Modelling and Gain Scheduled Control of Shape Memory Alloy Actuators*

J.Jayender R.V.PatelDepartment of Electrical and Computer Engineering

University of Western OntarioLondon, ON N6A 5B9, Canada

jjagadee@uwo.ca, rajni@eng.uwo.ca

S.Nikumb M.OstojicNational Research Council of Canada

Integrated Manufacturing Technologies InstituteLondon, ON N6G 4X8, Canada

suwas.nikumb@nrc.gc.ca, mile.ostojic@nrc.gc.ca

Abstract— This paper describes a Gain-Scheduled controllerfor a Shape Memory alloy (SMA) actuator. For accurate controlof an SMA actuator, it is important to develop a precisemodel for the SMA. A model has been proposed based onconcepts from physics. The equations include Joules heating -convectional cooling to explain the dynamics of temperature,Fermi-Dirac statistics to explain the variation of mole fractionwith temperature, and a stress-strain constitutive equation torelate changes in mole fraction and temperature to changes instress and strain of the SMA. This model is then used to developa gain-scheduled controller to control the strain in the SMA.Simulation and experimental results show fast and accuratecontrol of the strain in the SMA actuator.

Index Terms— Shape Memory Alloy, Fermi-Dirac principle,gain-scheduled controller, LQR optmization

I. INTRODUCTION

There has been considerable interest of late in developingShape Memory Alloy (SMA) actuators because of theiradvantages in producing large plastic deformations, highpower-to-weight ratio and low driving voltages. These ad-vantages coupled with the fact that SMAs are biocompatiblemakes them good candidates for medical applications. SMAbased medical forceps and devices for Minimally InvasiveTherapy have been reported in the literature. The advantagesof low actuating voltages and high forces generated have alsoled to a greater use of SMA actuators for robotic applications[4],[8] and vibration control. The efficiency of an SMAactuator, however, depends on the accuracy of its control.

There have been a few papers on modelling and controlof SMA actuators. Most papers use a curve-fitting basedmethod to describe the behavior of the SMA and use thedynamic equations thus obtained to control it. A continuous-time model, developed by Arai et.al. [1], fits a differentialequation to experimental data. The model however does notguarantee a correct response over the entire operating regionand is also not based on the physics of the process.

Some papers, e.g., see [2],[3],[6], have used the Preisachmodel to model the hysteresis nature of the SMA. Though,they are able to model the hysteresis characteristic of theSMA and the minor loops in the hysteresis graph, the for-mulation is not convenient for developing a control strategy.

*This research was supported by the Natural Sciences and EngineeringResearch Council (NSERC) of Canada under grant RGPIN-1345 and by aninfrastructure grant from the Canada Foundation for Innovation.

Conventional PID control does not work well since theSMA demonstrates a nonlinear behaviour. Feedback lin-earization has been used to varying degrees of success[7],[8]. The problem with feedback linearization is that itdepends on accurate cancellation of the nonlinear compo-nents of the SMA response which in turn depends on a largenumber of parameters. In order to overcome these problems,variable structure control has been utilized since the controlstrategy is fairly insensitive to uncertainties in the parametersof the model [4],[5]. However, the problem with variablestructure control is that tracking is possible so long as theerror is small. The discontinuous nature of switching alsomay introduce ringing and high frequency components inthe system which are not desirable.

In this paper, we present a model of the SMA based on thelaws of physics. In order to model the phase transformationfrom Martensite to Austenite and vice versa, we use theFermi-Dirac principle from statistical physics to model thedynamics of the transformation. Modelling equations forthe temperature dynamics and the constitutive equationshave been used previously [4]. The temperature dynamics,mole fraction variation with temperature and the stress-strainconstitutive equation completely define the dynamics of theSMA and can be conveniently written in state-space form todevelop a control strategy.

II. THEORY AND MODELLING

The property of SMAs of demonstrating a change instrain by a change in temperature is due to a solid-statephase change from the Martensite form to the Austeniteform. Martensite is relatively softer and easily deformableand exists at a lower temperature. Heating by means ofcurrent or hot water results in a phase transition from theMartensite phase to the Austenite phase. In the Austenitephase, the SMA is more structured and is harder andstronger than in the Martensite phase.

