# Research on Semi-Active Vibration Control of Pipeline Based on Magneto-Rheological Damper
## 3 Research on Semi-Active Pipeline Vibration Control System
### Comparative Analysis of Simulation of Different Vibration Control Algorithm
#### 3.3.3 SMVS Vibration Control Simulation  
SMVS control method makes the system state slide along the sliding surface by switching control variables, so that the system is invariant when disturbed by parameter perturbation or disturbance. The sliding mode can be designed independently of object parameters and disturbances, which makes the variable structure control having the advantages of fast response, insensitivity to parameter changes and disturbances, no need for online identification of the system, and simple physical implementation. It is suitable for linear and nonlinear systems, continuous and discrete systems, deterministic and uncertain systems, centralized parameters and distributed parameter systems, centralized control and distributed control systems.  
Under the disturbance action, the output of the pipeline system is as shown in Equation (6). Since the pipeline system is an inertial system, \(\mathbf{D}=\mathbf{0}\). The controlled pipeline system can be described by the following state equation:  
\[\left\{\begin{array}{l}\dot{\mathbf{Z}}(t)=\mathbf{A}\mathbf{Z}(t)+\mathbf{B}\mathbf{U}(t)\\ \mathbf{Z}(t_{0})=\mathbf{Z}_{0}\end{array}\right. \tag{23}\]  
The output goal of the pipeline vibration control system based on MR damper is to solve the control force \(\mathbf{U}\) to keep the pipeline in equilibrium position under the external disturbance excitation.  
\[\lim_{t\rightarrow\infty}\lVert\mathbf{Z}(t)-\mathbf{0}\rVert=0 \tag{24}\]  
The pipeline vibration error function can be expressed as:  
\[\mathbf{e}(t)=\mathbf{Z}(t) \tag{25}\]  
According to Equation (23), error and the rate of change of error can be expressed as:  
\[\mathbf{e}=\mathbf{Z}=\frac{\dot{\mathbf{Z}}}{\mathbf{A}}+\frac{\mathbf{B}\mathbf{U}}{\mathbf{A}} \tag{26}\]  
\[\dot{\mathbf{e}}=\dot{\mathbf{Z}}=\mathbf{A}\mathbf{Z}+\mathbf{B}\mathbf{U} \tag{27}\]  
The switching function of pipeline system is defined as:  
\[S=k\mathbf{e}=\frac{k}{A}\Big{(}\dot{Z}+B\dot{U}\Big{)} \tag{28}\]  
\[\dot{S}=k\dot{\mathbf{e}}=k(AZ+B\dot{U}) \tag{29}\]  
Using the exponential reaching law, the switching function of the system can be expressed as:  
\[\dot{S}=-\varepsilon\text{sgn}(S)-\delta S\ \ \ \ \ \ \varepsilon>0,\delta>0 \tag{30}\]  
By substituting Equation (28) and the Equation (29) into Equation (30), the following equation can be obtained:  
\[\mathbf{U}=\frac{\varepsilon\mathbf{A}}{k\delta(\mathbf{A}\mathbf{B}-\mathbf{B})}\text{sgn}(\mathbf{ S})+\frac{\dot{\mathbf{Z}}}{\mathbf{A}\mathbf{B}-\mathbf{B}} \tag{31}\]  
The Simulink model of the SMVS pipeline vibration control system based on MR damper is shown in Figure 9a. When the sinusoidal excitation amplitude is 2 mm and the frequency is 5 Hz, the vibration displacement response and control force simulation results of the pipeline under the SMVS vibration control are shown in Figure 9b,c. The SMVS vibration control strategy can effectively control the pipeline vibration. The attenuation amplitude of the pipeline vibration of the SMVS vibration control in the steady state phase is 79.72%.  
From the simulation results above, it can be seen that the SMVS vibration control algorithm has the highest attenuation rate of the pipeline vibration amplitude in the steady state phase, which is 79.72%. However, because the SMVS control is progressively stable, the vibration amplitude of the PID vibration control system in the transient state is smaller than that of the SMVS vibration control system. The control effect of two-state vibration control system on the pipeline vibration is general.