A MORPHING WING USED SHAPE MEMORY ALLOY

ACTUATORS NEW CONTROL TECHNIQUE WITH

BI-POSITIONAL AND PI LAWS OPTIMUM COMBINATION

Part 1: Design Phase

Teodor Lucian Grigorie, Andrei Vladimir Popov, Ruxandra Mihaela Botez

École de Technologie Supérieure, Montréal, Québec H3C 1K3, Canada

Mahmoud Mamou, Youssef Mébarki

National Research Council, Ottawa, Ontario K1A 0R6, Canada

Keywords: Morphing Wing, Shape Memory Alloy Actuators, Bi-positional and PI Control Design, Numerical

Simulations.

Abstract: The paper presents the design phase of the actuators control system development for a morphing wing

application. Some smart materials, like Shape Memory Alloy (SMA), are used as actuators to modify the

upper surface of the wing made of a flexible skin. The actuations lines control is designed and validated

using a numerical simulation model developed in Matlab/Simulink. The finally adopted control law is a

combination of a bi-positional law and a PI law; the control must behave like a switch between cooling

phase and heating phase, situations where the output current is 0 A, or is controlled by a law of PI type. The

PI controller, for the heating phase, is optimally tuned using the Ziegler-Nichols criterion and the linear

model obtained using the System Identification Toolbox of Matlab. The controlled linearized system for

heating phase is numerically tested in terms of time response, stability, controllability and the observability.

In the actuation control design final phase, numerical simulations, based on SMA non-linear analytical

model, were used for validation.

1 INTRODUCTION

Many researches are made around the world in the

new challenge field related to the morphing aircraft,

with the purpose to improve operational efficiency,

particularly by reducing fuel consumption (Chang,

2009, Smith, 2007, Hinshaw, 2009, Gonzalez, 2005,

Namgoong, 2006, Majji, 2007, and Ruotsalainen,

2009). Therefore, a lot of architecture were and are

still imagined, designed, studied and developed, for

this new concept application. One of these is our

team project including the numerical simulations

and experimental multidisciplinary studies using the

wind tunnel for a morphing wing equipped with a

flexible skin, smart material actuators and pressure

sensors. The aim of these studies is to develop an

automatic system that, based on the information

related to the pressure distribution along the wing

chord, moves the transition point from the laminar to

the turbulent regime closer to the trailing edge in

order to obtain a larger laminar flow region, and, as

a consequence, a drag reduction.

The objective of here presented research work

was to develop an actuation control concept for a

new morphing mechanism using smart materials,

like Shape Memory Alloy (SMA), as actuators.

These actuators modify the flexible upper surface of

the wing, changing the airfoil shape. The morphing

wing project was developed by Ecole de

Technologie Supérieure in Montréal, Canada, in

collaboration with Ecole Polytechnique in Montreal

and the Institute for Aerospace Research at the

National Research Council Canada (IAR-NRC).

To achieve the aerodynamic imposed purpose, a

first phase of the studies involved the determination

of some optimized airfoils available for 35 different

flow conditions (five Mach numbers and seven

angles of attack combinations). The optimized

Lucian Grigorie T., Popov A., Mihaela Botez R., Mamou M. and Mébarki Y. (2010).

A MORPHING WING USED SHAPE MEMORY ALLOY ACTUATORS NEW CONTROL TECHNIQUE WITH BI-POSITIONAL AND PI LAWS OPTIMUM

COMBINATION - Part 1: Design Phase.

In Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, pages 5-12

DOI: 10.5220/0002878400050012

 SciTePress

airfoils were derived from a laminar WTEA-TE1

reference airfoil (Khalid, 1993, and Khalid, 1993),

and were used as a starting point for the actuation

system design. The transition point position

estimation is made using the information received

from a pressure system sensors (optical and Kulite

types) equipping the upper face of the wing. Two

architectures were developed for morphing system:

open loop and closed loop. The difference between

the two architectures is given by using or not using

the position of transition point as a feedback signal

for the actuation lines control. Here described work

was developed in the open loop phase; in this phase

were made numerical and experimental studies

related to the aerodynamics of the morphed wing, to

the flexible skin realization, to the actuation system,

to the control of the actuation system, and, also, to

the real-time determination and visualization of the

transition point position using the pressure sensor

system. Here, the pressure sensors using is limited to

the monitoring of the pressure distribution and of the

RMS pressure distribution in the boundary layer.

