Mechanical Systems and Signal Processing 75 (2016) 607–617
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Mechanical Systems and Signal Processing
journal homepage: www.elsevier.com/locate/ymssp
Sufficient conditions for rate-independent hysteresis
in autoregressive identified models
Samir Angelo Milani Martins a,b,1, Luis Antonio Aguirre b
a
Department of Electrical Engineering, Federal University of São João del-Rei, Praça Frei Orlando 170 – Centro, 36307-352 São João
del-Rei, Minas Gerais, Brazil
b
Programa de Pós-Graduação em Engenharia Elétrica, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, 31270 901,
Belo Horizonte, MG, Brazil
a r t i c l e i n f o
abstract
Article history:
Received 31 December 2014
Received in revised form
28 September 2015
Accepted 29 December 2015
Available online 14 January 2016
This paper shows how hysteresis can be described using polynomial models and what are
the sufficient conditions to be met by the model in order to have hysteresis. Such conditions are related to the model equilibria, to the forcing function and to certain term
clusters in the polynomial models. The main results of the paper are used in the identification and analysis of nonlinear models estimated from data produced by a magnetorheological damper (MRD) model with Bouc–Wen rate-independent hysteresis. A striking
feature of the identified model is its simplicity and this could turn out to be a key factor in
controller design.
& 2016 Elsevier Ltd. All rights reserved.
Keywords:
Rate-independent hysteresis modelling
Bouc–Wen model
Magneto-rheological damper modelling
Nonlinear system identification
1. Introduction
Hysteresis is a quasi-static nonlinear characteristic found in several mechanical and magnetic systems as in magnetorheological dampers (MRD), electro-mechanical actuators and sensors, involving memory effects between input and output
[13,5]. In mechanical systems, hysteresis is directly related to the inelastic behaviour of joints and materials. Considering
magneto-rheological dampers, the hysteresis is related to the magnetic characteristics of the magneto-rheological fluid and
also to its mechanical structure.
Due to its strong nonlinearity, hysteresis is not easy to model. Traditionally, phenomenological models have been proposed for modelling systems with hysteresis [13]. One of them is the Bouc–Wen model which was initially proposed by Bouc
[7] and extended by Wen [27]. The Bouc–Wen model structure is composed by a first-order nonlinear differential equation,
built from physical laws [12] and is able to reproduce several hysteresis loops [10,24].
One of the advantages of phenomenological models is that they are able to incorporate non-local memory effects by
including in their structure information about the reversion points [25], as the Preisach model. In systems with non-local
memory, the next state on the hysteresis loop depends not only on the current state, but also on the past (non-local) path of
the loop.
An important application of models for systems with hysteresis is in control system design. The central idea is to design a
controller that implements a model for the system which is able to compensate for the main features of the specific hysteresis loop. In this type of application, it is known that the use of phenomenological models is quite awkward [21] and this
E-mail addresses: [email protected] (S.A.M. Martins), [email protected] (L.A. Aguirre).
Tel.: þ55 31 9 9206 5239, þ55 32 3379 2394.
1
http://dx.doi.org/10.1016/j.ymssp.2015.12.031
0888-3270/& 2016 Elsevier Ltd. All rights reserved.
608
S.A.M. Martins, L.A. Aguirre / Mechanical Systems and Signal Processing 75 (2016) 607–617
has served as a strong motivation for developing techniques for identification and control of system with hysteresis
[22,5,26,17,14].
In this respect, the use of data-driven models is particularly appealing because the model parameters can be estimated to
obtain a model that closely fits the main aspects of a particular hysteresis loop. However, when compared to their phenomenological counterparts, black-box models are generally unable to reproduce certain features as, for instance, the nonlocal memory effect. Nonetheless to estimate, possibly online and adaptively, models for hysteretic systems to represent the
main features of the loop is still a noble goal to pursue. Hence NARX polynomial and neural network models have been
proposed for hysteretic systems in [16] and [10], respectively.
The identification of black-box models for systems with hysteresis is not without its own challenges. Probably the
greatest one is to determine a convenient model structure in order to stand a chance to represent the main aspects of the
hysteresis loop. General algorithms for model structure selection, as the ERR (Error Reduction Ratio), have been developed
in the late eighties [15,6]). Unfortunately such algorithms typically do not work well on their own for models of systems
with hysteresis.
The main aims of this paper are to discuss the existence of hysteresis in autoregressive models, and to provide sufficient
conditions for hysteresis in such models. In order to address these issues, in this paper we introduce the concept of
bounding structure of equilibria. The paper also proposes a technique for structure selection for hysteresis models based on
term clustering as an effective aid to other regressor selection criteria. Our results show that a very simple autoregressive
structure (obtained by the proposed approach) is able to reproduce the main features of an MRD hysteresis loop. Such
models can be used, in the future, in model-based control of systems with hysteresis.
