13/31

Implicit State-Space Framework

The control algorithm is described with

1 J.Tk+1 = M.Xk + N.Uk 2 Xk+1 = K.Tk+1 + P.Xk + Q.Uk 3 Yk = L.Tk+1 + R.Xk + S.Uk

State-vector computation Implicit State-Space Framework   J −K I −L I     Tk+1 Xk+1 Yk   =   M N P Q R S     Tk Xk Uk  

slide-28
SLIDE 28

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 13/31

Implicit State-Space Framework

The control algorithm is described with

1 J.Tk+1 = M.Xk + N.Uk 2 Xk+1 = K.Tk+1 + P.Xk + Q.Uk 3 Yk = L.Tk+1 + R.Xk + S.Uk

Output computation Implicit State-Space Framework   J −K I −L I     Tk+1 Xk+1 Yk   =   M N P Q R S     Tk Xk Uk  

slide-29
SLIDE 29

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 13/31

Implicit State-Space Framework

The control algorithm is described with

1 J.Tk+1 = M.Xk + N.Uk 2 Xk+1 = K.Tk+1 + P.Xk + Q.Uk 3 Yk = L.Tk+1 + R.Xk + S.Uk

Implicit State-Space Framework   J −K I −L I     Tk+1 Xk+1 Yk   =   M N P Q R S     Tk Xk Uk  

slide-30
SLIDE 30

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 14/31

Intermediate variables

The intermediate variables introduced allow to make explicit all the computations done show the order of the computations express a larger parameterization All the coefficients used in the implicit framework can be regrouped in a generalized system matrix Z Z   −J M N K P Q L R S  

slide-31
SLIDE 31

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 14/31

Intermediate variables

The intermediate variables introduced allow to make explicit all the computations done show the order of the computations express a larger parameterization All the coefficients used in the implicit framework can be regrouped in a generalized system matrix Z Z   −J M N K P Q L R S  

slide-32
SLIDE 32

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 15/31

Implicit form

This form is implicit (but non singular) : the state or the output may be computed from intermediate variables an intermediate variable may be computed from another intermediate variable previously computed (in the same step) computation of Tk+1 is J.Tk+1 = M.Xk + N.Uk with J =          1 . . . . . . ⋆ ... . . . . . . ⋆ 1 . . . . . . ⋆ ... ⋆ . . . . . . ⋆ 1         

slide-33
SLIDE 33

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 16/31

Examples

A realization with the δ-operator is described by : δXk = AδXk + BδUk Yk = CδXk + DδUk δ q−1

∆

and it corresponds to the following implicit state-space :   I − ∆I I I     Tk+1 Xk+1 Yk   =   Aδ Bδ I Cδ Dδ     Tk Xk Uk  

slide-34
SLIDE 34

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 16/31

Examples

A realization with the δ-operator is described by : δXk = AδXk + BδUk Yk = CδXk + DδUk δ q−1

∆

and it corresponds to the following implicit state-space :   I − ∆I I I     Tk+1 Xk+1 Yk   =   Aδ Bδ I Cδ Dδ     Tk Xk Uk  

slide-35
SLIDE 35

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 17/31

Examples

The Observer State-Feedback ˆ Xk+1 = Ap ˆ Xk + BpUk + Kf (Yk − Cp ˆ Xk) Uk = − Kc ˆ Xk + Q(Yk − Cp ˆ Xk) where (Ap, Bp, Cp) corresponds to the plant system and Kc, Kf and Q are the controller’s parameters. A first parametrization

B B @ „ I − Q I « „ « „ « `−Kf −Bp ´ I `0 −I´ I 1 C C A B B B B @ @T (1)

k+1

T (2)

k+1

1 A ˆ Xk+1 Uk 1 C C C C A = B B @ „ « „ − Cp − Kc « „ I « `0 0´ Ap `0 0´ 1 C C A B B B @ T (1)

k

T (2)

k

! ˆ Xk Yk 1 C C C A

slide-36
SLIDE 36

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 17/31

Examples

The Observer State-Feedback ˆ Xk+1 = Ap ˆ Xk + BpUk + Kf (Yk − Cp ˆ Xk) Uk = − Kc ˆ Xk + Q(Yk − Cp ˆ Xk) where (Ap, Bp, Cp) corresponds to the plant system and Kc, Kf and Q are the controller’s parameters. A first parametrization

