## 3 Semi-Active Cable Damper
### Control Law  
The control law is semi-active linear viscous damping with negative stiffness (clipped VDNS), which can be realized by collocated control and is proven to be a very efficient damping algorithm (Figures 1b and 5, [24; 38; 39]). The high damping efficiency of clipped VDNS is explained by the fact that VDNS combines energy dissipation by the viscous force while the negative stiffness force increases the relative motion of the damper which leads to increased cycle energy within the damper. In addition, the control force characteristics of VDNS are similar to those resulting from optimal control as shown in [24]. For these two reasons, i.e., enlarged damper relative motion due to the negative stiffness force and optimal control similarity, VDNS is a very effective damping strategy. Further benefits are that VDNS does not require a state observer as optimal control does and that it works for any type of excitation, i.e., for single harmonic, narrow and broad band excitations because the desired semi-active control force can be computed in real time based on the measured actual damper motion only. In the following, the most relevant steps of the control law computation are explained. The viscous damping coefficient \(c_{frequency}^{opt}\) is adjusted in real time to the actual frequency of vibration in order to generate optimum modal damping (15). The negative stiffness coefficient \(k\) is formulated based on the known parameters \(T\) and \(a\)[46]. The two tuning parameters \(g_{1}\) and \(g_{2}\) are needed to compensate for the damping increase and stiffness loss due to the clipping of the active desired control force; more information on undesired damping increase and stiffness loss due to clipping can be found in [47]. Reasonable values for \(g_{1}\) and \(g_{2}\) are 0.80 and 1.20 [46]. The main modules of the control algorithm are shortly described here (Figure 5):  
* Bandpass filtering of damper relative motion (displacement sensor signal, Figure 3a) to remove offset and noise in the displacement signal. * Derivation of damper relative velocity by numeric differentiation with subsequent low pass filtering to attenuate noise due to the numeric differentiation. * Frequency detection from the peaks of the displacement signal resulting in a maximum time delay of half a period. This time delay is more than acceptable considering that resonant vibrations are long standing vibrations, see cable vibration measurements presented in [30]. * Computing the desired active control force \(f_{des}^{active}\) (17) and the desired semi-active control force \(f_{des}^{semi-active}\) (18). * The control force tracking task is solved in two steps. First, the desired viscous coefficient \(c_{des}(t)\) of the controllable damper is computed based on the desired semi  
Figure 3: (**a**) Semi-active hydraulic damper; (**b**) measured steady-state relation between electromagnetic valve current and resulting controllable viscous coefficient. active control force (19) and the actual damper relative velocity, then the desired current of the electromagnetic valve is computed based on the steady state relation between valve current and viscous damper coefficient (Figure 3a, (20)). Equation (19) shows that the controllable damper exerts quadratic viscous damping at constant bypass valve position, i.e., at constant valve current, because both the controlled bypass valve and the passive bypass act as throttles resulting in quadratic viscous damping. * The PID-based current controller including an anti-reset windup (ARW) due to the current limitations ensures that the actual valve current precisely tracks the desired current \(i_{des}\). * The state variables, which are given as function of time \(t\) in (15)-(20), are computed in real time at the controller frequency of 1000 Hz.