Section 2 presents some background material. Sufficient conditions for autoregressive models to reproduce rateindependent hysteresis are given in Section 3. In Section 4 a structure selection procedure is put forwards. Section 4.3
presents the numerical results for the Bouc–Wen model of a MRD. Concluding remarks and perspectives of future research
are given in Section 5.
2. Background
This section presents some background material which has been organized as a set of definitions.
Definition 2.1 (Rate-independent (RIH) hysteresis). Let x ¼ A sin ðωtÞ be the input of a system with hysteresis represented by
a closed curve Ht ðωÞ parametrized by the time in the input–output plane. H ¼ limω-0 Ht ðωÞ is here called the bounding
structure. If H delimits the hysteresis loop Ht ðωÞ, the system is said to display rate-independent hysteresis (RIH) (Fig. 1).
Remark 2.1. As it will be seen, in the case of RIH, H is formed by more than one set of equilibria that delimit a region in the
input–output plane in which the hysteresis loop Ht ðωÞ is observed. There are systems for which H will tend to a single set of
equilibria as ω-0. Such systems are said to have rate-dependent hysteresis (RDH).
One way of modelling RIH is using the Bouc–Wen model. Preisach, Prandtl–Ishlinskii, Duhem and other models can also
be used for modelling RIH, as presented in [11]. In this paper we will deal with rate-independent hysteresis (RIH) and use a
Bouc–Wen model as a bench test.
Fig. 1. Example of hysteresis curve, given a loading–unloading input signal.
S.A.M. Martins, L.A. Aguirre / Mechanical Systems and Signal Processing 75 (2016) 607–617
609
Definition 2.2 (Wen [27], Bouc–Wen model for RIH). The Bouc–Wen model is a phenomenological hysteresis model. A
general form of this model was initially proposed in [7]:
dz
dx dx
¼ g x; z; sign
;
ð1Þ
dt
dt dt
where z is the hysteretic variable, x the input and g½ a nonlinear function of x, z and signðdx=dtÞ.
An Euler discretization of Eq. (1) yields
zk zk 1 þ hg d ½xk 1 ; zk 1 ; signðxk xk 1 Þvk 1 ;
ð2Þ
where h is the integration step, xk is the displacement at k, vk is the velocity and g d ½ is a nonlinear function. The model
described in (2) indicates that a NARX polynomial model for a Bouc–Wen hysteresis (RIH) should include unit-delayed
regressors of xk 1 , signðxk xk 1 Þ and zk 1 . In the context of MRD modelling, in order to include damping effects, the model
should have regressors of E, the input voltage that determines the damping intensity.
The Bouc–Wen model was used by Spencer Jr. et al. [23] to model the hysteresis of an MRD (variable z of Eq. (3)):
f ¼ c1 ρ_ þ k1 ðx x0 Þ;
ρ_ ¼
1 αz þc0 x_ þ k0 ðx ρÞ ;
c0 þ c1
z_ ¼ γ ∣x_ ρ_ ∣z∣zjn 1 β ðx_ ρ_ Þ∣zjn þ Aðx_ ρ_ Þ;
α ¼ αa þ αb uMRD ;
c1 ¼ c1a þ c1b uMRD ;
c0 ¼ c0a þ c0b uMRD ;
u_ MRD ¼ ηðuMRD EÞ:
ð3Þ
The parameters of Eq. (3) can be obtained directly from [23,16] or [10] and will not be reproduced here.
Definition 2.3 (Chen and Billings [9], NARX models). A SISO NARMAX (Nonlinear AutoRegressive Moving Average model
with eXogenous inputs) model can be written as
yk ¼ F ℓ ½yk 1 ; …; yk ny ; xk d …xk nx ; ek 1 ; …; ek ne þ ek ;
ð4Þ
where xk and yk are respectively the input and output signals and ek accounts for uncertainties and possible noise. In this
work F ℓ ½ is assumed to be a polynomial with nonlinearity degree ℓ. If ℓ ¼ 1, the model is called ARMAX (Linear AutoRegressive Moving Average model with eXogenous inputs).
The moving average part of the model is required to avoid bias during parameter estimation due to the known fact that
the classical least squares estimator cannot handle coloured noise nor output noise. In practice, residual terms are used as
estimates for the noise, hence ξk ek and the extended least squares algorithm is used to estimate model parameters. Once
the parameters have been estimated, only the deterministic part of the model will be used in the simulations, it is in this
respect that the acronym NARX will be used.
