# Active Tendon Control of Stay Cable by a Giant Magnetotrstive Actuator Considering Time-Delay
## 3 Simplification and Verification of Bilinear Controlling Equations  
In Equation (11), the control matrix \(\mathbf{B}=\mathbf{B}_{1}\mathbf{Z}+\mathbf{B}_{2}\) is a linear function of the state variable \(\mathbf{Z}\), and the multiplication of \(\mathbf{B}\) and \(U(t)\) is a nonlinear term for the control system, which can be referred as a bilinear control system. [PERSON] et al. [21] employed the direct Lyapunov method to derive the optimal control algorithm for bilinear systems. The optimal control force in the algorithm is expressed as [21]:  
\[U(t)=-\frac{1}{R}\mathbf{Z}^{T}\mathbf{S}(\mathbf{B}_{1}\mathbf{Z}+\mathbf{B} _{2}) \tag{12}\]  
where \(R\) is the weight coefficient greater than zero, \(\mathbf{S}\) is a symmetric positive definite matrix and can be obtained from:  
\[\mathbf{SA+A}^{T}\mathbf{S=-Q} \tag{13}\]  
where \(\mathbf{Q}\) is also a symmetric positive definite matrix. In this paper, the research focuses on the influence of time delay on the controlled cable and the time delay compensation method. Due to the small-sag cable, bilinear control is considered as a weak nonlinear and simplified method for linearizing the bilinear system is adopted. A simplified linearization method is proposed for Equation (10), that is, \(\mathbf{B}_{1}=0\) is assumed, so that Equation (11) can be described as  
\[\dot{\mathbf{Z}}=\mathbf{A}\mathbf{Z}+\mathbf{B}_{2}U(t) \tag{14}\]  
Since the mode-based time delay compensation method is used in the follow-up study, the control force of Equation (14) is determined by the modal control of state equation. Let \(-\alpha_{j}\pm i\beta_{j}\) and \(\alpha_{j}\pm i\beta j\) (\(i^{2}=-1\), \(j=1\),2,3) be the \(j\)-th complex conjugate eigenvalue and \(j\)-th eigenvector of eigenmatrix \(\mathbf{A}\) in Equation (14), respectively. The following relationship can be obtained:  
\[\psi^{-1}\mathbf{A}\psi=\mathbf{\Lambda}=\left[ \begin{array}{ccc}\Lambda_{ 1}&0&0\\ 0&\Lambda_{2}&0\\ 0&0&\Lambda_{3}\end{array} \right] \tag{15}\]  
where  
\[\psi=\left[ \begin{array}{cccccc}a_{1}&b_{1}&a_{2}&b_{2}&a_{3}&b_{3}\end{array}  \right]\text{, }\mathbf{\Lambda}_{j}=\left[ \begin{array}{cc}\alpha_{j}&\beta_{j}\\ -\beta_{j}&\alpha_{j}\end{array} \right](j=1,2,3)\]  
The state vector of the system can be modally expressed as:  
\[\mathbf{Z}=\psi\xi(t) \tag{16}\]  
where \(\xi(t)\) denotes the vector of modal coordinates, and it has following forms:  
\[\xi(t)=\left[ \begin{array}{cc}\xi_{1}^{\text{T}}(t)&\xi_{2}^{\text{T}}(t)& \cdots&\xi_{n}^{\text{T}}(t)\end{array} \right]^{T}\text{, }\xi_{j}^{\text{T}}(t)=\left[ \begin{array}{cc}\xi_{j1}(t)&\xi_{j2}(t)\end{array}  \right]\text{, }(j=1,2,3)\]  
Substituting Equation (16) into Equation (14), the state equation expressed by the modal coordinates is:  
\[\dot{\xi}(t)=\mathbf{\Lambda}\xi(t)+\mathbf{U}_{m}(t) \tag{17}\]  
where  
\[\mathbf{U}_{m}(t)=\left[ \begin{array}{cc}\mathbf{U}_{m1}^{\text{T}}&\mathbf{ U}_{m2}^{\text{T}}&\mathbf{U}_{m3}^{\text{T}}\end{array} \right]^{\text{T}}=\psi^{-1} \mathbf{B}U(t)\text{, }\mathbf{U}_{mj}^{\text{T}}=\left[ \begin{array}{cc}u_{mj1}&u_{mj2}\end{array} \right] \tag{18}\]  
Equation (17) contains three pairs of independent equations, in which the \(j\)-th pair of equations describes the control of the \(j\)-th mode, namely:  
\[\dot{\mathbf{\xi}}_{j}(t)=\mathbf{\Lambda}_{j}\xi_{j}(t)+\mathbf{U}_{mj}(t) \tag{19}\]  
According to the LQR control algorithm, the optimal control force in Equation (17) can be obtained as follows:  
\[\mathbf{U}_{mj}(t)=-R^{-1}\mathbf{P}\xi_{j}(t)=-\mathbf{G}_{j}\xi_{j}(t) \tag{20}\]  
where, \(\mathbf{G}_{j}\) is the feedback gain matrix, matrix \(\mathbf{P}\) is the solution of the following Riccati equation [32]:  
\[-\mathbf{P}\mathbf{\Lambda}_{j}-\mathbf{\Lambda}_{j}^{\text{T}}\mathbf{P}+ \mathbf{P}R^{-1}\mathbf{P}-\mathbf{Q}_{1}=0 \tag{21}\]  
where \(\mathbf{Q}_{1}\) is the semi-positive definite weight matrix and \(R\) is the weight coefficient. So we have:  
\[\mathbf{U}_{m}(t)=\left[ \begin{array}{c}U_{m1}\\ U_{m2}\\ U_{m3}\end{array} \right]=-\left[