Structural monitoring of wind turbines often involves the use of a numerical model, which must be regularly updated to yield an adequate representation of the governing dynamics. The model updating can be cast as an inverse problem, in which selected model parameters are estimated by minimizing the discrepancy between experimental target poles and model-predicted ones. An issue that may prevail in this setting is that the problem will be ill-posed, because the number of model parameters to be updated exceeds the number of target poles. The noted issue can be remedied by a virtual implementation of output feedback, which allows for computation of multiple closed-loop (CL) eigenstructures through non-linear transformations of the open-loop transfer matrix. The present paper o ers an application study, in which virtual output feedback is used for updating numerical wind turbine models based on CL system poles. In particular, we explore the feasibility of employing the excitation stemming from the blade pitch control system as the input in the open-loop input-output realization required in the virtual output feedback. The methodological concept is outlined, and subsequently the applicability of the procedure is tested numerically in the context of simulations with a ﬁnite element model of a parked wind turbine. The preliminary ﬁndings suggest that procedures for customizing the pitch excitation to yield su cient system excitation are required for the CL model updating to be feasible. Keywords: Structural health monitoring, Model updating, Output feedback, Wind turbines, Jacket substructures 1. Introduction O shore wind turbine systems operate under vary- ing environmental and operational conditions, which are governed by, among other parameters, the wind, the waves, the temperature, and the control system. The shifts in these environmental and operational parame- ters (EOPs) render the structural properties time-variant and promote structural integrity degradation [1, 2], thus regular integrity assessments are required to ensure that the wind turbines operate adequately. A particular way of conducting the assessments is to use vibration- based structural health monitoring (VSHM) techniques, in which the structural condition is inferred on the ba- sis of processed vibration measurements [3]. Numer- ous VSHM techniques have been suggested for di er- ent levels of VSHM of wind turbine systems; cover- ing damage detection [4], localization [5], quantiﬁca- tion [6], and prognosis [7]. Numerical models are widely used in damage lo- calization, assessment, and prognosis of wind turbines [8, 9]. In order to ensure that the models provide an ad- equate representation of the governing system dynam-ics, they must be regularly calibrated through model up- dating [10]. Di erent studies have shown how indirect model updating—in which a subset of model parame- ters are estimated in an inverse problem setting—can allow for e cient calibration of numerical models of wind turbines and subsystems hereof [11, 12, 13, 14]. Indirect model updating aims to minimize the dis- crepancy between experimental target features and the corresponding model-predicted features by updating the selected model parameters. Let 2Cpdenote the discrepancy between the pexperimental and model- predicted features and let 2Rqbe the qmodel pa- rameters to be updated, then J; (1) where J2Cpqis the Jacobian matrix containing the sensitivities of with respect to . A commonly adopted approach in structural model updating is to cal- ibrate a ﬁnite element model by matching system poles from the model with target poles estimated from exper- iments [10]. When using this setting for updating wind turbine models, a likely scenario is that problem (1) will Preprint submitted to SHMII-10 January 10, 2021 be ill-posed, because the number of updating parame- ters, q, exceeds the number of identiﬁable poles, p. The ill-posedness of problem (1) can be resolved us- ing output feedback to design multiple closed-loop sys- tems and hence target features [15, 16, 17]. To imple- ment the feedback physically, the system input needs to be controllable, which will be infeasible in some appli- cations. Instead, a signal processing-based implemen- tation of the feedback can, as outlined in [18, 19, 20], be attained from available open-loop (OL) input-output realizations. As such, this implementation, which is referred to as virtual output feedback, eliminates the practical issues associated with closed-loop (CL) test- ing (including the physical constraints on the gain) and removes the constraint of CL system stability. The present paper addresses the use of virtual output feedback for CL updating of ﬁnite element models of o shore wind turbine systems in parked conditions. In particular, we explore the feasibility of using excitation from the blade pitch control system as deterministic in- put in the virtual output feedback during parked condi- tions. Here, the pitch control system refers to the tech- nology that is used to control the angle of the wind tur- bine blades in order to satisfy speciﬁc operational con- straints governed by the wind speed, power output, and so forth. As such, the proposed procedure does not re- quire any additions to the control system. This study is based on a numerical model of an 8 MW o shore wind turbine system, which is introduced to sti ness-related perturbations that emulate common structural damage in wind turbine substructures. The paper is organized as follows. In Section 2, the methodological principles of output feedback-based model updating are outlined. Section 3 describes the case study setup, including the use of pitch excitation input, while Section 4 presents the appertaining model updating results. The paper closes with some conclud- ing remarks in Section 5. 