Statement of Problem of Pareto Frontier Management and Its Solution in the Analysis and Synthesis of Optimal Systems

Abstract

The article research concerns the multi-criteria optimization (MCO), which assumes that operation quality criteria of the system are independent and specifies a way to improve values of these criteria. Mutual contradiction of some criteria is a major problem in MCO. One of the most important areas of research is to obtain the so-called Pareto - optimal options. The subject of research is Pareto front, also called the Pareto frontier. The article discusses front classifications by its geometric representation for the case of two-criterion task. It presents a mathematical description of the front characteristics using the gradients and their projections. A review of current domestic and foreign literature has revealed that the aim of works in constructing the Pareto frontier is to conduct research in conditions of uncertainty, in the stochastic statement, with no restrictions. A topology both in two- and in three-dimensional case is under consideration. The targets of modern applications are multi-agent systems and groups of players in differential games. However, all considered works have no task to provide an active management of the front. The objective of this article is to discuss the research problem the Pareto frontier in a new production, namely, with the active co-developers of the systems and (or) the decision makers (DM) in the management of the Pareto frontier. It notes that such formulation differs from the traditionally accepted approach based on the analysis of already existing solutions. The article discusses three ways to describe a quality of the object management system. The first way is to use the direct quality criteria for the model of a closed system as the vibrational level of the General form. The second one is to study a specific two-loop system of an aircraft control using the angular velocity and normal acceleration loops. The third is the use of the integrated quality criteria. In all three cases, the selected criteria are mutually contradictory and it is possible to use them for description in the Pareto frontier terms. Techniques for the active influence on the Pareto frontier have allowed us to define parameters not only of the system regulators, but the control object itself, which permit changing the front position. The article analyses the impact of these parameters on the angles of the front, calculated, using the second derivative of the criterion, by the first dJ2/dJ1. It notes that derivatives may act as an assessment of the balance of compromises. The work reveals that for a General form model the change in natural frequency (time constant) has inversely proportional impact on the tilt of Pareto frontier, i.e. the smaller the time constant, the steeper is front, i.e. it is possible to control the front tilt through changing the time constant. Thus, reducing the time constant leads to the left-hand shift of the Pareto frontier. As to the two-loop system, it shows that the increasing gain module of the angular velocity sensor causes compression of the Pareto frontier in overshoot and stretching time of the transition process. Here, a tilt of the Pareto frontier slightly changes. The increasing module of the sensor linear acceleration gain causes the left-hand shift of the Pareto frontier with simultaneously increasing angle of the front tilt, i.e. its sensitivity increases. The computations of the corresponding Pareto ranks showed that for two-loop system, each of the individual ranks corresponds to the variation of the coefficient of linear acceleration sensor at different gain values of the angular velocity sensor, and if the latter is modulo limited from below, it is necessary to move to the next Pareto rank. Found that the change of dynamic coefficient related to the efficiency of the elevator control has the greatest effect. Using the integral criteria gives the same effect. The influence of other factors on the Pareto frontier is insignificant for direct quality indicators. It is, however, far more effective for the integral criteria. For the latter, as noted, there is an equilibrium point on the graph of the derivatives. Taking into consideration the rather complex functions of integral parameters, the article presents the approximating dependences obtained and notes the usefulness of applying this approach to more extensive scope of research without additional modeling. Thus, the article offers a new formulation as applied to the study of the Pareto frontier, namely, active control of the front by changing the parameters of the controllers and the object properties and shows a particular implementation of the synthesis of control systems for aircrafts.