06132436.pdf

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 2, MARCH 2013 347

Rejection of Periodic Wind Disturbances on a SmartRotor Test Section Using Lifted Repetitive Control

Ivo Houtzager, Jan-Willem van Wingerden, and Michel Verhaegen

AbstractA repetitive control method is presented that isimplemented in real-time for periodic wind disturbance rejectionfor linear systems with multiple inputs and multiple outputs andwith both repetitive and non-repetitive disturbance components.The novel repetitive controller can reject the periodic wind dis-turbances for fixed-speed wind turbines and variable-speed windturbines operating above-rated and we will demonstrate this onan experimental smart rotor test section. The smart rotor isa rotor where the blades are equipped with a number of controldevices that locally change the lift profile on the blade, combinedwith appropriate sensors and controllers. The rotational speed ofwind turbines operating above-rated will vary around a definedreference speed, therefore methods are given to robustify therepetitive controllers for a mismatch in the period. The design ofthe repetitive controller is formulated as a lifted linear stochasticoutput-feedback problem on which the mature techniques of dis-crete linear control may be applied. For real-time implementation,the computational complexity can be reduced by exploiting thestructure in the lifted state-space matrices. With relatively slowchanging periodic disturbances it is shown that this repetitivecontrol method can significantly reduce the structural vibrationsof the smart rotor test section. The cost of additional wear andtear of the smart actuators are kept small, because a smoothcontrol action is generated as the controller mainly focuses on thereduction of periodic disturbances.

Index TermsReal time, repetitive control, smart rotor control,vibration control, wind turbine control.

I. INTRODUCTION

T HE trend in offshore wind turbines is to increase the rotordiameter as much as possible to reduce the cost per kWh.The increasing dimensions have led to a relative increase of theloads on the wind turbine structure, thus it is necessary to reactto disturbances in a more detailed way: each blade separatelywith individual pitch control (IPC) [1][4], or even at several ra-dial distances spanwise along each blade with smart rotor con-trol (SRC) [5][7]. The idea for IPC seems very attractive andalmost readily implementable, since many of the larger wind

Manuscript received February 28, 2011; revised July 11, 2011; acceptedNovember 06, 2011. Manuscript received in final form December 19, 2011.Date of publication January 16, 2012; date of current version February 14,2013. This work was supported under Contract TMR.5636 by the DutchTechnology Foundation STW, Applied Science Division of NWO, and thetechnology program of the Ministry of Economic Affairs. Recommended byAssociate Editor M. Lovera.The authors are with the Delft Center for Systems and Control, Faculty of

Mechanical, Maritime, and Materials Engineering, Delft University of Tech-nology, Delft 2628 CD, The Netherlands (e-mail: [email protected]; [email protected]; [email protected]).Digital Object Identifier 10.1109/TCST.2011.2181171

Fig. 1. Illustrative example of a wind turbines with the smart rotor concept.At the tip of the blade a number of additional control devices are placed; forexample, the translational tabs and trailing-edge flaps illustrated in (a) and (b),respectively. (a) Translational tab. (b) Trailing-edge flaps.

turbines already have individual pitch actuators, although con-trolled collectively, for the regulation of the rotor speed. How-ever, the performance of the IPCmethod is restricted by the lim-ited bandwidth of the pitch actuator and they only affect the loadon the whole blade [8]. The SRC concept is borrowed from thehelicopter industry [9], where active devices with significantlyhigher bandwidth, like trailing-edge flaps or translational tabs,are proposed to reduce the loads by manipulating the airflow lo-cally and consequently the aerodynamic forces. In Fig. 1(b) anillustrative example is presented of a wind turbine with smartrotor blades.The goal for both IPC and SRC is to increase the lifetime

of the wind turbine components and make the scaling to largerrotor diameters possible, and therefore improve the cost effec-tiveness of wind turbines. Currently, researchers have been in-vestigating mainly the use of feedback control after decouplingthe rotating frame to a fixed non-rotating frame [1][4], or theuse of distributed feedback controllers [10]. However, feedbackcontrollers will typically give a lot of control variations to theactuators, and therefore the manufactures are reluctant to imple-ment these concepts in practice. The reason is that in the case offeedback control, the lifetime of the wind turbine becomes de-pendent on the wear and tear of the pitch/smart actuators. A pre-liminary study in [11] investigated the effects of the turbulenceon pitch mechanism failures. It was observed from statistical

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348 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 2, MARCH 2013

Fig. 2. Left sideview of a modern wind turbine, showing the wind shear effect.

Fig. 3. Topview of a cross section from the tower, showing the tower shadoweffect.

