Piecewise Linear Estimate of Attractive Regions for Linear Systems with Saturating Control

Transcription

1 Pecewse Lnear stmate of Attractve Regons for Lnear Systems wth Saturatng Contro Yuzo Ohta Department of Computer and Systems, Kobe Unversty Nada, Kobe , Japan ABSTRACT We show how the combnaton of pecewse near Lyapunov functons and star-shaped poytopc Lyapunov functons can be used to estmate attractve regons for near systems wth saturaton We frst construct a pecewse near Lyapunov functon for a gven system by usng the near programmng method and the branch-and-boundng method, and compute the estmate of attractve regon accordng to the Lyapunov functon Then, we construct poytopes from obtaned regons, and appy star-shaped poytopc Lyapunov functon approach to mprove the estmate of attractve regon To demonstrate the usefuness of the proposed method we show some numerca exampes KY WORDS Attractve Regon, Saturaton, Lnear Programmng, Computatona Geometry 1 Introducton Lnear contro systems wth nput saturatons appears frequenty n practce snce most of actuators dspay saturaton characterstc Saturaton can have compcated effects on contro system performance and t therefore becomes essenta to determne the doman of attracton of the system There have been contnua efforts n addressng ths ssue In the ast decade, the ssue of computng estmates of attractve regons for near systems wth contro saturaton has been extensvey studed by many authors [1]- [10] and the references theren Most of resuts use a Lyapunov functon to estmate the attractve regons Attractve regons obtaned from quadratc, Luré-type, pecewse quadratc Lyapunov [3], [4], [6]-[10]) and pecewse-near Lyapunov functon [2], [5] See aso he survey by Geneso et a [1] and Hu and Ln [8] In order to take nto account the nonnear behavor of the cosed-oop system and obtan testabe condtons, the saturaton term was convenenty represented, but ths caused the conservatveness of the stabty resuts Conservatveness typcay comes from treatng the saturaton as an uncertan (sector-bounded) eement [3], [4], [7], [9] As proposed n [10], t shoud be modeed as pecewse near systems Then, homothetc type Lyapunov functon, such as quadratc Lyapunov functon and pecewse-near Lyapunov functon used n [5] and [2] mght oose ts power Non-homothetc type Lyapunov, such as Luré-type or pecewse quadratc Lyapunov functons[3], [4], [10] or pecewse-near Lyapunov functons used n [11]-[15], w be more usefu In ths paper, we use combnatons of pecewse-near Lyapunov functons (PLLFs) proposed n [11]-[15] and starshaped poytopc Lyapunov functons (SPLFs) proposed orgnay by Rosenbrock[16] PLLF s used to get an estmate of attractve regon and SPLF s used to enarge estmate of attractve regon and t guarantees that any souton statng from a pont n the enarged regon w get nto the regon obtaned by PLLF approach The usefuness of the proposng method s demonstrated by a numerca exampe 2 Probem Statement Consder a near system under saturated feedback sat (1) where R, R and sat! " denotes the unt saturaton defned by sat # f % f & &( (2) ) f ) We assume * +, s a stabe matrx Note that need not be a stabe matrx Let "-/0 ( be a souton of (1) wth nta condton 1 Let 2 be cosed set ncudng as ts nteror pont We say 2 s an estmate of attractve regon of the system (1) f for any 2 the souton 3-/4 stays n 2 for any -5% and converges to as Let< R be a poytope we are concern about the behavor of souton of (1) The ssue we consder n the foowng s to compute an estmate of attractve regon 2=+, whch s as arge as we can compute Let us defne poytopes?>a@ BC D 1& ) F BC D 1& BC D 1& F (3) (4) (5)

