sensors Article

Tracking Control of a Magnetic Shape Memory Actuator Using an Inverse Preisach Model with Modified Fuzzy Sliding Mode Control Jhih-Hong Lin and Mao-Hsiung Chiang * Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 106, Taiwan; [email protected] * Correspondence: [email protected]; Tel.: +886-2-3366-3730 Academic Editor: Vittorio M. N. Passaro Received: 29 June 2016; Accepted: 23 August 2016; Published: 25 August 2016

Abstract: Magnetic shape memory (MSM) alloys are a new class of smart materials with extraordinary strains up to 12% and frequencies in the range of 1 to 2 kHz. The MSM actuator is a potential device which can achieve high performance electromagnetic actuation by using the properties of MSM alloys. However, significant non-linear hysteresis behavior is a significant barrier to control the MSM actuator. In this paper, the Preisach model was used, by capturing experiments from different input signals and output responses, to model the hysteresis of MSM actuator, and the inverse Preisach model, as a feedforward control, provided compensational signals to the MSM actuator to linearize the hysteresis non-linearity. The control strategy for path tracking combined the hysteresis compensator and the modified fuzzy sliding mode control (MFSMC) which served as a path controller. Based on the experimental results, it was verified that a tracking error in the order of micrometers was achieved. Keywords: magnetic shape memory alloys; magnetic shape memory actuator; hysteresis compensator; Preisach model; inverse Preisach model; modified fuzzy sliding mode control; tracking control; non-linear control system

1. Introduction Magnetic shape memory (MSM) alloys are a new and important member of the class of smart materials that have been widely studied since 1996 [1,2]. MSM alloys are mainly composed of Ni-Mn-Ga composites, but there are also other alternative composites [3]. MSM alloys have the property called magnetic shape memory effect, which means its twin-variants in the phase martensite could be switched. The twin-variants can be induced to switch via the application of magnetic fields, and its maximum magnetic field induced strain is from 6% to 12%. The strain depends on the material composition and modulation [4–6]. Shape memory alloys (SMA) exhibit similar behavior, but create large strains up to 8% by changing their temperature, allowing the alloys to switch between the martensite and austensite phases [7]. Compared to SMAs, MSM alloys have the property of higher frequency response, since SMAs use heat transfer for phase change and MSM alloys inherently take advantage of the change of the twin-variants in the same phase with a much faster response. These properties of larger strain and faster response make MSM alloys potential candidate materials for many different applications. An MSM actuator is an application based on the properties of MSM alloys. Though several composites are found to possess the MSM effect, the material made of Ni-Mn-Ga is chosen here based on its largest magnetic field induced strains within the alternatives [2,3]. In addition, a MSM actuator consists of magnetic circuits, MSM elements and returning force springs so that the function of reversible actuation can be repeated through the elongation and contraction of the MSM elements. The

Sensors 2016, 16, 1368; doi:10.3390/s16091368

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operational principle of MSM demonstrated in Figurein1 Figure [8]. The1 strain is due to the switching The operational principle of alloys MSM is alloys is demonstrated [8]. The strain is due to the between the twin-variants of MSM alloys. In the beginning, MSM alloys are maintained in the presence switching between the twin-variants of MSM alloys. In the beginning, MSM alloys are maintained in of variant 1, which is shown in Figure 1a. The change from variant 1 to variant 2 is induced when the the presence of variant 1, which is shown in Figure 1a. The change from variant 1 to variant 2 is magnetic field the passes perpendicularly the MSM alloys the as shown in Figure 1b. The stronger induced when magnetic field passesthrough perpendicularly through MSM alloys as shown in Figure 1b. the magnetic field, the faster the variant 1 can transform to variant 2. The maximum strain is achieved The stronger the magnetic field, the faster the variant 1 can transform to variant 2. The maximum ifstrain variant 1 has fully changed1 to variant 2, as seentoinvariant the saturation inin Figure 1c. To contract the MSM is achieved if variant has fully changed 2, as seen the saturation in Figure 1c. To alloys, which means to reverse variant 2 to variant 1, there are two ways to achieve this goal. One is to contract the MSM alloys, which means to reverse variant 2 to variant 1, there are two ways to achieve use direction the magnetic field, perpendicular to the original, as shown in Figureas1d. The thisanother goal. One is to useofanother direction of the magnetic field, perpendicular to the original, shown other is to add a compressive force to the MSM alloys by inserting a mechanical spring, as shown in in Figure 1d. The other is to add a compressive force to the MSM alloys by inserting a mechanical Figure 1e. In Figure 1f, the MSM alloys contract to their original position if the magnetic field or the spring, as shown in Figure 1e. In Figure 1f, the MSM alloys contract to their original position if the compressive force is strong enoughforce [9]. is strong enough [9]. magnetic field or the compressive

Figure Figure1.1. Operational Operational principle principle of of MSM MSM alloys alloys (a) (a) MSM MSM alloys alloys have have only only variant variant 11 in in the the first first state; state; (b) With the strong magnetic field (>0.7 T) in the y-direction, the MSM element elongates (b) With the strong magnetic field (>0.7 T) in the y-direction, the MSM element elongates inin thethe xx-direction. Elongationwill willstop stopwhen whenall allthe thetwin-variants twin-variantsswitch switch to to variant variant 2; 2; (c) direction. Elongation (c) Under Under small small or or absence field, MSM MSMalloys alloyscan canhold holdtheir theirlength; length;(d)(d) MSM alloys contract with absence of of aa magnetic magnetic field, MSM alloys cancan contract with the the provision of magnetic field in the x-direction; (e) MSM alloys can contract with the provision provision of magnetic field in the x-direction; (e) MSM alloys can contract with the provision of of compressive compressiveforce; force;(f) (f)MSM MSMalloys alloyscan canreturn returntotoits itsoriginal originalposition positionthrough throughprocess process(d) (d)oror(e) (e)[10]. [10].

Since the MSM of of MSM alloys, it also inherits the drawbacks of MSM Since MSM actuator actuatoruses usesthe theproperties properties MSM alloys, it also inherits the drawbacks of alloys,alloys, namely the hysteresis non-linearity caused by the of twin-variants and MSM namely the hysteresis non-linearity caused by transformation the transformation of twin-variants temperature changes [11].[11]. Many researchers have tried different and temperature changes Many researchers have triedtotomodel modelthe the properties properties from different perspectives. Using the MSM alloys’ behavior by perspectives. the microscopic microscopicperspective, perspective,researchers researchershave havemodeled modeled MSM alloys’ behavior thethe variants’ structure andand thermodynamic freefree energy. Thermodynamics of the mechanical and by variants’ structure thermodynamic energy. Thermodynamics of the mechanical magnetic properties was was modelled by Ullakko and and Likhachev [12]. [12]. O’Handley et al. et[13] and magnetic properties modelled by Ullakko Likhachev O’Handley al. have [13] discussed the magnetic-field-induced strains of of MSM between have discussed the magnetic-field-induced strains MSMalloys alloysthrough through the the relationship between magnetization and and twin-boundary twin-boundary motion motion by by minimizing minimizing its its free free energy, energy, which which includes includes Zeeman Zeeman magnetization energy, magnetic anisotropy energy, internal elastic energy and external stress. Hirsinger and energy, magnetic anisotropy energy, internal elastic energy and external stress. Hirsinger and Lexcellent [14] [14] introduced introduced aa phenomenological phenomenological model model about about the the variants’ variants’ rearrangement rearrangement through through Lexcellent thermodynamicswith with internal variables, which contains chemical mechanical energy, thermodynamics internal variables, which contains chemical energy, energy, mechanical energy, magnetic magnetic and thermal energy. Gauthier et al. [15] combined thermodynamics with Lagrangian and thermal energy. Gauthier et al. [15] combined thermodynamics with Lagrangian formalism formalism and its Hamiltonian extension as a dynamical foralloy-based an MSM alloy-based actuator. and its Hamiltonian extension as a dynamical model for model an MSM actuator. Tan and Tan and[16] Elahinia [16]thedescribed the dynamic behavior of abyMSM actuator by combining its Elahinia described dynamic behavior of a MSM actuator combining its constitutive model, constitutive model, the reorientation the the kinematic model, theactuator. dynamicInmodel of the the reorientation kinetics, the kinematickinetics, model, and dynamic modeland of the these kinds actuator. these kinds of models, aofcomplete understanding the material energy of models,In a complete understanding the material structure andofenergy flow arestructure necessary.and However, flow are necessary. consider all of the to consider all of theHowever, influentialtofactors increases theinfluential complexityfactors of the increases model. the complexity of the model.

