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Most of the research so far on two wheeled mobile robots is on the control of the robot body. However when a robot is implemented in real environment the range of initial conditions of different states of the robot need to be known for safe operation. This paper presents the estimation of safe region of initial conditions of states for a statically unstable two wheeled mobile robot using a standard Lyapunov direct method. The method uses a known Lyapunov Function. Variation of stability region with the Lyapunov functions requires the selection of a good Lyapunov function for a system. Twotypes of Lyapunov functions, Quadratic and Energy based were formulated. The simulation results for the stability regions were evaluated using these Lyapunov functions and were compared with true region, determined using ode solver. A Lyapunov function that showed less conservative stability region was selected to show the theoretical stability region. The experiments verified the results.

The improved agility and performance of modern aircraft has led to a re-evaluation of many control techniques and the requisite for more contemporary methods capable of providing consistent handling qualities. Offering robust control in the face of both parametric and unstructured uncertainty, this paper details the application of a neural network-based controller to a twin ducted fan aircraft. Augmenting the existing pitch and altitude controller, the step response of the system is examined within the nominal flight regime and at the bounds of the aircraft's flight envelope. No a priori knowledge of the system dynamics is assumed.

A Two Wheeled Mobile Robot (TWMR) is a highly nonlinear and open-loop unstable system. Dynamic analysis and control design to keep the body of the robot balanced has been an area of research in the last decade. This problem has been dealt by many researchers for the robot motion on flat surfaces. A few studies have addressed the motion control on constant sloped path. This work studies the control of TWMRs on an uneven surface. With respect to previous results in the literature, the main contributions of this study are the dynamical modeling and control design of a two-wheeled mobile robot while traversing on uneven surface, in particular a single bump. A criterion has been proposed for the system performance evaluation. The system response to a baseline control and proposed Gain Scheduling control is quantified in simulation through proposed criteria. The results show an improvement in system performance on an uneven surface.

Wind turbines are complex, nonlinear, dynamic systems forced by aerodynamic, gravitational, centrifugal, and gyroscopic loads. The aerodynamics of wind turbines is nonlinear, unsteady, and complex. Turbine rotors are subjected to a complicated 3-D turbulent wind inflow field with imbedded coherent vortices that drive fatigue loads and reduce lifetime. Design of control algorithms for wind turbines must account for multiple control objectives. Future large multi-megawatt turbines must be designed with lighter weight structures, using active controls to mitigate fatigue loads while maximizing energy capture and adding active damping to maintain stability for these dynamically active structures operating in a complex environment. At the National Renewable Energy Laboratory, we are designing, implementing, and testing advanced controls to maximize energy extraction and reduce structural dynamic loads. These control designs are based on a linear model of the turbine that is generated by specialized modeling software. In this paper, we describe testing of a control algorithm to mitigate blade, tower and drivetrain loads using advanced state-space control methods. The controller uses independent blade pitch to regulate the turbine’s speed in Region 3, mitigate the effects of shear across the rotor disk, and add active damping to the tower’s first fore-aft bending mode. In addition, a separate generator torque control loop is designed to add active damping to the tower’s first side-to-side mode and the first drivetrain-torsion mode. This paper describes preliminary implementation and field tests of this controller in the Controls Advanced Research Turbine at the National Renewable Energy Laboratory. We make preliminary performance comparisons of this controller to results from a typical baseline proportional-integral-derivative controller designed with just Region 3 speed regulation as the goal.

