Output Feedback Adaptive Robust Control of Uncertain Linear Systems With Disturbances

Transcription

1 Bn Yao 1 School of Mechancal Engneerng, Purdue Unversty, West Lafayette, IN e-mal: byao@purdue.edu L Xu Research and Advanced Engneerng, Ford Motor Company, Dearborn, MI Output Feedback Adaptve Robust Control of Uncertan Lnear Systems Wth Dsturbances In ths paper, a dscontnuous projecton based output feedback adaptve robust control (ARC) scheme s constructed for a class of lnear systems subjected to both parametrc uncertantes and dsturbances that mght be output dependent. An observer s frst desgned to provde exponentally convergent estmates of the unmeasured states. Ths observer has an extended flter structure so that on-lne parameter adaptaton can be utlzed to reduce the effect of possble large dsturbances that have known shapes but unknown ampltudes. Estmaton errors that come from ntal state estmates and uncompensated dsturbances are dealt wth va certan robust feedback at each step of the backsteppng desgn. Compared to other exstng output feedback robust adaptve control schemes, the proposed method explctly takes nto account the effect of dsturbances and uses both parameter adaptaton and robust feedback to attenuate ther effects for an mproved trackng performance. Expermental results on the control of an ron core lnear motor are presented to llustrate the effectveness and achevable performance of the proposed scheme. DOI: / Keywords: adaptve control, robust control, output feedback, lnear system, uncertantes 1 Introducton A great deal of effort has been devoted to the control of uncertan nonlnear systems 1 7 and some of the results have been extended to the output feedback control. Specfcally, Kanellakopoulos et al. ntroduced backsteppng procedure to a class of nonlnear systems n whch nonlneartes depend only on the measured sgnals 8. In9, Krstć et al. presented an alternatve approach to the adaptve control of lnear systems wth parametrc uncertantes by usng nonlnear methods such as tunng functons and nonlnear dampng. The resultng controller possesses much better transent and steady state performance as compared to the tradtonal one. Parameter convergence propertes of ths controller were also analyzed by Zhang et al. 10. Recently, Ikhouane and Krstć showed that by usng a swtchng -modfcaton 11 or smooth parameter projecton 12 n the parameter adaptaton law, the robustness of ths scheme can be mproved wth respect to both unmodeled dynamcs and bounded dsturbances. A robust adaptve controller based on -modfcaton was presented by Jang and Praly n 13, n whch asymptotc trackng s lost even n the presence of parametrc uncertantes only. In ths paper, the adaptve backsteppng approach developed n 9 s combned wth the ARC desgn procedure 7 to construct a dscontnuous projecton based output feedback ARC controller for a class of lnear systems subjected to both parametrc uncertantes and bounded dsturbances. As only output sgnal s measured, a Kresselmeer observer 14 s frst desgned to provde exponentally convergent estmates of the unmeasured states. Ths observer has an extended flter structure so that parameter adaptaton can be used to reduce the effect of possble large dsturbances, whch s very mportant for practcal applcatons 15. The destablzng effect of estmaton errors s dealt wth usng robust feedback at each step of the desgn procedure. Compared to 1 Also a Kuang-pu Professor at the State Key Laboratory of Flud Power Transmsson and Control n Zhejang Unversty, Chna. Contrbuted by the Dynamc Systems, Measurement, and Control Dvson of ASME for publcaton n the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CON- TROL. Manuscrpt