The NARX part of model (4) can be expanded as the summation of terms with degrees of nonlinearity in the range ½1ℓ.
Each ðp þ mÞth-order term can contain a pth-order factor in y and an mth-order factor in x and is multiplied by a constant
parameter cp;m ðτ1 ; …; τm Þ as follows:
yk ¼
y ;nx
ℓ ℓX
m nX
X
m ¼ 0 p ¼ 0 τ 1 ;τ m
p
m
i¼1
i¼1
cp;m ðτ1 ; …; τp þ m Þ ∏ yk τi ∏ xk τi þ ek ;
ð5Þ
where the upper limit is ny if the summation refers to factors in y or nx for factors in x. The model structure can be chosen
using orthogonal techniques [9,18].
Steady-state analysis is accomplished by taking y ¼ yk τ ; 8 τ ¼ 0; …; ny , x ¼ xk τ ; 8 τ ¼ d; …; nx and in that case Eq. (5) can
be rewritten as
!
y ;nx
ℓ ℓX
m nX
X
y¼
cp;m ðτ1 ; …; τp þ m Þ y p x m :
ð6Þ
m¼0p¼0
τ 1 ;τ m
Definition 2.4 (Aguirre and Billings [2], Term cluster and cluster coefficients). The constant within the large parenthesis in
Eq. (6), denoted Σ yp xm , is the cluster coefficient of a set of terms, called term cluster, indicated by Ωyp xm . Terms of the form
p m
ypk i xm
k j A Ωy x for m þ p rℓ, where i and j are any time lags. In words, a term cluster is a set of terms of the same type and
the respective cluster coefficient is the summation of the coefficients of all the terms of the corresponding cluster.
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Definition 2.5 (Stability of equilibria). The solution of Eq. (6) will yield the equilibria, for a given input x [4]. The stability of
these equilibria depends on the eigenvalues of the Jacobian matrix D, that, in turn, depend on coefficients cp;m :
2
3
0
1
0
⋯
0
6 0
0
1
⋯
0 7
6
7
6
7
6
⋮
⋮
⋯
⋮ 7
D¼6 ⋮
;
ð7Þ
7
6
7
0
0
⋯
1 7
6 0
4 ∂F ℓ
5
ℓ
∂F ℓ
∂F ℓ
⋯ ∂y∂F
∂yk p
∂yk p þ 1
∂yk p þ 2
k1
yk ¼ y; xk ¼ x
ℓ
with F ½ the right side of Eq. (4), for a given constant input x.
Definition 2.6 (Loading-unloading quasi-static signal). A periodic signal xk with period N ¼ ðkf 2 ki1 þ 1Þ and frequency
ω ¼ 2π =N is called a loading–unloading signal if xk increases monotonically from xmin to xmax , for ki1 r k r kf 1 and decreases
monotonically from xmax to xmin , for ki2 r k rkf 2 (Fig. 1(b)). If the loading–unloading signal changes with ω-ϵ, 0 o ϵ⪡f s =2,
where f s is the sampling frequency, the signal is also called a quasi-static signal.
Definition 2.7 (Multi-valued functions). Let ϕðΔxk Þ: R-R, where Δxk ¼ xk xk 1 . ϕðΔxk Þ is a multi-valued function if:
8
>
< ϕ1 ; if Δxk 4 ϵ;
ð8Þ
ϕðΔxk Þ ¼ ϕ2 ; if Δxk o ϵ;
>
: ϕ ; if Δx ¼ 0;
k
3
where ϵ A R. For some inputs Δxk a 0 always, and the last value in (8) is not used.
Definition 2.8 (Sign function). A commonly used multi-valued function (Definition 2.7) is the signðxÞ: R-R function:
8
if x 40;
>
< 1;
signðxÞ ¼ 1; if x o0;
ð9Þ
>
: 0;
if x ¼ 0:
3. Representation of RIH with polynomial models
This section describes how polynomial models can be used to represent systems with rate-independent hysteresis (RIH).
Property 3.1. Consider a model excited by a loading–unloading quasi-static input. If such a model has at least one real and stable
equilibrium point for input values corresponding to the loading regime, and likewise for the unloading regime (the equilibria
depend on the history of the input), the model will display an RIH hysteresis loop Ht ðωÞ (Fig. 1(a)).
In what follows sufficient conditions for linear and nonlinear polynomial models to reproduce RIH will be presented and
illustrated using numerical simulations.
3.1. Hysteresis in autoregressive deterministic models
Simple autoregressive deterministic models with a specific input will display an RIH cycle as stated in the following
proposition. Here, by deterministic it is meant that, in the case of models that have been estimated with a moving average
part, such a part is not used nor is the white noise term considered.
