# Pareto Effect of LMI for Ship Propulsion
## 2 Object Control--Multivariable Object  
The model of the training ship \"Blue Lady\" being tested is owned by the Shipping Safety and Environmental Protection Foundation located in Ilawa-Kamionka (Poland). The ship is a model of a Very Large Crude Carrier (VLCC) class tanker in a 1:24 scale with all characteristics of the real vessel preserved. Verification of the test results was performed on a simulation mathematical 3D object (Figure 1) and not the actual ship model.  
The ship model is equipped with five propellers: a main fixed blade propeller, two rotary thrusters, two tunnel thrusters, and one fin rudder. This work will analyze the operation of propulsion and steering devices (Table 1).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Name of Signal** & **Range** & **Unit** \\ \hline ng & Main propeller rotational speed & \((-200;+480)\) & [rpm] \\ delta & Rudder deflection angle & \((-35;+35)\) & [deg] \\  & Relative thrust of the bow & & \\ sstd & tunnel thruster & \((-1,+1)\) & [-] \\  & Relative thrust of the stern & & \((-1,+1)\) & [-] \\  & tunnel thruster & & \\ ssod & Relative thrust of the bow & & \\  & rotary thruster & & \((0,+1)\) & [-] \\  & alfa d & Bow rotary thruster deflection angle & \((-120,+120)\) & [deg] \\  & Relative thrust of the stern & & \((0,+1)\) & [-] \\  & rotary thruster & & \((-120,+120)\) & [deg] \\  & alfa r & Stern rotary thruster deflection angle & \((-120,+120)\) & [deg] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Input signals of propulsion and control devices of the “Blue Lady” ship model.  
Figure 1: (**a**) Blue Lady 3D ship model. (**b**) Actual Blue Lady ship model on Silm lake. Ship parameters: Lenght 13.75 [m], width 2.38 [m].  
This paper mainly focuses on the description of \(H_{\infty}\) and \(H_{2}\) norms, and the multidimensional object used in simulations, the \"Blue Lady\" ship model, is described in detail in the papers below [11; 12; 13; 14; 15; 16]. State-space equations of the closed system are as follows:  
\[\left\{\begin{array}{rclcl}\hat{x}_{cl}&=&(\mathbf{A}_{cl}+\mathbf{B}_{u} \mathbf{K})\mathbf{x}_{cl}&+&\mathbf{B}_{w}w,\\ z&=&(\mathbf{C}_{z}+\mathbf{D}_{zu}K)x_{cl}&+&\mathbf{D}_{zw}w,\\ z_{\infty}&=&(\mathbf{C}_{\infty}+\mathbf{D}_{\infty u}\mathbf{K})x_{cl}&+& \mathbf{D}_{\infty z}w,\\ z_{2}&=&(\mathbf{C}_{2}+\mathbf{D}_{2u}K)x_{cl}&+&\mathbf{D}_{2w}w,\end{array}\right. \tag{1}\]  
where:  
* \(\mathbf{A}_{cl}\)--state matrix of the object, represents the dynamics of the system,
* \(\mathbf{B}_{u}\)--control matrix of control '\(u\)' signal, \(\mathbf{B}_{w}\)--control matrix of input '\(w\)',
* \(\mathbf{C}_{z}\)--output matrix of output signal outputs 'z', \(\mathbf{C}_{\infty}\)--output matrix of '\(z_{\infty}\)' signal,
* \(\mathbf{D}_{zu}\)--transition matrix of signals 'z' and '\(u\)', \(\mathbf{D}_{zw}\)--transition matrix of signals '\(w\)', \(\mathbf{D}_{\infty u}\)--transition matrix of signals '\(z_{\infty}\)' and '\(u\)', \(\mathbf{D}_{\infty u}\)--transition matrix of signals '\(w\)' and '\(z_{\infty}\)', \(\mathbf{D}_{2w}\)--transition matrix of signals '\(z_{2}\)' and '\(w\)', \(\mathbf{D}_{2u}\)--transition matrix of signals '\(z_{2}\)' and '\(w\)', \(\mathbf{D}_{2u}\)--transition matrix of signals '\(z_{2}\)' and '\(u\)',
* \(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ \mathbf{      \mathbf{   }}}}}}}}}}}}}\)  
The LMI notation for the determination of the controller matrix \(\mathbf{K}\) is as follows: \(\mathbf{Y}=\mathbf{K}\mathbf{X}\). There are two unknowns, \(\mathbf{K}\) and \(\mathbf{X}\), which are the conditions for Bilinear Matrix Inequalities (BMI). The gain matrix \(\mathbf{K}\) of the controller from state-space is described by the following relation:  
\[\mathbf{K}=\mathbf{Y}\mathbf{X}^{-1}. \tag{2}\]