Fractional model parameter estimation in the frequency domain using - - PowerPoint PPT Presentation

Fractional model parameter estimation in the frequency domain using set membership methods

Firas KHEMANE, Rachid MALTI, Xavier MOREAU SWIM June 15th 2011

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Objectives Fractional system modeling using uncertain frequency responses

◮ Extension of fractional derivatives and integrals to interval derivatives and integrals. ◮ Applying set membership approaches to estimate uncertain coefficients and uncertain derivative orders in fractional models ◮ Using tree set membership inclusion functions on frequency domain data based on rectangular, polar, and circular representations of complex intervals ◮ Merging all three solutions to obtain smaller intervals

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Table of contents

1

From fractional derivative to interval derivative

2

Fractional systems

3

Set membership estimation using uncertain frequency response

4

Numerical example

5

Application : Thermal diffusion system

6

Conclusion

From fractional derivative to interval derivative

1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion

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From fractional derivative to interval derivative

Fractional integration

Cauchy formula Inf (t) = Z t Z τ1 . . . Z τn−1 f(τn−2)dτn−2 . . . dτ1 Riemann-Liouville integral Iνf (t)

∆

= 1 Γ (ν) Z t f (τ) (t − τ)1−ν dτ, ∀ν ∈ R∗

+

Laplace transform L (Iνf(t)) = 1 sν L (f(t)) , Frequency characteristics of

1 sν

8 > > < > > : Gain (dB) : 20 log ˛ ˛ ˛

1 (jω)ν

˛ ˛ ˛ = −20ν log w Phase (rad) : arg “

1 (jω)ν

” = −ν π

2 .

10

−2

10

−1

10 10

1

10

2

−20 −10 10 20 Frequency (rad/s) Magnitude (dB) 10

−2

10

−1

10 10

1

10

2

−90 −45 Frequency (rad/s) Phase (deg)

−10dB/dec

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From fractional derivative to interval derivative

Fractional derivative

Gr¨ unwald definition Dνf(t) = lim

h→0

1 hν

∞

X

k=0

(−1)k k ν ! f(t − kh) ! , ν ∈ R+ Laplace transform L (Dνf(t)) = sνL (f(t)). Frequency characteristics of sν

8 < : Gain (dB) : 20 log |(jω)ν| = 20ν log(w), Phase (rad) : arg ((jω)ν) = ν π

2 .

10

−2

10

−1

10 10

1

10

2

−20 −10 10 20 Frequency (rad/s) Magnitude (dB) 10

−2

10

−1

10 10

1

10

2

45 90 Frequency (rad/s) Phase (deg)

10 dB/dec

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From fractional derivative to interval derivative

Interval integration

Definition I[ν]f (t)

∆

= n 1 Γ (ν) Z t f (τ) (t − τ)1−ν dτ, ν ∈ [ν]

• Example

f(t) = ( t2 if t ≥ 0, if t < 0 Integration by real interval [ν] = [0.5, 1.5] I[ν]f (t)

∆

= n 1 Γ (ν) Z t τ 2 (t − τ)1−ν dτ, ν ∈ [0.5, 1.5]

• 1

2 3 4 5 6 7 8 9 10 100 200 300 400 500 t I [ν] f(t) nu=0.5 nu=0.6 nu=0.7 nu=0.8 nu=0.9 nu=1 nu=1.1 nu=1.2 nu=1.3 nu=1.4 nu=1.5

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From fractional derivative to interval derivative

Frequency characteristics

Laplace transform L {I[ν](f(t))} = n 1 sν L {f(t)}, ν ∈ [ν]

• Frequency characteristics of

1 s[ν]

8 < : fG([ν]) = ˛ ˛

1 s[ν]

˛ ˛

dB

= 20 log ˛ ˛ ˛

1 (jω)[ν]

˛ ˛ ˛, fφ([ν]) = arg “

1 s[ν]

” = arg “

1 (jω)[ν]

”

10

−2

10

−1

10 10

1

10

2

−40 −20 20 40 Pulsation (rad/s) Gain (dB) 10

−2

10

−1

10 10

1

10

2

−80 −60 −40 −20 Pulsation (rad/s) Phase (°)

