Lorenzo Fagiano

Lorenzo Fagiano, Aldo U. Zgraggen, Manfred Morari et al.

An approach to control tethered wings for airborne wind energy is proposed. A fixed length of the lines is considered, and the aim of the control system is to obtain figure-eight crosswind trajectories. The proposed technique is based on the notion of the wing's "velocity angle" and, in contrast with most existing approaches, it does not require a measurement of the wind speed or of the effective wind at the wing's location. Moreover, the proposed approach features few parameters, whose effects on the system's behavior are very intuitive, hence simplifying tuning procedures. A simplified model of the steering dynamics of the wing is derived from first-principle laws, compared with experimental data and used for the control design. The control algorithm is divided into a low-level loop for the velocity angle and a high-level guidance strategy to achieve the desired flight patterns. The robustness of the inner loop is verified analytically, and the overall control system is tested experimentally on a small-scale prototype, with varying wind conditions and using different wings.

Lorenzo Fagiano, Andrew R. Teel

This manuscript contains technical results related to a particular approach for the design of Model Predictive Control (MPC) laws. The approach, named "generalized" terminal state constraint, induces the recursive feasibility of the underlying optimization problem and recursive satisfaction of state and input constraints, and it can be used for both tracking MPC (i.e. when the objective is to track a given steady state) and economic MPC (i.e. when the objective is to minimize a cost function which does not necessarily attains its minimum at a steady state). It is shown that the proposed technique provides, in general, a larger feasibility set with respect to existing approaches, given the same computational complexity. Moreover, a new receding horizon strategy is introduced, exploiting the generalized terminal state constraint. Under mild assumptions, the new strategy is guaranteed to converge in finite time, with arbitrarily good accuracy, to an MPC law with an optimally-chosen terminal state constraint, while still enjoying a larger feasibility set. The features of the new technique are illustrated by three examples.

Aldo U. Zgraggen, Lorenzo Fagiano, Manfred Morari

Airborne wind energy systems aim to harvest the power of winds blowing at altitudes higher than what conventional wind turbines reach. They employ a tethered flying structure, usually a wing, and exploit the aerodynamic lift to produce electrical power. In the case of ground-based systems, where the traction force on the tether is used to drive a generator on the ground, a two phase power cycle is carried out: one phase to produce power, where the tether is reeled out under high traction force, and a second phase where the tether is recoiled under minimal load. The problem of controlling a tethered wing in this second phase, the retraction phase, is addressed here, by proposing two possible control strategies. Theoretical analyses, numerical simulations, and experimental results are presented to show the performance of the two approaches. Finally, the experimental results of complete autonomous power generation cycles are reported and compared with first-principle models.

Giuseppe C. Calafiore, Lorenzo Fagiano

This paper discusses a novel probabilistic approach for the design of robust model predictive control (MPC) laws for discrete-time linear systems affected by parametric uncertainty and additive disturbances. The proposed technique is based on the iterated solution, at each step, of a finite-horizon optimal control problem (FHOCP) that takes into account a suitable number of randomly extracted scenarios of uncertainty and disturbances, followed by a specific command selection rule implemented in a receding horizon fashion. The scenario FHOCP is always convex, also when the uncertain parameters and disturbance belong to non-convex sets, and irrespective of how the model uncertainty influences the system's matrices. Moreover, the computational complexity of the proposed approach does not depend on the uncertainty/disturbance dimensions, and scales quadratically with the control horizon. The main result in this paper is related to the analysis of the closed loop system under receding-horizon implementation of the scenario FHOCP, and essentially states that the devised control law guarantees constraint satisfaction at each step with some a-priori assigned probability p, while the system's state reaches the target set either asymptotically, or in finite time with probability at least p. The proposed method may be a valid alternative when other existing techniques, either deterministic or stochastic, are not directly usable due to excessive conservatism or to numerical intractability caused by lack of convexity of the robust or chance-constrained optimization problem.

Lorenzo Fagiano, Khahn Huynh, Bassam Bamieh et al.

