Application of Multidimensional Linearization in Quasi-Optimal Controllers Synthesis in the Functional of Generalized Work

Abstract

The paper investigates the task of the analytically design of an optimal controllers (ADOC) as defined by A. A. Krasovskij for stable multidimensional objects, which are described by matrix differential equations with polynomial non-linearity from phase coordinates. The investigated class of polynomial control objects has a wide application: these models are used to describe the motion of systems with a very different nature — electromechanical equipment, chemical reactors, industrial recycling facilities, biological and ecological systems, etc. The most suitable task solution for the ADOC is the power series method, which in comparison with other methods, allows to finding control laws in widest range of the object’s phase space. However, its realization is dealt with a large amount of calculations and it is less formalized, so it comparatively hard for programming. In this paper the quasi-optimal controller’s synthesis method is suggested. It can reduce the disadvantages of the previously mentioned power series method. It uses the multidimensional linearization of polynomial objects procedure which implements extension of the object state space with new coordinates. These coordinates are the products of the original phase coordinates and the application of the matrix theory with the Kronecker product. The synthesis method can help to find an approximate ADOC task solution with a high degree of accuracy. The method is very easy to use, because it mainly based on uses of well-known software for the linear quadratic task solution in the optimal control. The ADOC task solution accuracy is defined by the accuracy of the corresponding degree (k = 2, 3...) that chosen for the object of the quasi-linearized model under study. It must be noted, that the k th power polynomial components of the control objects described, is considered in the k th power linearized model. Therefore, suggested synthesis method provides an accurate solution as a common power series method, holding its terms to the k th power inclusive. However, the devised synthesis method as a rule gives more accurate results, because it takes into account the functional matrix of the used quasilinear model of the object state augmented vector containing the original object’s phase coordinates products.