to- ( ).- Control and Stability Analysis of Cooperating Robots H. Kazerooni T. I. Tsay IS2-5,,.

Transcription

1 l( Amrican Control Confrnc Atlanta, Gorgia, Jun 1988 Control and Stability Analysis of Cooprating Robots H. Kazrooni T. I. Tsay Mchanical Enginring Dpartmnt Univrsity of Minnsota Minnapolis. Minnsota Abstract Th work prsntd hr is th dscription of th control stratgy of two cooprating robots. A twofingr hand is an xampl of such a systm. TIl control mthod allows for position control of th contact point by on of th robots whil th othr robot controls th contact forc. Th stability analysis of two robot manipulators has bn invstigatd using unstructurd modls for dynamic bhavior of robot manipulators. For th stability of two robots, thr must b som initial compliancy in ithr robot. Th initial compliancy in th roboti. can b obtaind by a nonzro snsitivity function for th tracking controllr or a passiv compliant lmnt such as an RCC. Introduction This papr dvlops th ssntial ruls in stability analysis of two cooprating robots. W assum th robots initially hav som typ of indpndnt tracking capabilitis. This assumption prmits IllS to xtnd th control analysis to covr industrial robot manipulators in addition to rsarch robots. Th tracking capability allows ach robot to follow its individual command indpndntly whn it is not constraind by ach othr. Onc th robots com in contact with ach othr, th contact forc btwn th two robots is fd back to on of th rlobots to dvlop compliancy (1,2,3,4). Th compliancy in on of th robots allows for control of th contact forc, whiil th othr robot govrns th position of th contact point. A stability bound has bn dvlopd on th siz of th forc fdback gain to stabiliz th closd loop systm of both robots. Th stability analysis has bn invstigatd using unstructurd modls for th dynamic bhavior of th robot m:anipulators. This unifid approach of modling robot dynamics is xprssd in trms of snsitivity functions as opposd to th Lagrangian approach. It allows us to incorporat th dynamic bhavior of all th lmnts of a robot manipulator (i.. actuators, snsors and th structural c:omplianc of th links) in addition to th rigid body dynamics(4). Damic Mod1 of th Robot In this sction, a gnral approach will b dvlopd to dscrib th dynamic bhavior of a larg class of industrial and rsarch robot manipulators having positioning (tracking) controllrs. Th fact that most industrial manipulators alrady hav som kind of positioning controllr is th motivation bhind our approach. Also, a numbr of mthodologis xist for th dvlopmnt of robust positioning controllrs for dirct and nondirct robot manipulators (6 ). In gnral, th ndpoint position of a robot manipulator that has a positioning controllr is a dynamic function of its input trajctory vctor,, and th xtrnal forc, f. Lt G and S b two lp stabl mappings that dscrib th robot ndpoint position, y. in a global coordinat fram. (f is masurd in th global coordinat fram also.) yg()+s(f) (1) Th assumption that linar suprposition (in quation 1) holds for th ffcts of f and is usful in lmdrstanding th natur of th intraction btwn two robots. This intraction is in a fdback foml and will b clarifid with th hlp of Figur 1. W will not latr that th rsults of th nonlinar analysis do not dpnd on this assumption, and on can xtnd th obtaind rsults to covr th cas whn G() and S(f) do not suprimpos. Th motion of th robot ndpoint in rspons to imposd forcs, f, is causd ithr by structural complianc in th robot or by th complianc of th positioning controllr. In a simpl xampl, if a Rmot Cntr Complianc (RCC) with a linar dynamic bhavior is installd at th ndpoint of th robot, thn S is qual to th rciprocal of stiffnss (impdanc in th dynamic sns) of th RCC. Robots with tracking controllrs ar not infinitly stiff in rspons to xtrnal forcs (also calld disturbancs). Evn though th positioning controllrs of robots ar usually dsignd to allow th robots to follow th trajctory commands and rjct disturbancs, th robot ndpoint will mov somwhat in rspons to imposd forcs on it. S is calld th snsitivity function and it maps th xtrnal forc to th robot ndpoint position. For a robot with a "good" positioning controllr, S is a mapping with small gain. No assumption on th intrnal structurs of G() and S(f) is mad. W dfin s' as invrs function of th S function: f S1(yG()) (2) Damics of Two Robots Suppos two manipulators with dynamic quation 1 ar in contact with ach othr. Equations 3 and 4 rprsnt th ntir dynamic bhavior of two intracting robots. y,g,(,)+s,(f,) (3) f2 S2'(Y2G2(2)) (4) whr: y, Y2 and f, f2 Figur 1 shows th block diagram of th intraction of two robots. Not that th blocks in Figur 1 ar in gnral non. linar oprators, howvr, in th linar cas on can trat ths blocks as transfr function matrics. z Yt to ( ). 4 IS2 5,,. f2 Figur 1: Intraction f1 l of Two Robots If all th oprators of th block diagrams in Figur 1 wr transfr function matrics, thn th contact forc, f2' could b calculatd from quation 5. f2 (5,+52)' (G,,G2 2) (5) Equation 5 motivats th block diagram of Figur 2 for rprsntation of th contact forc in th systm whr V, and V2 ar givn by quations 6 and 7.

