Page 1

Sufficient conditions for dissipativity on Duhem hysteresis model

Bayu Jayawardhana, Vincent Andrieu

Abstract—This paper presents sufficient conditions for dis-

sipativity on the Duhem hysteresis model. The result of this

paper describes the dissipativity property of several standard

hysteresis models, including the backlash and Prandtl operator.

It also allows the curve in the hysteresis diagram (the phase plot

between the input and the output) to have negative gradient.

I. INTRODUCTION

Hysteresis is a nonlinear element with memory and is a

common phenomenon in physical systems. Although there

are hysteretic elements that can be explained and understood

based on its underlying physical law (such as, gear train

or switch), many mathematical models of hysteresis are

based on phenomenological modeling. Numerous models

have been proposed to describe hysteresis, see for example

[1], [11], [12], [13]. Based on specific properties inherent

in these models, stability analysis of systems with hysteretic

components has been carried out (e.g., [7], [10]) and con-

troller designs have been proposed for such systems (see, for

example, [2], [4], [5], [6]).

Hysteretic phenomenon in an electrical inductor and in an

engine is known to dissipate energy by heat emission. If the

hysteretic element completes a loop (in the phase plot), then

the dissipated energy is defined by the area enclosed by the

loop [1]. The energy loss can be described by constructing

an ’energy’ function whose rate is less than or equal to the

quantity of the power transferred from an energy source [1],

[3]. The constructions are not unique (c.f., the hysteresis

potential function in [1] and the storage function in [3]). In

the systems theory literature, the ’energy’ function is called

storage function [15], [16].

The existence of the storage function for a hysteretic

component can be useful in the stability analysis of systems

which contains such element. In Gorbet et al [3], a storage

function is constructed for Preisach operator with non-

negative weighting function, and is employed to show the

stability of systems that use a hysteretic actuator. For relay

and backlash operator, the corresponding storage function

has been proposed in Brokate and Sprekels [1].

In nonlinear systems theory, dissipative nonlinear systems

are characterized by the existence of a storage function [16].

More precisely, nonlinear systems defined by

˙ x = f(x,u),y = h(x),x(0) = x0∈ Rn

B. Jayawardhana is with Dept. Discrete Technology and Production

Automation, University of Groningen, Groningen 9747AG, The Netherlands

e-mail: bayujw@ieee.org

V. Andrieu is with LAAS-CNRS, Universit´ e de Toulouse, France

and with LAGEP, CNRS, CPE, Universit´ e Lyon 1, France. e-mail: vin-

cent.andrieu@gmail.com

with a locally Lipschitz f : Rn× Rm→ Rnand h : Rn→

Rp, is called dissipative with supply rate (y,u) ?→ s(y,u), if

there exists a continuously differentiable function H : Rn→

R+such that

of dissipativity is interesting since the storage function pro-

vides an appropriate Lyapunov function candidate for the

stability analysis of nonlinear systems. Moreover it is an

efficient tool in nonlinear control design [8], [14], [15].

In this article, we present sufficient conditions for dis-

sipativity on the Duhem hysteresis operator Φ : u ?→

∂H(x)

∂xf(x,u) ≤ s(y,u). This characterization

Φ(u),C1(R+) → C1(R+) with supply rate ?

precise definition of the Duhem hysteresis operator is given

in Section II). In particular, we show the existence of a

storage function (Φ(u),u) ?→ H(Φ(u),u) such that

˙

????

Φ(u),u? (the

dH((Φ(u))(t),u(t))

dt

≤ ?(

˙

????

Φ(u))(t),u(t)?.

(1)

The motivation of this dissipativity property stems from the

physical law governing an electrical inductor. The magnetic

flux φ and the electric current I in an inductor can be related

by an operator Φ, i.e., φ = Φ(I) (for instance, with a linear

inductor model, Φ(I) = LI where L is the inductance).

