Exponentially Convergent Controllers for n-dimensional. Nonholonomic Systems in Power Form. Jihao Luo and Panagiotis Tsiotras

Transcription

1 997 Aerican Control Conference Albuquerque, NM, June 4-6, 997 Exponentially Convergent Controller for n-dienional Nonholonoic Syte in Power For Jihao Luo and Panagioti Tiotra Departent of Mechanical, Aeropace and Nuclear Engineering Univerity of Virginia Charlotteville, VA Abtract Thi paper introduce a new ethod for contructing exponentially convergent control law for n-dienional nonholonoic yte in power for. The ethodology i baed on the contruction of a erie of invariant anifold for the cloedloop yte under a linear control law. A recurive algorith i preented to derive a feedback controller which drive the yte exponentially to the origin. A nuerical exaple illutrate the propoed theoretical developent.. Introduction Nonholonoic control yte coonly arie fro echanical yte when non-integrable contraint are ipoed on the otion, e.g., velocity contraint, which can not be integrated to generate contraint on the conguration pace. Exaple include a rolling dik [], obile robot [2] and underactuated yetric rigid pacecraft [3, 4]. One of the ain reaon thee yte have attracted uch attention in the pat few year i that they repreent \inherently nonlinear" yte in a certain ene. For exaple, thee yte are controllable but not tabilizable by a ooth tatic or dynaic tate feedback control law []. A nuber of approache have been propoed to olve the tabilization proble for nonholonoic yte. Thee ethodologie can be broadly claied a dicontinuou, tieinvariant tabilization and tie-varying (uually ooth) tabilization. The non-oothne of tie-invariant feedback control i a conequence of the tructural propertie of the yte []. Stabilization reult uing non-ooth, tie-invariant control law have been propoed in [3, 4, 6, 7, 8]. Reference [3, 4] deal with the attitude tabilization of underactuated pacecraft by developing non-ooth, tie-invariant control law. Piecewie continuou tabilization controller have been reported in [6, 7]. A nonooth tranforation wa ued to develop tieinvariant, exponential convergent controller in [8]. Saon in [9] howed how to ayptotically tabilize a obile robot to a point uing tie-varying, ooth tate feedback. Coron in [0] proved that all controllable driftle yte could be tabilized to an equilibriu point uing ooth, periodic, tievarying feedback. Reference [, 2] and [3] deal with the contruction of tie-varying control law for everal nonholonoic yte. Hybrid feedback tie-varying control law are contructed for a cla of cacade nonlinear yte in [4], which could alo be ued for tabilizing a cla of nonholonoic yte, a well a for olving tracking proble. Reference [] and [6] develop tie-varying control law of exponen- Thi work wa upported in part by the National Science Foundation under Grant CMS tial convergence with repect to hoogeneou nor. Finally, [7] develop nonooth, tie-varying feedback control law which guarantee global, ayptotic tability with exponential convergence about an arbitrary conguration. For a ore coprehenive review of all the recent advance in the control of nonholonoic yte the intereted reader ay conult [8]. The analyi of dynaic yte i often iplied by the introduction of canonical or noral for, that i, yte of equation which all yte in a given faily are \equivalent" to. For nonholonoic yte there are two noral for which have been ued extenively in the pat, naely, the chained for and the power for. The atheatical odel of an n-dienional nonholonoic yte in power for with two input can be decribed a [7, 8] _x = u _x j = (j ; 2)! xj;2 u 2 j =2 3 ::: n () Although not all nonholonoic yte can be tranfored into chained or power for, a large nuber of echanical yte encountered in practice can be converted into thee for. Wheeled robot, ulti-trailer, underactuated yetric rigid pacecraft are only but a few exaple. In thi paper we derive feedback control law for nonholonoic yte in power for with two input. We rthow that a et of invariant anifold can be contructed for n- dienional nonholonoic yte in power