同步誤差係指各軸間運動軌跡之差異。直覺上如果可使每一軸都追蹤同一軌跡，即可藉由消除追蹤誤差的方式來達到同步控制的目的，但由於各軸的負載未必相同，且外部干擾和內部未知量的多寡亦頗有差異，使得各軸的追蹤性能不易掌握，進而影響了同步控制行為。本文提出一適應控制方式，藉由改善各軸未知量之影響，提升各軸之追蹤性能，並藉由Lyapunov穩定性理論確保同步誤差處於穩態為u.u.b.(uniformly ultimately bounded)，在暫態亦規範於一可調整之邊界內。

然而在某些工業的應用中，各軸間的同步性遠較單一軸的追蹤誤差重要，此時上述的控制方式顯然無法提供有效的性能。因此，本文提出一虛擬軸法，企圖以控制方式在各軸間加上耦合效應，使得某一軸的各別行為，也會影響到他軸的運動軌跡，在適當的設計下，即可達到同步的目的。

本文以此虛擬軸法，配合前述之適應控制器，成功設計出一同步控制架構，並亦以Lyapunov穩定性理論驗證得穩態的u.u.b.性能。最後為驗證所提出之理論，本文實做出一具雙馬達同步控制平台，進行多項實驗驗證。

Synchronization error can be regarded as the error between the output trajectories of mechanical systems. Intuitively , if the desired trajectories for all systems are the same , then synchronization can be achieved by eliminating the tracking error of individual system output. However , the loading condition for each system is not necessarily the same , and external disturbances as well as the internal uncertainties might be quite different , the synchronization performance would not be satisfactory. In this thesis , a function approximation technique based adaptive controller is designed to give proper performance of synchronization among system outputs. The closed loop system is proved to be uniformly ultimately bounded (u.u.b.) by using the Lyapunov stability theory and the transient state will be limited within an adjustable bound.

In some industrial applications , output synchronization is more important than the individual tracking performance , and hence a more sophisticated control strategy is require. A virtual axis method is proposed in this thesis to couple the output of systems to be synchronized so that variation of one output trajectory will give rise to the variation of other’s. A Lyapunov stability theory based analysis is also used to prove that the synchronization error is u.u.b. with guaranteed transient performance.

A two-motor experimental setup is built in this study. Several experiments are conducted to justify the feasibility of the proposed strategies.

[1]J. G. Bollinger and Y. Koren , ”Design Parameters For Sampled-Data Drivers For CNC Machine Tools,”IEEE Transaction on Industry Applications,Vol. IA-14, No.3, pp. 255-264, May/June, 1978.

[2]T. C. Tsao, ”Optimal Feed-Forward Digital Tracking Controller Design”, Transactions Of the ASME Journal of Dynamic Systems, Measurement , and Control, Vol.116, pp.583-592, 1994.

[3]M. Tomizuka, ”Zero phase Error Tracking Algorithm for Digital Control”, Transaction of the ASME Journal of Dynamic Systems, Measurement, and Control, Vol.109, pp.65-68.

[4]K. S. Narendra, Stable Adaptive System, Prentice-Hall, Inc., 1987.

[5]Bruce A. Francis and Pramod P. Khargonekar, Robust control theory, New York：Springer-Verlag, 1995.

[6]P. A. Ioannou and J. Sun, Robust Adaptive Control, Upper Saddle River, NJ： Prentice Hall, 1996.

[7]V. I. Utkin, ”Variable Structure System with Sliding Mode”, IEEE Transaction Automatic Control, Vol.AC-22, No.2, pp.212-222, April 1977.

[8]V. I. Utkin, ”Sliding Mode and Their Application in Variable Structure System”, MIR Publishers, Moscow, 1978.

[9]A. C. Huang and Y. S. Kuo, ”Sliding Control of Nonlinear Systems Containing Time-varying Uncertainties with Unknown Bounds”, International Journal of Control, Vol.74, No.3, pp.252-264, 2001

[10]Koren,Y.,”Cross-coupled biaxial computer control for manufacturing systems”,ASME Journal Dynamic System Measurement and Control, Vol.102, pp.265-272, 1980.

[11]Anderson R. G., Lorenz R., et al., ”Web Machine Coordinated Motion Control via Electronic Line-Shafting”, IEEE, Transactions on Industry Applications, Vol. 37, No.1, pp.247-254, January/February, 2001.

[12]Valenzuela A. , Lorenz R., ”Electronic Line-Shafting Control for Paper Machine Drives”, IEEE Transactions on Industry Applications, Vol. 37., No.1, pp.158-164, January/February, 2001.

[13]Perez-Pinal, F.J.; Calderon, G.; Araujo-Vargas, I., ”Relative Coupling Strategy”, IEMDC'03. IEEE International, Electric Machines and Drives Conference, Vol. 2, pp.1162 – 1166, 2003.

[14]郭有順”不確定時變系統之適應性控制研究”，國立台灣科技大學機械工程技術研究所，博士學位論文，2002.

[15]黃安橋，”適應控制理論”，上課講義，2005.

[16]黃安橋，”非線性控制系統”，上課講義，2005.

[17]黃安橋，” Comparison Lemma”，講義，2007.

[18]黃安橋，” Uniformly Ultimately Boundedness”，講義，2007.

[19]黃安橋，” FAT-based Adaptive Control of SISO Non-autonomous System”，講義，2007.

[20]P. C. Chen, ”FAT-based Adaptive Control Design for Uncertain Non-autonomous Systems with Applications to Hydraulic Active Suspension Systems”, Ph.D. Dissertation, NTUST, 2005.

[21]B. Chu, S. Kim, D. Hong，H. K. Park and J. Park, ”Optimal Cross-Coupled Synchronizing Control of Dual-Drive Gantry System for a SMD Assembly Machine”, JSME International Journal Series C, Vol.47, No3., pp.939-945, 2004.