# Pareto Effect of LMI for Ship Propulsion
## 1 Introduction  
In the last decade, numerous publications describing the use of Pareto curves in the field of economics have been published. This article, however, will present several papers where the Pareto curve is used in the fields of electronics and automation. For example, the authors of [1] describe the use of popular Linear Matrix Inequality (LMI) controllers, along with the use of \(H_{\infty}\), \(H_{2}\) norms, as a not very accurate approximation of Pareto optimal solutions. In order to obtain a correct Pareto curve, they study the BMI (Bilinear Matrix Inequalities) controller, which is designed for nonlinear objects. This controller obtains better results than the LMI controller only for a certain range. The use of a Multi-Objective Genetic Algorithm (MGA) based on the above-mentioned controllers results in better solutions for the Pareto curve (below, the Pareto curve for the LMI controller). In addition, the use of MGA for an LMI controller can significantly improve the accuracy of obtaining a (close to true) Pareto front. In contrast, the authors of [2] continue their research on obtaining a correct Pareto curve. They put forward the thesis that the use of multi-criteria genetic algorithm (MGA) is the best existing method (in 2004) for obtaining a close-to-true Pareto curve. In addition, they noted that MGA can be applied not only to mixed-norm \(H_{2}/H_{\infty}\) optimization, but to any multi-criteria control problem. However, in the publication [3], instead of using the well-known method of Multi-Objective Evolutionary Algorithms (MOEA), the authors propose to use Multi-Objective Pole Placement with Evolutionary Algorithms (MOPPEA) along with optimization of \(H_{2}/H_{\infty}\) norms. The authors point out that the main advantage of the method used is the absence of system stability issues during optimization. However, a definite disadvantage of the method is the high order of the obtained controller. The authors point out the problem of pole placement, where for higher order models, some pole localizations result in very high system gain. In [4], the authors compare the performance of passive suspension (vehicle dampers) with two types of active suspension based on a Linear Matrix Inequalities (LMI) controllerand a quasi-Linear Parameter Varying (qLPV) controller together with the application of \(H_{2}/H_{\infty}\) norms, finding the Pareto optimal solution for mixed norms used. Simulations were performed on a non-linear half-vehicle model (single vehicle axis). In their study, they prove the advantage of using the optimal solution of mixed \(H_{2}/H_{\infty}\) norms over the \(H_{\infty}\) norm alone (e.g., vehicle roll angle reduction). They then compare the performance of the LMI controller for the two Pareto optimal points together with passive suspension. They show that for passive suspension, there are large overshoots and oscillations, where for the LMI controller for the two values \([\gamma_{2},\gamma_{\infty}]\), the waveforms slightly differ between each other and show high robustness to disturbances. They then compare the performance of the LMI controller, qLPV controller, and passive suspension. The authors show that the qLPV controller exhibits more robust control over the LMI controller, although the qLPV controller is more complex than the LMI controller. The authors of [5], instead of using Dynamic Multi-objective Optimization (DMO), used the Dynamic Multi-objective Immune Optimization Algorithm (DMIOA). The results of the work were compared with three other algorithms (CSADMO, DNSGAII-A, and DNSGAII-B), which are used to solve DMO problems. The authors showed that the use of DMIOA is characterised by robustness of control and the achievement of the best optimal solution with respect to the compared algorithms (where the quality of their solutions was worse).