receved October 28, 2005; fnal manuscrpt receved Aprl 26, Assoc. Edtor: Santosh Devasa. Paper presented at the 1999 Amercan Control Conference. the RAC approaches 11,12, the proposed scheme explctly takes nto account the effect of dsturbances and uses both parameter adaptaton and robust feedback to attenuate the effect of dsturbances for an mproved trackng performance. In addton, the desgn puts more emphass on the robust control law desgn for a guaranteed robust performance n general. In fact, when the parameter adaptaton law n ARC s swtched off, the resultng controller becomes a determnstc robust controller. Furthermore, the proposed controller acheves a guaranteed transent performance and a prescrbed fnal trackng accuracy n the sense that the upper bound on the trackng error over the entre tme-hstory s gven and related to certan controller desgn parameters n a known form, whch s more transparent than those n RAC desgns 16. At the same tme, proof of achevable performance s made smpler. The precson moton control of an ron core lnear motor s used as an applcaton case study and the expermental results are obtaned to llustrate the effectveness of the proposed method. 2 Problem Statement The followng sngle-nput sngle-output SISO plant s consdered: yt = Bs Ds ut + As As y,t + d yt 1 n whch As=s n +a n 1 s n 1 + +a 1 s+a 0, Bs=b m s m + +b 1 s +b 0, Ds=d l s l + +d 1 s+d 0, and mln. The plant parameters a and b are unknown constants. For smplcty, d are assumed to be known but the results can be extended to the case where d are unknown constants wthout much dffculty. d y t represents the output dsturbance, and y,t represents any dsturbance comng from the ntermedate channels of the plant. Unlke the RAC approaches 11,12, n the followng, the dsturbance y,t wll be explctly taken nto account and handled as follows: we frst use the pror nformaton about the nature of the dsturbance to construct a nomnal dsturbance model n y,t=qy,t T c, n whch qy,t=q p y,t,...,q 1 y,t T R p represents the vector of known bass shape functons, c =c p,...,c 1 T represents the vector of unknown magntudes. Ths 938 / Vol. 128, DECEMBER 2006 Copyrght 2006 by ASME Transactons of the ASME

2 nomnal dsturbance model wll be used explctly n the followng controller desgn to mprove the achevable output trackng performance, whle the dsturbance modelng error, = n, wll be dealt wth va certan robust feedback for a robust performance. A state space realzaton of the plant 1 s thus gven by ẋ 1 = x 2 a n 1 x 1 ẋ n l 1 = x n l a l+1 x 1 ẋ n l = x n l+1 a l x 1 + d l q T y,tc + d l ẋ 1 = x a m+1 x 1 + d m+1 q T y,tc + d m+1 ẋ = x +1 a m x 1 + d m q T y,tc + d m + b m ut ẋ n = a 0 x 1 + d 0 q T y,tc + d 0 + b 0 u y = x 1 + d y where =n m s the relatve degree of the system. For smplcty, the followng notatons are used: for the th component of the vector, mn for the mnmum value of, and max for the maxmum value of. The operaton for two vectors s performed n terms of the correspondng elements of the vectors. Defnng a vector of unknown parameters = a n 1,..., a 0,b m,...,b 0,c p,...,c 1 T R m+n+p+1, the followng standard assumptons are made: 1. The nomnal plant s of mnmum phase,.e., the polynomal Bs s Hurwtz. The plant order n, relatve degree =n m, and the sgn of the hgh frequency gan sgnb m are known. 2. The extent of parametrc uncertantes, dsturbance modelng error t, output dsturbance d y t as well as ts dervatve ḋ y t satsfy : mn max : y,t t d y d d y :d y t d t ḋ y f ḋ y :ḋ y t f t where mn and max are known, and t, d t and f t are unknown but bounded functons. Gven the reference trajectory y r t, the objectve s to synthesze a control nput u such that the output y tracks y r t as closely as possble n spte of varous model uncertantes. The reference sgnal y r and ts frst dervatves are assumed to be known and bounded. In addton, y r s pecewse contnuous. 