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Researchers have used a mathematical model to describe the MSM alloys’ hysteresis phenomenon from the macroscopic perspective, which means the MSM alloys are regarded as a black box and the consideration is only its input as source signals and output as strain. The Preisach model is a method used to describe hysteresis and it is suitable for those smart materials possessing this property, such as piezoelectric, SMA, MSM alloys, etc. [17–21]. Inverting the hysteresis models can be used to compensate for tracking control. The MSM actuator is described as a Preisach-like Krasnosel’skii–Pokrovskii model, combining the inverse model as compensation and adaptive control for positioning control by Riccardi et al. [22,23]. In addition to the Preisach model, Sadeghzadeh et al. [24] used phase shift hysteresis compensation to describe the MSM actuator and combined with PID control and gain scheduling for position controller. Zhou et al. [25–27] used an inverse Prandtl-Ishlinskii model to eliminate the influence of the hysteresis non-linearity and a PID neural network as hybrid control for the MSM actuator. In previous studies, fuzzy sliding mode control (FSMC) had been used for MSM actuators to achieve a model-free and robust controller [28], and the modified fuzzy sliding mode control (MFSMC) is proposed by adding an additional integrator in FSMC to prevent the control signal from approaching zero and track the desired position in the steady state [10]. The purpose of this paper is follow-up research based on previous studies. There are some improvements in this paper. First, the test rig is an advanced design, so that the MSM actuator is suitable to work for the purpose of a tracking in the order of micrometers. Second, to enhance tracking ability, the accuracy of tracking error was improved through our novel controller. In the following sections, firstly, the MSM actuator testbed and the main components are introduced. Section 3 adopts the Preisach model with verified data from experiments to simulate the response of a MSM actuator, and then the inverse Preisach model by numerical recursive implementation in discrete time is applied as hysteresis compensator for the MSM actuator. Section 4 explains the concept of fuzzy sliding mode control. Therefore, the inverse Preisach model, as a position prediction, is a hysteresis compensator and path controller MFSMC is integrated in this paper. Section 5 shows the results of experiments, including results of using inverse Preisach solely, the results of using MFSMC solely, and the results of combining both controllers. Finally, Section 6 is presents the conclusions of the paper. 2. MSM Actuator Overview The MSM actuator experimental setup is shown in Figure 2a, and the photo of the test rig is shown in Figure 2b. The experimental setup is composed of a MSM actuator, bearing slides and a loading platform. The MSM actuator is the product made by AdaptaMat Ltd. (Helsinki, Finland), and its specifications are listed in Table 1. The MSM alloy elongation is induced by the strength of a magnetic field which is passing through it perpendicularly. Increasing the magnetic field elongates MSM alloys, and decreasing the magnetic field compresses MSM alloys by the return spring, which can release the elastic potential energy that was stored from the elongation of MSM alloys. The MSM actuator is driven from a voltage controlled current source that can linearly covert voltages to currents. The position sensor, an optical encoder made by NUMERIK JENA GmbH (Jena, Germany) with resolution of 20 nm, measures the stroke of the MSM actuator. All the signals are sent out and fed back to a PC-based controller via a multifunction DAQ card, made by National Instruments (Austin, TX, USA). The sampling frequency is set as 1000 Hz.

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(a)

(b) Figure Rig; (b) (b) Photo Photo of of the the MSM MSM actuator. actuator. Figure 2. 2. (a) (a) Configuration Configuration of of Test Test Rig; Table 1. Specifications of the MSM actuator. Table 1. Specifications of the MSM actuator. Items Items Actuating element Actuating element Element size: Element size: Fatigue life Fatigue life Rise time Rise time Frequency response Frequency response Maximum power dissipation Maximum power dissipation Actuator dimensions Actuator dimensions Weight

Speciafication Speciafication 5M Ni-Mn-Ga MSM element 5M Ni-Mn-Ga MSM element 20 × 2.5 × 1.0 mm3 20 × 2.5 × 1.0 mm3 200 million cycles (minimum) 200 million cycles (minimum) 1 ms 1 ms ≥250 Hz ≥250 Hz 3.5 W 3.5 W 25 × 25 × 66 mm3 25 × 25 160 × 66g mm3

Weight Data Acquisition Card: output Data Acquisition Card:counters output Data Acquisition Card:

160 g 16-bit analog outputs (833 kS/s) 16-bit analog outputs (833 32-bit counters kS/s)

Data Acquisition Card: counters Optical Encoder

32-bit counters Resolution: 20 nm Resolution: 20 nm

Optical Encoder

Company Company

AdaptaMat Ltd. AdaptaMat Ltd.

National Instruments Ltd. National Instruments Ltd. NUMERIK JENA GmbH NUMERIK JENA GmbH

The hysteresis phenomenon of the MSM actuator is shown in Figure 3. Triangular waves with 2 s of rising time and 2 s of falling time are used to drive the MSM actuator. Through the whole experiment, the highest voltage for the MSM actuator is 4 V, which equivalently means that the MSM

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actuator is driven by a 4 A current. To test hysteresis, the highest voltage of triangle wave was lowered 0.5 V each time until the highest voltage reaches 0.5 V. In the figure it is shown that the longest position is around 88 µm. Frequency response had also been tested as shown in Figure 4. Its frequency response isSensors up to more than Sensors 2016, 16, 5 5ofof2020 2016, 16,1368 1368 200 Hz, and the largest displacement sits at around 30 Hz. 9090 8080 7070

Position (um) Position (um)

6060 5050 4040 3030 2020 1010 00 -10 -10 00

0.5 0.5

11

1.5 22 2.5 1.5 2.5 voltage (V) voltage (V)

33

3.5 3.5

44

Hysteresis phenomenon ofofthe the MSM actuator. Triangle wave voltages range from to V. Figure Figure3.3.Hysteresis Hysteresisphenomenon phenomenonof theMSM MSMactuator. actuator.Triangle Trianglewave wavevoltages voltagesrange rangefrom from000VVVto to444V. V. After each experiment, subtract 0.5 V each time from the previous voltage until the highest voltage After each experiment, subtract 0.5 V each time from the previous voltage until the highest voltage reaches V.V. reaches0.5 0.5V. 1010 00 -10 -10

magnitude (dB) magnitude (dB)

-20 -20 -30 -30 -40 -40 -50 -50 -60 -60 -70 -70 -80 -80 -90 -90-1 -1 1010

0 0

1010

1 1

1010 frequency frequency(Hz) (Hz)

2 2

1010

Figure 4. 4. Frequency response response of the MSM MSM actuator. Figure Figure 4.Frequency Frequency responseofofthe the MSMactuator. actuator.

3.3.Model Model of Hysteresis and Its Compensator 3. Modelof ofHysteresis Hysteresisand andIts ItsCompensator Compensator InInthe the previous section, the hysteresis phenomenon isisobserved observed through experiments. Therefore, In theprevious previoussection, section,the thehysteresis hysteresisphenomenon phenomenonis observedthrough throughexperiments. experiments.Therefore, Therefore, modeling the hysteresis helps to predict the output position of the MSM actuator. The Preisach model modeling modelingthe thehysteresis hysteresishelps helpstotopredict predictthe theoutput outputposition positionofofthe theMSM MSMactuator. actuator.The ThePreisach Preisachmodel model is used in this paper for modeling the MSM actuator. Once the model is constructed, we can use the isisused usedininthis thispaper paperfor formodeling modelingthe theMSM MSMactuator. actuator.Once Oncethe themodel modelisisconstructed, constructed,we wecan canuse usethe the inverse model to predict the input voltage for tracking specific reference. inverse inversemodel modeltotopredict predictthe theinput inputvoltage voltagefor fortracking trackingspecific specificreference. reference.

