inversion of selected structures of block matrices of chosen mechatronic systems

BULLETIN OF THE POLISH ACADEMY OFSCIENCES SCIENCES BULLETIN OF THE POLISH ACADEMY OF TECHNICAL SCIENCES, Vol 64, No 4, 2016 TECHNICAL SCIENCES, Vol XX, No Y, 2016 DOI: 10.1515/bpasts-2016-0093 DOI: 10.1515/bpasts-2016-00ZZ Inversion of selected structures of block matrices of chosen Inversion of selected structures ofsystems block matrices of chosen mechatronic mechatronic systems Inversion of selected structures block matrices of Kurzyk chosen 1∗ and Dariusz Tomasz Trawi´nski , Adam Kochan 1of , Paweł Kielan Inversion of selected structures of block matrices of chosen T TRAWIŃSKI1*, A KOCHAN1, P KIELAN1, and D KURZYK2 Department mechatronic systems of Mechatronics, Silesian University of Technology, Akademicka 10A, 44-100 Gliwice, Poland 2Department of Mechatronics, Silesian University of Technology, 10AKaszubska Akademicka23, St.,44-100 44-100 Gliwice, Gliwice, Poland mechatronic systems Institute of Mathematics, Silesian University of Technology, Poland Institute of Mathematics, St., 44-100 Gliwice, Poland 1∗ Silesian University 1of Technology, 23 Kaszubska BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol XX, No Y, 2016 BULLETIN BULLETIN OF OF THE THE POLISH POLISH ACADEMY ACADEMY OF OF SCIENCES SCIENCES DOI: 10.1515/bpasts-2016-00ZZ TECHNICAL SCIENCES, Vol Vol XX, XX, No No Y, Y, 2016 2016 TECHNICAL SCIENCES, DOI: DOI: 10.1515/bpasts-2016-00ZZ 10.1515/bpasts-2016-00ZZ Tomasz Trawi´nski , Adam Kochan , Paweł Kielan and Dariusz Kurzyk 1∗ 1∗, Adam Kochan 1 , Paweł Kielan1 and Dariusz Kurzyk2 Tomasz ThisDepartment article describes hownski to calculate theUniversity number ofofalgebraic operations necessary implement block matrix inversion that occur, ofTrawi´ Mechatronics, Silesian Technology, Akademicka 10A, to 44-100 Gliwice, Poland Abstract Abstract This paper describes how to calculate the number of algebraic operations necessary to implement block matrix inversion that occurs, 11 Department among others, in2 mathematical models of Silesian modern positioning of mass storage23, devices The inversion method of block matrices is Institute of of Mechatronics, Mathematics, University of Technology, Kaszubska 44-100 Gliwice, Poland Silesian University of Technology, Akademicka 10A, 44-100 Gliwice, Poland Department of Mechatronics, Universitysystems ofsystems Technology, Akademicka Gliwice, among others, in mathematical models of Silesian modern positioning of mass storage devices.10A, The 44-100 inversion method Poland of block matrices is pre22 Institute presented as well Presented form of general formulas describing the calculation complexity of inverted form of were prepared of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland of Mathematics, Silesian University of Technology, Kaszubska 23,of44-100 Gliwice, sented as well.Institute The presented form of general formulas describing the calculation complexity inverted form ofPoland blockblock matrixmatrix were prepared for three different cases of their division into internal blocks The obtained results are compared with a standard Gaussian method for three different cases of division into internal blocks The obtained results are compared with a standard Gaussian method and the and “inv”the "inv" Abstract This article describes how to calculate thefor number ofinversion algebraicisoperations necessary to in implement block matrix inversion that occur, method used in Matlab The proposed method matrix much more effective comparison in standard Matlab matrix inversion method used in Matlab The proposed method for matrix inversion is much more effective in comparison in standard Matlab matrix inversion among inarticle mathematical models modern positioning systems mass storage devices The inversion block matrices is Abstract This describes how to calculate number algebraic necessary to implement block matrix inversion that Abstract This article describes how to of calculate the number of algebraicofoperations operations necessary toGauss implement blockmethod matrix of inversion that occur, occur, "inv"others, function (almost two two times faster) andand isthe much less numerically complex than standard Gauss method “inv” function (almost times faster) is much lessof numerically complex than standard method presented as well Presented form of general formulas describing the calculation complexity of inverted form of method block matrix were prepared among in models of positioning systems of devices The of matrices is among others, others, in mathematical mathematical models of modern modern positioning systems of mass mass storage storage devices The inversion inversion method of block block matrices is for three different cases of their division into internal blocks The obtained results are compared with a standard Gaussian method and the "inv" Key words: arrowhead matrices, mechatronic systems, matrix inversion, computational complexity presented as well Presented form of general formulas describing the calculation complexity of inverted form of block matrix were prepared Key words: arrowhead matrices, mechatronic systems, matrix inversion, computational complexity presented as well Presented form of general formulas describing the calculation complexity of inverted form of block matrix were prepared method in Matlab The proposed for matrix inversion is muchresults more are effective in comparison in standard Matlab matrix for different cases their division into blocks The compared with Gaussian method and the for three threeused different cases of of their divisionmethod into internal internal blocks The obtained obtained results are compared with aa standard standard Gaussian method and inversion the "inv" "inv" Introduction "inv" function (almost two times faster) and is much less numerically complex than standard Gauss method method used in Matlab The proposed method for matrix inversion is much more effective in comparison in standard Matlab matrix inversion  inversion method used in Matlab The proposed method for matrix inversion is much more effective in comparisonin standard Matlab matrix matrices Introduction "inv" function (almost times and much numerically complex than Gauss method Ls example, Msr an electromechan"inv"words: function (almost two two times faster) faster) and is is systems, much less lessmatrix numerically complexelementary than standard standard Gauss(blocks), method for Key arrowhead matrices, mechatronic inversion,  matrix [5]  T motors) inertia ical systemcomplexity (squirrel cage induction Mathematical models of physical objects are formulated using,computational  M Lr  Key words: words: arrowhead models matrices, mechatronic systems, matrix inversion, inversion, computational complexity (2) Key arrowhead matrices, systems, matrix complexity Mathematical ofmechatronic physical objects are formulated using,computational may take the form of:D =  sr  among others, the Lagrangian formalism It can be represented Introduction   others,form the Lagrangian formalism It canwritten be represented by theamong following of differential equations in maLs M.sr   Introduction by the following form of differential equations written in matrix    LsT Msr  trix form [1]: Mathematical models of physical objects are formulated using, Lsr M  Lrsr of M form [1]: s denotes matrices inductances(2)of stator where Ls , Lr ,DM= (2) sr  self  Mathematical of physical objects are formulated using,  among others, models the Lagrangian formalism It can be represented   T Mathematical models of physical objects are formulated using, T  L rr matrix M sr   = L  M windings andD rotor windings, of mutual stator (2) - rotor in(2) D = sr   among others, the Lagrangian formalism It can be represented by the following form of differential equations written in ma   among others, the Lagrangian be represented (1) q + Kq + GIt=can (1) Dqă +Cformalism matrices in electromechanductances respectively The inertia by following form trix form [1]: by the the following form of of differential differential equations equations written written in in mama , L , M denotes matrices of self inductances of stator where L ical systems may have structural features allowing them to be s r sr trixwhere formwhere [1]: trix form [1]: D - denotes - denotes matrix of cen-where D denotesinertial inertial matrix, matrix, CCdenotes matrix of centrifugal matricesofofself selfinductances inductances of of stator stator where Ls,, LMr, Mdenotes sr denotematrices L Lss ,, and windings rotor windings, matrix of mutual stator rotor inrr , into sr divided a number of submatrices, for the most elementary L M denotes matrices of self inductances of stator where L andand Coriolis forces, Kq˙denotes G denotes vectorG windings andsr rotor windings, matrix of mutual stator – rotor trifugal Coriolis forces, -Gdenotes stiffness matrix, Dqă +C + KqK+stiffness = τ matrix, (1) windings and rotor windings, matrix of mutual stator rotor inductances respectively The inertia matrices in electromechanwindings and rotor matrix mutual stator rotorSymmetric inmatrices -respectively thewindings, blocks have the of size of 2×2 [2,- 5] of gravitational denotes vector forces, inductances - denotes vectorD gravitational - denotes(1) vector q +C q + = ττ ofτ generalized Dof qăă forces, +C q + + Kq Kq +G G forces, = (1) ductances respectively The inertia matrices in electromechanical systems may have structural features allowing them to beof head ductances respectively The inertia matrices in electromechaninertia matrices encountered in mathematical models of generalized The inertia matrices in electromechanical systems may have where Dq -denotes denotesvector inertial C -displacements denotes matrix of cen- disof generalized forces, qmatrix, - denotes vector of generalized ical structural systems may haveallowing structural features allowing them to be be into amay number ofstructural submatrices, fordrives the most elementary ical systems have features allowing them to positioning systems of hard disk [3,into 4], have very difThe inertial matrix, (1), predominant cases,divided features them to be divided a number where D denotes inertial matrix, C indenotes matrix of trifugal Coriolis forces, Kpresent - denotes stiffness G where Dand denotes inertial matrix, Cpresent denotes matrix of cencenplacements The inertial matrix, inin(1), inmatrix, predominant divided into a number of submatrices, for the most elementary matrices the blocks have the size of 2×2 [2, 5] Symmetric divided into a number of submatrices, for the most elementary may be regarded as symmetrical, but τstiffness for- mathematical models offerent submatrices, the most on elementary matrices the blocks chains forms, for depending structures of its –kinematic Coriolis forces, denotes G -trifugal denotes vector gravitational forces, vector trifugal and Coriolis forces, Ksymmetrical, denotes stiffness matrix, G matrices cases,and may be of regarded asK butdenotes formatrix, mathematical the blocks have the of 2×2 [2, 5] Symmetric inertia matrices encountered insize mathematical models of head matrices the blocks have the size of 2×2 [2, 5] Symmetric of wide set of physical objects its elements are indirect function have the size of 2£2 [2, 5] Symmetric inertia matrices encounExemplary forms of these matrices are as follows: -of denotes vector of gravitational forces, τ denotes vector generalized forces, q physical - denotesobjects vector its generalized dis- denotes vector ofsetgravitational forces, τof -elements denotes are vector models of wide of indirect inertia matrices encountered in mathematical models of positioning systems of hard disk drives [3, 4], have very difof time The elements of inertial matrix ondisangularinertia tered in mathematical models of head positioning systems of matrices encountered in mathematical models of head head of generalized generalized forces, q matrix, denotes vector ofmay generalized placements The inertial present in (1), in depend predominant of forces, q elements denotes vector of generalized dis  function of time The of inertial matrix may depend positioning systems of hard disk drives [3, 4], have very different forms, depending on structures of its kinematic chains or linear displacements It happens very often in mathematical hard disk drives [3, 4], have very different forms, depending positioning systems of hard disk drives [3, 4], have very difplacements The inertial as matrix, present in in (1), inmathematical predominant d11 0 cases, may be regarded symmetrical, but forin placements The inertial matrix, present (1), predominant on angular or linear displacements It happens very often in ferent forms, depending onmatrices structures of its kinematic chains modelling of advanced mechanical systems such as robot maon forms, structures ofof itsthese kinematic chains forms of these Exemplary forms areExemplary as its follows: ferent depending on