Simple Methods for Stability Analysis of Nonlinear Control Systems
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1 Poeeig of he Wol Coge o Egieeig Coe Siee 009 Vol II WCECS 009, Ooe 0-, 009, S Fio, USA Sile Meho fo Sili Ali of Nolie Cool Se R. Moek, Mee, IAENG, I. Sv, P. Pivoňk, P. Oe, M. Se A Thee eho fo ili li of olie ool e e ioe i hi oiio: eho of lieizio, Lov ie eho Poov ieio. Sie ili li of olie ool e i iffil k i egieeig ie, hee eho e e eie le. Meho of lieizio: The le ile he olie eqio hei lie oiio. Lov ie eho: The le oi Lov fio fo ll e eqio eo oe. Poov ieio: The le will llow o iel eeie he ili of he olie ii wih he fe fio G( he oliei h ifie he loe k. Ie Te Glol oi ili (GAS, Phe-le jeo, Moifie feqe eoe. I. METHOD OF LINEARIZATION Coie he olie ooo -oe e. Thi e igh e eie oe olie -oe eqio o e o fi-oe olie iffeeil eqio [] f (,,..., f (,,..., (... (,,..., f o i eqio f ( ( The olio of he e ( i give he-le jeo i he -ieiol e e. The oi of he e i whih i f ( f(... f ( e igl oi of he e ee i he eqilii oi e ee.... The i eeeio fo he lie e (, whee f ( i lie fio we wie A. O oig h e A 0 i olio. The lie ie-ivi e h eqilii oi he oigi. A olie e hve oe eqilii oi ee f hve oe olio oe igl ( 0 oi. The eqilii oi e le o le. I ee o he he-le jeo. The e le if he jeo ohe he eqilii oi e o ifii he e le if he jeo eee. Sili heo l el ole i e heo Mi eeive Jl, 009. The el eee hve ee hieve ig i of he Mii of Eio, Yoh So of he Czeh li, eeh l MSM 0009: Iellige Se i Aoio. The ho e wih Bo Uivei of Teholog, Czeh li. The e ow wih he Iie of Aoio Coe Siee, Fl of Mehil Egieeig, Tehik, 9 Bo (e-il: oek@fe.v.z, v@fe.v.z. egieeig. Sili of eqilii oi we fi o lieizio of he eqio ( i he eighohoo of eh eqilii oi he we hve o fi o ili of oge e. If he lieizio i llowle he he olie e ehve iill he lieize e i he eighohoo of eqilii oi. If we ee fio f i (i,,..., i e ( i Tlo eie i he eighohoo of eh igl oi he we wie fo hi igl oi fi fi ( i i0 ( 0... ( 0 ( o i w ( 0 J( 0 ( 0 ( whee f f f... J ( 0 : : : ( f f f... i Joi i h i efie he i of il eivive wih eil vle give igl oi. The eqio ( i e of lie iffeeil eqio h ie he oigil e (. A ee ffiie oiio of ili of e i h he heii eqio h ll he oo i he lef hlf-le. If he heii eqio h oe o oe oo i he igh hlf-le, he e i le. If he igle o lile oo e loe o he igi i, we fi o ili ig lieizio. B hi ili h w fo o lieizio i lile ol i ll eogh egio i he eighohoo of eqilii oi. Now we will olih he il lieizio of eo oe e e he eqio g f ( ( ( 0 If hi i ege wo fi-oe eqio, hooig he he vile he e vile, h i ;, he eqio ( e wie (7 g f ( ( Sigl oi of hi e we will oi olvig he e he oi o he el i [ 0 ; 0]. Whe we ioe fo oiee he ol ISBN: WCECS 009
2 Poeeig of he Wol Coge o Egieeig Coe Siee 009 Vol II WCECS 009, Ooe 0-, 009, S Fio, USA Tle I: Lieize eqio of olie e No Eqio of olie e ψ Sigl oi Lieize eqio [ 0 ; 0 ] 0 0 [ 0 ; 0 ] ; e 0 e 0 [ ; 0 ] ; 0 e 7 e e 0 8 ( ( 0 q 9 ( ( q 0 ( f ( ψ g (8 he he Joi i of hi e i (whee we hve o give fo ; he ooie of igl oi ; 0 J ( 0 (9 he followig eqio eee he lieizio of he olie eqio o he eqilii oi ( 0 ( 0 (0 0 Thee eqio oeo o he lieize eo oe ISBN: WCECS 009
