Predictive Control of Autonomous Kites in Tow Test Experiments?

Tony A. Wood1, Henrik Hesse2, and Roy S. Smith1

Abstract—In this paper we present a model-based control ap- operating conditions and time delay which make tuning dif- proach for autonomous flight of kites for wind power generation. ficult and effectively limit the overall system power output, Predictive models are considered to compensate for delay in especially for ground-based systems. the kite dynamics. We apply Model Predictive Control (MPC), with the objective of guiding the kite to follow a figure-of-eight To improve the control performance in experimental im- trajectory, in the outer loop of a two level control cascade. plementations of kite power prototypes, model-based control The tracking capabilities of the inner-loop controller depend approaches have been proposed in [5]–[8]. The underlying on the operating conditions and are assessed via a frequency kinematic models link the kite heading angle to the overall kite domain robustness analysis. We take the limitations of the inner motion. In [6] the estimated kite heading angle was further tracking controller into account by encoding them as optimisation constraints in the outer MPC. The method is validated on a kite related to the steering input with a model that includes an system in tow test experiments. input delay. State estimation for ground-based (AWE) system with output delay is addressed in [9]. In [6]–[8], [10], [11] guidance strategies have been developed for experimental kite I.INTRODUCTION power systems to allow path following. In particular, [7] Airborne Wind Energy (AWE) generators have been pro- used a kinematic model including input delay for a figure- posed as a mobile, cost-effective, and more sustainable alter- of-eight path planning and tracking strategy. To ensure robust native to conventional wind turbines. In this work we focus on performance of the control approach we further considered kite power systems which generate power by flying a multi- limitations on the tracking bandwidth imposed by the input line tethered wing or kite in crosswind motion following a delay. The robustness of the cascaded control architecture figure-of-eight pattern. The tethers are connected to winches of [6], [7] was further improved following a Model Predictive on the ground which generate power by unreeling the lines in Control (MPC) approach in [11] by formulating the path the so-called traction phase. The traction phase is alternated following problem as an optimisation problem with constraints with a retraction phase to reel in the tethers using only a on the heading angle, its rate of change, and the kite position. fraction of the energy generated during traction, leading to MPC is naturally suited to address the complexities in the a net positive cycle power. control of kites, e.g. constraint satisfaction and minimisation of Since the original inception of the kite power concept [1], the deviation from a reference path, and has been extensively several groups have developed prototype systems and the explored in simulation in [12]–[15]. The application of MPC reference book [2] provides a detailed overview of the field approaches to kite power systems, however, tends to be sen- of (AWE). In this work we focus on the control of kites during sitive to unmodelled dynamics and hindered by limitations in the traction phase of a ground-based system, as developed processing power for real-time operation. In [11] we therefore in [3], with ground-based measurements of line angles and use the kinematic model introduced in [7] which allows for length, steering actuation, and power generation at the so- online adaptation of model parameters to reduce the model called ground station (GS). The experimental implementation mismatch. The approach was demonstrated in simulation to of autonomous kite power systems requires stabilising control achieve path following while satisfying constraints imposed arXiv:1703.07025v2 [math.OC] 22 May 2017 approaches which can handle the unstable, nonlinear dynam- by the limitations of a lower-level tracking controller that are ics. Model-free guidance approaches based on a switching subject to model uncertainty and input delay. The ability to point strategy and using the kite heading angle as feedback account for constraints is the main benefit over computation- variable for tracking [4], [5] provide a successful starting ally more efficient guidance methods that are purely based on point for further control development. The performance of geometry such as in [16]. such model-free approaches, however, is affected by varying In this paper we extend the kinematic model to account for variations of the kite velocity within a figure-of-eight cycle ?This manuscript is a preprint of a paper accepted for the IEEE Control Sys- tems Letters and is subject to IEEE Control Systems Society copyright. Upon which significantly improves the tracking capabilities of the publication, the copy of record will be available at http://ieeexplore.ieee.org. path following controller developed in [11]. Moreover, as main 1 T. A. Wood and R. S. Smith are with the Automatic Control Labora- contribution, we demonstrate the performance of the MPC tory (IfA), ETH Zurich, Physikstrasse 3, 8092 Zurich, Switzerland. E-mail: {woodt, rsmith}@control.ee.ethz.ch. Tel: +41 44 632 3629. approach for varying line length during tow test experiments 2H. Hesse is with the Aerospace Sciences Division, University of Glasgow with a prototype kite power system. (Singapore), 510 Dover Road, Singapore 139660 (formerly at IfA, ETH In Section II we describe the cascaded control architecture Zurich). Email: [email protected] This research was supported by the Swiss National Science Foundation with delay compensation and constrained outer-loop guidance (Synergia) No. 141836. that accounts for limitations of the inner-loop controller. In ez C0 Limitation lr analyser