structures of kinematic chains  cases, may be regarded as symmetrical, but for mathematical models of wide set of physical objects its elements are indirect cases, may be regarded symmetrical, but for mathematical d22 are d23as follows:    mathematical modelling ofInadvanced mechanical systems suchExemplary nipulators [1, 23, as 24] mathematical models of electro-mematrices are as follows: forms of these matrices Exemplary forms  matrices are as follows:  models of of objects its are indirect  of these function ofwide time.set The elements of inertial may depend models of wide set of physical physical its elements elements aremutual indirect chanical systems matrix parameters ofmatrix and induc 0  d.33 as robot manipulators [1], objects [23], [24] Inselfmathematical modd 0 D = (3) 11    function of or time Thedisplacements elements of inertial inertial matrix may depend on angular linear It happens very in dis d  function of time The elements of matrix may depend tances of stator and rotorsystems windings also depend onoften angular   d00 d00 00 00   els of electro-mechanical matrix parameters of self11 d 11 on angular angular or modelling linear displacements It happens happens very often in  mathematical advanced mechanical systems such  22 23  d on or linear It very often in placements of displacements the of rotor Similarly, in mathematical models  k−1,k and mutual inductances of[2] stator and rotor windings also ded d 22 23 d mathematical modelling of advanced mechanical systems such   d as robot manipulators [1], [23], [24] In mathematical mod22 23 mathematical of systems advancedofmechanical systems such D= (3) of headmodelling positioning mass devices sym 33   dk,k pend on angular displacements of modern the rotor [2].storage Similarly, in 00  dd33 as manipulators [1], [23], In els of electro-mechanical systems parameters of modself 33 (hard disk drives), ofmatrix inertial matrices (which are rep- (3) D= (3) = as robot robot manipulators [1],elements [23], [24] [24] In mathematical mathematical modD (3)    dk−1,k    mathematical modelsofofsystems head of positioning systemsalso ofselfmodern     els electro-mechanical matrix of and mutual inductances stator and rotorparameters deresented by mass moments inertia orwindings by masses) on   els of of electro-mechanical systems matrix parameters ofdepend self sym d d11 d12 d 13 dddk−1,k  1,k storage devices (hard disk elements of inertial k−1,k k,k andmass mutual inductances of statorof and rotor windings also despatial of configurations of thewindings links of its kinematic pend ontemporary angular displacements thedrives), rotor [2] Similarly, in and mutual inductances stator and rotor also de matrices (which are represented by mass moments of inertia or d  sym d 22 k,k and therefore the angular (orrotor linear) displacement ofin  dk,k pend angular displacements of [2] Similarly, mathematical models of head positioning systems of modern pend on onchain angular displacements of the the rotor [2] Similarly, injoints  sym    by masses) depend on temporary spatial configurations of the [3, 4] A thorough analysis of the structures of the inertia mad d d d   11 12 13 mathematical models of head positioning systems of modern 1,k d mass storage devices (hard disk drives), elements of inertial 33 mathematical models of head positioning systems of modern (4) D=        trix of mechanical systems, electromechanical systems, robot d d d d links of its kinematic chain and therefore the angular (or linear) 11 12 13 1,k mass storage devices (hard disk drives), elements of inertial d d d d   matrices (which are represented mass moments or 013 01,k  12 mass storage devices (hard diskbydrives), elementsofofinertia inertial  11 22  manipulators, head positioning systems ofanalysis hardofdisk and  displacement of joints [3, 4].by Aspatial thorough ofdrives, matrices (which are represented by mass moments inertia orstruc033 00  by masses) depend on temporary configurations ofthe the matrices (which are represented mass moments of inertia or dd22   22 d0 (4) D = many others, shows thatofthey have configurations oftensystems, block structure  of kinematic the inertia matrix mechanical electromesym d33  dk,k bytures masses) depend on temporary spatial of the the The links of its chain and therefore the angular (or linear)   by masses) depend on temporary spatial configurations of   (4) D = d 33 (4) D= inertia matricesrobot of these systems can behead divided internally into (4)      chanical systems, manipulators, positioning links of and angular displacement of jointschain [3, 4] Atherefore thoroughthe analysis of (or thelinear) struc- sys  links of its its kinematic kinematic chain and therefore the angular (or linear)   A square matrices with entries equal zero except for their    sym d00k,k  tems drives, many others, shows that they have displacement of joints [3, A thorough analysis of the tures of of thehard inertia matrix ofand mechanical systems, displacement ofdisk joints [3, 4] 4] A thorough analysis ofelectromethe strucstrucmain diagonal, one row and one column have many applicatures of *block the inertia matrixThe of inertia mechanical systems, electromeoften structure matrices of these systems sym chanical systems, robot manipulators, head positioning sys- can tures of the inertia matrix of mechanical systems, electromesym ddk,k k,k e-mail: tomasz.trawinski@polsl.pl in wireless communication systemsfor[19], squaree.g matrices with entries equal zero except theirneuralchanical systems, robotinto manipulators, head positioning sysbe of divided internally elementary matrices (blocks), for ex- A tions tems hard disk drives, and many others,head shows that they have chanical systems, robot manipulators, positioning sysA square matrices with entries equal zero except for their main diagonal, one row and one column have many applicanetwork models [20] as well as issues related with chemistry A square matrices with entries equal zero except for their tems of hard disk drives, many others, shows they have often block Theand inertia of thesethat systems can ample, anstructure electromechanical system cage tems of hard disk drives, and manymatrices others, (squirrel shows that they induction have main diagonal, one row and one column have many applications e.g in wireless communication systems [19], neural[21] or phisics [22] In this paper we propose a method of often block structure The inertia these systems be divided into elementary matrices (blocks), for can ex- main diagonal, one row and one column have many applicamotors) inertia matrix [5] may take theof of: Bull.internally Pol Ac.: Tech 2016matrices 853 often block structure The64(4) inertia matrices ofform these systems can tions e.g in wireless communication systems [19], neuralnetwork models [20] as well as issues related with chemistry of symetric matrices containing non-zero blocks in e.g in wireless communication systems [19], neuralbe internally for ample, an electromechanical system matrices (squirrel(blocks), cage induction be divided divided internally into into elementary elementary matrices (blocks), for exex- tionsinversion Unauthenticated network models [20] as well as issues related with chemistry [21] or phisics [22] In this paper we propose a method ofremaintheir main diagonal, one column, one row and zeros in network models [20] as well as issues related with chemistry ample, an an electromechanical system (squirrel cage induction induction Download Date | 1/27/17 4:49 PM motors) inertia matrix [5] maysystem take the(squirrel form of:cage ample, electromechanical ∗ e-mail: tomasz.trawinski@polsl.pl [21] or phisics [22] In this paper we propose a method of inversion of symetric matrices containing non-zero blocks in [21] or phisics [22] In this paper we propose a method of ing entries This kind of matrices is significant in modeling motors) inertia inertia matrix matrix [5] [5] may may take take the the form form of: of: motors) inversion of symetric matrices containing non-zero blocks their main diagonal, one column, one row and zeros in remaininversion of symetric matrices containing non-zero blocks in in T Trawiński, A Kochan, P Kielan, and D Kurzyk Square matrices with entries equal zero except for their of proposed method which exhibits the smallest increase of main diagonal, one row and one column have many applica- number of algebraic operation due to block matrix dimension tions e.g in wireless communication systems [19], neural-net- increase work models [20] as well Tomasz as issues related chemistry Trawi´ nski, Adam Kochan, Tomasz Trawi´ nski,with Adam Kochan,Paweł PawełKielan Kielanand andDariusz DariuszKurzyk Kurzyk Tomasz Trawi´nski, Adam Kochan, Paweł Kielan and Dariusz Kurzyk [21] or phisics [22] In this paper we propose a method of inversionsystems of symetric matricesstructures containing non-zero blocks 2.2.2 mechatronic Considered block matriInverting the block matrix using its internal mechatronic systems Considered structures block matriInverting theblock blockmatrix matrix usingits itsinternal internal Invertingthe the block matrix using ofofof mechatronic systems Considered structures ofofof block matri2 Inverting using inbebe their main diagonal, one column, one row and zeros in ces can used to describe an inertia matrix in mathematical ces can used to describe an inertia matrix in mathematical its internal structure structure structure ces can be used to describe an inertia matrix in mathematical structure remaining entries Thissystem kind ofofof matrices is significant in model of a head positioning a hard disk drive It model of a head positioning system a hard disk drive It model of a head of positioning system of aConsidered hard disk drive It Suppose that the inertia matrix object can bebe repSuppose that the inertia matrix of aphysical physical object can repmodeling mechatronic systems structures Suppose that the inertia matrix of physical object can be repseems interesting how fast may agroup ofof symmetric inertia seems interesting how fast may agroup group symmetric inertiaofSuppose that the inertia matrix ofof aaaphysical object can be repseems interesting how fast may a of symmetric inertia block matrices can be used to describe an inertia matrix in resented in the following block form: resented in the following block form: resented in the following block form: matrices, encountered models hard disk drive head pomatrices, encountered models hard disk drive head po- resented in the following block form: matrices, asasas encountered ininin models ofofof hard disk drive head mathematical model of a head positioning system ofpoa hard sitioning system, be inverted considering their block structure sitioning system, be inverted considering their block structure  a0a0 b1b1 b2b2 bkbk sitioningdisk system, consideringhow theirfast block drive.beItinverted seems interesting maystructure a group of a b b b and their reasonable block dimension also worth inand their reasonable block dimension.ItItIt also worth inb0TbT a1a 200 k00 and their reasonable block dimension isisis also tototo insymmetric inertia matrices, as encountered inworth models of hard  T 11    b vestigate influence an internal structure of block matrices has vestigate influence an internal structure of block matrices has  11T 1T a1   vestigatedisk influence an internal structure of be block matrices has drive head positioning system, inverted considering       a b a b T 2 (5) (5) onon inversion time asas well onon numerical complexity ininversion time well as numerical complexity inb22 a2   their block structure and their reasonable block dimension (5) (5) DDD === on inversion time as well asas on numerical complexity ofofof in      version process The above mentioned issues motivated the version process The above mentioned issues motivated the   is also worth investigate influence internal structure version Itprocess The to above mentioned issuesanmotivated the  000 Authors investigate the problem numerical complexity Authors to investigate the problem numerical complexity oftoto block matrices has on inversion time as complexity well as on numerT T Authors investigate the problem ofofof numerical ofofof inversion block matrices with different structures Similar inversion of block matrices with different structures Similar bbTkkbk 000 000 000 aakkak ical of inversion process.structures The aboveSimilar mentioned inversion ofofcomplexity block matrices with different problem was investigated [5], where effective method (based problem was investigated [5], where effective method (basedof The issues motivated the Authors toeffective investigate the problem division into elementary matrices the above matrix The division into elementary matrices of the above matrix problem was investigated ininin [5], where method (based The division into elementary matrices ofof the above matrix isisis T Tdecomposition) numerical complexity of