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Abstract --The power kite is a kind of high altitude wind

energy (HAWE), which is a still untapped source of renewable

energy and has received an increasing attention in the last

decade. Automatic control of power kites is a key aspect of

HAWE generators and it is a complex issue, since the system at

hand is open-loop unstable, difficult to model and subject to

significant external disturbances. In order to deal with this issue,

a new kind of adaptive predictive functional controller (APFC) is

presented in this paper. With subspace identification forpredictive model of kites, the maximum generation controller is

designed to control kites using PFC principles. The APFC, which

is a combination of on-line identification, learning mechanism

and predictive controller, is presented to solve the nonlinear real-

time receding horizon optimization. The stability of control

system is guaranteed by closed-loop subspace identification. The

implementation of closed loop control system is given, and the

proposed APFC approach for kite control results to be quite

effective, as it is shown via numerical simulation tests.

Index Terms--High altitude wind energy; kite; predictive

functional control; subspace identification

I. I NTRODUCTION

N recent years, High Altitude Wind Energy (HAWE)

technologies have emerged to harness the power of wind

blowing up to 1000 m above the ground, exploiting the

controlled flight of tethered airfoils [1], [2], [3], [4], [5], [6].

The main advantage of reaching higher altitudes lies in the

fact that the wind speed grows with the elevation from the

ground, and that the power that can be extracted by a wind

flow grows with the cube of its speed. As an example, at the

height of 500-1000 meters the mean wind power density is

about 4 times the one at 50-150 meters, and at 10000 meters it

is 40 times. Moreover, wind at higher altitude is also less

variable, thus providing a more reliable source. The height and

the size of wind turbines have increased in the past years tocapture the more energetic winds at higher elevations;

however, actually the limits of such a dimension growth have

been almost reached, from both the economical and

This work was supported by Research Fund for the Doctoral Program of

higher Education of China (No. 20120006110013).Q. Sun is with the School of Automation and Electrical Engineering,

University of Science and Technology Beijing, Beijing, 100083, P. R. China

(e-mail: [email protected]).

Y. Y. Wang is with the Century College, Beijing University of Posts and

Telecommunications, Beijing, 100876, P. R. China (e-mail: [email protected]).

technological points of views [6], [7]. Control of tethered

airfoils is investigated in [1], in order to devise a new class of

wind generators to overcome the main limitations of the

present wind technology, based on wind turbines. In the

typical kite generator of yo-yo structure, indicated as KiteGen

in [3], energy is generated by a cycle composed of two phases,

indicated as the traction and the drag one. The kite control unit

is connected to an electric drive. In the traction phase the

control is designed such that the kite pulls the electric drive,maximizing the amount of generated energy. When energy

cannot be generated any more, the control enters the drag

phase and the kite is driven to a region where the energy spent

to drag the electric drive is a small fraction of the energy

generated in the traction phase, until a new traction phase is

undertaken [3].

Automatic control is the core of KiteGen and it is indeed a

complex aspect, since the power kite has nonlinear dynamics

which are open-loop unstable and it is affected by unmeasured

wind turbulence[6], [7]. The aim of the control system is to

maximize the generated energy, while at the same time

satisfying operating constraints, for example, to keep the kite

sufficiently far from the ground and to avoid wrapping of the

two lines. Nonlinear Model Predictive Control (NMPC)

techniques have been mainly applied so far to tackle this

problem [1], [2], [3], [4], [5], with promising results.

However, NMPC requires that a sufficiently accurate model of

the system is available, which is not easy to be obtained for

KiteGen. Moreover, the computation burden for on-line

receding horizon optimization are of main concern.

The predictive functional control is developed on the

principle of model predictive control with the advantages that

the obtained control law is linear combination of some known

base function and the computation burden can be effectively

reduced. In this paper, a new kind of adaptive predictivefunctional controller (APFC) is presented to solve the

difficulties in accurate modeling of non-rigid kites and real-

time problems encountered when nonlinear model predictive

control methods are applied to control kites. This paper is

organized as follows. In section II, the subspace identification

of LPV system is reviewed. The APFC based on subspace

identification is addressed in detail in section III. In section

IV, the proposed APFC is used to control the KiteGen. The

conclusions are drawn in the last section.