3 State Estmaton As only the output y s measured, the K-flters 2 are utlzed to provde exponentally convergent estmates of the unmeasured states as follows. Rewrte 2 n the form where ẋ = A 0 x + k ax 1 + dq T y,tc + bu + d y = x 1 + d y A 0 = k1 k n I n 1 0 k 0 b = b m b 0 = k1 k n a = an 1 d = 0n l 11 d a 0 b = 0 11 b d = dl d 0 By sutably choosng k, the observer matrx A 0 wll be stable. Thus, there exsts a symmetrc postve defnte s.p.d. matrx P such that PA 0 + A T 0 P = I P= P T 0 6 Followng the desgn procedure of 2, the K-flters are gven by = A 0 + e n y n = A 0 n + k y = A 0 + e n u 0 n 1 0 m = A 0 + dq y,t 1 p where e denotes the th standard bass vector n R n. Let and be the n-dmensonal state vectors of the fltered nput and output by = A 0 + e n y 8 = A 0 + e n u Then, the flter states and n 7 can be obtaned from the followng algebrac expressons 2 n = A 0 n 5 7 = A 0 0 n 1 9 = A 0 0 m It s noted that the last equaton of 7 s ntroduced so that parameter adaptaton can be used to reduce the parametrc uncertanty comng from c to acheve a better dsturbance rejecton capablty. The state estmates can thus be represented by n 1 xˆ = n a + =0 b + =0 c 10 =1 Let x =x xˆ be the estmaton error. From 4, 7, and 10, t can be verfed that the observer error dynamcs s gven by x = A 0 x + a k d y + d 11 The soluton of ths equaton can be wrtten as x = + u 12 where s the zero nput response satsfyng =A 0, and u =0 t m e A 0 t a k d y + d y,d t 0 13 s the zero state response. Notng assumpton 2 and that matrx A 0 s stable, one has u u : u t t 14 where t s a vector of unknown but bounded functons. In the followng controller desgn, and u wll be treated as dsturbances and dealt wth usng robust control functons to acheve a guaranteed robust performance. p Journal of Dynamc Systems, Measurement, and Control DECEMBER 2006, Vol. 128 / 939

3 4 Dscontnuous Projecton Based ARC Backsteppng Desgn 4.1 Parameter Projecton. Let ˆ denote the estmate of and the estmaton error.e., =ˆ. From assumpton 2, the dscontnuous projecton based ARC desgn 7 can be generalzed to solve the robust trackng control problem for 1. Specfcally, the parameter estmate ˆ s updated through a parameter adaptaton law havng the form of ˆ = Proj ˆ 15 where 0 s a dagonal matrx, and s an adaptaton functon to be syntheszed later. The projecton mappng Proj ˆ =Proj ˆ1 1,...,Proj ˆm+n+p+1 m+n+p+1 T s defned n 17,5 as f ˆ = max and 0 Proj ˆ =0 0 f ˆ = mn and 0 otherwse 16 It can be shown that for any adaptaton functon, the projecton mappng 16 guarantees P1 ˆ = ˆ: mn ˆ max P2 T 1 Proj ˆ 0, Controller Desgn. The desgn combnes the adaptve backsteppng desgn 9 wth the ARC desgn procedure 7,6. In the followng, the states of the system are replaced by ther estmates and the estmaton errors are dealt wth at each step va robust feedback to acheve a guaranteed robust performance. Step 1: From 2, the dervatve of the output trackng error z 1 =y y r s ż 1 = x 2 a n 1 y + a n 1 d y + ḋ y ẏ r 18 Snce x 2 s not measured, t s replaced by ts expresson from 10 x 2 = n,2 2 a + 2 b + 2 c + x2 where x2 = 2 + u2 s the estmaton error of x 2, and 2 = n 1,2,..., 0,2 2 = m,2,..., 0, = p,2,..., 1,2 n whch,j represents the jth element of. Substtutng 19 nto 18 gves ż 1 = b m m,2 + n,2 + T y r where T = 2, 2, 2 +e *T * 1 y, = e n+1 m,2, 1=a n 1 d y +ḋ y u2, and e * s the th standard bass vector n R n+m+p+1.if m,2 were the nput, one would synthesze a vrtual control law 1 for m,2 such that