2019 16th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE)Mexico City, Mexico September 11-13, 2019

Robust Nonlinear flight Control of aPower-Generating tethered kite

1st J. Alvarez-GallegosDep. of Electrical Engineering

CINVESTAV-IPNMexico City, Mexicojalvarez@cinvestav.mx

2nd R. Castro-LinaresDep. of Electrical Engineering

CINVESTAV-IPNMexico City, Mexicorcastro@cinvestav.mx

3rd M.A. Zempoalteca-JimenezDep. of Electrical Engineering

CINVESTAV-IPNMexico City, Mexico

miguel.zempoalteca@cinvestav.mx

Abstract—The motion description of an airborne wind energy(AWE) system is very complex and is not suitable to design acontroller. In recent years, simplified models have been developedfor this kind of systems. In this paper a controller is designedusing a simplified model and is based on the sliding modetechnique. The designed controller is robust against modeluncertainties and disturbances in wind velocity. Some simulationresults are presented with a complete generation cycle where thewind velocity and the kite glide ratio are perturbed.

Index Terms—renewable energy, wind energy, tethered kite,robust control.

I. INTRODUCTION

In recent years, the searching for alternatives in powergeneration has increased. A young technology around thisissue is the so-called airborne wind energy (AWE) systems.For a survey in the development of these systems from theLoyd seminal paper [1] in the 80s until this decade, see [2].Also, the interested reader can review [3] in order to get recentinformation about the subject.

Power generation with an AWE system, generally consistsof three stages [4]. In the power stage, the kite flies in winddirection which induces tension in kite ropes. This tension isused to move an electric generator. A form to increase thetension, and in consequence the produced power, is to makethe kite to track an eight-shaped trajectory [5]. When the ropegets a certain maximal length, the power stage is stopped andthe transfer phase starts. In the transfer stage there is not ropevelocity and the kite is maneuvered to the equilibrium position.In this point the rope tension is minimal. When the equilibriumpoint is reached the return stage starts. In return stage the ropeis rolled until a certain minimal length. When such length isreached the power stage restarts and the cycle is completed.This is the so-called pumping cycle [2].

A good representation of the system dynamics is presentedin [6]. The model considers the kite as a one-mass-pointinteracting with aerodynamic, gravitational and tether forcesamong others. A simplified model is presented in [4]. Theprincipal simplification is to consider the kite massless, whichreduces the equations significantly. One of the most completerepresentations recently published is a four-point-mass model[7], this representation considers the system as a rigid body.

A very important subject is the kite control to track tra-jectories in the power stage, since the amount of energy that

can be harvested depends on it. Also it is important becausethe kite operation in this region is unstable without a suitablecontrol and can cause the kite crash rapidly. The whole powergeneration cycle appears in some works such as [4], [9],[10], [11], [12], [13]. Several publications are focused onlyin generation and trajectory tracking in the power stage, theydo not consider the complete cycle [14], [15], [16], [17], [18],[19], [20], [21], [22], [23].

In general, the employing of controllers as P, PI, PD andPID is common to trajectory tracking in AWE systems. Othersapplied techniques to these systems are nonlinear model pre-dictive control (NMPC), linear-quadratic regulator (LQR) andnonlinear dynamic inversion (NDI). However, since an AWEsystem is affected by several disturbances (changes in the windvelocity) and parameter variations (modifications of the liftand drag coefficients), a robust controller which is designedon a model that involves the AWE’s nonlinear dynamics, isneeded. The main contribution of this paper is the design ofa robust controller that allows the kite orientation of a kitemodel to track an specific trajectory in order to harvest energy.Also, the performance of the proposed controller is evaluatedthrough numerical simulation using a complete full model ofthe kite. The numerical simulation is designed to reproduce acomplete power generation cycle and in the presence of severaldisturbances.

II. KITE MODELS

This work makes use of two models; one is used to representthe system behavior while the other one is used to design acontroller. The first one describes in detail the kite dynamicsand it was developed by Fagiano et al. in [6]. The second oneis a simplified model which is the most simple representationof a kite dynamics and was deduced by Erhard et al. in [24].