2 ARCHITECTURE OF THE

CONTROLLED MORPHING

WING SYSTEM

The chosen wing model was a rectangular one, with

a reference airfoil WTEA-TE1, a chord of 0.5 m and

a span of 0.9 m. The model was equipped with a

flexible skin made of composite materials (layers of

carbon and Kevlar fibers in a resin matrix) morphed

by two actuation lines (Fig. 1). Each actuation line

uses SMA wires as actuators. In the same time, 32

pressure sensors (16 optical sensors and 16 Kulite

sensors), were disposed on the flexible skin in

different positions along of the chord. The sensors

are positioned on two diagonal lines at an angle of

15 degrees from centerline. The rigid lower structure

was made from Aluminum, and was designed to

allow space for the actuation system and wiring.

Actuation

lines

Cavities for

instrumentation

Flexible skin

(morphed extrados)

Rigid

intrados

Rigid part of

the extrados

Support plate for

actuation system

Leading

edge

Trailing

edge

Figure 1: General architecture of the mechanical model.

Starting from the reference airfoil, depending on

different flow conditions, 35 optimized airfoils were

calculated for the desired morphed positions of the

airfoil. The flow conditions were established as

combinations of seven incidence angles (-1˚, -0.5˚,

0˚, 0.5˚, 1˚, 1.5˚, 2˚) and five Mach numbers (0.2,

0.225, 0.25, 0.275, 0.3). Each of the calculated

optimized airfoils must be able to keep the transition

point as much as possible near the trailing edge.

The SMA actuator wires are made of nickel-

titanium, and contract like muscles when electrically

driven. Also, these have the ability to personalize the

association of deflections with the applied forces,

providing in this way a variety of shapes and sizes

extremely useful to achieve actuation system goals.

How the SMA wires provide high forces with the

price of small strains, to achieve the right balance

between the forces and the deformations, required

by the actuation system, a compromise must be

established. Therefore, the structural components of

the actuation system must be designed to respect the

capabilities of actuators to accommodate the

required deflections and forces.

Each of our actuation lines uses three shape

memory alloys wires (1.8 m in length) as actuators,

and contains a cam, which moves in translation

relative to the structure (on the x-axis in Fig. 2). The

cam causes the movement of a rod related on the

roller and on the skin (on the z-axis). The recall used

is a gas spring. So, when the SMA is heating the

actuator contracts and the cam moves to the right,

resulting in the rise of the roller and the

displacement of the skin upwards. In contrast, the

cooling of the SMA results in a movement of the

cam to the left, and thus a movement of the skin

down. The horizontal displacement of each actuator

is converted into a vertical displacement at a rate 3:1

(results a cam factor c

=1/3). From the optimized

airfoils, an approximately 8 mm maximum vertical

displacement was obtained for the rods, so, a 24 mm

maximum horizontal displacement must be actuated.

3 SMA ACTUATORS CONTROL

DESIGN AND NUMERICAL

SIMULATION

The control of SMA actuators can be achieved, in

principle, using any method of position control, but

the specific properties of SMA actuators, such as

hysteresis, the first cycle effect and the long term

changes must always be considered. Starting from

the established concept of the actuation system the

operating schema of the controller can be organized

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

as is presented in Fig. 3. Based on the 35 studied

flight conditions a database of the 35 optimized

airfoils can be built. Therefore, for each flight

condition results a pair of optimal vertical

deflections (dY

1opt

, dY

2opt

) for the two actuation lines.

The SMA actuators morph the airfoil until the

obtained vertical deflections of the two actuation

lines (dY

1real

, dY

2real

) become equals with the

required deflections (dY

1opt

, dY

2opt

). The morphed

airfoil vertical deflections in the actuation points are

measured using two position transducers. The role of

the controller is to elaborate an electrical current

command signal for the SMA actuators on the base

of the error signals (e) between the required vertical

displacements and obtained displacements. Because

the two actuation lines are identical the designed

controller will be valid for both of them.

