羅斯π引理

羅斯π引理（Ross'π lemma），得名自以撒·麥克·羅斯[1][2][3]，是計算最优控制的結果。以產生反馈控制的Caratheodory-π解為基礎，羅斯π引理提到存在基本的时间常数，是一控制系統需要針對其可控制性及穩定性進行計算的。此時間常數稱為羅斯時間常數（Ross' time constant）[4][5]，和統御非線性控制系統之向量場的利普希茨連續成反比[6][7]。

理論內涵

在定義羅斯時間常數T時的比例因子和受控制的擾動大小以及回授控制的規格有關。若沒有擾動，羅斯π-引理會證明開迴路的最佳解和閉迴路的相關。若有擾動，比例因子可以寫成朗伯W函数的形式。

實務應用

在實際應用中，羅斯時間常數可以用DIDO的數值實驗來求得。羅斯等人證明此時間常和Caratheodory-π解的實際實現方式有關[6]。羅斯等人證明，若回授解只由零階保持產生，則若要保持可控制性及穩定性，需要快很多的取樣率。另一方面，另回授解是由Caratheodory-π技術所產生，用較慢的取樣率即可。這表示產生回授解的計算負擔遠小於標準實現方式的計算負擔。此一概念已用在机器人学的避免碰撞演算法中。處理有關靜止或是移動障礙物，且資訊不完整，或是有不確定性的情形[8]。

相關條目

參考資料
  1. B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Vol. 330 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Series, Springer, Berlin, 2005.
  2. W. Kang, "Rate of Convergence for the Legendre Pseudospectral Optimal Control of Feedback Linearizable Systems", Journal of Control Theory and Application, Vol.8, No.4, 2010. pp. 391-405.
  3. Jr-S Li, J. Ruths, T.-Y. Yu, H. Arthanari and G. Wagner, "Optimal Pulse Design in Quantum Control: A Unified Computational Method", Proceedings of the National Academy of Sciences, Vol.108, No.5, Feb 2011, pp.1879-1884.
  4. N. Bedrossian, M. Karpenko, and S. Bhatt, "Overclock My Satellite: Sophisticated Algorithms Boost Satellite Performance on the Cheap" IEEE Spectrum, November 2012.
  5. R. E. Stevens and W. Wiesel, "Large Time Scale Optimal Control of an Electrodynamic Tether Satellite", Journal of Guidance, Control and Dynamics, Vol. 32, No. 6, pp. 1716–1727, 2008.
  6. I. M. Ross, P. Sekhavat, A. Fleming and Q. Gong, "Optimal Feedback Control: Foundations, Examples, and Experimental Results for a New Approach", Journal of Guidance, Control, and Dynamics, vol. 31 no. 2, pp. 307–321, 2008.
  7. I. M. Ross, Q. Gong, F. Fahroo, and W. Kang, "Practical Stabilization Through Real-Time Optimal Control", 2006 American Control Conference, 电气电子工程师学会, Piscataway, NJ, 14–16 June 2006.
  8. M. Hurni, P. Sekhavat, and I. M. Ross, "An Info-Centric Trajectory Planner for Unmanned Ground Vehicles 页面存档备份，存于", Chapter 11 in Dynamics of Information Systems: Theory and Applications, Springer, 2010, pp. 213–232.
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