The kite position is sketched in the Fig. 1a using sphericalcoordinates [θ, φ, r] and can be expressed in Cartesian coor-dinates (X,Y, Z) as⎛

⎝ XYZ

⎞⎠ = r

⎛⎝ cosφ cos θ

sinφ cos θsin θ

⎞⎠ , (1)

where r is the rope length (see Fig. 1a), (the wind direction isconsidered pointing to X). In order to describe the two models,

978-1-7281-4840-3/19/$31.00 ©2019 IEEE

the assignments x = (θ, θ, φ, φ) and u = Δl are made, whereΔl is the length difference between the two steering ropes (seeFig. 1b). As presented in [6] the kite motion can be described

100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

Fig. 1. a. System coordinates. b. Difference between steering ropes.

by the state equation

x = f(x, u) =

⎛⎜⎜⎝

x2F2(x, u)x4

F4(x, u)

⎞⎟⎟⎠ , (2)

where the functions F2(x, u) and F4(x, u) are composed bythe gravity force, the apparent force, the kite aerodynamicforce, the ropes drag force and the traction force exerted bythe ropes on the kite, and are given by

F2(x, u) = β[cos ξ cosψ cos η sinΔα

− sin ξ cosψ sin η sinΔα− sin ξ sinψ cosΔα]

− β

E[cos ξ cosΔα]− g cosx1

r− sinx1 cosx1x

24, (3)

F4(x, u) = β[sin ξ cosψ cos η sinΔα

− cos ξ cosψ sin η sinΔα− cos ξ sinψ cosΔα]

− β

E cos(x1)[sin ξ cosΔα] + 2x2x4 tan(x1). (4)

ξ(x) and Δα(x) represent the kite orientation and the variationof the angle of attack respectively (see Figs. 2a and 2c), whileψ(x, u) is the angle shown in Fig. 2b. These functions togetherwith functions β(x) and η(x) are given by the followingequations

ξ(x) = arctan

(Vw sinx3 + rx4 cosx1Vw cosx3 sinx1 + rx2

), (5)

Δα(x) = arcsin

(Vw cosx3 cosx1

‖Wa(x)‖), (6)

ψ(x, u) = arcsin

(u− d cosx1 sinx3

d

), (7)

β(x) =ρACL‖Wa(x)‖2

2rm, (8)

η(x) = arcsin(tan(Δα(x)) tan(ψ(x, u))). (9)

Also, E is the glide ratio (it depends on the lift coefficient CL

and the drag coefficient CD), m is the system mass includingthe kite and the ropes masses, d is the wingspan, A is the kitearea, ρ is the air density, g is the gravitational acceleration,

100 200 300 400 500 600 700 800 900

50

100

150

200

250

300

350

Fig. 2. (a) Δα(x), (b) ψ(x, u) and (c) ξ(x) . (�xb, �yb, �zb) are body axes and(�xw, �yw, �zw) are wind axes. The kite orientation ξ shown in (c) is only forreference; this figure is valid when the plane (�xb, �yb) is parallel to the plane(�LN , �LE).

Vw is the wind velocity and ‖·‖ denotes the two-norm. Animportant signal is the apparent kite velocity Wa(x) given by

Wa(x) =

⎛⎝ −Vw cosx3 sinx1 − rx2

−Vw sinx3 − rx4 cosx1−Vw cosx3 cosx1

⎞⎠ . (10)

In addition, the kite orientation is chosen as the output signal,this is

y = h(x) = ξ(x). (11)

As it is shown in [4], [14], [12] and [25], such an outputchoice allows to generate mechanical power when it tracksan specific trajectory. More precisely, the output trajectorytracking induced permits to generate mechanical power bymeans of the following expression (see [25] for more details)

Pm =1

2ρACR‖Wa(x)‖2r, (12)

where CR =√C2

D + C2L and r is the rope velocity. The

specific trajectory that the output signal (11) should followin order to generate mechanical power is described in thefollowing.

From kite position in the plane (φ, θ) (see Fig. 3) a referenceyTP is generated to reach any of the target points TPi (withi = 1, 2). When the distance μ between the kite and the currenttarget point is less than the radius σ, the current target point isswitched to the next target point. This cycle is repeated severaltimes in the power phase.