B B @ „ I − Q I « „ « „ « `−Kf −Bp ´ I `0 −I´ I 1 C C A B B B B @ @T (1)

k+1

T (2)

k+1

1 A ˆ Xk+1 Uk 1 C C C C A = B B @ „ « „ − Cp − Kc « „ I « `0 0´ Ap `0 0´ 1 C C A B B B @ T (1)

k

T (2)

k

! ˆ Xk Yk 1 C C C A

slide-37
SLIDE 37

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 17/31

Examples

The Observer State-Feedback ˆ Xk+1 = Ap ˆ Xk + BpUk + Kf (Yk − Cp ˆ Xk) Uk = − Kc ˆ Xk + Q(Yk − Cp ˆ Xk) where (Ap, Bp, Cp) corresponds to the plant system and Kc, Kf and Q are the controller’s parameters. A first parametrization

B B @ „ I − Q I « „ « „ « `−Kf −Bp ´ I `0 −I´ I 1 C C A B B B B @ @T (1)

k+1

T (2)

k+1

1 A ˆ Xk+1 Uk 1 C C C C A = B B @ „ « „ − Cp − Kc « „ I « `0 0´ Ap `0 0´ 1 C C A B B B @ T (1)

k

T (2)

k

! ˆ Xk Yk 1 C C C A

slide-38
SLIDE 38

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 17/31

Examples

The Observer State-Feedback ˆ Xk+1 = Ap ˆ Xk + BpUk + Kf (Yk − Cp ˆ Xk) Uk = − Kc ˆ Xk + Q(Yk − Cp ˆ Xk) where (Ap, Bp, Cp) corresponds to the plant system and Kc, Kf and Q are the controller’s parameters. A first parametrization

B B @ „ I − Q I « „ « „ « `−Kf −Bp ´ I `0 −I´ I 1 C C A B B B B @ @T (1)

k+1

T (2)

k+1

1 A ˆ Xk+1 Uk 1 C C C C A = B B @ „ « „ − Cp − Kc « „ I « `0 0´ Ap `0 0´ 1 C C A B B B @ T (1)

k

T (2)

k

! ˆ Xk Yk 1 C C C A

slide-39
SLIDE 39

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 17/31

Examples

The Observer State-Feedback ˆ Xk+1 = Ap ˆ Xk + BpUk + Kf (Yk − Cp ˆ Xk) Uk = − Kc ˆ Xk + Q(Yk − Cp ˆ Xk) where (Ap, Bp, Cp) corresponds to the plant system and Kc, Kf and Q are the controller’s parameters. A first parametrization

B B @ „ I − Q I « „ « „ « `−Kf −Bp ´ I `0 −I´ I 1 C C A B B B B @ @T (1)

k+1

T (2)

k+1

1 A ˆ Xk+1 Uk 1 C C C C A = B B @ „ « „ − Cp − Kc « „ I « `0 0´ Ap `0 0´ 1 C C A B B B @ T (1)

k

T (2)

k

! ˆ Xk Yk 1 C C C A

slide-40
SLIDE 40

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 17/31

Examples

The Observer State-Feedback ˆ Xk+1 = Ap ˆ Xk + BpUk + Kf (Yk − Cp ˆ Xk) Uk = − Kc ˆ Xk + Q(Yk − Cp ˆ Xk) where (Ap, Bp, Cp) corresponds to the plant system and Kc, Kf and Q are the controller’s parameters. An other possible parametrization

@ I − Bp I −I I 1 A @ Tk+1 ˆ Xk+1 Uk 1 A = @ − (QCp + Kc) Q (Ap − Kf C) Kf 1 A @ Tk ˆ Xk Yk 1 A

slide-41
SLIDE 41

ROCOND’06 —

  • T. Hilaire,
  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 18/31

Outline

1

A pole sensitivity stability related measure

2

Macroscopic representation of algorithms through the implicit state-space framework

3

Extension of the pole-sensitivity stability related measure

4

Optimal realization

5

Conclusion and Perspectives

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 19/31

Pole-sensibility stability related measure

The pole-sensitivity measure can be extended in implicit state-space framework. µ(Z) = min

1kr

1 − |λk| √NΨk whith N represents the number of non trivial elements, and Ψk =

  • ∂ |λk|

∂Z × WZ

  • 2

F

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 20/31

Pole-sensibility stability related measure

Ψk can be easily computed : Proposition 1 ∂ |λk| ∂Z = ¯ M⊤

1

∂ |λk| ∂¯ A ¯ M⊤

2

with ¯ M1

  • BpLJ−1

Bp KJ−1 In

  • ¯

M2

 J−1NCp J−1M In Cp  

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 21/31

Pole-sensibility stability related measure

Proposition 2 (Wu, Chen, Li, ... 2001) Let (λk)1kr be the eigenvalue of a matrix M ∈ Rr×r and (xk)1kr the corresponding right eigenvectors. Denote Mx

