Positive minimal realizations of continuousdiscrete linear systems with transfer function with separable denominator or numerator



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Pov ro of coodcr r wh rfr fco wh pr door or ror Ł Sw Fc of crc grg ło vr of choog Pod NK rc: h pov ro pro for coo-dcr r g-p g-op SISO ford. wo pc c of h coodcr r d. Mhod d o h vr dgr for fdg pov ro of gv propr rfr fco propod. Sffc codo for h c of pov ro of gv propr rfr fco wh pr ror or rfr fco wh pr door r hd. wo procdr for copo of pov ro r propod d rd rc p. Kword: coo-dcr pov ro c copo I pov p vr d op pov p vr d op o o-gv v. p of pov o o-gv v. p of pov r: dr proc vovg chc r: dr proc vovg chc rcor h chgr d do co org rcor h chgr d do co org copr wr d ophrc copr wr d ophrc poo od. vr of od hvg pov poo od. vr of od hvg pov r hvor c fod grg r hvor c fod grg g cc cooc oc cc oog g cc cooc oc cc oog d dc c. Pov r r dfd o d dc c. Pov r r dfd o co d o o r pc. hrfor h hor of co d o o r pc. hrfor h hor of pov or copcd d dvcd. pov or copcd d dvcd. ovrvw of of h r pov hor ovrvw of of h r pov hor gv h oogrph [ ]. h ro pro gv h oogrph [ ]. h ro pro for pov dcr- d coo- for pov dcr- d coo- who d wh d w codrd [3 7]. who d wh d w codrd [3 7]. Coo-dcr D r dc Coo-dcr D r dc h corpor oh coo- d h corpor oh coo- d dcr- dc. I h p d dcr- dc. I h p d op vcor of D dpd o coo d op vcor of D dpd o coo d dcr vr. p of coo-dcr dcr vr. p of coo-dcr cd wh r wch d hr cd wh r wch d hr ro d ohr oo coror cord ro d ohr oo coror cord rooc od hghw fgh coro rooc od hghw fgh coro d g og/dg crc. Wd d g og/dg crc. Wd vr of o o D p c fod.g. vr of o o D p c fod.g. oo [8]. h pov coo-dcr D r oo [8]. h pov coo-dcr D r hv rodcd [9] pov hrd r [] d h pov frco D hrd []. Dffr hod of ov of D hrd r hv dcd [] d h oo o gr D hrd r h drvd [3]. h ro pro for pov D hrd hv ddrd [ 4 6] d h ro pro for h rfr fco wh pr door d h rfr fco wh pr ror of D h ddrd [7 8]. Pov ro pro for D w-ow []. h pro for D coodcr ch or copcd d hr o fw pco cocrg pov ro. h prd ppr focd o dg h dgr hod [ 7] prvo d o ov pov ro pro o- [3 9] o ro. I h ppr w how h h vr dgr hod c d o cop h pov ro for pc c of D rfr fco propr rfr fco wh pr door or pr ror. o h ffc codo for h c of pov ro of h gv propr rfr fco w hd d procdr for copo of pov ro for h wo c of rfr fco w propod. h ppr dvdd 3 co. I co o prr cocrg h pov coo-dcr D r d ro r rcd d h pov ro pro ford. wo pc c of coo-dcr r d co. I h co h oo o h pov ro pro for wo c of rfr fco r prd d h ffc codo for c of pov ro r hd. Cocdg rr r gv co 3. I h ppr h foowg oo w d: h of r rc w dod R d R R. h of r rc wh ogv r w dod R d R R. M h of Mr rc r rc wh ogv off-dgo r. h d r w dod I d h rpo w dod. Por o Roo r /3 333

NK.Prr d pro foro Codr coo-dcr r dcrd h qo []: whr: 5 C C D c whr [ ] Z p d M D p r r rc. p C R C odr codo for d hv h for: p Z d. No h h coo-dcr r h r rcr h Ror od [ ]. Dfo. h coo-dcr r cd r pov f p d Z for rrr odr codo Z d p Z. hor. [ ] h coo-dcr r r pov f d o f: M p C p C p D. 3 h rfr r of h gv h for: p [ C C ] D I I whr R h of r rc d wh r coffc. For h -p d p-op coo-dcr r h propr rfr r h h foowg for: 4 p 5 p p for p; [ ] { [ ] } d d r h Z-rfor d Lpc opror. whr { } Mpg h ror d door of rfr r 5 w o h rfr r h pc for g. for whch drd o drw h pc dgr [9 5 ]: 6 for p;. Dfo. h rc 3 r cd h pov ro of h rfr r f h f h q 4. h ro f h r h h ow po do og ro. h pov ro pro c d foow: gv propr ro r p fd pov d ro 3. Rr. For D h ro h o wh h r of do whr h dgr of h chrcrc poo of h [9]. h w pcd coro d orv of h D. For D gr c h rohp o r [9] d h orv d coro of h D do o pc h of ro. Rr. h ro for D h o wh h r of do whr d r h dgr of h chrcrc poo d of h [9]..Pro oo for SISO h oo o h pov ro pro w prd o wo pc c of h D rfr fco. Propod hod w d o h vr dgr [ 7 5]. wo c of h rfr fco of coodcr r w codrd. 334