Due to the switching between target points, the referenceyTP has sudden changes. For this reason, the trajectory issmoothed using a Bezier polynomial. This modified trajectoryis the desired trajectory y∗ (see figure 4).

−1 −0.5 0 0.5 1

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Varphi

σ

θ

TP1

TP2

φμ1

μ2

Kite trajectory

Fig. 3. σ is the radius of circle with center at the target point TPi withi = 1, 2. This representation is in the plane (φ, θ). When the distance betweenthe kite and the target point is less than σ, then the target point is switchedto another target point.

30 31 32 33 34 35 36 37 38 39 400.3

0.4

0.5

0.6

0.7

0.8

0.9

30 31 32 33 34 35 36 37 38 39 40−3

−2

−1

0

1

2

3

σ

−yTP

+yTP

μ2

If μ1 ≤ σ If μ

2 ≤ σ y*

Δt

μ1

Fig. 4. μ1 and μ2 are the distance between the kite and the correspondingtarget point (see figure 3). yTP is the desired velocity angle to achieve thetarget point TPi from the kite’s position and y∗ is the desired trajectoryobtained.

A. Simplified Model

As explained in detail in [24] and [26], the principalassumption that allows to obtain a simplified model is toconsider a massless kite. As a consequence, the accelerationsterms in the model (2) are simplified. Therefore, the equationsof motion describe the kite’s velocities, this is(

ϑϕ

)=

(Va

1r

[cos ξ − 1

E tanϑ]

−Va 1r cscϑ sin ξ

), (13)

where ϑ(x) is the elevation angle and ϕ(x) is the deviationwith respect to the direction of the wind velocity, it is alsoconsidered that the wind direction points to the X direction.Va is the apparent kite velocity which is considered as a mea-surement. This model is enunciated in spherical coordinates(ϑ, ϕ) defined as in Fig. 5a. In order to have a concordance

100 200 300 400 500 600 700

50

100

150

200

250

300

350

400

Fig. 5. Auxiliary system coordinates.

with the coordinates (x1, x3) of the model (2), the simplifiedmodel is enunciated as in [28], i.e.(

x1x3

)=

(r−1vk(x) cos ξ(x)

(r cosx1)−1vk(x) sin ξ(x)

), (14)

where vk(x) is the kite velocity. The transformation betweencoordinates is defined in [27] as(

ϑ(x)ϕ(x)

)=

(arccos (cosx3 cosx1)arctan (sinx3 cotx1)

), (15)

Note that in [24], [26] and [27] the kite orientation appearsas an additional coordinate but here it is used as the systemoutput. Under the assumption that the kite is in crosswindflight, an approximation of the time derivative of the outputfunction y was found in [14] which the authors prove to be

valid in the power phase of the kite. That time derivative wasalso experimentally found in [24] and given by

ξ = Vagkuδ + ϕ cosϑ, (16)

in (ϑ, ϕ) coordinates, and

ξ = ‖Wa(x)‖gkuδ, (17)

in (x1, x3) coordinates, where uδ is an angular deflectionin the actuator; the rope length difference Δl in Fig. 1b isproportional to the angular deflection uδ in Fig. 5b. gk is theso-called proportional gain of the turn rate law and is givenby

gk =ρACL

2rmd. (18)

In accordance to [29], one can thus say that the system (13)with output y = ξ has relative degree one since Vagk �= 0in the power phase operation of the kite. Then, the internaldynamics of the system is represented by (13). In order tostudy the stability of these dynamics one sets y = 0 in (13)leading to zero dynamics

ϑ =Var

[1− 1

Etanϑ

], (19)

ϕ = 0. (20)

Let us propose a Lyapunov function candidate

V =1

2ϑ2 +

1

2ϕ2, (21)

whose time derivative along the dynamics (19)-(20) takes theform

V =ϑVar

[1− 1

Etanϑ

]. (22)

Since Va > 0, r is positive and the elevation angle ϑ is suchthat 0 < ϑ < π

2 , one has that the additional inequality

ϑ > arctan(E), (23)

holds, then V ≤ 0. This, is the zero dynamics (19) − (20)and thus the internal dynamics (13) are locally stable aroundthe point where y = ξ = 0 when the elevation angle ϑ isconstrained to evolve in accordance to

arctan(E) < ϑ <π

2. (24)

In the power phase operation of the kite such a conclusionon the evolution of the kite angle ϑ in order to assure anstable operation coincides with the experimental results foundin [24].