Basic electrical law yields that˙φ = V where V is the

voltage across the inductor. Moreover, the electrical power

(defined by ?V (t),I(t)?) transferred to the inductor is equal

˙

????

element and there is energy loss due to hysteresis, the power

being stored in the inductor has to be less than or equal to

the amount of power being transferred into the inductor. In

this case, (1) holds with u = I.

to ?(

Φ(I))(t),I(t)?. Since inductor is a passive electrical

II. DUHEM HYSTERESIS OPERATOR

We denote by C1(R+) the space of continuously dif-

ferentiable functions f : R+ → R. The Duhem operator

Φ : C1(R+) → C1(R+),u ?→ Φ(u) =: y is described by

[11], [13], [17]

˙ y(t) = f1(y(t),u(t))˙ u+(t) + f2(y(t),u(t))˙ u−(t),

y(0) = y0,

(2)

where ˙ u+(t) := max{0, ˙ u(t)}, ˙ u−(t) := min{0, ˙ u(t)}. The

functions f1 and f2 are defined appropriately according to

the hysteresis curve obtained from experimental data.

Oh and Bernstein have shown that the Duhem model

described by (2) is rate-independent ([13, Proposition 3.1]).

This characterizes the fact that for every function ρ : R+→

R+continuous, non-decreasingand such that limt→∞ρ(t) =

hal-00554874, version 1 - 11 Jan 2011

Author manuscript, published in "48th IEEE Conference on Decision and Control, 2009, China (2009)"

Page 2

∞, the Duhem operator Φ satisfies

(Φ(u ◦ ρ))(t) = (Φ(u))(ρ(t)),

∀u ∈ C1(R+), ∀t ∈ R+.

An operator Ψ : C(R+) → C(R+) is said to be causal

if, for all τ ≥ 0 and all v1,v2∈ C(R+), v1= v2on [0,τ]

implies that Ψ(v1) = Ψ(v2) on [0,τ]. With this definition,

the Duhem model is causal if the solutions of ODE in (2)

are unique for every u ∈ C1(R+) and for every initial

conditions. This is guaranteed, for example, if f1 and f2

are locally Lipschitz functions.

Following [10], the operator Φ : C(R+) → C(R+) is

said to be a hysteresis operator if Φ is causal and rate

independent. The Duhem operator Φ : C1(R+) → C1(R+)

is called Duhem hysteresis operator if (2) has unique solution

for every u ∈ C1(R+) and for every initial conditions

y0,u0∈ R2.

In the following subsections, we describe several standard

hysteresis operators that can be described by (2).

A. Backlash operator

The backlash (or play) operator is widely used in me-

chanical models, for example, gear trains or of hydraulic

servovalves. The mathematical analysis of backlash operator

can be found in [1], [9], [12].

In order to relate the model used in these articles with the

Duhem model (2), we describe the Backlash operator used

in [1], [9], [12]. For all h ∈ R+and all ξ ∈ R, we introduce

a backlash operator Bh,ξ defined on the space Cpm(R+) of

piecewise monotone functions, by defining, for every u ∈

Cpm(R+),

(Bh,ξ(u))(0) := bh(u(0),ξ)

(Bh,ξ(u))(t) := bh(u(t),(Bh,ξ(u))(ti))

t ∈ (ti−1,ti], i ∈ N

(3)

where 0 = t0 < t1 < t2 < ... is a partition of R+, such

that u is monotone on each of the intervals [ti−1,ti], i ∈ N

and where for each h ∈ R+, the function bh: R2→ R is

defined by

bh(v,w) := max{v − h,min{v + h,w}}.

Here ξ plays the role of an “initial state”. It is well known,

see, for example, [1, page 42], that the operator Bh,ξ :

Cpm(R+) → C(R+) can be extended uniquely to an operator

Bh,ξ: C(R+) → C(R+). The action of a backlash operator

is illustrated in Figure 1.

The backlash operator Bh,ξ: C1(R+) → C1(R+) can be

defined by the Duhem hysteresis operator (2) with

?

?

and with y0= max{u(0) − h,min{u(0) + h,ξ}}.

The Duhem model of backlash operator can also be easily

extended to a generalized backlash operator. For instance,

the generalized backlash operator Bµ1,µ2,h,ξ : C1(R+) →

f1(a,b) =

1

0

if a = b − h

elsewhere,

(4)

f2(a,b) =

1

0

if a = b + h

elsewhere,

(5)

−5

0

0

10

10

t

Bh,ξ(u)

u

−55

−4

0

0

10

10

Bh,ξ(u)

u

Fig. 1.Backlash operator Bh,ξ(h = 2, ξ = 1)

C1(R+) with µ1> µ2≥ 0, h > 0 and ξ ∈ R can be defined

by (2) where

?