Proposition 3.1. Given a SISO autoregressive asymptotically stable model:
yk ¼
p
X
i¼1
ciy yk i þ
q
X
j¼1
cjϕ ϕðΔxk j Þ;
ð10Þ
i
where cy and cjϕ are constant parameters. If ϕðΔxk Þ ¼ signðxk xk 1 Þ where xk is a loading–unloading quasi-static input signal of
frequency ω (Definition 2.6), then there will be a rate-independent hysteresis loop Ht ðωÞ in the x y plane confined by the
bounding structure H defined by the model equilibria.
Proof of Proposition 3.1. According to Property 3.1, a model reproduces RIH if at least one equilibrium point depends on
the history of the input.
S.A.M. Martins, L.A. Aguirre / Mechanical Systems and Signal Processing 75 (2016) 607–617
611
Because ϕðΔxk Þ ¼ signðxk xk 1 Þ, during loading (unloading) ϕðΔxk Þ will be a constant ϕ that depends on the signal xk.
The equilibria of Eq. (10) are
8
Σϕ
>
>
>
; for ϕ ¼ 1ðloadingÞ;
>
< 1 Σy
ð11Þ
y ϕ ¼
Σϕ
>
>
>
;
for
ϕ
¼
1ðunloadingÞ;
>
: 1 Σy
where the situation xk ¼ xk 1 has been ruled out for the loading–unloading input. From Eq. (11) it can be seen that because
signðÞ is a multi-valued function, model (10) has a single equilibrium point that depends on the history of the input. Because
the input xk is a loading–unloading signal, then the first difference will satisfy (see Fig. 1)
(
40 if ki1 r k rkf ;
ð12Þ
Δxk ¼ o0 if k r k rk 1 ;
i2
f2
hence, because the model is assumed stable, the output will converge to Σ ϕ =ð1 Σ y Þ when xk is loading and to
Σ ϕ =ð1 Σ y Þ when xk is unloading, thus forming the Ht ðωÞ in the x y plane. Further, the equilibria given by Eq. (11) are the
bounding structure H. The model output does not have to reach the equilibria in order to form Ht ðωÞ.□
Remark 3.1. If ϕðΔxk Þ is taken to be the input, say x~ k , (10) becomes an ARX model. The hysteresis loop Ht ðωÞ is formed in
the x y plane, not in the x~ y plane, as expected.
Remark 3.2. Equation (10) can be interpreted as Hammerstein deterministic model (see Fig. 2).
Remark 3.3. The bounding structure H confines the hysteresis loop Ht ðωÞ. As the frequency of the loading–unloading input
signal becomes smaller (ω-ϵ; 0 o ϵ⪡f s =2), the hysteresis loop Ht ðωÞ converges to the bounding structure H, as illustrated in
the following example.
Example 3.1. Consider the model yk ¼ 0:9yk 1 þ 0:5ϕk 1 for which ϕk ¼ signðΔxk Þ and xk ¼ sin ðωkÞ. The equilibria are
y ¼ 5, for ϕk ¼ 1 (loading) and y ¼ 5, for ϕk ¼ 1 (unloading) and are the bounding structure H that confines the hysteresis loop. Notice that how the input frequency influences Ht ðωÞ but not the bounding structure (Fig. 3).
3.2. Hysteresis in NARX deterministic models
According to Property 3.1, in order for a model to reproduce RIH, only one equilibrium state is required during the
loading regime of the input and another one is required for the unloading regime. In view of this, terms from the cluster
Ωyp ; p 4 1 will be automatically excluded from the following analysis. Also, higher powers of ϕk ¼ signðxk xk 1 Þ need not be
considered.
Fig. 2. Example of a Hammerstein model.
Fig. 3. The black dots are on Ht ðωÞ (see Remark 2.1) for model yk ¼ 0:9yk 1 þ 0:5ϕk 1 , with ϕk ¼ signðΔxk Þ and xk ¼ sin ðωkÞ. The bounding structure H in
red will always confine Ht ðωÞ. The top limit corresponds to a loading situation, whereas the bottom limit, to unloading. (a) ω ¼ 1. (b) ω ¼ 0:1. (For
interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
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S.A.M. Martins, L.A. Aguirre / Mechanical Systems and Signal Processing 75 (2016) 607–617
Proposition 3.2. Given a particular SISO nonlinear deterministic autoregressive stable model:
yk ¼ Σ 0 þ
p
X
i¼1
ciy yk i þ
q
X
j¼1
cjϕ ϕðΔxk j Þ þ
p X
q
X
i¼1j¼1
cijyϕ yk i ϕðΔxk j Þ:
ð13Þ
where Σ0, cy, cjϕ and cijyϕ are constant parameters. If ϕðΔxk Þ ¼ signðxk xk 1 Þ, where xk is a loading–unloading quasi-static input
signal of frequency ω (Definition 2.6), then there will be a rate-independent hysteresis loop Ht ðωÞ in the x y plane confined by
the bounding structure H defined by the model equilibria, which are assumed to be asymptotically stable.
