# Pareto Effect of LMI for Ship Propulsion
## 3 The Norm \(H_{\infty}\) in Linear Matrix Inequalities  
The controller determined from the second limiting tool, called the \(H_{\infty}\) norm, has an indirect effect on obtaining robust stability for the system, which is needed to evaluate the control deviation in linear matrix inequalities. The derived system of Equation (1) together with the relation (2) enables synthesis of a state controller using linear matrix inequalities. The signal \(z_{\infty}\) of specified transmittance of the closed system \(G(s)_{\infty}\) for norm \(H_{\infty}\) described by the output signal \(z(t)\) and input (setpoint) signal \(w(t)\), hereafter referred to as lag, is as follows:  
\[z_{\infty}=w-z. \tag{3}\]  
The author aimed to lead the closed control system to minimize deviation, which is indirectly related to the minimization of \(H_{\infty}\) norm, that is, to estimate it from above by means of a scalar quantity, denoted \(\gamma_{\infty}\). The determination of the \(H_{\infty}\) norm by means of linear matrix inequalities for the multidimensional object \"Blue Lady\" is first related to the description (in state-space equations) of the signal denoted as \"\(z_{\infty}\)\" (based on the Equation (1)), as shown below:  
\[z_{\infty}=(\mathbf{C}_{\infty}+\mathbf{D}_{\infty u}\mathbf{K}_{\infty})x_{ cl}+\mathbf{D}_{\infty u}w. \tag{4}\]  
Referring to [17; 18] LMI condition for \(H_{\infty}\) norm consists of the notation:  
1. If a positively symmetric matrix exists \(\mathbf{X}_{\infty}\) (\(\mathbf{X}_{\infty}=\mathbf{X}_{\infty}^{T}\), \(\mathbf{X}_{\infty}\succ 0\)), the form of LMI condition for the norm \(H_{\infty}\) has the following form: \[\left[\begin{array}{cccc}\mathbf{A}_{cl}\mathbf{X}_{\infty}+\mathbf{B}_{u} \mathbf{Y}_{\infty}+\mathbf{X}_{\infty}\mathbf{A}_{cl}^{T}+\mathbf{Y}_{ \infty}^{T}\mathbf{B}_{u}^{T}&\mathbf{B}_{w}&\mathbf{X}_{\infty}\mathbf{C}_{ \infty}^{T}+\mathbf{Y}_{\infty}^{T}\mathbf{D}_{u\infty}^{T}\\ \mathbf{B}_{w}^{T}&-\gamma_{\infty}\mathbf{I}&\mathbf{D}_{\infty u}^{T}\\ \mathbf{C}_{\infty}\mathbf{X}_{\infty}+\mathbf{D}_{u\infty}\mathbf{Y}_{ \infty}&\mathbf{D}_{\infty uw}&-\gamma_{\infty}\matrix size \(\mathbf{C}_{\infty}\) based on the relation (3) is the same as object matrix size \(\mathbf{C}_{ship}\). Matrix \(\mathbf{C}_{\infty}\) includes: \[\mathbf{C}_{\infty}=-\big{[}\ \ \mathbf{C}_{ship}\ \ \big{]};\] (6) \[\bullet\] matrix size \(\mathbf{D}_{\infty u}\) is the same as matrix \(\mathbf{D}_{ship}\) size. Matrix \(\mathbf{D}_{\infty}\) includes: \[\mathbf{D}_{\infty}=-\big{[}\ \ \mathbf{D}_{ship}\ \ \big{]};\] (7) \[\bullet\] matrix \(\mathbf{D}_{\infty u}\) is derived from relation (3) and has the form: \[\mathbf{D}_{\infty}w=diag[1]_{3x3}-\big{[}\ \ \mathbf{D}_{ship}\ \ \big{]}.\] (8)
2. For the transmittance matrix \(\mathbf{G}(s)_{\infty}\) is an estimate that is the upper boundary, described by a scalar value \(\gamma_{\infty}\) as stated: \[\|\mathbf{G}(s)\|_{\infty}^{2}<\gamma_{\infty}\] (9) For the calculation of norm \(H_{\infty}\), transmittance matrix \(\mathbf{G}(s)_{\infty}\) for the closed system can be determined: \[\mathbf{G}(s)_{\infty}=(\mathbf{C}_{\infty}+\mathbf{D}_{\infty u}\mathbf{K}_ {\infty})(sI-(\mathbf{A}_{cl}+\mathbf{B}_{u}\mathbf{K}_{\infty})^{-1}) \mathbf{B}_{cl}w+\mathbf{D}_{\infty u}.\] (10) In the design process, value \(\gamma_{\infty}\) is an estimate that is the upper boundary of norm \(H_{\infty}\). According to the authors, this value can be minimized or taken as a constant.