An explicit solution to polynomial matrix right coprime factorization with application in eigenstructure assignment

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1 Journal of Control Theory Applcatons () An explct soluton to polynomal matrx rght coprme factorzaton wth applcaton n egenstructure assgnment Bn ZHOU Guangren DUAN ( Center for Control Theory Gudance Technology Harbn Insttute of Technology Harbn Helongjang 11 Chna ) Abstract: In ths paper an explct soluton to polynomal matrx rght coprme factorzaton of nput-state transfer functon s obtaned n terms of the Krylov matrx the Pseudo-controllablty ndces of the par of coeffcent matrces The proposed approach only needs to solve a seres of lnear equatons Applcatons of ths soluton to a type of generalzed Sylvester matrx equatons the problem of parametrc egenstructure assgnment by state feedback are nvestgated These new solutons are smple they possess better structural propertes are very convenent to use An example shows the effect of the proposed results Keywords: Pseudo-Controllablty Indces; Krylov matrx; Controllablty; Polynomal matrx rght coprme factorzaton; Sylvester matrx equaton; Egenstructure assgnment 1 Introducton Polynomal matrx rght coprme factorzaton of lnear systems s a basc problem n control systems theory has mportant applcatons n many analyss desgn problems such as robust stablzaton [1] mnmal state space realzaton [] speed servo system desgn [4] synthess of H-nfnty controllers [] etc Furthermore t has been shown by Duan hs co-authors that a factorzaton can be used to parameterze all the solutons to a generalzed-type of Sylvester matrx equatons [] the solutons to the problems of egenstructure assgnment [7] robust pole assgnment [8] observer-based robust fault detecton [9 1] For solutons to polynomal matrx rght coprme factorzaton most researchers have concentrated to numercal algorthms (see eg [11 14]) [11] proposes an algorthm to compute polynomal matrx rght coprme factorzaton wth the generalzed Sylvester Resultant matrces such a method has a dsadvantage that t works n an teratve way A numercal relable algorthm s proposed n [1] wth system n an upper Hesenberg forms recently another very smple teraton formula s proposed n [1] for polynomal matrx rght coprme factorzatons of mult-nput lnear systems whch s also n system upper Hesenberg forms The formula proposed n [1] gves drectly the coeffcent matrces of the par of solutons to the rght coprme factorzaton of the system Hesenberg form nvolves only manpulatons of nverses of a few trangular matrces some matrx productons summatons A generalzaton of ths result to the case of descrptor lnear systems wth sngle nput has recently been gven n [14] In ths paper a very neat smple explct soluton to polynomal matrx rght coprme factorzatons of lnear systems s frst establshed Ths soluton s expressed n an explct form by the Krylov matrx together wth a generalzed symmetrc operator matrx Due to ts neatness smplcty specal form the soluton may have mportant applcatons n many analyss desgn problems n control systems theory It has been shown n [] that polynomal matrx rght coprme factorzaton has drect applcatons n the soluton to a type of generalzed Sylvester matrx equatons the problem of egenstructure assgnment by state feedback n lnear systems By applyng the establshed explct soluton to polynomal matrx rght coprme factorzaton of nputstate transfer functon