−10dB/dec −14dB/dec

Monotonicity

◮ Gain 8 < : ˛ ˛

1 s[ν]

˛ ˛

dB

= [−20ν log ω, −20ν log ω], ω ∈]0, 1], ˛ ˛

1 s[ν]

˛ ˛

dB

= [−20ν log ω, −20ν log ω], ω ∈ [1, +∞[ ◮ Phase fφ([ν]) = [−ν π 2 , −ν π 2 ]

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From fractional derivative to interval derivative

interval derivative

Gr¨ unwald D[ν]f(t) = n lim

h→0

1 hν

∞

X

k=0

(−1)k k ν ! f(t − kh) ! , ν ∈ [ν]

• .

Example f(t) = ( t2 if t ≥ 0, if t < 0,

1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 t D [ν] f(t) nu=0.5 nu=0.6 nu=0.7 nu=0.8 nu=0.9 nu=1 nu=1.1 nu=1.2 nu=1.3 nu=1.4 nu=1.5

Derivative by real interval D[ν]f(t) = n lim

h→0

1 hν

∞

X

k=0

(−1)k k ν ! (t − kh)2 ! , ν ∈ [0.5, 1.5]

• .

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From fractional derivative to interval derivative

Frequency characteristics

Laplace transform L {D[ν](f(t))} = n sνL {f(t)}, ν ∈ [ν]

• .

Frequency characteristics of s[ν] 8 < : fG([ν]) = ˛ ˛s[ν]˛ ˛

dB

= 20 log ˛ ˛ ˛(jω)[ν]˛ ˛ ˛, fφ([ν]) = arg “ s[ν]” = arg “ (jω)[ν]” .

10

−2

10

−1

10 10

1

10

2

−40 −20 20 40 Pulsation (rad/s) Gain (dB) 10

−2

10

−1

10 10

1

10

2

20 40 60 80 Pulsation (rad/s) Phase (°)

10dB/dec 14dB/dec

Monotonicity

◮ Gain 8 < : ˛ ˛s[ν]˛ ˛

dB

= [20ν log ω, 20ν log ω], when ω ∈]0+, 1−], ˛ ˛s[ν]˛ ˛

dB

= [20ν log ω, 20ν log ω], when ω ∈ [1+, +∞[ ◮ Phase fφ([ν]) = [ν π 2 , ν π 2 ]

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From fractional derivative to interval derivative

Monotonicity of time domain response with respect to derivative order

Fractional differential equation (FDE) Dνy(t) = f(y(t), u(t)) Uncertain fractional derivative equation (FDE) D[ν]y(t) = f(y(t), u(t)) Solution : Set of admissible trajectories Y(t) [y(t)] = [y(t), y(t)] Objective : framing Y(t) by Dνy(t) = f(y(t), u(t)), Dνy(t) = f(y(t), u(t)).

It is a tough problem which cannot be solved in the general case

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From fractional derivative to interval derivative

Monotonicity of time domain response with respect to derivative order

Example

◮ Differential equation Dνy(t) + λy(t) = u(t), λ = 1, with u(t) : Heaviside step function ◮ Transfer function Y (s) U(s) = 1 s (sν + λ) ◮ Inverse Laplace transform of Y (s)

U(s)

fν(t) =

∞

X

k=1

(−1)k+1λk−1 tνk Γ(νk + 1) ◮ Derivative of fν(t) d dν fν(t) =

∞

X

k=1

(−1)k+1 λk−1tνkk Γ(νk + 1) (log(t) − Ψ(νk + 1))

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From fractional derivative to interval derivative

Monotonicity of time domain response with respect to derivative order

Derivtive of fv(t) with ν = 0.9

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 Temps fν(t)’

Step response y(t)

1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Temps y(t) ν=0.2 ν=0.3 ν=0.4 ν=0.5 ν=0.6 ν=0.7 ν=0.8 ν=0.9

Solving uncertain differential equation with uncertain differential

• rders still an open problem.. !