2. Methodology We consider a linear, time-invariant system of order 2nwith input u(k)2Rrand output y(k)2Rmat time instant k. The discrete-time state-space formulation of the system is x(k+1)=Adx(k)+Bdu(k); (2a) y(k)=Cdx(k); (2b) where Ad2R2n2n,Bd2R2nr, and Cd2Rm2nare the state, input, and output matrices. In system (2), it is assumed that the direct transmission term is zero orsubtracted from the output. Let system (2) operate under static output feedback given as u(k)=Gy(k)+v(k); (3) in which G2Rrmis the feedback gain and v(k)2Rr is a disturbance vector. By inserting (3) into system (2), we get the CL state equation x(k+1)=(AdBdGC d)x(k)+Bdv(k) (4) with ¯A=AdBdGC d (5) being the CL state matrix. 2.1. Virtual output feedback The virtual implementation of output feedback fol- lows from OL input-output realizations of system (2). In particular, let H(z)=Cd(zIAd)1Bd (6) be the OL transfer matrix of system (2), then, with feed- back law (3), we establish y(z)=H(z)(Gy(z)+v(k)): (7) Hereby, the CL transfer matrix can be found as ¯H(z)=(I+H(z)G)1H(z); (8) which implies that multiple CL systems can be com- puted based on a single OL input-output realization by System+  System+¯1 G1 System+¯N GNOpen-loop Closed-loop Figure 1: The concept of virtual output feedback with a total of NCL systems based on a single OL input-output system realization. 2 plugging in di erent gains, Gj2Rrm. This concept is illustrated in Figure 1, where 2C2nis the full set of OL system poles while ¯j2C2ncontains the poles of the CL system formed using Gj. 2.2. Model updating formulation and gain design In practice, only a subset of will be identiﬁable. To increase the dimensionality of the target feature vec- tor, one can simply expand it with poles from ¯j(for j=1;:::; N) until the inverse problem to be solved is (over)-determined with pq. With the expanded target feature vector, the model updating can be conducted in full analogy to the conventional OL inverse problem for- mulations [10]. In the present study, the optimal model parameter set, ˆ, is estimated as ˆ=arg min 2Rq T ; (9) whereis speciﬁed as the di erence between the ex- perimental target features, T, and the model-predicted features, M(). The Ngains to be used in the updating can be de- signed using di erent approaches [18, 21, 22, 23]. Here, we employ a gain design aiming to decrease the condi- tion number of the Jacobian matrix in (1). Let G=h GT 1GT 2:::GT NiT2RNrm(10) be the compound gain matrix, then ˆG=arg min G2RNrmK(J) (11) withKdenoting the condition number operator. Worth of explicit note is that the gain design, for necessary practicality, is conducted on the basis of a model of the structural reference state, which does not take into ac- count the current realization of the system parameters. 3. Case study setup The o shore wind turbine model used in the numer- ical example is shown in Figure 2. The system, which is depicted without its rotor, consists of a three-legged jacket substructure with four brace levels connected to the soil with piles and an 8 MW turbine. Details about the modeling, including dimensions, soil conditions, and damping properties, can be found in [24]. We use the terms nominal model and simulation model to refer to, respectively, the model to be updated and the model used to simulate experiments. The two models are shown in Figure 3, and it is noted that they Figure 2: Wind turbine model used in the case study. The arrows in- dicate the wind load direction, which is co-directional with the waves. both have 1200 degrees of freedom (DOF). The mod- els have identical mass and damping matrices, denoted M,C2R12001200, while the sti ness, as outlined in Subsection 3.1, di ers. The nominal model—whose sti ness, K2R12001200, is to be updated—provides the model-predicted features, M(). Additionally, this model is used for designing the gains in accordance with the procedure outlined in Subsection 2.2. In the simulations, the turbine is, as speciﬁed in Ta- ble 1, in a parked condition with the impact from the wind and waves at a minimum. Hereby, the time-variant Table 1: Load case used in the case study. Parameter Value Turbine state Parked Wind speed, U[m/s] 4 Wind proﬁle NWP Wave height, Hs[m] 0.45 Wave period, Tp[s] 3.76 Wind and wave direction N Normal wind proﬁle (NWP) North (N) 3 λM(θ)K,M,C(a) Nominal model. λTy(k),u(k)fb (b) Simulation model. Figure 3: Models