γ(t) cmd Tethered kite δ(t) Heading angle γ (t) Predictive guidance p(t) system tracking controller controller γ(t) r(t) θ(t), φ(t) Fig. 2: Cascaded control architecture with inner-loop controller θ(t) tracking a commanded heading angle, γcmd(t), given by the

ey outer-loop guidance controller which incorporates the limita- cmd ϕ(t) tions of the inner loop as constraints, |γ˙ (t)| < lr.

ex A. Control Models Fig. 1: Coordinate system, with x-axis aligned with the wind The evolution of the kite position expressed in terms of line direction, and kite trajectory showing kite position, p(t), and angles, ξ(t) := (θ(t), φ(t)), can be modelled by the following heading angle, γ(t). kinematic unicycle model, [7], v (t) θ˙(t) = θφ cos (γ(t)) , (2a) Section III we describe the implementation of the control r v (t) scheme for tow test experiments. We present experimental φ˙(t) = θφ sin (γ(t)) , (2b) results in Section IV before concluding in Section V. r cos(θ(t)) where the velocity perpendicular to the tethers is modelled as II.CONTROL SCHEME a static function of the position, ξ(t), and orientation, γ(t), We consider a two line kite power system in the traction phase where the kite is flown in crosswind conditions. The vθφ(t) = rαL cos (θ(t)) cos (φ(t)) − rαG cos (γ(t)) , (3) motion of the kite perpendicular to the tethers is actuated by with parameters αL, αG > 0 representing velocity compo- the difference in lengths of the two lines, δ(t). The position of nents arising from lift and gravitational forces. Modelling the the kite in Cartesian coordinates, p(t), can be expressed as a velocity to vary within a figure-of-eight cycle captures the function of the elevation angle, θ(t), the azimuth angle, φ(t), true behaviour more closely than the assumption of constant and the line length, r(t), which are illustrated in Figure 1 and velocity made in the control model of [7], [11] as can be seen are all measured from the GS, in the experimental results shown in Figure 6a.     px(t) r(t) cos(θ(t)) cos(φ(t)) Based on experimental observations in [6], we relate the p(t) = py(t) = r(t) cos(θ(t)) sin(φ(t)) . heading angle, γ(t), to the steering input, δ(t), by integrator pz(t) r(t) sin(θ(t)) dynamics with a time delay td,

A cascaded architecture, illustrated in Figure 2, is used to γ˙ (t) = Kδ(t − td) . (4) control the kite along a figure-of-eight path. Controlling the The parameters, r, α , α , K, and t are considered constant heading angle defined as L G d within the control horizon and are re-identified online based ! cos(θ(t))φ˙(t) on updated measurements. γ(t) := arctan , (1) θ˙(t) B. Tracking Controller has been shown to be an effective approach for autonomous crosswind flight control of kites [4]–[6]. The outer loop of To control the heading angle we account for the delay in the cascade is controlled by the guidance controller which (4) in a model-based approach as described in [6]. We predict td produces a commanded heading angle trajectory, γcmd(t), that the orientation after the delay time, γ (t), and apply a pro- is tracked in the inner loop by the tracking controller. portional gain controller to the difference of the commanded The steering behaviour of the kite is affected by line and the predicted output, cmd td dynamics due to the indirect tether actuation. This effect can be δ(t) = C0(γ (t) − γ (t)) . (5) modelled as an input delay and taken into account in a model- based delay compensation scheme [6]. The presence of delay Controlling delayed systems based on model predictions is and model uncertainty imposes fundamental limitations on the referred to as predictor feedback [17]. To assess the perfor- tracking performance. As in [11] we assess the limitations mance of the tracking controller we consider the regulation of the time-shifted tracking error, e (t) := γcmd(t) − γ(t + t ). of the tracking controller, based on estimates of the current td d operating conditions, with a frequency domain robustness The control model in (4) is simple but there is a considerable analysis. We parametrise the limitations by an upper bound on degree of uncertainty in the steering gain parameter, K, and delay, td. In the frequency domain we model the uncertainty the rate of change of the commanded signal. The rate limit, lr, is communicated to the optimisation-based guidance controller of the plant, γ = G(s)δ, which takes the current inner-loop tracking capabilities into K G(s) = G (s)e−std = e−std , account as constraints. 0 s with a multiplicative perturbation with the perturbation weight 0.4 for delayed first-order systems introduced in [18],