z 1 s as small as possble 1 y,, m+1,,ˆ,t = 1a + 1s 1a = 1 n,2 + ˆ T ẏ r bˆ m 22 where = 1,..., T. In 22, 1a functons as an adaptve model compensaton law used to acheve an mproved model compensaton through on-lne parameter adaptaton gven by 15, and 1s s a robust control law to be syntheszed later. Snce the sgn of b m s known see assumpton 1, wthout loss of generalty, one can assume b m 0. It s thus reasonable to expect that the lower bound of b m s postve,.e., b m mn = n+1 mn 0. Then, due to the use of the projecton 15, from P1 of 17, bˆ m b m mn 0, whch mples that the control functon 22 s well defned. Let z 2 = m,2 1 denote the nput dscrepancy. Substtutng 22 nto 21 gves ż 1 = b m z 2 + 1s T * where 1 +e n+1 1a. In the tunng functon based backsteppng adaptve control 2, one of the key ponts s to ncorporate the adaptaton functon or tunng functon n the constructon of control functons to compensate for the destablzng effect of the tme-varyng parameter estmates. Here, due to the use of dscontnuous projecton 16, the adaptaton law 15 s dscontnuous and thus cannot be used n the control law desgn at each step. Snce the backsteppng desgn needs the control functon syntheszed at each step to be suffcently smooth n order to obtan ts partal dervatves. In the followng, t wll be shown that ths desgn dffculty can be overcome by strengthenng the robust control law desgn. The robust control functon 1s conssts of three terms gven by 1 1s = 1s1 + 1s2 + 1s3 1s1 = k 1s z 1 24 b m mn where 1s2 and 1s3 are robust control desgned n the followng and k 1s s any nonlnear feedback gan satsfyng k 1s g 1 + C g n whch C 1 s a postve defnte constant dagonal matrx to be specfed later. Substtutng 24 nto 23 gves b m ż 1 = b m z 2 k 1s z 1 + b m 1s2 + 1s3 T b m mn Defne a postve semdefnte p.s.d. functon V 1 as V 1 = 1 2 z 1 2 In vew of 26, ts tme dervatve satsfes V 1 b m z 1 z 2 k 1s z z 1 b m 1s2 T 1 + z 1 b m 1s From assumpton 2, one has T 1 M 1 29 where M = max mn. Thus T 1 s bounded by a known functon, whch ensures that there exsts a robust control functon satsfyng the followng condtons 18: z 1 b m 1s2 T z 1 1s2 0 where 11 s a postve desgn parameter. Essentally, condton of 30 shows that 1s2 s syntheszed to attenuate the effect of parametrc uncertantes wth the level of control accuracy beng measured by 11, and condton s to make sure that 1s2 s dsspatng n nature so that t does not nterfere wth the functonalty of the adaptve control law 1a. Smlarly, notng assumpton 2 and 14, one may obtan 1 1t a n 1 d t + f t t 31 Note that 1 s an unknown but bounded functon. In prncple, the same strategy as n 30 can be used to desgn a robust control functon 1s3 to attenuate 1. However, snce the bound of 1 s unknown, t s mpossble to prespecfy the level of control accuracy. So a more relaxed requrement compared to the condton of 30 s gven z 1 b m 1s / Vol. 128, DECEMBER 2006 Transactons of the ASME

4 where 12 s a postve desgn parameter. Remark 1. One smooth example of 1s2 satsfyng 30 can be found n the followng way. Let h 1 be any smooth functon satsfyng h 1 M Then, 1s2 can be chosen as 6,7 h 1 1s2 = z 1 4b m mn 11 An example of 1s3 satsfyng 32 s gven by s3 = z b m mn 12 Other smooth or contnuous examples of the robust control functons satsfyng condtons 30 can be worked out n the same wayasn6,7,19. Step 2: From 22, 9, and 8, and notng the rearrangements of 18 21, the dervatve of 1 s computed as 1 = 1c + 1u 1c = 1 y n,2 + ˆ T m+1 1 j j p 1 + j,2 + 1 j,2 t 1u = 1 y T ˆ ˆ 36 Snce,, and j are gven n 8 and 7, 1c s calculable and can be used n the desgn of control functons. However, 1u s not due to varous uncertantes. Therefore, 1u has to be dealt wth va