3.1. 3.1.The ThePreisach PreisachModel Modelfor forthe theMSM MSMActuator Actuator The ThePreisach Preisachmodel modelcan canbebeformulated formulatedasasfollows follows[29]: [29]:

f (ft()t  )

((, , )) d d d   [u[(ut()]t )] d

(1)

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3.1. The Preisach Model for the MSM Actuator The Preisach model can be formulated as follows [29]: x Sensors 2016, 16, 1368 f (t) = α≥ β µ ( α, β ) γαβ [ u ( t )] dαdβ

6 of (1) 20

where where f ((t)) is the elongation of the the MSM MSM actuator; actuator; µ((α,, β)) is a weighting weighting function function in in the the Preisach Preisach model; is the hysteresis operator; ( ) is the input signal; is a function of ( ), its model; γαβ is the hysteresis operator; u (t) is the input signal; γαβ is a function of u (t), and itsand trigger trigger byβ. and . values values are set are by αset and In hastotobebegreater greater equal . The relationship between In the the Preisach Preisach model, α has oror equal to to β. The relationship between γαβ and and u (t) ) is shown in Figure 5a.uAs ) is monotonically increasing is shown in orange is is( shown in Figure 5a. As increasing that that is shown in orange line,line, γαβ is zero (t) is( monotonically untiluntil u (t) exceeds α, then γαβ will switch directly to one afterwards. OnOn thethe contrary, asasu (t() is will switch directly to one afterwards. contrary, ) zero ( ) exceeds , then monotonically decreasing that is shown in green lines, γ will change to zero from one afterwards is monotonically decreasing that is shown in green lines, will change to zero from one αβ when u (t) when is under(β.) is under . afterwards



  [u(t )]



umax

1

umin  0

0



u(t)



umax

(a)



(b)

Figure 5. (a) Hysteresis operator; (b) The Preisach plane. Figure 5. (a) Hysteresis operator; (b) The Preisach plane.

The Preisach model in Equation (1) integrates multiple hysteresis operators. All the hysteresis The Preisach model in Equation integrates multiple hysteresis operators. hysteresis operators can be demonstrated on the(1)Preisach plane as shown in Figure 5b. All the indicates the operators can be demonstrated on the Preisach plane as shown in Figure 5b. u indicates the max maximum input value for the MSM actuator, and is zero voltage. The x-axis represents all the maximum value for represents the MSM actuator, and zero voltage. in The x-axis represents all the values of input and the y-axis the values of umin . Asisthe definition Equation (1), is always values of β and the y-axis represents the values of α. As the definition in Equation (1), α is always equal or larger than . Therefore, the working area is limited within the upper right triangle as equal or shown inlarger Figurethan 5b. β. Therefore, the working area is limited within the upper right triangle as shown in Figure 5b.model, hysteresis is not only influenced by local extrema, which means it is not only In this In this hysteresis is notbut only by localby extrema, means it is not related related to themodel, current input signal, it influenced is also influenced the pastwhich position. Hence, it isonly important toknow the current input signal, but it isisalso influenced by the position. Hence, it is important to know to that saving past extrema necessary in using thepast Preisach model. that Next, savingtopast extrema is necessary in using the Preisach model. implement the Preisach model, a new function F(α1,1) is defined as: Next, to implement the Preisach model, a new function F(α1 ,β1 ) is defined as: F (1 , 1 )  f1  f11  x  ( ,  )d  d  (2) F (α1 , β 1 ) = f α1 − f α1 β1 =T (1 , 1 ) µ(α, β)dαdβ (2) T (α ,β )

where is the output as the input signal increases to 1 1 ; is the output as the input signal first reaches and then to , and ( to , α) ; indicates the area between and where f α1 is the output as decreases the input signal increases 1 f α1 β 1 is the output as the input signal first on the Preisach plane, as shown in Figure 6. reaches α1 and then decreases to β 1 , and T (α1 , β 1 ) indicates the area between f α and f α β on the 1

Preisach plane, as shown in Figure 6.



 

1 S  (t ) T (1 , 1 )





1 1

F (1 , 1 )  f1  f11 



 ( ,  )d d 

(2)

T (1 , 1 )

where is the output as the input signal increases to ; is the output as the input signal first reaches and then decreases to , and ( , ) indicates the area between and Sensors 16, 1368plane, as shown in Figure 6. 7 of 20 on the2016, Preisach



 

1 S  (t ) T (1 , 1 )

1



Figure Figure 6. 6. The explanation explanation of of area area T ((α1 ,, β 1)) on the Preisach plane. Sensors 2016, 16, 1368

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We derive rising and falling processes of input signals with similar analytically discrete methods. We derive andthe falling processes of input signalsbywith analytically discrete It is now able torising calculate output of the MSM actuator the similar following two equations, as methods. the time It is now able to calculate the output of the MSM actuator by the following two equations, as time for input is increasing, Equation (3) is used; as the time for input is decreasing, Equation (4) isthe used. for input is increasing, Equation ( (3) is used; as the time for input ) is decreasing, Equation (4) is used. n −1

f (t) = ∑ Fn(1Mk , mk−1 ) − F ( Mk , mk )  + F (u(t), mn−1 ) f (t )    F ( M k , mk 1 )  F ( M k , mk )   F (u (t ), mn 1 ) k =1  k 1  ( )

(3) (3)

n −1

1 F (n M , m ) − F ( Mk , mk ) + F ( Mn , mn−1 ) − F ( Mn , u(t)) (4) f (tk∑ )=1   Fk ( Mkk−, 1mk 1 )  F ( M (4) k , mk )   F ( M n , mn 1 )  F  M n , u ( t )   k 1  In order to implement the numerical calculation, it is necessary to record the experimental data. In order to implement the numerical calculation, it is necessary to record the experimental data. The experimental data are saved for each voltage value. The voltage steps up from 0 V to 4 V in 0.5 V The experimental data are saved for each voltage value. The voltage steps up from 0 V to 4 V in 0.5 V steps. There are in total 36 points on the Preisach plane in Figure 7a, and the experimental values are steps. There are in total 36 points on the Preisach plane in Figure 7a, and the experimental values are shown in Figure 7b. Hence, all the points on the Preisach plane can be derived as Equation (5): shown in Figure 7b. Hence, all the points on the Preisach plane can be derived as Equation (5):

f (t) =

αβ αβ αβ  c1  f αβ =f c0αβc + αβ  c1 α+cc22β +  cc33  0 

(5) (5)

αβ the points on where thethe output at (at the parameters c0 αβ , c1 αβ , c, 2 αβ and α, β()., For output ). calculating For calculating the parameters , c3 ,and , the where f αβ is is the Preisach which are lying any within of squares, bilinear the spline interpolation is used; the points on theplane Preisach plane whichwithin are lying any the of squares, bilinear spline interpolation points thepoints Preisach plane whichplane are lying within any of triangles, c3 αβ is set to zero is and setatolinear zero is used;onthe on the Preisach which are lying within any of triangles, interpolation is used. and a linear interpolation is used.

(a)

(b)

Figure 7. Experimental data recorded for the numerical calculation (a) Set points within the working Figure 7. Experimental data recorded for the numerical calculation (a) Set points within the working area for data capture; (b) the output value for each point. area for data capture; (b) the output value for each point.