2 L o j Then, the system (1) can be represented by +GH3 ) I,,3 3 PLLF and Stabty Theorem >A@?@ Let BKJML ON be the set of a facets of For each JML, we defne convex conca hu cc J L and Pyramd Pd J L by cc J L QBCRS & RT% UV DWJ L Pd JMLX co BYBZ JML cc JML >A@ Fgure 1 [, []\_^, [a`, []^ and bcd/e We suppose that D>H@fg fgd@ s dvded nto smaer poytopes by hyperpanes Some of hyperpanes ncude a facet of Pd J L for some h, whch ntersect So far, we suppose that s dvded (6) sma poytopes Bk Fm/n by hyperpanes BCp3qsrC#tVrZ rum/v, p3qsrk#tvrz = Bw & q r T x xtvrg y, where z {B ZZ j and B!}_Zw o, and each k satsfes k ]~ cc J L for some h and B ) ƒu F In Fg 1, we ustrate, D>H@,, D@, and BSk, where the regon s dvded nto D>H@, and D@ by hyperpanes denoted by sod nes and s panted by gray coor The tranguar panted by back coor s a Pyramd Pd J L Dashed nes are hyperpanes whch ntersect Dotted nes are addtona hyperpanes The regon (and, hence, >H@, ) s dvde sma poytopes by these hyperpanes For each k and p"q(rk#tvrk, et us defne a number f r q r tvr T % ) f q r tvr _ tvr T f q r t r _ t rxˆ U T where s an nteror pont of k r by Then, a canddate of Pecewse Lnear Lyapunov Functon (PLLF) s gven by _" +Š TL r rƒm/v r 3q r T tvr Dk cc JML (7) where Š L s the normazed norma vector of the facet J L ( Š L Œ T for a +J L ), rea number r s the parameter correspondng to the hyperpane p"q r t r It can be shown that the functon _" defned by (7) s Lpschtz contnuous and satsfes Ž +, and t s easy to get the foowng stabty resut [13] Theorem 1 Let node k be the set of a nodes (0-facets) of k Assume that there exst %, %, and B r R o such that for any S node k, k cc JML, j z, S ˆ, the foowng two condtons hod Š L T r rum/vx r "q r T S t r % (8) and Š TL rƒm/v r r q T r #GH" S ) (9) where GH! " s gven by (6) Defne "RH7 Bw & _" R, Rs ũ T Qšœ BwR &s "RH Then, for any "R( ũ (, the souton 3-/4 of (1) stays n "R ũ and converges to exponentay Lex ž Ÿ QBw S node k & k cc JMLg S? denotes the boundary of Then, Rs ũ n Theorem 1 s gven by R ũ xšœ " HBF " F & S ž Ÿ where Therefore, to construct PLLF, we consder the foowng near programmng probem (LP) (LP1) š " ) sub to " S F ) r rƒm/v r 3q r T S 5 tvrz ) Š L T S S node k k cc JML r rum/v r q rtgh" F Š LTGH" S F S node k k cc r JML rum/v r "q r T t r xš TL S )Iª S node k k cc JML % _ % U r5 R o where " = f S ž«ÿ ž«ÿ (3 S = ª f &, and ª % s a gven constant In practce, we frst sove (LP0) whose objectve functon

3 s ) and whose constrants are same wth those of (LP1) If the optma vaue ) (LP1) wth addtona constrant that % of (LP0) s negatve, then we sove % If the optma vaue ) of (LP0) s non negatve, then we need to add dvdng hyperpanes, whch are denoted by dotted nes n Fg 1, to ncrease the freedom of the canddate of PLLF A scheme to determne addtona hyperpanes to strcty decrease the optma vaue of (LP0) was proposed n [15] The scheme conssts from sovng near programmng probems generated by the branch-andboundng scheme Ths process s repeated unt the optma vaue of (LP0) becomes negatve or t turns out that we can not determne an effectve hyperpane to decrease the the optma vaue of (LP0) Snce we assume that Q Q, s a stabe matrx, and snce GH" n (6) s contnuous, we can termnate the repeatng process of