Por o Roo r /3 335 vr w choo h op of gror d of d. g h vr dgr fg. w c wr h foowg dffr d dffrc qo: whr:. Sg o w o: whr for ;. Dfg vcor h for: 3 door: 7 C. h rfr fco wh pr ror: 8..C Dfg: 9 fro 9 d 7 w o:. g w drw h vr dgr how fg.. Fg.. S vr dgr for rfr fco 7 wh pr door R.. Sch ch d rc 7 prow ow C. h rfr fco wh pr door: 7 C. h rfr fco wh pr ror: 8..C Dfg: 9 fro 9 d 7 w o:. g w drw h vr dgr how fg..

NK w c wr h qo h for: [ C C ] D whr: D [ ] R..6.5 4.4.3 8 3.. 4 5 R R R R R R [ ] C R C [ ] R hrfor h coq hor h provd. hor. hr pov ro of do of rfr fco 7 f h pr door wh ogv coffc for ; d rrr ogv ror coffc for. If h po of hor r fd h pov ro 3 of 7 c fod h of h foowg procdr: Procdr. Sp. g h rfr fco 7 wr. Sp. g drw h vr dgr how Fg.. Sp 3. Choo h vr h op of gror d of d d wr qo. Sp 4. g fd h drd ro 5. p. Fd pov ro 3 of h coo-dcr wh propr rfr fco: 6 I h c d rfr fco h pr door c: d 4 8 3 3 4 6 g Procdr w o h foowg: Sp. g rfr fco 6 w c wr: 3.6.5.4.3.. 4. 7 Sp. S vr dgr h h for how fg.. Fg.. S pc dgr for rfr fco 6 for: 4 3. 6. 5. 4.3.. R.. Sch ch d rc 6 pr c: 4 3. 6. 5.4. 3.. Sp 3. g vr dgr w c wr h foowg qo: d Sg 8 o 8 w hv:.6 Sp 4. h drd ro of 6 h h for:.5.3. 4.6.4. 3. 7.7 3 3 9.6 [.9] C [.6 4.6 [ 7.7 9.6 ] ] [4] 8 8.9 9 C [] D [.6]. 336

Por o Roo r /3 337 whr:. 3 Sg 3 o 3 w o:. 4 Dfg vcor h for 3 w c wr h qo 4 h r for 4 whr: 5 Dfg vcor h for 3 w c wr h qo 4 h r for 4 whr: 5. ] [ ] [ ] [ D C C Od ro h o ogv r d do...c Dfg: fro 8 d w o:. g w drw h vr dgr how fg. 3. Fg. 3. S pc dgr for rfr fco 8 wh pr ror Fg. 3. Sch ch d rc 8 prow c Sr co. vr w choo h op of gror d of d. g vr dgr fg. 3 w c wr h foowg dffr d dffrc qo: 3

NK hrfor h coq hor h provd. hor 3. hr pov ro of do of rfr fco 8 f h pr ror wh ogv coffc for d rrr ogv door coffc for d. If h po of hor 3 r fd h pov ro 3 of 8 c fod h of h foowg procdr: Procdr. Sp. g h rfr fco 8 wr. Sp. g drw h vr dgr how fg. 3. Sp 3. Choo h vr h op of gror d of d d wr qo 4. Sp 4. g 4 fd h drd ro 5. p. Fd pov ro 3 of h coo-dcr wh propr rfr fco: 3 4 6 8 5.5.4.3.. 6 I h c d rfr fco h pr ror c: 3 4 6 8 5 6 3 4 5. g Procdr w o h foowg: Sp. g h rfr fco 6 w c wr: 3 4 5.5.4.3... 7 Sp. S vr dgr h h for how fg.4. Sp 3. g vr dgr w c wr h foowg qo: d.5.3 4 8.. 4.4 3. Sg 8 o 8 w hv: 8.4 3.5.5 5.5.8. 5.6.4 3. Sp 4. h drd ro of 6 h h for: 9.5.5 [.4] [ ].8. 5.5 [3] [.4] [ ] [3]. 5.6 C C D 3 Od ro h o ogv r d do. Prd pproch c d for rfr rc 5 of -p d -op. I MIMO c h rfr r 5 hod hv h for: D N or N D 3 whr D dgo r of coo door for ch row co of rfr r [4]. 3.Cocdg rr Fg. 4. S pc dgr for rfr fco 6 for 3 4 5. 5.4. 3.. R. 4. Sch ch d rc 6 pr c: 3 4 5.5. 4. 3.. hod for copo of pov ro of gv propr rfr fco wh pr ror d wh pr door of coodcr r h propod. Sffc codo for h c of pov ro of gv propr rfr fco hv hd. wo procdr for copo of pov ro hv propod. h ffcv of h procdr hv rd rc p. o of h codro for D coodcr r dcrd cod For- Mrch od [5] po. op pro foro of h cr d ffc codo for h c of oo of h pov ro pro for D coodcr r h gr for []. 338

