# Pareto Effect of LMI for Ship Propulsion
## 5 Compromise Solution of \(H_{2}\) and \(H_{\infty}\) Norms in LMI  
In recent years, linear matrix inequalities (LMIs) have gained importance as a tool for solving control problems that seemed impossible to solve analytically. This tool has become popular due to its ability to accommodate convex constraints. When implementing a mixed norm \(H_{2}/H_{\infty}\), an optimal solution of the norm is sought. Yet, when optimising these norms, there is a contradiction in the requirements of the individual norms. In order to solve this problem, among other things, additional scalar values are used. For the norm \(H_{\infty}\), a scalar value \(\gamma_{\infty}\) is used, which can be regarded as an upper boundary on this norm. Minimization of the parameter \(\gamma_{\infty}\) is one of the objectives of multi-criteria optimization. For a system described by equations in the state-space (22) with transmittance \(G(s)\), the dependence of the norm \(H_{\infty}\) on the parameter \(\gamma_{\infty}\) is as follows:  
\[\|\mathbf{G}(s)\|_{\infty}^{2}<\gamma_{\infty}^{2} \tag{22}\]  
The second condition for multi-criteria optimization is to minimize the scalar variable \(\gamma_{2}\) for the norm \(H_{2}\). Dependence of this variable for transmittance \(G(s)\) is described by the inequality (23):  
\[\|\mathbf{G}(s)\|_{2}^{2}\prec\gamma_{2}^{2}, \tag{23}\]  
Identical to the implementation of mixed norm \(H_{2}/H_{\infty}\), it is not possible to minimize both scalar values \(H_{2}\) and \(H_{\infty}\). In the literature, the mixed problem is usually solved by minimizing the convex combination of \(H_{2}\) and \(H_{\infty}\), which represents a compromise between the two norms. Such minimization can take the following form:  
\[min(\alpha_{1}H_{\infty}+\alpha_{2}H_{2}) \tag{24}\]  
The resulting objective function: \(J=\|H\|_{2}^{2}+\|H\|_{\infty}^{2}\) is optimized using a Pareto curve in many sources, called the Pareto front [19; 20; 21; 22; 23]. An optimal solution, in the sense of _Pareto_, is a non-dominated set of solutions (a non-dominated solution (Pareto-optimal) occurs when it is not possible to find a solution that is better with respect to at least one criterion without worsening the others), located in the entire admissible space of sought answers. Plotting the _Pareto curve_ for the two criteria, defined as the \(H_{\infty}\) norm and the \(H_{2}\) norm, requires several steps. 1. The first step is to compute the constraint from above the scalar variable \(\gamma_{\infty}\). First, the area of the pole placement in the left half-plane of the complex variable s [24] is determined. If there is a symmetric, positively definite matrix \(\mathbf{X}_{\infty}=\mathbf{X}_{\infty}^{\mathsf{T}}\), based on this, using the LMI condition for the norm \(H_{\infty}\), the scalar variable \(\gamma_{\infty}\) is determined. The designer assumes that the calculated value \(\gamma_{\infty}\) = is the upper limit of the mentioned norm. 2. The second step is to estimate the scalar variable \(\gamma_{\infty}\) from above, more precisely to minimize the norm \(H_{2}\). Determining the relationship (Pareto curve) between the minimum value of the \(H_{2}\) norm and the upper bound on the value of the \(H_{\infty}\) norm determined in the first step involves determining the range of scalar variables \(\gamma_{2}\) that are the upper bound for the norm under study. In this example, the scalar variable \(\gamma_{2}\) was tested in the range from [1.05:2.5] with a step of 0.05. Below, a flow Figure 2 is presented that shows steps necessary for controller synthesis. Points determined this way form the Pareto curve, where anything above the curve is a solution for which Criterion One \(H_{\infty}\) does not deteriorate Criterion Two \(H_{2}\). It is up to the user to decide which point in the region of acceptable solutions satisfies the intermediate assumptions regarding the control signal or static lag.