Data-driven Predictive Functional Control of

Power Kites for High Altitude Wind Energy

GenerationQu Sun, Member, IEEE , and Yong-yu Wang

I

2012 IEEE Electrical Power and Energy Conference

978-1-4673-2080-1/12/$31.00 ©2012 IEEE 274



II. LPV SYSTEM IDENTIFICATION

A. Problem Formulation

Linear Parameter-Varying (LPV) systems are a particular

class of nonlinear systems which can be thought of as time-

varying systems, of which the variation depends explicitly on

a time-varying parameter referred to as the scheduling or

weight sequence [8]. Many aeroelastic systems including kites

can be described as LPV systems in which the dynamic pressure or wind speed forms the scheduling [9], [10], [11]. In

this paper, we consider the following LPV model.

( )∑=

+ ++=

m

i

k i

k i

k ii

k k e K u B x A x

1

)()()()(1 µ , (1)

k k k k e DuCx y ++= , (2)

Where nk x R ∈ , r

k u R ∈ , l k y R ∈ , are the state, input and

output vectors. l k e R ∈ denotes the zero mean white noise.

The matrices nni A ×∈R )( , r ni B ×

∈R )( , nl C ×∈R , ∈ D r l ×R ,

l ni K ×∈R )( are the local system, input, output, feed-forward,

and observer gain matrices; and R ∈

)(ik µ the local weights.

The index m is referred to as the number of local models or

scheduling parameters. Note that the system, input, and

observer matrices depend linearly on the time-varying

scheduling vector as:

∑=

=

m

i

iik k A A

1

)()( µ , ∑=

=

m

i

iik k B B

1

)()( µ (3)

We assume that we have an affine dependence and the

scheduling is given by

[ ]T mk k k

)()2( ,,,1 µ µ µ ⋅⋅⋅= . (4)

We can rewrite (1) and (2) in the predictor form as:

( )∑=

+ ++=

m

i

k i

k i

k ii

k k y K u B x A x

1

)()()()(1

~~ µ (5)

k k k k e DuCx y ++= (6)

with

C K A A iii )()()(~−+ , D K B B iii )()()(~

−= . (7)

The identification problem can now be formulated as: given

the input sequence k u the output sequence k y and the

scheduling sequence k µ over a time },,1{ N k ⋅⋅⋅= ; find, if

they exit, the LPV system matrices)(i A ,

)(i B ,)(i K , C and

D for all },,2,1{ mi ⋅⋅⋅∈ up to a global similarity

transformation [9].

B. Identification Algorithm

Firstly, we reconstruct the state sequence up to a similarity

transformation by: pk

pk

pk k p pk z N x x κ φ +=

+ , (8)

Where k p,φ is the transition matrix

k k pk k p A A A~~~

11, +−+ ⋅⋅⋅=φ (9)

pκ is the time-invariant LPV controllability matrix; i.e.

[ ]11,, L L L p p p

⋅⋅⋅=−

κ (10)

with

1)(

1)1( ~

,,~

−− ⋅⋅⋅= pm

p p L A L A L

)()1(1 ,, m B B L ⋅⋅⋅=

)()()( ,~ iii K B B = .

and the matrix pk N is a matrix solely composed of the

scheduling sequence;

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅=

−+

+

1|

1|

|

pk p

k p

k p

pk

P

P

P

N (11)

with

l r k pk k p I P +−+ ⊗⊗⋅⋅⋅⊗= µ µ 1| .

and the following stacked vector is defined as:

⎥⎥⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎢⎢

⎣

⎡

=

−+

+

1

1

pk

k

k

p

k

z

z

z

z

(12)

withT T

k T k k yu z ],[= .

