9 Operational Vibration-Based Response Estimation for Offshore Wind Lattice Structures 89 The magnitude of the distributed wave force is calculated at mean sea level. This force is assumed to act within a wave impact zone of ˙5.0 m with respect to this level. After integration over this wave impact zone, a concentrated wave force signal at mean sea level is derived. In deriving this force signal, of which a 100 s window is presented in Fig. 9.4b, a 1.0 m/s current is added to the wave particle velocity. In correspondence to the wind force generation, the signal has a length of 546 s, while a cut-off frequency of 3 Hz has been adopted. The force location is chosen at the K-joint, connecting the first and second jacket frame, see Fig. 9.4c. Since the Morison equation relates the drag force nonlinearly to the wave particle velocity, the wave force contains frequency content above the cut-off frequency of the wave elevation energy spectrum. The added damping resulting from the hydrodynamic action is neglected. This hydrodynamic damping would result from the response velocity of the jacket members and its contribution can be assumed to be small. 9.2.6 Sensor Network Given the hostile environmental offshore conditions, the measurement of the structural motion requires a robust network of sensors. This robustness is first pronounced in the number of required sensors: if only a limited number of sensors is needed, the costs to build in sufficient redundancy remain low. Second, the positioning of the sensors affects the robustness of the system. This implies that no sensors should be placed under the water level, because these locations are not easily reached when maintenance is necessary. Moreover, to prevent sensors from early failure, sensors within the wave splash zone should be avoided. Therefore, the estimation of the dynamic response will be based on a network consisting of sensors attached to the turbine tower only. The design of a sensor network for optimal estimation of inputs and states is considered by Maes et al. [23]. In this contribution, the criteria for the invertibility of a system, and therefore the possible application of the joint input-state estimator, are derived in terms of stability and identifiability. For the joint input-state estimator to be stable, two requirements need to be fulfilled. First, the number of acceleration and/or velocity sensors, nd;a and nd;v, respectively, needs to be equal to or larger than the number of input forces: ndIa CndIv np (9.13) Second, the number of displacement/strain sensors nd;d needs to be equal to or larger than the number of input forces: ndId np (9.14) In this particular case, the separate estimation of the wind and wave force requires a network consisting of two acceleration and/or velocity sensors and two displacement/strain sensors. Identifiability relates to the controllability of the input forces, the observability of the states and the direct invertibility of the measurement outputs towards states and input. The controllability of the input forces can by assessed by determining the rank of the controllability matrix C2R2nm .np 2nm/: C BAB: : : A2nm 1B (9.15) If Cis of full rank, i.e. rank.C/ D2nm, the system is controllable, implying that the assumed forces are positioned such, that all modes can contribute to the response. Alternatively, the controllability can be assessed by determining the rank of the modal projections of the force selection matrix Sp®j, for which it should apply that rank Sp®j D1 for j D1; : : : ;nm. In a similar manner, the observability of the system can be assessed by determining the rank of the observability matrix H2R.2nm nd/ 2nm: H 2 66 64 G GA : : : GA2nm 1 3 77 75 (9.16)
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