A Direct Transformation of a Matrix Spectrum

Transcription

1 dvnces in Liner lgebr & Mtri heory Published Online September 05 in SciRes. Direct rnsformtion of Mtri Spectrum lbert Iskhkov Sergey Skovpen VNIIEM Corportion JSC Moscow Russi Northern (rctic) Federl University Severodvinsk Russi Emil: simth@bk.ru skovpensm@mil.ru Received 4 July 05; ccepted September 05; published 4 September 05 Copyright 05 by uthors nd Scientific Reserch Publishing Inc. his work is licensed under the Cretive Commons ttribution Interntionl License (CC BY). bstrct method is presented for clculting mtri spectrum with given set of eigenvlues. It cn be used to build systems with different spectrums with the im of choosing desired lterntive. It enbles prcticl implementtion of control lgorithms without resorting to trnsformtion of vribles. Keywords Mtri Spectrum Frobenius Mtri Frobenius rnsformtion Spectrl Eqution. Introduction In lgebr the problems deling with eigenvlues belong to spectrl ones. mtri spectrum is chnged vi its elements. his procedure cn be implemented in vrious wys. For emple in computing mthemtics mtri is multiplied by other mtri for solving systems of liner lgebric equtions. he problem of trget trnsforming spectrum is the subject of control theory. It is clled s the method of chrcteristic eqution setting rrngement of eigenvlues spectrum control nd modl control []-[4]. o chnge spectrum the reltionships between coefficients of chrcteristic polynomil nd its roots re used. hey re known s Viet s formuls. Such spectrum trnsformtion which is pertinently clled Frobenius hs cler theoreticl bsis; moreover it specifies n obvious wy for its prcticl ppliction which implies supplementing the elements of the row of Frobenius mtri to the vlues mking mtri spectrum equl to given set of numbers. he reson for serching new method of spectrum trnsformtion is the requirement for obtining desired spectrum for concrete technicl systems using rel-time control lgorithms. more detiled eplntion of the necessity of other pproch for solving this problem is given in ppendi. How to cite this pper: Iskhkov. nd Skovpen S. (05) Direct rnsformtion of Mtri Spectrum. dvnces in Liner lgebr & Mtri heory

2 he method for clculting desired spectrum for which the uthors find possible to use the definition in the hedline is not bsed on Frobenius mtri. It cn be used to clculte the feedbck coefficients of control system with the im to obtin desired spectrum of closed-loop system without resorting to trnsformtion of vribles. his llows prcticl problems of control to be solved t the design phse of the system. By simulting the system behvior with different spectrums it is possible to find suitble lterntive which cn be further implemented s direct digitl control lgorithm. he pper is n outgrowth of the work [5].. Informl Resoning In the mtri Frobenius trnsformtion forms the row of elements reflecting the coefficients of the chrcteristic polynomil. n dditive influence of the feedbck onto these elements vries spectrum. he feedbck elements re clculted in the obvious wy s the differences of the row elements nd the coefficients of the polynomil with roots tht equl to the vlues of given spectrum. he proposed method is bsed on the reltionships between the elements nd the spectrum not for the trnsformed mtri but the originl one. hese reltions represent nother kind of Viet s formuls where the sums of the min minors of the mtri pper insted of chrcteristic polynomil coefficients. In contrst to the Frobenius form nd Viet s formuls these minors contin ll of the mtri elements. In order to chnge spectrum by given set of eigenvlues the mtri elements re replced by unknowns nd then the corresponding elements of minors nd combintions of the mtri eigenvlues re replced by the sme combintions of numbers from given set. In this cse the identities re trnsformed into system of equtions for the unknowns. s result fter the unknown will be replced by the solution of the obtined system of equtions the mtri gins desired spectrum. Feedbck elements cn lso be clculted in obvious wy s the differences between replced mtri elements nd elements of solution.. im of the Work Suppose i j i j k is given k k rel mtri σ(а) is its spectrum nd { λ i } Λ is set of rel numbers. By denote the mtri with k replced elements by unknowns. he objective is to consider rnge of issues relted to evlution of the unknowns which re substituted into σ Λ is stisfied. the mtri such tht the condition ( ) 4. Definitions ) Replcement is replcing the elements of the mtri (replced elements) by other elements (replcing elements). Replcing mtri (mtri with replcement) is mtri with replcing elements. ) Spectrl equtions of mtri (replcement system) re k equtions tht ws formed by replcing the coefficients of Viet s formule by the sums of min minors of the mtri nd by replcing the roots by the elements from given set Λ. ) Replcement of the i-th order is replcement leding to spectrl equtions of the i-th order. Liner replcement is replcement of the first order. Non-liner replcement is replcement of the second order or higher. 4) Spectrl trnsformtion of the mtri is replcing the elements of the mtri by the solution of spectrl equtions. 5. Frobenius rnsformtion of Spectrum nd Its lterntive For mtri it is known Viet s formuls k k k kk k () 0

