PRZEGLĄD ELEKTROTECHNICZNY, ISSN 0033-2097, R. 89 NR 2a/2013 195 Andriy LOZYNSKYY, Lyubomyr DEMKIV Lviv Polytechnic National University Investigation of multicriteria optimal control with time-variable weight coefficients Abstract. A new approach for construction of quality functional at multicriteria optimization is proposed. A two-mass dynamical system which consists of two subsystems is considered. A case when in the area of large errors unstable subsystem dominates is considered. Function L is chosen for switching between subsystems and its optimal parameters are found. Streszenie. W artykule zaproponowano nowe podejście do opracowania funkcji jakości w optymalizacji wielokryterialnej. Badania przeprowadzono na systemie dynamicznych dwóch mas, dla przypadku przeważającej niestabilności jednej z nich. Zoptymalizowano parametry funkcji L, wybranej do określania przejść między obiektami. (Badanie sterowania optimum wielokryterialnym o współczynnikach wagowych zmiennych w czasie). Keywords: optimal control, fuzzy logic, dynamic system. Slowa kluczowe: optymalna kontrola, logika rozmyta, system dynamiczny. Introduction Currently, for optimization of technical systems one traditionally uses approaches that are well known in the linear systems theory. In particular, there are analytical method for controllers designing [1], Pontryagin maximum principle, Bellman dynamic programming [2], [3] and node methods. The disadvantages of these approaches are that they do not take into account changes in the conditions of the system and object’s modification. Application of methods of nonlinear control theory, including feedback linearization [4] has not found widespread use because of difficulties in finding aggregated variables in technical systems. Also, today is not common to apply methods of geometric control theory [5]. An attempt of control influences adaptation to the object’s state and conditions of technological process flow by means of the switching systems forming and providing sliding modes along set trajectories leads to possibility of self-oscillations and to so called supercontrol. New possibilities of multicriterion optimal control problem solving gives an application of fuzzy set theory. In particular, Shin and Chang [6] proposed a global criterion approach on the basis of fuzzy logic to obtain solutions of multicriterion strict and fuzzy control synthesis. Loetamonphong [7] studied optimization problems which has multi-objective functions with fuzzy constraints of equality type. Huang [8] proposed a method of fuzzy multi- objective optimization decision-making method that can be applied to two and more objectives of system functioning. Among methods of optimal control synthesis one should also mention “piecewise Lyapunov function” method. It implies that for a set of subsystems a generalized Lyapunov function is formed from Lyapunov functions for each subsystem taken with some coefficient. Fuzzy control allows synthesizing control influences in the areas where system is functioning and provides the passes between these areas. So, one can speak of control influences synthesis for set of subsystems which form dynamic system. Similar approach is applied here. One of the possible ways of system optimization is application of Takagi-Sugeno fuzzy controller [9]. Output of this controller is a control influence which is typical for control systems with full state vector. So, for a particular rule, synthesis of control actions based on the classical theory of linear systems is possible. One uses an object model which is linearized in given area taking into account all imposed restrictions that will operate in this area. In particular, this technique is applied in the works of Mitsuishi T., Shidama Y. [10]. Using, for example, method of Bellman dynamic programming one can consider various restrictions that are necessary for normal functioning of the system such as, for example, limitations on performance that may be useful for systems with backlash, bins and conveyor systems etc. If one changes the workspace then another control signal, which is optimal for a given point of the region of state space system, is synthesized. Thus for its synthesis one can use model obtained by linearization of the nonlinear system at a given point. Problem statement In classical control theory generalized functional, at finding the optimal control for the entire system, has the following form i i i F F    , where i – index of individual subsystem model, i const   is defined basing on peer reviews or on the theory of Pareto optimal solutions. In particular, in Fig. 1 the output signals at different values of the coefficients 1 2 1     in the case when the controller of subsystem, which operates in the area of large errors, is set on a standard form of Butterworth         2 6 1 0 F xt x t xt      , and a controller that operates in the vicinity of small errors is set to binomial form            2 2 4 6 2 0 0 0 3 3 F xt x t xt xt xt              , is shown. Fig. 1. Output signal of the system in the case of configuration to 1) standard form of Butterworth; 2) 1 0.7   ; 3) 1 0.5   ; 4) 1 0.3   ; 5) binomial form.