A. Modelling of Phase Transformation

Since an SMA exists in two states, it can be modelled as atwo-state system. We draw a similarity between this systemand an electron, which can exist only in two states - thepositive spin or the negative spin. The Fermi-Dirac statisticsare used to describe the number of electrons in the two states

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

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depending on the energy of the electron. We have modelledthe state of an SMA in the Martensite and Austenite formsusing the same statistics.We use two modelling equations based on whether the alloyis being heated or cooled due to hysteresis with two differenttransition temperatures. Since the SMA is in the Martensiteform at lower temperatures, the phase transformation equa-tion during heating is described by analogy with the Fermi-Dirac statistics in the form:

ξ =ξm

1 + exp(Tfa−Tσa

+ Kaσ)(1)

where ξ is the fraction of the Austenite phase, ξm is thefraction of SMA in Martensite phase prior to the presenttransformation from Martensite to Austenite, T is the tem-perature, Tfa

is the transition temperature from Martensiteto Austenite, σa is an indication of the range of temperaturearound the transition temperature Tfa

during which thephase change occurs, σ is the stress and Ka is the stresscurve-fitting parameter which is obtained from the loadingplateau of the stress-strain characteristic with no change intemperature.On cooling, the Austenite phase gets converted to theMartensite phase and the modelling equation during coolingis described by analogy with the Fermi-Dirac statistics in theform:

ξ =ξa

1 + exp(Tfm−Tσm

+ Kmσ)(2)

where ξa is the fraction of SMA in Austenite phase priorto the present transformation from Austenite to Martensite,T is the temperature, Tfm

is the transition temperature fromAustenite to Martensite, σm is an indication of the rangeof temperature around the transition temperature Tfm

duringwhich the phase change occurs, σ is the stress and Km isthe stress curve-fitting parameter which is obtained from theunloading part of the stress-strain characteristic.Since the SMA is modelled as a two-component system, atany given time, the sum of the mole fractions of the Austeniteand Martensite phase is 1, i.e.,

ξa + ξm = 1 (3)

The time derivatives of Eqns.(1) and (2) are as follows:For heating:

ξ =ξ2

ξm[exp(

Tfa− T

σa+ Kaσ)][

T

σa− Kaσ] (4)

For cooling:

ξ =ξ2

ξa[exp(

Tfm− T

σm+ Kmσ)][

T

σm− Kmσ] (5)

The above equations model the phase transformations atthe microscopic level and therefore accurately describe thebehaviour of the SMA.

B. Modelling of Temperature Dynamics

An SMA actuator is heated by the process of Joules heat-ing by applying a voltage across the SMA. The loss of heatfrom the SMA is through natural convection. Mathematicallythe dynamics of the temperature are given by equation (6)and have also been used in [4].

T =1

mcp[V 2

R− hA(T − Ta)] (6)

where m is the mass per unit length, cp is the specific heatcapacity, V is the voltage applied across the SMA, R is theresistance per unit length, h is the coefficient of convection,A = πd is the circumferential area of cooling, d is thediameter of the wire, T is the temperature and Ta is theambient temperature. The coefficient h is assumed to havethe characteristics of a second-order polynomial to enhancethe rate of convection at higher temperatures as observed inopen-loop results:

h = h0 + h2T2 (7)

C. Constitutive Equation

The constitutive equation relating changes in stress, strain,temperature and mole fraction is given by the followingequation and has been previously used in [4].

σ = Dε + θtT + Ωξ (8)

where σ is the stress in the SMA, D is the average of theYoung’s Moduli for the Martensite and Austenite phases, ε isthe strain, θt is the thermal expansion factor and Ω = −Dε0is the phase transformation contribution factor. The modelexplains the hysteresis as well as the minor loops in thehysteresis, as illustrated in Figure 1.