3 σ+ σ + + σk σσ + σσ + + σ σ k k σσ σ k where σ i nd i re the i-th root of the chrcteristic polynomil nd the result of summtion in the i-th row which re the coefficients of the chrcteristic polynomil considering the sign. Frobenius trnsformtion of spectrum is bsed on obtining the elements on the left-hnd side (tking into ccount the sign) by non-singulr trnsformtion of the mtri nd by supplementing them to the vlues tht stisfy given set. his corresponds to the fct tht the sum on the right-hnd side () re replced by the sme reltionships between the numbers of given set Λ { λ i } nd the elements on the left-hnd side re supplemented by unknowns i. his leds the system to the equtions + λ + λ + + λ d k + λλ + λλ + + λ λ d k k + λλ λ d k k k k with n obvious solution d where d i is sum in the i-th row. Substituting the solution into Frobenius mtri one forms its spectrum with the vlues from given set Λ. he possibility to chnge mtri spectrum by supplementing the mtri elements to the vlues tht stisfy given set provides n lterntive to Frobenius trnsformtion of mtri. o perform this procedure we use the system () in the form of sums of min minors on the left-hnd side. he emple of such system for mtri of the -rd order is given by In this cse ll of the mtri elements re in the system. In prticulr this system enbles one to evlute how ech element influences on the spectrum. his cn t be done with the help of Frobenius trnsformtion. Now we supplement rbitrry elements of for emple the min digonl elements by unknowns nd. s result the mtri tkes the form nd we obtin the system of equtions for supplements the sme s (): d ( + )( + ) + ( + )( + ) + ( + )( + ) d ( )( )( ) ( ) ( ) ( ) d. k () () (4) By solving (4) we consider the gol hs been chieved. Indeed substituting the solutions into the mtri one mkes it equl to given set without resort to trnsforming the mtri. Further efforts re imed to simplifying the method of solving since just the solving this prticulr system fter opening the brckets is very complicted nd the solving compleity increses mny-fold when the dimension increses. Difficulties in solving prticulr system cn be even more enhnced when we need to solve multivrite problems ssocited with choice of complementry elements. he bove emple is illustrted by supplementing digonl elements. Besides this embodiment other vrints cn be used the number of which lso etremely increses with incresing size of the mtri. Frobenius trnsformtion of spectrum does not hve the vriety of lterntives s it hs the unique solution to () when n pproprite condition is stisfied.

4 6. Spectrl Equtions he bove computtionl difficulties cn be significntly reduced by choosing s the unknowns the elements together with its supplements insted of just the supplements. fter solving the equtions we cn determine the supplements s esy s in the Frobenius trnsformtion. For this purpose the k rbitrry elements of re replced by unknowns which re denoted for presenttion by the cpitl letter with the sme indees. For emple insted of the mtri with unknown supplements to the elements it is ssumed the replcing mtri where insted of the elements clled in definition 4. s replced the replcing elements considered s the unknowns re locted. he result is the system of equtions for of the form + + d + + d (6) + + d. In generl cse replcement of k elements of with combining the replcing elements ij into the vector ij nd building gives the system of equtions F( ) 0 (7) where F is the non-liner vector function with size of k clled by the spectrl eqution. In similr wy we cn choose N C (8) different replcing sets of elements nd obtin replcing mtrices in the form of (5) nd equtions in the form of (7). he number N very rpidly increses with the size of. For smll vlues of k it is given in ble. 7. ypes of Spectrl Equtions he type of the system (7) depends on the rrngement of replcing elements in. If we llocte the replcing elements in different rows nd columns s it is shown for the mtri (5) the system cn tkes the liner or ble. he growth rte in the number of replcing sets inconsistent equtions nd solvble systems for the different vlues of k. k N n M k k