HeatingCooling

Three SMA wires

Flexible skin

Cam

Roller

Support plate for

actuation system

Rod

Compression

spring

Figure 2: The actuation mechanism concept.

, M,

Pilot

Control

Flight

conditions

Optimised

airfoils

database

1op t

2op t

Real airfoil

Integrated

controller

1real

2real

SMA

actuators

e=dY

opt

- dY

real

Airflow

perturbations

Current

Position

transducers

1real

2real

Figure 3: Operating schema of the SMA actuators control.

The first phase of the controller design supposes

the numerical simulation of the controlled actuation

system. Therefore, a model of SMA actuator was

required. In our system a non-linear model was used

(a numerical finite element one) build by Prof. P.

Terriault using the theoretical model of Lickhatchev

(Terriault, 2006). The SMA model has as inputs the

initial temperature of the alloy, the electrical current

that heats the alloy and the applied force; the outputs

are the displacement of the actuator and the

temperature of the alloy during functioning.

According to this model, to use the shape-changing

characteristics the SMA needs to be initialized by an

external force, which obliges it to go initially

through the transformation phase and further to

revert to the initial phase through the cooling phase.

Before these two phases, the control can’t be

realized, due to the intrinsic behavior of the SMA

(Terriault, 2006, and Popov, 2008).

Looking the wing as an object moving through

the atmosphere, aerodynamic forces are generated

between the air and the wing; these forces vary in

function of the airflow characteristics (Mach

number, Reynolds number and α - angle of attack).

Since the aerodynamic forces are suction forces, it

tends to lift the skin and to shorten the SMA wire.

Against the aerodynamic forces action the elastic

force of the flexible skin. A gas spring is needed in

order to counteract the aerodynamic forces, so that

the resultant force that acts on the SMA wire is

given by equation

.)(

faeroskinspringSMA

cFFFF 





(1)

To have the premises necessary to initialize the

SMA actuators in any conditions, they are loaded by

the gas spring even if there are no aerodynamic

forces applied on the flexible skin. So, the equation

(1) becomes

()(),

SMA pretension spring h skin v aero f

Fk kFc







(2)

where

vskinskinhspringpretensionspring

kFkFF 









(3)

SMA

is the SMA resultant force, F

spring

- gas spring

elastic force, F

skin

- elastic force produced by the

flexible skin, F

aero

- aerodynamic force, F

pretension

pretension force of the spring, c

- cam factor (1/3),

spring

and k

skin

are the elastic coefficients of the

spring, and of the skin, respectively, δ

and δ

are the

horizontal and vertical actuated displacements.

Implementing the SMA actuators model in a

Matlab S-function, the simulation model in Fig. 4

was obtained. As can be observed, to control the

SMA actuators, an adequate electrical current must

supply it. The length of the SMA wires is a complex

function of the SMA load force and temperature, the

last one being influenced by the supplying current in

time and by the interaction of the wires with the

environment in theirs cooling phase (when the

electrical supply is removed) (Grigorie, 2009).

Skin deflection (mm)

T [K]

SMA elongation x [m]

t [C deg]

1.8

SMA Initial

length [m]

XY Graph

(y vs. t)

To Workspace3

To Workspace2

To Workspace1

Skin deflection [mm]

Scope 2

Scope 1

SMA elongation [m]

Current

Force

Displacement

Temperature

SMA

F aero

x (m)

F SMA

y (mm)

Mechanical system

273.15

Kelvin

1500

Desired

Force aero

Desired

Current [A]

Figure 4: SMA actuators Simulink model.

A MORPHING WING USED SHAPE MEMORY ALLOY ACTUATORS NEW CONTROL TECHNIQUE WITH

BI-POSITIONAL AND PI LAWS OPTIMUM COMBINATION - Part 1: Design Phase

The block “Mechanical system” in Fig. 4 was

modeled accordingly with the equations (1) to (3).

As is shown in Fig. 2, the horizontal and the vertical

actuated distances are correlated by using the “cam

factor” c

=1/3. Therefore, the aerodynamic and the

skin forces (F

aero

and F

skin

) are reflected in the SMA

load force (F

SMA

) with the same rate. The gas spring

has a preloaded force of 1500 N and a linear elastic

coefficient of 2.95 N/mm. In simulations a linear

elastic coefficient of approximate 100 N/mm was

considered for the skin.

