# Vibration Control with Reinforcement Learning Based on Multi-Reward Lightweight Networks
## 2 Problem Formulation  
### Dynamic Model  
Figure 1 shows the dynamic model of the high-dimensional high-frequency vibration control system considered in this paper. The vibration exciter \(m\) is installed on the equipment platform \(M\), which is supported on the foundation by four rubber dampers. The rubber dampers can be regarded as a spring damping system composed of linear spring \(k\) and linear damping \(c\). Four electromagnetic actuators are mounted on the equipment platform, which generate counteracting control forces to reduce the vibration of the equipment platform caused by the vibration exciter. The rotation angles of the equipment platform around its horizontal inertia principal axes paralleling to the \(x\)-axis and \(y\)-axis are \(\psi\) and \(\theta\). The vibration exciter generates excitation forces in three directions, denoted as \(F_{ex},F_{ey},F_{cz}\). However, since the actuators can only generate control forces in the \(z\)-direction (\(F_{c1},F_{c2},F_{c3},F_{c4}\)), only the translational motion along the \(z\)-axis and the rotational motion around the horizontal axis are considered. Thus, there are three generalized coordinates in the dynamic model: \(z,\psi,\theta\).  
Using the Lagrange equation, the dynamic equations of the vibration control system can be described as follows:  
\[\begin{split}(M+m)\ddot{z}&+c\Big{[}4\dot{z}+\big{(}4a- L_{y}\big{)}\dot{\psi}+(4b-L_{x})\dot{\theta}\Big{]}\\ &+k\big{[}4z+\big{(}4a-L_{y}\big{)}\psi+(4b-L_{x})\theta\big{]}\\ &=\ -F_{ez}-F_{c1}-F_{c2}-F_{c3}-F_{c4}\\ J_{x}\ddot{\psi}+c\big{[}(4a-2L_{y})\dot{z}&+ \Big{(}4a^{2}-L_{y}a+2L_{y}^{2}-2L_{y}a\Big{)}\dot{\psi}\\ &+\big{(}4 ab-L_{x}a+L_{y}L_{x}-2L_{y}b\big{)}\dot{\theta}\Big{]} \\ &+k\Big{[}(4a-2L_{y})z+\Big{(}4a^{2}-L_{y}a+2L_{y}^{2}-2L_{y}a \Big{)}\psi\\ &+\big{(}4 ab-L_{x}a+L_{y}L_{x}-2L_{y}b\big{)}\theta\Big{]}\\ &=\ F_{c1}(L_{x}-a)-F_{c2}a-F_{c3}a+F_{c4}\big{(}L_{x}-a)\\ \int_{y}\!\!\ddot{\phi}+c\big{[}(4b-2L_{x})\dot{z}& +\big{(}4 ab-L_{y}b+L_{x}L_{y}-2L_{x}a\big{)}\dot{\psi}\\ &+\big{(}4b^{2}-L_{x}b+2L_{x}^{2}-2L_{x}b\big{)}\dot{\theta}\Big{]} \\ &+k\big{[}(4b-2L_{x})z+\big{(}4 ab-L_{y}b+L_{x}L_{y}-2L_{x}a\big{)} \psi\\ &+\big{(}4b^{2}-L_{x}b+2L_{x}^{2}-2L_{x}b\big{)}\theta\big{]}\\ &=\ F_{c1}\big{(}L_{y}-b\big{)}+F_{c2}\big{(}L_{y}-b\big{)}-F_{ c3}b-F_{c4}b\end{split} \tag{3}\]  
where \(J_{x}\) and \(J_{y}\) represent the moment of inertia of the system with respect to the \(x\)-axis and \(y\)-axis, \(m\) = 100 kg, \(M\) = 913 kg, \(L_{x}\) = 1.6 m, \(L_{y}\) = 1.2 m, \(a\) = 0.8 m, \(b\) = 0.6 m, \(J_{x}\) = 76 kg\({}^{\star}\)m\({}^{2}\), \(J_{y}\) = 58 kg\({}^{\star}\)m\({}^{2}\), \(k\) = 600,000 N/m, and c = 100 Ns/m.  
Figure 1: Schematic of the vibration control system: (**a**) axonometric view; (**b**) front view; (**c**) right view.  
The equipment platform generates vibrations in the z-direction subjected to the excitation force \(F_{e2}\). It is evident from Equations (1)-(3) that by adjusting the value of the control forces \(F_{e1},F_{e2},F_{e3},F_{e4}\), the vibrations of the equipment platform can be reduced, achieving active vibration control. In this paper, the vibration exciter is supposed to generate random multi-frequency excitations, making this vibration system a typical high-dimensional high-frequency vibration control problem.