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Fractional systems

1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion

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Fractional systems

Application

Fractional systems are gaining more and more interest in the scientific and industrial communities Area of Control :

◮ CRONE controllers base on fractional calculus ◮ Fractional PID controllers

Mechanics :

◮ Performance of CRONE suspension in vibration isolation

Signal processing :

◮ Performance of fractional operators in modeling fractal noise

Thermal domain :

◮ Performance of fractional operators in modeling flux diffusion

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Fractional systems

Fractional system representation

Differential equation y(t) + a1Dα1y(t) + a2Dα2y(t) + . . . + aNDαN y(t) = b0Dβ0u(t) + b1Dβ1u(t) + b2Dβ2u(t) + . . . + bM DβM u(t), Commensurable order ν : Commensurable order ν is the biggest real number such that all derivative orders

• f the differential equation are its integer multiples

αi ν ∈ N, i = 1, . . . , N and βj ν ∈ N, j = 0, . . . , M Fractional transfer function G(s, θ) =

M

P

j=0

bjsβj 1 +

N

P

i=1

aisαi , θ = [b0 . . . bM, a1 . . . aN , β0 . . . βM, α1 . . . αN] with 2(N + M + 1) parameters Commensurable transfer function G(s, θ) =

M

P

j=0

bjsjν 1 +

N

P

i=1

aisiν , θ = [b0 . . . bM, a1 . . . aN, ν] with (N + M + 2) parameters

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Fractional systems

Fractional system representation

State space representation  x(ν)(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) Stability theorem Assume a fractional and commensurable transfer function and Rν = Qν/Pν its rational form. F is stable iff : 0 < ν < 2, and ∀sk ∈ C, Pν(sk) = 0 such | arg(sk)| > ν π 2 .

Re(sk) I m(sk)

ν π

2

Re(sk) I m(sk)

ν π

2

Re(sk) I m(sk)

ν π

2

◮ a) ν < 1 ◮ b) ν = 1 ◮ c) ν > 1

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Set membership estimation using uncertain frequency response

1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion

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Set membership estimation using uncertain frequency response

Objective

Having a set Gn of N complex frequency responses, find the fractional models G consistent with all these frequency responses Frequency response : G(jω, θ) =

M

P

j=0

bj(jω)jν 1 +

N

P

i=1

ai(jω)jν . Imaginary vs Real part

0.05 0.1 0.15 0.2 0.25 0.3 −0.25 −0.2 −0.15 −0.1 −0.05

Gd

n

Gr

n

Gp

n

Gn

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Set membership estimation using uncertain frequency response

Set membership estimation using rectangular representation

Rectangular inclusion function Gr

n :

Gr

n = [Re(Gr n) Re(Gr n)] + j[I

m(Gr

n) I

m(Gr

n)]

8 > > > > > < > > > > > : Re(Gr

n) =

min

i=1...I (Re(z1), Re(z2), . . . , Re(zI)) , ∀zi ∈ Gn,

Re(Gr

n) =

max

i=1...I (Re(z1), Re(z2), . . . , Re(zI)) , ∀zi ∈ Gn,

I m(Gr

n) =

min

i=1...I (I

m(z1), I m(z2), . . . , I m(zI)) , ∀zi ∈ Gn, I m(Gr

n) =

max

i=1...I (I

m(z1), I m(z2), . . . , I m(zI)) , ∀zi ∈ Gn. CSP : CSP : 8 > < > : Re(Gr

n) ≤ Re(G(jωn, θ)) ≤ Re(Gr n),

I m(Gr

n) ≤ I

m(G(jωn, θ)) ≤ I m(Gr

n),

θ ∈ Θ, Solution set S : S = ˘θ ∈ Θ | f(ωn, θ) ∈ [y(ωn)], n ∈ {1, . . . , N}¯. 8 > > > > > > < > > > > > > : f(ωn, θ) = Re(G(jωn, θ)) I m(G(jωn, θ)) ! [y(ωn)] = [Re(Gr

n), Re(Gr n)]

[I m(Gr

n), I

m(Gr

n)]

!

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Set membership estimation using uncertain frequency response

Set membership estimation using polar representation

Polar inclusion function Gp

n :

Gp

n = ˆGn Gn

˜ exp{ j [ϕn

ϕn] }

8 > > > > > < > > > > > : ϕn = min

i=1...I (arg(z1), arg(z2), . . . , arg(zI)) , ∀zi ∈ Gn,

ϕn = max

i=1...I (arg(z1), arg(z2), . . . , arg(zI)) , ∀zi ∈ Gn,

Gn = min

i=1...I (|z1|, |z2|, . . . , |zI|) , ∀zi ∈ Gn, ∀zi ∈ Gn,

Gn = max

i=1...I (|z1|, |z2|, . . . , |zI|) , ∀zi ∈ Gn.

