## 4 Hybrid H\({}_{\infty}\)/SMC Controller Design
Equation (29) and the upper subsystem:  
\[\dot{\mathbf{q}}_{1}(t)=\big{(}\overline{A}_{11}+\Delta\;\overline{A}_{11} \big{)}\mathbf{q}_{1}(t)+\big{(}\overline{A}_{12}+\Delta\;\overline{A}_{12} \big{)}\mathbf{q}_{2}(t) \tag{30}\]  
have to be represented in the Linear Fractional Transformation (LFT) [42] shown in Figure 5, in which the uncertainty is pulled out in the \(\Delta\) block, \(I\) denotes the identity matrix, \(B_{\infty}\), \(C_{\infty}\), \(D_{\infty}\) are matrices obtained by singular value decomposition, and \(\mathbf{z}_{\infty}\) and \(\mathbf{d}_{\infty}\) represent the disturbed output and input, respectively. Figure 4: Controller design and optimization. Upper LFT is applied to the system, so that the H\({}_{\infty}\) problem becomes one of finding the stabilizing feedback gain \(K_{\infty}\) that minimizes \(\gamma>0\) in the quadratic objective function:  
\[J_{\infty}=\frac{1}{2}\!\int_{0}^{\infty}\Big{[}\mathbf{z}_{\infty}^{\mathbf{T }}(t)\;\mathbf{z}_{\infty}(t)-\gamma^{2}\;\mathbf{d}_{\infty}^{\mathbf{T}}(t) \;\;\mathbf{d}_{\infty}(t)\Big{]}\;dt. \tag{31}\]  
From Figure 5,  
\[J_{\infty}=\frac{1}{2}\!\int_{0}^{\infty}\!\Big{[}\mathbf{q}_{1}^{\mathbf{T}}( t)C_{\infty}^{T}\;C_{\infty}\;\mathbf{q}_{1}(t)+2\mathbf{q}_{1}^{\mathbf{T}}(t)C_{ \infty}^{T}\;D_{\infty}\;\mathbf{q}_{2}(t)+\mathbf{q}_{2}^{\mathbf{T}}(t)D_{ \infty}^{T}\;D_{\infty}\;\mathbf{q}_{2}(t)-\boldsymbol{\gamma}^{2}\mathbf{d}_ {\infty}^{\mathbf{T}}(t)\mathbf{d}_{\infty}(t)\Big{]}dt. \tag{32}\]  
The Ricatti-like matrix equation for this linear quadratic problem is  
\[P_{\infty}\Big{(}\overline{A}_{11}-\overline{A}_{12}O_{22}^{-1}O_{12}^{T} \Big{)}+\Big{(}\overline{A}_{11}^{T}-O_{12}O_{22}^{-1}\overline{A}_{12}^{T} \Big{)}P_{\infty}-P_{\infty}\Big{(}\overline{A}_{12}O_{22}^{-1}\overline{A}_{ 12}^{T}-\gamma^{-2}B_{\infty}\;B_{\infty}^{T}\Big{)}P_{\infty}+\Big{(}O_{11}- O_{12}O_{22}^{-1}O_{12}^{T}\Big{)}=0. \tag{33}\]  
where \(O_{11}=C_{\infty}^{T}C_{\infty}\), \(O_{12}=C_{\infty}^{T}D_{\infty}\), \(O_{22}=D_{\infty}^{T}D_{\infty}\). The solution of the above equation (\(P_{\infty}\)) is used to find the feedback gain \(K_{\infty}\):  
\[K_{\infty}=O_{22}^{-1}\Big{(}\overline{A}_{12}^{T}P_{\infty}+O_{12}^{T}\Big{)}, \tag{34}\]  
Thus,  
\[K_{\infty}= \begin{bmatrix}-2447&-12790&14632&-6474\\ 392&2015&-2301&1024\end{bmatrix} , \tag{35}\]  
which can be used to define the sliding manifold \(\mathbf{s}(t)\):  
\[\mathbf{s}(t)=K_{\infty}\mathbf{q}_{1}(t)+\mathbf{q}_{2}(t) \tag{36}\]  
When the state variables reach and stay on the sliding manifold (\(\mathbf{s}(t)=0\)), then the system dynamics is governed by Equation (29). Equation (36) can be written as  
\[\mathbf{s}(t)= \begin{bmatrix}K_{\infty}&I\end{bmatrix} \mathbf{q}(t) \tag{37}\]  
From Equation (21), the sliding manifold is expressed in terms of the original states as  
\[\mathbf{s}(t)=G\;\mathbf{x}(t) \tag{38}\]  
and  
\[G= \begin{bmatrix}K_{\infty}&I\end{bmatrix} H \tag{39}\]  
Figure 5: LFT configuration of the system Equations (34) and (36). Next, the control law of the SMC is suggested to comprise a nominal term (\(\mathbf{u}_{\cdot}(t)\)) and a discontinuous term (\(\mathbf{u}_{\mathbf{d}}(t)\)):  
\[\mathbf{u}(t)=\mathbf{u}_{\cdot}(t)+\mathbf{u}_{\mathbf{d}}(t) \tag{40}\]  
The nominal term is responsible for driving the states towards the sliding manifold defined by Equation (38). It is obtained by equating the derivative of the sliding variable to zero, that is  
\[\dot{\mathbf{s}}(t)=0=G\;\dot{\mathbf{x}}(t) \tag{41}\]  
Substituting the nominal terms of Equation (19),  
\[\dot{\mathbf{s}}(t)=0=GA\;\mathbf{x}(t)+GB\;\mathbf{u}_{\cdot}(t) \tag{42}\]  
from which  
\[\mathbf{u}_{\cdot}(t)=-(GB)^{-1}GA\;\mathbf{x}(t). \tag{43}\]  
provided that the \(GB\) matrix is non-singular; thus, it is invertible.