5 non-liner form of degree from to k. However not ll of the systems hve solution. Using prticulr mtri we cn t once determine group of systems tht do not hve solution. Further for the ske of simplicity we will denote the replcing nd non-replced elements of mtrices by the numbers tht equl to the indees nd dots respectively. he right-hnd side of the first equtions of the system (6) is the fied sum nd the left-hnd side hs the unknown therefore the eqution is consistent with rbitrry vlues of а а nd d. But for the other mtri there re no unknowns on the left-hnd side of.... (9) d so the lst epression is inconsistent. It is strightforwrd to mke the following generliztion. he mtri (9) belongs to the fmily of mtrices which is formed by replcing k elements of tht lie outside of the min digonl in the two tringle res contining k k elements. his mens tht necessry condition to solve (7) is tht t lest one replcing element must be locted on the min digonl. It follows tht the number of inconsistent Eqution (7) is equl to the number of combintions he dependence (0) is lso given in ble. Subtrcting (0) from (8) we obtin n. (0) C k k k M N n () (given in ble ) tht is the number of solvble systems (7). Under pproprite conditions this number is the sum M k M () i i where M i is the number of the i-th order equtions. Determining the terms in () for generl cse s functions of k is the problem tht needs to be solved. Even clculting M i.e. determining the number of the liner systems (7) is unobvious procedure tht requires n nlysis of equtions of the form (6). We cn sy definitely (or rther we cn suggest since there is no rigorous proof) bout only the single term M i for i k. It is equl to. In other words there is only one wy to replce k elements of mtri tht llows spectrl eqution of order k to be obtined by replcing the elements on the min digonl. We consider net prticulr cse for mtri of the third order. By nlyzing 64 consistent Eqution (7) we estblish 8 45 nd vrints to replce elements ccording to (). Replcements re described by two types of the first order equtions si types of the second order equtions nd one type of third order eqution. Equtions of ll types re resulted. 8. Liner Replcements t first we discuss the vrints with evident solving the problem of choosing elements for liner replcement ssocited with replcement of rows nd columns of. here is only one element of replcing rows nd columns in the summnds of minors. Ech of summnd contins one unknown nd the multipliers obtined from the remining elements give the coefficient t the summnd. ssembly of these coefficients forms mtri denoted by R. hese equtions belong to the type.. Some summnds on the left-hnd side s it cn be seen from the system (6) do not hve replcing elements. We combine these elements in the row i into the element b i. hen fter combining the elements b i nd d i into the

6 vectors b nd d respectively we cn represent the Eqution (7) in the liner form by replcing rows nd columns. he solution to () eists under condition he number of Eqution () is R d b () det R 0. (4) m k. (5) hey do not describe ll of possible liner systems but determine only obvious ones. If replcing elements re not rows or columns we cn lso get the liner system (7). In this cse the summnds of minors cn contin product of replcing elements. Indeed for emple for the mtri. (6).. the replcement system is + + d + + d + + d. In the third row we obtin the summnd with product of unknowns nd. In this cse we hve forml reson to ssign (7) to the second order system. However we cn find from the first eqution (i.e. is known) nd the system becomes liner. hese equtions belong to the type.. he given types of equtions ehust liner replcements. 9. he Second Order Replcements 9.. Replcing Single Digonl Element Replcements with single digonl elements led to different types of the second order equtions. Consider the mtri (7). (8).. which differs from (6) in single element. he second nd third equtions for (8) is given by + + d + + d. he lst eqution of (9) is like (7) only eternlly. In the product there is no vrible epressed from the first eqution. his type of replcement is denoted s.. he mtri (9)... (0) differs from (8) in single element nd contins by single product of elements in two equtions: 4

