4.1. Static Characteristics Modelling via the Generalised Prandtl-Ishlinskii Model
The MSMA is characterised by saturated hysteresis, which should be compensated in order to obtain good positioning accuracy. The developed approaches can be divided into physics-based models [
47] and phenomenological models. The first one uses hysteretic behavior and principles of physics effects. The example is the model, developed for hysteresis of ferromagnetic materials [
48]. Nowadays phenomenological models are most commonly used and applied. There are operator-based models, such as: Preisach [
49], Krasnosel’skii-Pokrovkii [
50], Prandtl–Ishlinskii [
51]. The earliest model of hysteresis is that developed by Preisach (1935), in which parallel connections of independent relay hysterons are applied. The hysteron input is a piecewise continuous monotonic function and its output
h varies between 1 and −1 (
Figure 9a). This model uses an infinite set of elementary hysteresis operators, which are similar to relay characteristics.
A modification of the Preisach model is proposed in [
52]. Both the generalized Prandtl-Ishlinskii model and a classical Prandtl-Ishlinskii model are well-known phenomenological hysteresis models [
53]. A clear mathematical description of different models of hysteresis such as: Preisach, Ishlinskii, Krasnoselskii and Duhem–Madelung are presented by Macki et al. in [
54]. The properties and the comparison of these models are also included. Hassani et al. in [
55] presented the survey on hysteresis modeling, including such various models as Preisach, Krasnosel’skii–Pokrovskii, Prandtl–Ishlinskii, Maxwell-Slip, Bouc–Wen and Duhem. In the paper also their applications in control and identification are presented. A similar solution is the Maxwell-slip model, which is designed for modeling of friction. It can be adapted for smart actuators modelling [
56]. Another group of models is based on differential equations. The examples are Duhem model [
57], Backlash-like [
58] and Bouc-Wen. The last model was introduced by Bouc [
59]. Wen extended this formula in paper [
60].
In recent years, different artificial neural networks have also been successfully used in modelling of different actuators’ hysteresis. Yu et al. in [
61] proposed the rate-dependent hysteresis NARMAX model based on a diagonal recurrent neural network. As the variable function of the NARMAX model, the adopted play operator is used. The experimental investigations show that the proposed model has a very good modeling precision. The proposed approach may enable the broadening of the application based on MSMA actuators in micro/nano devices. The investigations on compensation of MSMA-based actuator hysteresis is described in [
62]. To this end, a feed-forward neural network-based nonlinear autoregressive moving average with exogenous inputs model is applied. The modelling and control methods are investigated experimentally using an MSMA-based actuator. The results show that the proposed solution allows an accurate description of the hysteresis of the actuator and the developed compensator can effectively reduce the impact of hysteresis on the performance of the MSMA-based actuator.
In the oft-used Prandtl-Ishlinskii (PI) model, the relay hysterons are replaced by a play operator
Fr, which can be represented by backlash (a well-known phenomenon in mechanical systems), with the width of the backlash hysteresis defined by two thresholds
r (
Figure 9b). The generalised PI model (GPIM) is more suitable for the modelling of constant non-linear hysteresis. Detailed descriptions of hysteresis modelling techniques involving this model can be found in [
23,
54].
The mathematical function describing the model input, here the current signal
i(
t), must satisfy the condition of monotonicity in each subinterval [
tq,
tq+1] with
tq <
t ≤
tq+1 and 0 ≤
q ≤
N−1. A mathematical description of operator output
Fr for
t0 = 0,
t0 <
t1 <
t2 < … <
tN =
tE, can be formulated as
where:
Fr—current value of play operator in each time subinterval,
w(
tq)—play operator output in previous time moment,
i(
t)—monotonous input signal,
i(
tq)—value of input signal in previous time moment, and
r—threshold value. The operator must satisfy the initial condition expressed as
Fr(
i(0)) =
w(0).
The input-output relationship
yp in the Prandtl-Ishlinskii model is indicated by
where:
p(
r)—density function, which is a representation of the influence of each operator on the final shape of hysteresis and
R—upper integration limit.
In order to model the asymmetric and saturated shapes of the hysteresis characterising the MSMA actuators, the play operator described by Equation (6) was modified, becoming the generalised operator expressed as
Gr (
Figure 9b). In the
Gr operator, the increasing curve (function)
γl and the decreasing curve
γr are so-called envelope functions and must be continuous. The most suitable function with which to model both major and minor hysteresis loops in MSMA actuators is the hyperbolic tangent. The play operator
Gr for the generalised model must fulfil the same conditions as play operator
Fr. The mathematical formulation of
Gr is described by the following equation:
where:
Gr—current value of play operator in each time subinterval,
z(
tq)—play operator output in previous time moment,
i(
t)—monotonous input signal,
i(
tq)—value of input signal in previous time moment, and
r—threshold value.
Gr must also meet the same initial condition as the classical operator, i.e.,
Gr(
i(0)) =
z(0). In the hysteresis model, each play operator can be weighted by the density function
p(
r), which helps to match the model to the measured hysteresis and is described by Equation (9) as follows [
23,
47]:
where:
rj =
α·j (
j = 0, 1, …,
m).
The total number of used operators
j is equal to
m + 1. Parameters
α,
ρ (always positive) and
τ allow for the precise description of the density function for the
j-th operator. The output of the model described above is calculated as
However, for practical implementation in a control scenario this integral can be expressed as a superposition of a finite number of generalised play operators, where each one is weighted by a unique value of density function
p(
rj):
This equation can be simply implemented in a computer-based controller.
Hysteresis loops in the MSMA actuator are characterised by saturation. As their shape is very similar to that of the hyperbolic tangent function, this function can be used to describe the decreasing and increasing slopes of the hysteresis envelope curves via the following equations:
where:
a0,
a1,
a2,
a3,
b0,
b1,
b2,
b3 are the parameters of the hyperbolic tangent functions.
4.3. Actuator Dynamics Model
The MSMA actuator is an electromechanical device, the structure of which is presented schematically in
Figure 10; this structure is a simplified representation for dynamic modelling purposes.
The electric part consists of a coil connected serially to a resistor, as described by the following differential equation:
The transfer function of the actuator’s electrical part can thus be written as follows:
The current flow generates the magnetic field strength, which is proportional to this current and to the number of coil turns. Magnetic flux in the air gap acts on the MSMA, with the particles in the crystal lattice changing their orientation and thus the output force Fre is created.
Assuming linearity, the dynamics of the actuator’s mechanical part can be described using the force equilibrium formulated as the following differential equation:
where:
Fre—force generated by magnetically induced reorientation of crystal lattice,
m—mass of moving parts,
b—mechanical damping constant and
k—spring constant of system.
After Laplace transformation, this equation can be rewritten as the transfer function of the mechanical part:
Hysteresis is caused by nonlinear dependence between input current and the force that generates movement. The model of the actuator with hysteresis is shown in
Figure 11a. As preliminary laboratory research revealed that the above-described MSMA actuator is characterised by time delay, the model also includes a delay block. To compensate for the hysteresis, an additional element was applied in the form of the inversed GPI (IGPI) model of hysteresis. Thus, it can be assumed that the relationship between input current and output force is linear and can be expressed by transfer function (16), with gain factor
kF (
Figure 11b).
Based on
Figure 11b and omitting the nonlinearities, the transfer function of the linearized actuator is expressed as