i
Proof of Proposition 3.2. The proof follows the same line as the one for Proposition 3.1. The equilibria of Eq. (13) are
8
Σϕ þ Σ0
>
>
>
; for ϕ ¼ 1ðloadingÞ;
>
< 1 Σ y Σ yϕ
ð14Þ
y ϕ ¼
Σϕ þ Σ0
>
>
>
;
for
ϕ
¼
1ðunloadingÞ;
>
: 1 Σ y þ Σ yϕ
and because the equilibria are asymptotically stable, the model output will converge to these values when xk is loading or
unloading, respectively. □
Because hysteresis is a quasi-static behaviour, this phenomenon is frequently modelled using unit-delayed regressors
[16,10]. In this case, the Jacobian becomes D ¼ ∂f =∂yðk 1Þ ¼ Σ y þ Σ yϕ ϕ, and the stability is guaranteed if
1 o Σ y þ Σ yϕ ϕ o 1. Considering loading–unloading, the domain of validity for the hysteresis loop is
(
1 o Σ y þ Σ yϕ o 1;
ð15Þ
1 o Σ y Σ yϕ o 1:
Remark 3.4. In Propositions 3.1 and 3.2, the term cluster Ωϕ is directly responsible for Ht ðωÞ. Without this term cluster, H
would be a single curve regardless of the input and would not delimit a non-zero area region in the x y plane necessary
for RIH.
Remark 3.5. Besides Ωϕ , the term clusters indicated in (14) will change H and thus the shape of the hysteresis, but will not
interfere on its occurrence as long as the conditions established in Proposition 3.2 are met.
Remark 3.6. According to Eq. (17) (Proposition 3.3), using more inputs increases the model degrees of freedom and thus,
the dimension of the bounding structure. If three or more inputs are used, the model will be even more flexible, in such a
way that a dense set of equilibria may exist for a single value of x.
Remark 3.7. Polynomial autoregressive model structure typically do not carry information about reversal points and
therefore are models with local memory.
Proposition 3.3. Consider the nonlinear deterministic autoregressive model shown in Fig. 4 and represented as
yk ¼ Σ 0 þ
p
X
i¼1
þ
q
r
X
X
j¼1m¼1
þ
r
X
m1 ¼ 1
ciy yk i þ
q
X
j¼1
cjϕ ϕðΔxk j Þ þ
cjm
ϕx ϕðΔxk j Þxk m þ
r
X
⋯
mℓ ¼ mℓ 1
r
X
cm
x xk m þ
m¼1
p X
q
r
X
X
i¼1j¼1m¼1
p X
q
X
i¼1j¼1
cijyϕ yk i ϕðΔxk j Þ þ
cijm
y ϕðΔxk j Þxk m þ
yϕx k i
r
X
p X
r
X
i¼1m¼1
r
X
m1 ¼ 1 m2 ¼ m1
cim
yx yk i xk m
1 m2
cm
xk m1 xk m2 þ ⋯
x2
1 ;…;mℓ
cm
xk m1 ⋯xk ml
xℓ
ð16Þ
ijm
m1 m2
1 ;…;mℓ
where Σ0, cy, cjϕ , cx , cijyϕ , cyx , cjm
⋯cm
are constant parameters. If ϕðΔxk Þ ¼ signðxk xk 1 Þ, where xk is a
xℓ
ϕx , cyϕx , cx2
loading–unloading quasi-static input signal of frequency ω (Definition 2.6), then there will be a rate-independent hysteresis loop
Ht ðωÞ in the x y plane confined by the bounding structure H defined by the model equilibria, which are assumed to be
asymptotically stable.
i
m
im
Fig. 4. A SISO Hammerstein model that uses a MISO deterministic NARX block.
