## 4 Time-Delay Compensation Control Gain Design by GCC Algorithm
\begin{array}{cc}0\\ \frac{B_{u}\big{[}(\delta_{M}+1)^{-1}-1\big{]}}{C}\end{array} \right]\!. \tag{36}\]  
By substituting Equation (35) into Equation (34), and letting \(u(t)=-G\cdot Z\), the state-equation of a control system is:  
\[\dot{Z}(t)=[A-B_{2}G+DF(E_{1}-E_{2}G)]Z(t)+\big{(}\overline{A}+DFE_{d}\big{)}Z (t-d) \tag{37}\]  
The performance index of the system (37) can be defined as:  
\[J=\int_{0}^{\infty}\Big{(}Z^{T}QZ+U^{T}RU\Big{)}dt=\int_{0}^{\infty}\Big{\{}Z ^{T}\Big{[}Q+(-G)^{T}R(-G)\Big{]}Z\Big{\}}dt \tag{38}\]  
Figure 14: Time-varying delays of the experimental system. Supposing \(\widetilde{A}=A-B_{2}G+DF(E_{1}-E_{2}G)\). If there exists symmetric positive-definite matrices \(P,S\in R^{n\times n}\), then:  
\[\left[ \begin{array}{cc}\widetilde{A}^{T}P+P\widetilde{A}+S+Q+(-G)^{T}R(-G)&P \big{(}\overline{A}+DFE_{d}\big{)}\\ \big{(}\overline{A}+DFE_{d}\big{)}^{T}P&-S\end{array} \right]<0 \tag{39}\]  
Based on the [PERSON] stability theory [34], it proves that the system (37) is asymptotically stable. Defining:  
\[Y=\left[ \begin{array}{cc}\left(A-B_{2}G\right)^{T}P+P(A-B_{2}G)+S+Q+\left(-G \right)^{T}R(-G)&P\overline{A}\\ \overline{A}^{T}P&-S\end{array} \right] \tag{40}\]  
Equation (39) can be written as:  
\[Y+\left[ \begin{array}{c}PD\\ 0\end{array} \right]F[(E_{1}-E_{2}G)\quad E_{d}]+[(E_{1}-E_{2}G)\quad E_{d}]^{T }F^{T}\left[ \begin{array}{c}PD\\ 0\end{array} \right]^{T}<0 \tag{41}\]  
where the matrix \(F\) satisfies the condition \(F^{T}F<I\). If there exists a positive constant \(\mu\), inequality (41) can be written as:  
\[Y+\mu\Bigg{[} \begin{array}{c}PD\\ 0\end{array} \Bigg{]}\Bigg{[} \begin{array}{c}PD\\ 0\end{array} \Bigg{]}^{T}+\mu^{-1}[(E_{1}-E_{2}G)\quad E_{d}][(E_{1}-E_{2}G) \quad E_{d}]^{T}<0 \tag{42}\]  
The proof can be found in Reference [35]. From [PERSON]'s complement [36], inequality (42) can be expressed as:  
\[\left[ \begin{array}{cc}(A-B_{2}G)^{T}P+P(A-B_{2}G)+\mu PDD^{T}P+S+Q+(-G)^{T} R(-G)&P\overline{A}\quad(E_{1}-E_{2}G)^{T}\\ \overline{A}^{T}P&-S&E_{d}^{T}\\ (E_{1}-E_{2}G)&E_{d}&-\mu I\end{array} \right]<0 \tag{43}\]  
Inequality (43) is pre and post multiplying \(\text{diag}\{P^{-1},I,I\}\), and defining \(X=P^{-1}\):  
\[\left[ \begin{array}{ccc}H&\overline{A}&X(E_{1}-E_{2}G)^{T}\\ \overline{A}^{T}&-S&E_{d}^{T}\\ (E_{1}-E_{2}G)X&E_{d}&-\mu I\end{array} \right]<0 \tag{44}\]  
where \(H=X(A-B_{2}G)^{T}+(A-B_{2}G)X+\mu DD^{T}+XSX+X\Big{[}Q+(-G)^{T}R(-G)\Big{]}X\). From [PERSON]'s complement, and defining \(W=-GX\), inequality (44) can be described as:  
\[\left[ \begin{array}{cccc}\widetilde{H}&\overline{A}&(E_{1}X+E_{2}W)^{T}&X& W^{T}&X\\ \overline{A}^{T}&-S&E_{d}^{T}&0&0&0\\ E_{1}X+E_{2}W&E_{d}&-\mu I&0&0&0\\ X&0&0&-Q^{-1}&0&0\\ W&0&0&0&-R^{-1}&0\\ X&0&0&0&0&-S^{-1}\end{array} \right]<0 \tag{45}\]  
where \(\widetilde{H}=AX+B_{2}W+(AX+B_{2}W)^{T}+\mu DD^{T}\). Inequality (45) is pre and post multiplying \(\text{diag}\{I,\)\(S^{-1},I,I,I,I\}\), and defining \(V=S^{-1}\), it can be expressed as:\[\left[ \begin{array}{cccccccc}\widetilde{H}&\overline{A}V&(E_{1}X+E_{2}W)^{T}&X&W^{T }&X\\ V\overline{A}^{T}&-V&VE_{d}^{T}&0&0&0\\ E_{1}X+E_{2}W&E_{d}V&-\upmu I&0&0&0\\ X&0&0&-Q^{-1}&0&0\\ W&0&0&0&-R^{-1}&0\\ X&0&0&0&0&-V\end{array} \right]<0 \tag{46}\]  
The LMI toolbox of Matlab can be used to solve the optimal solutions of \(\upmu^{\prime}\), \(X^{\prime}\), \(W^{\prime}\), and \(V^{\prime}\). The state feedback control law for a time-delay system with parametric uncertainty is:  
\[U=-GZ=W^{\prime}(X^{\prime})^{-1}Z \tag{47}\]  
The GCC control gain shown in Equation (47) can be used to design a new compensation controller for AMD control systems with parametric uncertainties or time-varying delays caused by a higher-order CSI effect, which is different from those in the reference literature. To validate the efficiency of the developed method, the compensation controller with a state feedback GCC algorithm is applied to the experimental system. This paper focuses on the displacement control effect of the 4 th floor of the experimental system and the size of the AMD stroke.