欠驅動系統的特性為利用較少的致動器來驅動較多的系統自由度，這使得其在控制上頗為困難，因此目前學界欠缺一致性的方法來處理其控制問題。本文提出以Olfati轉換來將四階欠驅動系統表示成一特殊串接型式，此型式中的非匹配時變未知量，可用函數近似法來學習之，再設計一適應多面滑動控制，來處理非匹配問題。經由Lyapunov穩定度理論之應用，可證得輸出之穩定性，以及內部變數之有界性。另外，對穩態與暫態性能，亦有嚴格的分析。為了證實所提出的方法有足夠的一般性，本論文亦使用同一具控制器於TORA、倒單擺、天車、旋轉倒單擺等四種欠驅動系統，發現其均可獲致滿意的性能，這在現有文獻中，實屬少見！

Underactuated systems are mechanical systems using fewer actuators to drive its dynamics in a more degree of freedom space. This surely brings considerable challenges in controller designs. As of now, there are few unified approaches that are available to cope with the control problems in underactuated systems. In this paper, we propose to use Olfati’s transformation to represent a 4th order underactuated system into a special cascade form. Some time-varying uncertainties may enter the system in a mismatched fashion and the function approximation techniques together with the adaptive multiple-surface sliding controller will be applied to deal with this difficulty. The Lyapunov stability theory is employed to ensure system stability as well as the boundedness of the internal signals. In addition, both the steady state and the transient performance are evaluated with rigorous mathematical analysis. To prove the generality of the method, we apply the proposed controller to four different systems: TORA, inverted pendulum, crane system, and rotary inverted pendulum, and the results show that all these systems give satisfactory performance which is seldom seen in the literature.

[1] Abdel-Rahman, E. M., Nayfeh, A. H., and Masoud, Z. N., “Dynamics and control of cranes: a review,” Journal of Vibration and Control, vol. 9, no. 7, pp. 863-908, 2003.
[2] Acosta, J. A., “Furuta’s Pendulum: a conservative nonlinear model for theory and practise,” Mathematical Problems in Engineering, vol. 2010, article id. 742894, pp. 1-29, 2010.
[3] Aguilar, L. T., Boiko, I., Fridman, L., and Iriarte, R., “Generating self-excited oscillations via two-relay controller,” IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 416-420, Feb. 2009.
[4] Ahmad, M. A., “Active sway suppression techniques of a gantry crane system,” European Journal of Scientific Research, vol. 27, no. 3, pp. 322-333, 2009.
[5] Ashrafiuon, H., and Whitman, A. M., “Closed-loop dynamic analysis of a rotary inverted pendulum for control design,” Journal of Dynamic Systems, Measurement, and Control, vol. 134, pp. 024503-1-9, Mar. 2012.
[6] Astrom, K. J., Aracil, J., and Gordillo, F., “A family of smooth controller for swinging up a pendulum,” Automtica, vol. 36, no. 2, pp. 1841-1848, 2008.
[7] Balachandran, B., Li, Y. Y., and Fang, C. C., “A mechanical filter concept for control of nonlinear crane-load oscillations,” Journal of Sound and Vibration, vol. 228, no. 3, pp. 651-682, 1999.
[8] Bradshaw, A., and Shao, J., “Swing up control of inverted pendulum systems,” Robotica, vol. 14, pp. 397-405. 1996.
[9] Beeston, J.W., “Closed-loop time optimal control of a suspended load: a design study,” Proceedings of the IFAC 4th World Congress, Warsaw, Poland, pp. 85-99. 1969.
[10] Bugeja, M., “Non-linear swing up and stabilizing control of an inverted pendulum system,” EUROCON, Ljubljana, Slovenia, vol. 2, pp. 437-441, Sep. 2003.
[11] Cao, Y. Y., and Lin, Z., “Robust stability analysis and fuzzy scheduling control for nonlinear system subject to actuator saturation,” IEEE Transactions on Fuzzy Systems, vol. 11, no. 1, pp. 57-67, Feb. 2003.
[12] Carlos, A. I., Miguel S, S. C., and Oscar O, G. F., ”The direct Lyapunov method for the stabilisation of the Furuta pendulum,” International Journal of Control, vol. 83, no. 11, pp. 2285-2293, Nov. 2010.
[13] Chang, C. Y., “Adaptive fuzzy controller of the overhead cranes with nonlinear disturbance,” IEEE Transactions on Industrial Informatics, vol. 3, no. 2, May. 2007.