inversion of block matrices with The division into elementary matrices of the above matrix is on LDL of finding the inverse of this kind of on LDL decomposition) of finding the inverse of this kind of T essentially arbitrary, but there may occur practical reasons [6], essentially arbitrary, but there may occur practical reasons [6], on LDL decomposition) of finding the inverse of this kind of essentially arbitrary, but there may occur practical reasons [6], different structures Similar problem was investigated in [5], which essentially arbitrary, but there may occur practical reasons [6] matrix was proposed However, approximation ofofcomplexcomplexmatrixwas wasproposed proposed However, approximation complexdefine them It happens in branched head positioning which define them It happens in branched head positioning matrix However, approximation of T define them where effective method (based onWe LDL decomposition) which define them.It Ithappens happensininbranched branched head head positioning positioning ity the algorithm was very general propose adetailed ity the algorithm was very general We propose adetailed detailedofwhich systems of hard disk drives [4] and inin this case, division systems of hard disk drives [4] and this case,this this division ity ofofof the algorithm was very general We propose a systems of hard disk drives [4] and in this case, division finding the inverse of this kind of matrix was proposed systems of hard disk drives [4] and in this case, this this division analysis of the number of algebraic operations necessary to imanalysis of the number of algebraic operations necessary to imis correlated with the structure of the kinematic chain of head is correlated with the structure of the kinematic chain of head analysisHowever, of the number of algebraic operations necessary to im-wasis correlated with thethe structure head approximation ofblock complexity of theInIn algorithm is correlated with structureofofthe thekinematic kinematic chain chain of head plement inversion of considered matrices chapter plement inversion of considered block matrices chapter positioning system Due the ability toto give the physical inpositioning system Due the ability give the physical inplementvery inversion of We considered block matrices In of chapter positioning general propose a detailed analysis thematrinumber positioning system.Due Due the ability give the physical insystem totototo the ability toto give the physical ingeneral information about the internal structure ofof block general information about the internal structure block matriterpretation of the matrices of the structure of an arrow (arterpretation of the matrices of the structure of an arrow (argeneral information about the internal structure of block matriof algebraic operations necessary to implement inversion ofterpretation terpretation thematrices matrices of the an an arrow (arrowof ofthe thestructure structureofof arrow (arces, further considered inin the article presented.In this chapces, further considered the article presented.In this chap- rowhead), they became the subject ofofresearch research Arrowhead rowhead), they became thesubject subject research Arrowhead ces, further considered the article isisis presented.In this chapconsidered blockinmatrices head), they became thethe subject of research Arrowhead matrix rowhead), they became of Arrowhead ter the inverted form ofof block matrix, derived former works, ter the inverted form block matrix, derived former works, matrix representation the inertia matrix describing the equamatrix representation the inertia matrix describing the equater the inverted form of block matrix, derived ininin former works, In chapter general information about the internal struc-matrix representation of the inertia matrix describing the the equation representation ofofof the inertia matrix describing equais presented In former works, the authors have not investiis presented In former works, the authors have not investition of physical object under consideration, may also used tion of physical object under consideration, may also used ture ofIn block matrices, considered article istionofofphysical under consideration, maymay alsoalso usedused in the is presented former works,further the authors have in notthe investiphysicalobject object under consideration, ininin gated mutual interactions between internal structure of block gated mutual interactions between internal structure of block the description ofoperation operation of the wireless links [9] One the description of operation of the wireless links [9].One presented.In this chapter the internal invertedstructure form of block matrix,the description ofof ofof the wireless links [9].[9] One ofOne the gated mutual interactions between of block description operation the wireless links ofofof matrices (consisting more than 16 elements) and itsits numerical matrices (consisting more than elements) and numerical derived in former works, is16 presented In former works, the the problems associated withwith arrow matrices is effective determiproblems associated arrow matrices is effective dethe problems associated with arrow matrices is effective matrices (consisting more than 16 elements) and its numerical the problems associated with arrow matrices is effective de-decomplexity and resultant computation times In paper [18] the complexity and resultant computation times In paper [18] the authors have not investigated mutual interactions between nation of the eigenvalues of [7–9] Additionally, considered are termination the eigenvalues [7], [8], [9] Additionally, termination the eigenvalues [7],[8], [8],[9] [9] Additionally, complexity and resultant computation times In paper [18] the termination ofofof the eigenvalues ofofof [7], Additionally, number of algebraic operation has been calculated for an innumber of algebraic operation has been calculated for an ininternal structure of block matrices (consisting more than parallel matrix inversion methods as described in [10] In [11] considered are parallel matrix inversion methods as described considered are parallel matrix inversion methods as described number of algebraic operation has been calculated for an in- considered are parallel matrix inversion methods as described 16 elements) and itsdefined numerical complexity and resultant a quick method ofpresented solving asystems of linearofof equations of the version process strictly matrix (only one type), but version process strictly defined matrix (only one type), but inin[10] [11] isis presented method solving systems [10] In [11] aquick quick method solving systems version process ofofof strictly defined matrix (only one type), but in [10] InIn [11] isof presented a quick method of solving systems computation times.ofof Ininput paperblock [18] matrix, the number of algebraic arrow matrix coefficients is presented under different partition i.e into 4, under different partition input block matrix, i.e into 4, linear equations the arrow matrix coefficients linear equationsofof the arrow matrix coefficients under different partition of input block matrix, i.e into 4, ofofof arrow matrix ofofof coefficients operation hasmatrices been calculated for an showing inversion process Asequations shown in of [6]the inverted block matrix (5) can be repreand 1616 elementary Other papers the relaand elementary matrices Other papers showing the rela-of linear As shown in [6] inverted block matrix (5) can As shown in [6] inverted block matrix (5) canbeberepresented represented and 16 elementary matrices Other papers showing the relashown strictlyblock defined matrix (only one type), but of under different As sented as: in [6] inverted block matrix (5) can be represented tion between matrix structure and structure kinematic tion between block matrix structure and structure of kinematic as: as: tion between block matrixblock structure andi.e structure kinematic partition of(alternatively input matrix, into 4, 6of 16 posielemenchains ofof robots kinematic chains ofand head chains robots (alternatively kinematic chains of head posi- as:  chains of robots (alternatively kinematic chains of head posi−1 −1 −1 −1 tary matrices Other papers showing the relation between  c0c0 −c a1−1 −c a2−1 ξξ1ξ1 −c −c 0bb 0bb 1b 2b tioning systems) [3, 4], [6] and [18] oror structure ofof winding ofof tioning systems) [3, 4], [6] and [18] structure winding 1a1 2a2 −c a −c a c 0 tioning systems) [3, 4], [6] and [18] or structure of winding of block matrix structure and structure of kinematic chains of −1 −1 T T −1   electric machines [2] and [5] General formulas, describing electric machines [2] and [5].General General formulas, describing cb0 2baa2−1 a−1 ξ2ξ2    1a1bbT1bc1c00b electric machines [2] and [5] formulas, describing aa a cc11c1 aa−1 robots (alternatively kinematic chains of head positioning 22 ξ2   1   relation between block matrix’s structure and dimension, and relation between block matrix’sor structure and dimension,electric and   (6) relation between block and systems) [3, 4, 6]matrix’s and [18]structure structure ofdimension, winding of and r= r==  cc22c2 ξξ33ξ3(6) DDrD (6)(6) the number of algebraic operations have not been presented in the number of algebraic operations have not been presented in      machines [2] and operations [5] Generalhave formulas, describing a relation the number of algebraic not been presented in   authors’s former papers chapter this paper methods authors’s former papers chapter 3ofof this paper methods   between block matrix's structure and dimension, and the authors’s former papers InInIn chapter 33of this paper methods of accounting the number of algebraic operations, necessary of accounting the number of algebraic operations, necessary sym sym numberthe of algebraic not beennecessary presented in of accounting number ofoperations algebraichave operations, sym cckkck to make during inversion process are described Three difto make during inversion process are described Three difformer papers In chapter of this paper to makeauthors's during inversion process are described Threemethods dif−1 −1 T T−1 −1 k k −1 −1 T(c(c where (a(a (a(a bTTib(c ++ where ferent cases of block matrices internal portioning are considferent cases of block matrices internal portioning are considj a−1 i− 0−−∑k∑ ja i ==(a i−− of accounting the number of algebraic operations, neces-where wherecc00c0===(a jbb jj b j))−1 j ) , ,cci, ic= i −1 j jbbTb 00+ a b ferent cases of block matrices internal portioning are considi k− ∑ j j i j –1 T –1 T –1 –1 T –1 −1 –1−1 −1 −1 T −1 −1 T −1 −1 sary to make during inversion process areofof Three biba−1 c  = (a  ¡  bfor ci{1,  = (a  ¡ b aj −c b−c )kak , , ered Inversion times for allall chosen structures block matriered.Inversion Inversion times for chosen structures block matri{1, for k}, , k}, ja i= j bji)i ∈,∈ i (c0 ξ + b j )0 b i ) j for 0b kbia k−1 j ) bbi0)ib)−1 ered times for all chosen structures of described block matrib a i aj jbbTjb0j))−1 i ∈ {1, .i.,.,k}, ξ11ξ1== , –1 –1 −c T bk ak–1 different cases of block matrices internal portioning are ξi ξj==afor 1, …, kg, ξ  = ¡ , ξ  = a b c b a , −1 −1 −1 −1i 2 f −1 −1 TcT0 bk a k−1 −1 ces are compared with times of inversion using standard inverT ces are compared with times of inversion using standard inverT k 1 k b c b a , ξ = a b c b a a b c b a , ξ = a b c b a −1 −1 −1 ces are compared with times of inversion using standard inver- ξ 2= T T a−1 k k k k 1b 1c10–1bbkTac kbk, aξ–1 22 22 kk considered Inversion times for The all chosen structures 2 k0 k k3 = a2 b2 c0 bk ak 1ξ3 = a sion method inin Matlab (“inv” function) presented method sion method Matlab (“inv” function) The presented methodof The inverted block matrix of inertia (6) consist (as can The inverted block matrix of inertia (6) consist (as can sion method in Matlab (“inv” function) The presented method The inverted block matrix of inertia (6) consist ofofof (as can block matrices are compared with effective times of than inversion using of block matrix inversion is much more the one of block matrix inversion is much more effective than the one bebe observed) elementary matrices, which are calculated onon the observed) elementary matrices, which are calculated the of blockstandard matrix inversion ismethod much more effective than the oneThebe in Matlab (“inv” function) The inverted block matrices, matrix of which inertia (6) consist of (as can observed) elementary are calculated on the used Matlab Also, inin this article the number of algebraic used Matlab.inversion Also, this article the number of algebraic basis of the block matrix elements (5) before the inversion It basis of the block matrix elements (5) before the inversion It used ininin Matlab Also, in this article the number of algebraic presented method of block matrix inversion is much morebasis beof observed) matrices, arethe calculated on Itthe the blockelementary matrix elements (5)which