S.A.M. Martins, L.A. Aguirre / Mechanical Systems and Signal Processing 75 (2016) 607–617
613
Fig. 5. The black dots are on Ht ðωÞ (see Remark 2.1) for model yk ¼ 0:9yk 1 þ 0:5ϕk 1 þ 0:25x3k 1 þ 0:75 with ϕk ¼ signðΔxk Þ and xk ¼ sin ðωkÞ. The bounding
structure H in red confines Ht ðωÞ. The top (loading) and bottom (unloading) boundaries are sets that are dense in terms of x. (a) ω ¼ 1. (b) ω ¼ 0:1. (For
interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
Proof of Proposition 3.3. The proof follows the same line as the one for Propositions 3.1 and 3.2. The equilibria of Eq. (16)
are
8
>
Σ ϕ þ Σ 0 þ Σx x þ Σϕx x þ Σx2 x 2 þ ⋯ þ Σℓx x ℓ
>
>
>
>
;
>
>
>
1 Σ y Σ yϕ Σyx x Σyϕx x
<
y ϕ; x ¼
2
ℓ
>
>
> Σ ϕ þ Σ 0 þ Σx x Σϕx x þ Σx2 x þ ⋯ þ Σxℓ x
>
>
;
>
>
>
:
1 Σ y þ Σ Σyx x þ Σ x
yϕ
for
ϕ ¼ 1ðloadingÞ;
ð17Þ
for
ϕ ¼ 1ðunloadingÞ:
y ϕx
The bounding structure in this case depends not only on ϕ but also on the static value of the input x. Hence, the bounding
structure is formed by two sets of values (dense in terms of x) that depend on the input. Because the equilibria are
asymptotically stable, the output will converge to the Ht ðωÞ loop in the x y plane. The lower the frequency of the input, the
closer Ht ðωÞ will be to the bounding structure H.
The dependence of the bounding structure on the input is a consequence of the terms in bold face in Eq. (17). This shows
how the number of inputs and the type of regressors in the model influence the bounding structure and therefore the
hysteresis loop, as illustrated in the following example.□
Example 3.2. Consider the model yk ¼ 0:9yk 1 þ 0:5ϕk 1 þ 0:25x3k 1 þ 0:75 for which ϕk ¼ signðΔxk Þ and xk ¼ sin ðωkÞ. For
this model, Σ 0 ¼ 0:75, Σ y ¼ 0:9, Σ ϕ ¼ 0:5 and Σ x3 ¼ 0:25. The equilibria of this model are given by
8
>
0:5 þ 0:75 þ 0:25x 3
>
>
¼ 12:5 þ2:5x 3 ;
<
1 0:9
y ϕ; x ¼
>
0:5 þ 0:75 þ 0:25x 3
>
>
¼ 2:5 þ 2:5x 3 ;
:
1 0:9
for
ϕ ¼ 1ðloadingÞ;
for
ϕ ¼ 1ðunloadingÞ:
ð18Þ
Because the model is stable, the output will converge to the equilibria y. For a constant input at x ¼ 1 ¼ x, the equilibria is
y ¼ 15 for loading and y ¼ 5 for unloading. Likewise, for x ¼ 1 the equilibria are y ¼ 10 for loading and y ¼ 0 for unloading,
as seen in Eq. (18) and shown in Fig. 5.
The selection of the correct model structure is always a challenge in nonlinear system identification. In the case of
systems with hysteresis the challenge is even greater [28]. In what follows the results developed in this section will be used
to put forward a procedure for the identification of a system with hysteresis.
4. Numerical approach for structure selection
In this section, a numerical approach for structure selection of systems with RIH is proposed. Then, results are obtained
for the Bouc–Wen model described in Eq. (3). The signals were normalized as in [16,10] and a Gaussian random noise with
zero mean and standard deviation of 5% was added to the data.
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S.A.M. Martins, L.A. Aguirre / Mechanical Systems and Signal Processing 75 (2016) 607–617
4.1. “Classical” structure selection
The phenomenological model for a magneto-rheological damper, containing a Bouc-Wen hysteresis (Eq. (3)) was used as
the data generating system, with integration step equal to h¼ 0.002 s. The inputs (displacement, x, and the voltage input, E)
were generated by low-pass filtering two independent realizations of white Gaussian sequences. A specific data set was
generated for identification purposes.
1500
In our first approach, the data yk ; x1;k ; x2;k ; x3;k k ¼ 1 were used, where the output y is the hysteretic force (fk), and the
inputs are x1;k the voltage (Ek), x2;k the velocity (vk) and x3;k represents the sign of the velocity signðvk Þ ¼ signðxk xk 1 Þ,
where xk is the displacement. Choosing the meta parameters ny ¼ nx1;k ¼ nx2;k ¼ nx3;k ¼ ℓ ¼ 3, a pool of 455 candidate terms
was generated. That set contains a constant and all linear, quadratic and cubic combinations of the signals up to and
including lag 3.
Model structure selection was performed using the ERR criterion together with the AIC (Akaike Information Criterion)
[15,6]. After parameter estimation, all resulting models were unstable, which is often a consequence of an inadequate model
structure [1].
