Controller Design of High Order Nonholonomic System with Nonlinear Drifts

Transcription

1 Internatonal Journal of Automaton and Computng 6(3, August 9, 4-44 DOI:.7/s Controller Desgn of Hgh Order Nonholonomc System wth Nonlnear Drfts Xu-Yun Zheng Yu-Qang Wu Research Insttute of Automaton, Qufu Normal Unversty, Qufu 7365, PRC Abstract: A controller desgn s proposed for a class of hgh order nonholonomc systems wth nonlnear drfts. The purpose s to ensure a soluton for the closed-loop system regulated to zero. Addng a power ntegrator bacsteppng technque and the swtchng control strategy are employed to desgn the controller. The state scalng s appled to the recursve manpulaton. The smulaton example demonstrates the effectveness and robust features of the proposed method. Keywords: Nonholonomc systems, globally asymptotc stablty, addng a power ntegrator bacsteppng. Introducton The control of nonholonomc dynamc systems has receved consderable attenton durng the last few years. Ths partcularly nterestng class of nonlnear control systems arses from control problems related to mechancal systems wth nonholonomc or nonntegrable constrants. Among several practcal and techncal reasons to study nonholonomc control problems s the fact that such a class of nonlnear system cannot be asymptotcally stablzed by only usng contnuous state feedbac control sgnals. It s wellnown that many mechancal systems wth nonholonomc constrants can be locally or globally converted to the extended chaned form under coordnate change and state feedbac. Interestng examples of these mechancal systems nclude trcycle-type moble robots, cars towng several tralers, the nfe-edge, vertcal rollng wheels, rgd spacecrafts wth two torque actuators [ 7]. Several novel nonlnear control feedbac desgns have been proposed n the lterature to acheve the asymptotc stablzaton for these nonholonomc control systems. These methods nclude the use of smooth tme-varyng feedbac of form u = µ(x, dscontnuous feedbac technques, and nonsmooth tme-varyng homogeneous feedbac [8 4]. Based on the extended nonholomonc ntegrator, a varable structure exponental stablzaton of chaned systems was developed [5]. The robustness ssue for the asymptotc and exponental stablty propertes has been extensvely studed. A new class of nonholomonc systems n perturbed chaned form and wth drft uncertan nonlnearty was ntroduced n [6 7]. An adaptve global stablzaton controller was desgned for nonholomonc systems wth stronger nonlnear drfts [8]. Hgh order stablzaton for a class of nonholomonc perturbed systems was consdered n [9, ]. A class of hghorder nonholonomc systems n the so-called power chaned form was dscussed n [], for whch the standard chaned form systems were a specal case. The hgh order nonholomonc systems pose a new challenge to nonlnear feed- Manuscrpt receved September, 8; revsed Aprl 3, 9 Ths wor was supported by Natonal Natural Scence Foundaton of Chna (No *Correspondng author. E-mal address: wyq@qfnu.edu.cn bac synthess because t s nether stablzable by tmenvarant contnuous state feedbac nor affne n the control nput. The desgn procedure proposed n [] combned the dea of the σ process and addng a power ntegrator. A swtchng controller was also used. System global asymptotc regulaton was guaranteed by the controller desgned n []. Ths paper consders the desgn of controllers for a class of hgh order nonholomonc systems wth nonlnear drfts whch can guarantee the convergence of the closed-loop system. The controller desgn s also based on a combned applcaton of a state scalng technque, the σ process, and the addng a power ntegrator bacsteppng method. The proposed controller can