ELIMINATION OF FINITE EIGENVALUES OF STRONGLY SINGULAR SYSTEMS BY FEEDBACKS IN LINEAR SYSTEMS

Transcription

1 73 M>D Tadeuz azore Waraw Uiverity of Tehology, Faulty of Eletrial Egieerig Ititute of Cotrol ad Idutrial Eletroi EIMINATION OF FINITE EIENVAUES OF STONY SINUA SYSTEMS BY FEEDBACS IN INEA SYSTEMS Tadeuz azore A ew proble of dereaig of the degree of loed-loop harateriti polyoial by uitable hoie of tate feedba for trogly igular liear yte i forulated ad olved Neeary ad uffiiet oditio are etablihed uder whih it i poible to hooe tate feedba uh that the ozero loed-loop harateriti polyoial ha zero degree A proedure for oputatio of the feedba gai atrie i propoed Itrodutio Dai ha how [, ] that for igular deriptor liear yte Ex i Axi Bui, E, A, B, det E it i poible to hooe a atrix of the tate-feedba u x uh that the ozero loed-loop harateriti polyoial det[ Ez A B] ha zero degree It i eay to how that for tadard yte E I doe ot exit uh tate-feedba Mai ubjet of thi ote i to etablihed eeary ad uffiiet oditio for trogly igular liear yte uder whih it i poible to hooe tate feedba uh that the ozero loed-loop harateriti polyoial ha zero degree Thi proedure of dereaig of the degree of loed-loop harateriti polyoial by feedba will be alled the eliiatio of fiite eigevalue of atrie by feedba ie the loed-loop ha o fiite eigevalue pole Thi type of proble arie i deigig of the perfet oberver for liear tadard yte [,, ] To the bet owledge of the author thi eliiatio of fiite eigevalue of atrie by feedba i liear yte ha ot bee oidered yet Proble forulatio et be the et of real atrie ad : Coider the liear otiuou-tie yte E x Ax Bu where x x t ad u u t are the tate ad iput vetor, repetively ad E, A, B It i aued that det E, ra B, ra [ E A, for all C the field of oplex uber ad the peil E, A i ot regular, ie det [ E A] for all C 3 We are looig for a gai atrix of the tate-feedba u v x uh that det[ E A B] α 5 where α i a real uber idepedet of viv Polytehi Natioal Uiverity Ititutioal epoitory

2 7 The proble a be tated a follow ive E,A,B ad α, fid atifyig 5 We hall etablih eeary ad uffiiet oditio for the exitee of a olutio to the proble ad we hall give a proedure for oputatio of the gai atrix A o that Proble olutio If the proble a be olved a follow We a alway hooe det[ E A ] α The auptio ra B ad iplie det B ad fro A A B we obtai B [ A A] Thu, we aue tha < Theore There exit atifyig 5 if ad oly if the oditio i atified Proof Neeity Fro the equality it follow that 5 iplie Suffiiey et F i, i,, i [ E A, B ad i i,, atrix ] I [ E A B] [ E A, 6 be the ior opoed of the i, i,, i olu of the, i be the I atrix The fro the Biett Cauhy forula [3] we obtai ior opoed of the i, i,, i row of the HW> V $ Q * Q < < < 3 Q P If hold the there exit at leat oe ozero ior F,, 7, whih i idepedet of I Fro the truture of it follow that it i alway poible to hooe the etrie of o that the ior,,, i ozero ad all reaiig ior i, i,, orrepodig to ozero ior F i i,,, i are zero I thi ae fro 7 ad 5 we obtai,,,, α det[ E A B] F,, 8 Hee, it i alway poible to hooe o that α * N N N Q 9 N N N The hoie i geeral ae i ot uique To iplify the hoie of the followig proedure baed o eleetary operatio i reoeded The followig eleetary row ad olu operatio will be ued: Multipliatio of the ith row olu by alar Thi eleetary row olu operatio will be deoted by [ i ] [ i ] Additio the jth row olu ultiplied by a polyoial b b to the ith row olu Thi eleetary row olu operatio will be deoted by [ i j b] [ i j b] Q i 3 Iterhage the ith ad jth row olu Thi eleetary row olu operatio will be deoted by [ i, [ i, viv Polytehi Natioal Uiverity Ititutioal epoitory

3 75 If the oditio i atified the there exit a uiodular atrix U detu i idepedet of of eleetary olu operatio uh that E A, U [ I ] [ Fro 6 ad we have I [ E A B] [ E A, UU [ I ] ad [ E A B] det where I U, 3 To fid uig eleetary olu operatio we perfor the redutio ad we perfor I iultaeouly orrepodig eleetary row operatio of the atrix taig ito aout the followig orrepodee []: [ i ] [ i ], [ i j b] [ j i b], [ i, [ i, The etrie of are hoe o that det α If the oditio i atified the the atrix a be foud by the ue of the followig proedure Proedure Step Uig eleetary olu operatio perfor the redutio ad perforig I iultaeouly orrepodig eleetary row operatio defied by U o the atrix fid the atrix Step Chooe the etrie of o that det α Exaple Coider the yte with E, A, B Fid 3 uh that 5 i atified for α I thi ae 3 3,, ra[ E, ra E ra B ad det[ E A] for all C The oditio i atified ie ra [ E A, ra 3 for all C Hee proble ha a olutio ad uig the proedure we obtai viv Polytehi Natioal Uiverity Ititutioal epoitory

4 76 Step To redue the atrix ], [ B A E to the for ] [ I 6 we perfor the followig eleetary olu operatio ] 5 [ ], [, [,5] [3,], [,3], ], [ ], [ or equivaletly we potultiply the atrix 5 by the uiodular atrix U 7 O the atrix 3 3 I 8 we perfor iultaeouly the followig eleetary row operatio ] [5 ], [, [,5] [3,], [,3], ], [ ], [ or equivaletly we preultiply the atrix 8 by the uiodular atrix U 9 ad we obtai 3 3 I U viv Polytehi Natioal Uiverity Ititutioal epoitory

5 77 ad Step Fro we have 3 3 det We hooe, for exaple, 3, 3,, The det ad for we obtai 5 for α The deired atrix ha the for 3 3 Coludig rear A ew proble of dereaig of the degree of loed-loop harateriti polyoial by uitable hoie of tate feedba for tadard liear yte ha bee forulated ad olved Coditio have bee etablihed uder whih it i poible to hooe tate feedba for trogly igular yte uh that 5 hold It ha bee how that the proble ha a olutio if ad oly if the oditio i atified A proedure for oputatio of the atrix of ha bee propoed ad illutrated by a uerial exaple The oideratio preeted for tadard otiuou-tie liear yte are alo valid with light odifiatio for trogly igular direte-tie liear yte A exteio of thi proble for liear two-dieioal yte [, 5] i alo poible Dai Oberver for direte Sigular Syte // IEEE Tra Auto Cotr, AC-33 HEU±J±'D/6QJXODU&QWUO6\VWHPV6SUQJHU9HUODJ±%HUOQ± Toyo, ataher F Theory of Matrie, vol ad Chelea, New Yor, 959 azore T iear Cotrol Syte, vol ad, eearh Studie Pre ad J Wiley, New Yor, azore T Perfet oberver for igular D ad D liear yte // Pro Polih era Sypoiu Siee eearh Eduatio, 8 9 Sept Zieloa óra, 3ODQ±J± viv Polytehi Natioal Uiverity Ititutioal epoitory