used in the case study. The load vector, fb, is ex- plained in Subsection 3.2. and non-linear e ects induced by rotor operations are avoided, and the pitch excitation can be tailored to gov- ern a part of the output. Each simulated OL input-output realization has a duration of 600 s and a time increment of 0:01 s, hence totaling NS=60001 sample points. From the realizations, the state-space system matrices are estimated as ˜Ad,˜Bd, and ˜Cdsuch that the OL tar- get poles can be extracted as the eigenvalues of ˜Ad. The CL target poles are computed by plugging ˜Ad,˜Bd, and ˜Cdinto (5) with a gain from the compound gain in (10). These estimated OL and CL poles compose the target features, T. Subsections 3.1 to 3.3 provide further de- tails on the simulation model. 3.1. Input, output, and damage locations To accommodate the constraints imposed by practi- calities, it is chosen to place all output sensors in dry and accessible locations on the turbine. The sensors are taken as displacement sensors, which are distributed as shown in Figure 4. It is noted that at each out- put location, both x- and y-directional displacements are captured, thus the output vector is y(k)2R8for k=0;:::; 60001. The selected sensor distribution im- plies, as elaborated in [24], that the “local” brace modes of the system are not identiﬁable. The loads used as deterministic input in the model updating are computed based on the load iteration pro- Figure 4: Input, output, and damage /sti ness perturbation locations. cedure outlined in Subsection 3.2. The result is a set of interface loads, composed of x- and y-directional forces and moments around the x- and y-axes, at the location marked in Figure 4. Therefore, we get the input vector u(k)2R4fork=0;:::; 60001. In practice, the input should be taken as the pitch loads acting in the rotor, which are either to be measured directly or estimated based on the available pitch angle time histories. The procedure selected in this study is thus merely for com- putational convenience. The simulation model is, compared to the nominal model, perturbed by local sti ness changes at the lo- cations indicated in Figure 4. With the ﬁrst perturbation being the lowermost and the last the topmost, the pertur- bation vector is set to 920% 0 915%. The perturbations are introduced in three beam elements, which each has a length of approximately 3 m and is placed around a bolted connection in the tower; with the lowermost per- turbation being at the bolted interface point between the tower and the jacket. 3.2. Load iteration and response simulation The load iteration setup [25] depicted in Figure 5 is used to produce the required structural load input and 4 Interface point ub Craig-Bampton SE KrMrCrfr/bracketleftbiggub ui/bracketrightbiggAeroelastic simulation fb8 MWApply interface- loadsfb SEFigure 5: Load iteration used in input-output realizations. response output. The setup includes the following four steps: 1. Initial Ramboll o shore structural analysis (ROSA) [26] model containing a wave model. 2. Craig-Bampton condensation [27] of the ROSA model and wave loads to generate a superelement (SE) with reduced mass, damping, sti ness, and loading ( Mr,Cr,Kr, and fr) used in the subsequent aeroelastic simulation. 3. Aeroelastic simulation in LACﬂex [28], which computes the interface loads, fb. 4. Load recovery run in ROSA, with turbine loads from LACﬂex applied to the interface DOF, ub. The last step of the load iteration, namely, the load recovery run, computes the structural response of the jacket and tower. Here, it is afterwards possible to ex- tract displacements, stresses, and so forth. 3.3. Pitch excitation The interface loads from the pitch excitation are used as input in the OL system identiﬁcation, thus u(k)= 0 100 200 300 400 500 600-100001000 708090 Figure 6: Bending moment Mxat the interface point during pitching.fb(k). The pitching is programmed such that all three blades rotate between two speciﬁc pitch angles, namely, 90 deg and 75 deg. In Figure 6, the time history of one of the pitch excitation-induced loads, namely, the bending moment around the x-axis, Mx, is plotted along with the blade pitch angle. As expected, the temporal evolution of the moment correlates with the blade pitching. To quantify the impact of the pitch excitation on the structural response, we plot, with reference to Fig- ure 4, the displacements captured in the topmost sen- sor. The results are shown in Figure 7 for the system with and without the pitch excitation; with y1being 100 120 140 160 180 200102030 100 120 140 160 180 200-20020 Figure 7: Structural displacement response at the topmost sensor (with reference to Figure 4) with and without pitch excitation. y1andy2are thex- and y-directional displacements. 5 Table 2: OL continuous-time poles of the nominal model, M, plus the exact, T, and estimated, ˜T, poles of the simulation model. Pole M T˜T 10:04305:0153 i0:04314:9836 i0:04274:9878 i20:04305:0153 i0:04314:9837 i 30:11329:0915 i0:11349:0529 i0:11239:0845 i40:11329:0928 i0:11349:0542 i thex-directional displacements and y2they-directional displacements. Evidently, the pitching a ects the y- directional response, while the x-directional response is not a ected. The ﬁndings are a result of how the turbine loads are a ected by the wind and wave direction, and how the turbine is rotated with respect to the global axis and jacket setup, as depicted in Figure 2. 