drve the system states to the orgn wth exponental convergence. Problem formulaton Consder a class of hgh order nonholonomc systems wth nonlnear drfts ẋ = u p ẋ = x p u q φ (x, x ẋ = x p uq φ(x, x,, x ẋ n = u pn φ n(x, x,, x n, n ( x R, x = [x, x,, x n] T R n are the states, u and u are control nputs, φ (x, x,, x, n are nonlnear functons, p, =,,, n are odd ntegers, and q, =,,, n are ntegers. Assumpton. For n, there s a nown smooth nonnegatve functon ϕ that satsfes the followng nequaltes φ (x, u, x ( x x ϕ (x, u, x ( for all (t, x, u, x R R R R and x = (x,, x. In order to mae good use of the bacsteppng technque, we ntroduce the followng nput-state scalng dscontnuous transformaton defned by z = x,, z n = xn. (3 x r x rn

2 X. Y. Zheng and Y. Q. Wu / Controller Desgn of Hgh Order Nonholonomc System wth Nonlnear Drfts 4 Now, we gve the followng assumpton about the parameters r, n. Assumpton. r = q p r, r n =, n. Remar. It s worth pontng out that transformaton (3 has no defnton at x (t =. In order to guarantee the feasblty of the control desgn and performance analyss, wthout loss of generalty, we assume x (t. Let us consder x (t frst. The case of x (t = wll be dscussed n Secton 3. Under transformaton (3, system ( s transformed nto ż = z p ( u q u p φ(x, x z r x x x r ż n = u pn φ n(x, x n, n. (4 For smplfy, we denote u φ p = z r x φ(x, x x r φ n = φ n(x, x n, n. Lemma. For each n, there exsts a smooth nonnegatve functon γ such that φ (x, x (z,, z γ. (5 Proof. From (4 and Assumptons and, we have φ u p (x, x = z r x φ(x, x x r u p z r x ( x,, x ϕ (x, u, x x r ( z z γ. In the frst subsystem of (, we choose the control law Introducng the Lyapunov functon u = λ x. (6 V (x = x. Then, the tme dervatve of V (x along the frst equaton of ( satsfes V x p such that the x subsystem s asymptotcally stable. For the bacsteppng method, we defne a set of vrtual controllers,, z n and ξ = z, ξ = z p z p,, ξ n = z p p n n z p p n n. (7 Step. Let V = (nx z /. Then, the tme dervaton of V satsfes V (x, z = nx p z (( q z p z r ( (p zϕ(x, u, x x r nx p z ( q z p zγ = nx p z (( q (z p z p ( q z p zγ. (8 Let the contnuous vrtual controller be descrbed by z p = ( q (nz c z z γ (9 c > s a constant to be specfed later. Then V nx p nz c z ( q z (z p z p. ( Step. Let V = V Then, the tme dervaton of V satsfes (s p z p ( p ds. ( V nx p nz c z ( p z (z p z p (( q z p 3 ( p z r x p z φ (x, x x r W (( p z p z r z W = z x ( p x p φ(x, x x r x ( p x p ( (s p z p ( p ds, = ( z p p z = ( z p p x = ξ ( p ds, z (s p z p ( p ds, (s p z p ( p ds. Usng the nequalty x y p p x p y p, x, y R, odd nteger p >, and Yong s nequaltes x T y x p /p x q /q, p = p /(p, q = p, we have z (( q z p z r ( p x p φ(x, x x r a ξ z p (( q z p z r ( p x p z φ (x, x x r ξ 3 ξ σ 3, W ( p x p a ξ z p x p x x x p ξ σ 4, W = ( z p z p z ( p z ξ ( p a ξ z p z, (s p z p ( p ds z p z

3 4 Internatonal Journal of Automaton and Computng 6(3, August 9 ( q z (z p z p 3 ξ ξ, W ż (ξ ξ c (x, z, z, z W φ(x, x ( z x r ( p z r x p ξ ( p ( z z γ 3 ξ ξ σ, σ >. Substtutng the above nequaltes nto (, we obtan V nx p (n c z W z (( q z p 3 Choose ( p z r x p φ(x, x x r ( q z ξ W x ( p x p W z (( q z p z r ( p x p φ(x, x x r nx p nz c z ξ ξ ( σ σ σ 3 σ 4 x p ( q ξ ( p z p 3. (3 z p p 3 = ( ( q ξ (n c σ σ σ 3 σ 4. (4 Inequalty (3 can be rewrtten as V (n x p nz c z ξ (n c ξ ( q ξ ( p z p 3. (5 Step. Supposng that at, there s a postve defnte and proper Lyapunov functon V = V z z and V satsfes (s p p z p p ( p p ds (6 V (n x p we have ( q ξ ( (n c ξ p p (z p V (n x p W ż z ( q ξ ( z p (7 (n c ξ W ż W ẋ z x p p (z p z p (8 W = z z W = ξ ( z W z W x (s p p z p p ( p p ds, p p, z p p = ( p p z z z = ( z z (s p p z p p ( p p ds, z p p p p x Usng the followng nequaltes: ( q ξ ( (s p p z p p ( p p ds. p p (z p z p ξ ( p p ( p p p z z p 3 ξ ξσ, = 3,, n, W u p ( z r φ (x, x z x x r ξ ( p p 3 ξ ξσ, W (z p z ( u ( z z γ q u p z r x x a ξ z p p (z p z ( u q x u p z r x 3 ξ ξ σ 3, φ(x, x x r W z p p ẋ a ξ x p x x x p ξσ 4 we have V (n x p ( q ξ ( p p φ(x, x x r p p (n c ξ ξ σ ξ σ 3 ξ σ 4 x p 3 ξ 3 ξ (n x p ( p ξ ( p p z p 3 ξ ξ σ (n c ξ z p

