About One Approach to Determine the Weights of the State Space Method

• Main Author: I. K. Romanova
• Format: Article
• Language: Russian
• Published: MGTU im. N.È. Baumana 2015-01-01
• Series: Nauka i Obrazovanie
• Subjects: the method of state space; weights; multi-objective optimization; Pareto - optimal solutions; synthesis of control systems; aircraft; missile
• Online Access: http://technomag.edu.ru/jour/article/view/315

<p>The article studies methods of determining weight coefficients, also called coefficients of criteria importance in multiobjective optimization (MOO). It is assumed that these coefficients indicate a degree of individual criteria influence on the final selection (final or summary assessment): the more is coefficient, the greater is contribution of its corresponding criterion.</p><p>Today in the framework of modern information systems to support decision making for various purposes a number of methods for determining relative importance of criteria has been developed. Among those methods we can distinguish a utility method, method of weighted power average; weighted median; method of matching clustered rankings, method of paired comparison of importance, etc.</p><p>However, it should be noted that different techniques available for calculating weights does not eliminate the main problem of multicriteria optimization namely, the inconsistency of individual criteria. The basis for solving multicriteria problems is a fundamental principle of multi-criteria selection i.e. Edgeworth - Pareto principle.</p><p>Despite a large number of methods to determine the weights, the task remains relevant not only for reasons of evaluations subjectivity, but also because of the mathematical aspects. Today, recognized is the fact that, for example, such a popular method as linear convolution of private criteria, essentially, represents one of the heuristic approaches and, applying it, you can have got not the best final choice. Carlin lemma reflects the limits of the method application.</p><p>The aim of this work is to offer one of the methods to calculate the weights applied to the problem of dynamic system optimization, the quality of which is determined by the criterion of a special type, namely integral quadratic quality criterion. The main challenge relates to the method of state space, which in the literature also is called the method of analytical design of optimal controllers.</p><p>Despite the features of the problem, allowing us to obtain an analytical solution, it should be recognized that this criterion is, essentially, another form of a convolution of individual criteria. The problem to determine the weighting criteria in this quadratic convolution is still relevant and one of the main problems of the method.</p><p>The author traces the obvious connection between the interactive methods for finding Pareto-optimal solutions of the MOO problem and the classical method for analytical design of optimal controllers (ADOC), in which decision-maker, essentially, specifies the same conditions: assigns the weights of private optimality criteria; imposes restrictions on the values of private optimality criteria; evaluates proposed MOO, using the system of alternatives. An important feature of interactive methods is that during optimization process the decision-makers' preferences may change.</p><p>The article aims to link both approaches to the MOO problem. The expected advantage is the combination of analytical solutions based on the formulas of ADOC method and overcoming the criteria convolution shortcomings because of the difficult choice of weights. The novelty of the article is that the obvious idea of finding relationships between the weights of criteria by creating an indifference curve (Pareto frontier) has been already used for the special type of quadratic integral criteria rather than for non-linear convolution of criteria that have their drawbacks.</p><p>A modification of the quadratic criterion by breaking it into several components is made. Splitting into two criteria, which allowed us to obtain a graphical interpretation on a plane in the coordinates of the criterion describing the management costs and the criterion of the phase coordinates of the control object turned to be convenient. Since the method is often used for stabilization relative to the reference trajectory, representation of x and u as deviations of the phase coordinates and management costs was visual.</p><p>The article abandoned a traditionally taken provision that the total contribution of the maximum tolerated deviations of phase coordinates should be approximately equal to the total contribution of the maximum tolerated deviations of the control signals. The optimal solution was treated as a compromise and was in accordance with the Edgeworth - Pareto principle.</p><p>A weight ratio between the management costs and penalties for deviations of the phase coordinates was determined through constructing the Pareto front lines and analysis of the Pareto front. The optimal solution was determined according to theory of cooperative games. The source data to apply this approach to the optimization were specified as maximum tolerated values for each of the criteria as well as for the values of an ideal point. The obtained formulas and technique were applied to the synthesis of the aircraft movement control. An optimal solution was determined according to Cayley - Smorodinskii approach. Significantly reduced management costs were found through the specified optimal equilibrium solution.</p><p>It is noted that a similar approach can be also applied to the separation of private criteria into pairs within vector criterion related to restrictions on phase coordinates, as well as in the case of multidimensional control with individual control divisions. Thus, there is a possibility for renunciation of the principle of the equal weighted contribution of individual deviations, which will make the search tool for compromise solutions more flexible.</p>