7 + + d + + d. () his type of replcement is denoted s.. he mtri.... ().. differs from (6) in single element nd lso contins the product of elements in two equtions + + d + + d but the third equtions hs the product of three elements. his type is denoted s.. he mtri ()... (4) which differs from (6) in single element is chrcterized by the equtions + + d + + d with two products of two unknowns in the third row. his type is denoted s Replcement of wo Digonl Elements When two digonl elements re replced the type of equtions depends on choosing the third element. Consider two mtrices.. ).. b).. with identicl replced digonl elements nd common first eqution + + d. For the mtri ) the equtions + + d + + d differ from. in the first eqution. hey re denoted s.5. For the mtri b) the equtions (5) (6) (7) + + d + + d (8) with single product in the second row nd two products in the third row re denoted s Replcement of the k Order he lst eqution of (7) contins the k! summnds with products of k elements while the single summnd hs ll unknown multipliers. Replcement of k elements using vrints of (0) gives spectrl equtions of the k order only for unique cse when the min digonl of mtri is replced. Other vrints of replcement led to equ- 5

8 tions of the lower order. his conclusion is done without proving due to nlysis of ll spectrl equtions of the third order mtri. 0. Spectrl rnsformtion of the Second Order Mtri For the mtri we write the system () s σ + σ σσ. wo of fore elements cn be replced in si wys. Only one of those with replcing the elements а leds to incomplince equtions in the form (6). From five remining wys fore re replcements of rows nd columns nd led to liner spectrl equtions. Replcing the min digonl elements а leds to the second order eqution. 0.. Liner Equtions Replcement of rows nd columns gives the replcing mtrices.... ) ) ) 4).... nd liner equtions + d + d ) ) d d + d + d ) 4) d d. (9) We present them in the form of () s where R i i d bi (0) [ ] [ ] [ ] [ ] [ ] b b [ ] d d d 0 R 0 b b [ ] For the mtrices (9) the conditions (4) re given by R R R 4. ) 0 ) 0 ) 0 4) 0. If these conditions re stisfied the equtions tke the form ( ) ( ) d d d d d d ( ) ( ) d d d d d d. 4 6

9 Substituting the equtions into (9) one gives the mtrices ( ) ) d d d ) ( ) d d d ) d ( d d ) 4) ( d d ) d with spectrum tht equl to given set { λ λ } Λ. () 0.. he Second Order Eqution Replcing the digonl elements а gives the mtri with the eqution Its solution where d α d 4 ( d ) with given spectrum... + d d. α ± β α β () β + forms mtrices Emple. Given set { } Λ nd mtri fter clculting the mtrices () we obtin α ± β α β () 4. 0 ) ) ) 4) hey hve the spectrum tht contins the elements from the given set. With these results the solution () s well s the mtrices () Х.5 + i.98; Х.5 i.98 re comple..5 + i.98.5 i.98.5 i i.98. Spectrl rnsformtion of the hird Order Mtri From ble we hve the 84 vrints for choosing three elements. mong them the 0 vrints led to incomplint systems. he remining 64 vrints represent systems of the first second nd third order. hese sys- 7