2 v, (51+ 52)1 61 V2 (51+52)162 f2 Vl1 V22 2 V2 f2..=()., Figur 2: V1 and V2 dscrib th contributions both inputs,, and 2 (6) W assum Figur 2 is valid for rprsntation of th nonlinar cas also. In othr words, considring quations 3 and 4 as original quations for dynami: bhavior of th robots, on can arriv at oprators VI and V2 to show th contributions of l and 2 on th contact forc. W assum V I and V2 ar two Lpstabl oprators" in othr words VIl,): LpnLpn and V2 (2): Lpn Lp" and also thr xsit positiv scalars (XI, (X2, I and 2 su:h that: IIVI(l)lIp(Xlllllip + I (9) IIV2(2)llp(X21I2I1p + 2 (10) S Appndix A for som dfinitions on th lp stability. Th C1osdloop!;stm for Two Rob:Q1s. Th control architctur in FigurE' 3 shows how w dvlop compliancy in th systm. is a compnsator to b dsignd for th scond robot. Th input to this compnsator is th contact forc, f2. Th compnsator output signal is bing addd vctorially with th input command vctor, r2, rsulting in th rror signal, 2 for th scond robot manipulator. On can think of this architctur as a systm that allows th scond robot to "control" th forc and th first robot to" control" th position...c) ()8 H2 81 f2 fl Figur 3: Th Closdloop Systm, th first robot controls th position and th scond robot controls th forc. Thr ar two fdback loops in th systm; th first loop (which is th natural fdback loop). iis th sam as th on shown in Figur 1. This loop shows how th contact forc affcts th robots in a natural way whn two robots ar in contact with ach othr. Th scond fdback loop is th controlld fdback loop. If two robots ar not in contact, hn th dynamic bhavior of ach robot rducs to th on rprsntd by quation 1 (with fo). which is a simpl tracking systm. Whn th robots ar in contact with ach othr. thn th contact forcs and th ndpoint positions (If robots ar givn by f,. f2. Y, and Y2 whr th following quations ar tru: y,g,(,j+s,(f,j (11) f2 S2'(Y2G2(2JJ (12) y, Y2 (13) f,+f2o (14) 2r2+H2(f2) (15) If all th oprators ar considrd linar transfr (7) (8) of function matrics, thn: f2 [5,+52+ G2H2)'[G,,G2r2) (16) W plan to choos a class of compnsators, H2' to control th contact forc with th input command r2' This controllr must also guarant th stability of th closdloop systm shown in Figur 3. Not that th robot snsitivity functions and th lctronic compliancy, H2' add togthr to form th total snsitivity of th systm. If Q, thn only th snsitivity functions of two robots add togthr to form th compliancy for th systm. By closing th loop via H2, on can not only add to th total ssitivity but also shap th snsitivity of th systm. Whn two robots ar not in contact with ach othr, th actuaj ndpoint position of ach robot is almost qual to its input trajctory command govrnd by quation 1 (with fq). Whn th robots ar in contact with ach othr, th contact forc on th scond robot follows r2 according to quations Th input command vctor, r2, is usd diffrntly for th two catgoris of manuvrings of th scond robot; as an input trajctory command in unconstraind spac (quation 1 with fq) and as a command to control th forc in constraind spac. Stabilit Th objctiv of this sction is to arriv at a sufficint condition for stability of th systm shown in Figurs 3. This sufficint condition lads to th introduction of a class of compnsators, H2' that can b usd to dvlop compliancy for th class of robot manipulators that hav positioning controllrs. Th following thorm (Small Gain Thorm) (7,8) stats th stability condition of th closdloop systm shown in Figur 4. A corollary is givn to rprsnt th siz of to guarant th