before inversion isisis operations (necessary toto invert the matrices) has been calcuoperations (necessary invert the matrices) has been calcupossible to calculate chosen elementary submatrix (6) without possible to calculate chosen elementary submatrix (6) without operations (necessary to invert the matrices) has been calcueffective than the one used in Matlab Also, in this articlepossible basis of block matrix (5) before the inversion It is to the calculate chosenelements elementary submatrix (6) without lated and compared with Gaussian method ofof matrix inversion, lated and compared with Gaussian method matrix inversion, the need of calculation of the remaining elements of the blocks the need of calculation of the remaining elements of the blocks lated and compared with Gaussian method of matrix inversion, the number of algebraic operations (necessary to invert thethe possible to calculate chosen elementary submatrix (6) without need of calculation of the remaining elements of the blocks and has shown the advantages proposed method which andititithas hasshown shownthe theadvantages advantagesofofofproposed proposedmethod methodwhich which matrix matrix and matrices) has been calculated and compared with Gaussianmatrix the need of calculation of the remaining elements of the blocks exhibits the smallest increase number ofof algebraic operation exhibits the smallest increase number algebraic operation method of matrix inversion, and itof has shown the advantages matrix exhibits the smallest increase ofofof number algebraic operation Σ due block matrix dimension increase due block matrix dimension increase due tototo block matrix dimension increase 854 The number ofofalgebraic algebraic operations during Thenumber numberof algebraicoperations operationsduring during 3.3.3.The the inversion ofofthe the block matrix theinversion inversionof theblock block matrix Bull Pol Ac.: Tech 64(4) 2016 the matrix Unauthenticated For the sake consequence further course the discussion, For the sake consequence further course of the discussion, For the sake ofofof consequence ofofof further ofof the discussion, Download Datecourse | 1/27/17 4:49 PM definition of block dimension is formulated definition of block dimension is formulated definition of block dimension is formulated tion as self inertia matrices, have following forms: −1 −1 T −1 ci = (ai − bTi (c−1 + bi a j b j ) bi ) (11) Inversion of selected structures of block matrices of chosen mechatronic systems for i ∈ {1, , k} By introducing the above indications of elementary matri- By introducing the above indications of elementary The number of algebraic operations during ces, inverted block matrix (6) takes the following form: matrices, inverted block matrix (6) takes the following form: the inversion of the block matrix  matrix creasi of an and fi Gen matrix for th follow  dk For the sake of consequence of further course of the discussion,  e1 dk  definition of block dimension is formulated      e d k (12) (12) for n = D = Inversion of selected structures r  Inversion of selected structures  Definition If the symmetric block matrixInversion D Inversion has been ofdivided selected structures  of selected structures  The   elementary matrices using as k vertical lines pair (into of k+1 colTable 11 n of of into the block block matrix is defined defined an ordered ordered numTable Inversion of selected structures n the matrix is as an pair of nummine Table umns) and k horizontal lines (into k+1 rows) the block size Number n of the(k+1, block matrix is defined ordered pairpair numNumber of of algebraic algebraicsym operations necessary necessary to calculate calculate the leading leading element element cc0 Table to ck the nbers of the block matrix issize defined asblock an ordered of numoperations k+1) Block ofas thean matrix isofwritten written as:Number bers (k+1, k+1) Block size of the block matrix is as: of algebraic operations necessary to calculate the leading element c0 c00 the ca Inversion of selected structures Number of algebraic operations necessary to calculate the leading element n of the block matrix is defined as an ordered pair of numbers (k+1, k+1) Block size of the block matrix is written as: bers (k+1, k+1) Block size of the block matrix is written as: Block dimension n Table n=(k+1)×(k+1), or briefly by n=(k+1) n of bers the block matrix is defined ordered pair of numBlock dimension n operations required negati in this article the number ofof algebraic n=(k+1)×(k+1), or briefly by n=(k+1) (k+1, Block size ofas theanblock matrix is written as:Later Later inofthis article the number Block dimension n algebraic required n=(k+1)×(k+1), ork+1) briefly by by n=(k+1) Block dimension n2 243 operations 374 the10 4leading element c0 Number algebraic operations necessary n=(k+1)×(k+1), or briefly n=(k+1) 13 Sum of of algebraic algebraic operations loI oIto calculate 10 13 Sum operations l bers (k+1, k+1) Block size of the block matrix is written as: Table to implement in order to calculate the inverted block matrix type o n of the block matrix is asn = (k+1) an of ordered pair of numn = (k+1)£(k+1), or defined briefly by to implement in algebraic order tooperations calculate inverted block SumSum of algebraic operations loI loI4the 710 10 13 13 In order order to demonstrate demonstrate effectiveness the computing computing algo- Number of In to effectiveness of the algoBlock dimension n of algebraic operations necessary to calculate the leading element c n=(k+1)×(k+1), or briefly by n=(k+1) In order to demonstrate effectiveness of the computing algofor for three different cases ofofthe culati three different cases theinternal internalstructure structure of of input bers (k+1, k+1) Block size the block is written as: (5) (5) In order to demonstrate effectiveness of matrix the computing algorithm of matrix matrix inversion, theofnumber number of algebraic operations rithm of inversion, the of algebraic operations Sum of calculated algebraic Blockoperations dimension lnoI 24 37 410 513 rithm of matrix inversion, theby number of algebraic In order to demonstrate effectiveness of theoperations computing al-matrices matrices will be will be calculated matric n=(k+1)×(k+1), or briefly n=(k+1) rithm of matrix inversion, the number of algebraic operations needed to be be performed will will be calculated calculated forcomputing block matrices matrices In order to demonstrate effectiveness of the algoTable 22 needed to performed be for block Table gorithm ofperformed matrix inversion, the numberfor offor algebraic operations 10 13 Sum of algebraic operations needed to be performed willwill be be calculated block matrices Table loI2 to calculate needed to be calculated block matrices Number of algebraic operations necessary the negative negative feedback feedback Table with different internal divisions into elementary blocks subrithm of matrix the number of operations Number of algebraic operations necessary tomatrices calculate the with internal divisions into elementary blocks -algosub- Number In different order to effectiveness of algebraic the computing needed todemonstrate beinversion, performed will beelementary calculated for block matrices 3.1 Case – one-piece elementary – a partition of algebraic operations necessary to calculate the negative feedback with different internal divisions into blocks submatrix ddii to calculate the negative Number of algebraic operations necessary feedback Bull Pol Ac.: Tech XX(Y) 2016 with different internal divisions into elementary blocks submatrix matrices Allperformed ofinversion, the analyzed cases ofof the block matrix will needed tomatrix be will be calculated for block matrices Table di d2matrix matrices All of the analyzed cases of the block matrix rithmwith ofAll the number algebraic operations different internal divisions elementary blocks –will sub- in the 1‒1‒1‒1 order Ifmatrix the matrix block can be divided into matrices of of the analyzed cases ofinto the block matrix will i n Block dimension dimension matrices All the analyzed cases ofwill the block matrix will Number of algebraic operations necessary calculate the44negative have a structure such as matrix (5), it differ only in size Block n2totake 223 the334form 55 feedback with different internal divisions into elementary blocks subhave a structure such as matrix (5), it will differ only in size needed to be performed will be calculated for block matrices matrices All of the analyzed cases of the block matrix will one-piece elementary matrices (it will Table Block dimension n of a matrix have a structure such as matrix (5),(5), it will differ onlyonly in size Blockoperations dimension n matrix d 12 Sum of algebraic l have a structure such as matrix it will differ in size i oII of elementary matrices Asmatrix mentioned above, each ofinthe the in3 related the 9negative Sum of algebraic operations loII matrices All internal of thesuch analyzed cases of the differ block matrix will of algebraic operations necessary to3 calculate have a structure as (5),above, it will only of Number (5)), then the number of algebraic to12thefeedback calwith different divisions into elementary blocks -size subof elementary matrices As mentioned above, each of inSum of algebraic operations loII operations of of elementary matrices As mentioned each of the in326 639 9412 12 Sum of algebraic operations loII elementary matrices As mentioned above, each of the inBlock dimension n block matrix d i verse elementary matrices (6) can be calculated individually A have aelementary structure such as As matrix (5), itofabove, will differ only size elementary matrices mentioned each of thein inverse culation of the leading element c0 (7) is determined by: matrices All of the analyzed cases the block matrix will verse matrices (6) can be calculated individually A verse elementary matrices (6) (6) cancan be calculated individually A A verse elementary matrices becalculated calculated individually 36 4operations 512 Sum of algebraic loII detailed analysis of the theasstructure structure of the matrix (6) reveals that Blockoperations dimension n 2process, elementary matrices (6) can be individually size n of the block matrix, the inversion of of elementary matrices As mentioned above, each of in theA deinhave aanalysis structure such matrix (5), it will differ only size detailed analysis of of the matrix (6) reveals that detailed of the structure of the matrix (6) reveals that –1 T - a partition 3.1 Case one-piece elementary matrices detailed analysis of the structure of the matrix (6) reveals that tailed analysis of the structure of the matrix (6) reveals that addition (subtraction) in triples of matrices b a b , multipli3 12 Sum of algebraic operations l 3.1 Case one-piece elementary matrices a partition there are four different types of items elementary submatrices verse elementary matrices (6) can be calculated individually A j oII j j of are elementary matrices As mentioned above, submatrices each of the in- 3.1.3.1 there are four different types of items - elementary submatrices Case - 1one-piece elementary matrices - a -partition there four different types of items -items elementary –1 T matrix Case - one-piece elementary matrices a partition in the 1-1-1-1 order If the the block can be beinversion divided into there are four different ofcan items -calculation elementary submatrices there are four different types ofof –matrix elementary submatrices cation in triples of matrices bj ablock bj , and aj matrix in the 1-1-1-1 order If matrix can divided into of inverted block matrix requiring thethe These arethat asin the j matrix detailed analysis of thetypes structure (6)These reveals verse elementary matrices (6) be calculated individually A of inverted block matrix requiring the calculation are as 1-1-1-1 order If the block can be divided intointo of inverted block matrix requiring the calculation These are as in the 1-1-1-1 order If the block matrix can be divided one-piece elementary matrices (itblock will matrices take the form of aa mamaof block requiring the calculation These are By convention, division of the m atrix forform of inverted block matrix requiring thethe are as one-piece 3.1 Case - such one-piece elementary -one-piece a partition elementary matrices (it will take the of follows: there areinverted four different types of items -calculation elementary submatrices detailed analysis of thematrix structure of matrix (6)These reveals that one-piece elementary matrices (it will taketake the the form of aof mafollows: follows: one-piece elementary matrices (it will form a maas follows: elementary matrices will be referred as division in 1‒1–1‒1 trix (5)), then the number of algebraic operations related to follows: in the 1-1-1-1 order If the block matrix can be divided into 3.1 Case one-piece elementary matrices a partition (5)), then the number of algebraic operations related to of inverted block matrixtypes