The Duhem model and hysteresis:
a symbiotic relationship?
Fayçal Ikhouane
Universitat Politècnica de Catalunya
Department of Mathematics
[email protected]
June 16, 2016
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
A brief history of the Duhem model
“These curves exhibit, in a striking manner, a persistence of previous state,
such as might be caused by molecular friction. The curves for the back
and forth twists are irreversible, and include a wide area between them.
The change of polarization lags behind the change of torsion. To this
action . . . the author now gives the name Hysterēsis (. . . to be behind)”
J. A. Ewing. On the production of transient electric currents in iron and
steel conductors by twisting them when magnetised or by magnetising
them when twisted. Proceedings of the Royal Society of London, volume
33, pages 21–23, 1881.
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
2 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
A brief history of the Duhem model
P. Duhem. Sur les déformations
permanentes et l’hystérésis.
Premier Mémoire, tome LIV,
Mémoires couronnés et Mémoires des
savants étrangers, l’Académie royale
des sciences, des lettres et des
beaux–arts de Belgique, 1896.
Pierre-Maurice-Marie Duhem
9 June 1861 – 14 September 1916
S. L. Jaki. Uneasy genius: The Life and
Work of Pierre Duhem. Martinus Nijhoff
Publishers, The Netherland, 1987.
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
3 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
A brief history of the Duhem model
u = input
x = state or output
The Duhem model
(
f1 x(t), u(t) u̇(t)
ẋ(t) =
f2 x(t), u(t) u̇(t)
for u̇(t) ≤ 0,
for u̇(t) ≥ 0,
x(0) = x0 .
x = H(u, x0 )
Rate independent: H u ◦ φ, x0 = H u, x0 ◦ φ
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
4 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
What is hysteresis?
Visintin: “Definition. Hysteresis = Rate Independent Memory Effect.”
However, “this definition excludes any viscous-type memory” because it
leads to rate-dependent effects that increase with velocity. A definition
based on rate independence assumes that “the presence of hysteresis loops
is not . . . an essential feature of hysteresis.”
A. Visintin. Differential Models of Hysteresis. Springer–Verlag, Berlin, Heidelberg, 1994.
(Pages 13-14)
Oh and Bernstein: Hysteresis is a “nontrivial quasi-dc input-output closed
curve”
J. Oh and D. S. Bernstein. Semilinear Duhem model for rate-independent and
rate-dependent hysteresis. IEEE Transactions on Automatic Control, volume 50, pages
631–645, 2005.
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
5 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
What is hysteresis?
?
hysteresis = memory effect
Linear system:
ẋ = Ax +
R tBu and y = Cx
y (t) = C exp(tA)x0 + 0 exp (t − τ )A Bu(τ )dτ
The linear system does have memory. However, “hysteresis is a genuinely
nonlinear phenomenon”
M. Brokate and J. Sprekels. Hysteresis and Phase Transitions, Springer–Verlag, New
York, USA, 1996. (Page vii)
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
6 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
What is hysteresis?
A. Visintin. Differential Models of Hysteresis. Springer–Verlag, Berlin, Heidelberg, 1994.
I. Mayergoyz. Mathematical Models of Hysteresis, Elsevier Series in Electromagnetism,
New-York, 2003.
J. Oh and D. S. Bernstein. Semilinear Duhem model for rate-independent and
rate-dependent hysteresis. IEEE Transactions on Automatic Control, volume 50, pages
631–645, 2005.
K. A. Morris. What is Hysteresis? Applied Mechanics Reviews, volume 64, Article
Number: 050801, 14 pages, 2011.
F. Ikhouane. Characterization of hysteresis processes. Mathematics of Control, Signals,
and Systems, volume 25, pages 294–310, 2013.
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
7 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Table of contents
1
Summary of main results
M. A. Krasnosel’skiı̌ and A. V. Pokrovskiı̌
J. Oh and D. S. Bernstein
F. Ikhouane
M. F. M. Naser and F. Ikhouane
B. Jayawardhana, R. Ouyang, and V. Andrieu
A. Visintin
2
Case study, the semilinear Duhem model
Definition and global existence of solutions
Consistency
Hysteresis loop
Dissipativity
Numerical simulations
3
4
5
Relationships between concepts
Open problems
Minor loops
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
8 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
The generalized Duhem model
ẋ(t) = f x(t), u(t) g u̇(t) , for almost all t ∈ R+ ,
(1)
x(0) = x0 ,
(2)
y (t) = h x(t), u(t) , ∀t ∈ R+ .
(3)
u ∈ W 1,∞ (R+ , R)
0
f : Rn × R → Rn×n continuous
0
g : R → Rn continuous and g (0) = 0
h : Rn × R → R continuous
Assumption: ∀(u, x0 ) ∈ W 1,∞ (R+ , R) × Rn there exists a unique solution
Hs (u, x0 ) = x ∈ W 1,∞ (R+ , Rn ) that satisfies (1)–(2).