[14] Chatterjee, D., Patra, A., and Joglekar, H. K., “Swing up and stabilization of a car pendulum system under restricted cart track length,” Systems & Control Letters, vol. 47, issue. 4, pp. 355-364. Nov. 2002.
[15] Chen, B. S., Tseng, C. S., and Uang, H. J., “Robustness design of nonlinear dynamic systems via fuzzy linear control,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 5, PP. 571-585, Oct. 1999.
[16] Chen, C. S., and Chen, W. L., “Robust adaptive sliding mode control using fuzzy modeling for an inverted pendulum system,” IEEE Transactions on Industrial Electronics, vol. 45, no. 2, pp.297-306. Apr. 1998.
[17] Chen, Y. F., and Huang, A. C., “Controller design for a class of underactuated mechanical systems,” IET Control Theory & Applications, vol.6, issue 1, pp. 103-110, Jan. 2012.
[18] Chien, M. C., and Huang, A. C., “Adaptive control of flexible-joint electrically-driven robot with time-varying uncertainties,” IEEE Transactions on Industrial Electronics, vol. 54, no. 2, pp. 1032-1038, Apr. 2007.
[19] Collado, J., Lozano, R., and Fantoni, I., ”Control of convey-crane based on passivity,” Proceedings of American Control Conference, vol. 2, pp. 1260-1264, Jun. 2000.
[20] EI-Hawwary, M. I., Elshafei, A. L., Emara, H. M., and Abdel Fattah, H. A., “Adaptive fuzzy control of the inverted pendulum,” IEEE Transactions on Control Systems Technology, vol. 14, no. 6, pp. 1135-1144, Nov. 2006.
[21] Escobar, G., Ortega, R., and Sira-Ramirez, H., “Output-feedback global stabilization of a nonlinear benchmark system using a saturated passivity-based controller,” IEEE Transactions on Control Systems Technology, vol. 7, no. 2, pp.289-293, Mar. 1999.
[22] Fang, Y., Dixon, W. E., Dawson, D. M., and Zergeroglu, E., “Nonlinear coupling control laws for an underactuated overhead crane system,” IEEE/ASME Transactions on Mechatronics, vol. 8, no. 3, pp. 418-423, sep. 2003.
[23] Fantoni, I., and Lozano, R., Nonlinear Control for Underactuated Mechanical Systems, Springer Verlag, 2001.
[24] Fantoni, I., and Lozano, R., ”Stabilization of the Furuta pendulum around its homoclinic orbit,” International Journal of Control, vol. 75, no. 6, pp. 390-398, 2002.
[25] Franklin, G., Powell, J. and Emami-Naeini, A., Feedback Control of Dynamic Systems, Addison Wesley, 2nd ed., 1991.
[26] Freidovich, L., Shiriaev, A., Gordillo, F., Gomez-Estern, F., and Aracil, J., ”Partial energy shaping control for orbital stabilization of high-frequency oscillations of the Furuta pendulum,” IEEE Transactions on Control Systems Technology, vol. 17, no. 4, pp. 853-858, Jul. 2009.
[27] K. Furuta, Yamakita, M., and Kobayashi, S., “Swing up control of inverted pendulum using pseudo-feedback,” Journal of Systems and Control Engineering, vol. 206, no. 4, pp. 263-269, Nov. 1992.
[28] Furuta, K., Yamakita, M., and Kobayashi, S., “Swing-up control of inverted pendulum,” Proceedings of the International Conference on Industrial Electronics, Control and Instrumentation, vol. 3, pp. 2193-2198, 1991.
[29] Furuta, K., Yamakita, M., and Kobayashi, S., “Swing-up control of inverted pendulum using pseudo-state feedback,” Journal of Systems and Control Engineering, vol. 206, pp. 263-269, 1992.
[30] Gordillo, F., Acosta, J. A., and Aracil, J., “A new swing-up law for the Furuta pendulum,” International Journal of Control, vol. 76, no. 8, pp. 836-844, 2003.
[31] Hera, PX. L., Freidovich, LB., Shiriaev, AS., and Mettin, U., “New approach for swing up the Furuta pendulum: theory and experiments,” Mechatronics, vol. 19, pp. 1240-1250, Jul. 2009.
[32] Huang, A. C., and Chen, Y. C., “Adaptive sliding control for single-link flexible-joint robot with mismatched uncertainties,” IEEE Transactions on Control Systems Technology, vol. 12, no. 5, pp. 770-775, Sep. 2004a.
