## 4 Hybrid H\({}_{\infty}\)/SMC Controller Design  
The control of uncertain systems is required to provide robustness against parameter variations, external disturbances, and measurement noise. Nevertheless, seeking extreme robustness is not advised, because it leads to unnecessary conservativeness that affects the system's performance characteristics. Optimization can be a great aid in enhancing the controller design to calibrate the system's robustness/performance outcome. H\({}_{\infty}\) and SMC are well known robust controllers that have shown their effectiveness in many uncertain control systems, so the controller design process is composed of the steps shown in Figure 4 and detailed below:  
First, the H\({}_{\infty}\) control is used to design the sliding manifold of the SMC to provide a more powerful hybrid controller. To account for parameter variations, the system in Equation (15) is represented as  
\[\dot{\mathbf{x}}(t)=(A+\Delta A)\;\mathbf{x}(t)+B\;\mathbf{u}(t) \tag{19}\]  
where \(\Delta A\) represents the uncertainty matrix bounded by the Q matrix:  
\[\text{Q}= \begin{bmatrix}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0.2&0.2&0.2&0.2&0.2\\ 0.2&0.2&0.2&0.2&0.2\\ 0.2&0.2&0.2&0.2&0.2\\ \end{bmatrix}  \tag{20}\]  
Figure 3: PSO agent update. In order to apply decoupling for MIMO systems, a similarity transformation is required to transform the state variable \(\mathbf{x}(t)\) to a new state variable \(\mathbf{q}(t)\), defined as:  
\[\mathbf{q}(t)=H\;\mathbf{x}(t) \tag{21}\]  
where \(H\) is the \(n\times n\) similarity transformation matrix, obtained as  
\[H= \begin{bmatrix}N&B\end{bmatrix} ^{T} \tag{22}\]  
where the \(n\times(n-m)\) matrix \(N\) is the null space of the transpose of \(B\) matrix. Thus:  
\[H= \begin{bmatrix}0&0&1&0&0&0\\ 0.227&0.399&0&0.788&0.361&0.19\\ -0.793&-0.452&0&0.361&0.165&0.087\\ 0.564&-0.796&0&0.19&0.087&0.045\\ 0&0&0&-50&174.4&-124.2\\ 0&0&0&6.12&-38.93&48.62\end{bmatrix}  \tag{23}\]  
The transformed state space representation is  
\[\dot{\mathbf{q}}(t)=\overline{A}\;\mathbf{q}(t)+\Delta\overline{A}\;\mathbf{q }(t)+\overline{B}\;\mathbf{u}(t) \tag{24}\]  
where  
\[\overline{A}=H\;A\;H^{-1}, \tag{25}\]  
\[\Delta\overline{A}=H\;\Delta A\;H^{-1}, \tag{26}\]  
\[\overline{B}=H\;B, \tag{27}\]  
The new state variable representation becomes  
\[ \begin{bmatrix}\dot{\mathbf{q}}_{1}(\mathbf{t})\\ \dot{\mathbf{q}}_{2}(\mathbf{t})\end{bmatrix} = \begin{bmatrix}\overline{A}_{11 }&\overline{A}_{12}\\ \overline{A}_{21}&\overline{A}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{q}_{1} (\mathbf{t})\\ \mathbf{q}_{2}(\mathbf{t})\end{bmatrix} + \begin{bmatrix}\Delta\overline{A}_{11 }&\Delta\overline{A}_{12}\\ \Delta\overline{A}_{21}&\Delta\overline{A}_{22}\end{bmatrix}\begin{bmatrix} \mathbf{q}_{1}(\mathbf{t})\\ \mathbf{q}_{2}(\mathbf{t})\end{bmatrix} + \begin{bmatrix}0\\ \overline{B}\end{bmatrix} \mathbf{u}(t) \tag{28}\]  
In this representation, the \(m\) control signals \(\mathbf{u}(t)\) act as an input to the lower subsystem only (controlling the \(m\) new state variables \(\mathbf{q}_{2}(t)\) directly), then, during the sliding phase, \(\mathbf{q}_{2}(t)\) acts as an input to the upper subsystem (controlling the remaining \((n-m)\) new state variables \(\mathbf{q}_{1}(t)\) by feedback gain):  
\[\mathbf{q}_{2}(t)=-K_{\infty}\mathbf{q}_{1}(t) \tag{29}\]  
The \(m\times(n-m)\) feedback gain matrix \(K_{\infty}\), which is related to the sliding manifold, is to be designed using \(\mathrm{H}_{\infty}\) control to provide robustness to parametric uncertainty.