requiring the-calculation are as trixtrix there are four different of items elementary These submatrices (5)),(5)), then the the number ofofalgebraic operations related to to trix then number of algebraic operations related first elementary matrix, which later will be called the leading order Calculated number algebraic operations for various the calculation of the leading element c (7) is determined by: first elementary matrix, which later will be called the leading one-piece elementary matrices (it will take the form of a mathe 1-1-1-1 order If the block can divided the calculation of the leading element c00 is (7) isbe determined by: elementary matrix, which later will be called the leading follows: offirst inverted block matrix requiring the calculation are as the in calculation of the the leading element cmatrix determined by:into 1 first elementary matrix, later willwill be called theThese leading (7) element, has thewhich following form: dimensions of block matrices is the shown related the calculation of the leading element ctake (7)Table is determined by: first elementary matrix, which later be called the leading in block size n of the block matrix, inversion process, operaelement, has the following form: trix (5)), then the number of algebraic operations to one-piece elementary matrices (it will the form of a mablock size n of the block matrix, the inversion process, operaelement, has the following form: follows: has the following form: block sizesize n ofnthe block matrix, the the inversion process, operaelement, −1 T T, block ofthe the block matrix, inversion process, has thematrix, following form: −1 bby: tionscalculation of addition addition (subtraction) in triples triples of matrices b jj aaoperathe of the leading element c (7) is determined element, first elementary which later will be called the leading tions −1 T trix (5)), then number of algebraic operations related to j, b tions of (subtraction) in of matrices b j of addition (subtraction) in triples of matrices b j a jb ba−1 Table triples jj , bTj , tions of addition (subtraction) in of matrices −1 kk later will be called the leading T j j j block size n of the block matrix, the inversion process, operahas thematrix, following k form: −1 the calculation of the leading element c (7) is determined by: T element, first elementary which −1 T T −1 multiplication in triples triples of matrices matrices and matrix inina jcalculate multiplication in of bb jjto aaleading bbTjj ,, and k b j a−1 Number of in algebraic operations necessary (7)multiplication = (a (a00 − −∑ jj matrix T b −1)−1(7) triples of matrices b j a−1 baa0T−1 a jthe matrix in−1 inT, (7) cc00 = jj ,of j ab−1 b j abb−1 (7) (7) −∑ c0 = multiplication in triples ofelement matrices bj inversion band , and a j matrix matrix jjj )bTjj )−1 a b tions of addition (subtraction) in triples matrices b block size n of the block matrix, the process, operaelement, has the following form: j j j version By convention, such division of the block for j j j j c a b ) =0(a − c0 (a j of the version By convention, such division of the block matrix for j j ∑ j j version By convention, such division block matrix for k −1 for j T, −1 Tmatrices version By convention, such division of the block matrix a b tions of addition (subtraction) in triples of b j one-piece elementary matrices will be referred as division in j −1 T −1 multiplication in triples of matrices b , and a matrix inb a jin j in j jreferred one-piece elementary will be as j5division (a0 −in∑ b j a j b jinterpretation ) (7) c0 = that j as4 division Block dimensionmatrices n matrices elementary willwill be 2referred elementary matrices, physical can beone-piece −1 T , and one-piece elementary matrices be referred as division in elementary matrices, that in kjphysical interpretation can be 2 elementary matrices, that in physical interpretation can be 1-1-1-1 order Calculated number of algebraic operations for version By convention, such division of the block matrix −1 T −1 multiplication in triples of matrices b b a matrix ina elementary matrices, that in physical interpretation can be j j 1-1-1-1 order Calculated number of algebraic operations for j j order number elementary matrices, can (7) be 1-1-1-1 (a0feedback, −in∑physical b j a jhave b jinterpretation )following c0 = that responsible for negative feedback, have following forms: algebraic operations 10 operations 13 for for Sum of Calculated algebraic operations loI of algebraic 1-1-1-1 order Calculated number of responsible for negative forms: responsible for negative feedback, have following forms:forms: various various dimensions of matrices thesuch block matrices is shown shown in Table responsible for negative feedback, have following one-piece will beisofreferred division in version Byelementary convention, division the block matrix various dimensions of the block matrices is in Table dimensions of the block matrices shown inasTable for negative feedback, have following forms: responsible elementary for matrices, that in jphysical interpretation can be various dimensions ofmatrices thebetween block matrices is shown inthe Table General relationship the dimension of block 1-1-1-1 order Calculated number of algebraic operations for one-piece elementary will be referred as division in General relationship between dimension of the block −1 relationship between the the dimension of the block −1 =in−c −cphysical b ahave (8) for negative feedback, following forms: responsible elementary matrices, that interpretation can be General −1 General relationship between the dimension ofinthe block General relationship between the dimension of the block ddii−c = 00ibiia−1 (8) (8) di = matrix block dimension n and the number of algebraic operavarious dimensions of the block matrices is shown Table 1-1-1-1 order Calculated number of algebraic operations for ii (8) bi a matrix block dimension n and the number of algebraic operamatrix block dimension n and the number of algebraic opera= −c0 bi ahave di feedback, responsible for negative following forms: (8) various i matrix block dimension n and the number of algebraic operamatrix block dimension n and the number of algebraic operations that are needed for the calculation ofisleading the leading element dimensions thebetween block matrices shown in Table General relationship theofdimension of the block {1, for ∈ {1, k}, auxiliary matrices having forms: tions that are needed for the calculation of the leading element that are needed forof the calculation element ∈ ,, k}, auxiliary matrices having forms: forfor i ∈ii{1, , k}, auxiliary forms: = matrices −c0 bmatrices a−1 (8) tions dauxiliary having forms: tions are needed for calculation ofthe thethe leading element imatrices ihaving tions that are needed forthe the calculation of leading element i having {1,i 2 f for i ∈for ,1, …, kg, k}, auxiliary forms: (7), is as follows: c matrix block dimension n and the number of algebraic operaGeneral relationship between the dimension of the block isfollows: as follows: follows: cc0 (7), is asis −1 T T (7), = bTi ab−1 deeiii−a =−c −a0b−1 (9)c0 (7), −1 is as as ctions 00 (7),that (9) = −a matrix block dimension and the number operaei = (9) (8) arefollows: needed fornthe calculation of of thealgebraic leading element ii having forms: −1 iii biT for i ∈ {1, , k}, auxiliary matrices i ei = −ai bi (9) (9) tions that are needed for the calculation of the leading element (7), is as follows: c = 3(n 3(n −+1) 1)1(13) + 11 (13) matricesT having forms: oI3(n {1, for ∈ {1, − auxiliary 1} = + lloI − 1)− (13)(13) ∈ ,, kkk}, − 1} forfor i ∈ii{1, , k − 1} ei = −a−1 (9) c 0 (7), is as follows:loI = (13) loI = 3(n − 1) + i bi {1,i 2 f1, …, k ¡ 1g for i ∈for matrices, , k − 1} which elementary in physical interpretation can be −1 T interpretation elementary matrices, which in physical interpretation 3 elementary matrices, which cancan be be The calculations effort necessary to perform perform the designation designation eiin=physical −aphysical i calculations effort necessary to the i bare calculations effort necessary to1) perform designation elementary matrices, which in interpretation can(9) be TheThe 3(n − +perform the the (13) loI = responsible for positive couplings, forms of the products Thecalculations calculations effort necessary to designa{1, for i ∈ , k − 1} responsible for positive couplings, are forms of the products The effort necessary to perform the designation responsible for positive couplings, are forms of the products (8), is associated with the(13) imof negative feedback matrix d i (8), is associated with the imof negative feedback matrix d (8), is associated with the imof negative feedback matrix d responsible for positive couplings, are forms of the products i = 3(n − 1) + l i matrices, in physical interpretationcan canbe be of tion of negative feedback d(8), isassociated associatedwith with the the imoI matrix of the matrices (8) and (9)which {1, matrices, forthe i ∈elementary , k(8) − 1} elementary which in physical interpretation i (8), of matrices and (9) is negative feedback matrix d of the matrices (8) and (9) i plementation of algebraic operations necessary to: calculate The calculations effort necessary tonecessary perform theto: designation plementation of algebraic operations necessary calculate of algebraic operations necessary to:to: calculate theresponsible matrices (9) forand positive couplings, forms theproducts products implementation algebraic operations calculate of elementary matrices, which in physical interpretation can be plementation responsible for (8) positive couplings, areare forms ofofthe plementation ofofalgebraic operations necessary to: calculate The calculations effort necessary to perform the designation and b , calculate the inverse form of imthe the matrix product c (8), is associated with of negative feedback matrix d e d (10) i i i j and b , calculate the inverse form of the the matrix product c bi ,bcalculate the product c0 cand ei d j the matrices (8) (9) and the matrix product theinverse inverseform form of of the 0and ei d(9) (10)(10) the matrix i, icalculatethe responsible for (8) positive couplings, of theofmatrices and and boperations ,dcalculate the inverse form ofimthe the matrix product c0 0matrix ej i d j are forms of the products (10) matrix i (8), is associated with the of negative feedback (in present case dividing by an element of the mamatrix a plementation of algebraic necessary to: calculate i i (inpresent present case dividing by an element of mathe mamatrix matrix aaii (in case – -dividing anelement element of the the present case - dividing bybyan of (in of (8) and (9) {1, {1, for ∈matrices − 1}, 1}, ∈ {1, , k} k} present case -bof dividing by an inverse element of theofmamatrix (in {1, ii{1, ∈ ,, kk1}, − ∈ j ,(10) forfor i ∈the , k − j ∈ jj{1, e ,i dk} plementation of algebraic operations necessary to: calculate trix) Calculated numbers algebraic operations for different and , calculate the form the the matrix product c (10) trix) Calculated numbers of algebraic operations for different i trix) Calculated numbers of algebraic operations for different trix) Calculated numbers of algebraic operations for different {1, , k − for i ∈matrices, 1}, j can ∈ {1,be .called , k} in physical interpreta- trix) Calculated numbers of algebraic operations for different block which block matrices, which called in physical interpreta4 block matrices, which cancan be be called in physical interpretaand b , calculate the inverse form of the the matrix product c block dimension n of the block matrix, the negative feedback ei dcalled (10) case -i dividing by element offeedback the mamatrix (in present block dimension nnof block matrix, thean negative feedback the j block dimension of the block matrix, the negative dimension n of the block matrix, the negative feedback block matrices, which be in physical tion as self matrices, have following forms:interpreta- block {1, {1, for ∈for inertia inertia , kmatrices, − 1}, j can ∈have following , k} block dimension nshown ofcase the block matrix, theelement negative feedback as self matrices, have following forms: tiontion asiself inertia forms: (in present dividing by an of the matrix a i 2 f1, …, k ¡ 1g, j 2 f1, …, kg matrix d (8), are shown in Table It should be noted, that in-mamatrix d (8), are in Table It should be noted, that intrix) Calculated numbers of algebraic operations for different i i matrix d (8), are shown in Table It should be noted, indi (8), ini are shown in Table It should be noted, that that self matrices, have following forms:interpreta- matrix {1, inertia for ias ∈matrices, ,k − 1}, j can ∈ {1, called , k} tion block which be in physical matrix di size (8), of arethe shown inof Table Itone, should be noted, that intrix) Calculated numbers algebraic operations for different creasing block matrix by results in occurrence block dimension n of the block matrix, the negative feedback creasing of the block matrix by one, results in occurrence creasing sizesize of the block matrix by one, results in occurrence Tcan−1 −1 −1 T physical −1−1 −1 T −1 tion block matrices, which be called in T bT −1 T b −1 =i (a (a − (ccan + bcalled a−1 )physical bforms: )−1 interpreta(11) block self matrices, have i following creasing of matrix bymatrix results inthe occurrence asblock matrices, which be in Table 2matrix, = bTii−1 ccinertia (11)(11) − ci = nshown ofblock the the negative -the innoted, first row of additional an dimension additional negative feedback din ii (a iib− i ) bii)−1 interpretation i ab matrix disize (8),negative arethe inblock Table Itone, should thatrow ini (c i be −1 jjj )bTjj )b−1 j i ab−1 00 b+ (c+ in thefeedback first of an additional negative feedback matrix d first row of an feedback matrix d i i − bi (c0 have + bfollowing bforms: (11) ofNumber tion asasself have following i = (aimatrices, ia i) of algebraic operations necessary to calculate the negative j b j ) forms: selfcinertia inertia matrices, in the first row an additional negative feedback matrix d i matrix d (8), are shown in Table It should be noted, that inand first column of the inverted block matrix i column creasing size of of the the block matrix by one, results in occurrence andand firstfirst column inverted block matrix {1, for ∈ {1, k} of the inverted block matrix −1 T −1 −1 T −1 forfor i ∈ii{1, , k} ∈ ,, k} matrix di matrix (11) creasing and first column of block thefeedback inverted block size of the matrix by one, results inthe occurrence j bTj )−1 bi )−1 for i ∈ {1, c.i ,= k}.