*† ‡ Wind turbine components are sized according to their expected fatigue loads and any reduction in these would decrease costs and prolong working life. Wind turbine control is a multivariable problem with large interaction across multiple control channels (inputs to objectives). This article studies coupling and investigates a composite load controller for a wind turbine, consisting of a linear quadratic regulator for the loads and a baseline power controller for speed and torque regulation. The load objectives were to reduce tower foreaft, low speed shaft (LSS) tilt, and LSS yaw bending fatigue. Simulations in above rated wind conditions evaluated the composite design against the baseline controller alone, as well as a classical architecture with multiple single-input single-output (MSISO) loops. In this way, it is important to compare to a control design that is not aware of any coupling. Results showed significant mitigation of damage equivalent loads in the tower fore-aft, and LSS tilt and yaw directions across both designs when compared to baseline. The composite controller achieves better load reduction without the need for filtering out the effects of other channels, as would be required in the MSISO design.

Wind turbines are complex, nonlinear, dynamic systems forced by aerodynamic, gravitational, centrifugal, and gyroscopic loads. The aerodynamics of wind turbines is nonlinear, unsteady, and complex. Turbine rotors are subjected to a complicated three-dimensional (3-D) turbulent wind inflow field with imbedded coherent vortices that drive fatigue loads and reduce lifetime. Design of control algorithms for wind turbines must account for multiple control objectives. Future large multi-megawatt turbines must be designed with lighter weight structures, using active controls to mitigate fatigue loads, maximize energy capture, and add active damping to maintain stability for these dynamically active structures operating in a complex environment. Researchers at the National Renewable Energy Laboratory are designing, implementing, and testing advanced controls to maximize energy extraction and reduce structural dynamic loads. These control designs are based on a linear model of the turbine that is generated by specialized modeling software. This paper describes testing of a control algorithm to mitigate blade, tower, and drivetrain loads using advanced state-space control methods. The controller uses independent blade pitch to regulate the turbine's speed in Region 3, mitigate the effects of shear across the rotor disk, and add active damping to the tower's first fore-aft bending mode. Additionally, a separate generator torque control loop is designed to add active damping to the tower's first side-side mode and the first drivetraintorsion mode. This paper discusses preliminary implementation and field tests of this controller in the Controls Advanced Research Turbine at the National Renewable Energy Laboratory. Also included are preliminary comparisons of the performance of this controller to results from a typical baseline Proportional-Integral-Derivative controller designed with just Region 3 speed regulation as the goal.

Through high order aeroelastic simulations in turbulent wind, it has been shown that a Scheduled Model Predictive Controller (SMPC) can successfully control a wind turbine in above rated wind conditions. The SMPC shows the ability to control Multi-Input MultiOutput systems with multiple control objectives, allow for system input constraints and also adjust to the aerodynamic nonlinearities of the turbine system. The implementation of the SMPC is also only as complex as implementing a Linear Model Predictive Controller (LMPC) but provides the benefits of being able to control nonlinear systems with higher performance. Furthermore it has been shown that by using a SMPC the user has the ability to finely tune each controller the SMPC is comprised of giving more tailored performance to various operating regions.

The inherent periodic behavior of an operating wind turbine is not well accommodated by common time-invariant analysis and control techniques. A multi-blade coordinate transformation (MBC) helps to overcome this issue for rotors with three or more blades by mapping the dynamic state variables into a non-rotating reference frame. A number of researchers have applied MBC for modal analyses and individual blade pitch controller designs. They do so by assuming the transformed system model from MBC is time-invariant, which is not often the case. The paper explores the validity of the time-invariant assumption by comparison to direct periodic techniques, which retain all periodic system information. In a modal analysis study, eigenvalues of a system after MBC are compared to direct Floquet modes. In an individual blade pitch control design study, a linear quadratic regulation (LQR) design after MBC is compared to direct periodic LQR. A 5-MW three-bladed wind turbine model is used to quantify performance differences. Normal operating conditions are considered as well as conditions selected to increase the harmonics that are unfiltered by MBC. It is found that the direct periodic methods produce almost identical results to timeinvariant methods after MBC under all conditions studied. MBC is recommended for threebladed turbines, which can be followed by Floquet analysis or periodic control design methods if necessary.