The key approximation in this algorithm is that we assume

that 0, ≈k jφ for all p j ≥ . This approximation is commonly

used in N4SID, SSARX, PBSID [9]. For finite p , this

approximation might result in biased estimates. However, it

can be shown that, if the system in (5) and (6) is uniformly

exponentially stable, the approximation error can be made

arbitrarily small by choosing p large enough. With this

approximation, the state pk x + is approximated by :

pk

pk

p pk z N x κ ≈+

(13)

The input-output behavior is now approximately given by:)(

: p

pk pk pk pk

pk

p pk ye Du z N C y

++++ =++≈ κ (14)

Now we define the stacked matrices as follows:

],,[ 1 N p uuU ⋅⋅⋅= + , (15)

],,[ 1 N p y yY ⋅⋅⋅= + , (16)

[ ] p p N

p p N

p p z N z N Z −−

⋅⋅⋅= ,,11 . (17)

If T T T U Z ],[ has full row rank, pC κ and D can be estimated

by solving the following least squares problem:2||||min F

p DU Z C Y −− κ (18)

where F |||| • represents the Frobenius norm.

III. ADAPTIVE PREDICTIVE FUNCTIONAL CONTROL

Predictive functional control (PFC) as the third generation

of model predictive control (MPC) was firstly reported by

Richalet and Kuntze [12]. PFC considers the control input

structure as a key factor. It can avoid the problem caused by

275



L F

D F

mg

eW

X Y

Z

θ

φ

r

l

F

ψ l ∆d

L F

uncertain control input law, which can be seen in many MPC.

PFC has a good set-point following ability, robustness, and

accuracy in control performance but it requires accurate

system model in controller design. In this section, a new PFC

scheme is proposed, based on the subspace identification for a

LPV system.

A. Internal modelling

The LPV model given by (1) and (2) is used to model the

system dynamics. This idea is from the concept of internal

model, which aims to provide a trade-off between the accurate

system representation and minimal on-line computation effort.

In this paper, the LPV model for a MIMO system is chosen as

the internal model, which can be rewritten as:

))(),(()( k uk x F ik y =+ , P i ,,1 ⋅⋅⋅= (19)

B. Control Variable

The future control variable is assumed to be a linear

combination of priori known functions:

∑=

=+

N

n

nn i f ik u

1

)()( µ , 1,,0 −⋅⋅⋅= P i (20)

where n µ are the coefficients to be computed during the

optimization process, )(i f n is the base function (such as a step

function, ramp function, exponent function and so on) which is

selected beforehand, and N is the number of base functions.

The model output is composed of two parts, i.e.

)()()( iik yik y M P ω ++=+ , P i ,,1 ⋅⋅⋅= (21)

where ))(()( k x F ik y M =+ is the free (unforced) output

response when 0)( =k u in (19), ∑=

=

N

n

nn i g i

1

)()( µ ω is the

forced output response to the control variable given by (20).

C. Receding Horizon Optimization

The performance index may be a quadratic sum of the errors

between the predicted model output P y and the reference

trajectory r y . It is defined as follows:

∑=

+−+=

P

i

P r ik yik y J

1

2))()((min (22)

The performance index is optimized on line to determine the

coefficients of base functions (BF), and only the first term in

(20) is effectively applied for the control, as depicted in Fig.1.

Fig. 1. Scheme of adaptive predictive functional control based on subspace

identification.

The adaptive predictive functional control (APFC) consists

of three layers: the on-line model identification (MI) layer, the

receding horizon optimization (RHO) layer and the self-

learning (SL) layer. An immune optimization algorithm [13] is

adopted here to solve the nonlinear real-time receding horizon

optimization. Meanwhile, the characteristics of the

optimization problem can be memorized and recognized using

pattern recognition techniques in order to accelerate the

convergence of the searching procedure.

IV. APPLICATION OF APFC TO KITEGEN

The adaptive predictive functional controller developed insection III is applied to control the KiteGen. Control problem

and related objectives are firstly described.

