Stability Analysis of Discrete-Time Piecewise-Linear Systems: A Generating Function Approach

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1 Ieraioal Joural of Corol, Auomaio, ad Sysems (4) (5):5- DOI.7/s ISSN: eissn:5-49 hp:// Sabiliy Aalysis of Discree-Time Piecewise-Liear Sysems: A Geeraig Fucio Approach Kai Liu*, Jiaghai Hu, Yu Yao, Baoqig Yag, ad Xi Huo Absrac: This paper sudies he expoeial sabiliy of a class of discree-ime piecewise-liear sysems (DPLS). Some basic properies of he proposed DPLS are esablished, which eables he geeraig fucio approach o be used for he sysem sabiliy aalysis. By iroducig he geeraig fucios of DPLS ad showig heir properies, a sufficie ad ecessary codiio for he expoeial sabiliy of DPLS is derived. Furhermore, he maximum expoeial growh rae of sysem rajecories ca be obaied exacly by compuig he radii of covergece of he geeraig fucios. The algorihm for compuig he geeraig fucios is developed ad wo examples are give o demosrae he proposed approach. Keywords: Expoeial sabiliy, geeraig fucio, growh rae, piecewise-liear sysems.. INTRODUCTION Piecewise-liear sysems (PLS) have bee receivig icreasig aeio by corol commuiy, sice may pracical corol sysems ca be represeed by PLS, especially whe piecewise-liear compoes icludig deadzoe, sauraio, relays, ad hyseresis are ecouered, ad may highly oliear sysems ca also be approximaed by PLS []. I addiio, hey provide a equivale framewor o he well-ow liear complemeary sysems [] ad mixed logical dyamical sysems [3] from he model poi of view. Remarable progress has bee achieved o he sabiliy problem of PLS. Represeaive approaches o he sudy of sabiliy for PLS iclude he cosrucio of commo, muliple ad surface Lyapuov fucios [4-6], which have also bee exeded o he sabilizaio problem [7,8] ad robus corol problem [9,]. For umerical compuaio, hese approaches resric he available classes of Lyapuov fucios o he class of piecewise quadraic or higher-order fucios, which maes he resulig sabiliy codiios coservaive. Therefore, a ew research red i he rece wor [- 3] is o derive a compuable sufficie ad ecessary Mauscrip received February 8, 3; revised December, 3; acceped March 3, 4. Recommeded by Associae Edior Youg I So uder he direcio of Edior Hyugbo Shim. This wor was parially suppored by Naioal Naural Sciece Foudaio of Chia uder gras NSFC 6746, 6, 639 ad Kai Liu is wih Beijig Isiue of Aerospace Techology, Harbi, Chia ( liuaicarso@63.com). Jiaghai Hu is wih he School of Elecrical ad Compuer Egieerig, Purdue Uiversiy, USA ( jiaghai@purdue. edu). Yu Yao, Baoqig Yag, ad Xi Huo are wih he Corol ad Simulaio Ceer, Harbi Isiue of Techology, Chia ( s: {yaoyu, ybq, huoxi}@hi.edu.c). * Correspodig auhor. codiio for he sabiliy of PLS. For a class of plaar PLS (PPLS), Iwaai [] derived a explici ad exac sabiliy es, which was give i erms of he coefficies of he rasfer fucios of subsysems ad was compuaioally racable. Araposahis [] sudied he behavior of PPLS direcly, ad i was show ha he asympoic sabiliy of PPLS ca be fully modes. O he oher had, by iroducig he defiiio of iegral fucio, Liu [3] provided a ecessary ad sufficie sabiliy es for PPLS. However, he above resuls oly focus o he PPLS, here is sill a few seps away from beig effecive i he sabiliy aalysis of geeral PLS. I our previous wor [4], a mehod based o he oio of geeraig fucio is proposed o sudy he sabiliy of discree-ime swiched liear sysems uder hree ypes of swichig rules: arbirary swichig, opimal swichig ad radom swichig. I his paper, we will exed his mehod o he discree-ime piecewise liear sysems (D-PLS), i.e., swiched