# Automatic Decentralized Vibration Control System Based on Collaborative Stand-Alone Smart Dampers
## 4 Limitations on the Control Gain
### Actuator Dynamics Effect  
The problem of instability generated by the coupling between the proof-mass actuator and the hosting structure can be addressed by considering the complete controlled system in Laplace domain. A flexible structure can be modeled as a single DOF system, under the assumption that there is only one mode excited and controlled. For this reason, to simplify the discussion, the system is supposed to be a Single Input Single Output (SISO) system, where the input is the electromagnetic force \(F_{a}(s)\) and the output is the structure response due to the controlled vibration mode \(\hat{Y}(s)\). Into the block diagram in Figure 1, which represents the overall control system [20], the mobility transfer function \(M_{s}(s)\), the blocked response transfer function \(T_{a}(s)\) and the actuator mechanical impedance \(Z_{aa}(s)\) are defined as follows (\(m_{a}\), \(c_{a}\) and \(k_{a}\) are the mass, damping and stiffness of the actuator respectively):  
\[M_{s}(s) = \frac{\hat{Y}(s)}{F_{T}(s)}=R_{i,j}\,\frac{s}{s^{2}+2\xi_{i} \omega_{i}\,s+\omega_{i}^{2}}\] \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\] \[Z_{aa}(s) = \frac{F_{T}(s)}{\hat{Y}(s)}=\frac{m_{a}(c_{a}s+k_{a})s}{(m_{a}s^{ 2}+c_{a}s+k_{a})}\hat{Y}(s) \tag{13}\]  
we can derive the plant transfer function \(G(s)\) of the controlled system:  
\[G(s)=\frac{m_{a}\,R_{i,j}\,s^{3}}{(s^{2}+2\xi_{i}\omega_{i}\,s+\omega_{i}^{2}) (m_{a}\,s^{2}+c_{a}\,s+k_{a})+m_{a}\,R_{i,j}\,(c_{a}\,s+k_{a})\,s^{2}} \tag{14}\]  
and we can reduce the whole model to the canonical form containing the controller transfer function \(H(s)\) and the plant \(G(s)\), as shown in Figure 1.  
In \(H(s)\), expressed in Equation (15), we have applied the SNDF control logic, where we have centered the band-pass filter on the \(i\)-th vibration mode eigenfrequency (i.e. \(\omega_{c}=\omega_{i}\)) and considered \(\xi_{c}=\xi_{i}\). \[H(s)=-\frac{g_{i}\,(2\xi_{c}\omega_{c}+d_{c})\,s}{s^{2}+(2\xi_{c}\omega_{c}+d _{c})\,s+\omega_{c}^{2}} \tag{15}\]  
Hence, the open\(-\)loop transfer function can be easily defined and can be written in a simple way by grouping the same order denominators terms:  
\[GH(s)=\frac{g_{i}\,(2\xi_{c}\omega_{c}+d_{c})\,m_{a}\,R_{i,j}\,s^{4}}{B_{6}\, s^{6}+B_{5}\,s^{5}+B_{4}\,s^{4}+B_{3}\,s^{3}+B_{2}\,s^{2}+B_{1}\,s+B_{0}} \tag{16}\]  
where:  
Figure 1: Control system block diagram. \[B_{6} = A_{4}\] \[B_{5} = A_{3}+(2\xi_{c}\omega_{c}+d_{c})A_{4}\] \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\]0 \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\]1 \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\]2 \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\]3 \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\]4 \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\]5 \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\]6 \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\]7 \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\]8 \[T_{a}(s) = \frac{F_{T}(s)}{F_{a}(s)}=\frac{m_{a}s^{2}}{m_{a}s^{2}+c_{a}s+k_{ a}}\]9  
Looking at the Nyquist plot in Figure 2 obtained by Equation (16), there is an encirclement as in the case of the DVF, where \(H(s)=g\). The presence of this encirclement is due to the resonance of the proof-mass actuator and affects the system stability. In this case, the same gain has been applied on both control laws, ensuring the stability of the SNDF as opposed to the DVF. In order to define the maximum gain related to the actuator dynamics, Equation (16) is rewritten by substituting \(s=j\omega\) and by splitting the real part from the imaginary one.