Robust Stability Analysis of Discrete Uncertain Singularly Perturbed Time-Delay Systems

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1 Robut Stablty Aaly o Dcrete Ucerta Sgularly Perturbe Tme-Delay Sytem Shg-Ta Pa a Chg-Fa Che Departmet o Computer Scece a Iormato Egeerg, Shu-Te Uverty, Kaohug, Tawa 8, R.O.C. Departmet o Electroc Egeerg, Kao Yua Ittute o Techology, Kaohug, Tawa 8, R.O.C. Abtract I th paper, we vetgate the robut tablty problem o a cla o LTI crete ucerta gularly perturbe tme-elay ytem. I the gular perturbato parameter o thee ytem ucetly mall, they ca be eparate to two reuce-orer ubytem, the low ubytem a at ubytem. Two tablty crtera or the low a at ubytem o the ucerta ytem are the erve repectvely. The, the robut aymptotc tablty o the ull-orer ucerta ytem ubect to tructure perturbato ca be vetgate rom thoe o the ucerta low a at ubytem. Keywor: Sgularly Perturbe Sytem, Tme Delay, Robut Stablty, Structure Perturbato Notato S :et {,,, L, }, S :et {,, L, }, A :[ a, wth matrx A [ a, A B : a b or all,, wth A [ a a B [ b, ρ A :pectral rau o matrx A, A :pectral orm o matrx A, x :tate vector o low ubytem, x :tate vector o at ubytem.. Itroucto Multple tme-cale pheomea whch ext may phycal ytem, complcate the tablty aaly o the ytem. Fortuately, the gular perturbato metho prove u wth a powerul tool or the orer reucto a eparato o tme cale; ee, or example, Kokotovc et al. 986, Khall 989, Hao a Hwag 996, Su et al. 996, Sh et al. 998, a the reerece there. A mportat proceure o th metho that the multple tme-cale ytem are moele a gularly perturbe ytem. Typcal gularly perturbe ytem clue hgh-ga cotrol ytem, armature-cotrolle DC motor, tuel oe crcut, olear tme-varat RLC etwork, cotrol ytem o a arplae, etc Kokotovc et al A key to the aaly o multple tme-cale ytem le the cotructo o the reuce-orer ytem. The, the yamc o the ull-orer ytem ca be erre rom that o the reuce-orer ytem. O the other ha, the robut tablty o tme-elay ytem ha bee coere over the year becaue the tme elay commoly ecoutere may egeerg ytem. For example, the eeback ytem cotrolle by computer have tme elay ue to the executo o may umercal operato. Bee, chemcal procee, turboet ege, log tramo le, electrc etwork, hyraulc ytem, a peumatc ytem, ~ 7 ~

2 etc. all have tme elay. Ther extece requetly caue the tablty o the ytem. Moreover, mot o the yamc ytem cota ome ucertate that may are, or example, rom moelg error or learato approxmato Phooaruechaacha et al A thee ucertate are almot evtable or ay moelg o real ytem, the problem o matag the tablty o a ytem ubect to ucertate ha bee o coerato to reearcher over the year Wu a Mukam 99, Le et al Recetly, there are may relate reearch paper about aaly a cotrol eg o the two-tme-cale ytem. L a L 99 propoe a requecy-oma tablty crtero or crete two-tme-cale ytem whch the tme elay a ucerta perturbato have ot yet bee coere. Trh a Alee 995 propoe a crtero o aymptotc tablty or gularly perturbe ytem wth multple tme elay, but the elay term were treate a the perturbato o the omal ytem. Hao et al. vetgate the D-tablato problem o crete gularly perturbe ytem whch the actor o tme elay a ucertaty have ot bee o coerato. However, the actor o ucertate a tme elay o ext mot o the yamc ytem. Coequetly, t practcal to take them to coerato. It the purpoe o th paper to vetgate the robut tablty problem o ucerta crete tme-elay gularly perturbe ytem.. Problem Formulato Coer the ollowg ucerta crete tme-elay gularly perturbe ytem: x k [ A x A x k ε[ A x A x k a x k [ A x A x k ε[ A x A x k, b x R a x R. The mall potve parameter ε calle gular perturbato parameter, whch ote occur aturally ue to the preece o mall parameter varou phycal ytem. For tace, t may repreet the mache reactace or traet voltage regulator a power ytem moel, the tme cotat o the rver a the actuator a utral cotrol ytem, a t may be ue to at eutro a uclear reactor moel. The crete ytem reerre to a the C-moel Nau a Rao 985. Moreover, A, A,,,,,, are cotat matrce wth approprate meo, A, A,,,,, are the perturbato matrce wth the tructure perturbato A D, D whch D, D are cotat matrce wth potve elemet a are aume to be kow,, a potve teger. Beore the ma reult are erve, the omal ytem o the ucerta ytem rt trouce a ollow. x k A x A x k ε A x k A x k, a x k A x A x k ε A x k A x k. b Accorg to the tme-cale eparato metho Mahmou 98 a Sakea et al. 98, the low ubytem a at ubytem o ca be erve a ollow.. Slow Subytem o the Nomal Sytem I orer to obta the omal low ubytem, we replace x k h a k k, repectvely. [ [ [ [ x k by x x a The the omal crete gularly perturbe ytem ca be reuce to the ollowg low ubytem: x k [ A x A x k ε[ A x A x a ~ 7 ~