The envelope of the SMA actuator, obtained

through numerical simulation for different

aerodynamic load cases, is shown in Fig. 5. As can

be observed from Fig. 5, to obtain a skin maximum

vertical displacement (8 mm) in absence of

aerodynamic force, it is required a high temperature

(approximately 162˚C) in order to counteract the

spring force. Because the ability of the SMA wires

to contract is dependent upon Joule heating to

produce the transformation temperature required, the

need in higher temperature is reflected by a need in

higher electrical current. Due to the fact that the

aerodynamic forces reduce the actuators load the

required current and temperature values are

decreased; i.e. for F

aero

=1800 N the need in

temperature for the maximum vertical displacement

obtaining is approximately 90˚C. From other point

of view, the ability of the SMA wires to return to

their original configuration is dependant upon the

ability of the system to cool the wires. The simulated

SMA model offers just summary information about

this subject, the proper heating and cooling of the

wires being observed only in the moment of a

thermodynamic analysis of the physical morphing

wing. The system architecture play a big role in the

wire cooling by the convection process, and also the

performances of the system can be negatively

influenced by heat transfer from actuators to the

other components.

aero

= 0 N

aero

= 1800 N

Max 

= 8 mm

Figure 5: Simulated envelope of the SMA actuator.

According with Fig. 3, the integrated controller

purpose is to control the SMA actuators in terms of

supply electrical current so that to cancel the

deviation e between the required values for vertical

displacements (corresponding to the optimized

airfoils) and the real values, obtained from position

transducers. As mentioned previously, the design of

such controller is difficult considering the strong

nonlinearities of the SMA actuators characteristics,

nonlinearities significantly influenced by the forces

with which they are tense. The chosen design

procedure consisted of the following steps:

Step 1: numerical simulation of the SMA model

actuators for certain values of the forces in the system;

Step 2: approximation of the system with linear

systems in the heating and cooling phases using the

System Identification Toolbox of Matlab and the

numerical values obtained at the Step 1;

Step 3: the choice of the controller type and its

tuning for each of the two SMA actuators phase –

heating and cooling;

Step 4: the integration of the two obtained controller

in a single one followed by its validation for the

general model of the system (non-linear).

Because the team that established the actuation

line architecture (Georges, 2009) suggested that the

pretension force of the gas spring must have the

value F

pretension

=1500 N, F

aero

= 1500 N value was

chosen in numerical simulations for the aerodynamic

force. Simulating a cooling phase followed by a

heating phase with the model in Fig. 4, the blue

characteristics depicted in Fig. 6 were obtained. In

the first graphical window of the figure is presented

the SMA wire length changing in time (δ

), while in

the second window the SMA wire temperature

values in the two phases are shown. One observes

that a SMA wire dilatation results in the cooling

phase, and a wire contraction is obtained in the

heating phase. For a horizontal actuation distance of

approximately 24 mm the wire temperature reaches

a value near by 100˚C. Note are the transient time to

reach the steady-state values for the two phases:

approximate 60 s for the cooling phase and

approximate 40 s for the heating phase. For the

steady-state, after the cooling phase, the numerical

simulation obtained forces were: F

SMA

=1000 N,

skin

=0 N and F

spring

=1500 N. In this steady-state the

system is relaxed in terms of mechanical and the

vertical displacement of the actuator is null. For the

steady-state, after the cooling phase, the numerical

simulation obtained forces were: F

SMA

=1337 N,

skin

=266.1 N and F

spring

=1571 N. This steady-state

corresponds to the actuation system maximal

vertical displacement of approximately 8 mm.

Using the Matlab System Identification Toolbox

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

and the numerical values characterizing the δ

response at a series of successive step inputs, two

transfer functions were found for the SMA phases:

0.0177388 s 0.004017 s 0.0241958

(s) ,

s 1.43582 s 0.64742 s 0.001018

0.3535 s 0.2672

(s) ,

s 1.9386 s 0.011242

 











(4)

where H

(s) and H

(s) are the transfer functions for

heating and cooling phases. The displacements δ

corresponding to the linear systems obtained through

the two phases identification, are depicted with red

line in Fig. 6. A very good approximation can be

also observed for the two phases through the

identification

in simulated conditions. The previously

established transfer functions help to the controller

type choice for each phase and to its tuning.