Although the ERR criterion has been used with great success in a number of cases [3,8], some of its shortcomings have
been well reported [2,20,18]. Alternatives are available such as the ad hoc procedure suggested in [16] for a system with
hysteresis or the more general approach based on the SRR (simulation reduction ratio) [20]. In the next subsection a simple
structure selection procedure which is far less computer intensive than the SRR and that proved helpful in the present study
will be described.
4.2. Cluster-based structure selection
In this procedure [2], the regressors are inserted sequentially according to the order determined by the ERR criterion. The
corresponding cluster coefficients are computed for the resulting family of nested model structures, with increasing number
of terms. If a cluster coefficient tends to zero or changes its algebraic sign as the number of model terms increases, this
suggests that the regressors of such a cluster are unnecessary for representing the static behaviour [19].
Although hysteresis is a quasi-static behaviour, in this paper we have shown that (1) the bounding structure confines the
hysteresis loop, and (2) the bounding structure is defined by the model equilibria under loading or unloading conditions.
Therefore, procedures for structure selection based on steady-state concepts can be applied to the present problem. Hence
term clusters with coefficients equal to zero or changing the algebraic sign were eliminated from the candidate set. Further,
discrete models of hysteresis usually have only unit-delayed regressors [16,10]. Hence, only unit-delayed regressors of the
genuine clusters are chosen to compose the final model. Afterwards, based on the AIC, the model size was chosen. The final
model structure is at the same time general and simple, as it will be illustrated shortly.
4.3. NARX identification of a Bouc–Wen model
Thirty five candidate clusters were generated. Using the approach described in Section 4.2, six of them were classified as
genuine: Ω0, Ωy, Ωx1 , Ωx3 , Ωx2 x1 , Ωx3 x2 y . This approach considerably reduces the amount of candidate regressors and
therefore decreases the computational cost. The AIC criterion was used to determine the model size, which turned out to be
dim ðθÞ ¼ 4, hence
yk ¼ 0:8536yk 1 þ0:0388x3;k 1 þ0:6143x2;k 1 x1;k 1 0:4407x3;k 1 x2;k 1 yk 1 þ Ψ yxξ θ^ yxξ þ ξk ;
ð19Þ
where Ψ yxξ θ^ yxξ þ ξk indicates the moving average part of the model which consisted of six nonlinear terms involving
inputs, output and residual regressors. These regressors were automatically chosen by the ERR criterion. Such terms were
used during parameter estimation only, and were not employed during simulations. It should be highlighted that, according
to Eq. (2), the obtained model should have a unit-delayed output regressor yk 1 followed by a nonlinear function
g^ d ½Ek 1 ; yk 1 ; signðxk xk 1 Þ. The model obtained by the proposed approach (Eq. (19)) has this structure, but much simpler
when compared with the Bouc–Wen model for the MRD (Eq. (3)).
Since the term x3;k 1 is a multi-valued function (Definition 2.7) of Δxk , the model will produce hysteresis as long as the
equilibria are stable (Proposition 3.2). Also, model (19) uses more than one input and therefore the RIH formed, limited by
the bounding structure, will be geometrically more complicated. This will be illustrated in the sequel.
In order to validate the model over a wider range of operating points, a second validation data set was generated with
sinusoidal signals of displacement xt ¼ sin ð2π 3tÞ cm and voltage Et ¼ 0:5 sin ð2π 0:5tÞ þ1:6 V (Fig. 7). Finally, a third validation test was performed, to check how the model reproduces the main hysteresis loop (Fig. 9(a)).
The stability of model (19) can be established by simulation and mathematically (Section 4.3.1). Since it is stable and it
uses a multi-valued function (Definition 2.7) of Δxk , the necessary conditions to have an RIH in the x y plane, as established
in Proposition 3.2, are met. Hysteresis is confirmed by the simulation results shown in Figs. 6, 7 and 9. Fig. 8 presents the
error plots corresponding to Figs. 6 and 7. The error was computed as the difference between the data and the free-run
simulation of the identified model. The local memory effect (see Remark 3.7) is confirmed by the higher values of error at
the points where the output reverses its direction.
S.A.M. Martins, L.A. Aguirre / Mechanical Systems and Signal Processing 75 (2016) 607–617
615
Fig. 6. Model validation – first data set. The inputs were generated by low-pass filtering a white Gaussian sequences.
Fig. 7. Model validation – second data set. Inputs were generated with sinusoidal signals of displacement xt ¼ sin ð2π3tÞ cm and voltage
Et ¼ 0:5 sin ð2π0:5tÞ þ 1:6 V.
Fig. 8. Error plots of Figs. 6 and 7.
