Predictive Control of Autonomous Kites in Tow Test Experiments?
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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: fwoodt, [email protected]. 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, jγ_ (t)j < 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. 2 3 2 3 px(t) r(t) cos(θ(t)) cos(φ(t)) Based on experimental observations in [6], we relate the p(t) = 4py(t)5 = 4r(t) cos(θ(t)) sin(φ(t))5 : 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].