cowdg h wor w ppord rop Soc Fd d Poh Govr dr chorhp No. WIM/POKL/MD/ III// of H Cp Progr. Rfrc. Fr L. Rd S. Pov Lr S [:] hor d ppco J. W Nw or.. Kcor. Pov D d D S Sprgr- Vrg Lodo. 3. Kcor. łowc M. M ro pro for pov vr r wh d I. J. pp. Mh. Cop. Sc. Vo. 4 No. 4 8 87. 4. Kcor. ro pro for pov coo- r wh rdcd r of d I. J. pp. Mh. Cop. Sc. Vo. 6 No. 3 6 35 33. 5. Kcor. Pov ro for gr dcr- wh d d d coro. Po. cd. Sc. ch. Vo. 53 No. 3 5 93 98. 6. Kcor. Ro pro for pov dcr wh d S Scc Vo. 3 No. 4 4 7 3. 7. Kcor. Ro pro for pov vr dcr- r wh d h vcor d p I. J. pp. Mh. Cop. Sc. Vo. 6 No. 6 6. 8. Gow K. Wood J. d Mdo Sg Crc d S or d Frc Lodo. 9. Kcor. Lr coro Vo. Rrch Sd Pr J. W Nw or 99.. Kcor. Pov D hrd r. Po. cd. Sc. ch. Vo. 55 No. 4 7 35 358.. Kcor. Pov frco D hrd r. Po. cd. Sc. ch. Vo. 56 No. 3 8 73 77.. Kcor. Mrcho V. Sw Ł. Sov of D hrd r - copro of h dffr hod c Mchc oc Vo. No. 8 59 66. 3. Sw Ł. Pov ro of coodcr r wh -po d -ro rfr fco c Mchc oc Pr. 4. Kcor. Ro pro for pov D hrd COMPL Vo. 7 No. 3 8 63 63. 5. Sw Ł. Kcor. Copo of pov ro of MIMO hrd r h for of cod For-Mrch od rchv of Coro Scc Vo. No. 3 53 7. 6. Sw Ł. Kcor. Copo of pov ro of gr SISO hrd r JMRIS Vo. 3 No. 4 9 8 4. 7. oo G.. Prvopoo P.N. Vrof S.J. M pc ro of fcor D rfr fco I r. o Crc. d S. Vo. 35 No. 8 988 55 58. 8. S- Kg Lv.C. Morf M. Kh. Nw r -D hor Pr II: -D -pc od-ro d h oo of coro orv d Proc. of h I Vo. 65 No. 6 977 945 96. 9. Sw Ł. Soo of D gr hrd r Kr Vo. 38 No. 7/8 9 79 9.. Ror R.. dcr -pc od for r g procg I r. o o. Cor. C- 975.. Kr J. h gr -pc od for wodo r dg I r. o o. Cor. C-3 985 6 6.. v L. Fr L. or o h pov ro pro I r. o o. Coro Vo. 49 No. 5 4 65 664. 3. Dov M. Gh I. Rogr. Głow K. Ow D.H. Coro hor for c of D coo-dcr r I. J. Coro Vo. 77 No. 9 4 847 86. 4. Vrof S.J. Prvopoo P.N. oo G.. O h -pc ro of -po d -ro -D I r. o Crc. d S. Vo. 34 No. 3 987 89 9. Wc dodch rc ch łdów cągło-drch o rc prow c ow Src: Sforłow oł pro wc dod rc d owch łdów cągłodrch. Prow oł dw prpd cgó łdów cągło-drch. Zpropoow oł od ąc chc ch wc dod rc podw rc oprorow łd. Orśoo wr wrcąc dod rc d rc oprorow prow c ow. Podo dw procdr wc dod rc órch fwość orowo prłd rc. Słow cow: cągło-dr dod rc wc Ł Sw PhD g. or o 8h Dcr 98 o. MSc crc grg rcvd J 6 o o vr of choog. h vr J 9 h dfdd h PhD h d od h PhD dgr crc grg. Crr h wh Fc of crc grg of o vr of choog. H cfc r r coro hor pc pov coo-dcr d frco w oc coro d croprocor chq. -:.w@p.d.p Por o Roo r /3 339