Stability Verification for Periodic Trajectories of Autonomous Kite Power Systems

Abstract

Model-based predictive controllers are frequently employed in Airborne Wind Energy (AWE) systems to follow reference flight paths. In this paper we compute the region of attraction (ROA) of path-stabilizing feedback gains around a closed trajectory as typically flown by a power generating kite. The feedback gains are obtained from applying a time-varying periodic Linear Quadratic Regulator to the system expressed in transversal coordinates. To compute the ROA we formulate Lyapunov stability conditions which we verify by employing semidefinite relaxations of the Positivstellensatz. This results in an optimization problem for which we compare various definitions of the cost function representing the size of the ROA. In a numeric case study we demonstrate that the maximization of an expanding ellipse inside of a quartic Lyapunov functions leads to significantly larger verified ROAs compare to the standard approach of optimizing over the scaling of the sublevel set of a quadratic Lyapunov function.

Transverse Contraction-Based Stability Analysis for Periodic Trajectories of Controlled Power Kites with Model Uncertainty

“…We call this subspace the transverse subspace for which we obtain the associated system dynamics by a transformation of the system into transverse coordinates. This transformation was introduced in [16] and [17] and employed for stability analysis of limit cycles in [18], and [6]. We briefly restate the transformation law for completeness.…”

“…We present an algorithm to efficiently compute an estimate of the ROC for polynomial systems with limit cycles. Similar to the computational approach in [6] we employ a numerical method proposed in [10], in which semidefinite relaxations of the Positivstellensatz [20] are used to formulate SOSprograms to test polynomial positivity as semidefinite program efficiently. Non-polynomial systems can be considered by a Taylor series approximation.…”

“…We fix the radius ρ of B to a positive constant as an optimization over both ρ and M ⊥ would not lead to larger ROCs but to a mere rescaling of the coefficients of M ⊥ . In order to still have a measure to increase the ROC we need to include an additional constraint in which we are optimizing over the radius of a ball B 2 = {x ⊥ : x T ⊥ x ⊥ ≤ ρ 2 } such that B 2 ⊂ B, similar to an approach employed in [6]. This encourages the algorithm to find metrics which increase the size of the ROC during the optimization over M ⊥ .…”

“…The algorithm (18) is dependent on the timeparametrization τ . We efficiently treat this dependency in a similar way as employed in [6] by solving the optimization (18) for N fixed and equally spaced values of τ over the whole period. For the metric M ⊥ we can translate this discretization of τ by casting the entries of M ⊥ as piecewise linear in τ as described in [6].…”

“…Typically while generating power the motion pattern of the kite can be considered periodic, allowing for the application of limit cycle stability analysis tools. In a first approach, such tools were used in [6] to investigate guarantees for the stability of a path-tracking feedback control design for a kite flying periodic trajectories. Therein, the stabilizing region of a periodic path-stabilizing Linear Quadratic Regulator (LQR) was investigated via Lyapunov methods for a nominal Eva Ahbe and Roy S. Smith are with the Automatic Control Laboratory, Swiss Federal Institute of Technology (ETH Zurich), Physikstrasse 3, 8092 Zurich, Switzerland, {ahbee,rsmith}@control.ee.ethz.ch.…”