for. Thee anifold are derived fro the exact cloed-loop yte olution ubject to a linear feedback law. The derivation of thee anifold for yte in power for rt appeared in [7] no controller were derived for the general cae, however. Here we ue thee anifold to introduce tate and input tranforation. The tranfored yte i till in power for but of reduced dienion. By repeating the proce we end up with a 3-dienional yte in power for which i eay to tabilize. We how that the tabilization of thi yte iplie the tabilization of the original n-dienional yte. The reulting controller are iilar in for to the one propoed in [8], dicontinuou controller for chained yte were contructed by a nonlinear tranforation ( proce). The approach propoed here, on the other hand, generate controller with ulti-tie cale convergent propertie, a a reult of the invariable anifold ethod, which i not preent in [8]. The paper i organized a follow. Section 2 derive the invariant anifold and dicue their propertie. In Section 3 we preent a recurive algorith for power for yte baed on the derived invariant anifold. We alo provide an ex-

2 ponentially convergent controller for a 3-dienional nonholonoic yte in power for. The ain reult of the paper i given in Section 4 (Theore 4.). We baically how that the feedback control law for the 3-dienional generated yte in power for can be ued to ake the original n-dienional yte exponentially converge to the origin with a proper choice of the control gain. A nuerical exaple in Section illutrate the theoretical developent. 2. Invariant Manifold and Their Propertie Conider the yte in Eq. () and the following linear feedback u = u ;kx = u 2 ;kx 2 k>0 (2) With thi linear control law, the cloed-loop equation are _x = ; kx _x j = ; (j ; 2)! kxj;2 x 2 j =2 3 ::: n (3) Equation (3) can be explicitly integrated to obtain x (t) = x 0 e ;kt x 2(t) = x 20 e ;kt (4) x j(t) = j;2(x 0)+ (j ; )! xj;2 0 x20 e;(j;)kt j =3 4 ::: n, x 0 =[x 0 x 20 x 30 ::: x n0] T 2< n i the initial tate of the yte, and j = x j+2 ; (j + )! xj x2 j = 2 ::: n; 2 () Each equation in Eq. () dene a ooth function in ter of x x 2 ::: x n. Therefore, Eq.() dene a erie of ooth anifold by j = fx 2< n : j(x) =0g j = 2 ::: n; 2 (6) each of dienion n ;. Conider the following ooth anifold \ n;2 = j= j = fx 2< n : j(x) =0 j j = 2 ::: n; 2g Since rank = n ; 2 i iatwo-dienional ooth anifold [9]. Lea 2.Conider the yte in Eq. () under the feedback control in Eq. (2) and the anifold. Then, for all initial condition x 0 2 the cloed-loop trajectorie of the yte will tend to the origin exponentially, with rate of decay k. Proof: Firt we how that each anifold j i invariant for the cloed-loop yte. Indeed, _ j = _x j+2 ; (j + )! (jxj; _x x 2 + x j _x 2) = ; k j! xj x2 + (j +)! i ; k (j +)x j x2 =0 Subequently, the anifold i invariant for Eq. (3). For x 0 2 the olution of the cloed-loop yte are given by x (t) = x 0 e ;kt x j(t) = (j ; )! xj;2 0 x20e;(j;)kt j =2 3 ::: n The aertion of the lea follow iediately. The idea of contructing invariant anifold by directly integrating a cloed-loop yte ubject to linear feedback ha been initially ued in [4] to derive controller for underactuated yetric pacecraft. Thi idea wa ubequently generalized to nonholonoic yte in power for in [7]. 3. A Recurive Algorith for Syte in Power For In thi ection we preent a recurive algorith to create a erie of yte which will be ued to contruct convergent feedback controller for the yte in Eq. (). All the yte generated by thi recurive proce (herein called the generated yte) can be put into power for through a linear tranforation. Thee generated yte are, however, of reduced dienion. The ethodology i baed on the idea that by contructing a et of (n ; 2) anifold for the n-dienional yte, the proble of contructing convergent controller for the initial yte becoe one of contructing convergent controller for a iilar yte in power for but of dienion (n ; ). By repeating thi proce, we end up with a yte in power for of 3-dienion. 