n o [rad] 0.35

θ G := (1+Wm(s)∆(s)) G(s) k∆(s)k∞ ≤ 1 , (6) ξtd+TH ( K+δK −jδtdω π 0.3 K e − 1 if ω < δt , d Elevation ξ td Wm(jω) = K+δK π ξ + 1 if ω ≥ , K δtd 0.25 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 where δK, δtd are the bounds on the deviations of the Azimuth φ [rad] parameters, and ∆(s) is an unknown but bounded perturbation. Fig. 3: Prediction of flight trajectory: delay compensation The controller in (5), parametrised by the control gain, C0, cmd (dashed), MPC prediction (solid); reference path (dotted). can be written as, δ = C(s; C0)(γ − γ), with

C0 C(s; C0) = −st . ref 1 + C0G0(s)(1 − e d ) online such that the corresponding reference orientation, γ , is of sinusoidal form and its rate of change satisfies the For perturbed plants, Gp(s) ∈ G, the relation between the p limitation given by the tracking controller as suggested in [7]. input and the shifted error is, et = S (s; C0)δ, with d td This implies that for lower limits on the rate of change, larger 1 + L(s; C )(1 − estd ) (1 + W (s)∆(s)) figure-of-eight paths are required. Sp (s; C ) = 0 m , td 0 We apply a MPC approach with the objective of minimising 1 + L(s; C0) (1 + Wm(s)∆(s)) the deviation of the delay compensating prediction, ξtd (t), where L(s; C0)=C(s; C0)G(s) is the loop transfer function. ref from the reference path, ξ , over a finite time horizon, TH , The tracking capabilities depend on the control gain, C0, while constraining the kite to remain in a predefined safety cmd and on the properties of the commanded signal, γ (t). We window, ξ ≤ ξtd (t) ≤ ξ, and satisfying the limit on the would like to set the control gain and constrain the commanded rate of change of the commanded heading angle imposed by signal such that the shifted error remains small, |et (t)| < le, cmd d the tracking controller, |γ˙ (t)| < lr. We also constrain the for all perturbed plants in the set given in (6). In particular, we magnitude of the commanded orientation, |γcmd(t)| < l , cmd m limit the magnitudes of the commanded signal, |γ (t)| < lm, to avoid commanding the kite to fly straight down towards cmd and its rate of change, |γ˙ (t)| < lr. As discussed in [11], the ground. More details on the optimisation problem that is these limits on the commanded signal and the shifted tracking formulated and solved to capture the guidance objective are error are approximately translated to the frequency domain presented in Section III-B where the kinematic model given with the robust performance condition in (2) and (3) is used in the formulation of state constraints and sup kW (jω; l )Sp (jω; C )k < 1 , the limitation on the turn rate, derived from steering model (4), p r td 0 ∞ (7) k∆k∞≤1 is implemented as input rate constraint. Figure 3 illustrates the guidance concept with the predicted for specified parameters l and C , where we design the r 0 path of the kite. The prediction consists of two parts: the first performance weight to be part is determined by the past inputs in the delay compensation ( lm lr l if ω < l , scheme; the further evolution is due to current and future Wp(jω; lr) = e m lr if ω ≥ lr . actuation and can be optimised by the MPC. leω lm Bounds for the left-hand side of (7) can be determined by a III.EXPERIMENTAL IMPLEMENTATION structured singular value analysis. Given values of the model MPC for autonomous kites has been studied extensively in parameters, K, td, the level of uncertainty considered, δK, δtd, literature [12]–[15] but demonstrations have been limited to and the bounds lm, le, the condition in (7) depends on the simulation. In this work we design and conduct experiments control gain, C0, and the rate limit, lr. We follow the scheme to validate the applicability of predictive crosswind flight suggested in [11], where C0 is chosen for robust stability control. We further demonstrate a novel experimental method and lr is selected such that robust tracking performance is to test kite controllers in low wind conditions using a tow test guaranteed, i.e. the condition in (7) is satisfied. configuration.