liear sysems wih sae-depede swichig. This paper is orgaized as follows: The sysem model is described ad some basic properies of DPLS are esablished i Secio. I Secio 3, he geeraig fucios are defied, aalyzed ad used o characerize he expoeial sabiliy of DPLS. Their umerical compuaio algorihm is also developed. Secio 4 shows wo examples o demosrae he geeraig fucios approach ad coclusios are draw i Secio 5. Noaio: - I(S) : The ierior of a se S; - CoS : The covex hull of a se S; - (L) : For a sae-space pariio { R i} i =, (L) deoes he idex se of he regios Ri coaiig a se L,i.e. (L) = { i (L) R i } - d(x,s) : The disace from a poi x o a se S [,m] : = {,,,m-,m} ICROS, KIEE ad Spriger 4

2 6 Kai Liu, Jiaghai Hu, Yu Yao, Baoqig Yag, ad Xi Huo. PROBLEM FORMULATION.. Problem saeme We cosider a class of discree-ime piecewise-liear sysems represeed by Ax if x () R Ax if x () R x ( + ) = f( x ( )): = AMx if x() RM, () where A i are osigular marices ad R i are polyhedro coes wih oempy ierior, of he forms R i { x E x } = () i M such ha, Ri Rj = Lij, Ri = R, where L ij deoes i= he commo boudary of R i ad R j. To express he sysem rajecories uiquely, we iroduce he followig defiiio of he rajecories of DPLS () from he view of auoomous swiched sysems. Defiiio : Le x (; zσ, ) deoes he soluio of DPLS () wih he iiial sae z ad swich sequece σ Σ(z), where Σ(z) deoes he se of all possible swich sequeces for DPLS wih he iiial sae z. Defiiio : A he equilibrium origi, he DPLS () is called: sable i he sese of Lyapuov if, for each ε >, here exiss δ ε >, such ha z < δε x(; z, σ) ε for all Z +. asympoically sable if i is sable ad all he soluios xzσ (;, ) coverge o origi. expoeially sable if here exis δκ>, ad r (,) such ha z < δ x(; z, σ) κr z for all Z + ad σ Σ(z). Our objecive are: (i) prese a compuable sufficie ad ecessary crierio of expoeial sabiliy for coiuous-ime DPLS (); (ii) compue he expoeial growh rae wih coservaism, i.e., he exac r... Refieme for sae-space pariio This subsecio proposes a procedure for recursively refiig he sae-space pariio. We assume he procedure ermiaes afer a fiie umber of seps, which esures ha some ice properies (see Lemma ad ) of DPLS ca be cocluded uder he obaied sae-space pariio. The ey idea of he procedure is o separae each regio R i io several subregios { D } m ij j = saisfyig D = { x R & Ax R }. (3) ij i i j Obiviously, he ew sae-space pariio ca be compued by solvig D = { x E x & AE x }. (4) ij i i j Assumpio : I is assumed ha afer fiie seps of refieme algorihm, he sae-space pariio cao be furher refied, i.e., he ew sae-space pariio is ideical o he old oe. The discussio abou Assumpio ca be foud i [5], a sufficie codiio for Assumpio o hold have bee provided. From Assumpio we ca lear ha he Algorihm will covergece i he fiie seps, ad we call he obaied sae-space pariio as he fial sae-space pariio. All he properies i he followig are preseed for he DPLS wih he fial sae-space pariio. Remar : I is observed from he refiig procedure ha he fial sae-space pariio saisfies ha, for each regio R i, here exiss a regio R j such ha, A x R, x R. (5) i j i We call R j he objecive rasiio regio of R i..3. Properies of DPLS This subsecio esablishes some basic properies of DPLS wih he fial sae-space pariio. Lemma : If R j is he objecive rasiio regio of R i, he for all z R i, Aiz Rj. Lemma : For all z, z R i, he possible swichig sequece se of z + z is coaied i he possible swichig sequece se of eiher z or z, i.e., ( z z ) ( z ) ( z ) Σ + Σ Σ (6). The proof of Lemma ad Lemma ca be foud i [9]. Based o Lemma ad, some properies of he