3.2. The Inverse Preisach Model In the previous section, the Preisach model is used to predict the MSM actuator’s output for a given input. In this part, the inverse Preisach model is discussed for path tracking control so that the input signals of the MSM actuator can be derived for a giving reference position. Let be the desired trajectory, which is linear to the input signal as shown in Equation (6). The compensation signals can

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3.2. The Inverse Preisach Model In the previous section, the Preisach model is used to predict the MSM actuator’s output for a given input. In this part, the inverse Preisach model is discussed for path tracking control so that the input signals of the MSM actuator can be derived for a giving reference position. Let f r be the desired trajectory, which is linear to the input signal as shown in Equation (6). The compensation signals can be calculated by the inverse Preisach model, and then inputted to the system for eliminating the hysteresis phenomena: f r = A + Bu (t) (6) Since the PC-based computation is giving signal in discrete time with a specific sampling time Ts , 8 of 20 it’s necessary to transfer the Preisach model into a recursive discrete-time model. Equation (7) stands for increasing input, and Figure 8 shows the corresponding area of the symbol. f  nTs s     ,  d  d  S   nT f (nTs ) = µs (α, β)dαdβ Sensors 2016, 16, 1368

S+ (nTs )

=

s 

,fοru  nT   u  ( n  1)T  d  d+ s µ (α,β,)dαdβ, d  d for µ (α,β,)dαdβ u (nTs ) > u ((n − 1) Ts ) s

S   tm 1  S + ( t m −1 )

∆S

s

S

(7) (7)

s f 1)tm+ d d  = f (tm− µ (α, β),  dαdβ 1    ∆S

S

= f (tm− f u(nT f 1)tm+ f s ) − f fu(nTs )u(tm−1 ) 1   u  nTs 

u  nTs  u  tm1 

where thethe present discrete time, sampling time.time. u (nTs )( is the (n −( 1)−Ts1)is the wherenTs indicates indicates present discrete time, is previous the previous sampling ) is last extrema input. t is the time of the previous extrema input, and f t is the output for the ( ) m −1 the last extrema input. is the time of the previous extrema input, andm−1( ) is the output for previous extrema input. If theIfinput decreases, it canit be by the method and and shown in the previous extrema input. the input decreases, canderived be derived by same the same method shown Equation (8). in Equation (8). f (nTf s)nT = f (tfm−t1 ) + f u(t −1 )u(nTs )f − f u(for uT((n − 1) Ts ) t u ) for nTsu (nT u s()n<  1) s m 1   f u  tm s u  tm 1  m−1  m 1  u  nTs 

(8) (8)

Figure8.8.The Therelationship relationshipbetween betweeninput inputsignals signalsand andthe theprocessing processingofofthe thePreisach Preisachplane. plane. Figure

Therefore,combing combingEquation Equation(5) (5)with withthe theexperimental experimentaldata dataand andEquations Equations(7) (7)and and(8), (8),we wecan can Therefore, get the compensational signals as shown in Equation (9) for increasing signals and Equation (10) for get the compensational signals as shown in Equation (9) for increasing signals and Equation (10) for decreasingsignals: signals: decreasing u For For u (nT nT ) s> u u((n( n− 11))TT s), , s

s





  v  nTs  v  nTs  v nT v t v nTs  v  tm1  v (nT A  Bup  nTs   f p  tmv(1nT c0s)vs(tm −m11 )  c2 v(nT  s )vc(0nT s ) v ( tvm−t1m)1  s) A +v Bu nT − f t − c − c − c v t ( ) ( ) ( ) s m − 1 m − 1 0 0 2  nTs   v  nTs  v  nTs  v  nTs  v  nTs  v  nTs  v  tm1  v  nTs  v  tm1  v (nTs ) = v−1tm) 1  v( nTcs1)v(tm−1 )  c3v(nTs )v(tm v(nTs )v(nTs )c1 v(nTs ) v(cnT 2 s) c1 + c2 − c1 − c3 v ( t m −1 )

(9) (9)

For u  nT   u  ( n  1)T  , For u (nTs ) s< u ((n − 1) Tss),  

v (nTs ) =

 A  Bu nT  f

v  tm1  v  nTs 

v  tm1  v  nTs 



s )mv1(tm−1 ) + . . .   1m)− 1  c v(0tm−1 )v(nTs ) − 1c v(tm−1 )v(nT A + Bu (nTs ) −s f p (tm− 0 1 2 2 v ( t ) v ( t ) v ( t ) v ( t ) v ( t ) v ( t ) ) v t v t v t v t v t v t vvt(t v t)v(t             m − 1 m − 1 m − 1 m − 1 m − 1 m − 1  c + mc11 m1  c mv1 (tmm1−v1 ) t+ c2  c m1 m1vv(tmt −1 ) + c0  cc3  mm1 −1 m1  mv−1t v (tm−1 ) v  nTs   

0

1

p

t

c

m 1

c

2

c2 v(tm−1v)vt (nTsv) + c v(tm−1v)vt(nTs v) vnT ( t m −1 ) nT 3

c2

 m1  

4. Controller Design 4.1. Concept of Modified Fuzzy Sliding Mode Control

s



 c3  m1 



s



v t

m 1

v  tm1 

 ...

3

m 1



 

(10) (10)

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4. Controller Design 4.1. Concept of Modified Fuzzy Sliding Mode Control Figure 9 depicts the block diagram of the modified fuzzy sliding mode control system. Fuzzy logic is used for fuzzy controller, which includes fuzzification, a fuzzy rule base, fuzzy inference, and defuzzification. However, the fuzzy inference rules will increase and the membership functions will be . complicated if the fuzzy control system needs to consider both control e and error rate e. For instance, . to construct seven rules of error e and seven rules of error rate e for fuzzy logic, 49 rules are needed. It is complex and memory-intensive for real-time computation. Fuzzy sliding mode control (FSMC) combines the advantages from fuzzy control and sliding mode control. The FSMC system uses a . . Sensors 2016, 16, 1368 fuzzy sliding surface σ = αe + e, which is a linear combination of error rate e and error e,9 of so20that the dimensions thethat input space and of the of fuzzy be reduced. error of, so the dimensions thenumber input space and theinference number ofrules fuzzycan inference rules canDue be to the fuzzy rules, the output u approaches zero if σ is close to zero. In modified fuzzy sliding mode reduced. Due to the fuzzy rules, the output approaches zero if is close to zero. In modified control fuzzy modeiscontrol (MFSMC), an integrator is inserted aftercontrol, output from the the fuzzy logicactuator (MFSMC), ansliding integrator inserted after output from the fuzzy logic so that MSM control, so that the MSM actuator can keep its position to the desired position as the error approaches can keep its position to the desired position as the error approaches zero. Lyapunov’s theory is used to zero. Lyapunov’s theory is used to guarantee the stability of the tracking error of the guarantee the stability and the bound of the tracking errorand of the thebound system. system.

yref



   e  e

Figure 9. 9. Block ofMFSMC MFSMC system. Figure Blockdiagram diagram of system. 4.2. Designing MFSMC for Path Controller

4.2. Designing MFSMC for Path Controller

A non-linear system can be described as follows [10]:

A non-linear system can be described as follows [10]:

where

dnx (11) x ( n ) (t )  f ( x, t )  d (t )  u, x ( n )  n dt dn x (n) (n) (11) x (t) = f (x, t) + d(t) + u, x = n dt function with a bound of where ( ) = [ , , … , ( ) ] is the state vector; ( , ) is a non-linear ( , ); h( ) is a disturbance i T bounded by ( ) and u is the control input. The tracking error of the . defined ( n − 1 ) state vector is as: x (t) = x, x, . . . , x is the state vector; f (x, t) is a non-linear function with a bound of

e(t ) by x (t )D  x(dt()t )and  [e, eu,..., (12) of the F ( x, t); d (t) is a disturbance bounded isethe] control input. The tracking error state vector is defined as: where ( ) is the desired output. T . e(tis) defined = x( t ) − xd (t) = [e, e, . . . , e(n−1) ] (12) The fuzzy sliding surface as follows: ( n 1) T

d where xd (t) is the desired output.  (x, t )  (   )n 1 e  0 dt The fuzzy sliding surface is defined as follows:

(13)

where is a positive constant [30]. To simply the system, set the fuzzy sliding surface as the second n −1 order system as: d

σ (x, t) = ( + α) e=0   (e  dt e)  ZERO

(13)

(14)

is the slope of the[30]. fuzzyTo sliding surface σ= . the fuzzy sliding surface as the second where αwhere is a positive constant simply the system, set The order system as:fuzzy sliding surface σ is divided by Φ before fuzzification, and Φ is the boundary layer of . the fuzzy sliding surface σ. ( ) =σNB, PM, PB is the membership function set of = (NM, e +NS, α eZR, ) =PS,ZERO (14) the fuzzy sliding surface σ, where NB, NM, NS, ZR, PS, PM and PB are negative big, negative

negative positive small, where αmedium, is the slope of thesmall, fuzzyzero, sliding surface σ = positive ZERO. medium and positive big. ( ) = NB, NM, NS, ZR, PS, PM, PB is the membership function set of control input . and Φ is the boundary The fuzzy sliding surface σ is divided by Φ before fuzzification, In the conventional fuzzy control, control error and error rate need to build a 7 × 7 fuzzy layer of the fuzzy sliding surface σ. M (σ ) = {NB, NM, NS, ZR, PS, PM, PB} is the membership rules if each variable is divided by seven fuzzy rules. Only seven fuzzy rules are necessary for fuzzy sliding mode control. The simplified fuzzy rule base is demonstrated in Table 2.