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
9 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Set Λ of inputs
u : R+ → [umin , umax ] such that
(i) u is T –periodic and continuous,
(ii) u is C 1 on ]0, α1 [ and ]α1 , T [, and ku̇k < ∞,
(iii) u strictly increasing on ]0, α1 [, strictly decreasing on ]α1 , T [,
(iv) u(0) = u(T ) = umin and u(α1 ) = umax .
1
Input u(t)
0.8
0.2
0
0
F. Ikhouane (UPC)
1
2
Time t
4
5
The Duhem model and hysteresis
6
MURPHYS-HSFS
10 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Existence of periodic solutions
Assumption: ∀u ∈ Λ there exists a unique x0,u ∈ Rn such that Hs (u, x0,u )
is T –periodic.
sγ (t) = t/γ.
Cu,γ =
u ◦ sγ (t), Ho (u ◦ sγ , x0,u◦sγ ) (t) , t ∈ [0, T γ] .
ẋ = g1 u̇ A1 x + B1 u + E1
+ g2 u̇ A2 x + B2 u + E2
0.7
Output
0.6
x(0) = x0
y = Cx + Du
0.2
0.1
0
0
0.2
0.8
Input
F. Ikhouane (UPC)
1
Semilinear Duhem model
The Duhem model and hysteresis
MURPHYS-HSFS
11 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
A definition of hysteresis
(
dk (S1 , S2 ) = max
sup
η1 ∈S1
)
inf |η1 − η2 | , sup
inf |η1 − η2 |
.
η2 ∈S2
η2 ∈S2
η1 ∈S1
Definition: Ho is a hysteresis if ∀(u, x0 ) ∈ Λ × Rn
(i) ∃ a closed curve Cu ⊂ R2 such that limγ→∞ d2 (Cu , Cu,γ ) = 0.
(ii) ∃a, b1 , b2 ∈ R with b1 6= b2 such that (a, b1 ) ∈ Cu and (a, b2 ) ∈ Cu .
0.7
0.6
Output
0.5
0.2
γ=1
γ=1
γ = 10
γ = 100
ξ ou
1
0.1
0
0
ξ2 o u
0.2
0.8
1
Input
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
12 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Normalized input
u ∈ W 1,∞ (R+ , Rp ) non constant
Rt
ρu (t) = 0 |u̇(τ )| dτ
Iu = Range(ρu )
1,∞ (I , Rp ) such that ψ ◦ ρ = u.
Lemma: ∃!ψu ∈ Wn
u
u
u
o
kψ̇u kIu = 1 and µ % ∈ Iu | ψ̇u (%) is not defined or |ψ̇u (%)| =
6 1
= 0.
Construction of ψu : % ∈ Iu ⇒ ∃t% ∈ R+ such that ρu (t% ) = %
u(t% ) independent of the particular choice of t%
ψu (%) = u(t% )
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
13 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Normalized input
2
1
1
u(t)
ψ u ()
2
−1
−2
0
0
−1
5
t
15
20
−2
0
10
20
30
sγ (t) = t/γ
Property: ∀γ ∈ ]0, ∞[, Iu◦sγ = Iu and ψu◦sγ = ψu .
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
14 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Class of operators
H : W 1,∞ (R+ , Rp ) × Ξ → L∞ (R+ , Rm ) causal
∀t ∈ [0, τ ], u1 (t) = u2 (t) ⇒ ∀t ∈ [0, τ ], [H (u1 , x0 )] (t) = [H (u2 , x0 )] (t)
Assumption CICO: u constant on [θ, ∞[⇒ H (u, x0 ) constant on [θ, ∞[
All rate-independent systems satisfy the assumption: Preisach, Prandtl,
backlash, Duhem model and its special cases: Dahl model, LuGre model,
Bouc-Wen model, etc.
Some rate-dependent models satisfy: the generalized Duhem model.
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
15 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Normalized output
Lemma: ∃!ϕu ∈ L∞ (Iu , Rm ) such that ϕu ◦ ρu = H (u, x0 )
kϕu kIu ≤ kH (u, x0 )k
If H (u, x0 ) is continuous on R+ , then ϕu is continuous on Iu and
kϕu kIu = kH (u, x0 )k
Construction of ϕu : Let % ∈ Iu , then ∃t% ∈ R+ such that ρu (t% ) = %.