analysis that the cross-correlations between failures and the tur-bulence intensity coefficients are much larger than to the meanwind speed deviation coefficients. This could indicate that thecurrent pitch control are tuned in such a way that the pitch actu-ators are acting to heavily on the turbulence. Thus the numberof control variations on these actuators should be kept to a min-imum.The new concepts could provide the designer with an in-

creased envelope which can be used to, for instance, mount therotor downwind, remove one blade, and allow lighter compo-nents or increase themean loading on the blades which increasesthe power conversion. One of the research trends in wind turbinedesigns is to equip wind turbines only with two blades and po-sition the rotor downwind instead [12]. The two-bladed down-wind designs have a number of significant advantages comparedto the current three-bladed upwind design. The use of only twoblades will give a lighter rotor that can operate at higher ro-tational speeds, is cheaper to produce, and will be easier to in-stall. Placing the rotor downwind will eliminate the tower clear-ance issues, such that more flexible blades and higher rotationalspeeds are possible. The success of this design will heavily de-pend on active control techniques to compensate the disadvan-tages; namely the increased periodic loads, due the increasedtower shadow (see Fig. 3) and the unbalance of a two-bladedrotor.The rejection of disturbances using SRC is already an ex-

perimental research topic for several years. The feasibility ofvibration reduction using feedback control with a trailing-edgeflap applied on a smart airfoil was demonstrated in [13] and[14]. In [15], the feasibility of vibration reduction using both

feedforward and feedback control with trailing-edge flaps on atwo-bladed rotor was demonstrated. The introduction of feed-forward control to the feedback control with inverse notchesimproved the load reduction and reduced the control action con-siderably. In this adaptive feedforward method, the amplitudesof 1P, 2P, 3P, and 4P sinusoidal basis functions are adaptivelyfitted using the blade-root bending moment measurements fromthe previous rotation, and the frequencies are obtained by mea-suring the rotational speed. We will extend the developmentof this adaptive feedforward control by proposing the use ofrepetitive control (RC) in the form of a lifted feedback con-troller, which takes in addition the non-sinusoidal periodic loaddisturbances into consideration. Furthermore, the proposed RCmethod is very general and can more easily be adapted to beused with an increasing number of sensors and actuators in thenew rotor concepts.The load disturbances acting on an individual wind turbine

blade are to a large extent deterministic, such as wind shear (seeFig. 2), tower shadow (see Fig. 3), yaw error, unbalance, andgravity, and they depend on the rotation angle and speed. Thecontribution of these periodic disturbances is becoming larger,due to the increasing length and mass of the rotor blades. Repet-itive controllers consist of a periodic signal generator, enablingrejection of these kind of periodic disturbances. General RCmethods for discrete multi-input and multi-output (MIMO) sys-tems that can potentially expand the application to wind tur-bines greatly have been developed in the time-domain, like in[16], and in the trial domain using lifted models, see [17]. A re-cent overview of past literature in RC can be found in [18]. RCmethods can achieve perfect reduction in the deterministic case,i.e., excluding non-repetitive disturbances. In [19], a lifted RCmethod is presented for MIMO linear systems with both repet-itive and non-repetitive disturbance components. This designmethod is formulated as a lifted linear stochastic output-feed-back problem on which the mature techniques of discrete linearcontrol may be applied. The computational complexity is evenconsiderably reduced by exploiting the structure in the liftedstate-space matrices.An important drawback of RC is that a small mismatch be-

tween the controller period and the actual period of the dis-turbance signal can decrease the performance of RC substan-tially. Several approaches have been proposed in the literatureto improve robustness and are summarized in [20]. These ap-proaches can be divided into two groups, depending if the truerotor speed is accurately measured or not known. In this paper,we focus on the latter as this is the case with the smart rotortest section. Although it is expected that RC performs better withthe approaches that require the rotor speed to be measured. In[19], a weight is included in the formulated lifted output-feed-back problem to specify the amount of variation in the peri-odic wind disturbances. In [20][23], the RC is extended withmore memory loops. This so-called high-order RC can be mademore robust for small changes in period time, but this comesat the cost of the amplification of noise at the non-harmonicfrequencies. Using these developments, the application of RCto wind turbines with small variations in rotational speed, suchas fixed-speed and variable-speed operating above-rated with apitch controller that will keep the rotational speed of the wind

HOUTZAGER et al.: REJECTION OF PERIODIC WIND DISTURBANCES ON A SMART ROTOR TEST SECTION 349

turbine close to the defined reference speed, becomesmore prac-tical.The contributions of this paper are as follows. First, we

propose an RC method which can deal with periodic wind dis-turbances, and leave the non-periodic disturbances (turbulence)mostly unaffected. This will generate a smooth control actionthat will keep the additional wear and tear on the smart actu-ators small. Second, we implement and study two extensions(random walk weight, multiple memory loops) in the liftedoutput-feedback problem to robustify the performance againstperiod mismatch. Third is the experimental verification of RCon a smart rotor test section with periodic and turbulentwind disturbances generated by a wind generator. In addition,we verify the capability of RC to operate with an additionalfeedback controller, which is present in modern wind turbinesfor stabilization. To the authors knowledge, this is the firstexperiment with an implemented lifted repetitive controller toreduce the periodic wind disturbances (with small variations inthe period time) in a turbulent wind field, although repetitivecontrollers in the time-domain for disturbance rejection havebeen implemented in real-time before in [22], [24], and [25].Note, that in [26] the RC method is also evaluated in simulationsoftware for the applicability on multi-MW wind turbines.The outline of this paper is as follows. In Section II the

problem formulation, the assumptions made, and some nota-tions are presented. In Section III the theoretical frameworkis presented for the RC problem and an algorithm for MIMOlinear time-invariant (LTI) systems is given. In Section IVthe effectiveness of the proposed algorithm is shown with anexperimental study of disturbance rejection on a smart rotortest section. Moreover, some tuning guidelines are derived. Inthe final section we present the conclusions of this paper.

II. SYSTEM DESCRIPTION, ASSUMPTIONS AND NOTATIONS

In this section the system description, the assumptions made,and some notation is presented.