CSP : CSP : 8 > < > : Gn ≤ |G(jωn, θ)| ≤ Gn, ϕn ≤ ϕ(ωn, θ) ≤ ϕn, θ ∈ Θ, Solution set S : S = ˘θ ∈ Θ | f(ωn, θ) ∈ [y(ωn)], n ∈ {1, . . . , N}¯. 8 > > > > > > < > > > > > > : f(ωn, θ) = |G(jωn, θ)| ϕ(ωn, θ) ! [y(ωn)] = [ Gn Gn] [ϕ, ϕ] !

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Set membership estimation using uncertain frequency response

Set membership estimation using circular representation

Circular inclusion function Gd

n :

Gd

n = {c(Gn), r(Gn)},

CSP : CSP : ( G (jωn, θ) ∈ Gd

n

θ ∈ Θ, Solution set S : S = ˘ θ ∈ Θ | f(ωn, θ) ∈ [y(ωn)], n ∈ {1, . . . , N} ¯ . 8 > < > : f(ωn, θ) = X(c(G(jωn, θ)), r(G(jωn, θ))) [y(ωn)] = Gd

n

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Set membership estimation using uncertain frequency response

Set Inversion Via Interval Analysis (SIVIA) Jaulin and Walter, (1993)

SIVIA allows to obtain an inner S and an outer S enclosures of the solution set S, based on partitioning the parameter set into three regions : feasible, indeterminate and unfeasible : S ⊆ S ⊆ S. Uses an inclusion test [t] which is a function allowing to prove if [θ] is

◮ unfeasible [θ] is ignored, ◮ feasible [θ] is added to the set S, ◮ undetermined [θ] is bisected and tested again, unless its size is less than a precision parameter η tuned by the user and which ensures that the algorithm terminates after a finite number of iterations. ◮ if w[θ] < η [θ] is added to the set S

Algorithm SIVIA (in : [t], [θ], η ; out : S, S )

1 If [t]([θ]) = [0], return ; 2 If [t]([θ]) = [1], then

S := S ∪ [θ]; S := S ∪ [θ], return ;

3 If w([θ]) ≤ η, S := S ∪ [θ] ;

Else bisect [θ] into [θ1] and [θ2] ;

4 SIVIA (in : [t], [θ1], η ; out : S, S) ; 5 SIVIA (in : [t], [θ2], η ; out : S, S).

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Numerical example

1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion

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Numerical example

Fractional transfer function of the first kind

Transfer function G1(s, θ) = k sν + b. Uncertain linear model G1(s, θi) = ki sνi + bi , i = 1, . . . I, Uncertainties 8 > < > : ki = 3 + ρ(i)

k ,

bi = 2 + ρ(i)

b ,

νi = 0.5 + ρ(i)

ν

8 > < > : ρ(i)

k

= [−0.2, 0.2], ρ(i)

b

= [−0.2, 0.2], ρ(i)

ν

= [−0.05, 0.05] Vector of nominal parameters G1(jωn, θi) = ki (jωn)νi + bi , ( i = 1, . . . 20, n = 1, . . . 36.

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Numerical example

Uncertain frequency response

Bode

10

−2

10

−1

10 10

1

10

2

10

3

−30 −20 −10 10 Pulsation (rad/sec) Gain (dB) 10

−2

10

−1

10 10

1

10

2

10

3

−60 −40 −20 Pulsation (rad/sec) Phase (°)