C. Predictive Guidance Control A. Tow Test Experiments For the design of the guidance controller we assume that The experimental flight control tests are performed on a the tracking controller is able to follow the commanded signal prototype (AWE) system [3] which has been developed at cmd well but delayed by the steering delay, i.e., γ(t) ≈ γ (t−td). Fachhochschule Nordwestschweiz (FHNW) and modified to We apply the model given in (2) and (3) to compensate for allow for tow test experiments as depicted in Figure 4. By the delayed tracking. Given a prediction of the kite position mounting the GS on the back of a truck we can create relative t t t td ahead of time, ξ d (t) := (θ d (t), φ d (t)), we find a value wind when driving the vehicle on an airfield runway. The for the commanded signal at the current time, γcmd(t), that origin of the coordinate frame shown in Figure 1 moves with controls the kite to follow a reference figure-of-eight path, the truck and the x-axis points in the opposite direction to the ξref := (θref, φref). The reference path is generated and updated vehicle velocity. prediction from the reference path corresponds to controlling the deviation system to zero. We linearise the kite kinematics around the reference path, starting at the reference point closest to the prediction of the kite position after the delay time, to obtain the linear time vary- ∂fd ref ref ing system χi+1 ≈ Aiχi + Biui, with Ai := ∂ξ (ξj+i, γj+i) ∂fd ref ref and Bi := ∂γ (ξj+i, γj+i), for i = 0, 1,...,H − 1, where H is the number of discrete time steps in the prediction horizon. To encode the input rate constraint into the optimisation Fig. 4: Tow test configuration with the kite connected to GS problem we augment the deviation state such that its dynamics, control system (right) mounted on a moving vehicle (left). χˆi+1 = Aˆiχˆi + Bˆi∆ui,       χi Ai Bi Bi 2 χˆi := , Aˆi := , Bˆi := , For the results presented here, an HQ Apex III 5m ram air ui−1 0 1 1 kite was connected to the GS via two tethers of maximum are actuated by the change in input, ∆u := u − u . length 150 m. The forces on the tethers are controlled by i i i−1 The following optimisation solves the guidance task, regulating the reel-out speed of the winches. The winch control is independent of the steering control. The entire control and H−1 X > ˆ >
> ˆ > min χˆi Qχˆi +i Si +χ ˆH QH χˆH +H SH H , (8) estimation architecture is implemented in Matlab Simulink and ∆u, runs on a Speedgoat Real-Time Target Machine with a fixed i=0 ˆ ˆ sampling time, T = 0.01 s. With this mobile configuration s. t. χˆi+1 = Aiχˆi + Bi∆ui , we can test in still wind conditions and create reproducible χ ≤ χ +  , χ −  ≤ χ ,  ≥ 0 , i i i i i i i relative wind scenarios to develop controllers for crosswind u ≤ u ≤ u , ∆u ≤ ∆u ≤ ∆u , flight of kites with variable tether length. i i i i i i with actuation rate sequence ∆u := (∆u0, ∆u1,..., ∆H−1), B. Controller Implementation slack variables sequence  := (0, 1, . . . , H ), state limits χ := ξ −ξref , χ := ξ −ξref , input limits u := −l −γref , i j+i i j+i i m j+i As part of the predictive control approach in this work, ref ref ref we use real-time measurement data to identify and update the ui := lm − γj+i, rate limits ∆ui := −lrT − γj+i + γj+i−1, ref ref control model parameters online in an adaptive fashion. Given ∆ui := lrT − γj+i + γj+i−1, and positive-definite weighting ˆ ˆ estimates of the model parameters, the tracking control gain, matrices S, SH , Q := diag(Q, R), QH := diag(QH ,R), C0, and the rate limit on the commanded heading angle, lr, are where (Q, QH ) penalise the position deviation, R penalises the set such that robust tracking performance is guaranteed. We orientation deviation, and (S, SH ) penalise the slack variables. use a sufficient analytic condition derived in [11] to determine The inclusion of the slack variables with high penalisation suitable values for C0 and lr. The condition involves an S  Q and SH  QH prevents the constraint optimisation upper bound on the left-hand side of (7) which is explicitly problem from becoming infeasible in situations where there parametrised by the tuning variables C0 and lr, allowing for is no possibility of keeping the kite in the desired position simple evaluation online. window. This is relevant when initialising the controller from The kinematic model given in (2) and (3) is discretised with arbitrary positions. The optimisation problem (8) is imple- the forward Euler method, mented and solved online with the optimisation software 2 generation tool FORCES Pro [19]. θk+1 =θk +T αL cos(θk) cos(φk) cos(γk)−T αG cos (γk) , The value of the current commanded heading angle at time ∗ T αG sin(2γk) step k is determined by the first element of the sequence ∆u φk+1 =φk +T αL cos(φk) sin(γk)− . cmd ∗ cmd ref 2 cos(θk) that obtains the minimum of (8), γk = ∆u0 +γk−1 −γj−1 + ref With the short sampling period of T = 0.01 s, no signifi- γj . The optimisation problem is re-solved in every time step cant error is introduced by the discretisation. We denote the with a receding horizon. discretised model, which is used in the guidance control, as IV. RESULTS ξk+1 = fd(ξk, γk), with ξk := (θk, φk). The figure-of-eight reference path is parametrised by a The predictive control scheme has been experimentally ref ref ref ref periodic sequence of positions, ξ = (ξ1 , ξ2 , . . . , ξN ), and tested in tow test experiments. In this section we present ref ref ref ref orientations, γ = (γ1 , γ2 , . . . , γN ), that satisfy the model the results of an experiment conducted on the runway of St. ref ref ref dynamics, ξi+1 = fd(ξi , γi ), for i = 0, 1,...,N − 1, and Stephan airport in Switzerland on 9 December 2016. Data ref ref ref ξ1 = fd(ξN , γN ). from two test flights (Flight 1 and Flight 2) are presented. To capture the guidance objective, we consider a system The number of stages in optimisation problem (8) was describing the deviation of the prediction of the position after selected to be H = 30, resulting in a MPC prediction horizon td the delay from the reference path with state χk := ξk − of TH = 0.3 s. Note that the overall prediction horizon ref cmd ref ref ref ξj and input uk := γk − γj , where (ξj , γj ) represent of the guidance controller consists of the sum of the delay the reference point closest to the prediction of the kite state compensation and the MPC horizon; with an average estimated after the delay time. Minimising the deviation of the position delay of approximately td = 0.7 s the overall prediction trajectories of the position and its prediction can be explained 0.6 by model mismatch and to a greater extend by the tracking