robust feedback n ths step. From 7 and 36, the dervatve of z 2 = m,2 1 s ż 2 = m,3 k 2 m,1 1c 1u Consder an augmented p.s.d. functon gven by V 2 = V z 2 2 From 28 and 37, t follows that V 2 V z 2 b m z 1 + m,3 k 2 m,1 1c 1u where V 1 1 s a shorthand notaton for k 1s z 2 1 +z 1 b m T 1s2 1 +z 1 b m 1s Smlar to 22, the ARC control functon 2 for the vrtue control nput m,3 n 37 conssts of 2 y,, m+2,,ˆ,t = 2a + 2s 2a = bˆ mz 1 + k 2 m,1 + 1c 2s = 2s1 + 2s2 + 2s3 2s1 = k 2s z 2 k 2s g 2 + C C ˆ 40 where g 2 0 s a constant, C 2 and C 2 are postve postve defnte constant dagonal matrces, 2s2 and 2s3 are robust control functons to be chosen later. Substtutng 40 and 36 nto 39 and usng smlar technques as n 23, one may have V 2 V z 2 z 3 k 2s z z 2 2s2 T 2 + z 2 2s z ˆ 2 ˆ n whch z 3 = m,3 2 represents the nput dscrepancy, and * 2 = e n+1 z 1 1 y 2 = 1 y From 31, t follows that 2 1 /y 1. Smlar to 30 and 32, the robust control functons 2s2 and 2s3 are chosen to satsfy z 2 2s2 T 2 21 z 2 2s z 2 2s2 0, z 2 2s3 0 where 21 and 22 are postve desgn parameters. As n Remark 2, examples of 2s2 and 2s3 are gven by 2s2 = h z 2 2s3 = z 2 y 44 n whch h 2 s any smooth functon satsfyng h 2 M From 28 and 41, the dervatve of V 2 satsfes the followng nequalty: 2 V 2 z 2 z 3 k js z 2 j + z 1 b m 1s2 T 1 + z 1 b m 1s z 2 2s2 T 2 + z 2 2s ˆ ˆ z 2 46 Step 3: Mathematcal nducton wll be used to prove the general results for all ntermedate steps of the desgn. At step, the same ARC desgn as n the above two steps wll be employed to construct an ARC control functon for the vrtual nput m,+1. For step j, 3 j 1, let z j = m,j j 1 and recursvely defne j = j 1 y j = j 1 y 1 47 LEMMA 1. At step, choose the desred ARC control functon as y,, m+,,ˆ,t = a + s a = z 1 + k m,1 + 1c s = s1 + s2 + s3 s1 = k s z k s g C 2 ˆ where g 0 s a constant, C and C are postve defnte constant dagonal matrces, s2 and s3 are robust control functons satsfyng z s2 T 1 Journal of Dynamc Systems, Measurement, and Control DECEMBER 2006, Vol. 128 / 941

5 z s and z s2 0, z s3 0 1c = 1 y n,2 + ˆ T j + 1 j t Then the th error subsystem s p m+ 1 1 j j ż = z +1 z 1 k s z + s2 T + s3 + 1 ˆ ˆ and the dervatve of the augmented p.s.d. functon V = V z 2 52 Fg. 1 Trackng errors for snusodal trajectory wthout load satsfes V z z +1 k js z 2 j + z 1 b m 1s2 T 1 + z j js2 T j + z 1 b m 1s z j js3 + j j 1 ˆ z j ˆ 53 The proof of the Lemma s gven n the Appendx. Step : Ths s the fnal desgn step, n whch the actual control u wll be syntheszed such that m, tracks the desred ARC control functon 1. The dervatve of z can be obtaned as ż = m,+1 + u k m,1 1c 1 y T ˆ ˆ 54 If m,+1 +u were the vrtual nput, 54 would have the same form as the ntermedate step. Therefore, the general form apples to Step. Snce u s the actual control nput, t can be chosen as u = m,+1 55 where s gven by 48. Then, z +1 =u+ m,+1 =0. THEOREM 1. Let the parameter estmates be updated by the adaptaton law 15 n whch s chosen as = j z j 56 If dagonal controller gan matrces C j and C k are chosen such 2 that c kr 4 1/c 2 jr, where c jr and c kr are the rth dagonal element of C j and C k, respectvely. Then, the control law 55 guarantees that A In general, the control nput and all nternal sgnals are bounded. Furthermore, V s bounded above by V t exp tv exp t 57 where =2 mng 1,...,g, 1 = j1, 2 = j2, and 1 stands for the nfnty norm of 1t. B If after a fnte tme t 0, =0 and d y =0.e., n the presence of parametrc uncertantes and modeled dsturbance n =qy,t T c only, then, n addton to results n A, asymptotc output trackng or zero fnal trackng error s also acheved. Proof of the Theorem s gven n the Appendx. 