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function set of the fuzzy sliding surface σ, where NB, NM, NS, ZR, PS, PM and PB are negative big, negative medium, negative small, zero, positive small, positive medium and positive big. M (u) = {NB, NM, NS, ZR, PS, PM, PB} is the membership function set of control input u f . . In the conventional fuzzy control, control error e and error rate e need to build a 7 × 7 fuzzy rules if each variable is divided by seven fuzzy rules. Only seven fuzzy rules are necessary for fuzzy sliding mode control. The simplified fuzzy rule base is demonstrated in Table 2. Table 2. Fuzzy Rules. Rules 1

Sensors 2016, 16, 1368

R R2 R3 R4 R5 R6 R7

IF

THEN

σ is PB σ is PM σ is PS σ is ZR σ is NS σ is NM σ is NB

u is PB u is PM u is PS u is ZR u is NS u is NM u is NB

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Table 2. Fuzzy Rules. IF THEN The Mamdani method is used forRules fuzzy inference and the is PB is PB R for defuzzification: R is PM is PM R µ ( u ) · uPS· du is PS is R u f =RGu ( x, t)is·ZR R is ZR µ(uis) NS · du is NS R is NM is NM R is NB R where µ (u) indicates the membership function ofis NB u.

center-of-area defuzzifier is used (15)

However, the control signal of FSMC will approach zero if the position control error approaches The Mamdani method is used for fuzzy inference and the center-of-area defuzzifier is used for zero in the steady state. defuzzification: In addition, the MSM actuator will become zero position output. MFSMC  (u )  u  du is proposed to integrate additional integrator toFSMC [10]. Therefore, the MSM actuator can keep u f  Gu ( x, t )  (15) (u )  du the desired position, since the control signal can be preserved in a steady output value as the position where μ( ) indicates membership error approaching zero in the the steady state.function of . However, the control signal of FSMC will approach zero if the position control error approaches zero in the steady state. In addition, the MSM actuator will become zero position output. MFSMC is 4.3. Novel Controller Design by Combining Path Controller and Hysteresis Compensator proposed to integrate additional integrator to FSMC [10]. Therefore, the MSM actuator can keep the

desired position, since the control signal can be preserved in a steady output value as the position The control strategy combines the inverse Preisach model as hysteresis compensator and path error approaching zero in the steady state. controller, which use PID controller or MFSMC. According to Section 3, the MSM actuator’s hysteresis Novel Design Combining Path Controller and Hysteresis Compensatorthe hysteresis compensator phenomenon4.3. can be Controller linearized bybythe hysteresis compensator. However, control strategy the error inverse based Preisachcontrol. model as hysteresis compensator anderrors path is a feedforwardThe control, whichcombines is not an Negative feedback can adjust controller, which use PID controller or MFSMC. According to Section 3, the MSM actuator’s tracking through path controller, i.e., PID controller and MFSMC in the experiments. Path controller’s hysteresis phenomenon can be linearized by the hysteresis compensator. However, the hysteresis performance compensator had beenistested in thecontrol, tracking [10]. However, the obvious delay a feedforward which path is not an error based control. Negative feedback errors of the MSM can adjust tracking through path controller, i.e., PID controller and MFSMC in the experiments. Path actuator was induced during the change of direction by the friction of the system. To enable a good controller’s performance had been tested in the tracking path [10]. However, the obvious delay of the path trackingMSM in this state is difficult for path controller. This research tackles this problem with the actuator was induced during the change of direction by the friction of the system. To enable a novel controller, which is able to reduce the influence of friction. To verify the performance of MFSMC, good path tracking in this state is difficult for path controller. This research tackles this problem with the novel controller, which is able to reduce the influence of friction. To verify the performance of PID controller is used for comparison. The control block diagram of the novel controller with the MFSMC, PID controller is used for comparison. The control block diagram of the novel controller combined path controller and the hysteresis compensator is shown in Figure 10. with the combined path controller and the hysteresis compensator is shown in Figure 10.

Block diagram of novel controller design by combining path controller and the hysteresis Figure 10. Figure Block10.diagram of novel controller design by combining path controller and the compensator. hysteresis compensator.

5. Experiments In this section, path tracking control of the MSM actuator is implemented. First, we test the performance of the forward controller, i.e., the hysteresis compensator. Then, path controllers, such

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5. Experiments In this Sensors 2016,section, 16, 1368 path tracking control of the MSM actuator is implemented. First, we11test of 20the performance of the forward controller, i.e., the hysteresis compensator. Then, path controllers, such as PID control and MFSMC, are also tested separately. Finally, novel controller combinedwith withthe PIDascontrol and MFSMC, are also tested separately. Finally, thethe novel controller combined the hysteresis compensator and controller path controller is implemented. hysteresis compensator and path is implemented. Hysteresis Compensatorbybythe theInverse InversePreisach Preisach Model Model 5.1.5.1. Hysteresis Compensator In the following experiments, the pathtracking trackingreference referenceisisaasinusoidal sinusoidal wave wave with with amplitude amplitude of In the following experiments, the path of 81.5 μm and frequency of 0.2 Hz. As mentioned previously the MSM actuator is sensitive to 81.5 µm and frequency of 0.2 Hz. As mentioned previously the MSM actuator is sensitive to disturbance ◦ disturbance so that theistemperature keptCaround °Cexperiment. during the experiment. The experimental so that the temperature kept aroundis 35 during35 the The experimental results with results with only the hysteresis compensator are shown in Figure 11. Figure 11a shows the path only the hysteresis compensator are shown in Figure 11. Figure 11a shows the path tracking control tracking control response. Nonlinearity of the MSM actuator can be observed clearly in Figure 11b, response. Nonlinearity of the MSM actuator can be observed clearly in Figure 11b, and rapid falling and rapid falling of input signal is required to return the position back to its original state with good of input signal is required to return the position back to its original state with good response. The response. The control error, caused mainly by friction force and temperature variation, can be control error, caused mainly by friction force and temperature variation, can be maintained within maintained within 10 μm as shown in Figure 11c, especially by the change of moving direction. 10 µm as shown in Figure 11c, especially by the change of moving direction. Besides, temperature Besides, temperature changes during the experiment should be another reason causing fluctuations. changes during the experiment be another reason fluctuations. magnetic circuit is The magnetic circuit is driven should by electric current such thatcausing the MSM alloys tend The to have temperature driven by electric current such that the MSM alloys tend to have temperature change. However, change. However, the hysteresis compensator can linearize the MSM actuator with limited effect, asthe hysteresis compensator shown in Figure 11d. can linearize the MSM actuator with limited effect, as shown in Figure 11d. (a) 120

reference experiment

100 80

3 voltage (V)

position (  m)

(b) 4

60 40

1

20 0 0

2

5

10 time (s)

15

0 0

20

(c)

15

20

experimental position ( m)

100

10 error (  m)

10 time (s) (d)

15

5 0 -5 -10 0

5

5

10 time (s)

15

20

Figure Pathtracking trackingofofsinusoidal sinusoidal wave wave by by Figure 11. 11.Path responses; (b) control signals; (c) control errors; responses; (b) control signals; (c) control errors; experimental positions. experimental positions.