[rad] error in the inner-loop controller. For the same flight interval

θ 0.5 (Flight 1), the velocity perpendicular to the tethers, vθφ(t), 0.4 obtained from the derivative of the line angle measurements,

Elevation and its prediction based on (3) are shown in Figure 6a. The 0.3 significant velocity variation within the cycle is captured by −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 the velocity model in (3) which leads to better predictions Azimuth φ [rad] compared to the assumption of constant velocity. The tracking Fig. 5: Trajectory tracking results (Flight 1): kite position tra- of the commanded heading angle during the cycle (Flight 1) jectory, ξ(t) (solid), following a reference figure-of-eight path, is illustrated in Figure 6b. The heading angle, γ(t) follows ξref (dotted), with delay compensation prediction trajectory, the commanded signal, γcmd(t), well with a time shift. Note ξtd (t) (dashed). The trajectories start at the position marked that the model of the steering dynamics in (4) is independent with crosses at time t = 0s. of the kite position but due to the feedback of the estimated heading angle good tracking can be achieved. Considering a longer flight duration (Flight 2), we can 14 observe the adaptation to changing operating conditions. Fig- [m/s] ure 7 shows a flight trajectory over 3 minutes. As the wind

θφ 12 v 10 speed and line length change, the parameter estimates vary, shown in Figure 8a, and the bound on the rate of change of