A. System Description and Assumptions

The dynamics of the discrete-time system to be controlledcan be written as the following state-space model:

(1)

where , , , , ,and , are the state, input, output, periodic disturbance,process noise, and measurement noise vectors. The state-spacematrices , , , ,and are also called the state, input, periodic input,output, and direct feedthrough matrices, respectively. For sim-plification and clarification, we consider an LTI system. How-ever, the lifted repetitive control method described in this papercan be extended to periodic time-varying systems.The process and measurement noise disturbances are con-

sidered Gaussian noise sequences with and, respectively. In the case of colored Gaussian

Fig. 4. Block diagram of the system operating in closed loop.

noise sequences, the state-space system can be augmentedwith the noise filter dynamics. The periodic disturbance mayconsist of an integrated randomwalk component as [17], [27],[28]

(2)

where is a Gaussian noise sequence with ,and is the trial window size (or period). This is a quiteversatile noise formulation; the disturbance contains not only astochastic and a periodic component, but also a random walknature, so the periodic disturbance can vary between trials. Inaddition, the noise and periodic disturbances can be correlatedin time.We consider also the possibility that the system already

operates in closed loop with a feedback controller for stabiliza-tion or load reduction due to turbulence. A condition is thatthe feedback controller does not have poles on the unit disk,thus the feedback controller should let the repetitive controllerdeal with the repetitive disturbances. In this paper, we con-sider the repetitive control input as an additional input signalfed to the system, see Fig. 4. As discussed in [29], the repeti-tive control input could also be fed as a reference signal tothe feedback controller, but this can require a redesign of thefeedback controller to include reference tracking capabilities.For a successful RC design, the state-space description in (1)should in both cases be replaced by its closed-loop state-spacedescription. Thus the system should also be augmented witha state-space description of the feedback controller .

B. Lifted System Description and Assumptions

With the trial window size , the following stacked vector canbe defined as:

(3)

The stacked vectors , , , , and are definedin a similar way. Using the definition of the stacked vectors, wecan lift the system in (1) to the following lifted description:

(4)

where , , and are the ex-tended controllability matrices, and are given by


and , , and are the impulsematrices with a lower block triangular structure, isthe extended observability matrix, and are given by

...

. . ....

.... . . . . .

. . ....

.... . . . . .

. . ....

.... . . . . .

These lifted system matrices are considered trial invariant. Notethat is used to represent an identity matrix of appropriate di-mensions.In Assumption 1, we adopt the mild conditions commonly

used for the lifted repetitive control problem [19].Assumption 1: The system is asymptotically stable (if not, stabilize withfeedback control), controllable on , observable on

, and does not contain any zeros at . The process noise and measurement noise are zero-mean white or colored (if so, augment dynamics to system)Gaussian sequences.

The random walk noise is a zero-mean whiteGaussian sequence.

When the number of inputs is smaller than the number of out-puts , the controllability/observability condition in As-sumption 1 is normally violated. In this case the matrix

does not have full row rank. It is proposed toonly reduce the error that lies in the column space of that matrix,thus we try not to achieve perfect rejection for all disturbancesignals (or all the harmonics). To achieve this, we use the sin-gular value decomposition (SVD) as

(5)

where the order is determined by detecting a gap that separatesthe largest singular values from the remaining ones. Now, wecan replace the output equation of the lifted system in (4) by aprojected output equation given by

(6)

Furthermore, we can select the number of singular values notonly on the controllability/observability condition, but also tofurther reduce the size of the RC output-feedback problem andlet the lifted repetitive controller focus on the main harmoniccomponents and therefore robustify the controller performancein some sense; see also Section IV-C.

III. REPETITIVE CONTROL WITH LIFTED LQG DESIGN

In this section the theoretical framework for the repetitivecontrol method to reject periodic disturbances is presented. Thegoal is to formulate the RC design problem into an output-feed-back problem. After the formulation, the state-space matrices ofthe lifted repetitive controller can be calculated by solving twoRiccati equations.

A. Output-Feedback Formulation

To let the repetitive controller respond to periodic distur-bances, a non-stochastic output from (4) is defined as

(7)

and a non-stochastic error is defined, which is independent ofthe noise sequence from trial as

(8)

where is considered trial invariant. The change betweentwo trials is denoted by the operator, such that for examplethe difference between the vector of the input from time andthe vector of the input from time is defined as

. Using this definition, it follows that the non-stochastic tracking error is defined as:

(9)

An expression for the current stochastic tracking error basedon the previous non-stochastic error can be obtained and isgiven by

(10)

By using the operator again on the state as ,the difference in the initial conditions of between trials isderived from (7), and (2) as

(11)

Now we can combine (9), (10), and (11) into a stochastic linearsystem description as

(12)

where denotes the system is differenced once, the vectors, and are given by


and the system matrices ,, ,

, , and aregiven by

B. Multiple Memory LoopsAs described in [21][23], additional memory loops can be

introduced to make the RC more robust for small changes inperiod time or less sensitive to disturbances at non-periodic fre-quencies. This high-order RC output-feedback problem can beformulated in an almost similar state-space system as in (12). In[20] it was shown that themultiplication of the multiple memoryloops with the repetitive controller is commutative in the liftedsetting. This means that an additional memory loop can be intro-duced in the output-feedback problem by double-differencingthe input. Lets denote the double-differenced input as

where and with . For example,in [22] it was shown that and gives the bestperformance against period mismatch, and that and

gives the lowest sensitivity to disturbances at non-pe-riodic frequencies. The equations in (9), and (11) are differencedagain as

