# Design and Control of Magnetic Shape Memory Alloy Actuators

^{*}

## Abstract

**:**

^{3}, guaranteeing an available operating range of up to 1 mm, despite its great deformation range and dynamics. In the paper its dynamics model is proposed and its transfer function is derived. Moreover, the generalised Prandtl-Ishlinskii model of MSMA-actuator hysteresis is proposed. This model is then inverted and used in the control system for hysteresis compensation. A special test stand was designed and built to test the MSMA actuator with compensation. The step responses are recorded, showing that the compensated MSMA actuator exhibits the positioning accuracy as ±2 µm. As a result, the authors decided to apply a control system based on an inverse hysteresis model. The paper concludes with a summary of the research results, with theoretical analysis compared with the registered actuator characteristics.

## 1. Introduction

#### 1.1. Preface

#### 1.2. Smart Materials

## 2. Magnetic Shape Memory Alloys and Their Use as Actuators

#### 2.1. Basics of MSMA—Reorientation of a Crystal Lattice

_{2}MnGa MSMA comprises cells with dimensions a = b = 0.594 nm, c = 0.562 nm. These cells can be set in three possible directions known as variants V

_{1}, V

_{2}and V

_{3}. The magnetic permeability of the cells is different in each direction, with the largest value being in the axis parallel to the direction of side c and perpendicular to the cell surface indicated by edges a and b, also known as the easy magnetisation axis. In the initial phase, all cells in the MSMA probe are in variant V

_{1}; the probe length is L

_{z}. When a magnetic field of strength H

_{ext}is applied to an MSMA, the cuboidal cells tend to move to set their easy magnetisation axis in the same direction as the magnetic field direction i.e., in variant V

_{2}. Furthermore, when the energy needed for cell rotation is greater than that needed for twin boundary motion, the magnetisation process causes rearrangements in the crystal lattice. As a result, the volumes of variants V

_{1}and V

_{2}change, which is visible as a difference in shape, and thus the material changes its length in a direction perpendicular to magnetic field lines and can be used for force generation and position regulation. The most commonly used material in MSMA actuators is Ni

_{2}MnGa, which offers a repeatable scope of elongation of up to 6% of the initial length.

_{1}). Because the dimensions of the martensite cell variants are different, after reorientation the MSMA shortens in length. The above-described process is presented schematically in Figure 1.

_{max}is the maximum length of the MSMA material, L

_{z}is the length of the MSMA before deformation and ε

_{0}is the dimensionless free strain expressed as 1–c/a.

#### 2.2. MSMA Actuator Design Survey

_{TW}can restore the original length. In practice (i.e., the easiest method of application) this force is produced by a compression coil spring, but the compressive force is not sufficient to return the sample to its original shape after the first cycle of operation. This represents a distinct disadvantage, as it reduces the maximum scope of strain in subsequent cycles and increases the movable mass.

^{3}is described, together with a prototype sensor for micro displacement, which uses an LC circuit. Elsewhere, the practical usage of MSMA actuators for vibration reduction in a rotor system is described in [39].

Actuator Design | MSMA [mm^{3}] | Excitation Current [A] | Strain [µm] | Actuator Design | MSMA [mm^{3}] | Excitation Current [A] | Strain [µm] |
---|---|---|---|---|---|---|---|

Spring-returned mode | Push—push/push—pull mode | ||||||

[29] | 2 × 5 × 20 | 2,2 | 200 | [26] | 2 × 3 × 10 | 5 | 150 |

[21,26] | 1 × 2.5 × 20 | 5 | 400 | [32] | 3 × 5 × 20 | 2 | 1000 |

[35] | 2 × 3 × 15 | 2 | 1000 | [40] | 3 × 5 × 20 | 2.5 | 260 |

[41] | 1 × 2.5 × 20 | 5 | 400 | [31] | 3 × 5 × 15 | 3.5 | 280 |

[42] | 1 × 2.5 × 20 | 1.4 | 900 | ||||

[27] | 1 × 2.5 × 20 | 1.2 | 350 | ||||

Mass placed above actuator | Compressive solenoid | ||||||

[32,43] | 3 × 5 × 20 | 1 | 1000 | [34] | 1 × 2.5 × 20 | 2 | 450 |

## 3. Basics of the MSMA Actuator Design Process

^{3}in a steel S215 magnetic circuit. On one side the actuator stand is equipped with an HBM U9B force transducer (6) with a measuring range of ± 50 N, and on the opposite side with a push rod and coil spring (1) of stiffness 0.43 N/mm. The push rod passes through a PTFE sleeve (2) associated with the table driven by a micrometre screw (3), enabling the compressive force to be adjusted from 0 N up to a blocking force of about 6–7 N. Two coils (5) are located on both sides of the MSMA. Displacement of the actuator is measured by an optical transducer (7).

_{cs}—cross-sectional area of the MSMA sample (surface perpendicular to the desired direction of deformation), F

_{b}—blocking push force, y—desired strain length, σ

_{TW}—twinning stress, σ

_{mag}—maximum value of stress induced by magnetic field (3 MPa [44]), L

_{z}—length of MSMAs element and ε

_{0}—dimensionless free strain expressed as 1–c/a, where c, a are cell dimensions.