(ai − biT (c0−1 + bi a−1 General relationship between block dimension of the block the firstblock row of an additional negative feedback matrix di -ofin relationship between block dimension block General relationship between block dimension of the ci = (a bi (c0indications + bi a j b jof ) elementary bi ) (11) (11) General i− 2operations di - 4in 5are Block dimension n inverted General relationship between block dimension of the block the first row of an additional negative feedback matrix By introducing the above matrimatrix and the number of algebraic that needed and first column of the block matrix Byfor introducing above indications of elementary matriBy above indications of elementary matri- matrix andand the the number of algebraic operations that that are needed matrix number of algebraic operations are needed iintroducing ∈ {1, ,the k}.the By the above indications of elementary matrix and the number of algebraic are needed first column ofthe the inverted block matrix ces, inverted block matrix (6) takes the following form: matri- for and 3operations matrix 9that 12 Sum ofrelationship algebraic operations loIIfeedback (8) is as as for the calculation of the negative feedback matrix General between block dimension of block {1, forinverted iintroducing ∈for , k} ces,ces, inverted block matrix (6) takes the following form: block matrix (6) takes the following form: (8) is as the calculation of negative matrix d (8) is for the calculation of the negative feedback ddiithe i i 2 f1, …, kg ces,Byinverted block matrix (6) takes the following form: (8) is as for the calculation of the negative feedback matrix d General relationship between block dimension of the block i follows:and the number of algebraic operations that are needed introducingthe above indications of elementary matri- follows: matrix follows:     the above indications of elementary follows: introducing matrimatrix the number of algebraic operations that are needed d d d cc00 d1 (6) ces,Byinverted block takes the following form: is as for the and calculation of the negative feedback matrix di (8) k d d  c0matrix d1 d2 k dk d d d c   ces, inverted block matrix (6) takes the following form: (8) is as for the calculation of the negative feedback matrix d k   i   follows: = 3(n 3(n1)− − 1) 1) (14) loII e d ek 11ddkk  Bull Pol Ac.:  Tech 64(4) 855(14) oII3(n − (14) loII l=   c1 cc2016 =   11e1 de211d22 .e1 de    follows: = 3(n − 1) (14) l c e d e d    d d d c oII 1c2   Unauthenticated   e1 dk kkk  (12)for for = D  (12)(12) =rr  Dr D =3,2, 2, .3, 3, = nfor=nn2,=   c0 dc c2edccd2222 e2 deekd222ddk   Download Date | 1/27/17 4:49 PM   = 3(n − 1) (14) l oII (12) forThe Dr =  1 k    n =number 2, 3,of algebraic of algebraic operations needed in order order to deterdeter e1 dk    TheThe number in order to deter  number of algebraic operations in to = 3(n −needed 1) needed (14) loperations   c e d oII   1   c2 e2 dk  The number algebraic operations order to deter(12) mine Dr =    is related with mine the of positive feedback for n= 2,matrix 3, of of positive i d j (10), (10), isin related with the matrix feedback e d eneeded   c0 d1 c1 d2 e1 d2 c2 0) a- 1) ri- 2) ed ix ut the matrix product c0 and bi , calculate the inverse form of thegebraic operations for different block dimensions n of block gebraic operations for different block dimensions n of block matrix (in present case - dividing by an element of the ma-matrix, necessary for calculation of self inertia matrices, are matrix, necessary for calculation of self inertia matrices, are trix) Calculated numbers of algebraic operations for differentshown in Table shown in Table block dimension n of the block matrix, theT.negative feedback General relationship between block dimension of the block Trawiński, A Kochan, General P Kielan, and D Kurzyk between block dimension of the block relationship matrix di (8), are shown in Table It should be noted, that in-matrix and the number of algebraic operations that are needed matrix and the number of algebraic operations that are needed creasing size of the block matrix by one, results in occurrencefor the calculation of the self inertia matrices ci (11) is as folfor the calculation of the self inertia matrices ci (11) is as foltheoccurrence first rowlows: of an additional negative feedback matrix di - in in creasing size of the block matrix by one, results for the calculation of the self inertia matrices ci (11) is as follows: and of first of negative the inverted blockmatrix matrix ancolumn additional feedback di – in the first row lows: General between (16) loIV = 8(n − 1) and firstrelationship column of the invertedblock blockdimension matrix of the block (16) loIV = 8(n − 1)(16) General between block dimension block matrix and the relationship number of algebraic operations that of arethe needed for n = 1, 2, matrix and the number of algebraic operations that dare needed 1, 2, is asfor nfor=n = 1, 2, … for the calculation of the negative feedback matrix i (8) total amount of algebraic operations needed to calculate for the calculation of the negative feedback matrix di (8) is as The The total amount of algebraic operations needed to calcuThe total amount of algebraic operations needed to calculate follows: the inverted form of block matrix of the block dimension n follows: the inverted of block matrix of the block dimension the late inverted form ofform block matrix of the block dimension n and nassuming that all matrices are one-piece, is given by the and assuming that all matrices are one-piece, is given by the and assuming that all matrices are one-piece, is given by the (14)following loII = 3(n − 1)(14) formula: following formula: following formula: Tomasz Trawi´nski, Adam Kochan, Paweł Kielan and Dariusz n Kurzyk for nfor =n = 2, 3, … 2, 3, n n l = + 19 n − 10 (17) (17) The number of algebraic operations needed in order to de o The number of algebraic operations needed in order to deterl = + 19 (17) 2 − 10 o Table 3feedback ei dj (10), is related termine the matrix of positive 2 d (10), is related with mine the matrix of positive feedback e i j Number algebraic operations necessary to calculate =n = 1, 2, … 1, 2, withofthe calculation of the auxiliary matrixthe e , positive and itsfeedback productfor nnfor = 1, 2, the calculation of the auxiliary ei , and iits product withforRelationship matrixmatrix ei d j showing thethe number algebraic with negative feedback matrix dj It should be emphasized that Relationship Relationship showing numberof algebraic operations operations showing the number ofofalgebraic operations negative feedback matrix d It should be emphasized that this j to be to to calculate thetheelementary Block dimension n dimension n ¸ 3 this type of matrices occurs for block Theseneeded needed to done be done calculate elementarymatrices matrices (with (with needed to be done to calculate the elementary matrices (with typecalculations of matrices occursoperations for blockl of dimension n 18 ≥, multiplication 30 These cal-the require: inversion the block dimension growth blockmatrix), matrix),can canbe be reprerepreSum of algebraic a block dimension growth of of thetheblock oIII the3 matrix i the block dimension growth of the block matrix), can be repreT ofsented culations require: of the by matrix i , multiplication of matrices a–1binversion , multiplication (¡1) aand multiplication by sented graphically in Fig graphically in Figure −1 T i i and operations multiplication bysented graphically in Figure matrices matrixaidj.bCalculated numbersby of(−1) algebraic for vari , multiplication eleme eleme agona agona symm symm trix in trix in matric matric lar ma lar ma sion o sion o dimen dimen the 1the 1tioned tioned 1-2-21-2-2The The the lea the lea of inp of inp b j a−1 j b j a−1 j culate culate multip multip matrix matrix calcul calcul calcul calcul Tablematrix, ious dimensions of the block for the positive feedback3.2 Fig 2The number of algebraic required to be imple-ments Case - elementary matricesoperations in partition in the 1-2ments Case 2in- order elementary matrices in partition in -the Number operations necessary to calculate the self inertia 33.2 mented matrix of ei dalgebraic to calculate the matrix matrix inverse: a) for 1-2a single- Gen j are shown in Table 2-2 order Assume that the block can be divided into matrices ci 2-2 order Assume that the block matrix can be divided into element matrix (special case) the number of operations - 1, b) - for Gen Block dimension Tablen Sum of of algebraic algebraic operations to 16calculate 24 the 32 positive Number operationsloIV necessary feedback matrix ei dj Block dimension n matrix d j Calculated numbers of algebraic operations for var9 18 30 Sum of algebraic operations loIII ious dimensions of the block matrix, for3 the positive feedback matrix ei d j are shown in Table General relationship between block dimensionofofthe the block block General relationship between block dimension matrix number algebraicoperations operationsthat thatare are needed needed matrix andand the the number of of algebraic for the calculation of the positive feedbackmatrix matrixeedi dj (10) (10) is is for the calculation of the positive feedback i j as follows (assuming that matrix d was calculated in previous i as follows (assuming that matrix di was calculated in previous step): step): (n − 2)(n − 1) (15) loIII = (15) for nfor=n = 3, 4, … 3, 4, The number of algebraic operations The number of algebraic operationsneeded neededininorder order to to determine selfself inertia matrices (11) termine inertia matrices (11)isisassociated associatedwith with following following −1 T T calculations: productofofmatrices matrices bbiai–1 b ; matrix a inversion b ; matrix invercalculations: thethe product i i i i case by division by thebyelement of this matrix); the sumsion(in (inthis this case by division the element of this matrix); mation in thein inner inversioninversion of the internal expression the summation thebrackets; inner brackets; of the internal (in parentheses) and multiplying it by the matrices bi and aiT , expression (in parentheses) and multiplying it by the matriexpressions in external parentheses ces subtraction bi and aTi , of subtraction of contained expressions contained in exterand its inversions Calculated numbers of algebraic operations nal parentheses and its inversions Calculated numbers of alfor different block dimensions n of block matrix, necessary for gebraic operations for different block dimensions n of block calculation of self inertia matrices, are shown in Table matrix, necessary for calculation of self inertia matrices, are shown in Table Table General relationship dimension of the block Number of algebraicbetween operationsblock necessary to calculate the self matrix and the number ofinertia algebraic operations that are needed matrices ci for the calculation of the ci (11) is 5as folBlock dimension n self inertia matrices lows: Sum of algebraic operations loIII loIV = 8(n − 1) 16 24 32 (16) General relationship between block dimension of the block for nmatrix = 1, 2, the number of algebraic operations that are needed and The total amount of algebraic operations needed to calculate the inverted form of block matrix of the block dimension n and 856 assuming that all matrices are one-piece, is given by the following formula: lo = n2 n + 19 − 10 2 (17) block matrices with block dimension the number of operations - 15, c) - for matrices with block dimension the number of operations 32, d) - for matrices with block dimension the number of operations - 52 Fig The number of algebraic operations required to be implemented in order to calculate the matrix inverse: a) – for a single-element matrix (special case) the number of operations – 1, b) – for block matrices with block dimension the number of operations – 15, c) – for matrices with block dimension the number of operations – 32, d) – for matrices with block dimension the number of operations – 52 3.2 Case – elementary matrices in partition in the 1‒2–2‒2 order Assume that the block matrix can be divided into elementary matrices, so could be present at the block main diagonal as Fig Matrix partitioning into an 1-2-2-2 order: a) - block dimento: one-piece matrix, 2£2 square sions of submatrices, b) - dimensional signs assignment toand the symmetric submatricesmatrices The effect of such division of input matrix into elementary matrices, is that in the first line rectangular matrices (1£2 dimenTable sional) and in the first column rectangular matrices (with dimenNumber of algebraic operations necessary calculate the leading sion 2£1) are located Such a divisionto of the block matrixelement into c0 n symmetric one-piece elementary andBlock by 2dimension dimensional matrices will Inversion be calledofthe division insubmatrices the 1‒2–2‒2 block45matrix 4-elementary a−1 15The30 60 j order T and 3) fulfils the aboveofmentioned conditions, partition Multiplication triple matrices b j a−1 its 18 27 into 36 j bj elementary matrices Additions in 1‒2–2‒2 order is shown / Subtractions in Fig 2 a) Inversions / Divisions Sum of algebraic operations loI b) 26 51 76 101 elementary matrices, so could be present at the block main diagonal as to: one-piece matrix, × dimensional square and symmetric matrices The effect of such division of input matrix into elementary matrices, is that in the first line rectangular matrices (1 × dimensional) and in the first column rectangular matrices (with dimension × 1) are located Such a division2.ofMatrix the block matrix into one-piece elementary by Fig partitioning into an 1‒2–2‒2 order: a) – block and dimensions of submatrices, b) –matrices signs assignment the submatrices dimensional symmetric will betocalled the division in the 1-2-2-2 order The block matrix (3) fulfils the above mentioned conditions, and its partition into elementary matrices in 1-2-2-2 order is shown in FigureBull Pol Ac.: Tech 64(4) 2016 Unauthenticated The number of algebraic operation necessary to calculate Download Date | 1/27/17 4:49 PM the leading element c0 (7) is determined by block dimension n of input matrices, but also by dimensions of matrices in triple for n = 1, 2, Calculated numbers of algebraic operations needed for a determination of the negative feedback matrix di (8), in relation Inversion of selected structures of block matrices of chosen mechatronic systems to different block dimensions of input matrices is presented in Table It is worth to underline, that increase of block dimenonecalculated results innumber appearing of additional negative feedThe number of algebraic operation necessary to calculatesion byThe of algebraic operations needed to (1 × and × dimensional) in the first row back matrices d i the leading element c0 (7) is determined by block dimension n obtain the positive couplings matrices ei dj (10), upon different first column of the block matrix of input matrices, but also by dimensions of matrices in tripleand block dimensions of inverted input block matrix, is shown in Table –1 T Inversion of selected structures bj aj bj The number of algebraic operations necessary to cal- The overall relationship between the block dimension n of a culate matrices bj aj–1bjT results from two rectangular matricesblock matrix (partitioned into submatrices Table in the 1-2-2-2 order) Table Table to calculate the positive Number of algebraic operations necessary multiplications (with dimensions 1£2 and 2£1) with squareand the number of algebraic operations needed to calculate the Number of algebraic operations necessary to calculate the negative feedback Number of algebraic operations necessary to calculate the self inertia feedback matrix ei dj matrix (with dimension 2£2) Furthermore it is necessary to (8), is as follows: negative feedback matrix d matrix di matrices ci i Fig inverse ber of the nu sion dimen and th matric for n = calculate the inverse form of elementary aj–1 matrix The calBlock dimension n Block dimension n Block dimension n The culation effort which should be carried out for leading elements 23(n −bi1) (19) 96 oII = Inversion of 4-elementary submatrices aj–1bTi 1524 45 48 90 72150 Inversion of 4-elementary submatrices a−1 15 30 45 60 Multiplication of ltriple matrices a−1 j i verse c0 calculations is summarized in Table Inversion of selected structures −1 −1 Addition with c0 Multiplication of triple matrices −c0 bi Inversion 18 27 of selected 36 forstructures n Matrix = 2, 3,multiplication 24 48 80 ¡ai–1biT −1 2-2 or −1 T Inverting of matrix (c + b a b ) Sum of algebraic operations l 23 46 69 92 i oII i i Table TheMatrix calculated number of algebraic operations needed to obTable Table T −1 b 12 16 24 2440 dj −1Table multiplication Table Calculation of matrix beTii (c + bi a8−1 i Number ofoperations algebraic operations necessary to negative calculatefeedback the leading tainNumber i bi ) dcalculate 0matrices Number of operations necessary self the positive couplings upon different 32 j (10), the Inversion structures Number of of algebraic algebraic operations necessary necessary to to calculate calculate the the negative feedbackof selected Number of algebraic algebraic operations necessary eto toi calculate the self inertia inertia Subtractions element c matrix cci 27 81in 162 of algebraic loIII matrix, blockSum dimensions ofoperations input matrices block is shown Table12270 16 matrix ddii matrices i Table Inversion of 4-elementary submatrices 15 30 45 Block 22 33 44 55 dimension nn8 22 of 33a block 44 matrix 55 60 Table n6n to calculate Table Block dimension n dimension Overall relationship Block between dimension Rel Block dimension Block dimension Number of algebraic operations necessary the positive feedback Sum of algebraic operations l 53 106 159 212 −1dimension T oIV Overall relationship between of48the a block matrix Number of algebraic operations necessary negative feedback Number of algebraic operations to calculate the self inertia Inversion of submatrices aa−1 15 60 Multiplication of triple matrices bbi aanecessary 24 72 96 −1 −1 border) –1 30 the 45 T matrix ej i d to calculate (partitioned into following 1-2-2-2 and number of i i neede j Inversion of 4-elementary 4-elementary submatrices 15 30 45 60 Multiplication of triple matrices b 24 48 72 96 15 30 45 60 Inversion of 4-elementary submatrices a i i j i corder) j di matrix−1 matrices i (partitioned into following 1‒2–2‒2 and2 the number of Addition cc−1 1the 33 Multiplication −c 99 18 27 36 i a −1 0b algebraic operations needed towith calculate e44i d j verted 0−1 Addition with submatrices Multiplication of of triple triple matrices matrices −c –1 T 18 27 536 Block dimension n bi aii algebraic operations needed to calculate the submatrices e 18 27 36 Multiplication of triple matrices b a b −1 −1 T Block dimension n Block dimension n j i j j of + bbi aai−1 bbiT )) 22 33 44 dj the in Sum of algebraic lloII −123 is asInverting follows: Inverting of matrix matrix (c (c0−1 Sum algebraic operations operations 23 1546 46 4569 69 9092 92 150 (10),(10), i Inversion of of 4-elementary submatrices oII a j −1 i −1 T1 + is as follows: −1 −1 Ti −1 T Inversion of 4-elementary submatrices a 15 30 45 60 Multiplication of triple matrices b a b 24 48 72 96 j 16 24 32 + bi ai−1 biT ) −1 bi i i i8 AdditionsMatrix / Subtractions 124 48 80 Calculation sented Calculation of of matrix matrix bbiTi (c (c0−1 16 24 32 multiplication −a−1 bT + bi bi ) bi −18 Addition with c Multiplication of triple matrices −c0i bi ai−1 18 27 36 (n − 2)(n −01) Subtractions 44 88 12 16 i Subtractions 12 16 matrix Inversions / Divisions Matrix multiplication ei d j 112 24 40 loIII = 27 (c−1 + bi a−1 bT ) (20) Inverting of matrix 130 45 3(20) Sum of algebraic operations loII 23 46 69 92 Table 77 i 15 i Inversion of 4-elementary submatrices 60 Table Inversion of 4-elementary submatrices 15 30 45 60 mensi of algebraic operationscalculate loIII 27 positive 81 51feedback 162 76 270 −1 −1 T −1 Number algebraic operations necessary ofSum algebraic operations 26 101 Calculation ofofmatrix bTi (coperations 8106 16159 24212 32 bi ) b53 i Number of of Sum algebraic operations necessaryloIto to calculate the the positive feedback + bi loIV Sum algebraic for n = 3, 4, Sum of algebraic operations l 53 106 159 212 oIV for n = 3, 4, … matrix matrix eeii dd jj Subtractions 12 16 The number of algebraic operations forfordifferent 3.3 C The number of algebraic operations differentblock block diTablen7 Block dimension Inversion of 4-elementary submatrices 15 30 45 60 Block dimension n 3block4 dimension Generally, relationship between n6 feedback of blockmensions mensions n block of block matrix (resultingfrom fromthe thepartitioning partitioning in n of matrix (resulting order matrix with its partition into elementary matrices with dimenNumber of algebraic operations necessary the positive −1 to calculate Inversion of submatrices aa−1 15 45 90 150 Sum of algebraic operations loIV 53 106 159 212 j Inversion of 4-elementary 4-elementary submatrices 45matrices 90 with 150 dimen-1-2-2-2 j eiand matrix d j 15numbers with its partition into 1‒2–2‒2 order) of self the inertia self inertia matrices ci (11), are shown (11), are shown in order) of the matrices c menta sionsmatrix according to 1-2-2-2 order, of algebraic op−1 elementary i T Matrix multiplication −a biT 88 24 48 multiplication −ai−1 24 48 of80 80 sionsMatrix according to calculate 1‒2–2‒2 and numbers algebraic in Table i i border, Table and sq erations necessary to the leading elements c (7), is Block dimension n4 312 424 400 Matrix multiplication eei dd j Matrix multiplication leading 12 24 40 c0 (7), i j operations necessary to calculate the elements −1 onal The overall relationship between the block dimension n of as follows: Inversion of 4-elementary submatrices a j 27 15 45 90 150 Sum of Sum of algebraic algebraic operations operations lloIII 27 81 81 162 162 270 270 is as follows: Table oIII −1 T rical Matrix multiplication −ai bi 24 48 80 the block matrix (partitioned in the following 1-2-2-2 order) Number of algebraic operations necessary to calculate the self (18) = 25(n − 1) + (18) Matrixlmultiplication e d 12 24 40 oI i j inertia matrices ci Sum of algebraic operations loIIImatrices 27 81 162 270 Bull Pol Ac.: Tech XX(Y) 2016 matrix with its partition into elementary with dimenmatrix its2,partition into elementary matrices with dimenfor with nfor =n = 1, 2, … 1, to4 calculate Block n of algebraic operations2required Fig dimension The number the sions to 1-2-2-2 order, and of opCalculated numbers of algebraic operations neededfor for sions according according tonumbers 1-2-2-2 order, and numbers numbers of algebraic algebraic op-aa deCalculated of algebraic operations needed de- inverse matrix: of a)triple - for amatrices single-element case) 96 the num24 (special 48 72 Multiplication bi ai–1biT matrix erations necessary to calculate the leading elements c (7), is termination of calculate the negative feedback matrix di i(8), (8), in relation relation erations necessary to the leadingmatrix elements c00 in (7), is termination of the negative feedback d ber of operations 1, b) for block matrices with block dimension matrix with its block partition into elementary matricesiswith dimenas to different dimensions of inputmatrices matrices presented in Addition with c0–1 as follows: follows: to different block dimensions of input is presented in the number of operations 102, c) for matrices with block dimensionsTable according 1-2-2-2 order, and of algebraic opis to worth to underline, thatnumbers increaseof ofblock block dimendimenTable necessary It6.isItworth to calculate underline, that increase sion theofnumber of0–1operations with block +bi ai–1biT ) - 230, d) -1 for matrices Inverting matrix (c erations to the leading elements c (7), is sion by one results in appearing of additional negative feedback l = 25(n − 1) + (18) of operations 385 loI = 25(n − 1) +of1 additional negative (18)feed- dimension the number sion by one results appearing oI in as follows: 16 24 32 Calculation of matrix biT (c0–1 +bi ai–1biT )–1bi matrices di (1£2 and 2£1 dimensional) in the first row and first (1 × and × dimensional) in the first row back matrices d i for n = 1, 2, column for n = 1, 2, of the inverted block matrix Fig The