  • x1x2 . . . xr
  • and My =
  • y1y2 . . . yr
  • M−H

x

. Then ∂λk ∂M = y∗

k x⊤ k

∀k = 1, . . . , r and ∂ |λk| ∂M = 1 |λk|Re

  • λ∗

k

∂λk ∂M

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  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 22/31

Outline

1

A pole sensitivity stability related measure

2

Macroscopic representation of algorithms through the implicit state-space framework

3

Extension of the pole-sensitivity stability related measure

4

Optimal realization

5

Conclusion and Perspectives

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  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 23/31

Optimal design problem

Denote RH the set of equivalent realizations with H as a transfer function. The Optimal design problem consists in finding a realization Ropt in RH that maximizes µ Ropt = arg max

R∈RH

µ(R) RH is too large, and practically, only realizations with special structure (classical state-space, δ-operator, cascade decomposition, ...) are considered.

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 23/31

Optimal design problem

Denote RH the set of equivalent realizations with H as a transfer function. The Optimal design problem consists in finding a realization Ropt in RH that maximizes µ Ropt = arg max

R∈RH

µ(R) RH is too large, and practically, only realizations with special structure (classical state-space, δ-operator, cascade decomposition, ...) are considered.

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 24/31

Optimal design problem

Some subsets of RH can be defined from an initial realization Z0 and a similarity Z = T1Z0T2 with general case T1 =   U T −1 Ip   , T2 =   V T Im   U,V,T non-singular matrices

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 24/31

Optimal design problem

Some subsets of RH can be defined from an initial realization Z0 and a similarity Z = T1Z0T2 with classical state-space T1 =   Il T −1 Ip   , T2 =   Il T Im   T non-singular matrix

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 24/31

Optimal design problem

Some subsets of RH can be defined from an initial realization Z0 and a similarity Z = T1Z0T2 with δ-operator T1 =   T −1 T −1 Ip   , T2 =   T T Im   T non-singular matrix

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 25/31

Example

The example used here is a single-input single-output fluid power control system (Njabeleke, Whidborne)

Ap = B B @ 9.9988e−1 1.9432e−5 5.9320e−5 −6.2286e−5 −4.9631e−7 2.3577e−2 2.3709e−5 2.3672e−5 −1.5151e−3 2.3709e−2 2.3751e−5 2.3898e−5 1.5908e−3 2.3672e−2 2.3898e−5 2.3667e−5 1 C C A Bp = B B @ 3.0504e−3 −1.2373e−2 −1.2375e−2 −8.8703e−2 1 C C A Cp = B B @ 1 1 C C A

⊤

Z0 = B B B B B @ −3.3071e−1 1 1 1.9869e+0 1 −3.9816e+0 1 3.3255ee+0 −1.6112e−3 −1.5998e−3 −1.5885e−3 −1.5773e−3 −8.0843e−4 1 C C C C C A

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 25/31

Example

The example used here is a single-input single-output fluid power control system (Njabeleke, Whidborne)

Ap = B B @ 9.9988e−1 1.9432e−5 5.9320e−5 −6.2286e−5 −4.9631e−7 2.3577e−2 2.3709e−5 2.3672e−5 −1.5151e−3 2.3709e−2 2.3751e−5 2.3898e−5 1.5908e−3 2.3672e−2 2.3898e−5 2.3667e−5 1 C C A Bp = B B @ 3.0504e−3 −1.2373e−2 −1.2375e−2 −8.8703e−2 1 C C A Cp = B B @ 1 1 C C A

⊤

Z0 = B B B B B @ −3.3071e−1 1 1 1.9869e+0 1 −3.9816e+0 1 3.3255ee+0 −1.6112e−3 −1.5998e−3 −1.5885e−3 −1.5773e−3 −8.0843e−4 1 C C C C C A

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 25/31

Example

The example used here is a single-input single-output fluid power control system (Njabeleke, Whidborne)

Ap = B B @ 9.9988e−1 1.9432e−5 5.9320e−5 −6.2286e−5 −4.9631e−7 2.3577e−2 2.3709e−5 2.3672e−5 −1.5151e−3 2.3709e−2 2.3751e−5 2.3898e−5 1.5908e−3 2.3672e−2 2.3898e−5 2.3667e−5 1 C C A Bp = B B @ 3.0504e−3 −1.2373e−2 −1.2375e−2 −8.8703e−2 1 C C A Cp = B B @ 1 1 C C A

⊤

Z0 = B B B B B @ −3.3071e−1 1 1 1.9869e+0 1 −3.9816e+0 1 3.3255ee+0 −1.6112e−3 −1.5998e−3 −1.5885e−3 −1.5773e−3 −8.0843e−4 1 C C C C C A