3 x [ A x A x k ε A x A x [. b From, we obta the low ubytem a x k Mx M x k, a M A ε A A I ε A A A M A A A I ε A A A. ε Let B A A, b S. 5 Takg the -traorm o the low ubytem, we have h I M M x [ I K x G x K,6a ε A A I ε A A B, B 6b G [ I K. 6c I the low ubytem aymptotcally table, the et[ G >,.. Fat Subytem o the Nomal Sytem Durg the at traet pero, the qua-teay tate x a x are aume to be cotat. Thu, we may approxmate x k h a x k by x a x, repectvely. Moreover, by vew o the omal ytem a the low ubytem, t ca be ee that x k approache x k a ε approache ero. Thereore, ε ucetly mall, t reaoable to approxmate x k by x k,.e., x x. Accorg to the above cuo, we are poto to erve the at ubytem o the omal ytem. Deote x a the at compoet o x. We have x x x a x k x k x k x k x. 7 Accorg to b a b, we obta the at ubytem rom 7 a x k ε [ A x A x k. 8 Takg the -traorm o the at ubytem 8, we get I ε [ A A x G x, 9a G [ I ε B. 9b I the at ubytem 8 aymptotcally table, the et[ G >,. I the ollowg ervato, we aume that the omal low ubytem a at ubytem 8 are both aymptotcally table.. The Slow a Fat Subytem o The Ucerta Sytem I th ecto, the tablty o the low a at ubytem o the ucerta ytem exame. Some ueul lemma or urther ervato are trouce a ollow. Deto : The multple tme-elay ytem x k Ψ x k aymptotcally table a oly et I Ψ >,, Ψ [ Ψ. Lemma Hao et al. : For ay m m matrx A R, ρ A <, the et I ± A >. Lemma Su a Shyr 99: The ero tate o the ytem x k M x k aymptotcally table a oly et I M >,, [ ~ 7 ~

4 M M Lemma Chou 99: For ay matrce A, B, a C, B < C, the a AB ρ A B ρ A C ρ.. m m b ρ A B ρ A B ρ A B ρ A C.. Slow Subytem o the Ucerta Sytem Pror to vetgate the tablty o the ucerta low ubytem, we trouce a lemma. Lemma : I ε D D η < hol, the we have I ε A A ε, ξ a ξ η [ I ε D D η b wth η I ε A A. c Proo: From c, we have I ε A A ε η [ ε η I. Sce ε η ε D D η <, we obta [ I ε η ε I η [ ε η [ ε L Hece we have I ε η [ η ε D D [ η I. Thu, the equalty a hol vew o a. Th complete the proo. Let B A A, S. 5 Smlarly to the ervato o 6a a accorg to, we get G x, 6a G { I [ B ε [ A A [ I ε A ε A B } 6b It obvou that the ucerta low ubytem 6 aymptotcally table the ollowg equalty hol: et[ G >,. 7 Theorem : I the omal low ubytem aymptotcally table,.e., et[ G >,, the the ucerta low ubytem 6 aymptotcally table the ollowg equalty hol: ρ[ Ξ β G β <, β. 8a Ξ β { ε A A D D. ξ.[ ε D D ε D D D D Proo: Sce I ε A }.[ η. B β η. D D β ε A A. η. D D [ ε A η I ε A A ε ε η β β. 8b accorg to 6c, 6b ca be rewrtte a G [ I K K [ G K, 9a K ε A A η { ε[ A A ε A ε [ I ε A A [ ε }. η.[ B, 9b ~ 75 ~

5 Coequetly, we have et G [ et [ I K G et G [. Sce et[ G >,, the examato o 7 equvalet to the vercato o the ollowg equalty: et I K G >,. [ Let β. The ca be rewrtte a et[ I K β G β >, β. Bae o 9b a Lemma, K β ate K β Ξ β, β. By Lemma, t yel ρ [ K β G β ρ Ξ β β [ G. The, accorg to Lemma, hol the equalty 8a ate. Hece, the ucerta low ubytem 6 aymptotcally table vew o 7 a Deto. Th complete the proo.. Fat Subytem o the Ucerta Sytem Smlarly to the ervato o 9 a accorg to, we have G x, a G [ I ε B. b Thereore, the ucerta at ubytem aymptotcally table, the ollowg equalty hol: et[ G >,. Theorem : I the omal at ubytem et[ G >, aymptotcally table,.e.,, the the ucerta at ubytem aymptotcally table the ollowg equalty hol: ρ{[ ε D D β. G β } <, β. 5 Proo: Makg ue o 9b a 5, G ate et G [ [ I ε G [ G et et. Sce et[ G >,, the ollowg equalty: et[ I ε G >,, et{ I ε β G β },.e. [ > β, 6 mple. Accorg to Lemma, t ca be ealy how that ρ ε β β G {[ } ρ {[ ε } D β G β D. By Lemma, 5 hol, the the equalty 6 ate a hece the ucerta at ubytem aymptotcally table vew o a Deto. Th complete the proo.. Stablty Crtera o The Orgal Sytem I th ecto, we wll cu the tablty o the orgal ucerta ytem. The ollowg theorem how that the tablty property o the orgal ytem ca be erre rom that o t reuce-orer moel,.e. the low a at ubytem. Theorem : Hao et al., 997 I the low a at ubytem 6 a, repectvely, are both aymptotcally table, the the orgal ytem alo aymptotcally table or ucetly mall ε. 5. Cocluo The robut aymptotcally tablty o ucerta crete gularly perturbe ytem wth tme elay ha bee coere th paper. The reuce-orer moel,.e. the low a at ubytem, o the ytem are rt erve. Uer the coto that the low a at ubytem o the omal ytem are both aymptotcally table, we have propoe a requecy-oma robut elay-epeet tablty crtero uch that the low a at ubytem o the orgal ucerta ytem ~ 76 ~

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