rajecories of DPLS are derived as follows: Proposiio : The sysem rajecories of DPLS () have he followig properies: ) (Homogeeiy): For all z Ri ad σ Σ ( z), x (; z, σ) = x( ; z, σ). ) (Piecewise-Addiiviy): For all z, z Ri ad σ Σ ( z+ z), x ( ; z+ z, σ) = x ( ; z, σ) + x ( ; z, σ). Proof: ) Assume he swich sequece σ = ( σ(), σ() σ( ) ), he we have x( ; z, σ) = A A z = x( ; z, σ). σ( ) σ() ) For all z, z Ri ad σ Σ ( z+ z), i ca be implied from Lemma ha σ Σ( z) Σ( z), he we have x(; z+ z, σ ) = Aσ( ) Aσ() ( z+ z) = xz (;, σ) + xz (;, σ). 3. GENERATIONG FUNCTIONS OF DPLS I his secio, we prese he geeraig fucio approach o he expoeial sabiliy aalysis of DPLS. Defiiio 3: We defie geeraig fucio G(, z) of he DPLS as: G(, z) = sup x( ; z, σ). (7) σ Σ ( z) =

3 Sabiliy Aalysis of Discree-Time Piecewise-Liear Sysems: A Geeraig Fucio Approach 7 For each fixed, G(, z) is a fucio of z oly: G ( z): = G(, z). (8) 3.. Properies of geeraig fucio Based o Defiiio of geeraig fucio ad Proposiio, we obai he followig properies. Proposiio : G ( z) has he followig properies: ) (Homogeeiy): I ( z) is homogeeous of degree wo i z, i.e., G( αz) = α G( z), α >. ) (Sub-Addiiviy): For all z, z R, G ( z + z ) G ( z ) + G ( z ). (9) 3) (Covexiy): For each, G ( z ) is a covex fucio, i.e., for all z, z R ad α, α, α + α =, G ( α z + α z ) α G ( z ) + α G ( z ). () 4) (Commo-Boud): For each, I ( z) < for all z R implies ha G ( z) < g z. Proof: ) The homogeeiy propery is a direc sequece of homogeeiy propery of sysem soluio. ) I is implied by Proposio ad Cauchy-Schwarz iequaliy, + = + σ σ Σ ( z+ z) = = sup xz ( ;, σ) + xz ( ;, σ) σ Σ ( z+ z) = = sup [ x ( ; z, σ) + x ( ; z, σ) σ Σ ( z+ z) = G ( z z ) sup x( ; z z, ) + x ( ; z, σ) x ( ; z, σ) ] d σ Σ ( z+ z) = + σ Σ ( z+ z) = sup x ( ; z, σ) sup σ Σ ( z ) = σ Σ ( z ) = ( G z G z ) x (; z, σ ) + sup xz ( ;, σ) sup xz ( ;, σ) = G ( z ) + G ( z ) + G ( z ) G ( z ) d = ( ) + ( ). This implies he resul (9). 3) Wih he help of Sub-Addiiviy ad Homogeeiy for α, α ad α + α =, we have G( αz+ αz) G( αz) + G( αz) () = α G ( z ) + α G ( z ). d d d This proves he Covexiy of G ( z). 4) Assume here exis such ha G ( z) <. Le { z } i i= deoe a sadard basis of R, he for ay z S (ui-ball of R ), here exis α j, α j =, j= such ha z = α z. () j= j j Apply he Sub-Addiiviy of G ( z) ad he Cauchy- Schwarz iequaliy i he summaio form o ge ha, for all z S ( ) = ( α j j) α j ( j) j= j= α j. G( zj) g, j= j= G z G z G z where (3) g : = max G( zj ). (4) j {,, } By homogeeiy, we have G ( z) < g z, z R. 3.. Expoeial sabiliy crierio of DPLS I his subsecio, a sufficie ad ecessary codiio of expoeial sabiliy is preseed for DPLS via he proposed geeraig fucios. Theorem : The DPLS () is srogly expoeially sable if ad oly if he correspodig geeraig fucios saisfy ha G ( z ) is fiie for all z R. Proof: For sufficiecy, suppose for he DPLS (), G ( z) <, z. The by propery 4 i Proposiio, here exiss a cosa g such ha G ( z) < g z. Thus from he defiiio of G ( z ) we have σ Σ ( z) = sup xz ( ;, σ ) g z, which implies ha x (; z, σ ) g z, Z + ad x (; zσ, ) as due o he covergece of ifiie posiive series. Cosequely, he DPLS is asympoically sable, hece expoeially sable. For ecessiy, assume he DPLS is srogly expoeially sable, i.e., here exiss cosas ad r (,) such ha xz (;, σ ) r z, Z + ad σ Σ ( z). The we have, G ( z) = sup x( ; z, σ ) σ Σ ( z) = = r z = z <. r Furhermore, usig he piecewise-covexiy of G (z), he followig corollary ca be cocluded o show ha he expoeial sabiliy of DPLS is fully deermied by he geeraig fucios o some fiie pois, hece compuaioally racable.