This paper presents the development of autonomous control of a passenger vehicle for vehicle following using a robotic driver. The aim is to have a modular and transferable device that can autonomously control a standard passenger car. The robot controls the car in the same way as a human driver, by rotating the steering wheel and pushing the pedals. No permanent modification is required for the host car to be driven by the robotic driver. The robot uses PID controllers for both steering and pedal actuators. Information for motion control is provided by a 6-axis inertial measurement unit with integrated GPS and information about the environment is provided by a laser scanner. The open source Player Project is used as the framework for the controlling software and for simulation of control algorithms. The robotic driver has been tested on a rolling road dynamometer for accelerator and braking response.

This paper describes the development of a mobile two-wheel balancing platform to facilitate the testing of autonomous mobile applications. Two-wheeled platforms offer the benefits of increased stability for tall robots with a smaller footprint, resulting in greater agility. The integration of software and hardware is an important aspect of the design, as well as the implementation of reference trajectory controllers. Sensors to measure the speed, tilt angle and tilt rate of the platform are incorporated with two embedded PCs. The first PC provides control commands to the system based on MATLAB and Simulink and the second PC is responsible for implementing advanced autonomous applications such as people following and obstacle avoidance using 'Player', a well known robot software interface used for robotics research. The LQR controller is tuned to yield better closed-loop performance and tested with the platform.

*Fundamental to wind turbine operation is keeping the rotor facing toward the prevailing wind. Doing so maximizes the power capture and reduces the asymmetric loads on the rotor blades. This study investigates a possible new yaw control technique through actively varying blade pitch angles. It focuses on the feasibility of active yaw control through periodic state-space individual pitch control on the WindPACT 1.5MW three-bladed upwind turbine. The non-linear wind turbine simulator FAST was used in Matlab/Simulink software. A low order controller model with two degrees-of-freedom was used during the control system design and simulations were performed on a high order nonlinear model. The periodic linear optimal control technique was used in this study. Results of the study proved that yaw control was achievable through periodic state-space individual pitch control, without affecting performances on power generation or speed regulation. Fatigue loads could be kept within an acceptable range with the assistance of a yaw brake and other tower load control algorithms. Fine tuning may further improve the performance of the controller.

Unmanned aerial vehicles (UAVs) comprise various types of aircrafts such as conventional fixed-wing aircraft, helicopters, blimps, and air- ships. Among these, helicopters are classified as planar vertical take off and landing (PVTOL) aircraft by which it means that unlike a fixed-wing aircraft, it can take off and land in a limited space, hover in the air, and move sideways and backwards. This superior maneuverability allows performing important roles in many areas, which conven- tional aircraft could not achieve. The type of their useful work includes: dangerous applications such as in a war, victim rescue and volcano monitoring, where other types of vehicle are inaccessible, commercial application such as film making, and agricultural applications, farm monitoring and spreading chemicals [1]. The demand for UAVs keeps increasing. Unfortunately these exclusive maneuverability advantages give a big complexity and instability in its dynamics, hence making it hard to control. The development of a UAV is challenging, and it is an emerging area in nonlinear control study among researchers.

The Controls Advanced Research Turbine (CART) is a modified Westinghouse WWG-0600 machine rated at 600 kW. It is located at the National Wind Technology Center (NWTC) in Boulder, Colorado, and has been installed to test new control schemes for power and load regulation. In its original configuration, the WWG-0600 uses a synchronous generator, fluid coupling, and hydraulic collective pitch actuation. However, the CART is fitted with an induction generator, rigid coupling, and individual electromechanical pitch actuators. The rotor runs upwind of the tower and consists of two blades and a teetering hub. In order to design advanced control schemes for the CART, representative computational models are essential.

This tutorial paper describes the field of wind turbine control. It begins with the simplest turbine dynamic models and progresses to more advanced models which incorporate structural resonances in the blades and supporting tower. Disturbance accommodating control techniques are used to provide power control in fluctuating wind fields.