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 26/31

Example

For example, it is possible to find the δ-optimal realization Z δ

  • pt = arg max

det(T)=0

µ(Z(T)) with Z(T) =   T −1 T −1 Ip   Z δ   T T Im  

Zδ

  • pt =

B B B B B B B B B B B @ 1 −4.3728e+0 2.7770e+0 1.5953e+1 2.1160e+1 3.5644e−2 1 2.3090e+0 −1.2959e+0 −6.6800e+0 −9.5796e+0 −2.6145e−2 1 6.4736e+0 −4.1528e+0 −2.4059e+1 −3.2103e+1 −1.0745e−2 1 −1.7320e+0 1.0786e+0 6.0998e+0 8.1425e+0 1.8563e−2 −∆ 0 1 0 −∆ 0 1 0 −∆ 0 1 0 −∆ 1 −2.8733e+0 5.6735e−1 −1.3643e+0 2.7498e+0 −8.0843e−4 1 C C C C C C C C C C C A

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 26/31

Example

For example, it is possible to find the δ-optimal realization Z δ

  • pt = arg max

det(T)=0

µ(Z(T)) with Z(T) =   T −1 T −1 Ip   Z δ   T T Im  

Zδ

  • pt =

B B B B B B B B B B B @ 1 −4.3728e+0 2.7770e+0 1.5953e+1 2.1160e+1 3.5644e−2 1 2.3090e+0 −1.2959e+0 −6.6800e+0 −9.5796e+0 −2.6145e−2 1 6.4736e+0 −4.1528e+0 −2.4059e+1 −3.2103e+1 −1.0745e−2 1 −1.7320e+0 1.0786e+0 6.0998e+0 8.1425e+0 1.8563e−2 −∆ 0 1 0 −∆ 0 1 0 −∆ 0 1 0 −∆ 1 −2.8733e+0 5.6735e−1 −1.3643e+0 2.7498e+0 −8.0843e−4 1 C C C C C C C C C C C A

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 27/31

Example : some results

realization µ(R) parameters canonical form q 4.4196e-12 9

  • ptimal q

6.8714e-5 25 canonical form δ 1.1699e-5 9

  • ptimal δ

1.7413e-3 25 cascade 1.0484e-4 18

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  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 27/31

Example : some results

realization µ(R) parameters canonical form q 4.4196e-12 9

  • ptimal q

6.8714e-5 25 canonical form δ 1.1699e-5 9

  • ptimal δ

1.7413e-3 25 cascade 1.0484e-4 18

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 27/31

Example : some results

realization µ(R) parameters canonical form q 4.4196e-12 9

  • ptimal q

6.8714e-5 25 canonical form δ 1.1699e-5 9

  • ptimal δ

1.7413e-3 25 cascade 1.0484e-4 18

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  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 28/31

Outline

1

A pole sensitivity stability related measure

2

Macroscopic representation of algorithms through the implicit state-space framework

3

Extension of the pole-sensitivity stability related measure

4

Optimal realization

5

Conclusion and Perspectives

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  • P. Chevrel,

J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 29/31

Conclusions and Perspectives

Conclusions Implicit State-Space as a Unifying Framework A pole-sensitivity (stability related) measure

  • ptimal design on various forms

Perspectives Other structurations to study (q/δ mixed realizations, ...) Multi-criteria optimization (Roundoff noise gain, stability related measure, nb non-trivial parameters, ...) Toolbox to solve theses problems

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J.P. Clauzel Introduction A pole sensitivity stability related measure Implicit State-Space Framework Pole Sensitivity Measure Optimal realization Conclusion 29/31

Conclusions and Perspectives

Conclusions Implicit State-Space as a Unifying Framework A pole-sensitivity (stability related) measure

  • ptimal design on various forms

Perspectives Other structurations to study (q/δ mixed realizations, ...) Multi-criteria optimization (Roundoff noise gain, stability related measure, nb non-trivial parameters, ...) Toolbox to solve theses problems

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J.P. Clauzel Appendix

Acknowledgement Bibliography

30/31

Acknowledgement

The authors wish to thank PSA Peugeot Citro¨ en for their interest and financial support and James Whidborne for its numerical example.

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J.P. Clauzel Appendix

Acknowledgement Bibliography

31/31

Bibliography

  • M. Gevers and G. Li.

Parametrizations in Control, Estimation and Filtering Probems. Springer-Verlag, 1993.

  • R. Istepanian and J.F. Whidborne, editors.

Digital Controller implementation and fragility. Springer, 2001.

  • T. Hilaire, P. Chevrel, and Y. Trinquet

Implicit state-space representation : a unifying framework for FWL implementation of LTI systems IFAC05 Wolrd Congress, July 2005.