?

and y0is defined properly inside the hysteresis domain, i.e.

µ1(u0− h) ≤ y0≤ µ1(u0+ h).

f1(a,b) =

µ1

µ2

if a = µ1(b − h)

elsewhere

(6)

f2(a,b) =

µ1

µ2

if a = µ1(b + h)

elsewhere

(7)

B. Prandtl operator

The Prandtl operator represents a more general type of

hysteresis which, for certain input functions, exhibits nested

loops in the corresponding input-output characteristics. Let

ζ : R+→ R be a compactly supported and globally Lipschitz

function with Lipschitz constant 1. and let µ be a signed

Borel measure on R+such that |µ|(K) < ∞ for all compact

sets K ⊂ R+, where |µ| denotes the total variation of µ. The

Prandtl operator can be defined by [1]

(Pζ(u))(t) =

?∞

0

(Bh,ζ(h)(u))(t)µ(dh),

∀u ∈ C(R+), ∀t ∈ R+.

(8)

In this case, the Duhem model of backlash operator can be

used in (8).

hal-00554874, version 1 - 11 Jan 2011

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To provide a concrete example of a Prandtl operator, we

consider the Prandtl operator (8) with ζ = 0 and defined by

(P0(u))(t) =

?∞

?l

0

(Bh,0(u))(t)χ[0,l]dh

=

0

(Bh,0(u))(t)dh,

∀u ∈ C(R+), ∀t ∈ R+,

(9)

where l > 0 is a positive constant and χ[0,l]is the indicator

function of the interval [0,l]. This operator exhibits nested

loops as depicted in Figure 2.

10

−20

0

0

40

t

P0(u)

u

−510

−20

0

40

P0(u)

u

Fig. 2.Behaviour of the hysteresis operator P0 with l = 5.

III. MAIN RESULT

In the following sections, we consider the Duhem hys-

teresis operator Φ defined in (2) with f1,f2: R2→ [0,α],

α > 0.

We will describe the hysteresis curve that originates from a

point in the hysteresis diagram. For every pair (y0,u0) ∈ R2

in the hysteresis diagram, let wΦ,1(·,y0,u0) : [u0,∞) → R

be the solution of

x(τ) − x(u0) =

?τ

u0

f1(x(σ),σ) dσ,

x(u0) = y0, ∀τ ∈ [u0,∞),

and let wΦ,2(·,y0,u0) : (−∞,u0] → R be the solution of

x(τ) − x(u0) =

?τ

u0

f2(x(σ),σ) dσ,

x(u0) = y0, ∀τ ∈ (−∞,u0].

Using the above definitions, for every pair (y0,u0) ∈ R2

in the hysteresis diagram, the curve wΦ(·,y0,u0) : R →

R is defined by the concatenation of wΦ,2(·,y0,u0) and

wΦ,1(·,y0,u0):

wΦ(τ,y0,u0) =

?

wΦ,2(τ,y0,u0)

wΦ,1(τ,y0,u0)

∀τ ∈ (−∞,u0)

∀τ ∈ [u0,∞)

(10)

The curve wΦ(·,y0,u0) is the (unique) hysteresis curve

where the curve defined in (−∞,u0] is obtained by applying

a monotone decreasing u ∈ C1(R+) to Φ with u(0) =

u0, limt→∞u(t) = −∞, Φ(u)(0) = y0 and, similarly,

the curve defined in [u0,∞) is produced by introducing a

monotone increasing u ∈ C1(R+) to Φ with u(0) = u0,

limt→∞u(t) = ∞ and Φ(u)(0) = y0.

We define the storage function H : R2→ R for the

hysteresis operator Φ by

H(σ,ξ) = σξ −

?ξ

0

wΦ(τ,σ,ξ) dτ.

(11)

Theorem 3.1: Consider the Duhem hysteresis operator Φ

defined in (2) with locally Lipschitz functions f1,f2: R2→

[0,α], α > 0. Suppose that the following condition holds:

f1(a,b) ≥ f2(a,b) for all (a,b) ∈ R × [0,∞) and

f1(a,b) ≤ f2(a,b) for all (a,b) ∈ R × (−∞,0).

