ANALYSIS OF POSITIVITY AND STABILITY OF DISCRETE-TIME AND CONTINUOUS-TIME NONLINEAR SYSTEMS

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1 Vol. 5, No. 1, 15 COMPUTATIONAL PROBLEMS OF ELECTRICAL ENGINEERING ANALYSIS OF POSITIVITY AND STABILITY OF DISCRETE-TIME AND CONTINUOUS-TIME NONLINEAR SYSTEMS Kaczorek T., 15 Bałystok Uversty of Techology, Bałystok, Polad Abstract: The ostvty ad asytotc stablty of dscrete-te ad cotuous-te olear systes are addressed. Suffcet codtos for the ostvty ad asytotc stablty of the olear systes are establshed. The roosed stablty tests are based o a exteso of the Lyauov ethod to the ostve olear systes. The effectveess of the tests are deostrated o exales. Key words: ostve, dscrete-te, asytotc stablty, olear, Lyauov ethod. 1. Itroducto A dyacal syste s called ostve f ts trajectory startg fro ay oegatve-tal-codto state reas forever the ostve orthat for all oegatve uts. A overvew of state of the art the ostve syste theory was gve oograhs [8, 9] ad aers [15 18]. Systes havg ostve behavor ca be foud egeerg, ecoocs, socal sceces, bology ad edce, etc. The Layuov, Bohl ad Perro exoets ad stablty of te-varyg dscrete-te lear systes were corehesvely vestgated [1 7]. Furtherore, ostve stadard ad descrtor systes ad ther stablty were also aalyzed [9, 15 19]. Postve lear systes of dfferet fractoal orders [16, ] ad descrtor dscrete-te lear systes [17] were addressed by the author revous ublcatos. Descrtor ostve dscrete-te ad cotuous-te olear systes [1, 13] were lkewse aalyzed as well as the ostvty ad learzato of olear dscrete-te systes by state-feedbacks were vestgated [15]. The roble of u eergy cotrol of ostve lear systes was adequately addressed [11, 1, 14]. The stablty ad robust stablzato of dscrete-te swtched systes were aalyzed [1, ]. I ths aer, the ostvty ad asytotc stablty of dscrete-te ad cotuous-te olear systes wll be vestgated. The aer s orgazed as follows. I the secto, the deftos ad theores cocerg the ostvty ad stablty of ostve dscrete-te ad cotuouste lear systes are recalled. Necessary ad suffcet codtos for the ostvty of dscrete-te olear systes are establshed the secto 3. The asytotc stablty of ostve olear systes s addressed the secto 4, wth codtos for ther stablty beg roosed. The codtos for the ostvty of cotuouste olear systes are gve the secto 5, ad those for the stablty of cotuous-te ostve olear systes are reseted the secto 6. Cocludg rearks are gve the secto 7. The followg otatos wll be used: R deotes the set of real ubers, R reresets the set of real atrces, R stads for the set of atrces 1 wth oegatve etres ad R =R, Z s the set of oegatve tegers, M reresets the set of Metzler atrces (wth oegatve off-dagoal etres), detty atrx, I stads for the detty atrx.. Postve dscrete-te ad cotuous-te lear systes ad ther stablty Cosder a dscrete-te lear syste x 1 = Ax Bu, Z = {,1,...} (.1a) x R, y = Cx Du (.1b) u R, y R are the state, ut ad outut vectors, resectvely, ad A R, B R, C R, D R. Defto.1. [8, 9] A dscrete-te lear syste (.1) s called (terally) ostve f x x R, y R, Z for ay tal codtos R ad all uts u R, Z. Theore.1. [8, 9] A dscrete-te lear syste (.1) s ostve f ad oly f A R, B R, C R, D R.(.) Defto.. [8, 9] A ostve dscrete-te lear syste (.1) s called asytotcally stable f l x = for ay x R. (.3) Lvv Polytechc Natoal Uversty Isttutoal Reostory htt://ea.l.edu.ua

