本論文針對PID控制器於非線性控制上的運用，進行增益排程的研究。由於如PID這般的線性控制器，僅能保證非線性系統於某操作區間上的穩定性，一旦系統離開此操作區間，即有控制性能劣化之虞。故本文根據增益排程的理念，將系統的操作範圍劃分為多個鄰近區間，並藉由各種PID控制增益的設計方法，給予各操作區間一組保證其區間穩定性的控制器，最終透過控制增益的切換來獲得全域的系統閉迴路穩定度；同時，對於增益切換時造成的不穩定現象，文中亦有更深入的探討。另外，本論文提出了一個基於滑動模式控制的排程控制器，試圖降低傳統滑動模式控制器在強健項作用之下，所產生的龐大能量消耗。其中所有的控制器推導，皆由一套自行架設的即時控制實驗迴路來驗證控制器效能。

It is well-known that the PID control algorithm is only valid locally. Wider ranges of operations are required in many industrial applications when PID controller is implemented; therefore, some modifications are to be designed. The gain scheduling strategy has widely used for decades whose theoretical justification has been investigated only recently. In this thesis, we look into the theoretical development of the PID controller design followed by its gain scheduling modifications. Various computer simulations and experimental results show their feasibility.
The second part of this thesis is focused on the gain scheduling of the traditional sliding control. For covering the uncertainties, the sliding controller uses the worst variation bound for its gain selection. Here we propose to schedule the uncertainty bound along the system trajectory such that instead of the worst case in the whole operation range we estimate the worst case in a smaller window. Significant improvement in the control effort reduction can be observed in the experimental study with a minor price of slower transient.

[1] Slotine, J.-J. E. and Li, W., Applied Nonlinear Control, Prentice Hall, 1991.

[2] Astrom, K. J. and Wittenmark, B., Adaptive control, 2nd ed., Addison-Wesley, 1995.

[3] Khalil, H. K., Nonlinear Systems, 3rd ed., Prentice Hall, 2002.

[4] Kallstrom, C. G., Astrom, K. J., Thorell, N. E., Eriksson, J. and Sten, L., “Adaptive autopilots for tankers,” Automatica, Vol. 15, pp. 241-254, 1979.

[5] Shamma, J. S. and Athans, M., “Analysis of gain scheduled control for nonlinear plants,” IEEE Transactions on Automatic Control, Vol. 35, No. 8, pp. 898-907, 1990.

[6] Rugh , W. J., “Analytical framework for gain-scheduling,” IEEE Control Systems Magazine, Vol. 11, No. 1, pp. 79-84, 1991.

[7] Shahruz, S. M. and Behtash, S. “Design of controllers for linear parameter- varying systems by the gain scheduling technique,” Journal of Mathematical Ayalysis and Applications, Vol. 168, No. 1, pp. 195-217, 1992.

[8] Shamma, J. S. and Athans, M., “Guaranteed properties of gain scheduled control for linear parameter-varying plants,” Automatica, Vol. 27, No. 3, pp. 559-564, 1991.

[9] Kaminer, I., Pascoal, A., Khargonekar, P. and Coleman, E., “A velocity algorithm for the implementation of gain-scheduled controllers,” Automatica, Vol. 31, No. 8, pp. 1185- 1191, 1995.

[10] Leith, D. J. and Leithead, W. E., “Gain-scheduled and nonlinear systems: dynamic analysis by velocity-based linearization families,” International Journal of Control, Vol. 70, No. 2, pp. 289-317, 1998.

[11] Hunt, K. J. and Johansen, T. A., “Design and analysis of gain-scheduled control using local controller networks,” International Journal of Control, Vol. 66, No. 5, pp. 619-651, 1997.

[12] Nichols, R. A., Reichert, R. T. and Rugh, W. J., “Gain scheduling for H-infinity controllers-A flight control example,” IEEE Transections on Control Systems Technology, Vol. 1, No. 2, pp. 69-79, 1993.

[13] Zhao, Z., Tomizuka, M. and Isaka, S., “Fuzzy gain scheduling of PID Controllers,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. 23, No. 5, pp. 1392-1398, 1993.

[14] Apkarian, P., Gahinet, P. and Becker, G., “Self-scheduled control of linear parameter-varying systems : a design example,” Automatica, Vol. 31, No. 9, pp. 1251-1261, 1995.

[15] Packard, A., “Gain scheduling via linear fractional transformations,” Systems and Control Letters, Vol. 22, pp. 79-92, 1994.

[16] Ziegler, J. G. and Nichols, N. B., “Optimum settings for automatic controllers,” Transactions of the ASME, 1942.

[17] Bartelt, T. L. M., Industrial Control Electronics : Devices , Systems and Applications, Delmar, 1997.

[18] Auslander, D. M. and Kempf, C. J., Mechatronics : Mechanical Systems Interfacing, Prentice Hall, 1996.

[19] 黃安橋，“非線性控制系統”，國立台灣科技大學機械工程研究所，上課講義，2009。

[20] Buckner, G. D., “Intelligent bounds on modeling uncertainty : applications to sliding mode control,” IEEE Transactions on Systems, Man, and Cybernetics- Part C: Applications and reviews, Vol. 32, No. 2, pp. 113-124, 2002.