## 2 System Modelling and Problem Formulation
### Problem Formulation  
The state variables of the driver-seat system are defined as  
\[\mathbf{x}=[x_{1},x_{2}]^{T}=\big{[}z_{s},\dot{z}_{s}\big{]}^{T}. \tag{14}\]  
Combining Equations (13) and (14), the state space equations of the selected state variables are obtained as  
\[\dot{x}_{1}=x_{2} \tag{15}\] \[\dot{x}_{2}=\tfrac{1}{M}\big{[}-k_{s}(z_{s}-z_{v})-c_{s}\big{(} \dot{z}_{s}-\dot{z}_{v}\big{)}-F_{c}sign\big{(}\dot{z}_{s}-\dot{z}_{v}\big{)}+ k_{a}u+f\big{]}.\]  
It is noted that system (15) is highly nonlinear. On one hand, the key dynamic parameters of the system are difficult to measure accurately. For instance, the total mass \(M\) varies across different drivers, the stiffness \(k_{s}\) changes with the position of the seat suspension, while the nominal value of spring stiffness \(\kappa_{s}\) also becomes uncertain due to system creep. On the other hand, internal friction disturbances and the non-ideal characteristics of the controller (\(\Delta F_{a}\)) further complicate precise system control.  
In order to overcome the above-mentioned issues, Equation (15) is rearranged through the introduction of new auxiliary uncertain parameters. As introduced in Remark 2, the controlled system has up to six uncertain parameters: the integrated mass, control gain, lumped disturbance term, Coulomb friction, equivalent stiffness, and damping coefficients. Notably, for the latter five parameters, their relative values with respect to the integrated mass are of greater interest than their absolute values. To simplify the controller development, these uncertain quantities are redefined as an uncertain vector along with two relative uncertain parameters, as follows:  
\[\mathbf{\theta}_{z}=\left[\frac{k_{s}}{M}\quad\frac{c_{s}}{M}\quad \frac{F_{c}}{M}\right]^{T} \tag{16}\] \[\theta_{f}=\frac{f}{M},\theta_{u}=\frac{k_{a}}{M}.\]  
where each element of \(\mathbf{\theta}_{z}\) represents a normalized uncertain parameter (stiffness, damping, and friction coefficient) linearly dependent on the system dynamics. In contrast, the terms \(\theta_{f}\) and \(\theta_{u}\) lump non-structured uncertainties including unmodeled disturbances and actuator errors.  
By substituting Equation (16) into the state space Equation (15), the controlled system equation can be rewritten as  
\[\dot{x}_{1}=x_{2} \tag{17}\] \[\dot{x}_{2}=-\theta_{z}^{T}\mathbf{\chi}+\theta_{u}u+\theta_{f},\]  
where \(u\) is the control signal and \(\mathbf{\chi}\) is the vector of suspension deformation, which can be measured directly, given as \[\chi=\Big{[}(z_{s}-z_{\text{\tiny{D}}})\quad\big{(}\dot{z}_{s}-\dot{z}_{\text{ \tiny{D}}}\big{)}\quad sign\big{(}\dot{z}_{s}-\dot{z}_{\text{\tiny{D}}}\big{)} \Big{]}^{T}. \tag{18}\]  
Here, the term \(-\theta_{z}^{T}\chi\) explicitly captures the structured uncertainty with known regressor \(\chi\) (measurable states) but unknown constant parameters \(\theta_{z}\). This structure enables adaptive compensation via parameter estimation.  
Therefore, the objective of this study is to develop a proper controller such that the CL system (17) is asymptotically stable and the state variable is retained within a reasonable range in the presence of system uncertainties.  
In addition, for safety consideration, the physical limit of the suspension structure and the actuator saturation should be taken into consideration as well.  
(i) The suspension deformation should be retained within the limit of the suspension mechanical structure, which is given by  
\[|\Delta z|\leq\Delta z_{max}, \tag{19}\]  
where \(\Delta z=z_{s}-z_{v}\) is the suspension deformation, \(\Delta z_{max}\) is the corresponding structural limit.  
(ii) The active control torque should not exceed the limit of the motor:  
\[|T_{a}|\leq T_{a-max}, \tag{20}\]  
where \(T_{a-max}\) is the maximum actuating torque.