## 2 Force Controller Design  
Consider an SDOF non-linear vibration isolator with the governing equation of motion represented by:  
\[m\ddot{x}+\psi_{1}\Big{(}x,\dot{x},f_{e}\Big{)}=f_{c} \tag{1}\]  
where \(m\) is the isolated mass, \(\ddot{x}\), \(\dot{x}\), and \(x\) are the acceleration, velocity, and displacement of the mass, respectively. \(\psi_{1}(\cdot)\) represents the total effect of the linear/non-linear stiffness restoring forces, damping forces, and the external disturbing forces \(f_{e}\) that exert on the mass. \(f_{c}\) is an active control force that is used to further suppress vibrations of the mass. In this article, it is assumed in Equation (1) that \(\psi_{1}(\cdot)\) is difficult to be modeled exactly or not easily identifiable and, thus, is deemed to be unknown. Situations where \(\psi_{1}(\cdot)\) cannot be exactly determined occurs in many practical applications, such as when models are not well developed and where there exist seldom-known system parameters or unpredictable parameter variations, etc. Additionally, the mass \(m\) in Equation (1) is considered to be uncertain or has its value in a certain interval. Let \(\hat{m}\) be an estimated or nominal value of the mass \(m\), Equation (1) is rearranged as follows:  
\[\hat{m}\ddot{x}+(m-\hat{m})\ddot{x}+\psi_{1}\big{(}x,\dot{x},f_{e}\big{)}=f_{c} \tag{2}\]  
Let \(\psi\big{(}x,\dot{x},\ \ddot{x},\ f_{e}\big{)}=-(m-\hat{m})\ddot{x}-\psi_{1} \big{(}x,\dot{x},\ f_{e}\big{)}\), which includes all the system non-linearities and uncertainties and is supposed to be bounded. Then Equation (1) has:  
\[\hat{m}\ddot{x}=\psi\big{(}x,\dot{x},\ \ddot{x},\ f_{e}\big{)}+f_{c} \tag{3}\]  
In this study, only measurements of the velocity \(\dot{x}\) were required. In order to make the non-linear vibration isolator more robust and improve its dynamics behavior, we proposed two SMCs to supplement the isolation performance. In what follows, the SMC methods were implemented, respectively, in this paper: a conventional SMC [29; 31] and a SMC based on a super-twisting algorithm [30; 32; 33]. For the vibration suppression problems discussed here, the desired displacement, velocity, and acceleration trajectory of the isolated mass can be predefined as zeros. Subsequently, a sliding surface (s) for the conventional SMC design was adopted as:  
\[s(t)=\dot{x}(t)+\lambda x(t) \tag{4}\]  
where \(\lambda>0\). Therefore, its first-time derivative is given, together with Equation (3), by:  
\[\dot{s}(t)=\ddot{x}(t)+\lambda\dot{x}(t)=\lambda\dot{x}(t)+\hat{m}^{-1}\psi+ \hat{m}^{-1}f_{c}(t) \tag{5}\]  
First, a conventional SMC is introduced to be directly as [34]:  
\[f_{c}(t)=-\hat{m}\Big{(}\lambda\dot{x}(t)+\mu s\Big{)}-Dtanh\Big{(}\frac{s}{ \varepsilon}\Big{)} \tag{6}\]  
where the constant design parameters are \(\varepsilon>0\), \(\mu>0\), and \(D\geq\big{|}\psi\big{|}\). In Equation (6), the continuous tanh function is used to replace the discontinuous signum function in order to alleviate the chattering phenomenon of the SMC [29]. The value \(\varepsilon\) of \(tanh(\cdot)\) determines how the tanh can approximate the signum function. A smaller \(\varepsilon\), a closer approximation to the signum function was obtained, as shown in Figure 1. The estimation of \(D\) can be determined by the estimation of \(\big{|}\hat{m}\ddot{x}\big{|}\) from Equation (3). If a large \(\mu\) is selected, the isolation performances will get better, but the closed-loop stability will be lost when the value of \(\mu\) is large enough. Additionally, for the stability analysis of the closed-loop control system, it has been proven with a Lyapunov function \(V=\frac{1}{2}s^{2}\) that [31]:  
\[\lim_{t\to\infty}V(t)\leq\frac{0.2785D\varepsilon}{2\mu\hat{m}} \tag{7}\]  
So, the trajectory of the system (3) is asymptotically convergent to a vicinity of zero under the control input (6), and the convergent precision is determined by \(D\), \(\mu\), and \(\varepsilon\).