Linear Perturbation Bounds of the Continuous-Time LMI-Based H Quadratic Stability Problem for Descriptor Systems

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Peregrine Rudolf Woods
5 years ago
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1 UGRN DE OF ENE ERNE ND NFORON EHNOOGE Volu No 4 ofa a ubao ouds of h ouous- -asd H uadac ably obl fo Dscpo yss dy ochv chcal Uvsy of ofa Faculy of uoacs Dpa of yss ad ool 756 ofa Eal ayochv@u-sofa.bg bsac hs pap w ps a appoach o oba la pubao bouds fo h couous- la a quals basd H quadac sably pobl fo dscpo syss. Usg h appoach poposd w a abl o oba gh la pubao bouds fo h s soluos o h H quadac sably pobl fo dscpo syss. s also show how h sas of h dvdual codo ubs of h cosdd s ca b calculad. ucal apl s psd as wll. Kywods Dscpo syss a pubao bouds H uadac ably obl basd syhss a syss.. oduco a a quals s a wdly usd o solv ffcly ay fudaal pobls cool hoy H syhss h la quadac gulao pobl quadac sably pobl boudd gy pobl c. [ 6 7] ad h lau h. dsg s valuabl paccal applcabl ad usful haks o h sc of ffc cov opzao algohs [3] ad sofwa [4] plus h packag alp ad Du solv [5]. Dscpo syss sos also fd o as Dffal-lgbac DE o sgula syss dscb a boad class of syss whch a o oly of hocal s bu also hav ga paccal sgfcac. cosdabl aou of suds cocg la dscpo syss has b cad ou []. h ssus of coollably obsvably sably syhss of a sa fdback hav alady b cosdd []. 3

2 hs pap w popos a appoach o oba la pubao bouds of h basd H quadac sably pobl va oducg a suabl gh had pa h cosdd a quals. f h cosdd pobl s solvd h suls obad ca b appld h followg ways. Fs s possbl o sa h os h calculad soluo of h H quadac sably pobl whch a du o oudg os ad paac dsubacs h cosdd daa. cod s possbl o sudy h obus sably ad obus pfoac of h closd loop sys wh ucas h pla ad h cooll. h ucas h cooll appa bcaus of h ssvy of h H quadac sably pobl. Fuh h followg oao s usd R s h spac of al acs; R R ; h dy a; h u vco; h aspos of ; h psudo vs of ; σ a h spcal o of wh σ a s h au sgula valu of ; vc R h colu-ws vco psao of R ; R h vcpuao a such ha vc vc ; h Kok poduc of h acs ad. h oao sads fo qual by dfo. h s of h pap s sucud as follows. co pss h pobl s up ad objcv. co 3 vals h pfod la pubao aalyss of h -basd couous H quadac sably pobl fo dscpo syss. co 4 a ucal apl s gv. d fally co 5 w coclud wh so fal aks.. obl sup ad objcv a couous- dscpo syss a gally dscbd by h followg s of dffal-algbac quaos E& u y wh R u R ad R a h sys dscpo sa pu ad al codos ad ad E a cosa acs of copabl sz. Dfo. ys quvalc. wo syss E ad E a sad o b sys quval dod by E E f h s osgula asfoao acs R R such ha h quaos E ER R R hold u. Dfo. Rgulay. h sys s gula f h polyoal d se sasfs d se. 3

3 33 Dfo.3 Wsass oal fo. Fo ay gula sys h s wo o-sgula acs R R such ha by R R h followg dcoposd psao ca b obad. u N u & & Dfo.4 d of lpoc. h d of lpoc ν.. { } q N q ν s sad o b a d of a la dscpo sys. yss wh ν a calld hgh d DE syss. h dscpo sys has a soluo fo ay al codo ad suffcly sooh pu u. s possbl ha h soluo gh show pulsv bhavo. ha s why cosd h sys Wsass oal fo ud suffcly sooh pu sag fo a al codo. h h sa voluo ca b dscbd accodg o [] 3. ν ν τ δ τ u N N d u h psso 3 fo sa voluo pls ha d o dscpo syss ν ad N wll hav o pulsv soluos. hs cas h sys s calld puls f ad d o. osd h la couous- dscpo sys wh h s o dc lao bw h pu ad h oupu sgal. houghou h pap w assu h dscpo sys s a d o sys. h ss a quval sys [ ] E Wsass caocal fo wh R s a sabl a. h asfod sys s gv as 4. y u & h asfod sys 4 s obad usg h psso 3b fo h sa voluo. W cosd a appoach o solv h H quadac sably pobl fo dscpo syss as sad []. Fo d o dscpo syss w a sd h soluo of h followg sys of s 5 >.