2 1 Theore.. A ostve dscrete-te lear syste (.1) s asytotcally stable f ad oly f oe of the followg equvalet codtos s satsfed: 1) All coeffcets of the olyoal ( z) = det[ I( z 1) A] = (.4) 1 = z a z... az a 1 1 are ostve,.e. a > for =,1,..., 1. ) All rcal ors of the atrx A= I A= [ a ] are ostve,.e. M j = a >, 1 11 a11 a1 M = >,..., M = det A> a a 1 (.5) The roof was gve [9]. Let us cosder a cotuous-te lear syste x& = Ax Bu, (.6a) y = Cx Du (.6b) x= xt () R, u = ut () R, y = yt () are the state, ut ad outut vectors ad R A R, B R, C R, D R. Defto.3. [8, 9] A cotuous-te lear syste (.6) s called (terally) ostve f y R, t for ay tal codtos x x R, R ad all uts u R, t. Theore.3. [8, 9] A cotuous-te lear syste (.6) s ostve f ad oly f A M, B R, C R, D R. (.7) Defto.4. [8, 9] A ostve cotuous-te lear syste (.6) s called asytotcally stable f l x = for ay x R. (.8) t Theore.4. A ostve cotuous-te lear syste (.6) s asytotcally stable f ad oly f oe of the followg equvalet codtos s satsfed: 1) All coeffcets of the olyoal () s = det[ Is A] = (.9) 1 = s aˆ s... as ˆ aˆ are ostve,.e. ˆ k 1 1 a > for k =,1,..., 1. ) All rcal ors of the atrx A ˆ = A= [ aˆ j ] are ostve,.e. Mˆ = aˆ >, 1 11 ˆ11 ˆ ˆ a a1 M ˆ ˆ = >,..., M = det A> aˆ aˆ 1 The roof was gve [9]. (.1) 3. Postvty of dscrete-te olear systes Followg revously set reasog [18], let us cosder a dscrete-te olear syste x 1 = Ax f( x, u), Z = {,1,...}, (3.1a) x R, u R, y = gx (, u), (3.1b) y R, Z are the state, ut ad outut vectors, resectvely; f( x, u ) R, gx (, u ) R are cotuous vector fuctos of x ad u satsfyg the codtos f (,) =, g (,) = ad A R. Defto 3.1. A dscrete-te olear syste (3.1) s called (terally) ostve f x R, y ay tal codtos x R ad all uts R, Z for u R. Theore 3.1. A dscrete-te olear syste (3.1) s ostve f ad oly f A R ad f( x, u ) R, gx (, u ) R for all x R ad u R, Z. (3.) Proof. Suffcecy. Fro (3.1) for = we have x = Ax f( x, u ) R,, (3.3) y gx u 1 = (, ) R sce (3.) holds ad x R, u R. Slarly, for = 1 we obta x = Ax f( x, u ) R,, (3.4) y = gx ( 1, u1) R sce (3.) ad (3.3 ) holds. Reeatg the rocedure for =,3, we obta x y R ad R for Z ; therefore, by Defto 3.1 the syste s ostve. Necessty. Assug that the syste (3.1) s ostve, we shall show that (3.) holds. Fro (3.3) for f( x, u ) = we have x1 = Ax ad ths les that A R sce by assuto 1 R x ad, addtoally, x R ca be arbtrary. I other case, f Ax =, the fro (3.3) we have x1 = f( x, u) ad ths les that f( x, u ) R sce by assuto x 1 R. Fro (3.3) we also have that y = gx (, u) ad x R, u R sce by assuto y R. Cotug the rocedure, we ca show that (3.) holds f the syste s ostve. Lvv Polytechc Natoal Uversty Isttutoal Reostory htt://ea.l.edu.ua

3 Aalyss of Postvty ad Stablty of Dscrete-Te 13 Fro Theore 3.1 we have the followg: Corollary 3.1. A dscrete-te olear syste (3.1) s ostve oly f the lear syste s ostve. x 1 = Ax, Z = {,1,...