## 3 Problem Research
### Ship Control Technology Problem
#### 3.2.1 Linear Matrix Inequalities in Use  
As a case-study, this paper shows the results of testing on lake Silm of a multidimensional controller for ship motions in a 3 DOF space with the use of LMI [48]. Linear matrix inequalities are a tool of convex optimization from a convex set of constraints to be formulated for the process of synthesizing a controller from a state. The canonical form of linear matrix inequalities has the form:  
\[\mathbf{F}(x)=\mathbf{F}_{0}+\sum_{i=1}^{m}\mathbf{F}_{i}x_{i}\succ 0 \tag{1}\]  
where \(x=[x_{1},x_{2},\ldots,x_{i}]^{T}\in\mathbf{R}m\) is a decision variable vector (unknown); \(\mathbf{F}_{0},\mathbf{F}_{i}\in\mathbf{R}^{n\times n}\) are real and symmetrical matrices. In Equation (1), the notation \"\(\succ 0\)\" means that matrix \(\mathbf{F}(x)\) is positively determined. [PERSON] noted that a necessary and sufficient condition for a linear system to be asymptotically stable is to find a symmetric and positively specified matrix \(\mathbf{P}=\mathbf{P}^{T},\mathbf{P}\succ 0\) (the problem of finding a symmetric, positively defined matrix \(\mathbf{P}\) is often referred to as the existence problem (_feasibility problem_). \(\mathbf{P}\) is the unknown variable when satisfying the inequality called **Lapunov's inequality**:  
\[\mathbf{A}^{T}\mathbf{P}+\mathbf{PA}\prec 0. \tag{2}\]  
The first condition related to the existence issue (2) and the second condition \(\mathbf{P}\succ 0\) can be written together as a composite of the two conditions into a single LMI condition:  
\[\left[ \begin{array}{cc}-\mathbf{A}^{T}\mathbf{P}-\mathbf{PA}&0\\ 0&\mathbf{P}\end{array} \right]\succ 0, \tag{3}\]  
where the matrix \(\mathbf{A}\) is the matrix of the object under study, and the matrix \(\mathbf{P}\) is the search matrix. The following is a methodology for synthesizing a state regulator using linear matrix inequalities. 1. The first stage of synthesis is concerned with determining the desired pole position region, located in the left half-plane of the complex variable s in order to specify the dynamic properties of the closed system under study. The zone (Figure 7) was bounded by two vertical strips \(\alpha_{1}\) and \(\alpha_{2}\), and a sector with an opening angle of \(\phi\). \[\mathbf{F}_{\mathbb{D}\phi}= \begin{bmatrix}\sin(\phi)&\cos(\phi)\\ -\cos(\phi)&\sin(\phi)\end{bmatrix} s+ \begin{bmatrix}\sin(\phi)&\cos(\phi)\\ -\cos(\phi)&\sin(\phi)\end{bmatrix} s,\] (4)  
Figure 6: LMI-based low-speed controller syntheses. Conceptualization of the scheme using the state controller determined by the K-margin with overriding trajectory controller [47]. The vertical stripe is defined by two vertical strips \(\alpha_{1}\) and \(\alpha_{2}\) on the assumption that the \(Re(s)=\frac{1}{2}(s+s)\) area is defined as follows:  
\[\mathbf{D}_{\alpha 1,A2}=\left\{s\;;\;\alpha_{1}<\frac{1}{2}(s+s)<\alpha_{2} \right\}, \tag{5}\]  
The first condition of the LMI:  
\[\mathbf{R}_{\mathbb{D}}(\mathbf{A}_{\
u}\mathbf{X}_{zone})=\mathbf{L}\otimes \mathbf{X}_{zone}+\mathbf{M}\otimes(\mathbf{AX}_{zone})+\mathbf{M}^{T} \otimes(\mathbf{AX}_{zone})^{T}\prec 0 \tag{6}\]  
is satisfied if and only if there is a symmetric and positively defined matrix \(\mathbf{X}_{zone}\), where: \(\mathbf{A}\)--matrix of the object (ship \"Blue Lady\"); \(\mathbf{X}_{zone}\)--the symmetric and positively defined matrix being the sought Lyapunov \(\mathbf{X}\succ 0,\mathbf{X}=\mathbf{X}^{T}\); \(\mathbf{L}\),\(\mathbf{M}\)--the corresponding matrices, which have been defined by the user of the zone according to the theorem. The LMI notation for determining the controller matrix is shown below:  
\[\mathbf{Y}_{zone}=\mathbf{K}_{zone}*\mathbf{X}_{zone} \tag{7}\]  
1. It is necessary to transform the form of the BMI (two unknowns X and Y) condition to the LMI condition; therefore, an additional matrix \(\mathbf{Y}_{zone}\) is introduced. The use of such a \"treatment\" is due to the assumption that, since the matrix \(\mathbf{X}_{zone}\) is positively defined, its inverse is present, e.g.