5 Experment Results To llustrate the above desgns, experment results are obtaned for the trackng control of the x axs of a two-axs postonng stage detaled n 20 whose smplfed model s ẋ 1 = x 2 a 1 x 1 ẋ 2 = b 0 u + y,t y = x 1 58 where x 1 and x 2 are the states, y s the poston of the nerta load of the lnear motor, u s the control nput, and y,t represents the lumped dsturbance consstng of the electromagnetc coggng force 21, frcton force, and any external dsturbances. The coggng force n y,t s a perodc functon of the poston y. Its perod depends on the motor magnet s ptch P=60 mm. The coggng force can be approxmated qute accurately by the frst several harmoncs 21,22. For smplcty, n the proposed controller, only the fundamental and the thrd harmoncs are used to approxmate the perodc coggng force. Hence, the nomnal dsturbance model s chosen as n =q T c, where qy =cos2y/p,sn2y/p,cos6y/p,sn6y/p T s a vector of the known bass functons and c=c 4,c 3,c 2,c 1 T s a vector of the unknown magntudes. Thus the unknown parameter vector to be adapted s = a 1,b 0,c 4,c 3,c 2,c 1 T. For reference, standard least-square dentfcaton s performed to obtan the parameters of the system, and the nomnal value of a 1 s 12.5 and b 0 s 50. To test the learnng capablty of the proposed ARC algorthms, a 9 kg load s mounted on the motor and the dentfed values of the parameters are a 1 =4.2 and b 0 =16.7. The bounds descrbng the uncertan ranges n 3 are chosen as mn = 30,10, 10, 10, 10, 10 T, and max = 1.5,80,10,10,10,10 T. The control system s mplemented usng a dspace DS1103 controller board. The controller executes programs at a samplng rate of f s =2.5 khz. The followng two controllers are compared: ARC1: The output feedback ARC law syntheszed n Sec. 4. All the roots of the observer polynomals are placed at s= 942 / Vol. 128, DECEMBER 2006 Transactons of the ASME

6 Fg. 2 Parameter estmaton of ARC1 Fg. 4 Trackng errors for snusodal trajectory wth load 200, whch leads to k 1 =400 and k 2 = The controller parameters are: g 1 =200, 11 = 12 =510 3, C 1 =10 5 dag5,0.5,10,10,10,10,10, g 2 =210, 21 =0.1, 22 =1, C 2 =C 1, C 2 =10 5 I 3. The adaptaton rate s =10 6 dag2.5,1,100,100,100,100. The ntal parameter estmates are: ˆ0= 6.25,25,0,0,0,0 T. ARC2: The same control law as the ARC1 but wthout coggng force compensaton,.e., lettng =10 6 dag2.5,1,0,0,0,0. The motor s frst commanded to track a snusodal trajectory y r =0.05 sn2t. The trackng errors are shown n Fg. 1. As seen, ARC1 acheves a better trackng performance than ARC2. Ths llustrates the effectveness of coggng force compensaton. Parameter estmaton of ARC1 s gven n Fg. 2. It shows that the adaptaton algorthm s able to pck up the actual values of the two unknown parameters very quckly. The control nputs of the two controllers, whch are well below the physcal lmts ±10 V, are gven n Fg. 3. To test the performance robustness of the algorthms to parameter varatons, the motor s then run wth the 9 kg payload mounted on t. The trackng errors are gven n Fg. 4. It shows that both controllers acheve good trackng performance n spte of the change of nerta load, and agan, ARC1 performs better than ARC2. The motor s then commanded to track a fast pont-to-pont moton trajectory shown n Fg. 5. Followng ths trajectory, the motor s expected to run back and forth between pont A and pont B, wth a top velocty of ±1 m/ s and a top acceleraton of ±12 m/s 2. The trackng errors of both controllers are shown n Fg. 6. As seen, ARC1 outperforms ARC2. Furthermore, when the velocty s zero, the trackng accuracy of ARC1 s close to the optcal encoders resoluton. Fnally, the motor s commanded to track a slow pont-to-pont moton trajectory smlar to the trajectory shown n Fg. 5, but wth a top velocty of ±0.1 m/ s and a top acceleraton of ±2 m/s 2. The trackng errors are shown n Fg. 7. ARC1 agan performs better than ARC2, whch further llustrates the effectveness of coggng force compensaton n low-speed applcatons. 