} (3.5) Exale 3.1. Let us cosder the followg dscretete olear syste (3.1) wth x1,..1 x =, A, x =,.3.4 x1, x, e f( x, u) =, x, 1 e cos (3.6) x1,.1e gx (, u) =. x, cos As follows fro (3.6), the atrx A has oegatve etres ad the vector fuctos f( x, u), gx (, u ) are also oegatve for all x R adu R, Z. Therefore, by Theore 3.1, the syste s ostve. The lear art of the syste s also asytotcally stable sce the coeffcets of the olyoal z.8.1 det[ I( z 1) A] = =.3 z.6 = z 1.4z.45 are ostve,.e. a =.45, a 1 = 1.4. (3.7) The sae result follows fro the codto of Theore. sce A I A = = ad M 1 =.8, M = deta=.45. (3.8) 4. Stablty of ostve dscrete-te olear systes Cosder a ostve dscrete-te olear syste x x 1 = Ax f( x), x R, (4.1) R, A R, f( x ) R s a cotuous ad bouded vector fucto. Defto 4.1. A ostve dscrete-te olear syste (4.1) s called asytotcally stable the rego D R f x R, Z ad l x = for ay fte x D R. (4.) To test the asytotc stablty of the ostve syste (4.1), the Lyauov ethod s used. As a caddate of Lyauov fucto we choose T V( x ) = c x > for x R (4.3) c R s a vector wth strctly ostve cooets c k > for k = 1,...,. Usg (4.3) ad (4.1), we obta V( x ) = V( x ) V( x ) = 1 T T T 1 = c x c x = c {[ A I ] x f( x )} < for sce [ I Ax ] f( x) <, x D c R s a strctly ostve vector. (4.4) R (4.5) Therefore, the followg theore has bee roved. Theore 4.1. A ostve dscrete-te olear syste (4.1) s asytotcally stable the rego D R f the codto (4.5) s satsfed. Exale 4.1. Let us cosder the followg olear syste (4.1) wth ad x1,.1. x1, x, x =, A, f( x). x = =,..3 (4.6) x, The olear syste s ostve sce A R f( x ) R for all x1, ad x,, Z. I ths case, the codto (4.5) s satsfed the rego D defed by x.3..1 D: = { x1,, x, } = [ I Ax ] f( x) =.9x1,.x, x1, x,. (4.7) = R.7x,.x1, x, The rego D s show Fg x 1 Fg Stablty rego (sde the curved le). Lvv Polytechc Natoal Uversty Isttutoal Reostory htt://ea.l.edu.ua

4 14 By Theore 4.1, the ostve olear syste (4.1) wth (4.6) s asytotcally stable the rego (4.7). 5. Postvty of cotuous-te olear syste Cosder a cotuous-te lear syste x& = Ax f( xu, ), (5.1a) y = gxu (, ) (5.1b) x= xt () R, u = ut () R, y = yt () R are the state, ut ad outut vectors, resectvely; A R ; f( xu, ) ad gxu (, ) are cotuous ad bouded vector fuctos of x ad u, resectvely, satsfyg f (,) = ad g (,) =. Defto 5.1. [8, 9] A cotuous-te lear syste (5.1) s called (terally) ostve f y R (for t ) for ay tal codtos x R, x R ad all uts u R, t. Theore 5.1. [8, 9] A cotuous-te lear syste (5.1) s ostve f ad oly f A M, f( xu, ) R, gxu (, ) R for all x R, u R, t. (5.) Proof. The soluto to the equato (5.1a) for a gve A ad f( xu, ) has the for t xt () =Φ () tx Φ( t τ) f[ x( τ), u( τ)] dτ (5.3) At Φ () t = e. (5.4) Usg the Pcard ethod, we obta fro (5.3a) the followg: t k 1 =Φ Φ x () t () tx ( t τ) f[ xk( τ), u( τ)] dτ, k = 1,,... (5.5) As follows fro (5.4), f the codtos (5.) are satsfed, the xk () t R (for t, k = 1,,... ) sce for A M the cluso holds Φ() t R (for t ) [9]. Fro (5.1b) we have y assuto (5.) gxu (, ) R for t. R (for t ) sce by the x R, u R, 6. Stablty of cotuous-te olear systes Cosder a ostve cotuous-te olear syste x& = Ax f( x), (6.1) x= xt () R, A M, f( x ) R s a cotuous ad bouded vector fucto ad f () =. Defto 6.1. A ostve cotuous-te olear syste (6.1) s called asytotcally stable the rego D R f xt () R, t ad l xt () = for ay fte x D R. (6.) t To test the asytotc stablty of the ostve syste (6.1), the Lyauov ethod s used. As a caddate of Lyauov fucto we choose T V( x) = c x> for x= xt () R, t (6.3) c R s a vector wth strctly ostve cooets c k > for k = 1,...,. for sce Usg (6.3) ad (6.1), we obta T T V& ( x) = c x& = c [ Ax f( x)] < (6.4) Ax f( x) < for x D R, t (6.5) c R s the strctly ostve vector. Therefore, the followg theore has bee roved. Theore 6.1. A ostve cotuous-te olear syste (6.1) s asytotcally stable the rego D R f the codto (6.5) s satsfed. Exale 6.1. Let us cosder the followg olear syste (6.1) wth x1 1 xx 1 x=, A, f( x). x = = 1 3 x (6.6) The olear syste (6.1) wth (6.6) s ostve sce A M ad f( x ) R for all x R, t. I ths case, the codto (6.4) s satsfed the rego D defed by x1 x xx 1 D: = { x1, x} = < x. (6.7) 1 3x x Fro (6.7) we have x1( x) > x > ad x1 < (3 x) x. (6.8) Lvv Polytechc Natoal Uversty Isttutoal Reostory htt://ea.l.edu.ua