Nyquist-Circular form

−0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2

Real part Imaginary part

Nyquist-rectangular form

−0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05

Real part Imaginary part

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Numerical example

Set membership estimation using rectangular representation

Complex frequency response G1(jωn, θ) = k ` b + cos(ν π

2 )ωn

´ν + j ` sin(ν π

2 )ων n

´, Real and imaginary part G1(jωn, θ) = k ` b + ων

n cos

` ν π

2

´´ b2 + 2bων

n cos

` ν π

2

´ + ω2ν

n

+ j −kων

n sin

` ν π

2

´ b2 + 2bων

n cos

` ν π

2

´ + ω2ν

n

. Estimation using real and complex CSP

◮ Outer approximation S Parameters [k] [b] [ν] Real CSP [2.58, 3.52] [1.68, 2.40] [0.46, 0.56] Complex CSP [2.56, 3.59] [1.66, 2.43] [0.45, 0.56] Final rectangular inclusion [2.58, 3.52] [1.68, 2.40] [0.46, 0.56]

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Numerical example

Results

(a) Projection on (b, k) plan (b) Projection on (b, ν) plan

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Numerical example

Set membership estimation using polar representation

Complex frequency response G1(jω, θ) = |G1(jω, θ)| expjϕ(ω,θ) Gain and phase |G1(jω, θ)| = |k| p (b + cos(ν π

2 )ων)2 + (sin(ν π 2 )ων)2

ϕ(ω, θ) = 8 > > > < > > > : − arctan “ sin(ν π

2 )ων

Den(ω)

” , if Den(ω) > 0, −π − arctan „

sin(ν π

2 )ων

Den(ω)

« , if Den(ωn) < 0 with Den(ω) = b + cos ` ν π

2

´ ων.

29 / 43

Numerical example

Set membership estimation using polar representation

Estimation using real and complex CSP

◮ Outer approximation S

Parameters [k] [b] [ν] Real CSP [2.56, 3.50] [1.69, 2.26] [0.45, 0.56] Complex CSP [2.40, 3.74] [1.55, 2.49] [0.44, 0.57] Final polar inclusion [2.56, 3.50] [1.69, 2.26] [0.45, 0.56]

Fig.: Projection on (b, k) plan Fig.: Projection on (b, ν) plan

10

−2

10

−1

10 10

1

10

2

10

3

−30 −20 −10 10 Pulsation (rad/sec) Gain (dB) 10

−2

10

−1

10 10

1

10

2

10

3

−60 −40 −20 Pulsation (rad/sec) Phase (°)

Fig.: Bode diagram

30 / 43

Numerical example

Set membership estimation using circular representation

Complex frequency response G1(jωn, θ) = k ` b + cos(ν π

2 )ωn

´ν + j ` sin(ν π

2 )ων n

´, Estimation using complex CSP

◮ Outer approximation S `[k], [b], [ν]´ = `[2.29, 3.94], [1.32, 2.88], [0.43, 0.57]´, Fig.: Projection on (b, k) plan Fig.: Projection on (b, ν) plan Fig.: Nyquist diagram

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Numerical example

Merging all solution sets

(a) Projection on (b, k) plan : rectan- gular (red line), polar (blue line) and circular (green line) (b) Projection on (b, ν) plan : rectan- gular (red line), polar (blue line) and circular (green line)

Parameters [k] [b] [ν] Rectangular inclusion [2.58, 3.52] [1.68, 2.39] [0.46, 0.56] Polar inclusion [2.56, 3.50] [1.69, 2.25] [0.45, 0.56] Circular inclusion [2.29, 3.94] [1.32, 2.88] [0.43, 0.56] Final inclusion [2.58, 3.50] [1.69, 2.25] [0.46, 0.56]

32 / 43

Application : Thermal diffusion system

1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion

33 / 43

Application : Thermal diffusion system

Description

5 10 20

l(mm) Isolation thermique

Characteristics

◮ Aluminium rod l = 40cm, r = 1cm et S = 3.14cm2 ◮ Heater 4.8Ω, λ = 237W m−1K−1 et α = 9975 × 10−8m2s−1 ◮ Input : Heat flux φ

34 / 43

Application : Thermal diffusion system

Input-Output signals

Flux step of 5kW m−2, l = 5mm

2000 4000 6000 8000 10000 25 30 35 40

Temps(s) Temp´ erature(◦)

PRBS of ±5kW m−2

200 400 600 800 1000 1200 −5000 5000 200 400 600 800 1000 1200 −2 −1 1 2

Temps(s) Temps(s) Temp´ erature(◦) Flux(kW.m−2) 35 / 43

Application : Thermal diffusion system

Non parametric identification of the frequency response

Frequency response b G(jω) = b Gyu(jω) b Guu(jω) .