0 50 100 150 200 250 300

120

0 50 100 150 200 250 300

-6

heating

cooling

heating

SMA model

Time [s]

Temperature [

C] Displacement 

[mm]

aero

= 1500 N

Time [s]

Figure 6: Actuator displacement and temperature vs time.

Considering the significance of physical

controlled phenomenon, that the SMA wire must be

heated to contract and then cooled to dilate by

providing an appropriate electrical current by the

control block, it is normal that in the cooling phase

the actuators not be powered. This phase of cooling

may occur in controlling not only a long-term phase,

when it ordered a switch between two values of the

actuator displacements, but also as a short-lived

phase, which occurs when the real value of the

deformation exceeds its desired value and is need to

cool the actuator wires. On the other hand, it is

imperative that in the heating phase actuators to be

controlled so that the stationary error of the

automatic system to be zero. Therefore, for this

phase one opted for a simple architecture of the

controller of PI type (proportional-integral). It

combines the advantages of proportional type

controller, which reduces substantially the overshoot

and lead to a short transient time, with the benefits

of the integral controller, which cancels the steady-

state system error. As a consequence, the resulted

controller must behave like a switch between

cooling phase and heating phase, situations where

the output current is 0 A, or is controlled by a law of

PI type. The two phase’s interconnection leads to an

integrated controller, which can be viewed as a

combination of a bi-positional controller (an on-off

one) and a PI (proportional-integral) controller.

The input-output characteristic of a bi-positional

(on-off) controller can be described by the equation













,0if,

)(

(5)

where i(t) is the command variable (electrical

current in our case) in time, i

reflects the value of

the command and e is the operating error (Fig. 3).

The PI controller law is given by

,d)()()(



 tteKteKti

(6)

with K

- the proportional gain, and K

- the integral

gain. Combining the two controllers in a single one,

based on the rules previously mentioned results the

control law of the integrated controller as the form



















.0if,d)()(

,0if,0

)(

etteKteK

(7)

The optimal tuning of the controller in heating

phase was realized using an integral criterion, the

error minimum surface criterion, very well known in

the literature as Ziegler-Nichols criterion (Mihoc,

1980); the tuning methodology is: a) the regulator is

considered as a proportional one (P) and it is tuned

with respect to the K

parameter; b) the

amplification factor K

is increased until the

response of the automatic system will be self-

sustained oscillatory. One memorizes the value K

of K

for which the system has an oscillatory

behavior and the value of oscillations semi-period

). The optimal values for the parameters of the PI

regulator are determined using the relations:

).85.0/(,45.0

TKKKK

PIPP









(8)

Follows the controller tuning steps the next

numerical values for the PI controller parameters

were obtained and/or were calculated:

.0061.787,8.1792

,s68.2,3984



(9)

As a consequence, the controlled system in hea-

A MORPHING WING USED SHAPE MEMORY ALLOY ACTUATORS NEW CONTROL TECHNIQUE WITH

BI-POSITIONAL AND PI LAWS OPTIMUM COMBINATION - Part 1: Design Phase

ting phase can be modeled with an approximate

linear system with the block schema in Fig. 7. The

parameters a

÷a

and b

÷b

in the schema are the

coefficients of the H

(s) transfer function nominator

and denominator in ascending power of s (eq. (4)).

The open loop transfer function of the controlled

heating phase is

3210

432

3210

sss

(s) (s) (s) ,

ssss

ol PI h

qqqq

HCH

bbbb







(10)

while the closed loop transfer function is

3210

432

43210

sss

(s) (s) (s) .

ssss

cl PI h

qqqq

HCH

rrrrr







(11)

Current

SMA heating

model

sss

bbbb

aaa





SMA real

elongation

Wished SMA

elongation

PI controller

K 

Figure 7: The block schema with transfer functions of the

heating phase linear model.