80 60 40 20 0 -20 0

the the (d) (d)

20 40 60 80 reference position (  m)

100

hysteresis compensator: tracking hysteresis compensator:(a)(a)path path tracking linear relationship between reference and

linear relationship between reference and

5.2. Path Tracking Control without the Hysteresis Compensator

5.2. Path Tracking Control without the Hysteresis Compensator

This section shows the experimental results of the path tracking control with two different path

This section the experimental results the path tracking control with twoand different path controllers, i.e.,shows PID controller and MFSMC. The of experimental results of PID controller MFSMC controllers, i.e., PID controller and MFSMC. The experimental results of PID controller and MFSMC are shown in Figures 12 and 13, respectively. Friction during direction change causes the maximumare shown inespecially Figures 12inand respectively. Friction during directionAs change causes the maximum error, error, the 13, transition from elongation to contraction. the MSM actuator has serious especially in the fromcannot elongation contraction. Aswell. the MSM has hysteresis, hysteresis, thetransition PID controller tackletothis nonlinearity It canactuator be noticed inserious Figure 12a that results in delay of the controllerwell. during the be transition MSM from thehysteresis PID controller cannot tackle thisPID nonlinearity It can noticedof in the Figure 12aactuator that hysteresis elongation to contraction which is due to the falling rates of input signals are not fast enough in PID controller. MFSMC has better performance at the transition period as shown in Figure 13a, which is attributed to the fast switching in the sliding surface of MFSMC. The maximum error at the transition

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results in delay of the PID controller during the transition of the MSM actuator from elongation to contraction which is due to the falling rates of input signals are not fast enough in PID controller. MFSMC better performance at the transition period as shown in Figure 13a, which is attributed to Sensorshas 2016, 16, 1368 1368 12 of of 20 20 Sensors 2016, 16, 12 the fast switching in the sliding surface of MFSMC. The maximum error at the transition is around is and around μm, and during the period of ofand elongation and contraction, contraction, thebeerror error can be be maintained maintained 2 µm, during the period of the elongation contraction, the error can maintained under 500 nm is around 22 μm, and during period elongation and the can under 500 nm as shown in Figure 13c. under in 500Figure nm as 13c. shown in Figure 13c. as shown (a) (a) reference reference experiment experiment

100 100 80 80

voltage voltage(V) (V)

position m) position( (  m)

120 120

60 60 40 40 20 20 0 0 0 0

5 5

10 10 time (s) time (s)

15 15

experimental experimentalposition position((m) m)

error error( (m) m)

5 5 0 0 -5 -5 -10 -10 0 0

5 5

10 10 time (s) time (s)

15 15

3 3 2 2 1 1 0 00 0

20 20

(c) (c)

10 10

(b) (b)

4 4

5 5

10 10 time (s) time (s)

15 15

20 20

(d) (d)

100 100 80 80 60 60 40 40 20 20 0 00 0

20 20

20 40 60 80 20 40 60 80 reference position (  m) reference position (  m)

100 100

Figure 12. 12. Path Path tracking tracking of of sinusoidal sinusoidal wave wave with with PID PID controller: controller: (a) (a) path path tracking tracking responses; responses; (b) (b) Figure

Figure 12. Path tracking of sinusoidal with PID controller: (a) pathreference tracking and responses; (b) control control signals; (c) control control errors; wave (d) linear linear relationship between experimental control signals; (c) errors; (d) relationship between reference and experimental signals; (c) control errors; (d) linear relationship between reference and experimental positions. positions. positions.

(a) (a) reference reference experiment experiment

100 100

voltage voltage(V) (V)

position position( ( m)  m)

120 120

80 80 60 60 40 40 20 20 0 0 0 0

5 5

10 10 time (s) time (s)

15 15

experimental experimentalposition position((m)  m)

error m) error( (  m)

5 5 0 0 -5 -5 -10 -10 0 0

5 5

10 10 time (s) time (s)

15 15

20 20

3 3 2 2 1 1 0 00 0

20 20

(c) (c)

10 10

(b) (b)

4 4

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10 10 time (s) time (s)

15 15

20 20

(d) (d)

80 80 60 60 40 40 20 20 0 0 -20 -20 0 0

20 40 60 80 20 40 60 80 reference position ( m) reference position ( m)

100 100

Figure 13. Path tracking of sinusoidal wave with MFSMC: (a) path tracking responses; (b) control

Figure Path tracking sinusoidal wave wave with with MFSMC: tracking responses; (b) control Figure 13. 13. Path tracking ofofsinusoidal MFSMC:(a)(a)path path tracking responses; (b) control signals; (c) (c) control control errors; errors; (d) (d) linear linear relationship relationship between between reference reference and and experimental experimental positions. positions. signals; signals; (c) control errors; (d) linear relationship between reference and experimental positions.

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In comparison with the maximum error by using PID controller of about 7 µm in Figure 12c, it is evident that the MFSMC has better tracking performance than the PID control. Besides, compared the experimental results with that of the hysteresis compensator in Section 5.1, the hysteresis compensator serves as forward control so that it cannot receive the errors to modify the tracking performance. Therefore, the tracking accuracy of the hysteresis compensator in Section 5.1 is limited. 5.3. Novel Controller by Combination of Path Controller with the Hysteresis Compensator In this section, control strategy experiments are performed by combining the hysteresis compensator and path controller. In order to analyze the path tracking results in an experiment, Sensorscontroller 2016, 16, 1368 of 20 path is applied first, and then the hysteresis compensator is added, i.e., path controller13with the hysteresis compensator, at the time point of 20 s. The experimental results of PID path controller In comparison with the maximum error by using PID controller of about 7 μm in Figure 12c, it with the hysteresis compensator are shown in Figure 14; the experimental results of MFSMC path is evident that the MFSMC has better tracking performance than the PID control. Besides, compared controller with the hysteresis compensator are shown in Figure 15. the experimental results with that of the hysteresis compensator in Section 5.1, the hysteresis The path tracking responses are shown in Figures 14a and 15a. In the first 20 s, only path compensator serves as forward control so that it cannot receive the errors to modify the tracking controller works for tracking control, and the hysteresis compensator is added after 20 s for achieving performance. Therefore, the tracking accuracy of the hysteresis compensator in Section 5.1 is limited. the combination of path control and the hysteresis compensator. The hysteresis compensator becomes the major control signals, and path controllers turn out to be the modifying signal for correcting 5.3. Novel Controller by Combination of Path Controller with the Hysteresis Compensator control signals as shown in Figures 14b and 15b. In both experimental results, it can be seen that this section, control experiments are performed by combining the hysteresis path In controller combining withstrategy the hysteresis compensator provides better control performance. In compensator and path controller. In order to analyze the path tracking results in an experiment, path Figure 14c, it can be seen that total errors can be reduced from 7 µm to 2 µm through the PID controller with is applied first, andcompensator. then the hysteresis compensator is added, i.e.,be path controller controller the hysteresis Furthermore, the total errors can reduced fromwith 2 µmthe to hysteresis compensator, at thepath timecontroller point of 20 s. The results of PID path controller 500 nm through the MFSMC with theexperimental hysteresis compensator as shown in Figurewith 15c. the linear hysteresis compensator are in and Figure 14; the experimental results of MFSMC path The relationship between theshown reference experimental position can be improved obviously as controller with the14d hysteresis are shown in Figure 15. with the hysteresis compensator. shown in Figures and 15dcompensator by the combination of path controller (a)

(b)

120 reference experiment

3

80 60 40

2 1 0

20 0 0

PID controller Compensator

4

voltage (V)

position (  m)

100

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time (s)

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time (s)

PID controller PID controller + Compensator

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80

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reference position (  m)

Figure 14. 14. Comparison PID path path control control with with and and without without the the hysteresis hysteresis compensator: compensator: (a) (a) path path Figure Comparison of of PID tracking responses; and thethe hysteresis compensator; (c) control errors; (d) tracking responses; (b) (b)signals signalsofofPID PIDcontroller controller and hysteresis compensator; (c) control errors; linear relationship between reference and experimental positions. (d) linear relationship between reference and experimental positions. (a)

(b)

120

80

reference experiment

4 3 (V)

(  m)

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2

MFSMC Compensator

-10 0

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(d) (  m) reference position

80

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experimental position (  m)

error (  m)