Velocity 8 the commanded orientation, shown in Figure 8b, is adapted 0 2 4 6 8 10 12 14 16 18 20 22 according to the changing limitations of the tracking controller. Time t [s] We observe that the controller is able to track figure-of-eights (a) Velocity trajectory, vθφ(t), estimated based on derivatives of q cycles while the wind speed changes between 2.7 m/s and position measurements, r(t) θ˙2(t) + cos2(θ(t))φ˙2(t) (solid), and 6.6 m/s. Note that the periods of the reference paths are based on model (3) (dashed). relatively high due to the large delay and the high uncertainty considered in the steering model. 2 Throughout the experiment the line length is reeled out from [rad]

γ 79 m to 100 m. For larger line lengths the estimate of the 0 delay increases in general. For fast increases of the wind speed, however, the tether forces increase resulting in less line sag and −2 a lower delay estimate despite the line reeling out as evident Heading angle 0 2 4 6 8 10 12 14 16 18 20 22 for 80-100 s in Figure 8a. In this event the model assumptions Time t [s] are inaccurate leading to a poor model fit and an outlier in the (b) Trajectory of heading angle, γ(t) (solid), tracking commanded parameter estimates. cmd By updating the model parameters every half cycle the signal, γ (t) (dotted), with shifted tracking error, etd (t) (dashed). system can react to slow disturbances and adapt to a varying Fig. 6: Trajectories of velocity and heading angle (Flight 1). environment. The results indicate that good tracking perfor- mance can be achieved by constraining the guidance based on the limitations of the tracking controller and by re-evaluating horizon stretches over approximately 1 s. The weights of these for changing operating conditions. the MPC objective were Q = diag(1, 2), QH = 5 · Q, −3 5 5 R = 5 · 10 , S = 10 · Q, SH = 10 · QH . The safety window was defined by the position bounds θ = 0.17 rad, V. CONCLUSION θ = 1.40 rad, φ = −0.70 rad, and φ = 0.70 rad. The limit We have presented an MPC approach to fly kites au- on the commanded heading angle was set to be lm = 2.5 rad tonomously in crosswind conditions for power generation. and the desired maximum tracking error was selected to be The control problem is split into two parts with a cascaded le = 0.9 rad. The model parameters were updated online twice control architecture. We apply predictor feedback to account per figure-of-eight cycle with least-squares fits to real-time for the input delay affecting the system. To ensure that the measurements. The uncertainty levels of the parameters of the commanded signal determined by the outer control loop is steering model were set to be 20% of the estimated parameter not too fast for the inner control loop to track, the limitations values, δK = 0.2 · K, δtd = 0.2 · td. of the inner-loop controller are determined using a robustness Figure 5 shows the tracking of a reference figure-of-eight analysis based on model parameter uncertainty. The limitations path over one cycle (Flight 1). We observe that the kite are parametrised by a bound on the rate of change of the position trajectory, ξ(t) := (θ(t), φ(t)), follows the reference commanded signal. This bound is taken into account as a path, ξref. The delay compensated prediction of the position, constraint in the predictive guidance control. ξtd (t), follows the reference path more closely as it is the The main benefit of applying MPC in this approach is signal used in the MPC objective. The difference between the constraint satisfaction. The optimisation framework, however, [m/s] 6 w 5 4 3 [rad] Wind speed

θ 0.6 [m]

r 95 0.4 90 Elevation 85 −0.5 80 Line length 160 140 0 120 0.8 100 [s] d

80 t 0.6 60 40 0.4 0.5 20 Time t [s] Azimuth φ [rad] 0 Delay 0.2 Fig. 7: Long duration position trajectory, ξ(t) (Flight 2).

[rad/sm] −5 K −5.5 also enables the consideration of new objectives. The approach −6 has been successfully tested in tow test flight experiments. The 0 20 40 60 80 100 120 140 160 optimisation-based guidance strategy is able to steer the kite Steering gain Time t [s] to follow figure-of-eight paths. By adapting model parameters (a) Measurements of the wind speed, w(t), and the line length, r(t); and control constraints online using updated measurements, estimates of the parameters in the steering model (4). autonomous flights under strongly varying operating condi- tions were achieved. 1

VI.ACKNOWLEDGEMENTS 0.5 The authors would to thank J. Heilmann, D. Aregger, I.