The included coefficients are

32 212

101 00

31.8021, 21.1622,

46.5396, 19.0422,

PPI

PI I

qKa qKaKa

qKaKa qKa

 

 

(12)

respectively

19.0422.46.5386,

21.8096,

30.3663,,1

001001

2112

22334







aKraKaKbr

aKaKbr

aKbrbr

IIP

(13)

(s) is the transfer function of the PI controller.

The poles of the close loop transfer function H

(s)

result with the values

.)Re( , i,1.1943 - -0.1187

,)Re( , i,1.1943 + -0.1187

-0.4453,,-29.6837,

333

222

1411









ppp

pppp

(14)

One can observe that all poles of the transfer

function are placed in the left-hand side of the

s-plane, and the obtained system is stable.

In the state-space representation

),()()(

tDtCt

tBtAt

uxy

uxx





(15)

the state matrix A, the input matrix B, the output

matrix C and the feed-forward matrix D, were

obtained by the forms

 

.0,1946.521.131.8,001

0100

0010

0001

19.0422-46.5386-21.8096-30.3663-

















DCB

(16)

Evaluating the controllability and observability of

the system (P and Q matrices) results

1000

30.3663-100

900.302530.3663-10

26723.119-900.302530.3663-1













P

(17)

;

533857.8-1286744.7-568090.6-832193.6-

17986.343352.419139.328035.4

605.5-1460.9-647-944.5-

1946.521.131.8













Q

(18)

4.order system)rank()rank( 



QP (19)

As a consequence, the system is completely

controllable and observable.

Based on the previously considerations, the final

form of the integrated controller law is



















.0if,d)(0061.787)(8.1792

,0if,0

)(

ettete

(20)

Introducing the controller in a general block

schema, with the non-linear SMA model, the

Simulink model in Fig. 8 was obtained for the SMA

actuators control (see Fig. 3). The input variable of

the schema is the desired skin deflection and the

output is the real skin deflection.

The “Integrated controller” block contains the

implementation of the law described by equation

(20) and of the preliminary observations related to

the SMA actuators physical limitations in terms of

temperature and supplying currents. Its schema is

shown in Fig. 9. The inputs of the block are the

control error (difference between the desired and the

obtained displacements) and the wires temperature,

and the output is controlled electrical current applied

on the SMA actuators. There are two switches in the

schema; the first one chooses one of the two options

in the control law (20) and the second one switching

the electrical current value to 0A when the SMA

temperature value is over the imposed limit. Also, a

current saturation block is used to prevent the

current increase over the physical limit supported by

the actuation SMA wires.

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

Temperature

[K]

skin deflection [mm]

skin

deflection

[mm]

desired deflection [mm]

SMA elongation

[m]

skin

deflection

[mm]

skin deflexion [mm]

14.5

desired

skin deflection

[mm]

3/1000

cam factor

mm to m

XY Graph

Scope

0.078

SMA max set

[m]

SMA elongation [m]

Current

Force

Displacement

Temperature

SMA Model

1.8

SMA Initial

length [m]

1.8

SMA Initial

length

Diff error

Temperature

Current out

Integrated controller

Memory

F aero [N]

x [m]

F SMA [N]

y [mm]

Mechanical system

Current [A]

273.15

Celsius to Kelvin

1500

Aerodynamic

force [N]

Figure 8: The simulation model for the controlled SMA actuator with non-linear model.

Current out

Temperature

limiter

Switch

-1

Gain

Current when

cooling

Current

saturation

Current when

reached limit

Controller

|u|

Abs

Temperature

Diff error

Current

Figure 9: “Integrated controller” block in Simulink.

Loading the simulated model with aerodynamic

force F

aero

= 1500N, the characteristics in Fig. 10 are

obtained for a 6 mm step desired skin deflection (δ

First of all, can be observed that the controller work

good, the transition to the desired steady-state being

significantly improved through the integration of the

two control law in the equation (20): 1) the

amplitudes of oscillations were reduced and the

observed oscillations in the SMA temperatures

around the steady-state are due only to the thermal

inertia of the smart material; 2) the values of the

transition time from 0mm to the steady-state values

decrease from 20÷25 to approximate 5 s.