[33] Huang, A. C., and Chen, Y. C., “Adaptive multiple-surface sliding control for non-autonomous systems with mismatched uncertainties,” Automatica, vol. 40, issue 11, pp. 1939-1945, Nov. 2004b.
[34] Huang, A. C., and Kuo, Y. S., “Sliding control of nonlinear systems containing time-varying uncertainties with unknown bounds,” International Journal of Control, vol. 74, no. 3, pp. 252-264, 2001.
[35] Hung, L. C., Lin, H. P., and Chung, H. Y., “ Design of self-tuning fuzzy sliding mode control for TORA system,” Expert Systems with Applications, vol. 32, issue.1, pp. 201-212, Jan. 2007.
[36] Hussein, I. I., and Bloch, A. M., “Optimal control of underactuated nonholonomic mechanical systems,” IEEE Transactions on Automatic Control, vol. 53, no. 3, pp. 668-682, Apr. 2008.
[37] Hwang, G. C., and Lin, S. C., “A stability approach to fuzzy control design for nonlinear systems,” Fuzzy Sets and Systems, vol. 48, pp. 79-287. 1992.
[38] Iraj, H., and Saleh, M., “Controller design for rotary inverted pendulum system using evolutionary algorithms,” Mathmatical Problems in Engineering, vol. 2011, 2011.
[39] Jankovic, M., Fontaine, D., and Kokotovic, P. V., “TORA example: cascade- and passivity-based control designs,” IEEE Transactions on Control Systems Technology, vol. 4, no. 3, pp.292-297, May 1996.
[40] Jiangdagger, Z. P., and Nijmeijer, H., “Tracking control of mobile roots: a case study in backstepping,” Automatica, vol. 33, issue. 7,pp.1393-1399, Jul. 1997.
[41] Jiang, Z. P., and Kanellakopoulos, I., “Global output-feedback tracking for a benchmark nonlinear system,” IEEE Transactions on Automatic Control, vol. 45, no. 5, pp.1023-1027, May. 2000.
[42] Karagiannis, D., Jiang, Z. P., Ortega, R., and Astolfi, A., “Output-feedback stabilization of a class of uncertain non-minimum-phase nonlinear systems,” Automatica, vol. 41, issue. 9, pp.1609-1615, Sep. 2005.
[43] Khanesar, M. A., Teshnehlab, M., and Shoorehdeli, M. A., “Fuzzy sliding mode control of rotary inverted pendulum,” IEEE International Conference on Computational Cybernetics, pp. 57-62, 2007.
[44] Krstic, M., Kanellakopoulos, I., and Kokotovic, P., Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., 1995.
[45] Kung, C. C., and Chen, T. H., “ tracking based adaptive fuzzy sliding mode controller design for nonlinear systems,” IET Control Theory & Applications, vol.1, no 1, pp. 82-89, Jan. 2007.
[46] Lee, C. H., “Stabilization of nonlinear nonminimum phase system: adaptive parallel approach using recurrnt fuzzy neural network,” IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, vol. 34, no. 2, pp.1075-1088, Jul. 2004.
[47] Li, C., Yi, J., and Zhao, D., “Control of the TORA system using SIRMs based type-2 fuzzy logic,” IEEE International Conference on Fuzzy Systems, pp. 694-699, Aug. 2009.
[48] Lozano, R., Fantoni, I., and Block, D. J., “Stabilization of the inverted pendulum around its homoclinic orbit,” System and Control Letters, vol. 40, issue. 3, pp. 197-204. Jul. 2000.
[49] Masoud, Z. N., Daqaq, M. F., and Nayfeh, N. A., “Pendulation reduction on small ship mounted telescopic cranes,” Journal of Sound and Vibration, vol. 10, pp. 1167-1179, 2004.
[50] Matsuda, N., Izutsu, M., Ishikawa, J., Furuta, K., and Artrom, K. J., “Swing-up and stabilization control based on natural frequency for pendulum systems,” Proceedings of American Control Conference, pp. 5291-5296, Jun. 2009.
[51] Muskinja, N., and Tovornik, B., “Swing up and stabilization of a real inverted pendulum,” IEEE Transactions on Industrial Electronics, vol. 53, no. 2, pp. 631-639, Apr. 2006.