10 tems re described by equtions of tow si nd one types respectively. Ech vlue M i in () is found vi nlysis of equtions... Liner Spectrl Equtions Replcing rows nd columns one gives the mtrices ) ) ) ).. 5).. 6) with pproprite equtions. For emple for the mtri ) we obtin the following equtions + + d + + d + + d. (4) (5) hey cn be presented in the form () s where R i i d bi (6) [ ] [ ] [ ] [ ] [ ] [ ] 4 [ ] d d d d f f + f + g g g g4 g5 g6 g7 g8 g9 [ 0] b b [ f g 0] b b [ f g 0] b b f g g g g 5 5 R f f 0 0 f g g g R R R f g4 g5 g 6 g7 g8 g R f R 6 f. g4 g5 g 6 g7 g8 g 9 s it mentioned bove choice of replcing elements in different rows leds to lener equtions of the type.. Mtrices nd equtions for ll of remining vrints for such types re resulted in ppendi. he first eqution is not cited becuse it remins identicl for replcements with other elements... Mtrices with the Second Order Replcement In ppendi we consider mtrices nd equtions of the types. -.4 when single digonl element is re- 8

11 plced. In ddition we result equtions of the type.5 nd.6 when two digonl elements re replced... Mtri with the hird Order Replcement For the third order mtri 6 of 64 vrints for choosing three elements led to equtions of the first nd second order. he remining mtri is chrcterized by the third order eqution + + d + + d.... (7) d. he result obtined cn be generlized for n rbitrry order mtri. Emple. With set Λ { 0 } consider vrints of trnsforming spectrum by replcing rows nd columns in the mtrices ) ) ) 4) he mtri ). Let s clculte d 0 d d 0 nd determine mtrices nd vectors (6): d [ 0 0] f 4 f 0 f 6 g g 6 g g 4 6 g 5 g 6 6 g 7 g 8 6 g 9 b [ ] b b [ ] b b [ ] b R R he mtrices re non-singulr hence there re ll the solutions: R 0 8 R R R R ( d b ) [ ] [ ] R ( d b ) [ ] [ ] R ( d b ) [ ] [ ] 4 R4 ( d b4 ) [ ] [ ] 5 R5 ( d b5 ) [ ] [ ] R ( d b ) [ ] [ ] With these solutions replcing mtrices (4) (8) 9

12 tke the spectrum ( ) { } ( ) { } ( ) { } σ.56 0 σ σ ( ) { } ( ) {.4 0 } ( ) { } σ σ σ tht is equl to given set with clculting ccurcy. he mtri ). Omitting intermedite clcultions here nd further we find mtrices R R R R R R mong ll the mtrices only R 5 is singulr nd hence the solution Х 5 does not eist. With the remining mtrices the solutions to the systems (6) re [ ] [ ] [ ] [ ] [ ] Substituting them into the mtrices (4) one forms the spectrum σ σ σ ( ) { i i.4 0 } ( ) { } σ ( ) { } ( ) { } σ ( ) { i.69 0 i.69 0 } 4 6 tht is equl to given set. he mtri ). Let s evlute the mtrices 0

13 R R R R R R he mtrices R nd R 5 re singulr nd hence the solutions Х nd Х 5 do not eist. With the remining solutions the mtrices tke the given spectrum [ 5 4 ] [ ] [ 5 ] [ 4 ] ( ) { } ( ) { } ( ) { } ( ) { } σ 0 σ 0 σ.48 0 σ he mtri 4). mong the mtrices R R R R R R R nd R 6 re non-singulr. With them the solutions re For replcing mtrices [ ] these solutions provide given spectrum. Emple. his emple describes spectrum trnsformtion with liner replcement of elements locted in different rows nd columns. Consider the mtri nd Eqution () given in ppendi. From the first eqution we define the unknown d t once. wo others re reduced to n eqution for Х nd Х with the mtri

14 nd the vector ( ) d d + With the set nd mtri ) from emple the solution for the mtri 5). [ ] [ ] forms the given spectrum σ ( ) { }. 5 ll clcultions were mde in MthCD.. Conclusions method for obtining mtri spectrum equl to given set of numbers without trnsformtion to Frobenius form is stted. Clculting tool is system of equtions which hs been obtined by replcement of rbitrry mtri elements by unknowns. heir number is equl to the size obtined from reltionships between mtri elements in the form of min minors nd elements of given set. he method hs mny vrints for choosing replcing elements nd equtions to clculte replcing elements from liner to non-liner with n order equl to the size. References [] Leonov G.. nd Shumfov M.M. (005) he Methods for Liner Controlled System Stbiliztion. St.-Petersburg University Publisher St.-Petersburg. [] Kuzovkov N.. (976) Modl Control nd Observe Devices. Mshinostroenie Moscow. [] Krsovsky.. (987) Control heory Reference Book. Nuk Moscow. [4] Islmov G.G. (987) On the Control of Dynmicl System Spectrum. Differentil Equtions [5] Iskhkov. Pospelov V. nd Skovpen S. (0) Non-Frobenius Spectrum-rnsformtion Method. pplied Mthemtics