A study on filtering aspects of airborne wind energy generators is presented. This class of renewable energy systems aims to convert the aerodynamic forces generated by tethered wings, flying in closed paths transverse to the wind flow, into electricity. The accurate reconstruction of the wing's position, velocity and heading is of fundamental importance for the automatic control of these kinds of systems. The difficulty of the estimation problem arises from the nonlinear dynamics, wide speed range, large accelerations and fast changes of direction that the wing experiences during operation. It is shown that the overall nonlinear system has a specific structure allowing its partitioning into sub-systems, hence leading to a series of simpler filtering problems. Different sensor setups are then considered, and the related sensor fusion algorithms are presented. The results of experimental tests carried out with a small-scale prototype and wings of different sizes are discussed. The designed filtering algorithms rely purely on kinematic laws, hence they are independent from features like wing area, aerodynamic efficiency, mass, etc. Therefore, the presented results are representative also of systems with larger size and different wing design, different number of tethers and/or rigid wings.

Enrico Terzi, Lorenzo Fagiano, Marcello Farina et al.

A comprehensive approach addressing identification and control for learningbased Model Predictive Control (MPC) for linear systems is presented. The design technique yields a data-driven MPC law, based on a dataset collected from the working plant. The method is indirect, i.e. it relies on a model learning phase and a model-based control design one, devised in an integrated manner. In the model learning phase, a twofold outcome is achieved: first, different optimal p-steps ahead prediction models are obtained, to be used in the MPC cost function; secondly, a perturbed state-space model is derived, to be used for robust constraint satisfaction. Resorting to Set Membership techniques, a characterization of the bounded model uncertainties is obtained, which is a key feature for a successful application of the robust control algorithm. In the control design phase, a robust MPC law is proposed, able to track piece-wise constant reference signals, with guaranteed recursive feasibility and convergence properties. The controller embeds multistep predictors in the cost function, it ensures robust constraints satisfaction thanks to the learnt uncertainty model, and it can deal with possibly unfeasible reference values. The proposed approach is finally tested in a numerical example.

Lorenzo Fagiano, Mustafa Khammash

Polynomial Chaos Expansions represent a powerful tool to simulate stochastic models of dynamical systems. Yet, deriving the expansion's coefficients for complex systems might require a significant and non-trivial manipulation of the model, or the computation of large numbers of simulation runs, rendering the approach too time consuming and impracticable for applications with more than a handful of random variables. We introduce a novel computationally tractable technique for computing the coefficients of polynomial chaos expansions. The approach exploits a regularization technique with a particular choice of weighting matrices, which allow to take into account the specific features of Polynomial Chaos expansions. The method, completely based on convex optimization, can be applied to problems with a large number of random variables and uses a modest number of Monte Carlo simulations, while avoiding model manipulations. Additional information on the stochastic process, when available, can be also incorporated in the approach by means of convex constraints. We show the effectiveness of the proposed technique in three applications in diverse fields, including the analysis of a nonlinear electric circuit, a chaotic model of organizational behavior, finally a chemical oscillator.

Lorenzo Fagiano, Eric Nguyen-Van, Felix Rager et al.

A control design approach to achieve fully autonomous take-off and flight maneuvers with a tethered aircraft is presented and demonstrated in real-world flight tests with a small-scale prototype. A ground station equipped with a controlled winch and a linear motion system accelerates the aircraft to take-off speed and controls the tether reeling in order to limit the pulling force. This setup corresponds to airborne wind energy systems with ground-based energy generation and rigid aircrafts. A simple model of the aircraft's dynamics is introduced and its parameters are identified from experimental data. A model-based, hierarchical feedback controller is then designed, whose aim is to manipulate the elevator, aileron and propeller inputs in order to stabilize the aircraft during the take-off and to achieve figure-of-eight flight patterns parallel to the ground. The controller operates in a fully decoupled mode with respect to the ground station. Parameter tuning and stability/robustness aspect are discussed, too. The experimental results indicate that the controller is able to achieve satisfactory performance and robustness, notwithstanding its simplicity, and confirm that the considered take-off approach is technically viable and solves the issue of launching this kind of airborne wind energy systems in a compact space and at low additional cost.

Enrico Terzi, Lorenzo Fagiano, Marcello Farina et al.

The derivation of multi-step-ahead prediction models from sampled data of a linear system is considered. A dedicated prediction model is built for each future time step of interest. In addition to a nominal model, the set of all models consistent with data and prior information is derived as well, making the approach suitable for robust control design within a Model Predictive Control framework. The resulting parameter identification problem is solved through a sequence of convex programs, overcoming the non-convexity arising when identifying 1-step prediction models with an output-error criterion. At the same time, the derived models guarantee a worst-case error which is always smaller than the one obtained by iterating models identified with a 1-step prediction error criterion.

Hafsa Farooqi, Lorenzo Fagiano, Patrizio Colaneri

This manuscript contains technical details of recent results developed by the authors on shrinking horizon predictive control with a move-blocking strategy.