stability of th systm. r2 2 =..r't ::l f2,"' l H2 Figur 4: Two Manipulators with Forc Fdback Compnsator. If conditions I, II and III hold: I. VI and V2 ar Lp stabl oprators, that is VI(,J: L np L np and V2(2J: L np L np and: oj II VI(IJllp(,II,llp+, (17) b) II V2(2Jllp( 211 2l1p+ 2 (18) n. H2 is chosn such that mapping (f2j is Lpstabl, that is 0) H2(f2J: LnpLnp (19) b) IIH2(f2Jllp(3I1f2I1p+3 (20) m. and23<1 (21) thn th closd loop systm in Figur 4 is stabl. Th proof is givn in Appndix A. Th following corollary dvlops a stability bound if H2 is slctd as a linar transfr function matrix. Coro1la Th ky paramtr in th proposition is th siz of 23. According to th proposition, to guarant th stability of th systm, H2 must b chosn such that 23<1. IfH2 is chosn as a linar oprator (th impuls rspons) whil all th othr oprators ar still nonlinar, thn: II H2(f2J lip (.,. II f2 lip (22) whr:.,. 0" max (N J (23)

3 D" maxi indicats th maximum singular valu, and N is a matrix whos ijth ntry is II H2(.J II,. In othr words, ach mmbr of N is th L, norm of ach corrsponding mmbr of H2(.) (puls rspons). Considring inquality 22, th tjiird stability condition, inquality 21, can b rwrittn as: 'i' «2< 1 (24) To guarant th closd loop stability, '/'«2 must b smallr tjian unity, or, quivalntly: 'i' <1 (25) «2 To guarant th stability of th closd loop systm, H2 must b chosn such its "siz" is smallr than t},. rciprocal of th "gain" of th forward loop mapping in FigllT 4. Not that '/' rprsnts a "siz" of H2 in th singular valu sns. Whn all th oprators ar linar transfr function matrics on can us Multivariabl Nyquist Critrion to arriv at th sufficint condition for sta!:,ility of th closd loop systm. This sufficint conditicill lads to th introduction of a class of transfr function matrics,, that stabiliz th family of linarly tratd robot manipulators. Th dtaild drivation for th stability condition is givn in Appndix B. Appndix C shows that th stability condition givn by Nyquist Critrion is a subst of tb critria givn by th Small Gain Thorm. Using th rsu:lts in Appndix B, th sufficint condition for stability is givn by inquality 26. (5 1 5 ) G ) V'CAJ, (0,00) (26) D"max(H2) < (r, '1" ' D"max,+ 2 2 Similar to th nonlinar cas, H2 must be' chosn such that its "siz" is smallr than th rciprocal of th "siz" of th forward loop mapping in Figur 5 to guarant th stability of th closd loop systm. Not that in inquality 26 D"max rprsnts a "siz" of in th singular valu sns. r2 T\ 2, 1 (51+ 52) G., Figur 5: f2 C In th Linar Cas, V2(S1 + S2)'G2 and V,(S1+ S2)'G1 Considr n=l (on dgr of frdom systm) for mor undrstanding about th stabilit:y critrion. Th stability critrion whn n=l is givn by in,quality 27. IG2H21 < \/oo(o,oo) (27) whr 1.1 dnots th magnitud of a transfr function. Sinc in many cass 1 within th bandwidth of th tracking controllr of ach robot, 000' thn H2 must b chosn such that: IH21 < \/oo(o,ooo) (28) Inquality 28 rvals som facts about th siz of H2. Th smallr th snsitivity functions of th robot manipulators ar, th smallr H2 must b chosn. In th "idal cas", no H2 can b found to allow two prfct; tracking robots (8182 OJ intract with ach othrs. In othr words, for tj1 stability of th systm shown in Figur 3, thr must b som compliancy in ithr first or s<:ond robot. RCC, structural dynamics, and th tracking controllr stiffnss form th compliancy on th robot. ITh maximum singular valu of a mat;rix A, O"max(A) is dfind as: O"max(A) mox 1 Azi TZT whr z is a nonzro vctor and 1.