A determinant criterion for stability analysis and design of linear discrete systems

Linearni vremenski nepromjenljivi diskretni sustavi (Linear Time Invariant Discrete Systems) mogu se opisati jednadžbama linearne razlike konstantnog koeficijenta. Te se jednadžbe mogu lako pretvoriti u funkciju složene varijable metodom z transforma. Dvije trokutaste matrice stvorene su pomoću jednadžbe karakteristika koeficijenata sustava zajedno s minimalnim pomakom koeficijenata bilo lijevo ili desno i metode eliminacije koeficijenata. Konstruirana je jedna kvadratna matrica dodavanjem dviju trokutastih matrica. Za predloženu metodu konstruiranja kvadratne matrice potrebno je manje aritmetičkih operacija poput pomicanja i eliminiranja koeficijenata u usporedbi s konstrukcijom kvadratne matrice metodom matrice Jury i Hurwitza. Ta se Kvadratna matrica koristi za testiranje dovoljnog uvjeta postupkom određivanja unutarnje determinante Jurya. Dalje se predlaže još jedan potreban uvjet uz Juryeve uvjete stabilnosti. Dodane su i ilustracije za prikaz primjene predložene scheme. Razvijen je i algoritam za pronalaženje konstrukcijskog parametra k-vrijednosti koja pomaže u konstrukciji stabilnog Linearnog vremenski nepromjenljivog diskretnog sustava.Linear time invariant discrete systems can be described by constant coefficient linear difference equations. These equations can be easily transformed into the function of the complex variable by the z transform method. Two triangular matrices are formed with the help of the coefficients of system characteristics equation along with the minimal shifting of coefficients either left or right and elimination of coefficient method. A single square matrix is constructed by adding the two triangular matrices. The proposed method of construction of square matrix consumes less arithmetic operations like shifting and eliminating of coefficients, when compared to the construction of Square matrix by Jury’s and Hurwitz matrix method. This Square matrix is used for testing the sufficient condition utilising Jury’s Inner determinant procedure. Further one more necessary condition is also suggested along with Jury’s conditions for stability. Illustrations are also included to show the applicability of the proposed scheme. Also an algorithm was developed for finding the design parameter k-value which helps to design a stable Linear Time Invariant Discrete System