3.. The Recurive Proce Conider the n-dienional yte a given in Eq. (), and contruct a et of (n ; 2) invariant anifold under the linear feedback u = ; kx, u 2 = ; kx 2 a in Eq. (). Dene the following linear tranforation x 2 = x x 2 j = j j; j =2 3 ::: n; (8) Then one obtain the following yte in ter of x 2 j, for j n ;. _x 2 = u 2 (9a) _x 2 j = (j ; 2)! xj;2 2 u2 2 j =2 3 ::: n; (9b) u 2 = u (0a) u 2 2 = x u 2 ; x 2u (0b) The yte in Eq. (9) will be called the generated yte of econd order and we ue the rt index in the ubcript of the tate eleent to denote thi. For conitency, we dene the rt generated yte to be iply the original yte in Eq. (), that i, we letx j = x j for j n. Notice that the yte in Eq. (9) i a yte in power for of dienion (n ; ). The ae proce can be therefore repeated. After repeating thi proce (i ; ) tie one obtain the ith generated yte (of dienion (n ; i +)) _x i = u i (a) _x i j = (j ; 2)! xj;2 i ui 2 j =2 3 ::: n; i + (b) 2

3 For the ith generated yte, one can contruct (n ; i ; ) invariant anifold uing the linear control law u i = ; kx i (2a) u i 2 = ; 2 i; kx i 2 (2b) and the ethodology decribed earlier. anifold are dened by The correponding i j = fx 2< n;i+ : i j(x) =0 j n ; i ; g (3) i j = x i j+2 ; j = 2 ::: n; i ;. Dening now x i+ = x i 2 i; j!(2 i; + j) xj i xi 2 (4) x i+ j = (2 i; + j) i j; j =2 3 ::: n; i the (i + )th generated yte can be decribed a follow _x i+ = u i+ _x i+ j = (j ; 2)! xj;2 i+ ui+ 2 j =2 3 ::: n; i u i+ = u i (a) u i+ 2 = x i u i 2 ; 2 i; u i x i 2 (b) Thi proce can be continued until the (n ; 2)th generated yte, which i the 3-dienional yte _x n;2 = u n;2 (6a) _x n;2 2 = u n;2 2 (6b) _x n;2 3 = x n;2 u n;2 2 (6c) By contruction, it i iediate that if x i+ = x i+ 2 = x i+ 3 = = x i+ n;i = 0 for the (i + )th generated yte, then for the ith generated yte we have thatx i = x i 3 = = x i n;i+ =0. Thu, any convergent feedback controller about the origin for the (i + )th generated yte, will alo ake the ith generated yte converge to the one-dienional anifold M i = fx 2< n;i+ : x i = x i 3 = = x i n;i+ = 0g. In particular, if the controller for (i + )th generated yte i choen uch that, in addition, it atie the property li t! x i 2 = 0 then the ae control law willdrivetheith generated yte converge to the origin The 3-dienional Syte The rt tep in the propoed derivation of the feedback controller i to contruct a tatic, tate-feedback controller for the (n ; 2)th generated yte in Eq. (6). Fro the dicuion in Section, thi controller ha to be necearily non-ooth. For notational convenience, let u redene z i = x n;2 i (i = 2 3) and v i = u n;2 i (i = 2). Then the yte in Eq. (6) can be rewritten a _z = v (7a) _z 2 = v 2 (7b) _z 3 = z v 2 (7c) The following lea will be ueful in the equel. Lea 3.Conider the calar, linear dierential equation _y = ;y+ N j= h j e ; jt >0 j > 0 (8) h j,(j = 2 ::: N)arecontant. Then, the olution y(t) of Eq. (8) decay exponentially to zero. If, in addition, j >for j = 2 ::: N, then the olution decay exponentially with rate of decay. The proof of thi lea can be eaily etablihed by direct integration of Eq. (8). The following theore provide an exponentially convergent controller for the yte in Eq. (7). Theore 3.Conider the yte in Eq. (7) and the feedback control v = ; kz (9a) v 2 = ; kz 2 ; ( +) z (9b) with k>0, >0 and >( +)k, and = z 3 ; zz2 (20) + Thi control law i bounded along the trajectorie of the yte and ha the property that, for the cloed-loop yte, li t!(z (t) z 2(t) z 3(t)) = 0 with exponential rate of convergence, for all initial condition uch that z (0) 6= 0. Proof: The proof i quite traightforward. Firt, notice that z = z (0) e ;kt, and z decreae exponentially with rate of decay k. The variable in Eq. (20) i the invariant anifold for the yte in Eq. (7) under the linear feedback v = ; kz (2a) v 2 = ; kz 2 (2b) Uing Eq. (9), the dierential equation for i _ =_z 3 ; (vz2 + zv2) =; (22) + and decreae exponentially with rate of decay. By denition, li t! (t) = 0 iplie that li t! z 3(t) = 0. The dierential equation for z 2 can be written a follow _z 2 = ; kz 2 ; ( +)(t) (23) the function i an exponentially decaying function with rate of decay ; k, ince (t) = (t) z (t) = 0 z 0 e ;(;k)t (24) Fro Lea 3. and the fact that >( +)k, one ha li t! z 2(t) = 0. Moreover, the rate of decay ofz 2 i equal to k. Therefore the cloed-loop trajectorie of the yte in Eq. (7) with the control law in Eq. (2) have the property that li t!(z (t) z 2(t) z 3(t)) = 0. The clai that the control law (9) i bounded follow iediately by virtue of Eq. (24). 3