\ dnots th Euclidan norm. Suppos, th first robot is an idal positioning systm. In othr words, 5, has a zro gain. Thrfor th contact forc and th position of contact point btwn two robots ar: f200 (52 + G2H2)' (G,, G2 r2 ) (29) Y'oo G,[l) (30) Th first robot controls th position of th contact point, whil th othr controls th contact forc. Gnralizing this concpt to n robots, on robot controls th position of th contact point whil th othr robots control th n1 contact forcs such that: fl+f2+f3+ +fn0 (31) Summa and Cnnclusinn A nw architctur for complianc control of two cooprating robots has bn invstigatd using unstructurd modls for dynamic bhavior of robots. Each robot ndpoint follows its position input command vctor "closly" whn th robots ar not in contact with ach othr. Whn two robots com in contact with ach othr, on robot controls th position of th contact point, whil th othr controls th contact forc. Th unifid approach of modling robots is xprssd in trms of snsitivity functions. A bound for th global stability of th manipulators has bn drivd. For th stability of two robots, thr must b som initial compliancy in ithr robot. Th initial compliancy in th robots can b obtaind by a nonzro snsitivity function for th tracking controllr or a passiv compliant lmnt such as an RCC. Exampl Considr two ondgr of frdom robots with G and 5 in quation 1 givn as: G, [ 8 ) [8/5 + 1)[8/9+ 1)[8/ )[8/240+ 1)[8/290+ 1) 1 G2(6) (6/6 + 1)(6/10+ 1)(6/ )(6/ )(6/300+ 1) 8 ( ) 0.1 ( ) (6/4+1)(6/8+1) 6 (6/5+1)(6/9+1) Both robots hav good positioning capability (small gain for 8). Th pols that ar locatd at 250, 300, 290, 240 show th high frquncy mods in th robots. Th stability of th robots whn thy ar in contact with ach othr is analyzd. If w considr H2 as a constant gain, thn inquality 27 yilds that for (0.08 th valu of I G21 is always smallr than 18,+ 821 for all OOE (0,00). Figur 6 shows th plots of I G2 1 and for thr valus of H2. For H th systm is stabl with th closd loop pols locatd at (456.71, :t172.37j, 9.41, 8.38, 5.62, 4.58) whil H21 rsults in unstabl systm with th closd loop pols locatd at (800.88, 9.03, 8.05, 5.05, 4.13, 23.98:t474.35j). Not that th stability condition drivd via inquality 27 is a sufficint condition for stability; many compnsators can b found to stabiliz th systm without satisfying inquality 27. Figur 6 shows an xampl (H20.25) that dos not satisfy inquality 27 howvr th systm is stabl with closd loop pols at (598.64, 76.87:t298.04j, 9.1, 8.15, 5.19, 4.36). If on uss root locus for stability analysis, for (O. 75 all th closd loop pols will b in th lft half plan. Onc a constant valu for stabilizing H2 stablishd, on can choos a dynamic compnsator to filtr out th high frquncy nois in th forc masurmnts. H

4 L np th rsulting output blongs to L np. Morovr. th norm of th output is not largr than (X2 tims th norm of th input plus th offst constant. Dfinition 6: Th smallst (X2 such that thr xists a so that inquality b of Dfinition 5 is satisfid is calld th gain of th oprator V2[o). Dfinition 7: Lt V2[o): Lnp Lnp. Th oprator V2[oj is said to b causal if: V2[21T V2(2T) \,'T<oo nd \,' 2ELnp Figur 6; I G2H21 < is a suffic:int condition for stability. AppndiX A Dfinitions 1 to 7 will b usd in th stability proof of th closdloop systm (7,8). Dfinition 1: For all p(1,oo), w labl as Lnp th st consisting of all functions f[f1,f2 fn)t: (O,oo)_n such that: 00 fl IP dt <00 for 1, 2,...,n Dfinition 2: For all T(O. 00), th fw1ction ft dfindby: ft OltlT T<t is calld th truncation of f to th intrval [0, TJ. Dfinition 3: Th st of all functions f[fl, f2'.'" f njt: [O.oo)lRn such thatft E L np for all finit T is dnotd by L np. f by itslf mayor may not blong to L np. Dfinition 4: Th norm on L npis dfind by: II fll p whr IIfllip is dfind as: II fl lip n }"' II f.llp2 "i":1 1/2 00 J WI! fl IP dt a whr WI is th wighting factor. WI is particularly for scaling forcs and torqus of diffrnt units. IIp usful Dfinition 5: Lt V2(.):L"pL"p. W say that th oprator V2(.) is L p' stabl, if: a) V2(.): L"p L"p b) thr xist finit ral constants (X2 and 112 such that: II V2(2) IIp( (X2 li2l1p+ 2 V 2E L"p According to this dfinition w first assum that th oprator maps L "p to L "p. It is clar that if on dos not show that V2(.):L"pL"p, th satisfacticln of condition (a) is impossibl sinc L "p contains L"p. Onc th mapping of V2(.) from L"p to L"p is stablishd, thn w say that th oprator V2(.) is lpstabl if whnvr th input blongs to Proof of th nonlinar stability proposition Dfm th closdloop mapping A:(,.r2) 2 (Figur 4). 2 r2 + H2 (V,(,)V2(2)) For ach finit T, inquality A2 is tru. (AI) Il2Tllp( Ilr2Tllp+ II H2(V,(,)V2(2))Tllp 'l/t<oo (A2) H2(V,(,)V2(2)) is Lp stabl, thrfor, using inqualitis 17, 18, and 20: Il2Tllp11 r2t II p+ tx3tx1 li1tllp+ tx3tx2 li2tllp+ tx31 + tx VT < 00 (A3) Sinc tx3tx2 is lss than unity: li II Il'Tllp + IIr2Tlip 2T p 1tX3tX2 1tX3tX2 + tx3(1+ 2)+ 3 VT<oo (A4) 1tX3tX2 Inquality A4 shows that 2(.) is boundd ovr (0, T). Bcaus this rasoning is valid for vry finit T, it follows that 2(.)L"p, i.., that A:L"pL"p' Nxt w show that th mapping A is Lpstabl in th sns of dfinition 5. Sinc IIr211p and Il,lIp< 00 (thy both blong to L"p spac), thn from inquality A4: II2Tllp < 00 VT<oo (A5) In th limit whn T +00: II211p < 00 (A6) Inquality A6 implis 2 blongs to L "p spac whnvr r2 and, blong to L "p spac. With th sam rasoning from quations AI to AS, it can b shown that inquality A7 is tru. II211p ( Il,llp+ IIr211p + (X3('+2)+3(A7) 1 (X3(X 2 1 (X3 (X 2 1 (X3(X2 Appndix B Th objctiv is to find a sufficint condition for stability of th closdloop systm in Figur 5 by Nyquist Critrion. Th block diagram in Figur 5 can b rducd to th block diagram in Figur Bl whn all th oprators ar linar transfr function matrics. r2 UI "2.;: j G2H Inquality A7 shows th linar bounddnss of 2. (Condition b of dfinition 5) Inquality A7 and A6 takn togthr, guarant that th closdloop mapping A is Lpstabl. :J.. Figur B1: Simplifid Block Diagram of th Systm in Figur 5 l

5 Thr ar two lmnts in th fdback loop; G2H28,' and 828,'.8281' shows th natural forc fdback whil G2H28,' rprsnts th controlld forc fdback in th systm. Th objctiv is to us Nyquist Critrion (5) to arriv at th sufficint condition for stability of th systm whn H2 o. Th following conditions ar rgardd: 1) Th closd loop systm in Figur Bl is stabl if H20. This condition simply stats th stability of two robot manipulators. (Figur 2 shows this configjllration.) 2) H2is chosn as a stabl linar transfr function matrix. Thrfor th augmntd loop transfr function (H28,' + 828,' ) has th sam numbr of unstabl pols that 8,, has. Not that in many cass 8,' is a stabl systm. 