4 8 Kai Liu, Jiaghai Hu, Yu Yao, Baoqig Yag, ad Xi Huo Corollary : The DPLS () is srogly expoeially sable if ad oly if he geeraig fucios are fiie o all he ui-verexes, i.e., G ( z ) < g z, i [, m], j [, ]. (5) ij 3.3. Characerizaio of maximum expoeial growh rae I he followig subsecio, we defie he radius of srog covergece of he geeraig fucios. I ca be show his quaiy characerizes he maximum expoeial growh rae of DPLS exacly. Defiiio 4: The radius of covergece of he geeraig fucio, deoed by *, is defied as { G z z } = sup ( ) <, R. (6) The, he followig heorem shows he relaioship of he radius of covergece * ad he maximum expoeial growh rae r *. Theorem : Give a DPLS wih a radius of covergece of geeraig fucio * /, for ay r > ( ), here exiss a cosa r such ha x (; z, σ ) r r z, Z + /, σ Σ ( z). Furhermore, ( ) is also he smalles value for he previous saeme o hold. I oher words, he maximum expoeial growh rae of DPLS / is r = ( ). The proof is ideical o ha i [[5], Corollary ] Compuaio of geeraig fucios All he aalysis mehods proposed i previous subsecios require he compuaio of he geeraig fucios of DPLS. I his subsecio, we develop a algorihm for compuig he rucaios of geeraig fucios as approximaios of G (z) defied as below. Defiiio 5: For each Z +, defie G ( z) = sup x( ; z, σ), z R. σ Σ ( z) = The followig proposiio liss properies of G ( z). Proposiio 3: For all (, ) ad Z +, G ( z) coverges expoeially fas o G ( z),, i.e. + G( z) G( z) g z, z R. g Proof: Le x (): = x (; zσ, ), where σ Σ ( z) deoes he swich sequece o achieve he supremum of G ( z), hus we have G ( x ( )) G ( x ( )) = x ( ). From propery 5 i Proposiio, his furher implies, G ( x( )) G( x( )) G( x( )), g which is equivale o i / g G( x( )) G( x( )). By iducio o his iequaliy, we have + G ( z) G ( z) = x( + + ) = + + / g = G( x( + )) G( x( )) + + / g G ( z) / g g z = g / g z. ( ) ( ) This complees he proof. + + Nex, o prese he algorihm for compuig he rucaios of geeraig fucios, we firs provide he followig Lemma. Lemma 3: For ay Z +, he G ( z) saisfies he Bellma Equaio, i.e., + G ( z) = z + max G( Ap z), (7) p ( z) where ( z) = { i z R i }. The proof ca be direcly implied by he defiiio of G ( z), hus omied. Lemma 3 acually provides a accurae approach o compue G ( z), which is summarized as Algorihm. From Proposiio 3 we lear ha, G ( z) coverges expoeially fas o G ( z) as, ad he approximaio error is bouded by formula (3). Therefore, Algorihm ca be applied o compue G ( z) wih ay precisio as permied by he umerical compuaio errors. By repeaedly applyig Algorihm o a icreasig sequece of, a uderesimaes of * ca be obaied. 4. ILLUSTRATIVE EXAMPLES I his secio, we will demosrae he proposed approach hrough wo umerical examples. Example : Cosider he followig DPLS A = A3 =, 4 ; 3.8 A = A =..8 R = { x x & x }, R = { x x & x }, R = x x & x, R = x x & x. { } { } 3 4 Firs, we eed o furher refie he origial sae-space pariio. The refieme process is show i Fig.. A () sep (), R splis io R () ad R () such ha ay () sae rajecory sarig from he iiial sae x R () will eer R i oe sep; While ay sae rajecory () sarig from he iiial sae x R will remai i () R i oe sep. The same procedure is applied for R () () R ad R () Thus, we obai oe-sep sae-, 3 4.