In this paper we investigate the use of Disturbance Accommodating Control (DAC) for periodic and non periodic control of two -bladed horizontal -axis wind turbines operat ing in Region III. R ealizab le DAC controller/estimator pair s are developed for use on an existing wind -turbine called the CART (Control s Advanced Research Turbine) located at the National Wind Testing Technology Center in Golden, CO. Previously, these designs were done using a gen eric model of the turbine . The DAC controller s developed are verified on a simulation of this CART model . T his model was created and validated in time -invariant control studies. Four different case s are presented: each with a different set of DOF’s. In each case, both a constant -gain and a periodic -gain Full -State Feed Back (FSFB) controller are developed. In addition to the FSFB controllers, a constant -gain and a periodic -gain realizable controller are developed . A total of 16 different simulations ha ve been run: 4 different controllers for each of the 4 cases. All simulations were run against a 5 DOF non -linear plant model. The results indicate that the periodic controllers are slightly better than the constant -gain controllers, but the less complic ated non -periodic controller s might be adequate for wind turbine applications. Also, the controllers with higher numbers of DOF’s did not perform any better than the simple controllers. However, the design of the non -periodic controller is based upon the periodic design; consequently periodic control is an essential part of the design methodology. NOMENCLATURE

A composite state-space controller was developed for a multi-objective problem in the variable-speed operation of wind turbines. Disturbance Tracking Control theory was applied to the design of a torque controller to optimize energy capture under the influence of persistent wind disturbances. A limitation in the theory for common multi-state models is described, which led to the design of a complementary pitch controller. The goal of the independent blade pitch design was to minimize blade root fatigue loads. Simulation results indicate an 11% reduction in fatigue damage using the proposed controllers, compared to a conventional torque-only design. Meanwhile, energy capture is almost identical, partly because of nonlinear effects.

Performance of a model-based periodic gain controller is presented using Disturbance Accommodating Control (DAC) techniques to estimate fluctuating wind disturbances. Operation is restricted to speed regulation using Independent Blade pitch Control (IBC) and by measuring only rotor angle and speed. The modeled turbine is a free-yaw, two-bladed, teetered rotor machine with simple blade and tower flexibility — 6 discrete degrees of freedom. A comparison is made to a time-invariant DAC controller, constructed by various approaches. Results indicate that the two controllers perform nearly identically, despite the inherent periodic dynamics in the system. The best time-invariant state estimator model was designed by freezing the periodic plant at a specified rotor azimuth position.Copyright © 2001 by The American Institute of Aeronautics and Astronautics Inc. and ASME

The use of individual pitch control (IPC) based on loads transformed into nonrotating coordinates is explored on a range of wind turbines with ratings between 5MW and 15MW. Turbine models are generated using classical upscaling based on properties of the NREL 5MW reference wind turbine. The Ziegler-Nichols method is used with a low order linear model of each turbine to objectively tune a gain-scheduled, proportional-integral individual pitch controller. The performance of IPC is assessed by measuring reductions in blade and tower root damage equivalent loads from simulations at several wind speeds spanning Region 3. It is observed that the load reductions obtained with individual pitch control are maintained on upscaled turbines, with minimal impact on tower root loads, while actuator usage scales at a rate lower than expected with classical scaling.

This paper examines the operating modes of a twobladed teetered wind turbine modeled with seven degrees of freedom: nacelle yaw, hub teeter, flapping of each blade, tower fore-aft bending, tower lateral bending, and tower twist. Because of the gyroscopic asymmetry of its rotor, this turbine’s dynamics can be quite distinct from that of a turbine with three or more blades. This asymmetry leads to system equations with periodic coefficients that must be solved by the Floquet approach to extract the correct modal parameters. Results illustrate how the turbine modes become more dominated by the centrifugal and gyroscopic effects as the rotor speed increases. Under certain design conditions, the gyroscopic coupling may cause phase locking or frequency coalescence of the yaw and teeter modes resulting in self-excited dynamic instabilities.

