To find optimal gain matrices of SOFCs in this paper, a meta-heuristic method is applied [32; 33]. The SOFCs with optimal gain matrices are called LQ SOF controllers (LQSOFCs). In addition to feedback, the other issues in the area of suspension control are robustness, ability to guarantee control performance under parameter uncertainty and nonlinearity in a plant, disturbance attenuation, and ability to reduce the effects of external disturbance on control performance. In suspension control, external disturbances have much more effects on control performance than parameter uncertainty and nonlinearity. To cope with the problem, several robust controller design methodologies such as LQ optimal control, \(H_{2}/H_{\infty}\) control, robust model predictive control, adaptive control has been applied to vehicle suspension control to date [1; 4; 5]. Among them, an adaptive control is selected in this paper because it can adaptively change a controller according to parameter uncertainty and nonlinearity in a plant and can effectively attenuate the effects of external disturbances [34; 35; 36; 37; 38]. In this paper, adaptive control means that the gains of SOFC are adaptively updated by a parameter adaptation algorithm. Let denote this controller as parameter adaptive controller (PAC). While LQSOFC has the constant gain matrix, PAC adaptively updates the gains in every step by a parameter adaptation algorithm [34; 35; 36; 37; 38]. In other words, PAC is a learning-based controller, which tries to minimize an error by a learning mechanism [36]. To date, a recursive least square (RLS) has been well-known as a parameter adaptation algorithm. In addition, it was shown that RLS is equivalent to Kalman filter [39]. The unique feature of the Kalman filter is that it can be used for both state and parameter estimation. Using this feature, the dual extended Kalman filter (DEKF) has been applied to vehicle control [40; 41; 42; 43; 44; 45]. For that reason, the extended Kalman filter (EKF) is selected for parameter adaptation in this paper because EKF can easily handle multiple outputs [40; 41; 42; 43; 44; 45]. Contrastingly to LQSOFC with constant gains, PAC can cope with parameter uncertainties, nonlinearities and external disturbances by using parameter adaptation with EKF [38]. To verify the performance of LQSOFCs and PACs, a simulation is performed on a vehicle simulation software. Based on simulation responses, the effect of PAC in respect of RC and MS is shown. The objective of this paper is to design LQSOFCs and PACs based on SOF structure for RC enhancement and MS reduction. The contributions of this paper can be summarized as follows:  
1. Two types of SOFCs are proposed and designed to reduce \(a_{zc}\) and \(\omega_{yc}\) over the frequency range from 0.8 to 8.0 Hz. Those SOFCs are optimized with LQOF and validated through simulation on a vehicle simulation package. 2. PACs are proposed with two SOF structures and EKF. Differently from LQSOFCs, PAC can adaptively update gains of SOFCs under parameter uncertainties, nonlinearities and external disturbances. 3. With LQSOFCs and PACs, the simulation is performed on CarSim. From simulation responses, it is identified which controller can give the best performance in relation to RC and MS. This paper is composed of five sections. A HCM is presented and its linear SSEQN is derived in Section 2. With the linear SSEQN, two SOFCs are proposed and optimized with LQOF. PAC is designed in Section 3. In Section 4, a simulation with LQSOFCs and PACs is performed and the simulation responses are analyzed and discussed. The conclusions are presented in Section 5.