Figure 14. Comparison of PID path control with and without the hysteresis compensator: (a) path 80 (b) signals of PID controller and the hysteresis compensator; (c) control errors; (d) 5 responses; Sensorstracking 2016, 16, 1368 14 of 20 60 linear relationship between reference and experimental positions. (a)

120 reference experiment

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20 4

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2 -20 0

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1

Figure 15. Comparison of MFSMC path control with and0without the hysteresis compensator: (a) path 20 tracking responses; (b) signals of PID controller and the hysteresis compensator; (c) control errors; 0 -1 Sensors 2016, 16, 1368 14 of 20 0 10 20 30 and experimental 40 0 positions. 10 20 30 40 (d) linear relationship between reference time (s)

time (s)

(c)

(d)

error (  m)

experimental position (  m)

10 100 Figure Cont. The path tracking responses are shown in 15. Figures 14a and 15a. In the first 20 s, only path controller works for tracking control, and the hysteresis80compensator is added after 20 s for achieving the combination 5of path control and the hysteresis60 compensator. The hysteresis compensator becomes the major control signals, and path controllers turn out to be the modifying signal for 0 40 correcting control signals as shown in Figures 14b and 15b. In both experimental results, it can be 20 seen that path controller combining with the hysteresis compensator provides better control -5 0 can be reduced performance. In Figure 14c, it can be seen that total errors from 7 μm to 2 μm through MFSMC MFSMC + Compensator the PID path controller with the hysteresis compensator. Furthermore, the total errors can be reduced -10 -20 0 10 20 30 40 0 20 40 60 80 100 from 2 μm to 500 nm through the MFSMC path controller with reference the hysteresis time (s) position (  m) compensator as shown in Figure 15c. The linear relationship between the reference and experimental position can be Figure 15. of path with and without the (a) 15.Comparison Comparison ofMFSMC MFSMC pathcontrol control with andby without thehysteresis hysteresis compensator: compensator: (a) path path with the improvedFigure obviously as shown in Figures 14d and 15d the combination of path controller tracking responses; (b) signals of PID controller and the hysteresis compensator; (c) control tracking responses; (b) signals of PID controller and the hysteresis compensator; (c) control errors; errors; hysteresis compensator. (d) (d) linear linearrelationship relationshipbetween betweenreference referenceand andexperimental experimentalpositions. positions. Figure 16 shows the comparisons of control signals with and without the hysteresis compensator. According Figure 14b, control signals14a from to 5 In s asthe PID control and that The path trackingtoresponses are the shown in Figures and0 15a. first 20 s, only path from Figure 16 shows the comparisons of control signals with and without the hysteresis compensator. works forcontrol tracking control, and the hysteresis compensator is addedfor after 20 s for achieving 20 tocontroller 25 s as PID path with the hysteresis compensator are selected comparison and shown According to Figure 14b, the control signals from 0 to 5 s as PID control and that from 20 to 25 s as PID the combination of path controlwith and Figure the hysteresis The hysteresis compensator in Figure 16a. Similarly, consistent 15b the compensator. control signals from to 5 sin asFigure MFSMC path control with the hysteresis compensator are selected for comparison and 0shown 16a.path becomes the major control signals, and path controllers turn out to be the modifying signal for control and that fromwith 20 to 25 s15b asthe MFSMC path control hysteresis compensator Similarly, consistent Figure control signals from 0 towith 5 s asthe MFSMC path control and that are correcting control signals as shown in Figures 14b and 15b. In both experimental results, it can be compared from 20intoFigure 25 s as 16b. MFSMC path control with the hysteresis compensator are compared in Figure 16b.

voltage (V)

voltage (V)

seen that path controller combining with the hysteresis compensator provides better control performance. In Figure 14c, (a) it can be seen that total errors can be reduced from (b) 7 μm to 2 μm through 3 3.5 path controller with the hysteresis compensator. the PID Furthermore, the total errors can be reduced from 2 3μm to 500 nm through the MFSMC path controller with the hysteresis compensator as shown 2.5 in Figure 15c. The linear relationship between the reference and experimental position can be 2.5 improved obviously as shown in Figures 14d and 15d by 2 the combination of path controller with the hysteresis compensator. 2 Figure 16 shows the comparisons of control 1.5 signals with and without the hysteresis 1.5 compensator. According to Figure 14b, the control signals from 0 to 5 s as PID control and that from 1 20 to 251s as PID path control with the hysteresis compensator are selected for comparison and shown in Figure 16a. Similarly, consistent with Figure 15b the control signals from 0 to 5 s as MFSMC path 0.5 MFSMC 0.5 and that PID from controller control 20 to 25 s as MFSMC path control with the hysteresis compensator are MFSMC + compensator PID controller + compensator compared in Figure 16b. 0 0 0

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Figure 16. Control signal comparisons pathcontroller controller solely and using controller Figure 16. Control signal comparisonsbetween between using using path solely and using pathpath controller 3 2.5 withwith the hysteresis compensator: controllersolely solely and using controller the hysteresis compensator:(a) (a)Using UsingPID PID path path controller and using PIDPID pathpath controller withwith the 2.5 hysteresis compensator; path controller solely and using MFSMC the hysteresis compensator;(b) (b)Using Using MFSMC MFSMC path controller solely and using MFSMC path path 2 controller with the hysteresis compensator. controller 2with the hysteresis compensator. 1.5

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15 Due of 20 that the control signal is especially modified during the changing of direction. to the friction, only using path controller cannot adjust the output fast enough so as to cause delay and and errors. errors. By By combining combining path path controller controller with with the the hysteresis hysteresis compensator, compensator, it it can can be be noticed noticed that that the the signals has fast response to overcome the delay. Moreover, MFSMC with the hysteresis compensator signals has fast response to overcome the delay. Moreover, MFSMC with the hysteresis compensator can than PID PID controller controller with with the the hysteresis hysteresis compensator. compensator. can perform perform faster faster response response than

5.4. Novel Controller in Different Trajectories Control Trajectories Control We have demonstrated demonstrated the tracking tracking performance performance with with our our novel novel controller. controller. The hysteresis compensator is provided the main input signal for path tracking, and it compensator is provided the main input signal for path tracking, and it is is able able to give the proper response as path changing. In In addition, addition, the the path path controller controller is is able able to to enhance enhance tracking tracking accuracy. accuracy. More experiments to verify the performance of the novel controller follow. follow. 5.4.1. Decaying Sinusoidal Trajectory Control experiments of decaying decaying sinusoidal sinusoidal wave wave with with 0.2 0.2 Hz Hz are shown shown in Figures 17 and 18, and The experiments −0.1t . sin (0.4πt − 0.5π)) + 82. It is shown that the tracking the tracking reference is set as f = 41e set as r = 41 sin(0.4 0.5 + 82. It is shown that performance by using using the novel controller is better than by only using path controller in both performance experiments. The maximum error occurs at the changing of direction of maximum amplitude. As the amplitude decays during the time, the error is also reduced. In Figure 17b, it is shown the maximum µm with PID controller, and the maximum error is reduced to 1.36 μm µm by adding the error is 5.4 μm error is 1.96 µmμm with MFSMC, andand the hysteresis compensator. compensator.Likewise, Likewise,ininFigure Figure18b, 18b,the themaximum maximum error is 1.96 with MFSMC, totaltotal errors can be kept enough withinwithin 450 nm. It can how the how inputthe signals modified the errors can be small kept small enough 450 nm.be It seen can be seen inputis signals is by addingby the hysteresis compensator. modified adding the hysteresis compensator.

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Figure 17. Tracking Figure 17. Tracking control control comparisons comparisons of of decaying decaying sinusoidal sinusoidal trajectories trajectories with with 0.2 0.2 Hz Hz frequency frequency between using PID path controller solely and PID path controller with the hysteresis between using PID path controller solely and PID path controller with the hysteresis compensator: compensator: (a) tracking responses; responses; (b) (b) control control errors; errors; (c) (c) control control signals. signals. (a) path path tracking

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Figure 18. Tracking Tracking control comparisons comparisons of decaying sinusoidal trajectories with 0.2 Hz frequency Figure of decaying decayingsinusoidal sinusoidaltrajectories trajectorieswith with0.2 0.2Hz Hzfrequency frequency Figure18. 18. Tracking control control comparisons of between using MFSMC path controller solely and MFSMC path controller with the hysteresis between solely and and MFSMC MFSMC path pathcontroller controllerwith withthe thehysteresis hysteresis between using using MFSMC MFSMC path path controller controller solely compensator:(a) (a) path path tracking tracking responses; responses; (b) control errors; (c) control signals. compensator: (b) control errors; (c) control signals. compensator: (a) path tracking responses; (b) control errors; (c) control signals.