5 Aalyss of Postvty ad Stablty of Dscrete-Te 15 The rego D s show Fg x x 1 Fg Stablty rego (sde the curved le). By Theore 6.1, the ostve olear syste (6.1) wth (6.6) s asytotcally stable the rego (6.7). 7. Cocludg rearks The ostvty ad asytotc stablty of the dscretete ad cotuous-te olear systes have bee addressed. The ecessary ad suffcet codtos for the ostvty of the dscrete-te olear systes have bee establshed (Theore 3.1). Usg the Lyauov drect ethod, the suffcet codtos for asytotc stablty of the dscrete-te olear systes have bee roosed (Theore 4.1). The effectveess of the codtos has bee deostrated o Exale 4.1. The suffcet codtos for the ostvty of cotuous-te olear systes have bee establshed secto 5 (Theore 5.1) ad for the asytotc stablty secto 6 (Theore 6.1). The stablty codtos for cotuous-te olear systes are llustrated o Exale 6.1. The cosderatos ca be exteded to fractoal dscrete-te olear systes. A oe roble s a exteso of the codtos to the descrtor fractoal dscrete-te ad cotuous-te olear systes. Ackowledget Ths work has bee suorted by Natoal Scece Cetre Polad uder work No. 14/13/B/ST7/3467. Refereces [1] A. Czork, Perturbato Theory for Lyauov Exoets of Dscrete Lear Systes. Krakow, Polad: Uversty of Scece ad Techology Press, 1. [] A. Czork, A. Newrat, M. Nezabtowsk, ad A. Szyda, O the Lyauov ad Bohl exoet of te-varyg dscrete lear systes, Proc. th Medterraea Cof. o Cotrol ad Autoato, , Barceloa, Sa, 1. [3] A. Czork ad M. Nezabtowsk, Lyauov Exoets for Systes wth Ubouded Coeffcets, Dyacal Systes: a Iteratoal Joural, vol. 8, o., , 13. [4] A. Czork, A. Newrat, ad M. Nezabtowsk, O the Lyauov exoets of a class of the secod order dscrete te lear systes wth bouded erturbatos, Dyacal Systes: a Iteratoal Joural, vol. 8, o. 4, , 13. [5] A. Czork ad M. Nezabtowsk, O the stablty of dscrete te-varyg lear systes, Nolear Aalyss: Hybrd Systes, vol. 9,. 7 41, 13. [6] Czork A., Nezabtowsk M., O the stablty of Lyauov exoets of dscrete lear syste, Proc. Euroea Cotrol Cof.,. 1 13, Zurch, Swtzerlad, 13. [7] A. Czork, J. Klaka, ad M. Nezabtowsk, O the set of Perro exoets of dscrete lear systes, Proc. World Cogress of the 19 th Iteratoal Federato of Autoatc Cotrol, , Cae Tow, South Afrca, 14. [8] L. Fara ad S. Rald, Postve Lear Systes; Theory ad Alcatos, New York, USA: J. Wley,. [9] T. Kaczorek, Postve 1D ad D systes, Srger Verlag, Lodo, UK, 1. [1] T. Kaczorek, Descrtor stadard ad ostve dscrete-te olear systes, Autoatyzacja Procesów Dyskretych, vol. 1, , 14. [11] T. Kaczorek, Mu eergy cotrol of descrtor ostve dscrete-te lear systes, COMPEL, vol. 33, o. 3, , 14. [1] T. Kaczorek, Mu eergy cotrol of fractoal descrtor ostve dscrete-te lear systes, It. I. Al. Math. Sc., vol. 4, o. 4, , 14. [13] T. Kaczorek, Descrtor ostve dscrete-te ad cotuous-te olear systes, Proc. SPIE, vol. 99, 14. [14] T. Kaczorek, Necessary ad suffcet codtos for u eergy cotrol of ostve dscretete lear systes wth bouded uts, Bull. Pol. Acad. Tech. Sc., vol. 6, o. 1, , 14. [15] T. Kaczorek, Postvty ad learzato of a class of olear dscrete-te systes by state feedbacks, Logstyka, vol. 6, , 14. [16] T. Kaczorek, Postve lear systes cosstg of subsystes wth dfferet fractoal orders, IEEE Tras. Crcuts ad Systes, vol. 58, o. 6, , 11. Lvv Polytechc Natoal Uversty Isttutoal Reostory htt://ea.l.edu.ua