5 Sabiliy Aalysis of Discree-Time Piecewise-Liear Sysems: A Geeraig Fucio Approach 9 space pariio. A sep (), R () splis io R (3) ad (3) (3) () R, R 3 = R. I he same way, we ca obai (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) ( R4, R5, R6 )( R7, R8, R9 )( R, R, R )( R, R, (3) R ). These obaied regios cosis of he wo-sep sae-space pariio. I is oed from he obaied regio (3) ha, every sae rajecory wih he iiial sae x R rasies o R (3) i oe sep. For coveiece, we (3) (3) wrie his as R R. Similarly, we have (3) R (3) (3) (3) R3 R R. These observaios imply ha if (4) ruig he hird sep of pariio, we have R (3) i = Ri. I oher words, he refiig procedures ermiae. Algorihm is used o compue geeraig fucio for differe values of :.,.4,.6,.8,.,.. Fig. (d) illusraes he plos of he compued / g. Sice g a = is fiie, he give DPLS is expoeially sable from Theorem. Acually, a esimaed value of =.6 ca be obaied by exrapolaio mehod, which shows ha he maximum expoeial rae / r = ( ).985 from Theorem. Furhermore, he followig example shows ha he proposed approach is also useful for he sabiliy aalysis of DPLS wih higher dimesios. Algorihm : Compuaio of G ( z). Iiialize =, G ( z) =, z = z;. Repea + ; 3. for each z, do + G ( z ) = z + max G ( A z ) ed for p ( z ) 4.Se p arg max G ( ) ( Ap z p ( z ) ) 5. Se z Ap z 6. = K K 7. Reur G ( z). = ; + = Example : Cosider he followig DPLS.5.6 A =.5, A.8 = ;.6 R = x x, R = x x. { } { } 3 3 Similar o Example, he origial sae-space pariio ca be furher refied. Fig. illusrae he refieme process. Afer wo seps, he obaied eigh subregios are exacly he eigh quadras i sadard hreedimesioal space. I is easy o verify ha every subregio has a uique objecive rasiio regio. I oher words, he refiig process ermiae. Therefore, he ui-verexes of he fial sae-space pariio are (,, )(,, )(,, )(,, )(,,)(,, ). Algorihm is used o compue G ( z ij ) for differe values of :.,.4,.6,.8,.. Fig. (d) illusraes he plos of he compued / g. Sice g a = is fiie, he give DPLS is expoeially sable from K p Fig.. Simulaio resuls of Example. Fig.. Simulaio resuls of Example. Theorem. Acually, a esimaed value of =.54 ca be obaied by exrapolaio mehod, which shows / ha he maximum expoeial rae r = ( ).974 from Theorem. 5. CONCLUSION A compuaioal approach o aalyzig he sabiliy for a class of DPLS was preseed via geeraig fucios ogeher wih some quaiies derived from geeraig fucios, such as radii of covergece ad quadraic bouds. These quaiies fully characerize he expoeial sabiliy of he DPLS. Our fuure wor will focus o how o exed he proposed approach o he robus sabiliy aalysis of DPLS. REFERENCES [] E. Soag, Noliear regulaio: he piecewise liear approach, IEEE Tras. Auomaic Corol, vol. 6, o., pp , 98. [] W. Heemels, B. D. Schuer, ad A. Bemporad, Equivalece of hybrid dyamical models, Auomaica, vol. 37, o. 7, pp. 85-9,. [3] A. Bemporad ad M. Morari, Corol of sysems