Fig. 1. Hysteresis Characteristics of SMA for varying sinusoidal input

III. EXPERIMENTAL SETUP

In the experimental setup, the SMA wire is attachedbetween a fixed support and a motor which provides a

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Fig. 2. Experimental setup

constant tension to keep the wire taut at all times. Thereadings from the encoder of motor give the change in strainin the SMA wire which is fed to a computer through anencoder card. The real-time implementation of the gain-scheduled controller was done in C++ on a Windows basedPC at a sampling rate of about 65Hz. The setup is shown inFigure 2.

IV. NO-LOAD SIMULATION OF THE MODEL

No-load simulation was done in MATLAB for the SMAactuator based on the above modelling equations. Since noload was assumed, σ and σ are equal to zero. A sinusoidalinput voltage V was applied to the SMA model and theresulting strain vs. temperature was plotted to demonstratethe hysteresis nature of SMA, as shown in Figure 1.

The simulation results were also matched with open-loopresults for a sinusoidal input to obtain the parameters of themodel, as shown in Figure 3 and Figure 4. The parametersused for modelling the SMA are given in Table 1.

Fig. 3. Comparison of simulation and open-loop results for a sinusoidalcurrent input

Fig. 4. Comparison of simulation and open-loop results for a sinusoidalcurrent input - time response

V. GAIN-SCHEDULED CONTROLLER

Equation (4) or (5) (heating or cooling), together withequations (6) and (8) completely define the dynamic charac-teristics of the SMA. We can also define σe as the integralof the error, i.e.,

σe = ε − εref (9)

where ε is the strain of the SMA actuator and εref is thereference trajectory. The dynamic equations of the SMAalong with equation (9) can be represented in the state-spaceform:

˙z = f(z, u, t) (10)

where

z =

⎡⎢⎢⎣

εTξσe

⎤⎥⎥⎦

and u is the input voltage to the SMA wire. The nonlinearequations are linearized about a set of operating points(ε0, T0, ξ0, u0) on the reference trajectory.

To obtain the operating points, ε0 is chosen as the valueof the reference strain at that instant of time. T0 and ξ0

are obtained by integrating equations (4) or (5) and (8),depending on whether the SMA is being heated or cooled,for a given value of ε0. The value of u0 is obtained fromequation (6) for a given value of T0, assuming steady stateconditions.

Fig. 5. Gain-scheduled controller for the SMA actuator

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The non-linear system represented by equation (10) islinearized about the calculated operating points and eachlinearized model is assumed to be time-invariant betweenthe operating points. A gain-scheduled controller is designedat each of the operating points to stabilize the linear, time-invariant models. The linear model is represented as

˙z = Az + Bu (11)

y = Cz (12)

where

A =[∂f

∂z

]

ε0,T0,ξ0,u0

B =[∂f

∂u

]

ε0,T0,ξ0,u0

For the no-load case, which is the case considered here, σand σ are equal to zero. In this case, the model given by (11)is not controllable since the number of controllable states isonly 2. Physically, for a given input current, the SMA reachesa desired temperature resulting in a phase transformationwhich causes a change in strain as given by the constitutiveequation. The three states (temperature, mole fraction, strain)cannot be independently controlled to a desired value. Onremoving the uncontrollable modes, the following state spacemodel is obtained:

˙x = A′x + B′u (13)

where

x =[

εσe

]

For a PI controller, the feedback is of the form

u = −KP (ε − εref ) − KIσe + u0 (14)

where KP is the proportional gain and KI is the integralgain. Writing

K = [Kp KI ]

the gains are computed such that the resulting controllerminimizes the quadratic cost function,

J(u) =∫ ∞

0

(xT Qx + uT Ru) dt (15)

The choice of gains for minimizing J(u) are obtained by firstsolving for S in the algebraic Riccati equation:

A′T S + SA′ − SB′R−1B′T S + Q = 0 (16)

The matrix K is given by:

K = R−1B′T S (17)

VI. SIMULATION AND EXPERIMENTAL RESULTS FOR THE

CONTROLLER

The simulation of the controller and the SMA was doneusing MATLAB. The values of Q and R were chosen as:

Q =[

7.2e6 00 500

]R = 0.003

The simulation result for a step reference input is shown inFigure 6 and that for a sinusoidal input is shown in Figure7.