(13)

and

(14)

Similarly as the system in (12) with single-differenced input,the system with double-differenced input can also be combinedinto two matrix equations. This results in the following double-differenced (or second-order) RC output-feedback problem:

(15)

where denotes the system with double-differenced input, thevector is given by

and the system matrices ,, ,

and are given by

The above computations can be repeated to develop a period-robust structure based on a higher number of memory loops.For example, the high-order RC output-feedback problem for

memory loops would have an times differencedinput as

(16)

C. Lifted LQG SolutionIn this subsection the repetitive control method for LTI sys-

tems with both repetitive and non-repetitive disturbances is pre-sented. Lets replace and by a more general notation of todenote the system is differenced with an arbitrary number. Now,the high-order RC problem can be formulated as follows.1) Problem Description 1 (Lifted Repetitive Control):

Given the system in the form of , like in (12) or (15), andin (16), design a strictly-proper dynamic repetitive controller.The controller is strictly proper, because current trial mea-

surements are assumed unavailable for feedback. The block di-agrams in Figs. 5 and 6 illustrate how the lifted repetitive con-trollers can be implemented with one memory loop and twomemory loops, respectively. Given Assumption 1, the infinite-horizon discrete-time LQG solution is well known, see [30], andit is desirable in our situation, as it stabilizes the system andminimizes the expectation of the loss function:

Compared to the minimum-variance criterion in [19], we intro-duced in the LQG criterion a weight to penalize the differencedinput, and therefore we are able to make a tradeoff between


the variance of the differenced input and output signals. By re-quiring that stabilizes , it can be shown to get exponen-tial convergence of the RC algorithm. This does not imply thatthe RC algorithm has monotonic decayas recommended in [31], but the experience is that the introduc-tion of a weight on the differenced input could give the engineersome means to prevent for the large overshoots conventionallyseen with some RC laws with non-monotonic decay in the ini-tial transient periods, see also Section IV-C. Using a steady-stateapproximation of the optimal LQG solution, the lifted controller

is given by

(17)

where

The matrices and are the positive-definite stabilizing solu-tions to the discrete algebraic riccati equations (DARE)

(18)

where , , , and thedisturbance covariance matrices are calculated as

(19)

In many cases, the solvers for the Riccati equations in (18)have difficulties to determine the solution numerically. For ex-ample the QZ-based DARE solver in MATLAB will fail, becausethe Hamiltonian or sympletic matrix has eigenvalues on and/orclose to the unit circle. To overcome this problem, the learningrate should be lowered by adding forgetting or leakage.With theinclusion of forgetting, the weights of the times differencedinput in (16) should be constrained with , wherethe scalar is the so-called forgetting factor.The biggest challenge of implementing this RC description

of the output-feedback problem is that the lifted matrices areenormous, and it requires large memory storage and will take along time to solve the DARE. These issues can be resolved byexploiting the matrix structure in (18). For matrices in denseform, the computational complexity for DARE solvers usingthe Schur method is . In [19], efficient methods forsolving Riccati equations using the so-called sequentially semi-separable (SSS) matrix structure [32] and its properties are dis-cussed. An SSS matrix toolbox for MATLAB has been devel-oped that includes iterative algorithms to check matrix stability,to solve Lyapunov and Riccati equations, and to compute LQGcontrollers in complexity. The matrices , , , ,, , and can be put into this structure for fast com-

putation of the similar structured LQG solutions and tothe RC output-feedback problem. Further, the structured liftedrepetitive controller in Figs. 5 and 6 can be implemented

Fig. 5. Block diagram of the closed-loop system with the RC loop with onememory loop.

Fig. 6. Block diagram of the closed-loop system with the RC loop with twomemory loops.

real-time using three SSS structured matrix-vector computa-tions in instead of . In Fig. 7, the m-func-tion of the SSS structured matrix-vector product is comparedto the inbuilt dense matrix-vector product in computation time.Even in m-code the SSS structured matrix-vector product isfaster for . Although not directly beneficial for thesmart rotor section experiment (with ), the structuredmatrix-vector algorithms are used to verify their capability forreal-time computation. For controller synthesis, which requiresthe offline computation of two Riccati matrix equations, the SSSRiccati solver using sign iterations is also considerably fasterfor , see [19]. It is noted that the SSS Riccati solver re-quires after some iterations a model reduction step, therefore theresult will be an approximation of the true solution up to a giventolerance. Increasing the tolerance value will speed up the com-putation of the SSS structured Riccati solution considerably.

IV. EXPERIMENTAL STUDY ON SMART ROTOR SECTION

In this section we compare the periodic wind disturbance re-jection of the proposed RC method with different settings on asmart airfoil section with trailing-edge flap actuator. First, wepresent the experimental setup used to show the feasibility ofperiodic wind disturbance rejection with the repetitive controlmethod. Second, we describe the identification experiment toobtain an accurate model for the RC design. Third, the proposedRC method is applied on the smart rotor test section, where


Fig. 7. Computation time averaged over 10 runs in MATLAB of matrix-vectorproduct. Light grey line is with dense matrix structure and dark grey line withSSS matrix structure.

the periodic and non-periodic disturbances are created using awind generator.