An investigation has been made into the performance of Disturbance Accommodating Control designed for a structural model with rotor speed as the only degree of freedom. A comparison has been made with the same controller attempting to regulate speed of a structural model with four degrees of freedom (including tower and blade deflection). The latter model was constructed using a symbolic modeling method, capable of providing the system equations of motion explicitly. For the current investigation, however, only simulation response was analyzed. The results show unstable closed-loop behavior of the multivariable system, due to excitation of collective blade flap-wise motion from the blade pitch controller.

A computational structural dynamics model for a Horizon,tal Axis Wind Turbine is presented, referred to as SymDyn. The model is capable of providing system dynamics in explicit form as opposed to only time responses. Validation of the model is made by comparison to a model built in ADAMS, a commercial dynamics simulation code. Properties for the models are taken from a generic two-bladed, constant speed, teetered-rotor turbine. Aerodynamic loads are neglected but provision is made for user defined external loading on the blades. Simulation results show almost exact agreement for a number of different test cases. Minor differences are attributed to the different numerical integration schemes employed.

This paper presents the development of autonomous control of a passenger vehicle for vehicle following using a robotic driver. The aim is to have a modular and transferable device that can autonomously control a standard passenger car. The robot controls the car in the same way as a human driver, by rotating the steering wheel and pushing the pedals. No permanent modification is required for the host car to be driven by the robotic driver. The robot uses PID controllers for both steering and pedal actuators. Information for motion control is provided by a 6-axis inertial measurement unit with integrated GPS and information about the environment is provided by a laser scanner. The open source Player Project is used as the framework for the controlling software and for simulation of control algorithms. The robotic driver has been tested on a rolling road dynamometer for accelerator and braking response.

Low-cost Autonomous Ground Vehicles (AGVs) can make effective tools for mechatronics education. By using a Field-Programmable Gate Array (FPGA) as the controller and central processor, students have a wide degree of flexibility in what sensors and algorithms are implemented. A vehicle has been developed at The University of Auckland, New Zealand, for this purpose using a toy car chassis and custom electronics, with an Altera FPGA development board. The completed platform cost approximately NZ$1000 and is capable of navigating a pre-defined path with errors of less than 2 m, using a low-end GPS and magnetic compass.

Floating offshore wind turbines offer a feasible solution for water depths greater than 60m. However, floating wind turbines will experience additional motion induced by incident wind and waves due to the lack of rigid foundations. Therefore, the control system becomes an important component that can be used to reduce the induced motions or, if not carefully taken into consideration, could exacerbate the motions. In our previous work, we demonstrated that using state space controllers with multiple objectives resulted in improved performance with respect to a gain scheduled PI controller. Further improvements were obtained by implementing individual blade pitching through periodic control as it overcomes the limitations of collective blade pitching. However, the periodic controller destabilized un-modeled degrees of freedom (DOFs) and these DOFs had to be included in the design of the controller to ensure stability of the system. Disturbance accommodating controllers can further improve the performance by canceling or reducing the effects of persistent disturbances: incident wind and waves. Currently, only wind speed is modeled as a disturbance to reject. Simulation results show that only when the platform yaw degree of freedom is included in the controller design the performance is improved through better power and speed regulation.

A disturbance accommodating controller (DAC) to reject wind speed perturbations is applied on a 5MW wind turbine mounted on a tension leg platform (TLP). Multi-blade coordinate (MBC) transformation is used to address the periodicity of the floating wind turbine system. A method to apply DAC after applying MBC transformation is developed and several implementation options are presented. Simulations were carried out to assess the fatigue loads of the system in accordance with IEC61400-3 standard design load case 1.2; however simulations were restricted to region 3 analysis due to the lack of region transition logic currently implemented for the DAC. A gain scheduled PI (GSPI) controller is used as a baseline/reference controller to compare the performance of the DAC. Simulation results show that the DAC significantly improves power and speed regulation, reduces tower fatigue loads, and maintains the fatigue loads of the blades and the shaft to a comparable level to the baseline controller applied on the TLP. This improvement is due to the use of individual blade pitching and rejecting wind speed variations. Relative to an onshore wind turbine with a GSPI controller, the DAC reduces tower side-side fatigue load and maintains the blade and shaft loads to a comparable level. Tower fore aft fatigue loads remain higher by 27% (instead of 45% increase by the baseline controller on the TLP).

