## 3 A Compensation Strategy Using an \(\mathbf{H_{\infty}}\) Control Law
### Design Principle
\right)^{T}\Omega\left( \begin{array}{c}Z(\sigma)\\ Z(\sigma-\tau_{1})\\ Z(\sigma-\tau_{2})\\ \vdots\\ Z(\sigma-\tau_{m})\\ w(\sigma)\end{array} \right)d\sigma \tag{28}\]  
where \(\Omega\) is the coefficient matrix which can be expressed as,  
\[\Omega=\left[ \begin{array}{ccccc}PA+(PA)^{T}+\sum\limits_{i=1}^{m}S_{i}+C^{T }C&-P(B_{1}G_{c1})-C^{T}(D_{1}G_{c1})\\ -(B_{1}G_{c1})^{T}P-(D_{1}G_{c1})^{T}C&(D_{1}G_{c1})^{T}(D_{1}G_{c1})-\sum \limits_{i=1}^{m}S_{i}\\ -(B_{2}G_{c2})^{T}P-(D_{2}G_{c2})^{T}C&(D_{2}G_{c2})^{T}(D_{1}G_{c1})\\ \vdots&\vdots\\ -(B_{m}G_{cm})^{T}P-(D_{m}G_{cm})^{T}C&(D_{m}G_{cm})^{T}(D_{1}G_{c1})\\ E^{T}P+F^{T}C&F^{T}(D_{1}G_{c1})\\ -P(B_{2}G_{c2})-C^{T}(D_{2}G_{c2})&\cdots&-P(B_{m}G_{cm})-C^{T}(D_{m}G_{cm})&PE+ C^{T}F\\ (D_{1}G_{c1})^{T}(D_{2}G_{c2})&\cdots&(D_{1}G_{c1})^{T}(D_{m}G_{cm})&(D_{1}G_{c1}) ^{T}F\\ (D_{2}G_{c2})^{T}(D_{2}G_{c2})-\sum\limits_{i=1}^{m}S_{i}&\cdots&(D_{2}G_{c2}) ^{T}(D_{m}G_{cm})&(D_{2}G_{c2})^{T}F\\ \vdots&\ddots&\vdots&\vdots\\ (D_{m}G_{cm})^{T}(D_{2}G_{c2})&\cdots&(D_{m}G_{cm})^{T}(D_{m}G_{cm})-\sum \limits_{i=1}^{m}S_{i}&(D_{m}G_{cm})^{T}F\\ F^{T}(D_{2}G_{c2})&\cdots&F^{T}(D_{m}G_{cm})&F^{T}F-\gamma^{2}I\end{array} \right] \tag{29}\]  
The closed loop system shown by Equation (13) can be stabilized by a state-feedback compensation controller with an \(\text{H}_{\infty}\) performance index according to the references [34; 35]. Only when there are symmetric positive-definite matrices \(P\) and \(S_{i}\) (\(i\) = 1, 2,..., \(m\)), the inequality \(\Omega<0\) is satisfied. Then,  
\[J_{z}\leq\int_{\sigma=0}^{t}\left( \begin{array}{c}Z(\sigma)\\ Z(\sigma-\tau_{1})\\ Z(\sigma-\tau_{2})\\ \vdots\\ Z(\sigma-\tau_{m})\\ w(\sigma)\end{array} \right)^{T}\Omega\left\{\begin{array}{c}Z(\sigma)\\ Z(\sigma-\tau_{1})\\ Z(\sigma-\tau_{2})\\ \vdots\\ Z(\sigma-\tau_\[\left[ \begin{array}{cccccccc}PA+(PA)^{T}+\sum\limits_{i=1}^{m}S_{i}&-P(B_{1}G_{c1} )&-P(B_{2}G_{c2})&\cdots&-P(B_{m}G_{cm})&PE&C^{T}\\ -(B_{1}G_{c1})^{T}P&-\sum\limits_{i=1}^{m}S_{i}&0&\cdots&0&0&-(D_{1}G_{c1})^{T} \\ -(B_{2}G_{c2})^{T}P&0&-\sum\limits_{i=1}^{m}S_{i}&\cdots&0&0&-(D_{2}G_{c2})^{T} \\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ -(B_{m}G_{cm})^{T}P&0&0&\cdots&-\sum\limits_{i=1}^{m}S_{i}&0&-(D_{m}G_{cm})^{ T}\\ E^{T}P&0&0&\cdots&0&-\gamma^{2}I&F^{T}\\ C&-(D_{1}G_{c1})&-(D_{2}G_{c2})&\cdots&-(D_{m}G_{cm})&F&-I\end{array} \right]<0 \tag{33}\]  
The variable substitution method [37] is referred for transforming the inequality (33) from a nonlinear matrix inequality into a linear matrix inequality. Supposing \(X_{i}=P(B_{i}G_{ci})\) and \(Y_{i}=D_{i}G_{ci}\). Only when there are symmetric positive-definite matrices \(P\), \(S_{i}\) and matrices \(X_{i}\), \(Y_{i}\) (\(i=1,2,\,\ldots,\,m\)), the following inequality is satisfied. \[\left[ \begin{array}{cccccccc}PA+(PA)^{T}+\sum\limits_{i=1}^{m}S_{i}&-X_{1}&-X_ {2}&\cdots&-X_{m}&PE&C^{T}\\ -X_{1}^{T}&-\sum\limits_{i=1}^{m}S_{i}&0&\cdots&0&0&-Y_{1}^{T}\\ -X_{2}^{T}&0&-\sum\limits_{i=1}^{m}S_{i}&\cdots&0&0&-Y_{2}^{T}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ -X_{m}^{T}&0&0&\cdots&-\sum\limits_{i=1}^{m}S_{i}&0&-Y_{m}^{T}\\ E^{T}P&0&0&\cdots&0&-\gamma^{2}I&F^{T}\\ C&-Y_{1}&-Y_{2}&\cdots&-Y_{m}&F&-I\end{array} \right]<0 \tag{34}\]  
The _ith_ control force is,  
\[u_{i}(t)=\Big{[}(PB_{i})^{-1}X_{i}\Big{]}Z(t) \tag{35}\]  
Then the _ith_ feedback gain matrix is,  
\[G_{ci}=-(PB_{i})^{-1}X_{i} \tag{36}\]