Then for every u ∈ C1(R+) and for every y0 ∈ R, the

function t ?→ H?(Φ(u))(t),u(t)?

PROOF.Let u ∈ C1(R+) and y0 ∈ R. First, we would

prove that for all t ∈ R+,˙H?(Φ(u))(t),u(t)?exists. Using

(Φ(u)), we have (if it exists)

(A)

with H as in (11) is

differentiable and satisfies (1).

(11) and with Leibniz derivative rule and denoting y =

dH?y(t),u(t)?

dt

= y(t)˙ u(t) + ˙ y(t)u(t)

−d

dt

?u(t)

0

wΦ(τ,y(t),u(t)) dτ

= ˙ y(t)u(t) −

?u(t)

0

d

dtwΦ(τ,y(t),u(t))dτ,

(12)

where the last equation is due to wΦ(u(t),y(t),u(t)) = y(t).

The first term in the RHS of (12) exist for all t ≥ 0 since

y(t) satisfies (2). Therefore, in order to get (1), it remains to

check whether the last term exists, is finite and satisfies

?u(t)

0

d

dtwΦ(τ,y(t),u(t))dτ ≥ 0.

(13)

Let t ≥ 0. In order to show the existence of the integrand

and to compute (13), it suffices to show that, for every τ ∈ R,

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the following limit exists:

lim

ǫց0+

1

ǫ[wΦ(τ,y(t+ǫ),u(t+ǫ))−wΦ(τ,y(t),u(t))], (14)

and the limit is greater or equal to zero for every τ ∈ R.

For any ǫ ≥ 0, let us introduce the continuous function

wǫ: R → R by

wǫ(τ) = wΦ(τ,y(t + ǫ),u(t + ǫ)) .

More precisely, for every ǫ ≥ 0, wǫis the unique solution of

Note that wǫ is C1on R \ u(t + ǫ). Moreover, we have

w0(τ) = wΦ(τ,y(t),u(t)) for all τ ∈ R and

wǫ(τ) =

y(t + ǫ) +

?τ

u(t+ǫ)

f1(wǫ(s),s)ds

∀τ ≥ u(t + ǫ),

y(t + ǫ) +

?τ

u(t+ǫ)

f2(wǫ(s),s)ds

∀τ ≤ u(t + ǫ),

(15)

wǫ(u(t + ǫ)) = y(t + ǫ)

,

∀ ǫ ∈ R+.

(16)

In order to show the existence of (14) and its limit being

greater or equal to zero, we consider several case. First,

we assume that ˙ u(t) > 0. This implies that there exists a

sufficiently small γ > 0 such that for every ǫ ∈ (0,γ], we

have u(t + ǫ) > u(t) and

w0(u(t + ǫ)) = y(t) +

?u(t+ǫ)

u(t)

f1(w0(s),s)ds.

Moreover, with the change of integration variable s = u(v)

1we obtain

?t+ǫ

for all ǫ ∈ [0,γ].

The functions ǫ ?→ w0(u(t + ǫ)) and ǫ ?→ y(t + ǫ) with

ǫ ∈ (0,γ] are two C1functions which are solutions of the

same locally Lipschitz ODE and with the same initial value.

By uniqueness of solution, we get w0(u(t + ǫ)) = y(t + ǫ).

This fact together with (16) shows that

w0(u(t + ǫ)) = y(t) +

t

f1(w0(u(v)),u(v)) ˙ u(v) dv

wǫ(u(t + ǫ)) = w0(u(t + ǫ)) ∀ǫ ∈ [0,γ].

Since for every ǫ ∈ (0,γ] the two functions wǫ(τ) and

w0(τ) satisfy the same ODE for2τ > u(t + ǫ), we have

wǫ(τ) = w0(τ)

,

∀ τ ≥ u(t + ǫ),

for all ǫ ∈ [0,γ]. This implies that

lim

ǫց0+

1

ǫ[wǫ(τ) − w0(τ)] = 0,

(17)

1This change is allowed since for every ǫ ∈ [0,γ], u is a strictly

increasing function from [t,t + ǫ] toward [u(t),u(t + ǫ)].