3 Poeeig of he Wol Coge o Egieeig Coe Siee 009 Vol II WCECS 009, Ooe 0-, 009, S Fio, USA eqio ( ( We ie he oigi olie eqio ( hi eqio ( o fi o ili of he olie e like ili of he lie e, ol i ll eogh egio i he eighohoo of eqilii oi. Noie h he ole ee i he eqio ( i o h oe iflee ili. A log igl oi lie he oigi 0, he o i zeo he eqio ( i ( The el of hi lieizio e i he le I. Fo ele o fi o ili of he e 0, Sie io hi ; he eqio will e 0, The e h iqe eqilii oi he oigi 0 ; 0. I hi oi i [ ] ( 0, 7, ( 0, 0 [ 0;0] he lieize eqio ( i 0 The heii eqio h oo 9,7; 0,. The oo e el egive, he eqilii oi i le. The e i le i he eighohoo of hi eqilii oi. We ol fi o h he eqilii oi i ko. I le I e he ool e he eqio of he eo oe, hei igl oi he lieize eqio fo he eighohoo of eqilii oi. The le hel o fi o ieiel if he eqilii oi i le o le. The ele ove h he eqio 0, oeo o No., whee i ; ; ;, ;. Thi eqio h oe igl oi [ 0 ; 0 ] i he eighohoo of eqilii oi i vli he eqio. The le hel o olve ili of he olie eo oe e. II. LYAPUNOV DIRECT METHOD Lov eho fo ili li i i iile ve geel owefl. The jo wk, whih eiol lii i i ie, i he iffil ofe oie wih oio of he Lov fio o V-fio eqie he eho. Coie gi he olie ooo -oe e. Thi e igh e eie oe olie -oe eqio o e o fi-oe olie iffeeil eqio ( o i eqio (. The veo i he e veo, i elee e e vile. The oigi (... of he e e will e e o e eqilii olio. The Lov fio V (,,..., i l fio of he e vile i i oiive efiie. Now le V V... ( e lle iig (. If V wee o e fo o e lw egive, he el V eee oiol, he e e i he oigi of he e e, ilig oi ili. To evelo hee oe, he followig efiiio e e fo he ig of V V : - A e i gloll oill le if V i egive efiie. - I i gloll le if V i egive eiefiie. - I i le if V i oiive efiie o eiefiie. (If V i iefiie he i i o oile o eie o ili. Lov eho i ve owefl ool fo ig he ili of eqilii oi. Howeve, hee e wo wk of he eho h we hol e we of. Fi, hee i o ei eho fo fiig Lov fio fo give e. Seo, he oiio of he eho e ol ffiie; he e o ee. File of Lov fio ie o if he oiio fo ili o oi ili oe o e h he oigi i o le o oill le. We wol like o eliie he fi wk of Lov eho. The le of Lov fio fo eo-oe e i he le II fo hi-oe e i he le III. Fo ele: Coie he eo-oe e ( T kig he e vile, w e oi he e eqio ( The e h eviel eqilii oi he oigi. Uig he le we will hooe Lov fio e ( q ; ; ( V, The eivive of V log he jeoie of he e i give ISBN: WCECS 009
4 Poeeig of he Wol Coge o Egieeig Coe Siee 009 Vol II WCECS 009, Ooe 0-, 009, S Fio, USA Tle II: Lov fio fo eo-oe e No Eqio of e iio Lov fio V - q eve q o q eve q, o q, eve q, o ( q k, eve k,q, o ( ( e f V V Tle III: Lov fio fo hi-oe e No Eqio of e iio Lov fio V,>0, o V h e h e 7 8 e h z z e h e f,,e>0, o,,h eve,,e>0 h,, o,,, eve,,,>0,,,z o,,, eve,,,e>0,,,h o,,, eve e V h e V h V z e V h h h z h z,,,,e>0 V z h,,,,z o,,, eve e h h,,>0,, o,,, V ( eve,,,,e>0 f<0 f f e V,, o,,, eve V (, [ ( ] ( The fio V i egive eiefiie (ee fo he fio V i fo i eql zeo he e i gloll le. III. POPOV CRITERION The Poov ieio i oiee oe of he o oie iei fo olie e i e oe wih he Nqi ieio fo lie e []. Howeve hee e eevio h ele o he ve eee, oee eliili of he ieio. I i ee o ehie h hi ieio i elile, he oiio of i l. hol e lel eifie i ve. ISBN: WCECS 009