sovng (LP0) n a fnte teraton f the regon s not so arger than, and f does not ncude unstabe equbra If, we can determne dvdng hyperpanes so that (LP0) has negatve optma vaue n one shot Suppose that we have an estmate "R ũ of attractve re- PSfrag gon In the next secton, we consder the ssue to enarge estmate of the attractve regon by Star-shaped Poytopc Lyapunov Functon (SPLF) 4 SPLF and Stabty Theorem A regon s sad to be star-shaped poytopc regon f the foowng condtons are satsfed 1) nt, where nt denotes the nteror of 2) for each the ne segment connected and s ncuded n An exampe of a star-shaped poytopc regon s shown n Fg 2 (a) Note that the regon shown n Fg 2 (b) s not a star-shaped poytopc regon, snce the ne segment denoted by dashed ne s not ncuded n >H@ D@ >A@ >H@ (a) (b) Fgure 2 (a) A star-shaped poytopc regon ² (b) A regon ²³ whch s not a star-shaped poytopc regon Let be a star-shaped poytopc regon We decompose nto pyramds Bk as shown n Fg 3 (b) A facet of each >H@?@ pyramd ncuded the boundary of s caed a facet of A canddate of SPLF s gven by " +Š T L µ D cc JML (10) where ŠsL s the normazed norma vector of the facet JML By a qute smar way n provng stabty theorems n [13], [14], [17], we can prove the foowng resut Theorem 2 Let be star-shaped poytopc regons Assume the foowng % B )!_ F S node J L #J L xw~ Š TL GH" ) Š TL (11) where JML s a facet of, and vector of JML Then, s an estmate of attractve regon D@ >H@ (a) ŠsL s the normazed norma Fgure 3 (a) A star-shaped poytopc regon ² (b) Decomposton of ² nto pyramds (b) Now we ustrate how we use and construct SPLF to get a arger estmate of attractve regon Suppose that we computed "R( ƒ ( by the method we stated n the prevous secton From the constructon scheme, 3Rs ũ s a starshaped poytopc regon satsfyng (11), n whch and šœ " HB ŠsLa *Š(Lw¹O rum/v Š T L S & S k cc r r q r (12) node k J LC¹? ˆ (13) Suppose that "R( ƒ s a regon shown n Fg 4 (a), and GH" S at each node S of 3Rs ũ are arrows shown there We want to construct a arger star-shaped poytopc regon shown n Fg 4 (b) satsfyng the condton of Theorem 2 Note that the condton of Theorem 2 requre that Š TL GH" S F for GH" S F at each node S of satsfes any facet ncudng, and, hence, GH" S F, whch s denoted by arrow n the fgure must has the drecton shown n the fgure

4 D@ >A@ >A@ (a) "R( ũ ( (b) Fgure 4 È >H@ >H@ (a) Ç Fgure 5 (a) Ponts to be added (denoted by É ) must be ocated n the ntersecton of a cone and [ Ê (b) nargng process of an estmate of attractve regon Another constrant requres n constructng a arger estmate of attractve regon s that s a star-shaped poytopc regon, that s, nodes of whch are added must be ocated n the ntersecton of a cone and ~ as shown n Fg 5 (a) Let be the shaded regon of Fg 5 (a) Smary, we defne º >H@ and º, where º Let M )Yª f ) ~ ª ~ I,ƒ f x M ª f Fx a postve constant % Let % be a constant } For each B ) ƒu n (13) and et, defne a near functon ¼ ~ #3 by ¼ ~ 3½H ¾À I ~ÂÁ >H@ 3½ ) ª ~ (14) When ~ 3Rs ũ H ~ s convex for any B ) ƒu F, we propose an agorthm to enarge attractve regon n the foowng Agorthm SFLP NLARG Step 1 for ÃB )!