2we have for all τ > u(t + ǫ) :

dwǫ(τ)

dτ

= f1(wǫ(τ),τ)

,

dw0(τ)

dτ

= f1(w0(τ),τ)

for all τ > u(t).

It remains to check (14) for τ ≤ u(t). Since ˙ u(t) > 0,

there exists γ > 0 such that we have τ ≤ u(t) < u(s) <

u(t + ǫ) and ˙ u(s) > 0 for all s in (t,t + ǫ), and all ǫ in

(0,γ). It follows from (15) and Assumption (A) that for

every ǫ ∈ (0,γ):

dwǫ(u(s))

ds

=

f2(wǫ(u(s)),u(s)) ˙ u(s) ∀s ∈ (t,t + ǫ),

≤

f1(wǫ(u(s)),u(s)) ˙ u(s) ∀s ∈ (t,t + ǫ),

(18)

and the function y satisfies

dy(s)

ds

Since wǫ(u(t + ǫ)) = y(t + ǫ) and using the comparison

principle (in reverse direction), we get that for every ǫ ∈

[0,γ):

= f1(y(s),u(s)) ˙ u(s)

,

∀s ∈ (t,t + ǫ).

wǫ(u(s)) ≥ y(s)

,

∀ s ∈ [t,t + ǫ].

Since the two functions wǫ(τ) and w0(τ) for τ ≤ u(t) are

two solutions of the same ODE, it follows that3wǫ(τ) ≥

w0(τ) for all τ ≤ u(t) and we get that (if exists) :

1

ǫ[wǫ(τ) − w0(τ)] ≥ 0

lim

ǫց0+

,

∀ τ ≤ u(t) . (19)

In the following, we compute the bound of (19) in order

to show the existence of (19). Note that for every ǫ ∈ [0,γ],

|wǫ(τ) − w0(τ)| ≤ |y(t + ǫ) − y(t)|

?????

+

u(t)

+

?u(t)

?????

u(t+ǫ)

f1(wǫ(s),s)ds

?????

?τ

f1(wǫ(s),s) − f1(w0(s),s)ds

?????

≤ |y(t + ǫ) − y(t)| +

?u(t+ǫ)

u(t)

|f1(wǫ(s),s)|ds

+

?u(t)

τ

|f1(wǫ(s),s) − f1(w0(s),s)|ds,

for all τ ≤ u(t). By the locally Lipschitz property of f1,

by the boundedness of f1and by the boundedness of wǫon

[τ,u(t)] for all ǫ ∈ [0,γ], we obtain

|wǫ(τ) − w0(τ)| ≤ |y(t + ǫ) − y(t)|

?u(t)

where L is the Lipschitz constant of f1on [wmin,wmax] ×

[τ,u(t)] with

+

τ

L|wǫ(s) − w0(s)|ds + α|u(t + ǫ) − u(t)| ,

wmin =min

(c,s)∈[0,γ]×[τ,u(t)]wc(s)

max

(c,s)∈[0,γ]×[τ,u(t)]wc(s) .wmax =

3Otherwise there exist τ1 < τ2 such that wǫ(τ1) = w0(u(τ1)) and

wǫ(τ2) > w0(u(τ2)) which contradict the uniqueness of the solution of

the locally Lipschitz ODE.

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Page 5

With Gronwall’s lemma, this implies that for every ǫ ∈ [0,γ]

|wǫ(τ) − w0(τ)| ≤ exp((u(t) − τ)L)

?

|y(t + ǫ) − y(t)|

+ α|u(t + ǫ) − u(t)|

?

,

for all τ ≤ u(t). Hence

lim

ǫց0+

1

ǫ|wǫ(τ) − w0(τ)|

≤ exp((u(t) − τ)L)

?

|f1(y(t),u(t))| + α

?

˙ u(t),

for all τ ≤ u(t).

We can use similar arguments to prove the case when

˙ u(t) < 0.

Finally, when ˙ u(t) = 0, we simply get

lim

ǫց0+

1

ǫ|wǫ(τ) − w0(τ)| = 0,

by continuity of the above bound.