Federico Maria Vitrò, Marco Leonesio, Lorenzo Fagiano

This paper proposes a novel approach to improve the training efficiency and the generalization performance of Feed Forward Neural Networks (FFNNs) resorting to an optimal rescaling of input features (OFR) carried out by a Genetic Algorithm (GA). The OFR reshapes the input space improving the conditioning of the gradient-based algorithm used for the training. Moreover, the scale factors exploration entailed by GA trials and selection corresponds to different initialization of the first layer weights at each training attempt, thus realizing a multi-start global search algorithm (even though restrained to few weights only) which fosters the achievement of a global minimum. The approach has been tested on a FFNN modeling the outcome of a real industrial process (centerless grinding).

Lorenzo Fagiano, Eric Nguyen-Van, Felix Rager et al.

The design of a prototype to carry out take-off and flight tests with tethered aircrafts is presented. The system features a ground station equipped with a winch and a linear motion system. The motion of these two components is regulated by an automatic control system, whose goal is to accelerate a tethered aircraft to take-off speed using the linear motion system, while reeling-out the tether from the winch with low pulling force and avoiding entanglement. The mechanical, electrical, measurement and control aspects of the prototype are described in detail. Experimental results with a manually-piloted aircraft are presented, showing a good matching with previous theoretical findings.

Lorenzo Fagiano, Stephan Schnez

The problem of launching a tethered aircraft to be used for airborne wind energy generation is investigated. Exploiting well-assessed physical principles, an analysis of three different take-off approaches is carried out. The approaches are then compared on the basis of quantitative and qualitative criteria introduced to assess their technical and economic viability. Finally, a deeper study of the concept that is deemed the most viable one, i.e. a linear take-off maneuver combined with on-board propellers, is performed by means of numerical simulations. The latter are used to refine the initial analysis in terms of power required for take-off, and further confirm the viability of the approach.

Eric Nguyen Van, Lorenzo Fagiano, Stephan Schnez

The problem of autonomous launch and landing of a tethered rigid aircraft for airborne wind energy generation is addressed. The system operates with ground-based power conversion and pumping cycles, where the tether is repeatedly reeled in and out of a winch installed on the ground and linked to an electric motor/generator. In order to accelerate the aircraft to take-off speed, the ground station is augmented with a linear motion system composed by a slide translating on rails and controlled by a second motor. An onboard propeller is used to sustain the forward velocity during the ascend of the aircraft. During landing, a slight tension on the line is kept, while the onboard control surfaces are used to align the aircraft with the rails and to land again on them. A model-based, decentralized control approach is proposed, capable to carry out a full cycle of launch, low-tension flight, and landing again on the rails. The derived controller is tested via numerical simulations with a realistic dynamical model of the system, in presence of different wind speeds and turbulence, and its performance in terms of landing accuracy is assessed. This study is part of a project aimed to experimentally verify the launch and landing approach on a small-scale prototype.

Marko Tanaskovic, Lorenzo Fagiano, Carlo Novara et al.

This manuscript contains technical details of recent results developed by the authors on the algorithm for direct design of controllers for nonlinear systems from data that has the ability to to automatically modify some of the tuning parameters in order to increase control performance over time.

Lorenzo Fagiano, Rudolf Gati

An approach to derive low-complexity models describing thermal radiation for the sake of simulating the behavior of electric arcs in switchgear systems is presented. The idea is to approximate the (high dimensional) full-order equations, modeling the propagation of the radiated intensity in space, with a model of much lower dimension, whose parameters are identified by means of nonlinear system identification techniques. The low-order model preserves the main structural aspects of the full-order one, and its parameters can be straightforwardly used in arc simulation tools based on computational fluid dynamics. In particular, the model parameters can be used together with the common approaches to resolve radiation in magnetohydrodynamic simulations, including the discrete-ordinate method, the P-N methods and photohydrodynamics. The proposed order reduction approach is able to systematically compute the partitioning of the electromagnetic spectrum in frequency bands, and the related absorption coefficients, that yield the best matching with respect to the finely resolved absorption spectrum of the considered gaseous medium. It is shown how the problem's structure can be exploited to improve the computational efficiency when solving the resulting nonlinear optimization problem. In addition to the order reduction approach and the related computational aspects, an analysis by means of Laplace transform is presented, providing a justification to the use of very low orders in the reduction procedure as compared with the full-order model. Finally, comparisons between the full-order model and the reduced-order ones are presented.