A. Description of the Experimental Setup

The experimental setup mainly consists of the followingcomponents: wind generator, blade, actuator and sensors, andreal-time environment. The wind generator blows air with con-siderable turbulence into a tunnel section with a blade sectionhanging inside. The diameter of the outlet of the wind generatoris 350 mm and the tunnel section is 400 mm by 400 mm. Theblade section that we use for our experimental verification is anairfoil with at the trailing edge a control surface, the so-calledtrailing-edge flap. The blade section is at the top connected toa half meter long aluminium plate with the other end fixed to arigid frame and has two degrees of freedom. The plate allowsthe blade only to move in the flapwise and torsional direction. Afull description and schematic illustrations of the experimentalsetup are given in [33], where it has already successfully beenused to study recursive subspace identification methods.For control purposes, the smart rotor section is equipped

with sensors which measure the dynamic behavior of the blade.Since the final goal for this experiment is to reduce vibrations ofthe structure, one of the three macro fiber composite (MFC) [34]patches that are adhered to the root at the frontside of the halfmeter long aluminium plate is used to measure the high strainsassociated with the first flapwise bending mode. The main ad-vantage of an MFC is that no amplification is required to have agood signal-to-noise ratio (SNR) ratio. However, with the MFCit is not possible to do static measurements due to the capaci-tance behavior of the MFC. This high-pass behavior is desirablefor this experiment, as we want to control the dynamic behaviorof the system, rather than the static deformations.The trailing-edge flap can reduce the structural vibrations

of the smart rotor section when this repetitive controller isenabled. The controller intelligence and data acquisition capa-bility are added with a dSPACE [35] system. The controller

and data acquisition scheme are fully developed in the MATLAB[36] and SIMULINK [37] environment and then compiled to thedSPACE [35] chip. On a separate computer, all the signals aremonitored using control desk [35] and also the control parame-ters can be adjusted in real-time. The input to the wind generatoris a reference wind speed profile with a given period and is gen-erated by a separate system. For this reason, the actual periodof the periodic wind disturbances is not directly available to therepetitive controller.

B. Experimental Modelling and ControlAmodel for modern model-based controller design is a math-

ematical model normally governed by (preferably linear) differ-ential equations. For controller synthesis this model should onlycontain the relevant dynamics between the input and the outputsand should be accurate over the bandwidth of the controller. An-alytical modelling has been performed in the design stage usingthe theory in [38], which strongly depends on a large numberof parameters that determine the (aero) dynamic behavior of thesystem. Most of these parameters can be roughly estimated orcalculated. Still, a large amount of uncertainty is present; thismakes it difficult to design a stable controller based on suchmodels. This motivates that identification can become a neces-sary building block for model-based controller design.The control loops in the prototyped smart rotor section are

the transfer functions between the trailing-edge flap , and thefirst frontside MFC sensor . Using the predictor-based sub-space identification (PBSID) method for systems with periodicdisturbances as described in [15], we identify the system to becontrolled in the innovation form as

(20)

where denotes the predicted state vector withand minimal,

contains the basis functions needed to express the periodic dis-turbances, and denotes the white innovation sequencewith , and denotes the Kalman gainmatrix. The identified Bode diagram of the fifth-order SISOmodel is given in Fig. 8. The first resonance frequency is locatedat 0.75 Hz and is related to the first flapwise bending mode. Thesecond resonance frequency is located at 4.5 Hz and is related tothe second flapwise bending mode. The resonance frequency lo-cated at 3 Hz related to the 1st torsional bending mode is hardlyobservable due to the sensor position and therefore is not iden-tified, however in Fig. 9 a small resonance peak is still visibleat that location.The subspace identification is done with a pink (1/f) general-

ized binary noise (GBN) signal with a bandwidth of 50 Hz andwith an amplitude of 200 V on the flap actuator. The data is fil-tered and resampled to a sampling rate of 20 Hz 0.05 s .The periodic wind disturbance is created by changing the speedof the fan in the wind generator. The reference signal given tothe wind generator fan is based on the bilinear windshear modeland is described by

ifif


Fig. 8. Identified Bode diagram. For .

Fig. 9. Square-root of the PSD of the output signal. The light grey line is withGBN excitation and the dark grey line is without excitation. For

m/s, and .

As the wind disturbance profile is not purely a sinusoid, multipleharmonics are expected to be excited. In Fig. 9, the square-rootof the power spectral density with and without GBN excita-tion is illustrated. With no excitation, we clearly see peaks atthe 1P6P (Per revolution) frequencies due to the wind distur-bance with a velocity ranging between 9 and 11.5 m/s, periodof 3 s 0.05 s , and no variation on the pe-riod time: . When we excite the system, we see thatespecially at lower frequencies a good SNR is obtained. How-ever, the 1P is dominantly present and that is the main reason weuse an identification scheme that can handle these periodic dis-turbances. Further, we can conclude that there is a good controlauthority at the 1P6P frequencies, because these harmonics arein the range in terms of amplitude and bandwidth of the actu-ator.

Fig. 10. Block diagram of generalized plant to synthesize the controller.

For the feedback control synthesis we use the so-calledcontroller synthesis method, which is illustrated in the well-known generalized plant setting in Fig. 10. The goal of the feed-back controller is to suppress the unknown disturbances asmuchas possible with the requirement that the system should remainstable and that the control input signal is bounded between 250V and 250 V. The method does this by minimizing thenorm between and and is mathematically given by

(21)

where

In these objective functions we can embed manually the loopshaping ideas in the weighting filters , , and . The windturbulence spectrum is obtained from identification. To letthe controller not act on the 1st torsion and the second flapwisebending mode frequencies in the error signal, the input sensi-tivity function is weighted by a first-order filter with highgains at those high frequencies. A second-order low-pass filter

is added in series with the controller, such that the final feed-back controller will roll-off with increasing frequency.