6 16 [17] T. Kaczorek, Postve descrtor dscrete-te lear systes, Probles of Nolear Aalyss Egeerg Systes, vol. 1, o. 7, , [18] T. Kaczorek, Postvty ad stablty of dscrete-te olear systes, Subtted to CYBCONF, 15. [19] T. Kaczorek, Postve sgular dscrete te lear systes, Bull. Pol. Acad. Tech. Sc., vol. 45, o. 4, , [] T. Kaczorek, Selected Probles of Fractoal Systes Theory, Srger-Verlag, Berl, Geray, 1. [1] H. Zhag, D. Xe, H. Zhag, ad G. Wag, Stablty aalyss for dscrete-te swtched systes wth ustable subsytes by a ode-deedet average dwell te aroach, ISA Trasactos, vol. 53, , 14. [] J. Zhag, Z. Ha, H. Wu, ad J. Hug, Robust stablzato of dscrete-te ostve swtched systes wth ucertates ad average dwell te swtchg, Crcuts Syst, Sgal Process., vol. 33, , 14. АНАЛІЗ ПОЗИТИВНОСТІ ТА СТІЙКОСТІ ДИСКРЕТНИХ ТА НЕПЕРЕРВНИХ В ЧАСІ НЕЛІНІЙНИХ СИСТЕМ Тадеуш Качорек Досліджено позитивність та асимптотична стійкість нелінійних систем, часові залежності яких є дискретними або неперервними. Встановлено достатні умови позитивності та асимптотичної стійкості нелінійних систем. Запропоновані тести на стійкість базуються на розширенні методу О. Ляпунова для позитивних нелінійних систем. Ефективність цих тестів демонструється на прикладах. D. Sc., Professor, bor 193 Polad, receved hs M.Sc., Ph.D. ad D.Sc. degrees Electrcal Egeerg fro Warsaw Uversty of Techology, Polad, 1956, 196 ad 1964, resectvely. I the erod he was the dea of Electrcal Egeerg Faculty ad the erod the rorector of Warsaw Uversty of Techology, Polad. Sce 1971 he has bee a rofessor ad sce 1974 a full rofessor at Warsaw Uversty of Techology, Polad. I 1986 he was elected a corresodg eber ad 1996 full eber of Polsh Acadey of Sceces. I the erod he was the drector of the Research Cetre of Polsh Acadey of Sceces Roe. I Jue 1999 he was elected the full eber of the Acadey of Egeerg Polad. I May 4 he was elected to the hoorary eber of the Hugara Acadey of Sceces. He was awarded by the Uversty of Zeloa Gora, Polad () by the ttle doctor hoors causa, the Techcal Uversty of Lubl, Polad (4), the Techcal Uversty of Szczec, Polad (4), Warsaw Uversty of Techology, Polad (4), Balystok Uversty of Techology, Polad (8), Lodz Uversty of Techology, Polad (9), Oole Uversty of Techology, Polad (9), Poza Uversty of Techology, Polad (11), ad Rzeszow Uversty of Techology, Polad (1). Hs research terests cover the theory of systes ad the autoatc cotrol systes theory, secally, sgular ultdesoal systes, ostve ultdesoal systes ad sgular ostve 1D ad D systes. He has tated the research the feld of sgular D, ostve D lear systes ad ostve fractoal 1D ad D systes. He has ublshed 5 books (7 Eglsh) ad over 1 scetfc aers. He suervsed 69 Ph.D. theses. More tha of these PhD studets becae rofessors USA, UK ad Jaa. He s the edtor--chef of the Bullet of the Polsh Acadey of Sceces, Techcal Sceces ad the edtoral eber of about te teratoal jourals. Lvv Polytechc Natoal Uversty Isttutoal Reostory htt://ea.l.edu.ua