[H (u, x0 )] (t% ) independent of t% ,
ϕu (%) = [H (u, x0 )] (t% )
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
16 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Consistency
u ◦ sγ (t), [H (u ◦ sγ , x0 )] (t) , t ∈ R+
= ψu (%), ϕu◦sγ (%) , % ∈ Iu
Su,γ =
Definition: H is consistent with respect to (u, x0 ) if ∃ϕ?u ∈ L∞ (Iu , Rm )
such that limγ→∞ kϕu◦sγ − ϕ?u kIu = 0
Consistency ⇒ limγ→∞ dp+m S̄u,γ , S̄u? = 0
Su? = ψu (%), ϕ?u (%) , % ∈ Iu
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
17 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Consistency
H : W 1,∞ (R+ , Rp ) × Ξ → L∞ (R+ , Rm ) ∩ C 0 (R+ , Rm ) causal, satisfies
Assumption CICO, is consistent w.r.t all (u, x0 ) ∈ W 1,∞ (R+ , Rp ) × Ξ
Proposition (canonical decomposition): H = H? + H†
H? (u, x0 ) = ϕ?u ◦ ρu satisfies H? (u ◦ sγ , x0 ) = H? (u, x0 ) ◦ sγ
lim H† (u ◦ sγ , x0 ) = 0
γ→∞
H? = rate-independent component of H
H† = nonhysteretic component of H
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
18 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Consistency
1
0.6
0.8
ϕu◦sγ (̺)
Input u(t)
0.5
γ=1
γ = 10
γ = 100
γ=∞
0.2
0.2
0.1
0
0
1
2
Time t
F. Ikhouane (UPC)
4
5
6
0
0
1
2
4
5
6
̺
The Duhem model and hysteresis
MURPHYS-HSFS
19 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Strong consistency
H is consistent w.r.t (u, x0 )
Define ϕ?u,k ∈ L∞ [0, ρu (T )] , Rm by
ϕ?u,k (%) = ϕ?u ρu (T ) k + % , ∀% ∈ [0, ρu (T )]
Definition: H is strongly consistent
with respect to (u, x0 ) if
∃ϕ◦u ∈ L∞ [0, ρu (T )] , Rm such that limk→∞ kϕ?u,k − ϕ◦u k[0,ρu (T )] = 0
hysteresis loop of H w.r.t (u, x0 )
Gu = ψu (%) , ϕ◦u (%) , % ∈ [0, ρu (T )]
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
20 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Strong consistency
0.6
0.6
ϕ⋆u,k(̺)
ϕ◦u (̺)
0.5
k=0
k=1
k=2
k=∞
0.2
0
0
0.5
F. Ikhouane (UPC)
̺
1.5
0.2
0.1
0
0
0.2
2
The Duhem model and hysteresis
0.8
1
ψu (̺)
MURPHYS-HSFS
21 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Non-trivial hysteresis loop
H has a nontrivial hysteresis loop w.r.t (u, x0 ) if (i) and (ii) hold.
(i) H strongly consistent w.r.t (u, x0 )
(ii) µ %1 ∈ Iu | ∃%2 ∈ Iu such that ψu (%1 ) = ψu (%2 ) and ϕ◦u (%1 ) 6=
ϕ◦u (%2 ) 6= 0.
H has a trivial hysteresis loop w.r.t (u, x0 ) if (i) holds and (ii) does not
0
H = H? + H†
lim H† (u ◦ sγ , x0 ) = 0
ϕ◦u (̺)
−0.2
γ→∞
H† =
nonhysteretic component of H
−0.8
−1
0
0.2
0.8
1
ψu (̺)
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
22 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Experimental observations
umax,2
Major loop
umax,1
ϕ◦u (̺7 )
ϕ◦u (̺2 )
umin,2
Minor loop
ϕ◦u (̺6 )
umin,1
0
̺6
̺1
̺2
̺5
̺3
̺7
̺4
ψu (̺2 ) = umin,2
ψu (̺1 ) = umax,1
M. Hamimid, S. M. Mimoune, M. Feliachi,
K. Atallah.
Physica B, 451 (2014), pp. 16–19.
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
23 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Formal definition
H : W 1,∞ (R+ , R) × Ξ → L∞ (R+ , Rm ) ∩ C 0 (R+ , Rm ) causal and satisfies
Assumption CICO.
H consistent w.r.t all (u, x0 ) ∈ W 1,∞ (R+ , R) × Ξ
H strongly consistent w.r.t all periodic inputs u ∈ W 1,∞ (R+ , R) and all
x0 ∈ Ξ.
Assumption: ∀(u, x0 ) ∈ Mumin,1 ,umin,2 ,umax,1 ,umax,2 ,α1 ,α2 ,α3 ,T × Ξ we have
ϕ◦u (%1 ) = ϕ◦u (%5 )
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
24 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Semi-linear Duhem model
1
0.8
0.8
ϕ◦u (̺)
ψu (̺)
0.7
0.5
0.2
0
0
2
4
6
̺
F. Ikhouane (UPC)
8
10
0.4
0
0.2
0.8
1
ψu (̺)
The Duhem model and hysteresis
MURPHYS-HSFS
25 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
Modeling versus Control
Conjecture: the Duhem model does not reproduce the behavior of
experimental minor loops
Control: a precise model is not important as long as the control law is
robust
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
26 / 27
Introduction
Outline
Oh and Bernstein
Ikhouane
Minor loops
Parasitic or symbiotic?
THANK YOU!
F. Ikhouane (UPC)
The Duhem model and hysteresis
MURPHYS-HSFS
27 / 27