nsghtful general complete parametrc solutons to the type of generalzed Sylvester matrx equatons parametrc egenstructure assgnment n lnear systems va state feedback are deduced Compared wth exstng solutons to these two problems for example those n [] these new parametrc solutons possess better structural propertes gve a gudance of choosng the free Receved September 4; Revsed March Ths work s supported by the Chnese Outstng Youth Foundaton (No 998) Program for Changjang Scholars Innovatve Research Team n Unversty

2 148 B ZHOU et al / Journal of Control Theory Applcatons () parameters are more convenent to be used An llustratve example demonstrates the advantage of the proposed results Soluton to polynomal matrx rght coprme factorzaton In ths secton we solve the followng problem Problem 1 (Polynomal Matrx Rght Coprme Factorzaton) Gven matrces A R n n B R n r C R p n a par of real coeffcent polynomal matrces N T (s) D T (s) of dmensons p r r r respectvely such that the followng factorzaton holds: C(sI A) 1 B = N T (s)d 1 T (s) (1) N T (s) D T (s) are rght coprme At frst we ntroduce the followng lemma Lemma 1 [1] Let (A B) be controllable (A C) be observable then N T (s) D T (s) satsfyng (1) are rght coprme f only f deg (det (D T (s))) = dm (A) =n () Accordng to the above lemma we have the followng proposton Proposton 1 Let (A B) be controllable (A C) be observable f N(s) D(s) satsfyng are rght coprme then (si A) 1 B = N(s)D 1 (s) () N T (s) =CN(s)D T (s) =D(s) are also rght coprme satsfy the polynomal matrx rght coprme factorzaton (1) The dual concluson of the above proposton s also easly to be obtaned Accordng to the above proposton n order to solve Problem 1 we need only to solve the polynomal matrx rght coprme factorzaton () If we denote D (s) =[D j (s)] t = max (deg (D j (s))) (4) 1 j r then N (s) D (s) can be rewrtten as N(s) = t 1 D(s) = = t = Substtutng () nto () we obtan N s N R n r D s D R r r t 1 N s +1 A t 1 N s = B t D s = = = Now both sdes are polynomals n s By equatng the coeffcents of the power of s on both sdes we obtan the () followng equatons: AN = BD AN N 1 = BD =1 t 1 N t 1 = BD t Now we ntroduce another lemma: Lemma Let N = 1 t 1 D = 1 t be determned by () Then they satsfy equaton () f only f D = 1 tsatsfy D [B AB A t 1 B A t B ] = (7) N = 1 t 1 satsfy [N N t 1 N t 1 ]=Q c (A B t) S (D t) (8) D 1 D t 1 D t S (D t) = (9) D t 1 D t s an upper block generalzed Hanker matrx Q c (A B t) =[B AB A t B A t 1 B ] (1) s the so-called Krylov matrx [1] Proof It s easy to see that the polynomal matrx equaton () s equvalent to the seres of matrx equatons () so we need only to show the equvalence between (7) (8) () Frstly f the seres of equatons () hold multplyng both sdes of the -th equaton by A = 1 tgves AN = BD A +1 N A N 1 = A BD =1 t 1 A t N t 1 = A t BD t Summng the above t +1equatons n both sdes yelds =BD + ABD A t 1 BD t 1 + A t BD t whch s obvous equvalent to equaton (7) Now we show equaton (8) hold The last equaton of () s equvalent to N t 1 = BD t Substtutng the above equaton nto the t-th equaton of () produces N t = AN t 1 + BD t 1 = ABD t + BD t 1 By repeatng the above process we can derve t equatons wth left sde beng N = 1 t 1 Combnng these equatons rewrtng them n matrx form gves the equaton (8) D t ()