4 4. The Feedback Controller The following theore contain the ain reult of thi paper. It how that the convergent control law in Eq. (9), can alo be ued to drive the original yte in Eq. () to the origin. Theore 4.Conider the yte in Eq. () (n 3) and the feedback controller u = u n;2 (2a) n;4 u 2 = ; `=0 2` k x+` 2 x` + un;2 2 x n;3 (2b) u n;2 = ;kx (26a) u n;2 2 = ;kx n;2 2 ; ( +) (26b) x n;2 k>0, >( +)k, =2 n;3, and = x n;2 3 ; xn;2 2 x (27) + and x n;2 2, x n;2 3 and x` 2 (` = 2 ::: n; 3) are derived through the recurive proce decribed in Section 3. Then thi control law i bounded along the trajectorie of the yte and ha the property that li t!(x (t) x 2(t) ::: x n(t)) = 0 with exponential rate of convergence, for all initial condition uch that x (0) 6= 0. Proof: The proof of thi theore require repeated ue of Lea 3.. We aue here that n 4 ince the cae when n = 3 ha been addreed in Theore 3.. Fro Theore 3. we have that the control law ineq.(26) achieve li t! x n;2 j(t) =0forj = 2 3. In addition, fro the ae theore we have that the function = x (28) decay exponentially with rate ; k. The ret of the proof i hown by induction. To thi end, let u aue that for the (i + )th order generated yte we have that li t! x i+ j (t) = 0, for j = 2 3 ::: n; i, which iplie that li t! x i j(t) = 0, for j = 3 ::: n;i+. It ha been hown previouly that with the control law in Eq. (26) x n;2 2 decay exponentially with rate 2 n;3 k and x n;3 2 decay exponentially with rate 2 n;4 k. Aue now that the function (` = n ; 2 ; i n ; 3 ; i ::: ) decay exponentially, x i+` 2 each with correponding rate 2 i+`; k.we will how that alo li t! x i 2(t) = 0 with rate 2 i; k. The dierential equation for x i 2 i given by n;2;i _x i 2 = ;2 i; kx i 2 ; 2 i;+` k xi+` 2 ;( +) `= x n;;i x` i = 2 ::: n; 3 (29) A traightforward calculation how that (n 4) 2 n;3 +2; n + i 2 i; for all i = 2 ::: n; 3 and 2 i+`; ; `>2 i; for all i =2 3 ::: n; 3and` = n ; 2 ; i n ; 3 ; i :::. Since >(2 n;3 +)k, the function n;;i = x n;;i i = 2 ::: n; 3 (30) decay exponentially with rate ;(n;;i) k>2 i; k, i = ::: n; 3. Moreover, ince by auption x i+` 2 decay exponentially with rate 2 i+`; k and x decay with rate k, one ha that the function ` = xi+` 2 x` ` = n ; 2 ; i n ; 3 ; i ::: (3) decay exponentially with rate (2 i+`; ; `) k>2 i; k for i = 2 3 ::: n; 3 and ` = ::: n ; 2 ; i. Ue of Lea 3. indicate that li xi 2(t) =0 i = 2 ::: n; 3 (32) t! with rate of decay at leat 2 i; k and the proof i coplete. The fact the the control law in Eq. (2) i bounded follow iediately fro the fact that x reache the origin only ayptotically (not in nite tie), and the fact that the the function `(t) in Eq. (3) and (t) in Eq. (30) are bounded. The control law in Eq. (2) ake the contructed invariant anifold in each tep attractive. It i clear that for thi procedure to work the attraction of the trajectorie to the correponding anifold at each tep hould take place on dierent tie cale. For the ake of iplicity, thi i achieved by taking the gain of the control law u i 2 twice a the one in the previou tep. Thi can be relaxed at the expene of ore coplicated control expreion. Reark 4.The control law in Eq. (2) will work a long a x (0) 6= 0. If initially x (0) = 0 one can ue any control law uch that x becoe nonzero. One poible choice i to ue u = u 0 u 2 =0,u 0 i oe nonzero contant.. Nuerical Exaple Siulation reult how that the recurive algorith provide an eective way to drive an n-dienional nonholonoic control yte in power for to the origin with exponential rate of convergence. Figure how the trajectorie of a - dienional yte with initial tate x(0) = (4 ; ;2 3) ubject to the feedback control in Eq. (2). The gain were choen a k = and = 6. Notice that the control law drive the tate to