An investigation of the performance of a model-based periodic gain controller is presented for a variable speed wind turbine. An attempt is made to quantify how periodic the chosen four degree of freedom linear system is, using modal properties. It was found that, while the system appears to be significantly periodic compared to other configurations, high gains were required for periodic control to outperform a fixed gain controller in speed regulation. INTRODUCTION Fixed gain control is a common technique used in performance regulation of wind turbines. Proportional- integral-derivative (FDD) control is one such method that has had widespread use because little knowledge of the plant dynamics is necessary prior to implementation. (If we assume a particular mode of turbine operation, gain-scheduling techniques are excluded from consideration.) As dynamics models of turbines are developed, more information can be incorporated into model-based controllers, with generally better performance. Optimal Control and Disturbance Accommodating Control (DAC) are two such methods of this type that have been investigated. If the controller model is assumed time-invariant as is invariably the case to take advantage of extensive control theory this method again involves constant gains. The periodic nature of wind turbine dynamics, both from aerodynamic loads and structural properties, suggests an improvement can be made to overall performance if a periodic control law, with periodic gains, is used. The goal of this paper is to make an assessment of how effective periodic control can be, compared to fixed gain control. Model-based linear control design requires a suitably accurate dynamics model of the system. For structural dynamics we may take advantage of SymDyn, a utility that derives the equations of motion of a simple wind Copyright © 2001 by the American Institute of Aeronautics and Astronautics Inc. and the American Society of Mechanical Engineers. All rights reserved turbine model explicitly. In SymDyn rigid bodies with single revolute joints and springs are used to model all flexible parts namely the tower, shaft, and blades. The result is a relatively simple set of discrete degrees of freedom. The equations of motion have been validated with ADAMS, a rigorously tested dynamics code, and have been exercised in applications such as operating modal analyses and preliminary control studies.*' For the analysis in this paper a simple four degree of freedom SymDyn model is used. An explicit set of governing equations for aerodynamic loads on turbine blades is currently not available. Instead, to complete the aeroelastic model we use aerodynamic subroutines from AeroDyn, developed at the University of Utah. AeroDyn has been coupled with many other structural dynamics models, including YawDyn, FAST_AD, and ADAMS. Details on how AeroDyn is used in this paper will be presented in a later section. Optimal periodic control has seen rigorous mathematical treatment in the last few decades and recently appeared in applications such as aircraft trajectory and spacecraft orbit optimization''. Hie application of this theory to wind turbines is straightforward provided familiar control issues such as stabilizability are addressed. The control objective in this paper is to regulate turbine speed in a fluctuating wind field near rated wind speed (region III or constant power turbine operation). We use a constant generator torque and control full-span blade pitch. The following section of this paper describes the structural and aerodynamic models in more detail and explains how the periodic linear model is derived. The remaining sections introduce methods for measuring how periodic a system is, describe the control system designs, and then present results comparing the fixed and periodic gain controllers. MODEL DESCRIPTION The turbine modeled is a downwind, free-yaw, variable speed machine with two blades and no hub teeter. Only (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. flap motion of each blade is considered, using a flap hinge and torsional spring to model the first flap bending mode. The tower, shaft, and blade sections were treated as rigid bodies. The result is four degrees of freedom: nacelle yaw (y), shaft rotation/azimuth position (vy), and flap of each blade (Pi, (J2). These degrees of freedom were chosen to give a periodic yet simple system to analyze. Properties were chosen to resemble the AWT-26 horizontal axis wind turbine. An accurate representation of a real turbine was not necessary in this study. See Figure 1 for an illustration and Table 1 for a list of the main geometric properties. The model may be recognized from previous papers by the authors as a reduced order SymDyn model.' SymDyn is a utility used to derive the equations of motion of a wind turbine symbolically. For the system described here the nonlinear equations of motion can be represented by the following vector equation. where f(q,q,q,L)=0, 0) Pi q= Pi IPJ and L is the vector of applied loads, described shortly. Description Hub height Blade length Yaw axis to flap hinge distance Shaft to flap hinge distance Precone angle Symbol - - d* 4 3o Value 25m 13m 2.8m 5.8m 7° Table 1: Geometric properties of the turbine model