5 Poeeig of he Wol Coge o Egieeig Coe Siee 009 Vol II WCECS 009, Ooe 0-, 009, S Fio, USA Fig. how he e ofigio wih oe olie elee lie wih he fe fio G( h ile ll lie elee. I how he oliei h i igle-vle, ie ivi oie o hh eo oe loe k h i e o if: fo he e whe ll ole of G( e iie he lef-hlf le f ( e 0 k <, e fo he e whe G( h ole o he igi i (he o-lle iil e f ( e 0 < k < e. G ( Fig.: No-lie ool ii heii of olie elee Fo he e h G( h ole ol iie he lef-hlf -le, he i heii e zeo oh i he egiig o of he egiig. Fo he e whe G( h ole o he igi i, i o e zeo o of he egiig. A olie ii wih fe fio of he lie G( wih he olie elee (wih he ove eie oliei i gloll oill le whe i el e q (> 0 o o < 0 ei whee fo eve ω 0 he followig ieqli i olee [( j ωq G( jω ] > 0 ( k The Poov ieio e fo oe oveiee lie ghill i he G (jω-le. Le l oifie feqe eoe fio G*(jω, efie G G ( jω G( jω ( jω ω. G( jω k e e w0 eqio ke loe: k fo ω 0 ( we oi he ghil ieeio of he Poov ieio fo glol oi ili (GAS: The ffiie oiio fo GAS of olie ii i h he lo of G*(jω hol lie eiel o he igh of he Poov lie whih oe he el i -/k loe /q (q i i el e. I hi oiio le III how he ool e olie ii (wih ili eig olve. The le h ee oe fo he ii wih iffee fe fio G(.Thee i lgei olio o he ieqli ( o oiio h: 0 k < (ol fo ole G( iie he lef-hlf of -le; 0 < k < (lo fo ole G( o he igi i;, > 0; ω... fo eve vle fo 0 o ; q...i. I he le IV, he feqe eoe fio of he lie of he ii i eee i he fo iee o i hei he ieqli (, he el elio i eee fe oifiio. If he ieqli i ifie fo k, q,,, ω, he ii i qeio i gloll oill le (GAS Tle IV: Moifie feqe eoe lo N G( G*(jω o ( ( ( ( ( ( ( -/ / / -/ -/ ( ISBN: WCECS 009
6 Poeeig of he Wol Coge o Egieeig Coe Siee 009 Vol II WCECS 009, Ooe 0-, 009, S Fio, USA Tle V: l of ili li Poov ieio G(jω i ieqli ( Coiio of No G( Solio of ( [( jωq G( jω ] > 0 ili k k > 0 GAS fo eve k jω - kω q > 0 GAS fo eve k 7 8 j qk > 0 GAS fo eve k ω k( ω q ω j ω ω ( ω j ω ω ω ω ( ( ( ω kω ω > 0 ( q > 0 k < ω ω ω kω ( q j ω ω ( ( k ( ω > 0 [ q ω ] ω ω ω kω ( j ω ω ( ω > 0 GAS fo eve k GAS fo eve k o le fo i k GAS fo eve k GAS fo eve k 9 ( ω j ω ω ω ( ( GAS fo k < 0 ( ( jω... GAS fo k < 8 ( jω jω ( jω... k > 0 8 ( ( ( jω ( jω (... k < 8 8 fo eve vle k. If i i o he e, he i i oile o eeie fo whih k ii i GAS o h i i o fo k (hi llio i ille i he le. Theefoe he le will llow o iel eeie he ili of he olie ii wih he fe fio G( he oliei h ifie he loe k. Tle V ille he oifie feqe eoe lo elig ghi olio o ili. IV. CONCLUSION Sili li of olie ool e i iffil k i egieeig ie. Thi e hel o olve he ole. The eie eho ee le gh fo hee eho of ili li. Fi eho i eho of lieizio. The le ile he olie eqio ie hei lie oiio. Seo eho i Lov ie eho. The le oi Lov fio fo ll e ool ii eo hi-oe. The l eho i Poov ieio. The le will llow o iel eeie he ili of he olie ii wih he fe fio G( he oliei h ifie he loe k. REFERENCES [] R. Ie, Digil Cool Se, Sige, Beli, 989 [] W. S. Levie, The Cool Hook, CRC Pe, I., Bo Ro, Floi 99 [] I. Sv, R. Moek, How whe o e Poov ile iei fo ili, I Poeeig of he ICCC 008, Sii (Roi, 008,. -. ISBN [] I. Sv, Sili li of olie ool e ig lieizio, I Poeeig of h ICCC 00, Zkoe, ge - 0 [] I. Sv, A ew oh o gh o ili of olie ool e, Jol Elekoik, Ww, No 8-9/00, ge 8- [] N. Sioov, B. Logiov, A. Sii, M. Flleev, Lov-Shi Meho i Nolie li Aliio, Klwe Aei Plihe, 00 ISBN: WCECS 009
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