4 X. Y. Zheng and Y. Q. Wu / Controller Desgn of Hgh Order Nonholonomc System wth Nonlnear Drfts 43 ξ (σ σ σ 3 σ 4. (9 Clearly, a contnuous vrtual controller can be chosen as z p p = ( q ξ (n c σ σ σ 3 σ 4. ( In the last step, choosng Lyapunov functon V n = V n zn z n (s p p n z p p n n ( p p n ds and usng the bacsteppng technque above, we can get V n x p (n c ξ n ξ ( p p n n u pn Choose the control law as ξ n(σ n σ n σ n3 σ n4. ( u = [ξ n(c n n σ n σ n σ n3 σ n4] pn. ( Then nequalty ( satsfes V n x p n (n c ξ. (3 Snce V n s a postve defnte and proper Lyapunove functon, by Kurzwel s theorem, the system s globally strongly stable, whch does not mean the global stablty of a dscontnuous closed-loop but the asymptotc convergence for all the solutons startng from any ntal condton n the open and dense set Ω = (x, z,, z n, due to the use of dscontnuous transformaton (3. 3 Swtchng controller Wthout loss of generalty, we assume that t =. When the ntal state x (, we have gven controller (6 and ( for u and u of system (. Now, we dscuss how to select the control laws u and u when x ( =. In the absence of the dsturbances, most of the commonly used control strateges use constant control u = u n tme nterval [, t s. In ths paper, we also use ths method when x ( =, wth u chosen as follows: u ( = u, u >. (4 Snce x ( =, from the frst subsystem of ( we now that ẋ ( = u p ( = u p. (5 Gven any fnte t s >, durng the tme perod [, t s, apply the bacsteppng-based nonlnear feedbac u = u (x, x and apply the procedure n Secton to the orgnal x system n ( wth u as defned n (4, nstead of (6. Then, from a dfferental nequalty smlar to (3, we conclude that the x state of ( cannot blow up durng the tme perod [, t s, f the desgn parameters are properly selected. Snce x(t s at t s, we swtch the control nputs u and u nto (6 and (, respectvely. We are now ready to state the man theorem of our paper. Theorem. For system (, under Assumptons and, f control strateges (6 and ( are appled, wth an approprate choce of the desgn parameters, the globally asymptotc convergence of the closed-loop system s acheved and the state x can be regulated at the orgn asymptotcally as long as the ntal condton x (t s nonzero. Moreover, applyng swtchng control strategy (4 to (, the state x converges to the orgn asymptotcally when the ntal condton x (t s zero. 4 Smulaton example We tae the followng example: ẋ = u p ẋ = x p u q x ẋ = u p x (6 x = [x, x, x ] T s the state of the system, the output of the system s y = [x, x ] T, u and u are the nputs of the system. For llustraton purposes, we choose p = 5, q = 3, and p = p = 3. In the frst subsystem of (6, we choose the control law u = x. (7 In order to mae use of the bacsteppng technque, ntroducng the followng nput-state scalng dscontnuous transformaton defned by we get z = x, z x 3 = x (8 ż = z 3 z 3z x 4 ż = u 3 z. (9 Usng the method gven n Secton, we gan z 3 = z ( c 3x 4 u = [ξ (c σ σ σ 3 σ 4] 3 (3 ξ = z 3 z 3, σ = σ = ( 6(3 c 3x σ 3 = 5 3 (3 c 3x4 ( 5 6 (3 c 3x 4 ( c σ 4 = ξ 4 ( 6 zx3 6. Ths smulaton results are shown n Fgs. and, wth the ntal condton (x (, x (, x ( = (,, 3. As s clear n these fgures, all of the states and the control laws converge to zero.