A. KiteGen Control Problem

The KiteGen in [3] aims to harvest High-Altitude Wind

Energy by using tethered power kites, connected to the ground

with two lines, made of strong composite fibre and wound

around two drums, kept at ground level and linked to

reversible electric motors. The system consists of the kite, the

lines, the onboard sensors, the drums, the generators and the

control hardware named Kite Steering Unit (KSU) [3]. The

model developed in [1] is employed to mimic the kite

dynamics in simulations below. A fixed cartesian coordinatesystem (X, Y, Z) is considered, with X axis aligned with the

nominal wind speed vector direction, as depicted in Fig. 2. A

spherical coordinate system is also considered, centered where

the kite lines are constrained to the ground. In this system, the

kite position is given by its distance r from the origin and by

the two angles θ and φ. FD is the drag force and FL is the lift

force, computed as:2

21 || e D D W AC F ρ −= (23)

2

21 || e L L W AC F ρ −= (24)

where ρ is the air density, A is the kite characteristic area, CL

and CD are the kite lift and drag coefficients. All of these

variables are supposed to be constant. We is the effective windspeed.

Fig. 2. Model of a power kite.

Applying Newton’s laws of motion in the local coordinatesystem, the following equations are obtained [1]:

m

F r r r θ θ φ θ θ θ =+− 2)cos()sin(

2 (25)

276



m

F r r r

φ φ θ θ φ θ φ θ =++ )sin(2)cos(2)sin( (26)

m

F r r r r

=−−222

)(sin φ θ θ (27)

withad F mg F θ θ θ += )sin(

ad F F φ φ =

l ad r r F F mg F −+−= )cos(θ

where m is the kite mass. The external forces Fθ, Fφ and Fr

include the contributions of gravitational force mg,

aerodynamic force Fad

and force Fl exerted by the kite on the

lines.

The control variable is defined by

)arcsin( d l ∆=ψ (28)

with d being the distance between the two lines fixing points

at the kite and Δl the length difference of the two lines. Angle

ψ influences the kite motion by changing the aerodynamic

force. Thus the system dynamics are of the form:

))(),(),(()( t wt ut g t xx = (29)

where T t r t t t r t t t )](),(),(),(),(),([)( φ θ φ θ =x , )()( t t u ψ = and

w(t) stands for the actual wind speed. All the model states are

supposed be measured, to be used for feedback control.

The main objective is to generate energy by a suitable

control action on the kite. Energy is generated by continuously

repeating a two-phase cycle. The two phases are referred to as

the traction phase, in which the lines are unrolled under high

pulling forces, thus generating power, and the recovery phase,

in which the lines are then rolled under low pulling forces [1].

In both phases, the kite can exert on the lines positive

forces only, so force Fl >0. Since the power is l F r P = , in the

traction phaseref

r >0 is chosen and a positive power is

generated. In the recovery phase, ref r <0 is required since the

lines length has to be reduced, and a negative power results.

For the whole cycle to be generative, the total amount of

energy produced in the first phase has to be greater than the

energy spent in the second one to recover the kite before

starting another cycle. In both phases, optimal controllers need

to be designed, which should guide the kite in order to

generate the maximum amount of power, while at the same

time satisfying operational constraints, since the kite has to be

kept above a minimal height from the ground and line

wrapping has to be avoided [1].

Control objective adopted in the traction phase is tomaximize the energy generated in the interval [tk , tk +TP], thus

the following cost is chosen to be minimized:

∫ +

−= pk

k

T t

t

l k d F r t J τ τ τ ))()(()( (30)

while satisfying constraints concerning state and input values,

i.e.