_ do begn Step 2 for J L Ä]~ do begn Step 3 for M node J L do begn Step 4 ½S, Ä j Step 5 do whe ( ½(a W~ º ) Step 6 ¼ ~ "½s( Step 7 ~ co B ~ f Step 8 j Step 9 end Step 10 end Step 11 end Step 12 end where co B ~Æf denotes the convex hu, whch s computed by appyng the Beneath-Beyond Method [18] In Fg 5 (b) we ustrate the process of executng SFLP NLARG Startng we enarge the attractve regon a tte bt (area panted sghty ght coor) then usng Ç we further enarge t repeatng ths process we fnay the attractve regon panted most ght coor, and of course, t ncudes areas panted darker coors A justfcaton of the above agorthm s the foowng Theorem 3 Suppose that s a star-shaped poytopc regon satsfyng the condton of Theorem 2 Let ), node JML, JML x ~, and et ~ co B ~ f B ¼ ~ " (15) ~ ¹ ~ ¹ ˆ (16) >H@ f If % s suffcenty sma, then s agan a star-shaped poytopc regon satsfyng the condton of Theorem 2 If we repace from ) to or, we have the same resut 5 An xampe In ths secton, we w compare the approaches on a exampe from the terature The method that we w compare s anayss methods based on the crce and Popov-crtera from [3], [4] and the pecewse quadratc approach n [10] xampe 1 [3], [10] Q Consder (1) wth ) IË ) )5Ì Fg 6 shows the estmate of attractve regon (panted dark coor) obtaned usng PLLF and enarged regon (panted ght coor) by appyng SFLP NLARG In Fg 7, estmates of attractve regons obtaned appyng severa methods, where the smaest epsoda area shown by the dotted ne s the estmate of attractve regon obtaned by the quadratc Lyapunov functon [3] the area denoted by the dashed ne s the estmate of attractve regon obtaned by appyng crce crtera[3] the area denoted by the dot dashed ne s the estmate of attractve regon obtaned by appyng Popov crtera[3] the area panted by dark coor s the estmate of attractve regon obtaned by appyng the pecewse quadratc approach [10], whch s amost concdes wth the regon obtaned usng PLLF the area panted by ght coor s the estmate of attractve regon obtaned by appyng SFLP NLARG The ast regon amost ncudes the regon panted by dark coor Roughy

5 speakng, our scheme usng PLLF and SFLP NLARG gvesí the argest estmate of the attractve regon and new proposas, I Trans Automatc Contro, vo AC-28, pp (1985) [2] B G Romanchuk, Computng Regons of attracton wth poytopes panar case, Automatca, vo 32, pp (1996) [3] C Pettet, S Tarbourech and C Burgat, Stabty regons for near systems wth saturatng contros va crce and Popov crtera, Proc 36th CDC, vo 5, pp (1997) ) ) [4] H Hnd and S Boyd, Anayss of near systems wth saturaton usng convex optmzaton, Proc 37th CDC, vo 1, pp (1998) Fgure 6 [5] J M Gomes da Sva Jr and S Tarbourech, Poyhedra regons of oca asymptotc stabty for dscretetme near systems wth saturatng contros, I Trans Automatc Contro, vo AC-44, pp (1999) ) ) [6] I K Fong and C C Hsu, State feedback stabzaton of snge nput systems through actuator saturaton and dead zone characterstcs, CD-ROM of 39th CDC (2000) [7] T Hu and Z Ln, On enargegng the basn of attracton for near systems under saturated near feedback, System & Contro Letters, vo 40, pp (2000) Fgure 7 [8] T Hu and Z Ln, Contro Systems wth Actuatr Saturaton Anayss and Desgn, Brkhäuser, Boston, Concudng Remarks In ths paper, we proposed a scheme for estmatng of attractve regon of near systems wth saturatng contro, and shows our approach gve a arger estmate of attractve regon than the prevous methods The approach uses combnatons of PLLFs and SPLFs Usng combnatons of PLLFs and SPLFs s a key pont of our method If the regon whch we are concerned s too arge, PLLF approach mght fa to construct PLLF, on the other hand, the estmaton of attractve regons are not so arge f s not so arge When a star-shaped poytopc estmate of attractve regon s obtaned by PLLF