BULLETINBULLETIN OF THE OF THE POLISH POLISH ACADEMY ACADEMY OF OF SCIENCES SCIENCES TECHNICALTECHNICAL SCIENCES, SCIENCES, Vol. Vol. XX, 64, No. No. Y, 4, 20162016 DOI: 10.1515/bpasts-2016-00ZZDOI: 10.1515/bpasts-2016-0093 BULLETIN OFInversion THE POLISH ACADEMY of OF selected SCIENCES structures of block matrices of chosen TECHNICAL SCIENCES, Vol. XX, No. Y, 2016 BULLETIN OF THE POLISH ACADEMY OF SCIENCES DOI: 10.1515/bpasts-2016-00ZZInversion of selected structures of block matrices of chosen TECHNICAL SCIENCES, Vol. XX, No. Y, 2016 mechatronic systems DOI: 10.1515/bpasts-2016-00ZZ mechatronic systems 1 1 1 2 InversionTomasz of Trawiselectednski´ ∗, structures Adam Kochan of, Paweł block Kielan matricesand Dariusz of Kurzyk chosen InversionT. of TRAWIŃSKI selected1* structures, A. KOCHAN1 of, P. KIELAN block1 matrices, and D. KURZYK of chosen2 1 Department of Mechatronics,mechatronic Silesian University of Technology, systems Akademicka 10A, 44-100 Gliwice, Poland 12DepartmentInstitute ofof Mathematics,Mechatronics,mechatronic Silesian Silesian University University of ofTechnology, Technology, systems 10A KaszubskaAkademicka 23, St., 44-100 44-100 Gliwice,Gliwice, Poland Poland 2Institute of Mathematics,1 Silesian University 1of Technology, 23 Kaszubska1 St., 44-100 Gliwice, Poland2 Tomasz Trawinski´ ∗, Adam Kochan , Paweł Kielan and Dariusz Kurzyk Tomasz Trawinski´ 1 , Adam Kochan 1, Paweł Kielan1 and Dariusz Kurzyk2 Abstract. This1 Department articleTomasz describes of Trawi Mechatronics, hownski´ to calculate∗, Silesian Adam the University number Kochan of of algebraic Technology,, Paweł operations Kielan Akademicka necessaryand 10A, Dariusz to 44-100 implement KurzykGliwice, block Poland matrix inversion that occur, Abstract. This paper describes how to calculate the number of algebraic operations necessary to implement block matrix inversion that occurs, among others,1 Department in2 mathematicalInstitute of of Mechatronics, Mathematics, models of Silesianmodern Silesian University positioningUniversity of of systems Technology, Technology, of mass Kaszubska Akademicka storage 23, devices. 10A, 44-100 44-100 The Gliwice, Gliwice, inversion Poland Poland method of block matrices is among others,Department in mathematical of Mechatronics, models of Silesianmodern positioning University systems of Technology, of mass storage Akademicka devices. 10A, The 44-100inversion Gliwice, method Polandof block matrices is pre- 2 presentedsented as as well. well.Institute PresentedThe presented of formMathematics, form of generalof general Silesian formulas formulas Universitydescribing describing of Technology, the the calculation calculation Kaszubska complexity complexity 23, of 44-100 inverted of inverted Gliwice, formform of Poland block of blockmatrixmatrix were prepared were prepared for threefor differentthree different cases cases of their of division division into into internal internal blocks. blocks. The The obtained obtained results results are compared are compared with a with standard a standard Gaussian Gaussian method methodand the and“inv” the "inv" Abstract. This article describes how to calculate the number of algebraic operations necessary to implement block matrix inversion that occur, methodmethod used used in Matlab. in Matlab. The The proposed proposed method method for for matrixmatrix inversion is is much much more more effective effective in comparison in comparison in standard in standard Matlab Matlab matrix matrixinversion inversion among others, in mathematical models of modern positioning systems of mass storage devices. The inversion method of block matrices is Abstract."inv" function“inv”This function article (almost (almost describes two two times how times faster) to faster) calculate and and is the is much much number less less of numerically numerically algebraic operations complex complex than thannecessary standard standard to Gauss implement Gauss method. method. block matrix inversion that occur, presentedamong others, as well. in mathematical Presented form models of general of modern formulas positioning describing systems the calculation of mass storage complexity devices. of inverted The inversion form of method block matrix of block were matrices prepared is forpresentedKey three words:Key different as