5.4.2. Sinusoidal Trajectory Control in Higher Frequency 5.4.2. Frequency 5.4.2.Sinusoidal Sinusoidal Trajectory Trajectory Control Control in Higher Frequency Sinusoidal wave with the amplitude of 50 μm and higher frequency at 1 Hz is tested and shown Sinusoidal 50 μm µm and andhigher higherfrequency frequencyatat1 1Hz Hzisistested testedand and shown Sinusoidalwave wave with with the the amplitude of 50 shown in Figures 19 and 20. ininFigures Figures19 19and and20. 20.

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Figure 19. Tracking control comparisons of sinusoidal trajectory with 1 Hz frequency and 50 μm Figure 19. Tracking control comparisons of sinusoidal trajectory with 1 Hz frequency and 50 μm Figure 19. Tracking comparisons of sinusoidal trajectory 1 Hz frequency and 50 µm amplitude between control using PID path controller solely and PID pathwith controller with the hysteresis amplitude between using PID path controller solely and PID path controller with the hysteresis amplitude between using PID responses; path controller solely and (c) PID path signals. controller with the hysteresis compensator: (a) path tracking (b) control errors; control compensator: (a) path tracking responses; (b) control errors; (c) control signals. compensator: (a) path tracking responses; (b) control errors; (c) control signals.

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It is evident thatthat higher frequency errors.InInFigure Figure 19b, maximum It is evident higher frequencysignals signalshave have larger larger errors. 19b, thethe maximum errorerror increases to 7.6 µmwith withPID PID controller, and it can be seen controller response isresponse much delayed increases to 7.6 μm controller, and it can be that seenthethat the controller is much withwith higher frequency signals as shownas inshown Figure 19c. Adding19c. the hysteresis compensator cancompensator improve delayed higher frequency signals in Figure Adding the hysteresis the tracking performance, the maximum error is able to be reduced to less than 2 µm. Similarly can improve the tracking performance, the maximum error is able to be reduced to less than as 2 μm. shown Figure in 20b, 3 µm is the3maximum with MFSMC, and itMFSMC, can be reduced less Similarly asinshown Figure 20b, μm is theerror maximum error with and itfurther can betoreduced than 1.7 µm by adding the hysteresis compensator. further to less than 1.7 μm by adding the hysteresis compensator. 5.4.3. 5th Order Trajectory Control in Higher Frequency (a)

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In these experiments, 5th order trajectory is used as a smooth tracking path for the MSM actuator. 40 5th order trajectory has continuous function in position, velocity and acceleration. Therefore, it Reference is evident that the errors are 20 small in Figures 21 and 22. In Figure 21b, the maximum error is PID 1.87 µm, which is controlled by PID controller solely, and it hasComp+PID large friction because the speed of the 0 MSM actuator is slow. Besides, maximum error can be reduced 0 2 4 6 8 to 660 10nm by adding the hysteresis time (s) compensator, and the steady state error is 280 nm. For MFSMC, the error is much less than PID (b) controller with the hysteresis compensator. It is able to be reduced to less than 280 nm for maximum PID error, and the steady state error2 is able to reach to 20 nm, which is the resolution of position sensor. Comp+PID Adding the hysteresis compensator for MFSMC, 134 nm is the maximum error, and the steady state 0 error is able to be kept as small as 20 nm. -2

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It is evident that higher frequency signals have larger errors. In Figure 19b, the maximum error increases to 7.6 μm with PID controller, and it can be seen that the controller response is much delayed with higher frequency signals as shown in Figure 19c. Adding the hysteresis compensator can improve the tracking performance, the maximum error is able to be reduced to less than 2 μm. Similarly in Figure 20b, 3 μm is the maximum error with MFSMC, and it can be reduced Sensors 2016,as 16,shown 1368 18 of 20 further to less than 1.7 μm by adding the hysteresis compensator. (a) position (  m)

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In these experiments, 5th order trajectory is used as a smooth tracking path for the MSM 6 actuator. 5th order trajectory has continuous function inPIDposition, velocity and acceleration. 4 Comp+PID Therefore, it is evident that the errors are small in Figures 21 and 22. In Figure 21b, the maximum 2 error is 1.87 μm, which is controlled by PID controller solely, and it has large friction because the 0 -2 speed of the MSM actuator is slow. Besides, maximum error can be reduced to 660 nm by adding the 0 2 4 6 8 10 hysteresis compensator, and the steady state errortime is 280 (s) nm. For MFSMC, the error is much less than PID controller with the hysteresis compensator. It is able to be reduced to less than 280 nm for Figure 21. control comparisons of 5th trajectory between using PID path Figureerror, 21. Tracking Tracking 5th order order trajectory between path controller controller maximum and thecontrol steadycomparisons state error isofable to reach to 20 nm, whichusing is thePID resolution of position solely and PID path controller with the hysteresis compensator: (a) path tracking responses; (b) control solely and PID path controller with the hysteresis compensator: (a) path tracking responses; (b) control sensor. Adding the hysteresis compensator for MFSMC, 134 nm is the maximum error, and the steady errors; (c) control signals. control signals. state errors; error is(c)able to be kept as small as 20 nm. (a) position (  m)

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Figure 22. Tracking Trackingcontrol control comparisons of order 5th order trajectory between using MFSMC path Figure 22. comparisons of 5th trajectory between using MFSMC path controller controller solely and MFSMC path controller with the hysteresis compensator: (a) path tracking solely and MFSMC path controller with the hysteresis compensator: (a) path tracking responses; responses; control errors; signals. (c) control signals. (b) control (b) errors; (c) control

6. 6. Conclusions Conclusions The The MSM MSM actuator actuator inherits inherits the the advantageous advantageous properties properties of of huge huge strain strain and and fast fast response response like like MSM alloys, however, they also exhibit a hysteresis phenomenon. This paper aims to develop MSM alloys, however, they also exhibit a hysteresis phenomenon. This paper aims to develop aa novel novel controller by combining a path controller with a hysteresis compensator for path tracking control of MSM actuators. The Preisach model was used to model the hysteresis phenomenon. The hysteresis of the MSM actuator was modeled on a Preisach plane by collecting experimental data. The desired position could be achieved by applying the inverse Preisach model, which was also used as a hysteresis compensator. The novel controller combined the hysteresis compensator and path controller. The

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controller by combining a path controller with a hysteresis compensator for path tracking control of MSM actuators. The Preisach model was used to model the hysteresis phenomenon. The hysteresis of the MSM actuator was modeled on a Preisach plane by collecting experimental data. The desired position could be achieved by applying the inverse Preisach model, which was also used as a hysteresis compensator. The novel controller combined the hysteresis compensator and path controller. The path controllers, including PID controller and MFSMC, were implemented for comparison. Different controller trajectories were tested, including sinusoidal wave, decaying sinusoidal wave, sinusoidal wave in higher frequency and 5th order trajectory. It was shown that the MFSMC path control can perform better than PID controller in the path tracking control experiments. Besides, the novel controller by combining path controllers with the hysteresis compensator was able to improve tracking accuracy. The path tracking control experiments verified that the best performance was the MFSMC path controller with the hysteresis compensator. Acknowledgments: The research was sponsored in part by the Ministry of Science and Technology, Taiwan under the grant MOST 105-3113-E-002-015-CC2. Author Contributions: Mao-Hsiung Chiang conceived and designed the experiments; Jhih-Hong Lin performed the experiments and analyzed the data. Mao-Hsiung Chiang and Jhih-Hong Lin wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations MSM FSMC MFSMC

Magnetic Shape Memory Fuzzy Sliding Mode Control Modified Fuzzy Sliding Mode Control

References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12.

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