C. Experimental Results

In this subsection, the main results of the experiments on thesmart rotor test section are presented. We present four casesthat will show the features of the proposed lifted RC algorithm.During the design of the proposed lifted RC method, a largenumber of parameters have to be tuned. We have the singularvalue order , the disturbance covariance matrices , , and, and the weight on the differenced input, and the number

of memory loops and its weighting factors (and ). In thefollowing four cases some tuning guidelines are provided byanalyzing the effects of these tuning parameters.1) Effect of the Singular Value Order : In Section II-B, we

proposed to select the number of singular values in (5) not onlyon the controllability/observability condition, but also to furtherreduce the size of the RC output-feedback problem. It turns outthat for even lower values of , i.e., reducing the error that liesin a smaller column space, the lifted repetitive controller willfocus more on the main harmonic components and therefore ro-bustify the controller performance in some sense. To analyze the


Fig. 11. Bode diagram of the two noise sensitivity functions. Dark grey line (partly under light grey) is designed

with and light grey line is designed with . Dotted line isthe noise sensitivity function . For , , , and

.

effect, we consider the frequency response of the lifted repeti-tive controller . From Figs. 5 and 6, we can derive the fol-lowing frequency-domain representation of the complete repet-itive controller as:

The lifted repetitive controller is in fact a periodic lineartime-varying system, thus this transformation to the frequencydomain does not generally hold. However during the design isobserved that for LTI systems, the lifted controller behavesalmost as a very high-order LTI system. This is observed fromthe frequency responses that are very similar toeach other for every step within the period. Plotting one of theBode diagrams of the sensitivity function is then also a usefulengineering tool for lifted RC design.In Fig. 11, the Bode diagrams of two noise sensitivity func-

tions with and are given. The two noise sen-sitivity functions are almost similar for the main (1P4P) har-monic components. Increasing the number of singular values to

introduces also notches at the (5P8P) harmonic compo-nents. Basically, reducing the number of singular values lets thelifted repetitive controller focus on the main harmonic compo-nents. In our case we selected the number , because firstthe repetitiveness of the higher harmonics (5P8P) are ques-tioned (especially if the period time varies over time) andsecond the system should not be excited around 23 Hz, due tothe badly damped (unobservable) torsion mode in the system.For the SISO case, an almost similar Bode diagram of the

noise sensitivity function can be created using control de-sign synthesis with inverse notches as weights. If the frequency

Fig. 12. Initial responses of the input . Dark grey line is designed with, grey line is designed with , and light grey line is designed

with . For , , , and .

response closely matches each other, the asymptotic behaviorwill be almost similar. However, the transient behavior will bevery different. The feedback controller based on inverse notcheswill immediately react on any harmonic variations in the errorsignal. Instead with lifted RC, the input signals are averagedover previous periods and are not directly influenced by the cur-rent error signal, therefore the input signal will be considerablesmoother, but it will take longer to converge.2) Effect of Parameter : As the identified state-space

model is considered to be a close estimate of the systemto be controlled, the disturbance covariance matrices can beestimated from the identification as: , .The strategy to pick the parameter is very similar to thetime-domain LQG problem, see [30]. It is common practiceto pick the parameter such that a smooth control signal isobtained and a large overshoot is avoided. In Fig. 12, the initialresponses of the input for three different weights aregiven. In [31] it is even recommended to have a monotonicdecrease of , which is almost (apart from the initialdifferenced state) directly related to the monotonic increase of

). By increasing the weight , the overshoot over thetrials can be lowered in most cases and improve the smoothnessof the generated control signal at the cost that it takes longerto converge to steady state. As shown in Fig. 12, an overshootduring the initial response of is clearly visible with

, and monotonic increase is visible with the highervalues of and .3) RC With Variations in Period Time: In this case we show

the performance of solely the repetitive controller using the re-sults of the experiments on the smart rotor test section. Wehave stochastic disturbances coming from the wind (turbulence)and periodic disturbances due to generated wind speed profile.Experimentally the implications of variations in period time tothe performance with respect to the periodic disturbance rejec-tion and the smoothness of control signal are investigated. Forthis reason, the experiments have been repeated with differentsettings for the deviation of the period time. The considered


maximum deviation 0.03 s (1%) is based on 10 minsimulations with the UPWIND 5MWwind turbine (see [39] and[40]) in above-rated conditions and with a turbulence intensityof 12% [41]. The maximum difference in rotational speed wasabout 3%.As suggested in the introduction, two ways (or better: a com-

bination of the two) are proposed to improve the performance ofthe repetitive controller in the case there is a small mismatch be-tween the controller period and the actual period of the periodicdisturbance signal. If the period is not measured it is commonpractice to formulate a high-order repetitive control problemwith multiple memory loops. The selection of the number ofmemory loops and its weighting factors (and ) is not aneasy task. In [22], [23] it is shown that there is a clear trade-offbetween improved suppression of periodic disturbances and am-plification of non-periodic noise. It was shown thatand gives the best performance against periodmismatch and that andgives the lowest sensitivity to non-periodic noise for two andthree memory loops respectively. The design of these param-eters highly depends on the periodic and non-periodic distur-bance characteristics. In [22] and [23] periodic and non-periodicperformance criteria are used for designing time-domain high-order repetitive controllers. As shown in Section III-B, multiplememory loops can be introduced in the lifted RC problem at thecost of larger matrices. However the design of the weightingfactors becomes even more difficult and in this paper we haveeven tuned manually, because they are implicitly related withthe dynamic output-feedback problem formulation in (15).Compared to the time-domain RC methods in [22], [23], the