3 B ZHOU et al / Journal of Control Theory Applcatons () Secondly f the two equatons (7) (8) hold we wll show that the seres of matrces N = 1 t 1 D = 1 t wll satsfy equaton () By drectly calculatng we have AN BD = A ( BD A t 1 ) BD t + ABD A t BD t = AN N 1 = BD =1 t 1 BD t N t 1 = Obvously the above two expressons are equvalent to () The proof s then completed Another deducton of the equaton (7) s presented n [1] But the equaton (8) the equvalence between () (7) (8) appear to be new The above lemma s the man foundaton of solvng Problem 1 n ths paper We fnd from equaton (7) that f matrces D = 1 t have been provded wth some methods then the coeffcent matrces N = 1 t 1 can be unquely determned by (8) Thus polynomal matrces D (s) N (s) can be expressed n the form of N (s) =Q c (A B t) S (D t)(l (t) I r ) (11) D (s) =[D D t ](L(t +1) I r ) L(k) =[1 s s k 1 ] T (1) In the followng we wll derve an explct expresson of rght coprme factorzaton () by usng the above lemma Frstly we ntroduce the followng so-called Pseudo- Controllablty Indces (PCI) Defnton 1 The set of PCI s any of ntegers {v }= 1 rsatsfyng the followng two condtons: 1) r v n =1 r; ) v = n =1 The above defnton s a generaton of the defnton n [17] the frst nequalty s substtuted by 1 v D = n r +1 The admssblty of any set of PCI can be defned as follows: Defnton [18] The set of PCI {v } =1 r s sad to be admssble f the followng n columns {b 1 A 1 b 1 A v1 1 b 1 b r A 1 b r A vr 1 b r } v 1 =1 r (1) form a set of lnearly ndependent columns Now we gve the man results n ths secton: Theorem 1 Let A R n n B R n r (A B) be controllable {v }=1 rbe an admssble PCI set then there exsts a group of scalars α jk such that A v k b k = r v 1 =1 j= A j b α jk k =1 r v 1 If we defne 1 j = v k β jk = α jk j v k 1k =1 r j > v k t = max(v p 1 p r) then [ t ] D(s) =[D k (s)] = s j β jk (1) j= satsfes the rght coprme factorzaton () Proof The general proof may be a lttle nvolved thus we gve an example nstead Frstly snce the n columns defned n (1) are lnearly ndependent there always exsts a group of scalars α jk (also unque) such that (14) holds We consder a system wth the followng parameter A R 7 7 B =[b 1 b b b 4 ] assume that an admssble PCI set s {v 1 v v v 4 } = { 4 1 } Accordng to Lemma the seres of matrces D = 1 t determned by the polynomal matrx rght coprme factorzaton () must satsfy the equaton (7) In fact accordng to (14) we obtan [b 1 b 4 A 4 b 1 A 4 b 4 ]D = T α 11 α 1 α 1 α 111 α 11 1 α 1 α 1 α 1 α α α 11 α 1 α α 1 α 1 α α α 11 α 1 1 α α α 14 β 4 α 4 1 α 114 v 14 α 4 α 4 By comparng (1) wth (1) we get D(s) as s 1 α 11 s 1 α 1 s 1 α 1 s 1 α 14 s = = = = D(s) = α 1 s s 4 α s α s α 4 s = = = = (17) α 1 α s α α 4 1 (14) (1)

4 1 B ZHOU et al / Journal of Control Theory Applcatons () It follows that r v det (D (s)) = s=1 + terms of lower degree n s r deg (det (D (s))) = v =7(=n) =1 Accordng to Lemma we conclude that D (s) satsfes the rght coprme factorzaton The proof s then done Remark 1 The scalars α jk determned by (14) can be obtaned by solvng the followng lnear equatons A v k b k =[b 1 A v1 1 b 1 b r A vr 1 b r ]α k k =1 r (18) α k =[α 1k α (v1 1)1k α rk α (vr 1)rk ]T Remark There s an unque factorzaton accordng to a PCI set obvously Specally f we search a PCI set by the column of the Yong-dagram ( [1] pp 4-47) then such D (s) determned by Theorem 1 wll have a specal structure whch s called Hermnte-form or upper trangular form see [1]; f we search a PCI set by the row of the Yong dagram then we obtan D (s) whch s expressed n polynomal echelon form (n ths case we call such a PCI set Controllablty Indces) respectvely See also [1] Based on ths fact we assert that the result provded n Theorem 1 s a general case provded n [1] From Theorem 1 f the system s n a sngle-nput case we have the followng corollary: Corollary 1 If r =1 (A B) s controllable then the polynomal matrx rght coprme factorzaton of nputstate transfer functon (si A) 1 B can be expressed as N(s) =Q c (A B n)s(c n)[1 s s n 1 ] T D(s) =[c c n 1 1][1 s s n ] T = c(s) (19) c = 1 n 1 are the coeffcents of the characterstc polynomal of A e c (λ) =det(λi A) =λ n + c n 1 λ n c () S (c n) s a symmetrc matrx n the form of c 1 c n 1 1 S (c n) = (1) c n 1 1 Remark Ths above corollary can be extended to the mult-nput case But n such case N (s) D(s) expressed as (19) are not rght coprme any more See [19] for detals Applcatons In ths secton we solve the followng two problems wth the help of Theorem 1 a techncal lemma Problem (Sylvester Matrx Equaton) Gven matrces A R n n B R n r a matrx J C n n n Jordan form fnd the parametrc expresson for all the matrces V C n n W C r n satsfyng the followng generalzed Sylvester matrx equaton AV + BW = VJ () The above problem s a very basc problem because the generalzed Sylvester matrx equaton () can be utlzed n many problems such as pole assgnment [] egenstructure assgnment [] robust pole assgnment [7 8] constrant control [1] observer desgn [] fault detecton [9 1] so on Problem (Egenstructure Assgnment) Gven matrces A R n n B R n r a matrx J C n n n Jordan form fnd all the matrx K R r n whch makes the Jordan form of the matrx A + BK to be dentcal wth the gven Jordan matrx J The soluton to the above Problem can be easly obtaned when a soluton to Problem s establshed Regardng the soluton to Problem we have the followng general result Lemma [] Let A R n n B R n r (A B) be controllable wth J = Blockdag(J 1 J J p ) C n n J = s 1 s 1 s C q q s =1 p are a set of self-conjugate complex numbers Further let N(s) D(s) be a par of rght coprme polynomal matrces satsfyng () then all the matrces V W satsfyng the generalzed Sylvester matrx equaton () are gven by V =[V 1 V p ] W =[W 1 W p ] V j =[v j1 v jqj ] () W j =[w j1 w jqj ] wth v j w j

5 B ZHOU et al / Journal of Control Theory Applcatons () = N (s ) f j + d N (s ) f j 1 D (s ) 1!ds D (s ) d N (s ) ( 1)!ds 1 f 1 D (s ) =1 p j =1 q (4) f j C r =1 p j =1 q are arbtrary parametrc vectors representng the degrees of freedom n the soluton Based on the above lemma we can now gve the desred solutons to Problems Theorem Let A R n n B R n r (A B) be controllable {v } =1 rbe an admssble PCI set of (A B) t= max (v p 1 p r) D(s) be determned by (1) then 1) All the matrces V W satsfyng the generalzed Sylvester matrx equaton () are gven by V = Q c (A B t) S (D t) Q o (F J t) () W = D F + D 1 FJ + + D t F t J t ( Q o (F J t) =Q c J T F T t ) s the generalzed observablty matrx F =[F 1 F p ] R r n () F j =[f j1 f jqj ] s an arbtrary parameter matrx ) All the matrx K R r n whch makes the Jordan form of A + BK dentcal wth the gven Jordan matrx J are parameterzed by K = WV 1 V W are gven by () (4) wth the parametrc matrx F satsfyng the followng constrants: C1) det[v (f j )] j =1 q =1 p; C) f j = f lj f s = s l The proof s gven n appendx Remark 4 Accordng to the second concluson of the above theorem we need matrx V to be nonsngular Accordng to the frst concluson a necessary condton for such requrement