## 3 A Compensation Strategy Using an \(\mathbf{H_{\infty}}\) Control Law  
### Design Principle  
The state-space equation of the multi-time-delay system is,  
\[\left\{ \begin{array}{l}\dot{Z}(t)=AZ(t)+\sum\limits_{i=1}^{m}B_{i}u_{i}(t- \tau_{i})+Ew(t)\\ Y(t)=CZ(t)+\sum\limits_{i=1}^{m}D_{i}u_{i}(t-\tau_{i})+Fw(t)\end{array} \right. \tag{13}\]  
where \(\tau_{i}\) denotes the time-delay of \(i\)th controller, \(u_{i}\) represents the \(i\)th control force. \(Y\) is the controlled vectors, which includes the dynamic responses and the control forces. \(C\) indicates the state output matrix, \(D_{i}\) indicates the direct transmission matrix of the control forces, and \(F\) indicates the direct transmission matrix of the external excitations. The expressions are as following,  
\[C=\left[ \begin{array}{cc}I&0\\ 0&I\\ -M^{-1}K&-M^{-1}C\\ 0&0\end{array} \right],D_{i}=\left[ \begin{array}{c}0\\ 0\\ M^{-1}B_{si}\\ 1\end{array} \right],F=\left[ \begin{array}{c}0\\ 0\\ M^{-1}B_{w}\\ 0\end{array} \right]. \tag{14}\]  
For the system (13), the negative feedback control forces are,  
\[u_{i}(t-\tau_{i})=-G_{ci}Z(t-\tau_{i}) \tag{15}\]  
where \(G_{ci}\) denotes the \(i\)th feedback gain. The system is,  
\[\left\{ \begin{array}{l}\dot{Z}(t)=AZ(t)+\sum\limits_{i=1}^{m}\left(-B_{i}G_{ ci}\right)Z(t-\tau_{i})+Ew(t)\\ Y(t)=CZ(t)+\sum\limits_{i=1}^{m}\left(-D_{i}G_{ci}\right)Z(t-\tau_{i})+Fw(t) \end{array} \right. \tag{16}\]  
According to the reference [33], this paper introduces the following definitions and lemmas to facilitate the analysis. Regarding the system (16), only when there are symmetric positive-definite matrices \(P\) and \(S_{i}\) (\(i=1,2,\,\ldots,\,m\)), Lyapunov function is described as,  
\[V(Z)=Z^{\mathsf{T}}PZ+\sum\limits_{i=1}^{m}\int_{t-\tau_{i}}^{t}Z^{\mathsf{T} }(\sigma)S_{i}Z(\sigma)d\sigma \tag{17}\]  
where \(V(Z)\) denotes a positive-definite matrix. When the system (16) is asymptotically stable, its derivative \(\dot{V}(Z)\) is a negative-definite function. Then,  
\[ \begin{array}{l}\dot{V}(Z)=\dot{Z}^{\mathsf{T}}(t)PZ(t)+Z^{\mathsf{T}}(t)P \dot{Z}(t)+\sum\limits_{i=1}^{m}\left[Z^{\mathsf{T}}(t)S_{i}Z(t)-Z^{\mathsf{ T}}(t-\tau_{i})S_{i}Z(t-\tau_{i})\right]\\ =\left[AZ(t)+\sum\limits_{i=1}^{m}\left(-B_{i}G_{ci}\right)Z(t-\tau_{i})+Ew(t) \right]^{\mathsf{T}}PZ(t)\\ +Z^{\mathsf{T}}(t)P\left[AZ(t)+\sum\limits_{i=1}^{m}\left(-B_{i}G_{ci}\right) Z(t-\tau_{i})+Ew(t)\right]+\sum\limits_{i=1}^{m}\left[Z^{\mathsf{T}}(t)S_{i}Z(t)-Z^{ \mathsf{T}}(t-\tau_{i})S_{i}Z(t-\tau_{i})\right]\end{array}  \tag{18}\]  
Then,  
\[ \begin{array}{l}\dot{V}(Z)=Z^{\mathsf{T}}(t)\left[PA+(PA)^{\mathsf{T}}+\sum \limits_{i=1}^{m}S_{i}\right]Z(t)-\sum\limits_{i=1}^{m}\left[Z^{\mathsf{T}}(t -\tau_{i})S_{i}Z(t-\tau_{i})\right]\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad -\sum\limits_{i=1}^{m}\left\{\left[Z^{\mathsf{T}}(t)P(B_{i}G_{ci})Z(t-\tau_{i} )\right]-\left[Z^{\mathsf{T}}(t)P(B_{i}G_{ci})Z(t-\tau_{i})\right]^{\mathsf{T}} \right\}\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad +w^{\mathsf{T}}(t)\left(E^{\mathsf{T}}P\right)Z(t)+Z^{\mathsf{T}}(t)(PE)w(t) \end{array}  \tag{19}\]  
\[\mathbf{\Psi}=\left[ \begin{array}{ccccc}C^{T}C&-C^{T}(D_{1}G_{c1})&-C^{T}(D_{2}G_{ c2})&\cdots&-C^{T}(D_{m}G_{cm})&C^{T}F\\ -(D_{1}G_{c1})^{T}C&(D_{1}G_{c1})^{T}(D_{1}G_{c1})&(D_{1}G_{c1})^{T}(D_{2}G_{c2 })&\cdots&(D_{1}G_{c1})^{T}(D_{m}G_{cm})&-(D_{1}G_{c1})^{T}F\\ -(D_{2}G_{c2})^{T}C&(D_{2}G_{c2})^{T}(D_{1}G_{c1})&(D_{2}G_{c2})^{T}(D_{2}G_{c2 })&\cdots&(D_{2}G_{c2})^{T}(D_{m}G_{cm})&-(D_{2}G_{c2})^{T}F\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ -(D_{m}G_{cm})^{T}C&(D_{m}G_{cm})^{T}(D_{1}G_{c1})&(D_{m}G_{cm})^{T}(D_{2}G_{c2 })&\cdots&(D_{m}G_{cm})^{T}(D_{m}G_{cm})&-(D_{m}G_{cm})^{T}F\\ F^{T}C&-F^{T}(D_{1}G_{c1})&-F^{T}(D_{2}G_{c2})&\cdots&-F^{T}(D_{m}G_{cm})&F^{T} F-\gamma^{2}I\end{array} \right] \tag{27}\]  
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}