approach, SPLF s used to enarge estmate of attractve regon so that any souton statng from a pont n the enarged regon w get nto the regon obtaned by PLLF approach References [1] R Geneso, M Tartaga and AVcno, On the estmate of asympttc stabty regons state of the art [9] J M Gomes da Sva Jr, S Tarbourech and R Regnatto, Concervatvty of epsoda stabty regons estmates for nput saturated near systems, CD- ROM of 15th IFAC Word Congress (2002) [10] M Johansson, Pecewse quadratc estmates of domans of attracton for near systems wth saturaton, CD-ROM of 15th IFAC Word Congress (2002) [11] Y Ohta and M Onsh, Stabty anayss by usng pecewse near Lyapunov functons, Proc of the 1999 IFAC Word Congress, Part D, pp , 1999 [12] M Johansson, Anayss of pecewse near system va convex optmzaton - a unfyng approach, Proc of the 1999 IFAC Word Congress, Part D, pp , 1999 [13] Y Ohta and K Yamamoto, Stabty anayss of nonnear systems va pecewse near Lyapunov functons, Proc of ISCAS 2000, Part II, pp , 2000

6 & J Š [14] Y Ohta, T Wada and D D Sjak, Stabty Anayss of Dscontnuous Nonnear Systems va Pecewse Lnear Lyapunov Functons, Proc of 2001 Amercan Contro Conference, pp , 2001 [15] Y Ohta, On the constructon of pecewse near Lyapunov functons, Proc of the 40th CDC, pp , 2001 [16] HH Rosenbrock, A method of nvestgatng stabty, Proc 2nd IFAC Word Congress, pp , 1963 [17] Y Ohta, H Imansh, L Gong and H Haneda, Computer generated Lyapunov functons for a cass of nonnear system, I Trans Crcuts and Systems, part I, vo 40, pp , 1993 [18] Y Ohta, Y Naga and L Gong, Beneath-Beyond Method and Constructon of Lyapunov Functons, Proc of NOLTA 97, pp (1997) [19] J M Ortega, Numerca Anaays A Second Course, SIAM 1900 Appendx The proof of Theorem 2 s qute smar to those of stabty theorems, and, hence, we omt the proof Proof of Theorem 3 We prove the case when ) Other cases are proved n the same way Let ½? ¼ ~ " ˾" I ~ Á >H@ " ) ª ~ Snce % s suffcenty sma, we have ¾" I ~Á >H@ I ) ) Î where Î denotes terms such that Î ~ _ gî 6 as 6 Therefore, ½ can be represented as ½ Ä ) # ~ ª ~ A xî (18) Let J be any facet of ~ such that ÏÐJ and Š be the normazed vector of J Then, by the assumpton, we have Š T Š T# ~ ª ~ Observe that Š T½ Š T" ) ~ ª ~ H Î! Š T ) Š T ~ ª ~ Î % ) #Š T Î xî % Therefore, ½ ~ Let ~ co B ~ f BC½ facets of ~ are not facets of ~, and (19) Then some ~ has new facets ncudng ½ Let J be a new facet of ~, whch ncudes ½, and et Š be the normazed norma vector of J If we show that Š T ~ ª ~ and node ~? J (20) Š T ~ ½ ª ~ Then, we have the concuson (21) Snce (20) s equvaent to the condton that M œ ~ ª ~ nt ~ for suffcenty sma postve number œ % [17] and ths s obvous snce ~ If we show that ~ ŠW +Š Š Î (22) for some Š, then we have Š T ~ ½] ª ~ Šœ Š Î! T # ~ " ) ~ ª ~ Î!A ª ~ ) Š T ) Š T & Š & & ~ T ª ~#& Î ) & Š & & ~ ª ~ƒ& xî and, hence, we have (21) Therefore, f we show (22), we have the resut Note that s determned by the equaton ½ T "½ ) T 3½ ) >H@ T Z, >A@ are nodes of both J and Ò Smary, Š s determned by the equaton rank ŠW where rank ÄÒ Let T " T " ) >H@ T (23) such that (24) WÓ Because of (18), ÓÔ +Õ Ó WÎ, where Ó s a constant matrx determned by ~ ª ~ Then, we have Š ) ŠW >A@ ) xó >H@ Q Óa >H@ ÓXŠ & Š ) Š & & Š & ÖK Óa >A@ ÖYÖCÓTÖ Ö >A@ Ö ÖwÓTÖ ) Ö >H@FÖSÖwÓTÖ Ö >H@ Ö ÖwÓ Ö xî ) Ö >H@ÖSÖwÓ Ö xî To derve above nequaty, we used Perturbaton Lemma (see, for exampe, [19], p 32) Therefore, we have (22) for some Š