## 3 Active Control of a Quasi-Zero-Stiffness Isolator
### Simulation Results and Analyses  
Two kinds of control forces \(f_{c}(t)\) from Equations (6) and (12) were used for numerical verification of the control schemes on the isolator. They were applied to five cases involving the set-up of a controller's parameter \(\hat{m}\) and an isolator's parameter \(k_{n}\): (i) \(\hat{m}=300\) kg and \(k_{n}\) was the nominal value (\(k_{n}=k_{v}\)/2.5); (ii) \(\hat{m}=360\) kg and \(k_{n}=k_{v}\)/2.5; (iii) \(\hat{m}=240\) kg and \(k_{n}=k_{v}\)/2.5; (iv) \(\hat{m}=300\) kg and \(k_{n}\) had an approximately 17% decrease, i.e., \(k_{n}=k_{v}\)/3; (v) \(\hat{m}=240\) kg and \(k_{n}=k_{v}\)/3. Other parameters of the isolator retained their nominal values. The algorithm parameters of the control forces \(f_{c}(t)\) were chosen to be \(\mu=160\), \(D=5.0\), \(\lambda=10\), and \(\varepsilon=0.5\) for Equation (6) and \(U=4.0\), \(D=5.0\), \(\lambda=1.0\), and \(b=16\) for Equation (12) throughout all cases. In case (i), Figures 3a-c and 4a-c show responses of displacement and acceleration of the mass under the base excitation of the white noise and the active control forces applied, respectively, for the control algorithms (6) and (12). Figure 5a,b presents responses of displacement and acceleration of the passive isolator with nominal parameter values. Their root mean square (RMS) values of the responses are presented in Table 1 for comparison. Two statements could be presented from the numerical experiments: (I) The conventional SMC provided the best vibration isolation performance at the cost of moderate magnitudes of control forces. But it led to higher frequency oscillations of the responses for displacements on a microscale and the intense chattering for control inputs on a macroscale, which is unacceptable in practical implementation. (II) The super-twisting control gave the second-best results with small magnitudes of control forces. Although only achieving a displacement reduction by 46.15% in the RMS values and 8.65% in the acceleration reduction compared with the corresponding passive isolator with QZS, it produced moderate oscillating responses and fewer chattering occurrences, which is also important and preferable in practical applications, compared with those given by the conventional SMC. Figure 3: The displacement and acceleration responses (**a**,**b**), and (**c**) control forces (6) of the hybrid isolator with quasi-zerostiffness (QZS) for case (i). For cases (ii) to (v), numerical experiments were conducted to investigate the robustness of the control schemes for the hybrid isolator. The RMS values of the responses are presented in Figure 6. Although the RMS values of the displacement responses with the super-twisting control showed some variations, as shown in Figure 6a, they were all smaller than that of the corresponding passive isolator. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Active Hybrid Isolator** & \multicolumn{3}{c}{**Response and Control Force (RMS)**} \\ \cline{2-4}
**and Passive Isolator** & **Acceleration (ms\({}^{-2}\))** & **Displacement (m)** & **Control Force (N)** \\ \hline With control force (6) & 0.0431 & 1.1326\(\times 10^{-5}\) & 30.02 \\ With control force (12) & 0.0845 & 0.0014 & 4.03 \\ Passive isolator & 0.0924 & 0.0026 & null \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RMS comparison of responses with the nominal parameters of the isolator. Figure 4: The displacement and acceleration responses (**a**,**b**), and (**c**) control forces (12) of the hybrid isolator with QZS for case (i). Figure 5: The displacement and acceleration responses of the passive isolator with QZS for the nominal parameter values (**a**,**b**). with QZS. It was demonstrated that the hybrid isolator with the proposed control methods had robust vibration isolation performances with respect to the estimation error of \(\hat{m}\) and the system parametric variations, and could overcome the essential defect of the non-linear passive isolators with QZS, i.e., their performances were very sensitive to the deviations in system parameters.