3.4. Updating features The input and output data is used to infer an estimate of the system and, as such, the employed target features through system identiﬁcation. In this study, the data- driven subspace identiﬁcation algorithm N4SID [29] is used under the precondition that the system is observ- able and controllable. The system identiﬁcation yields consistent estimates of two poles of the wind turbine, which are presented in Table 2 along with the exact ﬁrst and second bending mode poles from the simulation model and the nominal model. Evidently, the attained estimates match, with reasonable accuracy, the poles of the ﬁrst two sets of bending modes. The corresponding exact poles from the nominal model have, as expected, higher frequencies since no perturbations are present. 4. Results The updating is conducted in a setting where the three model parameters perturbed with 920% 0% 915%in the simulation model are to be estimated in the nominal model. With three model parameters, the two estimated OL poles are insu cient to render the updating problem well-posed, so CL systems are formed to expand the tar- get vector such pq. We design three CL systems from each of which we extract the two CL poles that “corre- spond” to the two OL poles. The target feature is thus a mixture of two OL and six CL poles. With the target vector discrepancy 2C8, the three model parameters are estimated via minimization of =T; (12)which is solved in MATLAB ®using “fmincon” with an interior-point algorithm [30]. The results for dif- ferent initial conﬁgurations, 0, are listed in Table 3. As can be seen, the CL updating scheme fails to pro- vide the correct perturbation conﬁguration, as it, even when initializing the optimization from the true conﬁg- uration, converges to smaller perturbations (in an ab- solute sense). The associated OL poles for this con- verged model conﬁguration are presented in Table 4, where it can be appreciated that the frequency of the ﬁrst pole is above the target value, while the frequency of the second pole is below the target value. It is noted that when conducting the updating with just the two OL poles, rendering the problem ill-posed, the results are 912% 92:2% 1:3%Tfor0= 0 0 0Tand 99:3% 1:3%95:8%Tfor0= 920% 0% 915%T. The error in the estimate of the second OL pole, see Table 2, is a likely cause of the erroneous ﬁndings in Tables 3 and 4. Therefore, we conduct a new update of the nominal model based solely on the ﬁrst OL pole and three “corresponding” CL modes. The appertain- ing results are listed in Table 5 for di erent initial con- ﬁgurations. When initializing the optimization from Table 3: CL updating results using eight target features with in percentage. Note that ’a’ is the initial evaluation of the cost function and ’b’ the converged. Setup T 0ˆ1ˆ2ˆ3 (1093) 1a 0 0 0Initial evaluation 5 :2 1b 0 0 0913 916 8:0 0:63 2a 920 0 915Initial evaluation 2 :8 2b 920 0 915914 3:293:9 0:70 Table 4: Estimated target poles and poles of the nominal model up- dated with the converged solution in conﬁguration 2b in Table 3. ˜T M(ˆ) 90:04274:9878 i90:04314:9978 i 90:11239:0845 i90:11339:0744 i 6 Table 5: CL updating results using only the ﬁrst OL pole and three “corresponding” CL poles with in percentage. Note that ’a’ is the initial evaluation of the cost function and ’b’ the converged. Setup T 0ˆ1ˆ2ˆ3 (1094) 1a 0 0 0Initial evaluation 36 1b 0 0 0922 97:0 96:8 1:1 2a 920 0 915Initial evaluation 1 :2 2b 920 0 915921 90:26 915 1:1 0= 0 0 0T, the three model parameters converge at values that do not coincide with the true values. In the case of0= 920% 0% 915%T, the three model param- eters converge in the vicinity of the theoretical values. The attained cost function is, however, equal to the one attained in setup 1, so although using only the ﬁrst OL pole and the “corresponding” CL poles improves the re- sults, there is still some clear room for improvement. It is contended that this can be achieved if the pitch excita- tion can be tailored trough input shaping [31, 32], such that more accurate estimates can be obtained of a subset of the OL poles. 5. Conclusion The paper explores the feasibility of employing blade pitch-induced excitation as input in CL updating of nu- merical models of parked o shore wind turbines. The merit of the CL implementation is that it can be formu- lated such that the inverse problem to be solved in the model updating is well-posed. A case study with a nu- merical wind turbine system is presented, and although the current, preliminary ﬁndings do not appear particu- larly encouraging, it is contended that the encountered issues can be resolved by tailoring the delivered pitch excitation to allow for improved estimation of the sys- tem poles. This is a part of the ongoing research.