s that (A B) s controllable (J F ) s observable wth observablty ndex smaller than t +1 Remark Dfferent from the orgnal result provded by Lemma () our soluton needs only the power of matrx J whle the orgnal result needs the dfferental of N (s) D (s) So bluntly speakng our soluton s easer to deal wth than the orgnal soluton (4) 4 A numercal example We now consder a system wth the followng parameters A = B =[b 1 b b ] 1 1 = Soluton to the polynomal matrx rght coprme factorzaton of (si A) 1 B It s easy to verfy that (A B) s controllable an admssble PCI set s {v 1 v v } = { } Accordng to (1) a set of lnearly ndependent columns can be gotten as {b 1 Ab 1 b Ab A b } From (18) we have the followng three denttes A b 1 =7b 1 +Ab 1 +4b Ab + A b A b =b 1 Ab 1 +b Ab +A b b = b 1 Ab 1 b + Ab +A b It follows from Theorem 1 that 7 1 D = 4 1 D 1 = D = 1 D = 1 Substtutng (8) nto (11) produces s s 1 s N (s) = s s + s s 1 s 1 s 1 (7) (8)

6 1 B ZHOU et al / Journal of Control Theory Applcatons () s 7 s s+ D (s) = s +s 4 s s +s s +1 1 whch s a soluton to polynomal matrx rght coprme factorzaton of nput-state transfer functon (si A) 1 B 4 Soluton to the generalzed Sylvester matrx equaton egenstructure assgnment Accordng to Theorem for arbtrary Jordan matrx J C the general explct soluton to the generalzed Sylvester matrx equaton () can be expressed as D 1 D D F V =[B AB A B ] D D FJ D FJ W = D F + D 1 FJ + D FJ + D FJ F C s an arbtrary parameter matrx D = 1 are shown as (8) When specally takng matrx J as 1+j 1 1+j J = 1 j 1 1 j matrx F as F = gves 1 j +j 1+j j 1 j 1 1+j 1 V = j 1 +j 1 4j 9 j 4j 9+j 18 +j j 7j j 4 1 j 1 1j 1 1 W = 7j j 4 1 j 1 1j K = WV 1 9/1 1/ / 4/ / = / 89/ / /1 / / / / References [1] E S Armstrong Coprme factorzaton approach to robust stablzaton of control-structures nteracton evolutonary model [J] J of Gudance Control Dynamcs (): [] N F Almuthar S Bngulac On coprme factorzaton mnmal-realzaton of transfer-functon matrces usng the pseudoobservablty concept [J] Int J of Systems Scence 1994 (7): [] S Bngulac N F Almuthar Novel-approach to coprme factorzaton mnmal-realzaton of transferfuncton matrces [J] J of the Unversty of Kuwat-Scence 199 (1): 4-4 [4] K Ohsh T Myazak Y Nakamura Hgh performance ultralow speed servo system based on doubly coprme factorzaton nstantaneous speed observer [J] IEEE-ASME Trans on Mechatroncs 199 1(1): [] M Green H-nfnty controller synthess by J-lossless coprme factorzaton [J] SIAM J on Control Optmsaton 199 (): -47 [] G Duan Solutons to matrx equaton AV + BW = VF ther applcaton to egenstructure assgnment n lnear systems [J] IEEE Trans on Automatc Control 199 8(): 7-8 [7] G Duan Robust egenstructure assgnment va dynamcal compensators [J] Automatca 199 9(): [8] G Duan N K Nchols G Lu Robust pole assgnment n descrptor lnear systems va state feedback [J] European Journal of Control 8(): [9] G Duan R J Patton Robust fault detecton n lnear systems usng Luenberger observers[c]// Proc of Int Conf on Control Swansea 1998: [1] G Duan R J Patton Robust fault detecton usng Luenbergertype unknown nput obsververs-a parametrc approach [J] Int J of Systems Scence 1 (4): -4 [11] J C Baslo B Kouvartaks An algorthm for coprme matrx fracton descrpton usng Sylvester matrces [J] Lnear Algebra ts Applcatons 1997 (1): 17-1 [1] V P Bajnkant Computaton of matrx fracton descrptons of lear tme-nvarant systems [J] IEEE Trans on Automatc Control 1981 (1): [1] G Duan Rght coprme factorzaton usng system upper Hessenberg forms- the mult-nput system case [J] IEE Proc Control Theory Applcatons 1 [14] G Duan Rght coprme factorzatons for mult-nput descrptor lnear systems A smple numercally stable soluton [J] Asan J of Control 4(): [1] T Kalath Lnear Systems [D] Englewood Clffs N J: Prentce- Hall Inc 198