well.words:arrowhead cases Presented arrowhead of their matrices, form matrices, division of mechatronicgeneral intomechatronic internal formulas systems, systems, blocks. describing matrix Thematrix obtained the inversion, calculation results computational computational are complexity compared complexity. ofcomplexity with inverted a standard form Gaussianof block matrix method were and theprepared "inv" methodfor three used different in Matlab. cases of The their proposed division method into internal for matrix blocks. inversion The obtained is much results more are effective compared in comparison with a standard in standard Gaussian Matlab method matrix and inversion the "inv" "inv"method1. function Introduction used in (almost Matlab. two The times proposed faster) method and is much for matrix less numerically inversion is complex much more than effective standard in Gauss comparison method. in standard Matlab matrix inversion "inv" function1. 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Mathematicalform [1]: models of physical objects are formulated using, where Ls, Lr,DM=sr denotessrs matricesrsr of self inductances(2) of stator amongMathematical others, models the Lagrangian of physical formalism. objects are It can formulated be represented using, T = M.sr L.r .... byamong the following others, the form Lagrangian of differential formalism. equations It can writtenbe represented in ma- windings andD rotor= windings,M.sr L.r matrix.. of mutual stator(2) - rotor in- among others, the LagrangianDq¨ +C formalism.q˙ + Kq+ G It= canτ be represented (1)(1) . ductances respectively. The. inertia.. matrices in electromechan- trixby the form following [1]: form of differential equations written in ma- . trix form [1]: whereicalLs, systemsLr, Msr denotes may have matrices structural of self features inductances allowing of stator them to be trixwhere formwhere [1]:D - denotesD denotes inertial inertial matrix, CC denotes- denotes matrix matrix of centrifugal of cen- where Ls, Lr, Msr denote matrices of self inductances of stator windingswheredividedLs, andLr, intoM rotorsr denotes a windings, number matrices of matrix submatrices, of of self mutual inductances for stator the most - rotorof stator elementary in- trifugaland andCoriolis Coriolis forces,Dq¨ + forces,C Kq˙ denotes+ KqK+ stiffness-G denotes= τ matrix, stiffness G denotes matrix,(1) vectorG windingss r andsr rotor windings, matrix of mutual stator – rotor ductanceswindingsmatrices and respectively. rotor - the windings, blocks The have inertia matrix the matrices of size mutual of in 2 statorelectromechan-2 [2, - 5].rotor Symmetric in- - denotesof gravitational vectorD ofq¨ forces,+ gravitationalCq˙ + τ Kqdenotes+ G forces,=vectorτ ofτ generalized- denotes (1)forces, vector inductances respectively. × Dq¨ +Cq˙ + Kq+ G = τ (1) icalductances systemsinertia respectively. matrices may have encountered structural The inertia features matrices in mathematical allowing in electromechan- them models to be of head whereof generalizedDq -denotes denotes vector forces, inertial of generalizedq matrix,- denotesC -displacements. denotes vector of matrix generalized of cen- dis- The inertia matrices in electromechanical systems may have dividedical systemspositioning into a may number systems have of structural submatrices, of hard features disk for drivesthe allowing most [3, 4],elementary them have to be very dif- trifugalwhereplacements.D and- denotesThe Coriolis Theinertial inertial inertial forces, matrix, matrix, matrix,K present- denotesC present -in denotes (1), stiffness inin (1), matrixpredominant in matrix, predominant of cen- Gcases, structural features allowing them to be divided into a number may be regarded as symmetrical, but for mathematical modelsmatricesdivided offerent submatrices, into - the forms, a number blocks for depending havethe of submatrices,most the onelementary size structures of for2 2 thematrices [2, of most 5]. its – elementarykinematicSymmetric the blocks chains. -trifugal denotescases, and may vector Coriolis be ofregarded gravitational forces, asK symmetrical,- denotes forces, τ stiffness but- denotes for matrix, mathematical vectorG × of wide set of physical objects its elements are indirect functioninertiamatrices have matrices the - the size blocks of encountered 2 2