6 Conclusons In ths paper, an output feedback ARC scheme based on dscontnuous projecton s presented for a class of lnear systems havng both parametrc uncertantes and dsturbances that mght be output dependent. In contrast to other exstng robust adaptve control Fg. 3 Control nputs for snusodal trajectory wthout load Fg. 5 Fast pont-to-pont moton trajectory Journal of Dynamc Systems, Measurement, and Control DECEMBER 2006, Vol. 128 / 943

7 Appendx Proof of Lemma 1 It s easy to check that the frst two steps satsfy the Lemma. So let us assume that the lemma s vald for step j, j 1, and show that t s also true for step to complete the nducton process. From A1 y Thus, there exst s2 and s3 satsfyng 49 7,19. The control law 48 can then be formed. The dervatve of z s gven by Fg. 6 Trackng errors for the fast pont-to-pont moton trajectory schemes, the proposed controller uses on-lne parameter adaptaton to compensate for the nonlnear dsturbances that can be modeled. The uncompensated dsturbances and the estmaton errors of unmeasured states are effectvely handled va certan robust feedback to acheve a robust performance. The resultng controller acheves a guaranteed transent performance and a prescrbed fnal trackng accuracy n the presence of both parametrc uncertantes and bounded dsturbances. In the presence of parametrc uncertantes only, asymptotc output trackng s acheved wthout usng swtchng or nfnte-gan feedback. Expermental results verfed the effectveness of the proposed scheme. Acknowledgment Ths work s supported n part by the US Natonal Scence Foundaton Grant No. CMS and n part by the Natonal Natural Scence Foundaton of Chna NSFC under the Jont Research Fund Grant for Overseas Chnese Young Scholars. Fg. 7 Trackng errors for the slow pont-to-pont moton trajectory ż = m,+1 k m,1 1c 1 y T ˆ ˆ A2 After substtutng m,+1 =z +1 + and 48 nto A2, t s straghtforward to verfy that 51 and 53 are satsfed for step. Ths completes the nducton process. Proof of Theorem 1 Notng z +1 =0, from 53, 30, 32, 43, and 49, the dervatve of the Lyapunov functon satsfes V g j + + j1 + j2 1 2 z j j 1 ˆ 2 j 1 C j2 + C j j ˆ z 2 j ˆ A3 n whch the fact 0 /ˆ =0 s used. By completon of square, t follows that j 1 z j ˆ ˆ z j j 1 C j C 1 j ˆ ˆ j 1 z 2 j + 1 ˆ 4 C 1 j ˆ 2 A4 Notng that C 1 j and are dagonal matrces, from 15 and 56, one further has 1 Cj ˆ 2 = C 1 j Proj ˆ 2 C 1 j 2 C 1 j k z k 2 k=1 C 1 j k 2 z k 2 k=1 A5 If C j and C k satsfy the condtons n the theorem, substtutng A5 nto A4 gves j 1 z j ˆ ˆ In vew of A6, A3 becomes j 1 C j2 z 2 j + ˆ 4 k=1 C 1 j k 2 2 z k j 1 C j2 z 2 j + ˆ C k k 2 2 z k k=1 A6 944 / Vol. 128, DECEMBER 2006 Transactons of the ASME

8 V g j z 2 j + j1 + j2 1 2 V A7 whch leads to 57. Snce Bs s Hurwtz, the zero dynamcs s stable. Followng the standard adaptve control arguments as n 2, t can be proved that all nternal sgnals are bounded. A of the theorem s thus proved. The followng s to prove B of the theorem. In the presence of parametrc uncertantes only.e., =0 and d y =0, 1= 2. From 43 and 49, t s easy to check that z 1 b m 1s and z j js3 + j j2 2 2,,...,. Thus, notng 56 and condton of 30, 43, and 49 from 53 and A6, one has V T j z j g j z 2 j + j2 2 2 = g j z 2 j T Defne a new p.d. functon V as V = V T 1 + T P A8 A9 where 2. Notng P2 of 17 and the fact that =A 0, from A8 and 6, the dervatve of V s V g j z 2 j T T 1 ˆ 2 2 g j z j A10 n whch P2 of 17 s