3) Numbr of pols on Jcu axis for both loop 828,' and (G2H28,'+828,') ar qual. Considring that th systm in Figur Bl is stabl whn O, w plan to find how robust th systm is whn. G2H281' is addd to th fdback loop. If th loop transfr function 8281' (without compnsator, H2) dvlops a stabl closdloop systm, thn w ar looking for a condition on H2 such that th augmn1;d loop transfr function (G2H28,' + 828,') guarants th stability of th closdloop systm. According to th NyqlJlist Critrion, th systm in Figur Bl rmains stabl if t:h anticlockwis ncirclmnt of th dt(g2h28,'+ 8,1+ In) around th cntr of th splan is qual to th numbr of unstabl pols of th loop transfr function (G2H28,'+ 82 8,'). According to conditions 2 and 3, th loop j;ransfr functions 828,' and (G2H28,'+828,') both hav th sam numbr of unstabl pols. Th closdlo.op systm whn H20 is stabl according to condition 1; th ncirclmnts ofdt(828,1+ln) is qual to unstabl pols of 828,'. Sinc th numbr of unstabl pols of (132H28,'+8281') and that of 828,' ar th sam, thrfor for stability of th systm dt(g2h28,' '+ In) must hav th sam numbr of ncirclmnts that dt(82 8,' + In) has. A sufficint condition to guarant th quality of th numbr of ncirclmnts of dt(g2h28,' + 528,' + In) and that of dt(82 5,' + In) is that th dt(g2h28,'. 525,' + In) dos not pass through th origin of th splan for all possibl nonzro but finit valus of H2' or dt (G2H25,'+ 525,'+ In) 0 Vcu(O,oo) (B1) If inquality B1 dos not hold thn thr must b a nonzro vctor z such that: (G2H28,' ,' + In) z 0 (B2) or: H28 1'Z (828,' + In) Z (B3) A sufficint condition to guarant that qllality B3 will not happn is givn by inquality B4. umax (G2H28,')<Umln (525,'+ln)Vcu(0,OO) (B4) or a mor consrvativ condition: 1 umax (H2)«(8 5 ), G ) VCUE:(O,oo) (B5) umax,+ 2 2 Not that ( )' is th transfr function matrix that maps 2 to th contact forc, f2 whn,o.. Figur 5 shows th closdloop systm. According to th rsult of th proposition, H2 must b chosn such that th siz of H2 is smallr than th rciprocal of th siz of th forward loop transfr function, (5,+52)'G2. Appndix C Th following inqualitis ar tru whn p2 and and V2 ar linar oprators. II H2(f2Jllp ("VII f211p II V2(2J lip ( }III whr: 2 lip (Cl) (C2) Ji O"mlx(QJ, and Q is th matrix whos ijth ntry is givn by (QJ1J SUP<.>I(V2J1JI, V O"max(RJ. and Ris th matrix whos ijth ntry is givn by (RJ1J sup..,lfh2j1ji According to th stability condition, to guarant th closd loop stability Jl. v < 1 or: V< 1. (C3) Jl. Not that th following ar tru: D"max(VV ( Jl. Voo (0, 00) (C4) D"max(H2) ( V Voo (0, 00) (C5) Substituting C4 and C5 into inquality C3 which guarants th stability of th systm, th following inquality is obtaind: 1 O'max(H2) < Voo (0,00) (C6) D"max(V2) I O"max (H2)«(S S )1 G ) Vw(O,oo) (C7) O"max Inquality C7 is idntical to inquality 26. This shows that th linar stability condition by th multivariabl Nyquist Critrion is a subst of th gnral condition givn by th Small Gain Thorm. Rfrncs 1) Kazrooni, H. "Fundamntals of Robust Compliant Motion for Manipulators", IEEE Journal of Robotics and Automation, N2, V2, Jun 1986b. 2) Kazrooni, H, "A Dsign Mthod for Robust Compliant Motion of Manipulators", IEEE Journal of Robotics and Automation, N2, V2, Jun 1986c. 3) Kazrooni, H., Guo, J., "Dsign and Construction of an Activ EndEffctor", In procdings of th IEEE Intrnational Confrnc on Robotics and Automation, Raligh, North Carolina, April 1987, and also IEEE Journal of Robotics and Automation. 4) Kazrooni, H, "On th Stability of th Robot Compliant Motion Control, "In procdings of th IEEE CDC, Dc ) Lhtomaki, N. A., Sandll, N. R., Athans, M., "Robustnss Rsults in LinarQuadratic Gaussian Basd Multivariabl Control Dsigns", IEEE Transaction on Automatic Control AC26(1):7592, Fbruary, ) Spong, M. W., Vidyasagar, M., "Robust Nonlinar Control of Robot Manipulators", IEEE Confrnc on Dcision and Control", Dcmbr ) Vidyasagar, M., "Nonlinar Systms Analysis", PrnticHall, ) Vidyasagar, M., Dsor, C. A., "Fdback Systms: InputOutput Proprtis", Acadmic Prss, 1975.