2)( π θ θ <≤t (31)

ψ ψ ≤)(t (32)

ψ ψ ≤)(t . (33)

The following initial state value ranges are considered to

start the traction phase:

⎪⎩

⎪⎨

⎧

≤≤

≤

≤≤

ΙΙ

Ι

ΙΙ

r t r r

t

t

)(

)(

)(

φ φ

θ θ θ

(34)

with

⎩⎨

⎧

<<

<<<

Ι

ΙΙ

2/0

2/0

π φ

π θ θ

.

And the traction phase ends when the following condition is

reached:

r t r =)( . (35)

where r is the max length of lines. Then the recovery phase

can start, which has been divided into three sub-phases. The

cost function in each sub-phase is respectively set as follows:

τ π τ φ τ θ d t J pk

k

T t

t k

22 )2/)(()()( −+= ∫ +

(36)

∫ +

= pk

k

T t

t

l k d F r t J τ τ τ )()()( (37)

τ τ φ θ τ θ d t J pk

k

T t

t k ))()(()( 1 +−= ∫

+

(38)

where 2/)(1 ΙΙ += θ θ θ . During the whole recovery phase the

state constraint (31) and the input constraints (32), (33) areconsidered in the control optimization problems [1].

B. Simulation Results

At any sampling time k t , control )( k t ψ results to be a

nonlinear static function of the system state )( k t x , the nominal

measured wind speed )( k x t W and the reference ref r :

))(()( k k t f t ω ψ = (39)

where T k ref k xk k t r t W t xt )](),(),([)( =ω . For a given )( k t ω , the

value of the function is typically computed by solving the

constrained optimization problem at each sampling time.

However, an online solution of the optimization problem is a

difficult task, which may not be finished at the sampling

period required for this application, of the order of 0.1 second.

To deal with this problem, the APFC in section III is

adopted. The control system depicted in Fig. 1 has been

implemented in Matlab/Simulink. The model expressed by (29)

is used as the “real” system, model and control parameters are

reported in Table I. Table II contains the state values which

identify each phase starting and ending conditions and the

values of state and input constraints.

TABLE I. MODEL AND CONTROL PARAMETERS

m 300 kite mass (kg)

A 500 kite area (m2)

ρ 1.2 air density (kg/m3)

w 12 wind speed (m/s)

CL 1.2 lift coefficient

CD 0.15 drag coefficient

277



r 2.14 traction phase reference speed (m/s)

r -4 recovery phase reference (m/s)

T 0.1 sample time (s)

P 6 control horizon

M 6 prediction horizon

TABLE II. CYCLES STARTING AND ENDING CONDITIONS, STATE AND

INPUT CONSTRAINTS

Ιθ 35

traction phase starting conditions

Ιθ 45

φ 5

Ιr 631m

Ιr 641m

r 831m traction phase ending conditions

ΙΙφ 65 2nd recovery sub-phase starting

conditionsΙΙ

θ 40

θ 85

state constraint

ψ 8.5 input constraints

ψ 20 /s

-2000

200400

600800

-500

0

500

0

200

400

600

800

x(m)y(m)

z ( m )

Fig. 3. Kite trajectory during a yo-yo cycle.

0 100 200 300 400 500 600-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

time(s)

u ( r a d )

Fig. 4. Control variable.

The simulation results are presented here. Fig. 3 shows the

trajectory of the kite, and Fig. 4 shows the control variable. The

power generated in the simulation is reported in Fig. 5, the

mean value is 1800 kW. The random disturbances in wind

speed do not cause system instability, showing the control

system robustness.

0 100 200 300 400 500 600-500

0

500

1000

1500

2000

2500

3000

3500

4000

time(s)

P ( k w )

Fig. 5. Instant power generated in the simulation

V. CONCLUSION

The paper presented a new kind of adaptive predictive

functional control (APFC) scheme to control a tethered kite,

employed to convert high altitude wind energy. The proposed

APFC is a data driven approach combining subspace

identification with predictive functional control, in which the

LPV model of a kite serves as the internal model of the

system. The closed loop structure of subspace identification

technique is used, since a feedback controller is required to

prevent the kite from becoming unstable under high dynamic

pressure or wind speed. And its effectiveness in the considered

application has been shown through numerical simulation

results.