2

Remark 3.2: The storage function H in the Theorem 3.1

is non-negative and H(y,0) = 0 for all y ∈ R. Indeed,

without loss of generality, let us consider the case when u ≥

0. In this case, it can be checked that for every y ∈ R,

wΦ(τ,y,u) ≤ y for all τ ∈ [0,u]. It follows that

?u

≥ 0,

H(y,u) =

0

y − wΦ(τ,y,u) dτ

for all u ≥ 0 and y ∈ R.

△

Remark 3.3: The non-negativity assumption imposed on

functions f1,f2in Theorem 3.1 can be relaxed into locally

Lipschitz function f1,f2 : R2→ [−α,α]. Using the same

proof of the theorem and using the same storage function

H, we can obtain the same result. However, other conditions

need to be imposed on f1,f2 if we want a lower bounded

H.

△

Remark 3.4: Related to the dissipativity concepts by

Willems [16], the storage function as constructed in the

Theorem 3.1 is equal to the available storage function as

defined in [16]. In order to show this, given (y0,u0) in R2,

let us integrate (12) from 0 to T > 0, along the solution y

of (2) with y(0) = y0and u ∈ C1(R+) with u(0) = u0:

H?y(T),u(T)?− H?y0,u0

−

00

or equivalently,

?=

?T

0

˙ y(τ)u(τ)dτ

?T

?u(σ)

d

dσwΦ(τ,y(σ),u(σ)) dτdσ,

−

?T

0

˙ y(τ)u(τ)dτ = H?y0,u0

−

00

?− H?y(T),u(T)?

?T

?u(σ)

d

dσwΦ(τ,y(σ),u(σ)) dτdσ.

Taking the supremum in both sides of this equation with

arguments T and u ∈ C1(R+) with u(0) = u0, we get

?T

?

sup

T>0

u∈C1(R+),u(0)=u0

−

0

˙ y(τ)u(τ)dτ = H?y0,u0

?

+ sup

T>0

u∈C1(R+),u(0)=u0

− H?y(T),u(T)?

−

?T

0

?u(σ)

0

d

dσwΦ(τ,y(σ),u(σ)) dτdσ

?

.

Note that H is non-negative according to Remark 3.2 and the

integrand on the RHS is also non-negative according to (13).

Therefore, we only need to check whether there exist T > 0

and u ∈ C1(R+) with u(0) = u0 such that the supremum

value in the RHS is equal to zero. Following the proof of

Theorem 3.1, we can choose arbitrary T > 0 and arbitrary

monotone function u ∈ C1(R+) with u(T) = 0 which give

the desired result. In other words,

sup

T>0

u∈C1(R+),u(0)=u0

−

?T

0

˙ y(τ)u(τ)dτ = H?y0,u0

?.

(20)

In [16], the LHS of (20) is called the available storage

function with respect to the supply rate ˙ y(t)u(t).

△

Remark 3.5: As described in Remark 3.4, the storage

function H as in the Theorem 3.1 corresponds to the maxi-

mum available energy that can be extracted from the system.

On the other hand, the supplied energy is given by

?t

0

?

˙

????

Φ(u)(τ),u(τ)

?

dτ.

Thus (12) shows that the rate of the available energy at

time t ≥ 0 is equal to the rate of supplied energy minus

?u(t)

t.

0

d

dtwΦ(τ,y(t),u(t))dτ. The latter component can have

physical interpretation as the rate of dissipated energy at time

△

IV. PASSIVITY FOR THE BACKLASH OPERATOR

The following theorem is used to describe passivity for

the backlash operator as described in Section II-A. Note that

the condition on f1 and f2 which is assumed in Theorem

3.1 (i.e. Assumption (A)) excludes the Duhem model for

backlash operator.

Theorem 4.1: Consider the Duhem hysteresis operator Φ

defined in (2) with f1,f2 as in (4)-(5). Then for every

u ∈ C1(R+) and for every y0 ∈ R, the function t ?→

H?(Φ(u))(t),u(t)?

PROOF.The proof is similar to that of Theorem 3.1. Notice

that H can be given explicitly as follows:

with H as in (11) is differentiable and

satisfies (1).

H(y(t),u(t)) =

?

0

1

2(y(t) − h)2

y(t) ∈ [−h,h]

elsewhere.

hal-00554874, version 1 - 11 Jan 2011