number of algebraic operations required to calculate the 4needed calculate 16 Fig and 3.Subtractions The of of algebraic operations required to the c and first column ofofthe block thenumber number algebraic operations to12the calculate Calculated i loIinverted = 25(noperations − 1) +matrix needed matrix: a) for a single-element matrix (special case) numCalculated numbers numbers of algebraic algebraic operations needed for for aa dede-(18)inverse inverse matrix: a) for a single-element matrix (special case) the numThe overall relationship between the block dimension n of a matrices (11), is as follows: Inversion of 4-elementary submatrices 15 30 45 60 termination of the negative feedback matrix d (8), in relation Table termination of the negative feedback matrix dii (8), in relation ber ber of of operations operations 1, 1, b) b) for for block block matrices matrices with with block block dimension dimension 22 for nNumber =matrix 1, 2, of (partitioned dimensions algebraic block into submatrices the 1-2-2-2 order)the operations necessary toincalculate the negative to different block of input matrices is presented in Fig The number of algebraic operations required to number of operations 102, c) for matrices with block dimento different block dimensions of input matrices is presented in the number 53 with 106block 159calculate 212 the Sum ofof algebraic operations operations - 102, loIV c) - for matrices dimenCalculated numbers of feedback algebraic operations needed for a dematrix dneeded and the number ofunderline, algebraic operations to calculate thesion inverse i of block = 53(n − 1) (21) l matrix: a) for a single-element matrix (special case) the numTable It is worth to that increase dimen3 the number of operations 230, d) for matrices with block oIV Table It is worth to underline, that increase of block dimen- sion the number of operations - 230, d) - for matrices with block termination of the negative matrixnegative di (8), infeedrelation (8), is as follows: negative feedback matrix di feedback ber of operations 1, b) for block matrices with block dimension dimension the number of operations 385 sion by one results in appearing of additional Block dimension n the - 385 sion by one results in appearing of additional negative feed- dimension for The n= 1, 2,number relationship of operations tomatrices differentdiblock dimensions of input matrices is presented in the overall between thematrices block dimension of number of operations - 102, c) - for with blockn dimen(1 × and × dimensional) back –1 in the first row and × 1submatrices dimensional) in15the30first45 row 60 back matrices d (1of×4-elementary Inversion aj The sum of algebraic operations needed to calculate inblock matrix (partitioned in the following order) Table It isi of worth to underline, increase of block dimension the number of operations - 230, d) - for 1‒2–2‒2 matrices withan loII =block 23(nthat − 1) (19)and the and first column the inverted matrix the number of algebraic operations needed to calculate ccblock ii 1-2and first column of the inverted block matrix verse block matrix with block dimension n (partitioned in –1 and the number of algebraic operations needed to calculate dimension the number of operations - 385 and the number of algebraic operations needed to calculate ci negative 18 27 afeed36 Multiplication of triple matrices ¡c sion by one results in appearing of0badditional i The relationship (11), is as follows: The overall relationship between the the block block dimension dimension nn of of a matrices foroverall n= 2, 3, d (1 × 2between 2-2 order) can be obtained by the following expression: matrices (11), is as follows: matrices (11), is as follows: and × dimensional) in the first row back matrices i block into in order) 23 46 69 to92 Sum(partitioned of algebraic operations loII block matrix matrix (partitioned into submatrices submatrices in the the 1-2-2-2 1-2-2-2 order) number of algebraic operations needed obandThe firstcalculated column of the inverted block matrix and the number of lalgebraic ci and the number of algebraic operations needed to calculate the = 53(n − 1) (21) n needed to calculate n2 operations and tain the number of algebraic operations needed to calculate the the positive couplings matrices e d (10), upon different 53(n −121 1)(21) (21)(22) lloIV i block j oIV==27 The overall relationship between the dimension n of a + − 73 matrices (11), is as follows: o is as follows: negative matrix ddii (8), (8),block is between as matrix, follows: negative feedback matrix 2 blockfeedback dimensions of input shown in Table The overall relationship theisin block dimension n7.offor n = 1, 2, block matrix (partitioned into submatrices the 1-2-2-2 order) for n = 1, 2, a blockrelationship matrix (partitioned into submatrices inblock the 1‒2–2‒2 for n = 1, 2, … Overall between dimension of a matrix Relations showing the number of algebraic operations of operations needed an and the number of algebraic operations needed to calculate the The The sum sum of algebraic algebraic operations needed to calculate calculate an inin= 53(n −needed 1) to (21) loIV lloII = 23(n − (19) order) andinto the following number algebraic operations needed to calcuThe sum of algebraic operations to calculate an 23(n − 1) 1) order) (19) (partitioned 1-2-2-2 and the number of needed to be done to calculate the elementary matrices of inoII = of verse block matrix with block dimension n (partitioned in 1-2negative feedback matrix di (8), is as follows: verse block matrix with block dimension n (partitioned in 1-2late the negative feedback di (8), the is assubmatrices follows: block with block dimension n (partitioned in n of for n =block 1, 2, matrix matrix algebraic operations needed matrix to calculate ei d j2-2 inverse verted (with an increase ofexpression: block dimension for order) be by for nn = = 2, 2, 3, 3, 2-2 1‒2–2‒2 order) can can be obtained obtained by the the following following expression: order) can be obtained by the following expression: The sum of algebraic operations needed to calculate in(10), is as follows: be represented graphically, asanpreThe algebraic needed = 23(noperations − 1)(19) The calculated calculated number number of ofloII algebraic operations needed to to obob-(19) the input block matrix) can verse matrix n that (partitioned 1-2nn nn2blockbedimension sentedblock in Figure with It should considered the inputinblock tain positive couplings eiidd jj (10), upon different tain the the for positive couplings matrices matrices (10), 2-2 order) can be (22) n = 2, 3, … lloo = + 121 − 73 (22) 27 (n −e2)(n − 1) upon different + 121 − 73 (22) = 27 for n = 2, 3, obtained by the following expression: matrix has been partitioned into elementary matrices with di22 22 block loIIIblock = 27 matrix, block dimensions dimensions of of input input block matrix,2is is shown shown in in Table Table 7 (20) The calculated number of algebraic operations needed to obmensions resulting from following 1-2-2-2 order Overall Relations of Overall relationship relationship between between dimension dimension of of aa block block matrix matrix Relations showing showing the the number number of nalgebraic algebraic operations operations n2 tain the positive couplings matrices ei dand upon different for nBull = 3, 4, j (10), (partitioned into following 1-2-2-2 order) the number of needed to be done to calculate the elementary of inlo = 27 the+elementary 121 − 73 matrices Pol Ac.: Tech 64(4) 2016 order) and the number of (partitioned into following 1-2-2-2 needed to be done to calculate matrices of857 in-(22) 2 block dimensions of inputtoblock matrix, issubmatrices shown in Table Theoperations number ofneeded algebraic operations for different block di3.3 Case elementary matrices in partition in the 1-2-1-2 d algebraic calculate the e block matrix (with an increase of block dimension nn of algebraic operations needed to calculate the submatrices eii d jj verted Unauthenticated verted block matrix (with an increase of block dimension of Overall between dimension of apartitioning block matrix Relations showing the number ofcan algebraic operations mensions nrelationship of block matrix (resulting from the inthe input order Assume thatcan the block matrix be divided into ele(10), Download Date | 1/27/17 4:49 PM block matrix) be represented graphically, as pre(10), is is as as follows: follows: the input block matrix) can be represented graphically, as pre(partitioned into 1-2-2-2 order)ciand needed to be done to calculate the elementary matrices of in(11),thearenumber shown of insented 1-2-2-2 order) of following the self inertia matrices mentary matrices in such a way that 1-element matrices (1 × 1) in in Figure Figure 3 It It should should be be considered considered that that the the input input block block (n − to 2)(n − 1) the submatrices ei d jsented algebraic calculate verted blockmatrices matrix (with an increase of block n of Table operations needed and square (2 × 2)occur alternately ondimension the main diag- Fig Matrix partitioning into 1-2-1-2 order: a) - block dimensions of submatrices, b) - marks assignment to the submatrices Inv T Trawiński, A Kochan, P Kielan, and D Kurzyk Table Number of algebraic operations necessary to calculate the leading element c0 Block dimension n a) Inversion of 4-elementary submatrices a−1 j T Multiplication of triple matrices b j a−1 j bj Additions / Subtractions Inversions / Divisions Sum of algebraic operations loI 15 b) 16 31 32 10 19 20 1 26 29 54 57 in the first row are, alternately, rectangular matrices (of dimension × 2) and 1-element matrices (1 × 1), and the first column is the block transposition of the first row Such partitioning of input block matrix will be referred to the partition in the 1-21-2 order Such conditions correspond to an exemplary block Fig.shown Matrix into the 1‒2–1‒2 order: a) –by block matrix inpartitioning Figure and matrix given (3).dimensions Number of submatrices, b) – marks assignment to the submatrices of algebraic operations necessary to be implemented in order Fig The number of algebraic operations required to calculate the in-to calculate the form of the leading element c0 (7), using the verse matrix: a) – for a single-element matrix (special case) the numberpartitioning of input matrices into elementary matrices in the Table of operations – 1, b) – for block matrices with block dimension the 1-2-1-2 order, the block dimension n ofthe input maNumber of depend algebraic on operations necessary to calculate leading number of operations – 102, c) – for matrices with block dimension n = 2,c0the input block matrix is the number of operations – 230, d) – for matrices with block dimensiontrix If a block dimension iselement the number of operations – 385 composed of four matrices (Fig.4.a) and the number of alge2 Block dimension n braic operations necessary to calculate –1the leading element c0 15 16 32 4-elementary submatrices (7) isInversion the sameofas in the case analyzed in Chapter 3.2 -31formula Relations showing the number of algebraic operations(18) Multiplication of triple matrices bi ai–1biT 10 19 20 needed to be done to calculate the elementary matrices of in- Total algebraic operations will be different for larger block Additions / Subtractions verted block matrix (with an increase of block dimension ndimensions n of the input block matrix, and will depend on the Inversions / Divisions on the 1main 1diof the input block matrix) can be represented graphically, asnumber of 1- and 4-element submatrices, lying presented in Fig It should be considered that the input blockagonal ofofthe input block matrix computation 29 54 effort 57 Sum algebraic operations loI Algebraic 26 matrix has been partitioned into elementary matrices with di-associated with the leading element c (7) is shown in Table mensions resulting from following 1‒2–2‒2 order In general, the relationship between the block dimension n In general, the relationship between the block dimension of the block matrix and the number of algebraic operations, 3.3 Case – elementary matrices in partition in the 1‒2–1‒2 n of the block matrix and the number of algebraic operations, the partitionto calculate the leading element c0 , using order Assume that the block matrix can be divided into ele-needed needed to calculate the leading element c0, using the partiing into elementary matrices following the 1-2-1-2 order,order, is as mentary matrices in such a way that 1-element matrices (1£1) tioning into elementary matrices following the 1‒2–1‒2 follows: and square matrices (2£2) occur alternately on the main di- is as follows: agonal All matrices appearing on the main diagonal are symloI = n + 24d + 2(n − d − 1)(23) (23) metrical The result of this partition is that the elementary matrices in the first row are, alternately, rectangular matrices n ∧ n = 2, 3, where d -dnumbers (of dimension 1£2) and 1-element matrices (1£1), and thefor dfor