10

−2

10

−1

10 −120 −100 −80 −60

Gain (dB) Pulsation (rad/s)

10

−2

10

−1

10 −140 −120 −100 −80 −60 −40

Phase (°) Pulsation (rad/s)

36 / 43

Application : Thermal diffusion system

Parametric estimation

Criterion J%(θ) = αJG + (1 − α)Jφ, with JG% = PN

n=1

„ | b G(jωn)|dB − |b b G(jωn)|dB «2 PN

n=1

“ | b G(jωn)|dB ”2 × 100, and Jφ% = PN

n=1

„ arg “ b G(jωn) ” − arg(b b G(jωn)) «2 PN

n=1

“ arg( b G(jωn)) ”2 × 100,

37 / 43

Application : Thermal diffusion system

Parametric estimation

Model obtained G(s) = k

1 p1 × s2ν + 1 p2 sν + 1 exp−2s

with 8 > > > < > > > : k = 0.8 × 10−3, p1 = 1 × 10−3, p2 = 7.7 × 10−3, ν = 0.73 J = 0.03%

10

−3

10

−2

10

−1

−120 −100 −80 −60 −40

Gain (dB) f (Hz)

10

−3

10

−2

10

−1

−140 −120 −100 −80 −60 −40

Phase (°) f (Hz)

38 / 43

Application : Thermal diffusion system

Uncertainties

Stochastic errors : Under assumption of normal distrubtion, errors due to time domain-frequency domain conversion are used to determine boundes

• n each frequency response.

Standard deviation on gain : σ[| b G|] | b G| ≈ ` 1 − γ2

uy

´1/2 |γuy|√2nd , Standard deviation on phase : σ[b ϕ] ≈ ` 1 − γ2

uy

´1/2 |γuy|√2nd , Coherence function : γ2

uy(ω) =

| b Guy(jω)|2 b Guu(jω) b Gyy(jω) , 0 ≤ γ2

uy(ω) ≤ 1

∀ω.

39 / 43

Application : Thermal diffusion system

Uncertain frequency response

Uncertain frequency response : Lower and upper boundes are obtained using confidance intervals with 3 standard deviation | b G| − 3σ[| b G|] ≤ |G| ≤ | b G| + 3σ[| b G|] b ϕ − 3σ[b ϕ] ≤ |ϕ| ≤ b ϕ + 3σ[b ϕ].

10

−2

10

−1

10 −120 −100 −80 −60

Gain (dB) Pulsation (rad/s)

10

−2

10

−1

10 −140 −120 −100 −80 −60 −40

Phase (°) Pulsation (rad/s)

Set membership approach is a bouded error approach 8 > > > < > > > : Gn : Lower bound on gain, Gn : Upper bound on gain, ϕn : Lower bound on phase, ϕn : Upper bound on phase

40 / 43

Application : Thermal diffusion system

Set membership estimation

Initial searching space ([k], [p1], [p2], [ν]) = ` [0.00, 5.00], [0.00, 5.00], [0.00, 5.00], [0.00, 2.00] ´ . Outer approximation ([k], [p1], [p2], [ν]) = ` [4.00, 12]×10−4, [4.5, 15]×10−4, [0.3, 2]×10−2, [0.62, 0.97] ´ ,

(c) Projection on (k, ν) plan (d) Projection on (p1, p2) plan

10

−2

10

−1

10 −120 −100 −80 −60 Pulsation(rad/sec) Gain (dB) 10

−1

10 −150 −100 −50 Pulsation(rad/sec) Phase (°)

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Conclusion

1 From fractional derivative to interval derivative 2 Fractional systems 3 Set membership estimation using uncertain frequency response 4 Numerical example 5 Application : Thermal diffusion system 6 Conclusion

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Conclusion

Conclusion

Definitions of fractional integrals and derivatives are extended to interval integrals and derivatives. Set membership parameter estimation is applied to estimate all feasible parameters and differentiation orders of fractional models Three different inclusion functions of the complex frequency response are used in the set membership estimation and the final results are merged to

• btain a smaller solution set

Application to a thermal diffusion system and a set of feasible parameters is obtained

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