VI. ACKNOWLEDGMENT

The authors thank Dr. J.W. van Wingerden for his help in

subspace identification algorithms.

VII. R EFERENCES

[1] M. Canale, L. Fagiano, and M. Milanese, "Control of tethered airfoils

for a new class of wind energy generator," in Proc. 2006 IEEE Conf. on Decision and Control , pp.4020-4026.

[2] A. Ilzhofer, B. Houska, M. Diehl, "Nonlinear MPC of kites undervarying wind conditions for a new class of large scale wind power

generators," International Journal of Robust Nonlinear Control , vol.17,

pp.1590–1599, 2007.[3] M. Canale, L. Fagiano, and M. Milanese, "Kitegen: A revolution in wind

energy generation," Energy, vol.34, pp. 355-361, 2009.[4] L. Fagiano, M. Milanese, and D. Piga, "High-altitude wind power

generation," IEEE Trans. Energy Conversion, vol.25, pp.168-180, Mar.

2010.[5] M. Canale, L. Fagiano, and M. Milanese, "High altitude wind energy

generation using controlled power kites," IEEE Trans. Control Systems

Technology, vol.18, pp.279-293, Mar. 2010.[6] C. Novara, L. Fagiano, and M. Milanese, "Direct data-driven inverse

control of a power kite for high altitude wind energy conversion," in Proc. 2011 IEEE Int. Conf. on Control Applications, pp.240–245.

[7] C. Novara, L. Fagiano, and M. Milanese, "Sparse Set Membershipidentification of nonlinear functions and application to control of power

kites for wind energy conversion," in Proc. 2011 IEEE Conference on

Decision and Control and European Control Conference, pp.3640–

3645.

[8] J. Shamma, and M. Athans, "Guaranteed properties of gain scheduledcontrol for linear parameter varying plants," Automatica, vol.27, pp.559-

564, 1991.

278



[9] J.W. van Wingerden, P. Gebraad, and M. Verhaegen, "LPV

identification of an aeroelastic flutter model," in Proc. 2010 IEEE Conf.on Decision and Control , pp.6839–6844.

[10] J.W. van Wingerden, and M. Verhaegen, "Subspace Identification of

MIMO LPV systems: The PBSID approach," in Proc. 2008 47 th IEEE

Conf. on Decision and Control , pp.4516–4521.

[11] J.W. van Wingerden, and M. Verhaegen, "Subspace identification ofmultivariable LPV systems: a novel approach," in Proc. 2008 IEEE

International Conference on Computer-Aided Control Systems, pp.840– 845.

[12] H.B. Kuntze, A. Jacubasch, J. Richalet, "On the Predictive Functional

Control of an Elastic Industrial Robot," in Proc. 1986 IEEE Conf. on Decision and Control , pp.1877-1881.

[13] Y.J. Li, D.J. Hill, T.J. Wu, "Nonlinear predictive control scheme withimmune optimization for voltage security control of power system,"

Automation of Electric Power Systems, vol.28, pp.25-31, Aug. 2004.

VIII. BIOGRAPHIES

Qu Sun (M’2003) was born in Xi’an in thePeople’s Republic of China, on July 10, 1971. He

received his Ph.D. degree in system engineering

from Xi’an Jiaotong University in 2000.His employment experience included Shanghai

Jiaotong University, Sichuan University. He has been a full professor in the School of Automation

and Electrical Engineering, University of Scienceand Technology Beijing since April 2008. His mainresearch interests include advanced control theories

and their applications, wind energy generation and grid integration.

Yong-yu Wang was born in Xining in the People’s

Republic of China, on July 2, 1973. She received her

Ph.D. degree from Beijing University of Posts andTelecommunications in 2008.

She has been an associated professor in theCentury College, Beijing University of Posts and

Telecommunications since June 2008. Her main

research interests include optimization methods,wind energy generation and grid integration.

279

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