Fig. 6. Simulation result: Closed loop SMA response to a step referenceinput

Fig. 7. Simulation result: Closed loop SMA response to a sinusoidalreference input

The strain of the SMA actuator, however, gets distortedwhen the reference changes too fast, as the rate of cooling isslow compared to the rate of heating and cannot be controlledto ensure a faster rate of cooling.

The experimental verification of the controller was doneon a 700MHz Windows based PC at a sampling rate ofabout 65Hz. Since the form of matrices A′ and B′ are fairlysimple, the solution for the Ricatti equation was obtainedin closed form using MAPLE. This also greatly reduced thecomputation time by avoiding matrix decompositions like theSchur decomposition [9]. The response of the SMA to a stepinput is shown in Figure 8.

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Fig. 8. Experimental result: SMA response to a step reference input

From the experimental results, it can be observed that theresponse time for heating is approximately 1.0 second andfor cooling is 2.1 seconds for a 0.012” diameter SMA wire.The rate of heating and cooling are much higher for a thinnerwire since the ratio of surface area to volume increases asthe wire diameter is reduced, thereby increasing the rate ofconvectional cooling.

The experimental result for a sinusoidal reference inputis shown in Figure 9 which shows an excellent tracking ofthe reference trajectory by the SMA in closed loop. The

Fig. 9. Experimental result: SMA response to a sinusoidal reference input

DC offset of the reference was also varied to check theperformance of the controller in the entire operating range.The results are shown in Figure 10. The graphs show anexcellent response over the entire operating region.

The frequency for the reference input was also varied tocheck the performance of the controller and the SMA. Theresults are shown in Figure 11. The response of the controlleris satisfactory for frequencies lower than 0.1Hz. However, theresponse of the SMA deteriorates if the reference input variesrapidly. A sinusoidal reference input of frequency 0.2Hz waschosen and the closed loop response of the SMA is shown

Fig. 10. Experimental results: Closed-loop responses for two sinusoidalreference inputs with different DC offsets

Fig. 11. Experimental results: Closed-loop responses for two sinusoidalreference inputs with different frequencies

in Figure 12.

VII. CONCLUSION

In this paper, we have shown the performance of anLQR optimization based PI controller for an SMA actuator.The experimental open-loop and closed-loop results matchclosely with the simulation results, thereby validating themodelling equations used in the simulations and controllerdesign. This clearly justifies the use of the model fordescribing the transformation between the Martensite andAustenite phases. The high accuracy achieved by the con-troller makes it possible to use SMA actuators safely forcritical applications such as in biomedical devices and tools.Further improvement in performance is possible if a thinnerSMA wire is used to enhance the rate of cooling. Anotherpossibility is to use narrow hysteresis SMA, since the amountof cooling required to transform from the Austenite phase toMartensite phase would be much lower.

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Fig. 12. Experimental results: Closed loop response for a 0.2Hz sinusoidalreference input

Parameter ValueMass per unit length (m in kg.m−1) 4.54e−4

Specific heat capacity (cp in J.kg−1.K−1) 320Resistance per unit length (R in Ω.m−1) 8.0677

Youngs Modulus(Austenite) (Da in N.m−2) 75e9

Youngs Modulus(Martensite) (Dm in N.m−2) 28e9

Thermal Expansion (θt in N.m−2.K−1) -11e−6

SMA initial strain (ε0) 0.03090Heat convection coefficient (h0 in J.m−2.s−1.K−1) 1.552Heat convection coefficient (h2 in J.m−2.s−1.K−3) 4.060e−4

Diameter of wire (in m) 304e−6

Length of wire (in m) 0.24Ambient temperature (Ta in oC) 20

Martensite to Austenitetransformation temperature (Tf a in oC) 85

Austenite to Martensitetransformation temperature (Tf m in oC) 42

Spread of temperature around Tf a (σa in oC) 6Spread of temperature around Tf m (σm in oC) 3.5

TABLE I

PARAMETERS OF THE SMA

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