used. Therefore, zl 2 2. It s also easy to check that ż s bounded. So, z 0 as t by the Barbalat s lemma, whch leads to B of Theorem 1. References 1 Polycarpou, M. M., and Ioannou, P. A., 1993, A Robust Adaptve Nonlnear Control Desgn, n Proceedngs of the Amercan Control Conference, pp Krstć, M., Kanellakopoulos, I., and Kokotovć, P. V., 1995, Nonlnear and Adaptve Control Desgn, Wley, New York. 3 Freeman, R. A., Krstć, M., and Kokotovć, P. V., 1996, Robustness of Adaptve Nonlnear Control to Bounded Uncertantes, n IFAC World Congress, Vol. F, pp Pan, Z., and Basar, T., 1996, Adaptve Controller Desgn for Trackng and Dsturbance Attenuaton n Parametrc-Strct-Feedback Nonlnear Systems, n IFAC World Congress, Vol. F, pp Yao, B., and Tomzuka, M., 1996, Smooth Robust Adaptve Sldng Mode Control of Manpulators Wth Guaranteed Transent Performance, ASME J. Dyn. Syst., Meas., Control, 1184, pp Yao, B., and Tomzuka, M., 1997, Adaptve Robust Control of SISO Nonlnear Systems n a Sem-Strct Feedback Form, Automatca, 335, pp Yao, B., 1997, Hgh Performance Adaptve Robust Control of Nonlnear Systems: A General Framework and New Schemes, n Proceedngs of the IEEE Conference on Decson and Control, pp Kanellakopoulos, I., Kokotovć, P. V., and Morse, A. S., 1991, Adaptve Output-Feedback Control of Systems Wth Output Nonlneartes, n Foundatons of Adaptve Control, P. V. Kokotovć, ed., Sprnger-Verlag, Berln, pp Krstć, M., Kanellakopoulos, I., and Kokotovć, P. V., 1994, Nonlnear Desgn of Adaptve Controllers for Lnear Systems, IEEE Trans. Autom. Control, 39, pp Zhang, Y., Ioannou, P. A., and Chen, C. C., 1996, Parameter Convergence of a New Class of Adaptve Controllers, IEEE Trans. Autom. Control, 4110, pp Ikhouane, F., and Krstć, M., 1995, Robustness of the Tunng Functons Adaptve Backsteppng Desgn for Lnear Systems, n Proceedngs of IEEE Conference on Decson and Control, pp Ikhouane, F., and Krstć, M., 1998, Adaptve Backsteppng wth Parameter Projecton: Robustness and Asymptotc Performance, Automatca, 344, pp Jang, Z. P., and Praly, L., 1998, Desgn of Robust Adaptve Controllers for Nonlnear Systems wth Dynamc Uncertantes, Automatca, 347, pp Kresselmeer, G., 1977, Adaptve Observers wth Exponental Rate of Convergence, IEEE Trans. Autom. Control, 224, pp Yao, B., Al-Majed, M.and Tomzuka, M., 1997, Hgh Performance Robust Moton Control of Machne Tools: An Adaptve Robust Control Approach and Comparatve Experments, IEEE/ASME Trans. Mechatron., 22, pp Ioannou, P. A., and Sun, J., 1996, Robust Adaptve Control, Prentce-Hall, New Jersey. 17 Sastry, S., and Bodson, M., 1989, Adaptve Control: Stablty, Convergence and Robustness, Prentce-Hall, Englewood Clffs, NJ. 18 Yao, B., and Tomzuka, M., 2001, Adaptve Robust Control of MIMO Nonlnear Systems n Sem-Strct Feedback Forms, Automatca, 379, pp Parts of the paper were presented n the IEEE Conference on Decson and Control, pp , 1995, and the IFAC World Congress, Vol. F, pp , Yao, B., and Tomzuka, M., 1997, Adaptve Robust Control of Nonlnear Systems: Effectve Use of Informaton, n Proceedngs of the 11th IFAC Symposum on System Identfcaton, pp nvted. 20 Xu, L., and Yao, B., 2001, Adaptve Robust Precson Moton Control of Lnear Motors wth Neglgble Electrcal Dynamcs: Theory and Experments, n IEEE/ASME Trans. Mechatron., 64, pp Braembussche, P. V., Swevers, J., Brussel, H. V., and Vanherck, P., 1996, Accurate Trackng Control of Lnear Synchronous Motor Machne Tool Axes, Mechatroncs, 65, pp Yao, B., and Xu, L., 1999, Adaptve Robust Control of Lnear Motors for Precson Manufacturng, n Proceedngs of the IFAC 99 World Congress, Vol. A, pp Journal of Dynamc Systems, Measurement, and Control DECEMBER 2006, Vol. 128 / 945