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

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.