III. CONTROLLER DESIGN

The controller is designed to output track a specific trajec-tory (see section II). To this end the equation (17) is rewrittenas

ξ = b(x)uδ, (25)

where

b(x) = | �Wa(x)|gk. (26)

Then, the output tracking error is defined as

e = y − y∗, (27)

where y∗ is the desired trajectory.The kind of systems considered in this work typically

operate under several disturbances. In this case two strongimprecisions are present, first the model simplifications andsecond the variations in the wind velocity. Specifically, thiswork considers only an estimation of the wind velocity inthe kite position, this difference must be compensated bythe controller. An option to deal with these uncertainties isto apply techniques of robust control. A simple approach torobust control is the so-called sliding control methodology.Using this approach, a switching function ς which is functionof the error, is proposed as in [30], this is

ς = e+ C0

∫ t

0

e(τ)dτ, (28)

where C0 is a proportional gain. In order to attract the motionof the system to the sliding surface defined by ς = 0, the so-called sliding condition ςς < 0 should hold. To assure this, itis proposed that

ς = e+ C0e = −Γsign(ς). (29)

Substituting the time derivative e into (29), one obtains

ς = (b(x)uδ − y∗) + C0e = −Γsign(ς). (30)

And the control law uδ take the form

uδ = b(x)−1(y∗ − C0e− Γsign(ς)) (31)

This control law is represented in Fig. 6. Note that it can beobtained u = kuδ with k being a constant (see section II).Notice that when the dynamics of the system is constrainedto evolve on a sliding surface ς = 0, one has a linear timeinvariant dynamics of the output tracking error given by e +C0e = 0 which implies that the e → 0 when t → ∞ thusgetting trajectory output tracking

100 200 300 400 500 600 700 800

50

100

150

200

250

Fig. 6. Scheme control for trajectory tracking.

The control law is implemented by mean of software, fromthe system states and the reference signal (see Fig. 4), alsothe sign function is approximated by the saturation function.This law was designed to compensate the variations in windvelocity, in system parameters and changes in the rope length.

IV. CONTROL STRATEGY

The designed controller in the previous section allows tomanage the kite orientation and in consequence the spaceposition. Nevertheless, in order to practically generate power,additional components are necessary. The blocks diagram inthe Fig. 7 is used to reproduce a power generation cycle. Inthat diagram the kite system dynamics is given by equations(2). The block is also fed with a perturbed signal on thewind velocity. The output tracking control block computes thecontrol action in order to track the desired trajectory (see Fig.6). Note, that the controller glide ratio E′ is different from thesystem glide ratio E, this in order to represent a parameteruncertainty.

The trajectory generator produces the trajectory that musttrack the system output. This block guides the system to trackan eight-shaped trajectory in the space in power phase and alsoguides the system to go to the equilibrium point in transfer andreturn phases (see Fig. 8). Note, that the trajectory tracked bythe output system (kite orientation) and the trajectory trackedby the kite position in the space are different.

Fig. 7. Control scheme for a complete power generation cycle.

The control cycle is designed using triggering conditionsand it is responsible to indicate to the trajectory generatorblock when the eight-shaped trajectories must be generated orto send the system to the equilibrium point (see Fig. 8). Thisblock is also responsible of generating the rope velocity, apositive value in the power phase, a null value in the transferphase and a negative value in the return phase.

−10

0

10

20

30

1. Power phase

2. Transfer phase

3. Return phaseWind direction

Fig. 8. Control scheme for a complete power generation cycle.

V. SIMULATION RESULTS

Some numerical simulations were carried out to evaluatethe proposed controller in the power phase. Additionally, a

complete power generation cycle is performed, in order to esti-mate the amount of energy harvested in the complete process.The simulation is accomplished using data reported in [4],therefore the results are comparable with this publication. Thekite parameters, the trigger conditions and the gain controllersare reported in table I. Also, it is necessary to mention thatinstead of using the “sign function” the “saturation function”with ε1 is used in the output trajectory controller.