5 44 Internatonal Journal of Automaton and Computng 6(3, August 9 Fg. Fg. 5 Conclusons System control u(t and state x (t, x (t System control u (t and state x (t The feedbac control law of globally asymptotcal stablzaton has been proposed for hgh order uncertan nonholonomc systems. The swtchng scheme stablzes the systems when the ntal condton fals. In ths procedure, we employ addng a power ntegrator bacsteppng technque and nput-state scalng method. The system s regulated at the orgn asymptotcally. The smulaton example proves the effeteness of the method. References [] R. M. Murray, S. S. Sastry. Nonholonomc Moton Plannng: Steerng Usng Snusods. IEEE Transactons on Automatc Control, vol. 38, no. 5, pp. 7 76, 993. [] I. Kolmanovsy, N. H. McClamroch. Developments n Nonholonomc Control Problems. IEEE Control Systems Magazne, vol. 5, no. 6, pp. 36, 995. [3] C. C. de Wt, G. Bastn, B. Sclano. Theory of Robot Control, Sprnger-Verlag, New Yor, USA, 996. [4] A. Astolf. Asymptotc Stablzaton of Nonholonomc Systems wth Dscontnuous Control, Ph. D. dssertaton, Edgensssche Technsch Hochschule, Zurch, 995. [5] C. Samson. Control of Chaned Systems Applcaton to Path Follow n and Tme-varyng Pont-stablzaton of Moble Robots. IEEE Transactons on Automatc Control, vol. 4, no., pp , 995. [6] L. Yang, X. P. Guan, C. N. Long, X. Y. Luo. Feedbac Stablzaton over Wreless Networ Usng Adaptve Coded Modulaton. Internatonal Journal of Automaton and Computng, vol. 5, no. 4, pp , 8. [7] Y. L, H. Kashwag. Hgh-order Volterra Model Predctve Control and Its Applcaton to a Nonlnear Polymersaton Process. Internatonal Journal of Automaton and Computng, vol., no., pp. 8 4, 5. [8] J. M. Coron. Global Asymptotc Stablzaton for Controllable System wthout Drft. Mathematcs of Control, Sgnals, and Systems, vol. 5, no. 3, pp. 95 3, 99. [9] J. B. Pomet. Explct Desgn of Tme-varyng Stablzng Control Laws for a Class of Controllable System wth out Drft. System and Control Letter, vol. 8, no., pp , 99. [] R. M. Murray, S. S. Sastry. Nonholonomc Moton Plannng: Steerng Usng Snusods. IEEE Transactons on Automatc Control, vol. 38, no. 5, pp. 7 76, 993. [] W. Ln. Tme-varyng Feedbac Control of Nonaffne Nonlnear System wthout Drft. Systems & Control Letters, vol. 9, no., pp., 996. [] R. M Closey, R. M. Murray. Exponental Stablzaton of Drftless Nonlnear Control Systems Usng Homogeneous Feedbac. IEEE Transactons on Automatc Control, vol. 4, no. 5, pp , 997. [3] Z. P. Jang. Iteratve Desgn of Tme Varyng Stablzers for Mult-nput Systems n Chaned Form. Systems & Control Letters, vol. 8, no. 5, pp. 55 6, 996. [4] Z. P. Jang, H. Njmejer. A Recursve Technque for Tracng Control of Nonholonomc Systems n Chaned Form. IEEE Transactons on Automatc Control, vol. 44, no., pp , 999. [5] W. L. Xu, W. Huo. Varable Structure Exponental Stablzaton of Chaned Systems Based on the Extended Nonholonomc Integrator. Systems & Control Letters, vol. 4, no. 4, pp. 5 35,. [6] Z. P. Jang. Robust Exponental Regulaton of Nonholonomc Systems wth Uncertantes. Automatca, vol. 36, no., pp. 89 9,. [7] Z. R. X, G. Feng, Z. P. Jang, D. Z. Cheng. Output Feedbac Exponental Stablzaton of Uncertan Chaned Systems. Journal of Franln Insttute, vol. 344, no., pp , 7. [8] K. D. Do, J. Pan. Adaptve Global Stablzaton of Nonholonomc Systems wth Strong Nonlnear Drfts. Systems & Control Letters, vol. 46, no. 3, pp. 95 5,. [9] W. Ln, R. Pongvuththum. Dscontnuous Controllers for Hgh-order Nonholonomc Systems n Power Chaned Form. In Proceedngs of the Amercan Control Conference, IEEE Press, Chcago, Illnos, USA, vol. 5, pp ,. [] W. Ln, R. Pongvuththum, C. Qan. Control of Hghorder Nonholonomc Systems n Power Chaned Form Usng Dscontnuous Feedbac. IEEE Transactons on Automatc Control, vol. 47, no., pp. 8 5,. Xu-Yun Zheng graduated from Qufu Normal Unversty, PRC n 3. She receved the M. Sc. degree from Qufu Normal Unversty n 6. She s currently a Ph. D. canddate n the Research Insttute of Automaton at Qufu Normal Unversty. Her research nterests nclude nonlnear control and adaptve control. Yu-Qang Wu receved the Ph. D. degree n automatc control from Southeast Unversty, PRC n 994. From 996 to 997, he was wth the Department of Mathematcs and Computng, Central Queensland Unversty, Australa, as a vstng research fellow. In 998, he was wth Department of Electrcal Engneerng, Unversty of Texas at Dallas as a research assocate. Currently, he s a professor n the Insttute of Automaton, Qufu Normal Unversty, PRC. Hs research nterests ncludes varable structure control, adaptve control, performance evaluaton of computer communcatons networs, and wreless communcatons.