4 34 hs s a Egvalu obl EV wh spc o h vaabls ad. H w assu ha h opal closd-loop pfoac op of h sys 4 s alady obad. od o achv quadac H sably ad o su closd-loop pfoac s cssay o dsg a sa-fdback cool uk. Fo chu copl agu [8] h abov qualy s quval o 6 > K K wh spc o h vaabls K ad. s obvous ha h qualy 6 s o a wh spc o h dcso vaabls K ad. ha s why w pfo h subsuo > ad K o oba h followg sys of s 7. > h pap s ad a obag la pubao bouds of h sys 7 a h opal valu of dd o solv h H quadac sably pobl fo d o dscpo syss. uppos ha h acs a subjc o pubaos ad assu ha hy do o chag h sg of h sys a pubao bouds calculao W cay ou ssvy aalyss of h 7 fo h d o dscpo sys gv Wsass oal fo 8 wh. h ffc of h pubaos ad o h pubd soluos ad has o b sudd. H ad a h oal soluo of h qualy 8 ad h pubaos spcvly. f oducg a slghly pubd suabl gh had pa w ca oba

5 35 9 op op op op wh ad s calculad usg h so calld oal. op op h a foao s ak o accou wh spc o h daa ad closd-loop pfoac pubaos h oudg os ad h ssvy of h o po hod ha s usd o solv h cosdd s. pplyg psso h pubd quao 9 ca b w h followg way wh. op op λ λ Du o h fac ha w hav o calcula la pubao bouds h h s of scod ad hgh od a glcd. hus w ps h psso a vcozd fo vc vc vc wh [ ] vc vc q

6 36 vc [ ]. vc vc vc vc op f pfog h ahacal asfoaos w oba 3. vc vc vc vc vc op q Fally h lav pubao boud fo h soluo of h 7 s obad 4 op op vc vc vc vc vc vc vc vc vc vc vc vc vc q wh op vc vc vc vc vc vc vc vc vc vc vc vc vc vc vc vc vc

7 37 a h sas of h dvdual lav codo ubs of 7 wh spc o h pubaos ad. W apply a pocdu lk psd od o copu h lav pubao bouds fo h soluo of h wh. Du o h fac ha w hav o calcula la pubao bouds h h s of scod ad hgh od a glcd. h lao 5 a vcozd fo wll look h followg way 6 vc vc vc wh [ ] vc vc y vc

8 vc vc vc [ 3 4 5]. vc op hus o calcula h la pubao bouds h followg lao s usd 7 y vc vc 3vc 4vc 5 op vc. low h lav pubao boud fo h soluo of h 7 s show y vc vc 8 vc h vc vc vc vc vc vc vc vc vc vc vc vc vc 5 vc vc 3 vc vc op vc 5 op vc vc vc vc vc vc vc vc 5 op vc a h h sas of h dvdual lav codo ubs of 7 wh spc o h pubaos ad. 4. Nucal apl [] osd h couous- d o dscpo sys gv Wsass oal fo.. E 38

9 [ ]. c w would lk o calcula h la bouds h pubaos h sys acs a chos such a way as o la h scod ad hgh od s h dvao pocdu.. op op fo h pubd soluos ad a calculad applyg h hod psd [9] ad usg h sofwa [4]. ayg ou h poposd appoach h la lav pubao bouds fo h soluos ad of h sys 7 a calculad usg pssos 4 ad 8 spcvly. W hav cosdd dff sz of pubaos calculad h la pubao bouds ad h obad suls a show h followg abl abl q vc oud 4 y vc oud asd o h poposd soluo appoach o pfo ssvy aalyss of h couous- basd H quadac sably pobl fo dscpo syss w oba h pubao bouds 4 ad 8. hs bouds a clos o h al q y lav pubao bouds ad hs as ha hy vc vc a good ss ha hy a gh. 39

10 5. ocluso hs pap w popos a appoach o copu h la pubao bouds of h couous- basd H quadac sably pobl fo dscpo syss. W also show how h sas of h dvdual codo ubs of h s cosdd ca b calculad. gh la pubao bouds a obad fo h a quals dg h pobl soluo. asd o ahacal dvaos w hav obad hocal suls ha appld o a ucal apl show h poac of h poposd soluo appoach o aalyz h ssvy of h basd H quadac sably pobl fo dscpo syss. R f c s. o y d.. E l. G h a o u F. F o V. a l a k s h a. a a quals yss ad ool hoy. hladlpha o c h v.. K o s a o v. k o v. a a quals ool hoy. ofa Da 5. N ulgaa. 3. N s o o v.. G a h. o-o olyoal lgohs ov ogag. hladlpha G a h.. N o v s k. a u b. h l a l. ool oolbo fo Us wh. h ahwoks c.. 5. a u c l l D. D. H o. a b K. a z. Us s Gud fo Du fac.4. -NR. 6. D o y l J.. K. G l o v.. K h a g o k a.. F a c s. a-pac oluos o adad H ad H ool obls. EEE asacos o uoac ool s o. R.. D. O. d s o E.. J o k h. Fs cpls oluo o h No-gula H ool obl.. J. Robus Nola ool D u l l u d G. E. F. a g a. ous Robus ool hoy. Nw ok pg- Vlag. 9. G a h.. p k a a. a a qualy ppoach o H ool.. J. Robus ad Nola ool D a. gula ool yss. l pg-vlag R h. ool of Dscpo yss ppoach. hd Dssao. 4

k of the incident wave) will be greater t is too small to satisfy the required kinematics boundary condition, (19)

k of the incident wave) will be greater t is too small to satisfy the required kinematics boundary condition, (19) TOTAL INTRNAL RFLTION Kmacs pops Sc h vcos a coplaa, l s cosd h cd pla cocds wh h X pla; hc 0. y y y osd h cas whch h lgh s cd fom h mdum of hgh dx of faco >. Fo cd agls ga ha h ccal agl s 1 ( /, h hooal