7 B ZHOU et al / Journal of Control Theory Applcatons () [1] M Volker H Xu An analyss of the pole assgnment problem II: the mult-nput case [J] Electronc Trans on Numercal Analyss 1997 (1): [17] S P Bngulac R V Kptolca On admssblty of Pseudoobservablty Pseudo-controllablty ndexes [J] IEEE Trans on Automatc Control 1987 (1): 9-9 [18] B Stanoje F Al-Muthar Naser On the equvalence between MFD models pseudo-observable forms of MIMO systems [J] Computers & Electrcal Engneerng Volume (): 9-14 [19] G Duan B Zhou An explct soluton to rght factorzaton wth applcaton n egenstructure assgnment [J] J of Control Theory Applcatons : 7-79 [] S P Bhattacharyya E de Souza Pole assgnment va Sylvester equaton [J] Systems & Control Letters 198 1: 1- [1] A Saber A A Stoorvogel P Sannut Control of Lnear Systems wth Regulaton Input Constrants n Seres of Communcatons Control Engneerng [M] New York: Sprng-Verlag 1997 [] J Chen R Patton H Zhang Desgn unknown nput observers robust fault detecton flters [J] Int J of Control 199 (1): 8-1 Appendx We frstly prove the frst statement of Theorem Substtutng N (s) nto (4) produces v j = Q c (A B t) S (D t) Θ (1) Θ = 4 f jc s f jc 1 s 1 f jc s f jc t s t f jc t 1s t f j 1C 1 1 s f j 1C 1 s 1 f j 1C 1 t s t f j 1C 1 t 1s t f 1C j 1 j 1 s f 1C j 1 t 1 st j 7 By lettng F =[f 1 f f j f q ] we have v j = Q c (A B t) S (D t) [(F T j) T (F T jt 1) T ] T 8! >< j!( j)! > j C j = 1 = j or j =or = >: < j T jk k = 1 t 1 are gven by C j 1 C1 1 s T j = C s T j1 = C1 s 1 T jt 1 = t 1 st j C 1 t 1s t C t 1s t 1 7 So accordng to () we have V =[v 1 v q ] F T F T = Q c (A B t) S (D t) () 4 7 F T Tj =[T 1j T j T q j ] j = 1 t 1 Notng that [T 1 T T q ]=I q = J C s C s = 4 C s 7 C s =1 p [T 11 T 1 T q 1 ]=J C1 s 1 C1 1 s C1 s 1 = C 1 1 s 4 C1 s 1 C1 1 s 7 C1 s 1 =1 p we have [T 1k T k T q k ]=J k k = 1 t; =1 p So () can be expressed as

8 14 B ZHOU et al / Journal of Control Theory Applcatons () F I q F J V = Q c (A B t) S (D t) =1 p 4 7 F J t Notng () t follows that V =[V 1 V p ] F 1I q1 F pi qp F 1J 1 F pj p = Q c (A B t) S (D t) 4 7 F 1J1 t F pjp t = Q c (A B t) S (D t) Q o (F J t) Secondly we note that W n () can be arranged as W = [D D 1 D t ]Q o (J F t +1) Then the second expresson of () can also be proved wth the above approach We thus complete the proof Bn ZHOU was born n HuBe Provnce Chna n 1981 He receved the Bachelor s degree from the Department of Control Scence Engneerng at Harbn Insttute of Technology Harbn Chna n 4 He s now a graduate student n the Center for Control Systems Gudance Technology n Harbn Insttute of Technology Hs current research nterests nclude lnear systems theory constraned control systems Emal: bnzhou@hteducn Guangren DUAN Guangren DUAN receved the BS degree n appled mathematcs the MS PhD degrees n control engneerng from Harbn Insttute of Technology Harbn Chna He s currently a Professor of Control Systems Theory at Harbn Insttute of Technology Hs research nterests nclude robust control egenstructure assgnment descrptor systems mssle autoplot desgn magnetc bearng control E-mal: grduan@hteducn