The offshore wind energy potential is huge and to capture that energy, turbines have to be placed further offshore. Floating wind turbines offer a solution for deep waters. However, a floating wind turbine has extra motions that will affect the turbine in power production and structural loads. Therefore, the turbine control system has to be able to regulate power production and maintain safe operation of the turbine under incident wind and wave conditions. The work presented here discusses the application of state-space optimal controllers to regulate rotor speed and platform pitch above rated wind speed. A gain scheduled PI controller is used as a baseline to gauge the performance of the new controllers developed. A collective pitch linear quadratic regulator, designed to only regulate rotor speed and platform pitch, improves system performance but this improvement is thought to be due to better controller tuning as both controllers use the same mechanism to restore platform pitch and regulate speed. Individual blade pitch control using periodic control theory is applied because it uses a different mechanism to regulate platform pitch. Preliminary simulation results show that individual blade pitch control has platform pitch regulation over collective pitch controllers. However, unintended excitation of platform roll indicate that a more complicated controller may be required to ensure closed loop stability of the entire floating turbine.

Modal analysis and the future design of control strategies for a wind turbine rely heavily on the development of system descriptive dynamics models. Until now, wind turbine models have focused only on turbine simulations and not system characterization, which is necessary for the evaluation of operating modes, parameterization of stability, and controI designs. This paper presents an explicit, descriptive model of a rigid component, generic horizontal axis wind turbine, from which the system parameters and operating modes are extracted. A four degree-offreedom two-bladed turbine under free vibration is analyzed in parts and then as a complete system, presenting physical insight into the dynamic coupling behavior. A study perfonned at different rotor azimuth positions exposes the importance of Floquet analysis to better understand rotor-nacelle dynamic interactions. Mode shapes are identified and results are presented for modal frequencies at different rotor speeds.

This paper introduces the use of time-varying modern control theory for wind turbine applications. The dynamics of an operating wind turbine are periodic in time due to the rotational motion of its blades. One cannot then use common linear time-variant control principles to guarantee closed-loop stability or desirable performance. Periodic state-space control can be employed, whereby both the feedback gains and state estimator are periodic. Such ideas have been utilized in simulation studies to yield superior results for multi-objective problems.

The present study applies an approximate output regulator for controlling a MIMO nonlinear non-minimum phase system, considering the example of a four degree of freedom spherical inverted pendulum. The spherical pendulum consists of a slim cylinder attached to a universal joint upon which the planar control force acts. The aim in output tracking is to control the pendulum such that the base follows a desired reference trajectory as closely as possible while maintaining the upright position. The standard output regulator requires the solution to a mixed algebraic partial differential equation, which entails finding the manifold on which the system state trajectories result in exact tracking of the reference signal. This can be very difficult to solve in practice for the non-minimum phase case thus motivating the use of approximation methods. A local approximation method, based on a Taylor series expansion of the system dynamics is used, such that an output regulator may be applied to the spherical inverted pendulum. This gives the first application of output regulation for the output tracking of this system.