lifted RC problem allows a more versatile disturbance formu-lation. It includes a periodic component with a random walknature , such that the periodic disturbance can vary betweentrials. In addition, the noise and periodic disturbances can beeven correlated and varying in time. In Fig. 13, the Bode dia-gram of the noise sensitivity function for three different weights

are illustrated. With an increasing value the invertednotches at 1P and 2P frequencies become wider, but less deep. Itcan be concluded that the increasing weight provides better dis-turbance rejection at frequencies near the harmonics at the costof performance degradation at the exact repetitive frequencyand its harmonics. When the system is identified (without exactperiod time) in the innovation form in (20), where basis func-tions in are used to express the periodic disturbances, it iseven possible to relate a lifted covariance matrix with thedeviation of the period time by a first-order approximationas

where is the stacked vector of . As the lifted covariancematrix has a time-varying structure, the start of the repeti-tive controller should be matched with the phases of the basisfunctions.In Table I, the manually tuned values for the parameters ofand are given with an increasing value for the deviation

of the period time and an increasing value for the number of

Fig. 13. Bode diagram of the noise sensitivity functionat the 1P and 2P frequencies. Dark grey line is designed

with , grey line is designed with , and light grey lineis designed with . For , , , and .

TABLE IVALUES FOR AND FOR DIFFERENT DEVIATIONS OF PERIOD TIME. FOR

, AND (MULTIPLY WITH )

memory loops . These values have been used during the ex-periments on the smart rotor test section; see the results inSection IV-C-IV. The parameters have been tuned by lookingat the cost after thecontrol signal has converged to steady state. By including an ap-proximation for the acceleration of the generated input signal tothe cost, we try to achieve a smooth input signal which shouldreduce the wear and tear of the actuator. From the table it isobserved that with increasing deviation of the period-time, theweight should be increased, although at a slower rate ifmore memory loops are used. The memory loop weights aremoving clearly in the direction of the values and

which should give the best performance againstperiod mismatch for two and three memory loops, respectively.It is noted that in most wind turbines and in the rotating

smart rotor in [15], the period time could be estimated fromthe measured rotational speed or from the measured azimuth an-gles of the blades. For future research it would be interesting toformulate a real-time RC output feedback problem, where wecan make the parameters and/or adaptive to changes inthe measured data.Table II summarizes the variance values of the output in per-

centage with respect to the output without any controland the approximated input ac-

celeration after the control signal hasconverged to steady state. The variance of the correspondingsignals can be seen as a rough measure for fatigue. We see that


TABLE IIREDUCTION OF VARIANCE FOR OUTPUT AND APPROXIMATED INPUTACCELERATION FOR DIFFERENT DEVIATIONS OF PERIOD TIME. FOR

, , AND

Fig. 14. Square-root of the PSD of the output signal. Dark grey line is the outputspectrum with RC and light grey line is without. For 911.5 m/s, ,

and 0 s.

we have a reduction in variance of at least 34% if only repet-itive disturbance signals are compensated. This increases to avariance reduction of around 42% when the actual period of thewind profile matches the period of the controller. The use ofmore memory loops gives an increase between 1%2%, whichis a small increase compared to the improvements made in ear-lier experiments with high-order time-domain repetitive con-trollers in [22] and [25]. The tuning of the weight alreadyconsiderably improves the robust performance for variations inperiod time, such that incorporating more memory loops willnot make that much difference anymore. However with morememory loops, the smoothness of the input signal is most ofthe time much better, probably due to the averaging effect overmultiple periods, at the cost that the convergence is consider-ably slower.The steady-state performance of any repetitive controller is

bounded by an inherent trade-off between the suppression ofperiodic disturbances and the amplification of non-periodic in-puts [25]. In Figs. 14 and 15 the square-root of the power-spec-tral density (PSD) is given for the system output without anycontrol and with the proposed repetitive control for both 0%and 1% deviation in period time, respectively. In both figures,we clearly see an improvement with respect to the uncontrolledcase. For 0% case, the spikes at the 1P4P harmonic frequenciesare almost completely removed, and the noise amplifications atsome lower and higher intermediate frequencies are small. Forthe 1% case, the spikes at the 1P4P frequencies are only partly

Fig. 15. Square-root of the PSD of the output signal. Dark grey line is the outputspectrum with RC and light grey line is without. For 911.5 m/s, ,

and 0.03 s.