4 CONCLUSIONS

The objective of here presented research work is to

develop an actuation control concept for a new mor-

phing mechanism using smart materials, like Shape

Memory Alloy (SMA), as actuators. These smart

actuators modify the upper surface of a wing made

of a flexible skin so the laminar to turbulent

transition point moves close to the wing airfoil

trailing edge.

The designed controller must controls the SMA

actuators in terms of supply electrical current so that

to cancel the deviation between the required values

for vertical displacements (corresponding to the

optimized airfoils) and the real values, obtained

from position transducers. The envelope of the SMA

actuator in Fig. 5, obtained through numerical

simulation using model in Fig. 4 for different

aerodynamic load cases, confirms that the length of

the SMA wires is a complex function of the SMA

load force and temperature, the last one being

influenced by the supplying current in time and by

the interaction of the wires with the environment in

theirs cooling phase (when the electrical supply is

removed).

0 102030405060708090100

-2

100

120

10 20 30 40 50 60 70 80 90 100

20 30 40 50 60 70 80 90 100 110 120

Time [s]

Temperature [

C] Displacement 

[mm]Displacement 

[mm]



required



obtained

SMA

initialisation

SMA initialisation

SMA actuation

SMA

initialisation

Figure 10: Response for a step input and F

aero

=1500 N.

As can be observed from Fig. 5, to obtain a skin

maximum vertical displacement (8 mm) in absence

of aerodynamic force, it is required a high

temperature (approximately 162˚C) in order to

counteract the spring force. Because the ability of

the SMA wires to contract is dependent upon Joule

heating to produce the transformation temperature

required, the need in higher temperature is reflected

by a need in higher electrical current. Due to the fact

that the aerodynamic forces reduce the actuators

load the required current and temperature values are

A MORPHING WING USED SHAPE MEMORY ALLOY ACTUATORS NEW CONTROL TECHNIQUE WITH

BI-POSITIONAL AND PI LAWS OPTIMUM COMBINATION - Part 1: Design Phase

decreased; i.e. for F

aero

= 1800 N the need in

temperature for the maximum vertical displacement

obtaining is approximately 90˚C.

The final configuration of the integrated

controller was a combination of a bi-positional

controller (particularly an on-off one) and a PI

(proportional-integral) controller, due to the two

phases (heating and cooling) of the SMA wires

interconnection. The resulted controller must behave

like a switch between cooling phase and heating

phase, situations where the output current is 0 A, or

is controlled by a law of PI type.

Using an integral criterion, the error minimum

surface criterion (Ziegler-Nichols), the PI controller

for the heating phase was optimal tuned, the resulted

values are K

=1792.8 and K

=787.0061. Evaluating

the systems’ performances one observed that the

poles of closed loop transfer function of the

controlled heating phase resulted with the values

(14) are all placed in the left-hand side of the s-

plane, so the obtained system is stable. On the other

way, the system was found to be completely

controllable and observable based on the values

established in equations (17)÷(19). So, the final form

of the integrated controller law was (20).

Loading the numerically simulated general

model (the non-linear one with F

pretension

=1500N) in

Fig. 8 with aerodynamic force F

aero

= 1500N, the

obtained characteristics in Fig. 10 confirm that the

controller works good, the transition to the desired

steady-state being significantly improved through

the integration of the two control law in the equation

(20): 1) the amplitudes of oscillations were reduced

and the observed oscillations in the SMA

temperatures around the steady-state are due only to

the thermal inertia of the smart material; 2) the

values of the transition time from 0mm to the

steady-state values decrease from 20÷25 to

approximate 5 s.

As second and third validation methods a bench

test and a wind tunnel test were performed and will

be presented in the second part of the paper, related

to the experimental validation.

ACKNOWLEDGEMENTS

We would like to thank the Consortium of Research

in the Aerospatial Industry in Quebec (CRIAQ),

Thales Avionics, Bombardier Aerospace, and the

National Sciences and Engineering Research

Council (NSERC) for the support that made this

research possible. We would also like to thank

George Henri Simon for initiating the CRIAQ 7.1

project and Philippe Molaret from Thales Avionics

and Eric Laurendeau from Bombardier Aeronautics

for their collaboration on this work.

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