## 3 Problem Research
### Ship Control Technology Problem
#### 3.2.1 Linear Matrix Inequalities in Use
\(\mathbf{Y}_{zone}=\mathbf{X}_{zone}^{-1}\). Therefore, the gain matrix \(\mathbf{K}_{zone}\) of the controller from the state is described by the following equation: \[\mathbf{K}_{zone}=\mathbf{Y}_{zone}\cdot\mathbf{X}_{zone}^{-1}.\] (8)
2. The second stage is the minimization of the \(H_{\infty}\) norm, associated with the estimation of the scalar value \(\gamma_{\infty}\), which is the upper limit of the norm. When determining the smallest value of the norm \(\gamma_{\infty}\) for a given area of the position of the poles of a closed system, the obtained scalar value \(\gamma_{\infty}\) is assumed to be a constant value. The scalar value of \(\gamma_{\infty}\) can be calculated on the basis of the minimization of the norm, or it can be assumed that \(\gamma_{\infty}\) is a constant value constraint. The controller determined from the second step of the so-called second limiting tool, called the \(H_{\infty}\) norm, has an indirect effect on obtaining robust stability for the system. It is needed to evaluate the control lag in linear matrix inequalities. The definition of the \(H_{\infty}\) norm is derived from [PERSON]'s algebraic inequality proof in [48; 49]: \[(\mathbf{X}_{\infty}\mathbf{A}+\mathbf{A}^{T}\mathbf{X}_{\infty}+\mathbf{C}^ {T}\mathbf{C})+(\mathbf{X}_{\infty}\mathbf{B}+\mathbf{C}^{T}\mathbf{D})( \gamma_{\infty}^{2}\mathbf{I}-\mathbf{D}^{T}\mathbf{D})^{-1}(\mathbf{B}^{T} \mathbf{X}_{\infty}+\mathbf{D}^{T}\mathbf{C})\prec 0\] (9)Referring to the lemma of [PERSON]'s complement for a matrix \(\mathbf{X}\) of dimension \([3\times 3]\) and based on the inequality (9), the following theorem that is useful for controller synthesis can be quoted. **Theorem 1** (Theorem 5.4 [49]). : _The notation \(\mathbf{G}(s)_{\infty}<\gamma_{\infty}\) is true if and only if there exists a matrix \(\mathbf{X}\succ 0\) for which one of the following three inequalities is satisfied:_  
\[\left[ \begin{array}{ccc}\mathbf{X}_{\infty}\mathbf{A}^{T}+\mathbf{A}\mathbf{ X}_{\infty}&\mathbf{B}&\mathbf{X}\mathbf{C}^{T}\\ \mathbf{B}^{T}&-\gamma_{\infty}\mathbf{I}&\mathbf{D}^{T}\\ \mathbf{C}\mathbf{X}&\mathbf{D}&-\gamma_{\infty}\mathbf{I}\end{array} \right] \prec 0 \tag{10}\]  
_or_  
\[\left[ \begin{array}{ccc}\mathbf{A}\mathbf{X}_{\infty}+\mathbf{X}_{\infty} \mathbf{A}^{T}&\mathbf{B}&\mathbf{X}\mathbf{C}^{T}\\ \mathbf{B}^{T}&-\gamma_{\infty}^{2}\mathbf{I}&\mathbf{D}^{T}\\ \mathbf{C}\mathbf{X}&\mathbf{D}&-\mathbf{I}\end{array} \right]\prec 0, \tag{11}\]  
_or_  
\[\left[ \begin{array}{ccc}\mathbf{A}\mathbf{X}_{\infty}+\mathbf{X}_{\infty} \mathbf{A}^{T}+\mathbf{B}\mathbf{B}^{T}&\mathbf{X}_{\infty}\ \mathbf{C}^{T}+\mathbf{B}\mathbf{D}^{T}\\ \mathbf{C}\mathbf{X}_{\infty}+\mathbf{D}\mathbf{B}^{T}&-\gamma_{\infty}^{2} \mathbf{I}+\mathbf{D}\mathbf{D}^{T}\end{array} \right]\prec 0. \tag{12}\]  
Theorem 1 has three conditions. The paper uses condition 2 and makes use of the Yalmip [50] and SeDuMi [51] library. For a system described by equations in their own state space with transmittance \(\mathbf{G}(s)\), the dependence of the norm \(H_{\infty}\) on the parameter \(\gamma_{\infty}\) is as follows:  
\[\|\mathbf{G}(s)\|_{\infty}^{2}<\gamma_{\infty}^{2} \tag{13}\]  
The state controller synthesis problem has a solution if the second LMI condition is satisfied. Referring to the property about the existence of the inverse of the matrix \(\mathbf{Y}_{\infty}=\mathbf{X}_{\infty}^{-1}\), the matrix \(\mathbf{K}_{\infty}\) is written by the following relation:  
\[\mathbf{K}_{\infty}=\mathbf{X}_{\infty}^{-1}\cdot\mathbf{Y}_{\infty}. \tag{14}\]  
If matrix \(\mathbf{X}_{\infty}\) (\(\mathbf{X}_{\infty}=\mathbf{X}_{\infty}^{T},\mathbf{X}_{\infty}\succ 0\)), we can write:  
\[\mathbf{K}_{\infty}=\mathbf{Y}_{\infty}\cdot\mathbf{X}_{\infty}^{-1}. \tag{15}\]  
For a proof of the Theorem 1, see the paper [49]. 3. The third step is the minimization of the \(H_{2}\) norm. For a fixed value of \[\gamma_{\infty}>\gamma_{\text{\it{comin}}}\] (16) the value of \(\gamma_{2}\) is sought.