TABLE IIIREDUCTION OF VARIANCE FOR OUTPUT AND INPUT ACCELERATION (ANDRC ONLY) FOR DIFFERENT DEVIATIONS OF PERIOD TIME DESCRIBED INPERCENTAGE OF PERIOD TIME. FOR , , AND

reduced, and this reduction comes even at the cost that someamplifications are visible at some intermediate frequencies andhigher harmonic frequencies.4) RC With Feedback Control: In this case we show the per-

formance of the repetitive controller and feedback controller to-gether using the results of the experiments on the smart rotortest section. For this case, the repetitive controller has been re-designed with the model of the closed-loop system in Fig. 4.Table III again summarizes the variance values of the outputand approximated input acceleration after the control signal hasconverged to steady state. We see that we have an improved re-duction in variance of at least 64% on the sensor channel if boththe feedback and the repetitive control is active. This increasesto a reduction of around 72%when the period of the wind profilematches the period of the controller. The use of more memoryloops gives again an increase between 1%2%. The introduc-tion of feedback makes the variance of the input acceleration48 times larger than with RC only. Thus the input signal is con-siderably less smooth and can cause in practice a considerablereduction in actuator lifetime. Most of this increase is caused bythe feedback controller.In Figs. 16 and 17 the square-root of the power-spectral den-

sity (PSD) are given for the system output without any controland with the feedback and repetitive control together for both0% and 1% deviation in period time. In both figures, we againsee a clear improvement with respect to the uncontrolled case.


Fig. 16. Square-root of the PSD of the output signal. Dark grey line is the outputspectrum with and light grey line is without. For 911.5 m/s,

, and .

Fig. 17. Square-root of the PSD of the output signal. Dark grey line is the outputspectrum with and light grey line is without. For 911.5 m/s,

, and 0.03 s.

For both cases, there is an additional 10 dB reduction at the firstnatural frequency caused by the feedback control action. Thiscomes at the cost by some small amplification of the lower fre-quencies and the frequencies between the two natural frequen-cies. Compared to RC only, the spikes at the 2P3P frequen-cies are reduced further to the same level of the surroundingfrequencies. Thus feedback and repetitive control could be usedtogether for the reduction of both harmonic and natural frequen-cies, if the dynamics of the feedback controller are taken into theRC design.

V. CONCLUSION

In this paper we presented a novel repetitive control methodthat is implemented in real-time for periodic wind disturbancerejection in linear systems with multiple inputs and multiple

outputs and with both repetitive and non-repetitive disturbancecomponents. The design of the repetitive controller is formu-lated as a lifted linear stochastic output-feedback problem onwhich the mature techniques of discrete-time linear controlmay be applied. The formulation of the lifted repetitive controlproblem can be made more robust to small changes in periodtime by using multiple memory loops, but also by including aperiodic component with a random walk nature, so that the pe-riodic disturbance can vary between trials. Efficient algorithmsexist for controller synthesis and for real-time implementationto reduce the computational complexity and memory usageby exploiting the structure in the lifted state-space matrices.Moreover, the paper provides some guidelines on how topick the free parameters in the algorithm. The novel repeti-tive controller could learn the periodic wind disturbances forfixed-speed wind turbines and variable-speed wind turbinesoperating above-rated and we have demonstrated this on anexperimental smart rotor test section. For relatively slowchanging periodic and turbulent wind disturbances created by awind generator it was shown that this repetitive control methodcould reduce the variance of the load signals of the smartrotor test section up to 42%. The cost of additional wear andtear of the smart actuators are kept small, because a smoothcontrol action is generated as the controller mainly focuses onthe reduction of periodic disturbances.

ACKNOWLEDGMENT

The authors would like to thank J. K. Rice, G. van der Veen,and the reviewers for their helpful comments.

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Ivo Houtzager received the B.Eng. degree inmechanical engineering from Rotterdam Universityof Applied Sciences, Rotterdam, The Netherlands,in 2003, and the M.Sc. degree in systems andcontrol engineering from Delft University of Tech-nology, Delft, The Netherlands, in 2007, where heis currently pursuing the Ph.D. degree in controlengineering.His research interest includes the development

of identification methods and inversion methods forlarge-scale dynamical systems and the use of the

identification and inversion results in tuning controllers, as well as towardsapplications of the developed methods and techniques in the light weightstructures and the wind energy.

Jan-Willem vanWingerdenwas born on December9, 1980, in Ridderkerk, The Netherlands. He receivedthe B.Sc. degree cum laude and the Ph.D. degreecum laude from the Delft Center of Systems andControl, the Delft University of Technology, Delft,The Netherlands, in 2004 and 2008, respectively.His Ph.D. project was entitled smart dynamic

rotor control for large offshore wind turbines. Hisgraduation project was carried out at Philips Ap-plied Technologies, Eindhoven, The Netherlands.Currently, he is an Assistant Professor with the

Delft University of Technology. His main research interests include LPVidentification, subspace identification, smart structures, and control andidentification of wind turbines.

Michel Verhaegen received the engineering degreein aeronautics from the Delft University of Tech-nology, Delft, The Netherlands, in 1982, and thedoctoral degree in applied sciences from the CatholicUniversity, Leuven, Belgium, in 1985.From 1985 to 1994, he has been a Research Fellow

of the U.S. National Research Council (NRC) andthe Dutch Academy of Arts and Sciences. In the pe-riod 19941999 he was an Associate Professor of theControl Laboratory, Delft University of Technologyand became a full Professor at the faculty of Applied

Physics, University of Twente, Twente, The Netherlands, in 1999. From 2001on, he moved back to the University of Delft and joined the Delft Center forSystems and Control. His main research directions include system identifica-tion, distributed and fault tolerant control, and data driven controller designmethodologies. Application areas include smart structures, swarms of satel-lites, adaptive optics, wind energy, and vehicle mechatronics.

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