Turn rate [rad/s] 0 Lymperopoulos, and S. Diwale (A2WE project [3]) for their 0 20 40 60 80 100 120 140 160 contribution in conducting the flight experiments. Support Time t [s] from embotech GmbH and in particular A. Hempel is grate- (b) Bound on the rate of change of the commanded heading angle, fully acknowledged. lr (solid), and resulting magnitude of the rate of the commanded orientation, |γ˙ cmd(t)| (dotted). REFERENCES Fig. 8: Evolution of parameters (Flight 2): operating conditions [1] M. L. Loyd, “Crosswind kite power,” J. of Energy, vol. 4, 1980. and parameter estimates (8a), resulting limit on the rate of [2] U. Ahrens, M. Diehl, and R. Schmehl, Airborne Wind Energy. Springer, change of the commanded signal (8b). 2013. [3] “Autonomous Airborne Wind Energy project (A2WE),” http://a2we. skpwiki.ch, Oct. 2015. [4] L. Fagiano, A. U. Zgraggen, M. Morari, and M. Khammash, “Automatic [13] M. Canale, L. Fagiano, and M. Milanese, “High altitude wind energy crosswind flight of tethered wings for airborne wind energy: Modeling, generation using controlled power kites,” IEEE Trans. on Control Syst. control design, and experimental results,” IEEE Trans. Control Syst. Technol., vol. 18, no. 2, pp. 279–293, 2010. Technol., vol. 22, no. 4, pp. 1433–1447, 2014. [14] A. Ilzhofer,¨ B. Houska, and M. Diehl, “Nonlinear MPC of kites under [5] M. Erhard and H. Strauch, “Flight control of tethered kites in au- varying wind conditions for a new class of large-scale wind power tonomous pumping cycles for airborne wind energy,” Control Eng. generators,” Int. J. of Robust and Nonlinear Control, vol. 17, no. 17, Practice, vol. 40, pp. 13–26, 2015. pp. 1590–1599, 2007. [6] T. A. Wood, H. Hesse, A. U. Zgraggen, and R. S. Smith, “Model- [15] M. Zanon, S. Gros, and M. Diehl, “Model predictive control of based identification and control of the velocity vector orientation for rigid-airfoil airborne wind energy systems,” in Airborne Wind Energy. autonomous kites,” in American Control Conf., 2015, pp. 2377–2382. Springer, 2013. [7] ——, “Model-based flight path planning and tracking for tethered [16] C. Jehle and R. Schmehl, “Applied tracking control for kite power wings,” in IEEE Conf. on Decision and Control, 2015, pp. 6712–6717. systems,” J. of Guidance, Control, and Dynamics, vol. 37, no. 4, pp. [8] N. Rontsis, S. Costello, I. Lymperopoulos, and C. N. Jones, “Improved 1211–1222, 2014. path following for kites with input delay compensation,” in IEEE Conf. [17] M. Krstic, Delay Compensation for Nonlinear, Adaptive, and PDE on Decision and Control, Dec. 2015, pp. 656–663. Systems. Springer, 2009. [9] M. Polzin, T. A. Wood, H. Hesse, and R. S. Smith, “State estimation for [18] D. L. Laughlin, D. E. Rivera, and M. Morari, “Smith predictor design kite power systems with delayed sensor measurements,” in IFAC World for robust performance,” Int. J. of Control, vol. 46, no. 2, pp. 477–504, Congr., 2017. 1987. [10] S. Costello, G. Franc¸ois, and D. Bonvin, “Directional real-time opti- [19] A. Domahidi and J. Jerez, “FORCES Professional,” embotech GmbH mization applied to a kite-control simulation benchmark,” in European (http://embotech.com/FORCES-Pro), Jul. 2014. Control Conf., 2015, pp. 1594–1601. [11] T. A. Wood, E. Ahbe, H. Hesse, and R. S. Smith, “Predictive guidance control for autonomous kites with input delay,” in IFAC World Congr., 2017. [12] M. Diehl, H. G. Bock, and J. P. Schloder,¨ “A real-time iteration scheme for nonlinear optimization in optimal feedback control,” SIAM J. on Control and Optimization, vol. 43, no. 5, pp. 1714–1736, 2005.