15 ppendi. Proving the Method of Direct Spectrum rnsformtion In technicl systems vribles hving certin physicl sense re used. hese vribles chrcterize energy stores such tht speed of moving mss solenoid current cpcity voltge nd similr prmeters which re mesured by sensors. o obtin Frobenius mtri it is required both direct nd reverse trnsformtion of vribles in the feedbck loop. he reverse trnsformtion is eplined by the fct tht combintion of physicl vribles must enter into the system input. Firmwre implementing the trnsformtion needs dditionl hrdwre epenses nd softwre reliztion needs ependiture of time. his leds to dely in the feedbck loop nd to deteriorte dynmicl properties of system. s result the dvntge of control method bsed on the vrition of system spectrum is used not to the full owing to the fetures of the method using to implement spectrum trnsformtion. ppendi. Replcing Mtrices nd Spectrl Equtions for the hird Order Mtri... Liner Replcements For the type. replcement of the element а (number 7) refers to the mtri (6) nd Eqution (7): ) 9).. 0).. 8) + + d + + d 9) + + d + + d 0) + + d + + d. Replcing the element gives... ).. ). ).. 4).... ) + + d + + d ) ) + + d + + d + + d + + d 4) + + d + + d. Replcing the element а gives... 5) 6) 7).. 8)

16 5) 6) 7) 8) + + d + + d + + d + + d + + d + + d + + d + + d.... he Second Order Replcement with Single Digonl Element For the type. replcement of the element а (number ) refers to the mtri (8) nd Eqution (9): ) ) Replcing the element а gives.. ). 4)... ) 4) d d + + d + + d + + d. Replcing the element а gives.. 5) 6).... 5) 6) + + d + + d + + d + + d. d. For the type. replcement of the element а (number 7) refers to the mtri (0) nd Eqution (): 8) 8) d + + d. 4

17 Replcing the element а gives.. 9). 0)... 9) 0) + + d + + d + + d + + d. Replcing the element а gives.. ).. ).. ) ) + + d + + d + + d + + d. he type. (number ) refers to the mtri () nd Eqution ():.... 4).. 5) ) 5) + + d + + d + + d + + d. For the type.4 replcement of the element а (number 6) refers to the mtri (4) nd Eqution (5): ) 8).. 9) ) 8) 9) + + d + + d + + d + + d + + d + + d. Replcing the element а gives ).. ). ).. )

18 0) ) ) + + d + + d + + d + + d + + d + + d ) + + d + + d. Replcing the element а gives ).. 5) 6).. 7) ) + + d + + d 5) 6) 7) + + d + + d + + d + + d + + d + + d.... he Second Order Replcement with wo Digonl Elements For the type.5 replcement of the elements (number 8) refers to the mtri () (6) nd Eqution (7) 9) 9) d + + d. Replcing the elements gives... 0) )... 0) ) + + d + + d + + d + + d. Replcing the elements gives 6

19 ). )..... ) ) + + d + + d + + d + + d. For the type.6 replcement of the element (number 4) refers to the mtri (b) (6) nd Eqution (8): ).. 6). 7) ) 6) 7) + + d + + d + + d + + d + + d + + d. Replcing the elements gives ) 9) 40).. 4) ) + + d + + d 9) + + d + + d 40) 4) + + d + + d + + d + + d. Replcing the elements gives.... 4).. 4). 44).. 45) ) 4) + + d + + d + + d + + d 7

20 44) 45) + + d + + d + + d + + d. 8