[52] Nair, S., “A normal form for energy shaping: application to the Furuta pendulum,” Proceedings of IEEE Conference on Decision and Control, vol. 1, pp. 516-521, Dec. 2002.
[53] Noroozi, N., Roopaei, M., and Jahromi, M. Z., “Adaptive fuzzy sliding mode control scheme for uncertain systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, issue. 11, pp.3978-3992. Nov. 2009.
[54] Oguamanam, D. C. D., Hansen, J. S., and Heppler, G. R., “Dynamics of a three-dimensional overhead crane system,” Journal of sound and vibration, vol. 242, no.3, pp. 411-426, 2001.
[55] Oh, S. K., Pedrycz, W., Rho, S. B. and Ahn, T. C., “Parameter estimation of fuzzy controller and its applicatiuon to inverted pendulum,” Engineering Applications of Artificial Intelligence, vol. 17, pp. 37-60. 2004.
[56] Olfati-Saber, R., “Normal forms for underactuated mechanical systems with symmetry,” IEEE Transactions on Automatic Control, vol. 47, no. 2, pp. 305-308, 2002.
[57] Olfati-Saber, R., “Fixed point controllers and stabilization of the cart-pole and rotating pendulum,” Proceedings of the 38th Conference on Decision and Control, vol. 1, pp. 1174–1181, Dec. 1999.
[58] Olfati-Saber, R., and Megretasi, A., “Controller design for a class of underactuated nonlinear systems,” Proceedings of the 37th Conference on Decision and Control, vol. 4, pp. 4182-4187, Dec. 1998.
[59] Olfati-Saber, R., ”Cascade normal form for underactuated mechanical systems,” Proceedings of the 39th Conference on Decision and Control, vol. 3, pp. 2162-2167, Dec. 2000.
[60] Pavlov, A., Janssen, B. N., Wouw, V. D. and Nijmeijer, H., “Experimental output regulation for a nonlinear benchmark system,” IEEE Transactions on Control Systems Technology, vol. 15, no. 4, pp.786-793, Jul. 2007.
[61] Park, M. S., and Chwa, D., “Swing up and stabilization control of inverted pendulum systems via coupled sliding-mode control method,” IEEE Transactions. on Industrial Electronics, vol. 56, no. 9, pp. 3541-3555, Sep. 2009.
[62] Petres Z., Baranyi, P., Korondi, P., and Hashimoto, H., “Trajectory tracking by TP model transformation: case study of a benchmark problem,” IEEE Transactions on Industrial Electronics, vol. 54, no. 3, pp.1654-1663, Jun. 2007.
[63] Piazzi, A., and Visioli, A., ”Optimal dynamic-inversion-based control of an overhead crane,” IEE Proceedings-Control Theory Applications, vol. 149, no. 5, pp. 405-411, Sep. 2002.
[64] Riachy, S., Orlov, Y., Floquet, T., Santiesteban, R., and Richard, J. P., “Second-order sliding mode control of underactuated mechanical system I: local stabilization with application to an inverted pendulum,” International Journal of Robust and Nonlinear Control, vol. 18, pp. 529-543, 2008
[65] Santiesteban, R., Floquet, T., Olov, Y., Riachy, S., and Richard, J. P., “Second order sliding mode control of underactuated mechanical systems II: orbital stabilization with of an inverted pendulum with application on swing up/balancing control,” International Journal of Robust and Nonlinear Control, vol. 18, issue. 4-5, pp. 544-556, Apr. 2008.
[66] Shiriaev, A. S., Freidovich, L. B., Robertsson, A., Johansson, R., and Sandberg, A., “Virtual-holonomic-constraints-based design of stable oscillations of Furuta pendulum: theory and experiments,” IEEE Transactions on Robotics, vol. 23, no. 4, pp. 827-832, Aug. 2007.
[67] Slotine, J. E., and Li, W., Applied nonlinear control, Englewood Cliffs, NJ: Prentice-Hall, 1991.
[68] Spong, M. W., “The swing up control for the acrobat,” IEEE Control Systems, vol. 15, issue. 1, pp. 49-55. Feb. 1995.
[69] Spong, M. W., “Energy based control of a class of underactuated mechanical systems,” IFAC World Congress, Jul. 1996.
[70] Spong, M. W., ”Underactuated mechanical systems,” In: B. Sciliano and K. P. Valavanis (eds), Control Problems in Robotics and Automation, Lecture Notes in Control and Information Sciences, 230, Springer, London, UK. 1997.