the origin at dierent tie cale. Figure 2 how the tie hitory of the logarith of the euclidean nor of the tate. The linear lope indicate the exponential rate of convergence. The control eort i hown in Fig Concluion We preent a new technique for contructing exponentially convergent controller for nonholonoic yte in power for. The contruction of the propoed control law i baed on a recurive algorith which ue a erie of invariant anifold in order to contruct a equence of generated yte in power for of reduced dienion. Uing thi proce, one end up with a 3-dienional yte in power for. The propoed control law for the n-dienional yte i the one that tabilize thi 3-dienional yte by proper choice of the gain. Finally, becaue of the equivalence between chained and power for yte, the control law propoed here can alo be ued for nonholonoic yte in chained for a well. Reference [] T. R. Kane, Dynaic: Theory and Application. New York: McGraw-Hill Inc., 98. 4

5 U 2 Control Effort 0 U Tie Figure : Trajectorie of cloed-loop yte Tie Figure 3: Control hitory. Logarith of x Tie Figure 2: Logarithic plot of kxk 2 v. tie. [2] R. M. Murray, Z. Li, and S. S. Satry, A Matheatical Introduction to Robotic Manipulation. Boca Raton, Florida: CRC Pre, 994. [3] H. Krihnan, M. Reyhanoglu, and H. McClaroch, \Attitude tabilization of a rigid pacecraft uing two control torque: A nonlinear control approach baed on the pacecraft attitude dynaic," Autoatica, vol. 30, pp. 023{027, 994. [4] P. Tiotra, M. Corle, and M. Longuki, \A novel approach for the attitude control of an axiyetric pacecraft ubject to two control torque," Autoatica, vol. 3, no. 8, pp. 099{2, 99. [] R. W. Brockett, \Ayptotic tability and feedback tabilization," in Dierential Geoetric Control Theory (R. W. Brockett, R. S. Millan, and H. J. Suan, ed.), pp. 8{ 208, Birkhauer, 983. [6] C. Canuda de Wit and O. J. Srdalen, \Exaple of piecewie ooth tabilization of driftle NL yte with le input than tate," in Proc. IFAC Nonlinear Control Syte Deign Sypoiu, 992. Bordeaux, France. [7] H. Khennouf and C. Canuda de Wit, \On the contruction of tabilizing dicontinuou controller for nonholonoic yte," in IFAC Nonlinear Control Syte Deign Sypoiu, pp. 747{72, 99. Tahoe City, CA. [8] A. Atol, \Dicontinuou control of nonholonoic yte," Syte and Control Letter, vol. 27, pp. 37{4, 996. [9] C. Saon, \Velocity and torque feedback control of a nonholonoic cart," in Int. Workhop on Adaptive and Nonlinear Control: Iue in Robotic, pp. 2{, 990. Grenoble, France. [0] J. M. Coron, \Global ayptotic tabilization for controllable yte without drift," Math. Cont. Signal and Syt., vol., pp. 29{32, 992. [] R. Murray, \Control of nonholonoic yte uing chained for," Field Intitute Counication, vol., pp. 29{24, 993. [2] A. Teel, R. Murray, and G. Walh, \Nonholonoic control yte: Fro teering to tabilization with inuoid," in Proc. 3t IEEE Conference ondeciion and Control, pp. 603{609, 992. [3] J. P. Poet, \Explicit deign of tie-varying tabilizing control law for a cla of controllable yte without drift," Syte and Control Letter, vol. 8, pp. 47{8, 992. [4] I. Kolanovky and H. McClaroch, \Hybrid feedback law for a cla of cacade nonlinear control yte," IEEE Tranaction on Autoatic Control, vol. 4, pp. 27{282, 996. [] R. T. M'Clokey andr. M. Murray, \Exponential tabilization of driftle nonlinear control yte uing hoogeneou feedback," in Proceeding, 33rd Conference ondeciion and Control, pp. 37{322, 994. Lake Buena Vita, FL. [6] P. Morin and C. Saon, \Tie-varying exponential tabilization of the attitude of a rigid pacecraft with two control," in Proceeding of the 34th Conference ondeciion and Control, pp. 3988{3993, 99. New Orlean, LA. [7] O. J. Srdalen and O. Egeland, \Exponential tabilization of nonholonoic chained yte," IEEE Tranaction on Autoatic Control, vol. 40, no., pp. 3{49, 99. [8] I. Kolanovky and H. McClaroch, \Developent in nonholonoic control proble," IEEE Control Syte Magazine, vol., no. 6, pp. 20{36, 99. [9] W. M. Boothby, An Introduction to Dierentiable Manifold and Rieannian Geoetry. San Diego, California: Acadeic Pre, 2nd ed., 986.