TABLE ISIMULATION PARAMETERS I

Kite Parameters Trigger conditions Controller Gainsm = 7 kg θTP = 0.65 rad KP = 1A = 30 m2 φTP = 1 rad Γ = 4d = 6 m σ2 = 0.33 ε1 = 0.05E = 5 θr = 1.35 rad

CL = 0.85 ltransfer = 190 mE′ = 4 lrestart = 25 m

The initial condition and other system parameters are givenin table II.

TABLE IISIMULATION PARAMETERS II

Initial Conditions Othersθ0 = 1 rad Vw = 7.5 m/sφ0 = 0 ulim = ±3 m

θ0 = 0.2 rad/sφ0 = 0.2 rad/s Vin = −2.5 m/sr0 = 25 m Vout = 1.5 m/s

The disturbance that is added to the wind velocity Vw attime t = 10s (see figure 9) was chosen as

D = 2.5 + 0.25 cos(3t) + sin(0.5t+ 0.5) (32)

A. Results

In Fig. 9 the wind velocity is reported. The controllerreceives a constant value, while in the system this value isaffected by the addition of D given by (32).

0 20 40 60 80 100 120 140 160 1804

5

6

7

8

9Wind Velocity

Time (s)

Vel

ocity

(m

/s)

V

w

Vw

+ D

Fig. 9. Wind velocity used by the controller Vw and the wind velocity appliedto the system Vw +D.

In Fig. 10 it is shown how the reference ξref is trackedby the kite orientation. Note, that the controller is capable todeal with the disturbances and make the system follow thereference trajectory.

0 20 40 60 80 100 120 140 160 180−2

0

2Kite orientation ξ

ξ (r

ad)

ξ

ref

ξ

54 55 56 57 58 59 60 61−2

0

2

Time (s)

ξ (r

ad)

Fig. 10. Trajectory tracking.

The control effort, the tacking error and the sliding surfaceare presented in Fig. 11. The signals are active in power phaseand are almost null in transfer and return phase. Note thatthe control effort is limited to the range of ±3m, due tophysical restrictions, however the system is capable to track thereference. In practice, the actuator works as a filter avoidingthe chattering generated by the control law, therefore the signaldelivered to the ropes will be smooth.

0 20 40 60 80 100 120 140 160 180−4

−2

0

2

4Control Action

Δ l (

m)

0 20 40 60 80 100 120 140 160 180−0.1

0

0.1Tracking error

0 20 40 60 80 100 120 140 160 180

−0.1

0

0.1

Surface

Time (s)

Fig. 11. Control effort, tracking error and sliding surface.

All results are obtained under the assumption that signalsof the system states, the output and the wind velocity areavailable and it is not introduced noise to the signals.

The most important result is the harvested energy in thecycle, Fig. 12 shows the rope length, the rope tension and theinstant power during whole process. The produced energy wascompared with the results in [4] noticing that:

• The designed controller is capable to keep the systemworking in presence of the disturbances introduced.Mainly, the difference between the value of the windvelocity using in the controller and the value presentin the system. Also, the uncertainties introduced by themodel simplification were diminished when using thecontroller designed in this work.

• As a consequence, good trajectory tracking was obtainedwhile lead to a greater energy production compared withthe results published in [4].

0 20 40 60 80 100 120 140 160 1800

10

20Rope Tension

Ten

sion

(kN

)

0 20 40 60 80 100 120 140 160 180−10

0

10

20

Instant Power

Time (s)

Pow

er (

kW)

0 20 40 60 80 100 120 140 160 1800

100

200Rope length

Leng

th (

m)

Fig. 12. The area under the curve in instant power graph represents the energy.Green line is the produced energy and red line is the consumed energy. Inthis test the produced energy is Eout = 1304kJ and the consumed energy isEin = 100kJ .

VI. CONCLUSION

In this work a sliding mode controller was designed using asimplified model and was tested in numerical simulation usinga complete model to represent the system. The results indicatethat this technique is suitable to control an AWE system, sincethe produced energy is greater than the consumed energy inthe whole process; also when compares to previous resultspublished in the literature. As a future work it is proposedto minimize the control effort in power phase, in order todecrease the energy consumed. Also, it is proposed to provethe controller robustness and its stability against disturbances.Moreover, the obtained results are planed to be evaluated in areal platform.

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