_{b}= 70 N and minimum repeatable strain y = 1 mm. It was assumed that in future research this actuator will also be tested in an electrohydraulic valve, with the assumed parameter values adequate for application with an EPO-45 electromagnet (produced by Fanina), which are widely used in flow control proportional valves. Based on Equations (2) and (3), MSMA probe geometric dimensions of A

_{cs}= 30 mm

^{2}and L

_{z}= 32 mm were calculated, resulting in a sample size of 3 × 10 × 32 mm

^{3}. The probe was commissioned and prepared by Adaptamat Comp.

_{M}

_{1,}R

_{M}

_{2,}R

_{M}

_{3}—magnetic reluctances of magnetic core parts (Figure 5.), R

_{MAG}—magnetic reluctance of air gap and in MSMA probe, Φ

_{AG}—magnetic flux in air gap, i—coil current, n—number of turns and p—number of coils.

_{0}, µ

_{r}—magnetic permeability of vacuum and relative of path in core.

_{AG}should be 0.7 T, assumed current for single coil was 2 A, thus based on (4) turns were calculated.

_{r}≈ 1). Thus, in the presented air gap of width 3.2 mm and area 348 mm

^{2}(total clearance between sample and core of 0.2 mm) and for a 2 A excitation current, the calculated magnetic induction is 0.7 T, a value that provides a full range of MSMA deformation. The width of the air gap also plays a key role in coil design, because it has the largest and most significant reluctance. MSMA samples were produced with one side significantly shorter than the other; this shortening should reduce the size of the required magnetic core, especially the coils. In the developed actuator the core is solid, but for improvement of dynamic properties the core should be made of transformer steel sheets (Fe-Si), which would reduce the influence of eddy currents [30].

^{2}generates about 2.4 N of force [46]. Because the cross section of the used MSMA sample is 30 mm

^{2}(3 × 10 mm

^{2}), the theoretical maximum output force is equal to about 72 N and thus the spring was pretensioned to about 36 N. Middle value of pretension force provides that maximum displacement is obtained in spring returned mode. A photograph of the assembled actuator is presented in Figure 8.

## 4. Modelling and Control of MSMA Actuator

#### 4.1. Static Characteristics Modelling via the Generalised Prandtl-Ishlinskii Model

_{r}, 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].

_{q}, t

_{q}

_{+1}] with t

_{q}< t ≤ t

_{q}

_{+1}and 0 ≤ q ≤ N−1. A mathematical description of operator output F

_{r}for t

_{0}= 0, t

_{0}< t

_{1}< t

_{2}< … < t

_{N}= t

_{E}, can be formulated as

_{r}—current value of play operator in each time subinterval, w(t

_{q})—play operator output in previous time moment, i(t)—monotonous input signal, i(t

_{q})—value of input signal in previous time moment, and r—threshold value. The operator must satisfy the initial condition expressed as F

_{r}(i(0)) = w(0).

_{p}in the Prandtl-Ishlinskii model is indicated by

_{r}(Figure 9b). In the G

_{r}operator, the increasing curve (function) γ

_{l}and the decreasing curve γ

_{r}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 G

_{r}for the generalised model must fulfil the same conditions as play operator F

_{r}. The mathematical formulation of G

_{r}is described by the following equation:

_{r}—current value of play operator in each time subinterval, z(t

_{q})—play operator output in previous time moment, i(t)—monotonous input signal, i(t

_{q})—value of input signal in previous time moment, and r—threshold value.

_{r}must also meet the same initial condition as the classical operator, i.e., G

_{r}(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]:

_{j}= α·j (j = 0, 1, …, m).

_{j}):

_{0}, a

_{1}, a

_{2}, a

_{3}, b

_{0}, b

_{1}, b

_{2}, b

_{3}are the parameters of the hyperbolic tangent functions.

#### 4.2. Inverse of Generalised Prandtl-Ishlinskii Model

_{pγ}

^{−}

^{1}can be written

_{r}—position reference for the actuator control system, γ

_{l}

^{−1}, γ

_{r}

^{−1}—inverse envelop functions, $\widehat{p}\left({r}_{j}\right)$—density function for inverse model, ${F}_{{\stackrel{\u2322}{r}}_{j}}$—play operator for redefined threshold ${\stackrel{\u2322}{r}}_{j}$.

#### 4.3. Actuator Dynamics Model

_{re}is created.

_{re}—force generated by magnetically induced reorientation of crystal lattice, m—mass of moving parts, b—mechanical damping constant and k—spring constant of system.

_{F}(Figure 11b).

## 5. Test Stand

## 6. Hysteresis Compensation and MSMA Actuator Control

#### PID Control

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Deformation of a crystal lattice in a magnetic field, where a and c are the lengths of a single martensite cell (c = 0.94a).

**Figure 2.**Spring operating mode—theoretical curve strain vs. current, where L

_{z}is the length of the MSMA element, ∆L is MSMA deformation, H

_{ext}is the strength of the external magnetic field, F

_{ret}is the compressive force, and σ

_{TW}is the twinning stress.