The model with four unit-delayed regressors is able to describe the system hysteresis in the operating regions analyzed,
having as a great advantage its very low complexity, when compared with the model in [16,10] and the Bouc–Wen model of
the magneto-rheological damper (Eq. (3)). Also, the obtained model is able to represent several hysteresis curves by simply
616
S.A.M. Martins, L.A. Aguirre / Mechanical Systems and Signal Processing 75 (2016) 607–617
Fig. 9. Model validation – main loop and example of hysteresis. xt ¼ sin ð2πtÞ cm and voltage Et ¼ sin ð2πtÞ þ 1:5 V.
changing parameter values. Fig. 9(b) shows some examples of hysteresis curves that can be obtained by varying the model
parameters.
An important practical situation is when the hysteretic system is of order higher than one. In this case one would expect
to need regressors with lags larger than one. It should be noted that Proposition 3.3 is not limited to first-order dynamics.
The results in this paper restrict the term clusters that can be present in the model but do not limit the maximum lags used.
4.3.1. Bounding structure and operating range for the hysteresis loop
Performing a steady-state analysis of Eq. (19) yields y ¼ Σ y y þ Σ ϕ x 3 þ Σ x1 x2 x 2 x 1 þ Σ ϕx2 y x 3 x 2 y or
y ðx 1 ; x 2 ; x 3 Þ ¼
Σ ϕ x 3 þ Σ x1 x2 x 2 x 1
;
1 Σ y Σ ϕx2 y x 3 x 2
ð20Þ
where Σ y ¼ 0:8536, Σ ϕ ¼ 0:0388, Σ x1 x2 ¼ 0:6143, Σ ϕx2 y ¼ 0:4407. Eq. (20) presents the equilibria that defines the
bounding structure H, whose location depends on the history of one of the inputs ðx3 ¼ ϕðΔxk Þ ¼ signðΔxk ÞÞ, described by
8
Σ ϕ þ Σ x1 x2 x 2 x 1
>
>
>
; for ϕ ¼ 1ðloadingÞ;
>
< 1 Σ y Σ ϕx2 y x 2
ð21Þ
y ðx 1 ; x 2 ; x 3 Þ ¼
Σ ϕ þ Σ x1 x2 x 2 x 1
>
>
>
;
for
ϕ
¼
1ðunloadingÞ:
>
: 1 Σ y þ Σ ϕx y x 2
2
Addition of x2, x3 changes the equilibria y and therefore the bounding structure which depends on x1 (a multi-valued
function of Δx) and also on x2, x3. Therefore the bounding structure is of greater complexity in this case, as can be confirmed
from Figs. 7(b) and 9(b).
The local stability condition for these equilibria is 1 o Σ y þ Σ ϕx2 y x2 x3 o1, or rearranging and using Σ y ¼ 0:8536 and
Σ ϕx2 y ¼ 0:4407 yields
4:2060 o x 2 x 3 o 0:3322:
ð22Þ
There are two situations to be analyzed: for loading, ðx3 ¼ 1Þ, the stability condition becomes x2 o 4:2060, meaning that
this equilibrium point is locally stable for velocity less than 4.2060 times the nominal velocity, since the data were normalized. For unloading ðx3 ¼ 1Þ, a similar analysis can be done, yielding the same local stability condition. Then, in the
region analyzed, the model (19) is stable, as required by Property 3.1.
S.A.M. Martins, L.A. Aguirre / Mechanical Systems and Signal Processing 75 (2016) 607–617
617
5. Conclusions
This paper presented sufficient conditions for polynomial models to be able to reproduce rate-independent hysteresis.
The main results are given in Propositions 3.1–3.3. A key concept in the discussion is that of a bounding structure H which is
defined by the model equilibria. It has been shown that the hysteresis loop converges to H as the frequency of the loading–
unloading input decreases. Hence, for a model to display a hysteresis loop, H should be formed by at least two equilibria or
sets of equilibria. The use of a specific class of inputs and of certain term clusters in the models guarantee hysteresis.
The identification of a Bouc–Wen model of a magneto-rheological damper was used as a bench test. Commonly used
structure selection techniques fail in producing stable models and it was argued that term-cluster structure selection is one
way of overcoming this difficulty. In particular, the results given in Propositions 3.1–3.3 indicate which term clusters if
present in the model will enable the model to reproduce the hysteresis loop. The identified model is rather general and
much simpler, when compared with the Bouc–Wen model. Hence the estimated polynomial model could be used in a feedforward inverse control of systems with hysteresis and model-based control.
Acknowledgement
The authors are thankful to Brazilian agencies CAPES and CNPq for financial support.
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