# A Systematic Literature Review of Various Control Techniques for Active Seat Suspension Systems
## 8 Control Methods Used in the Active Seating Suspension System
### Robust \(H_{\infty}\) Controller  
In the early 1980s, a new controller strategy was proposed by [PERSON] and [PERSON] called H\({}_{\infty}\)[72]. The H\({}_{\infty}\) controller is an optimal control theory developed in the frequency domain which addresses the feedback design issues related to robustness [73]. It has been used extensively in many research fields and applications due to the mathematical optimization solutions and has a superior advantage over classical controllers. It is applicable to provide high stability performance of the plant in present of disturbances and noises. However, it requires a high level of mathematical understanding and large control efforts [74]. Figure 13 illustrates the standard H\({}_{\infty}\) configuration where \(P(s)\) is the generalized plant; \(w\) is the external input; \(z\) is the output signal, which contains the performance and robustness measurements; \(u\) is the vector control signal; \(y\) is the plant output vector to the controller \(K\). The main objective of the H\({}_{\infty}\) is to find a feedback controller gain \(K\) to minimize the output \(z\). Generally speaking, the controller gain \(K\) should minimize the H\({}_{\infty}\) norm that is \(\min\lVert T_{zw}\rVert_{\infty}\). The mathematical model of the above closed loop system is represented as follows  
\[\left[ \begin{array}{c}z\\ y\end{array} \right]=\left[ \begin{array}{cc}p_{11}&p_{12}\\ p_{21}&p_{22}\end{array} \right]\left[ \begin{array}{c}w\\ u\end{array} \right] \tag{4}\]  
Since \(z\) must be minimized, therefore, it can be written as  
\[z=\left[P_{11}+P_{12}K(I-P_{22}K)^{-1}P_{21}\right]w=\mathcal{F}_{l}(P,K)w, \tag{5}\]  
where \(\mathcal{F}_{l}(P,K)w\) is the _lower linear fractional transformation_ of P and K. The design objective becomes  
\[K_{\text{stabilizing}}^{min}\lVert\mathcal{F}_{l}(P,K)w\rVert_{\infty} \tag{6}\]  
and is referred to as the \(H_{\infty}\) optimization problem. \(H_{\infty}\) strategy has been intensively used in the active seating suspension system control [75]. In [76] the author used the static feedback \(H_{\infty}\) controller with friction compensation based on the  
Figure 13: Standard H\({}_{\infty}\) configuration [73]. seat acceleration, suspension relative displacement and suspension relative velocity. An acceleration measurement-based observer was designed to estimate the friction. The state feedback controller with friction estimation was built as  
\[u=KX+\overline{f_{r}}, \tag{7}\]  
where \(K\) is the state feedback gain, and \(\overline{f_{r}}\) is the estimated friction. The experiment was conducted under the sinusoidal excitation in the frequency range of 1-4.5 Hz, bump road excitation and random road excitation. The acceleration Power Spectral Density (PSD) and ISO 2631-1 were used to evaluate the seating suspension system comfort performance. The experimental results indicated that the seating suspension system comfort performance was improved from a little uncomfortable of 0.6267 m/s\({}^{2}\) RMS acceleration to a not uncomfortable of 0.2615 m/s\({}^{2}\) RMS acceleration. \(H_{\infty}\) controller was used in the active vibration cancellation to reduce the vertical vibration of a vehicle seat and guarantee the stability [71]. The external disturbances were constructed as additive uncertainty and unmodelled dynamics. A hybrid structure consists of feedforward and feedback controller were designed to increase the robustness of the active seating suspension system. The active seating suspension system was tested under a random broadband excitation. The experimental results showed a vibration attenuation in the real time control by 3.3 dB. However, the active seating suspension system comfort performance became degraded when the random excitation was applied. Plant uncertainty such as actuator saturation, suspension deflection and tire deflection were considered when designing a novel optimal approach of the active seating suspension system based on output feedback \(H_{\infty}\) controller [77].