**Figure 3.**Design of the MSMA actuator used for preliminary studies: 1—coil spring, 2—PTFE sleeve, 3—table, 4—air gap with MSMA, 5—coils, 6—force transducer, 7—displacement transducer.

**Figure 7.**Magnetic induction measured in air gap vs. current for low-carbon steel 04J, steel S215, and model which represents equivalent circuit.

**Figure 8.**Second actuator design (1)—push rod, (2)—wave spring, (3)—coils, (4)—magnetic pole, (5)—aluminium body, (6)—screws, made of non-magnetic austenitic steel.

**Figure 9.**Preisach relay operator (

**a**), classical play operator from PIM (

**b**) and generalised play operator from GPIM (

**c**) for input i(t).

**Figure 10.**Schematic representation of the electromechanical structure of the MSMA actuator, where: u(t)—supply voltage, R

_{c}—coil resistance, L

_{c}—coil inductance, i(t)—current, y(t)—mass displacement, H—magnetic field strength, m—mass, b—resultant damping coefficient, and k—resultant spring constant.

**Figure 13.**Comparison of hysteresis curves obtained via measurement and simulation for damped sine input (black curve is measured, red curve is output of simulation model).

**Figure 14.**Modelling error between curves presented in Figure 13.

**Figure 15.**IGPI model output showing the relationship between position reference and compensation current.

**Figure 21.**Open-loop control for a series of rectangular position references with increasing amplitude.

**Figure 23.**Block diagram of closed-loop control system with direct hysteresis compensation by the inverse model.

**Figure 25.**Positioning error for the step responses presented in Figure 24: (

**a**) 1000 µm, (

**b**) 750 µm and (

**c**) 500 µm.

GPIM Parameters | ||
---|---|---|

Model order m | 10 | |

Operator scope | 11 (0, 1…10) | |

Envelope functions γ | a_{0} = 1.906 | b_{0} = 1.9931 |

a_{1} = 1.1530 | b_{1} = 1.2839 | |

a_{2}= −1.9515 | b_{2}= −0.8924 | |

a_{3} = 1.5358 | b_{3} = 1.2934 | |

Density function parameters p(r) | α = 0.1255 | |

ρ = 110.5488 | ||

τ = 2.9377 |

Error Type/Error Value | (µm) | % |
---|---|---|

MSMA actuator strain | 1074.5 | 100 |

Max. positive error | 39.73 | 3.70 |

Max. negative error | −37.45 | 3.49 |

Max. absolute error | 39.73 | 3.70 |

Mean error | 2.58 | 0.24 |

Mean absolute error | 11.71 | 1.09 |

RMSE | 13.9 | 1.29 |

Reference y_{r}(t) | ||
---|---|---|

(µm) | % | |

MSMA actuator strain | 1000 | 100 |

Max. positive error | 86.37 | 8.64 |

Max. negative error | −24.98 | 2.50 |

Max. positive error + |Max. negative error| | 111.35 | 11.14 |

Max. absolute error | 86.37 | 8.64 |

Mean error | 26.25 | 2.63 |

Mean absolute error | 33.75 | 3.38 |

RMSE | 39.62 | 3.96 |

Controller | Gains | ||
---|---|---|---|

kp | ki | kd | |

PID1 | 0.3 | 6 | 0.01 |

PID2 | 0.5 | 8 | 0.01 |

PID3 | 0.55 | 10 | 0.015 |

PID4 | 0.57 | 11,3 | 0.02 |

Rise Time t_{u} [ms] for y_{r} ±5% | ||||||
---|---|---|---|---|---|---|

Controller | Position reference y_{r} [µm] | |||||

500 | +25 | 750 | +37.5 | 1000 | +50 | |

−25 | −37.5 | −50 | ||||

PID_{1} | 770 | 510 | 480 | |||

PID_{2} | 470 | 370 | 340 | |||

PID_{3} | 301 | 295 | 250 | |||

PID_{4} | 230 | 202 | 180 |

Integral Absolute Error | |||
---|---|---|---|

Controller | Position reference y_{r} [µm] | ||

500 | 750 | 1000 | |

PID_{1} | 126.3 | 154.9 | 187.1 |

PID_{2} | 92.4 | 126.3 | 142.8 |

PID_{3} | 70.25 | 99.02 | 116.9 |

PID_{4} | 72.35 | 89.44 | 108.4 |

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**MDPI and ACS Style**

Minorowicz, B.; Milecki, A.
Design and Control of Magnetic Shape Memory Alloy Actuators. *Materials* **2022**, *15*, 4400.
https://doi.org/10.3390/ma15134400

**AMA Style**

Minorowicz B, Milecki A.
Design and Control of Magnetic Shape Memory Alloy Actuators. *Materials*. 2022; 15(13):4400.
https://doi.org/10.3390/ma15134400

**Chicago/Turabian Style**

Minorowicz, Bartosz, and Andrzej Milecki.
2022. "Design and Control of Magnetic Shape Memory Alloy Actuators" *Materials* 15, no. 13: 4400.
https://doi.org/10.3390/ma15134400