[71] Spong, M. W., and Praly, L., “Control of underactuated mechanical systems using switching and saturations,” Lecture Notes in Control and Information Sciences, New York: Springer-Verlag, vol. 222, 1997.
[72] Sukontanakarn, V., and Parnichkun, M., “Real-time optimal control for rotary inverted pendulum,” Americam Journal of Applied Sciences, vol. 6, pp. 1106-1115, 2009.
[73] Tao, C. W., Chan, M. L., and Lee, T. T., “Adaptive fuzzy sliding mode controller for linear systems with mismatched time-varying uncertainties,” IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, vol. 33, no. 2, pp.283-294. Apr. 2003.
[74] Tong, S., Wang, T., and Li, H. X., “Fuzzy robust control for uncertain nonlinear systems,” International Journal of Approximate Reasoning, vol. 30, issue. 2, pp. 73-90. Jun. 2002.
[75] Tong, S., and Li, H. H., “Observer-based robust fuzzy control of nonlinear systems with parametric uncertainties,“ Fuzzy Sets and Systems, vol. 131, issue. 2, pp. 165-184. Oct. 2002.
[76]. Tsai, Y. C., and Huang, A. C., “FAT based adaptive control for pneumatic servo system with mismatched uncertainties,” Mechanical Systems and Signal Processing, vol. 22, no. 6, pp.1263-1273, Aug. 2008.
[77] Turker, T., Gorgun, H., and Cansever, G., “Lyapunov’s direct method for stabilization of the Furuta pendulum,” Turkish Journal of Electrical Engineering & Computer Sciences, vol. 120, no. 1, pp. 99-110, 2012.
[78] Uchiyama, N., “Robust control of rotary crane by partial-state feedback with integrator,” Mechatronics, vol. 19, issue 8, pp. 1294-1302, Dec. 2009.
[79] Wai, R. J., and Chang, L. J., “Adaptive stabilizing and tracking control for a nonlinear inverted pendulum system via sliding mode technique,” IEEE Transactions on Industrial Electronics, vol. 53, issue. 2, pp. 674-692, Apr. 2006.
[80] Wan, C. J., Bernstein, D. S., and Coppola, V. T., “Global stabilization of the oscillation eccentric rotor,” Nonlinear Dynamics, vol. 10, issue. 1, pp. 49-62, May. 1996.
[81] Wang, L. X., “Stable adaptive fuzzy controllers with application to inverted pendulum tracking,” IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, vol. 26, no. 5, pp.677-691. Oct. 1996.
[82] Wang, W., Yi, J., Zhao, D., and Liu, D., “Design of a stable sliding-mode controller for a class of second-order underactuated systems,” IEE Proceedings -Control Theory Appl., vol. 151, no. 6, pp. 683-690, Nov. 2004.
[83] Won, M., and Hedrick, J. K., “Multiple-surface sliding control of a class of uncertain nonlinear systems,” International Journal of Control, vol. 64, no. 4, pp. 693-706, 1996.
[84] Xu, R., and Ozguner, U., “Sliding mode control of a class of underactuated systems,” Automatica, vol. 44, pp. 233-241, 2008.
[85] Yamakawa, T., “Stabilization of an inverted pendulum by a high speed fuzzy logic controller hardware system,” Fuzzy Sets and Systems, vol. 32, pp. 161-180. 1989.
[86] Yi, J., and Yubazaki, N., “Stabilization fuzzy control of inverted pendulum systems,” Artificial Intelligence in Engineering, vol. 14, issue. 2, pp.153-163, Apr. 2000.
[87] Yoo, B., and Ham, W., “Adaptive fuzzy sliding mode control of nonlinear system,” IEEE Transactions on Fuzzy Systems, vol. 6, no. 2, pp.315-321, May. 1998.
[88] Yu, C. H., Wang, F. C., and Lu, Y. J., “Robust control of a Furuta pendulum,” Proceedings of SICE Annual Conference, pp. 2559-2563, Aug. 2010.
[89] Zhang, H. Z., Li, M., Yang, J., and Yang, D., “Fuzzy model-based robust networked control for a class of nonlinear systems,” IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, vol. 39, no. 2, pp.437-447. Mar. 2009.
[90] Zhang, M., and Tarn, T. J., “A hybrid switching control strategy for nonlinear and underactuated mechanical systems,” IEEE Transactions on Automatic Control, vol. 48, no. 10, pp. 1777-1782, Oct. 2003