6 Kai Liu, Jiaghai Hu, Yu Yao, Baoqig Yag, ad Xi Huo Iegraig logic, dyamics ad cosrais, Auomaica, vol. 35, o. 3, pp. 4-47, 999. [4] M. Johasso ad A. Raer, Compuaio of piecewise quadraic Lyapuov fucios for hybrid sysems, IEEE Tras. Auomaic Corol, vol. 43, o. 4, pp , 998. [5] J. Gocalves, A. Megresi, ad M. Dahleh, Global aalysis of piecewise liear sysems usig impac maps ad surface Lyapuov fucios, IEEE Tras. o Auomaic Corol, vol. 48, o., pp. 89-6, 3. [6] M. Lazar ad W. Heemels, Global ipu-o-sae sabiliy ad sabilizaio of discree ime piecewise affie sysems, Noliear Aalysis: Hybrid Sysems, vol. 6, o., pp , 8. [7] Y. Gao, Z. Y. Liu, ad H. Che, Observer based coroller desig of discree piecewise affie sysems, Asia Joural of Corol, vol., o. 4, pp ,. [8] J. Zhag ad W. Tag, Aalysis ad corol for a ew chaoic sysem via piecewise liear feedbac, Chaos Soluios ad Fracals, vol. 4, o. 4, pp. 8-9, 9. [9] K. Liu, Y. Yao, D. Su, ad V. Balarisha, Improved sae feedbac coroller syhesis for piecewise liear sysems, Ieraioal Joural of Iovaive Compuig, Iformaio ad Corol, vol., o. 8, pp ,. [] K. Liu, Y. Yao, ad H. Ma, Adapive H corol of piecewise liear sysems wih parameric ucerai-y ad exeral disurbace, Asia Joural of Corol, vol. 5, o. 4, pp , 3. [] Y. Iwaai ad S. Hara, Sabiliy ess ad sabilizaio for piecewise liear sysems based o poles ad zeros of subsysem, Auomaica, vol. 4, o. 5, pp , 6. [] A. Araposahisa ad M. Brouce, Sabiliy ad corollabiliy of plaar coewise liear sysems, Sysems & Corol Leers, vol. 56, o. 6, pp. 5-58, 7. [3] K. Liu, Y. Yao, B. Yag, ad Y. Guo, Expoeial sabiliy aalysis of plaar piecewise liear sysems: a iegral fucio approach, Ieraioal Joural of Corol, Auomaio ad Sysems, vol., o., pp. -,. [4] J. Hu, J. She, ad W. Zhag, Geeraig fucios of swiched liear sysems: aalysis, compuaio ad sabiliy applicaios, IEEE Tras. o Auomaic Corol, vol. 5, o. 5, pp ,. [5] K. Liu, Sabiliy Aalysis ad Corol Desig of Piecewise-liear Sysems, Ph.D. Thesis, Harbi Isiue of Techology, 3. Kai Liu received his Ph.D. degree i Corol ad Simulaio Ceer, Harbi Isiue of Techology, Chia. He is currely a egieer of Beijig Aerospace Isiue of Techology His research ieress iclude piecewise-liear sysems ad fligh corol. Jiaghai Hu received his Ph.D. degree from Uiversiy of Califoria a Bereley, USA, 8. He is currely a associae professor i School of ECE, Purdue Uiversiy, USA. His research ieress iclude hybrid sysems ad muli-age sysems. Yu Yao received his Ph.D. degree i Auomaic Corol from Harbi Isiue of Techology, Chia i 99. He is currely a professor i Harbi Isiue of Techology, Chia. His research ieress iclude oliear corol ad robus corol. Baoqig Yag received his Ph.D. degree i Auomaic Corol from Harbi Isiue of Techology, Chia i 9. He is currely a lecure i Harbi Isiue of Techology, Chia. His research ieress iclude predicive corol ad fligh corol. Xi Huo received his M.S. ad Ph.D. degrees i Corol Sciece ad Egieerig from Harbi Isiue of Techology, Chia, i 6 ad, respecively. He he joied he faculy of he Corol ad Simulaio Ceer, School of Asroauics, a Harbi Isiue of Techology as a lecurer. His research ieress iclude osmooh dyamical sysems, moio corol, hybrid sysem modelig, simulaio ad corol.